Representation Formulas and Stability Analysis for Hilfer–Hadamard Proportional Fractional Differential Equations

Abstract

This paper introduces a novel version of the Gronwall inequality specifically related to the Hilfer–Hadamard proportional fractional derivative. By utilizing Picard’s method of successive approximations along with the definition of Mittag–Leffler functions, we derive a representation formula for the solution of the Hilfer–Hadamard proportional fractional differential equation featuring constant coefficients, expressed in the form of the Mittag–Leffler kernel. We establish the uniqueness of the solution through the application of Banach’s fixed-point theorem, leveraging several properties of the Mittag–Leffler kernel. The current study outlines optimal stability, a new Ulam-type concept based on classical special functions. It aims to improve approximation accuracy by optimizing perturbation stability, offering flexible solutions to various fractional systems. While existing Ulam stability concepts have gained interest, extending and optimizing them for control and stability analysis in science and engineering remains a new challenge. The proposed approach not only encompasses previous ideas but also emphasizes the enhancement and optimization of model stability. The numerical results, presented in tables and charts, are provided in the application section to facilitate a better understanding.

1. Introduction

The concept of stability has been a central theme in the development of control theory and dynamical systems for centuries. Its roots trace back to classical control methodologies where ensuring that a system’s output remains bounded and behaves predictably over time became a fundamental objective. As the field evolved, the focus shifted toward more rigorous mathematical criteria to characterize system behavior, leading to the emergence of various stability concepts suited for different types of systems. Among these, Ulam–Hyers stability garnered particular interest, initially introduced in the context of nonlinear functional equations, especially within the scope of stability under small perturbations [1].

Ulam–Hyers stability fundamentally addresses the question of how approximate solutions relate to exact solutions, emphasizing the robustness of solutions when subjected to minor inaccuracies or uncertainties. Its significance was magnified in controlling real-world systems where perfect models are rare, and imperfections are inevitable. Researchers recognized early on that this form of stability could be instrumental in guaranteeing system reliability amid unpredictable disturbances. Over time, the notion was extended to various classes of equations and systems, including fractional differential equations, which naturally model phenomena with memory and hereditary properties [2,3].

The extension of the Ulam–Hyers concept to fractional calculus significantly enhanced its relevance, especially in the context of engineering and applied sciences. Fractional models are renowned for their ability to accurately describe anomalous diffusion, viscoelasticity, and diverse biological processes. Their inherent complexity and non-local nature pose unique challenges in stability analysis. Consequently, the adaptation of Ulam–Hyers stability principles to fractional systems provided a powerful framework for understanding how solutions respond to perturbations or modeling inaccuracies within these non-integer-order systems [4].

In particular, the application of Ulam–Hyers stability concepts has proven invaluable in nonlinear fractional systems, where classical stability criteria may fall short. Many real-world problems rely on fractional derivatives to incorporate memory effects, making the stability of solutions under perturbations a critical concern. This form of stability assures practitioners that approximate numerical solutions or models affected by uncertainties remain close to the true solutions, a property essential for both theoretical analysis and practical implementation [5].

The relevance of Ulam–Hyers stability in fractional calculus extends well into the domain of fractional integral and differential equations commonly encountered in the calculus of variations, optimal control, and optimization. As these methods often involve inherently approximate computational techniques, the guarantees provided by Ulam–Hyers stability become crucial for ensuring the credibility and robustness of solutions. Consequently, stability analysis grounded in this concept underpins many modern approaches to solving fractional variational problems, facilitating the development of more reliable and efficient numerical schemes [6].

Moreover, the importance of Ulam–Hyers stability extends beyond theoretical interest; it influences the design of control strategies and algorithms for systems modeled by fractional dynamics. Engineers and scientists rely heavily on such stability properties to develop controllers capable of maintaining desired system behavior despite uncertainties. The insights gained from this approach also foster advances in fractional optimal control, where balancing efficiency and stability often hinges on robustness assessments rooted in Ulam–Hyers principles [7].

In summary, the evolution of stability concepts, particularly Ulam–Hyers stability, has played a transformative role in the advancement of fractional calculus and its applications. Its development reflects a deeper understanding of how systems respond to imperfections, uncertainties, and disturbances—factors that are unavoidable in practical scenarios. The importance of this stability notion is particularly pronounced in solving fractional problems in calculus of variations and fractional optimization, where the non-local, memory-dependent nature of these systems demands a more nuanced approach to stability analysis. As research continues to expand in this area, Ulam–Hyers stability remains a cornerstone, guiding the design of resilient, reliable, and accurate methods for tackling the complexities inherent in fractional systems [8,9].

Inspired by the insights from these foundational papers [10,11,12,13,14,15,16,17], this paper presents an innovative adaptation of the Gronwall inequality tailored to the Hilfer–Hadamard proportional fractional derivative. By applying Picard’s successive approximation method and leveraging the definition of Mittag–Leffler functions, we derive a formula that represents the solution of the Hilfer–Hadamard proportional fractional differential equation with constant coefficients, formulated in terms of the Mittag–Leffler kernel. We prove the uniqueness of this solution using Banach’s fixed-point theorem, drawing on various properties of the Mittag–Leffler kernel. Additionally, we investigate optimal stability results linked to certain special functions.

This paper aims to introduce a novel perspective on Ulam-type stability, termed optimal stability, which leverages classical special functions to establish a new framework. The approach focuses on deriving the most accurate approximation error bounds through an innovative form of perturbation-based stability analysis. This concept enables the development of various approximation schemes depending on the initially selected special functions, facilitating the assessment of maximal stability and minimal error. Consequently, it allows for the determination of a unique, optimal solution to functional equations, inequalities, and fractional equations. Although Ulam stability and its variants have attracted substantial scholarly interest, extending these concepts to more effective generalizations and optimizing controllability and stability criteria remain challenging, largely unexplored issues. The proposed notion of optimal stability not only encompasses existing stability concepts but also emphasizes the importance of optimization within the stability framework. It offers a comprehensive approach for improving the robustness of mathematical models applied in natural sciences and engineering, aiming to optimize their stability properties across various scenarios.

2. Preliminaries

Throughout this paper, consider a finite interval $[a,b]\subset {\mathbb{R}}^{+}$ in which $0<a<b<+\infty .$ Let ${\mathfrak{P}}_1,{\mathfrak{P}}_3\in (n-1,n],0\le {\mathfrak{P}}_2\le 1,$ and ${\mathfrak{P}}_3\ge {\mathfrak{P}}_1,$${\mathfrak{P}}_3>{\mathfrak{P}}_2,$ $n-{\mathfrak{P}}_3<n-{\mathfrak{P}}_3$$(n-{\mathfrak{P}}_1),$ where ${\mathfrak{P}}_3={\mathfrak{P}}_1+{\mathfrak{P}}_2(n-{\mathfrak{P}}_1)$ and $n\in \mathbb{N}.$ Let $\varpi =C\left(\right[a,b]\times \mathbb{R},\mathbb{R})$ be the Banach space of continuous functions $\varphi$ on $[a,b]$ with the norm ${\parallel \varphi \parallel }_{\varpi }={sup}_{\sigma \in [a,b]}\left\{\right|\varphi \left(\sigma \right)\left|\right\}.$ Let $A{C}^n\left([a,b]\right)=\{\varphi :[a,b]⟶\mathbb{R};{\varphi }^{(n-1)}\in AC\left([a,b]\right)\}$ be the space of the n-times absolutely continuous functions $\varphi$ on $[a,b].$ Also, we consider the Banach space $L^p([a,b],\mathbb{R})$ of all Lebesgue measurable functions $\varphi :[a,b]⟶\mathbb{R}$ with the norm ${\parallel \varphi \parallel }_{L^p\left([a,b]\right)}<+\infty .$

2.1. Special Functions

Definition 1

([12]).

(D1) 

For every ${\mathfrak{P}}_1,{\mathfrak{P}}_2\in \mathbb{C},$ $\Re \left({\mathfrak{P}}_1\right),\Re \left({\mathfrak{P}}_2\right)>0,$ and $\sigma \in \mathbb{R},$ the one-parameter Mittag–Leffler function and two-parameter Mittag–Leffler function are, respectively, given by

$$\begin{array}{ccc}& & {\mathrm{\Xi }}_{{\mathfrak{P}}_1}\left(\sigma \right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{\sigma }^n}{\Gamma ({\mathfrak{P}}_{1}n+1)}},\hfill \\ & & {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_2}\left(\sigma \right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{\sigma }^n}{\Gamma ({\mathfrak{P}}_{1}n+{\mathfrak{P}}_2)}}.\hfill \end{array}$$

(D2) 

For every $\alpha ,\beta \in \mathbb{C},$ and $\Re \left(\alpha \right),\Re \left(\beta \right)>0,$ the Wright function is defined as

$$W_{\alpha ,\beta }(\sigma )=\sum _{n=0}^{\infty }{\displaystyle \frac{{\sigma }^n}{n!\Gamma (\alpha n+\beta )}}.$$

(D3) 

For every $\alpha ,\beta ,\gamma \in \mathbb{C},$ and $\Re \left(\alpha \right),\Re \left(\beta \right),\Re \left(\gamma \right)>0,$ the Hypergeometric function is given by

$$H_{\alpha ,\beta ,\gamma }(\sigma )=\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(\alpha \right)}_n{\left(\beta \right)}_n}{{\left(\gamma \right)}_n}}{\displaystyle \frac{{\sigma }^n}{n!}}.$$

Note that the symbol $\Re \left(\alpha \right)$ represents the real part of the complex number $\alpha$.

Lemma 1.

([18]). Let ${\mathfrak{P}}_1\in (0,1),{\mathfrak{P}}_2>0.$ Then, ${\mathrm{\Xi }}_{{\mathfrak{P}}_1}$ and ${\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_2}$ are non-negative functions, and for $\sigma <0,$ ${\mathrm{\Xi }}_{{\mathfrak{P}}_1}\left(\sigma \right)\le 1$ and ${\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_2}\le {\displaystyle \frac{1}{\Gamma \left({\mathfrak{P}}_2\right)}}.$ In addition, for $\rho <0,$ and ${\sigma }_1,{\sigma }_2\in [a,b],$ we have that

$${\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1+{\mathfrak{P}}_2}\left(\rho {[\ln ({\displaystyle \frac{{\sigma }_2}{a}})]}^{{\mathfrak{P}}_1}\right)⟶{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1+{\mathfrak{P}}_2}\left(\rho {[\ln ({\displaystyle \frac{{\sigma }_1}{a}})]}^{{\mathfrak{P}}_1}\right),as{\sigma }_2⟶{\sigma }_1.$$

2.2. Fractional Calculus

Definition 2

([19,20,21]). We note the following concepts:

• Let ${\mathfrak{P}}_1\in \mathbb{C},$ $\Re \left({\mathfrak{P}}_1\right)>0,$ ${\mathfrak{P}}_0\in (0,1].$ The Hadamard–Riemann–Liouville proportional fractional integral of order ${\mathfrak{P}}_1$ of a function $\mathrm{\Phi }\in L^1\left([a,b]\right)$ is defined by

  $${}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_1,\ln (\sigma )}\mathrm{\Phi }(\sigma )={\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}{\displaystyle \frac{\mathrm{\Phi }\left(s\right)}{s}}ds,$$
  where $\Gamma \left({\mathfrak{P}}_1\right)={\int }_a^{\infty }{z}^{{\mathfrak{P}}_1-1}\exp (-z)dz,$ and $z>0.$
• Let ${\mathfrak{P}}_0\in [0,1]$ and ${\lambda }_0,{\lambda }_1:[0,1]\times \mathbb{R}⟶[0,\infty )$ be continuous such that, for $\sigma \in \mathbb{R},$ we have ${lim}_{{\mathfrak{P}}_0⟶0^{+}}{\lambda }_1({\mathfrak{P}}_0,\sigma )=1,$ and ${lim}_{{\mathfrak{P}}_0⟶0^{+}}{\lambda }_0({\mathfrak{P}}_0,\sigma )=0,{lim}_{{\mathfrak{P}}_0⟶1^{-}}{\lambda }_1({\mathfrak{P}}_0,\sigma )=0,{lim}_{{\mathfrak{P}}_0⟶1^{-}}{\lambda }_0({\mathfrak{P}}_0,\sigma )=1,$ and ${\lambda }_1({\mathfrak{P}}_0,\sigma )\ne 0,$ ${\lambda }_0({\mathfrak{P}}_0,\sigma )\ne 0,$ for ${\mathfrak{P}}_0\in (0,1].$ The proportional derivative operator of order ${\mathfrak{P}}_0$ with respect to the increasing function $\ln (\sigma )$ is given by

  $${}_{{\mathfrak{P}}_0}D^{\ln \left(\sigma \right)}\mathrm{\Phi }(\sigma )={\lambda }_1({\mathfrak{P}}_0,\sigma )\mathrm{\Phi }(\sigma )+{\lambda }_0({\mathfrak{P}}_0,\sigma ){\displaystyle \frac{{\mathrm{\Phi }}^{\prime }(\sigma )}{\sigma }}.$$

  (1)
• Let ${\mathfrak{P}}_1\in \mathbb{C},\Re \left({\mathfrak{P}}_1\right)>0,0<{\mathfrak{P}}_0\le 1.$ The Hadamard–Riemann–Liouville proportional fractional derivative of order ${\mathfrak{P}}_1$ of a function $\mathrm{\Phi }\in C^n\left([a,b]\right)$ is given by

  $${}_{{\mathfrak{P}}_0}D_{a^{+}}^{{\mathfrak{P}}_1,\ln \left(\sigma \right)}\mathrm{\Phi }(\sigma )={}_{{\mathfrak{P}}_0}D^{n,\ln \left(\sigma \right)}{}_{{\mathfrak{P}}_0}I_{a^{+}}^{n-{\mathfrak{P}}_1,\ln \left(\sigma \right)}\mathrm{\Phi }(\sigma ),$$

  (2)
  where $n=[\Re \left({\mathfrak{P}}_1\right)]+1$ and ${}_{{\mathfrak{P}}_0}D^{n,\ln \left(\sigma \right)}=\underset{n}{\underbrace{{}_{{\mathfrak{P}}_0}D^{\ln \left(\sigma \right)}\cdots {}_{{\mathfrak{P}}_0}D^{\ln \left(\sigma \right)}}}.$
• Let ${\mathfrak{P}}_1\in \mathbb{C},\Re \left({\mathfrak{P}}_1\right)>0,0<{\mathfrak{P}}_0\le 1.$ The Hadamard–Caputo proportional fractional derivative of order ${\mathfrak{P}}_1$ is given as

  $${}_{{\mathfrak{P}}_0}{}^{HC}D_{a^{+}}^{{\mathfrak{P}}_1,\ln \left(\sigma \right)}\mathrm{\Phi }(\sigma )={}_{{\mathfrak{P}}_0}I_{a^{+}}^{n-\ln \left(\sigma \right)}{}_{{\mathfrak{P}}_0}D^{n,\ln \left(\sigma \right)}\mathrm{\Phi }(\sigma ).$$

  (3)
• Let $n-1<{\mathfrak{P}}_1<n,n\in \mathbb{N},0<{\mathfrak{P}}_0\le 1,0\le {\mathfrak{P}}_2\le 1,$ and $\mathrm{\Phi }\in C^n\left([a,b]\right),-\infty <a<b<+\infty .$ The Hadamard–Hilfer proportional fractional derivative of order ${\mathfrak{P}}_1$ and type ${\mathfrak{P}}_2$ is given by

  $$\begin{array}{ccc}\hfill {}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\mathrm{\Phi }(\sigma )& =& {}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_2(n-{\mathfrak{P}}_1),\ln \left(\sigma \right)}({}_{{\mathfrak{P}}_0}D^{n,\ln \left(\sigma \right)}){}_{{\mathfrak{P}}_0}I_{a^{+}}^{(1-{\mathfrak{P}}_2)(n-{\mathfrak{P}}_1),\ln \left(\sigma \right)}\mathrm{\Phi }(\sigma )\hfill \\ & =& {}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_2(n-{\mathfrak{P}}_1),\ln \left(\sigma \right)}{}_{{\mathfrak{P}}_0}D_{a^{+}}^{{\mathfrak{P}}_3,\ln \left(\sigma \right)}\mathrm{\Phi }(\sigma ).\hfill \end{array}$$

  (4)

Lemma 2.

([11,19,20,21]). We note the following results:

• Let $0<\Re \left({\mathfrak{P}}_2\right)<\Re \left({\mathfrak{P}}_1\right),\Re \left({\mathfrak{P}}_1\right),\Re \left({\mathfrak{P}}_2\right)\in (n-1,n],n\in \mathbb{N},$ and $0<{\mathfrak{P}}_0\le 1.$ Then,

  $$\begin{array}{c}\hfill {}_{{\mathfrak{P}}_0}D_{a^{+}}^{{\mathfrak{P}}_2,\ln \left(\sigma \right)}{}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_1,\ln \left(\sigma \right)}\mathrm{\Phi }(\sigma )={}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_1-{\mathfrak{P}}_2,\ln \left(\sigma \right)}\mathrm{\Phi }(\sigma ).\end{array}$$
• Let $0<{\mathfrak{P}}_0\le 1,$ $n-1<{\mathfrak{P}}_1<n,n\in \mathbb{N},0\le {\mathfrak{P}}_2\le 1,{\mathfrak{P}}_3={\mathfrak{P}}_1+{\mathfrak{P}}_2(n-{\mathfrak{P}}_1),$ such that $n-1<{\mathfrak{P}}_3<n.$ If $\mathrm{\Phi }\in C_{{\mathfrak{P}}_3}\left([a,b]\right)$ and ${}_{{\mathfrak{P}}_0}I_{a^{+}}^{n-{\mathfrak{P}}_3,\ln \left(\sigma \right)}\mathrm{\Phi }\in C_{{\mathfrak{P}}_3,\ln (\sigma )}^n\left([a,b]\right),$ then,

  $$\begin{array}{c}\hfill \\ & & {}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_1,\ln \left(\sigma \right)}{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\mathrm{\Phi }(\sigma )\hfill \\ & =& \mathrm{\Phi }(\sigma )-\sum _{j=1}^n{\displaystyle \frac{{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{\left[\ln \left(\frac{\sigma }{a}\right)\right]}^{{\mathfrak{P}}_3-j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-j}\Gamma ({\mathfrak{P}}_3-j+1)}}({}_{{\mathfrak{P}}_0}I_{a^{+}}^{j-{\mathfrak{P}}_3,\ln \left(\sigma \right)}\mathrm{\Phi }(a)).\hfill \end{array}$$

  (5)
• Let ${\mathfrak{P}}_4,{\mathfrak{P}}_1\in (n-1,n),n\in \mathbb{N},{\mathfrak{P}}_2\in [0,1],{\mathfrak{P}}_0\in (0,1],$ and ${\mathfrak{P}}_4\ge {\mathfrak{P}}_1+{\mathfrak{P}}_2(n-{\mathfrak{P}}_1).$ If $\mathrm{\Phi }\in C^n\left([a,b]\right),$ then,

  $$\begin{array}{c}\hfill {}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}{}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_4,\ln \left(\sigma \right)}\mathrm{\Phi }(\sigma )={}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_4-{\mathfrak{P}}_1,\ln \left(\sigma \right)}\mathrm{\Phi }(\sigma ).\end{array}$$

  (6)
• Let ${\mathfrak{P}}_1\ge 0,$ and ${\mathfrak{P}}_2>0.$ Then, for ${\mathfrak{P}}_0\in (0,1]$ and $n=\left[{\mathfrak{P}}_1\right]+1,$ we obtain

  $$\begin{array}{c}\hfill {}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_1,\ln \left(\sigma \right)}\left({({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_2-1}\right)={\displaystyle \frac{\Gamma \left({\mathfrak{P}}_2\right)}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_1+{\mathfrak{P}}_2)}}{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1+{\mathfrak{P}}_2-1}.\end{array}$$

  (7)
• Let $n-1<{\mathfrak{P}}_1<n,n=\left[{\mathfrak{P}}_1\right]+1,{\mathfrak{P}}_2\in [0,1],{\mathfrak{P}}_0\in (0,1],{\mathfrak{P}}_3={\mathfrak{P}}_1+{\mathfrak{P}}_2(n-{\mathfrak{P}}_1).$ Then, for ${\mathfrak{P}}_4\in \mathbb{R},$ with ${\mathfrak{P}}_4>n,$ one has

  $$\begin{array}{c}\hfill {}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\left({({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_4-1}\right)={\displaystyle \frac{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_4\right)}{\Gamma ({\mathfrak{P}}_4-{\mathfrak{P}}_1)}}{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_4-{\mathfrak{P}}_1-1}.\end{array}$$

  (8)

2.3. Some New Fractional Calculus Connections Between Mittag–Leffler Functions

Lemma 3.

([11,18,22]). We have the following results:

(L1) 

Let $0<{\mathfrak{P}}_0\le 1,{\mathfrak{P}}_1,{\mathfrak{P}}_2,{\mathfrak{P}}_3,{\mathfrak{P}}_4\in {\mathbb{R}}^{+},$ and $\rho \in \mathbb{R}.$ Then,

$$\begin{array}{ccc}\hfill & & {}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\left[{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_4-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_3,{\mathfrak{P}}_4}\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_3}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3}\right)\right]\hfill \\ & & ={\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{\left[\ln ({\displaystyle \frac{\sigma }{a}})\right]}^{{\mathfrak{P}}_4-{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_3,{\mathfrak{P}}_4-{\mathfrak{P}}_1}\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_3}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3}\right).\hfill \end{array}$$

(9)

(L2) 

Let $0<{\mathfrak{P}}_0\le 1,{\mathfrak{P}}_1,{\mathfrak{P}}_2,{\mathfrak{P}}_3,{\mathfrak{P}}_4\in {\mathbb{R}}^{+},$ and $\rho \in \mathbb{R}.$ Then,

$$\begin{array}{ccc}\hfill & & {}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_1,\ln \left(\sigma \right)}\left[{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_4-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_3,{\mathfrak{P}}_4}\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_3}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3}\right)\right]\hfill \\ & & ={\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{\left[\ln ({\displaystyle \frac{\sigma }{a}})\right]}^{{\mathfrak{P}}_4+{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_3,{\mathfrak{P}}_4+{\mathfrak{P}}_1}\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_3}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3}\right).\hfill \end{array}$$

(10)

(L3) 

Let ${\mathfrak{P}}_1>0,{\mathfrak{P}}_0\in (0,1],\theta >0,\rho ,x\in \mathbb{R},$ and $\mathrm{\Phi }\in C\left(\right[a,b\left]\right).$ Then,

$$\begin{array}{ccc}\hfill & & {}_{{\mathfrak{P}}_0}I_{a^{+}}^{\theta ,\ln \left(\sigma \right)}\left[{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(\omega ){\displaystyle \frac{d\omega }{\omega }}\right]\hfill \\ & & ={\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{\theta }}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{\left[\ln ({\displaystyle \frac{\sigma }{\omega }})\right]}^{{\mathfrak{P}}_1+\theta -1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1+\theta }\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }\left(\omega \right){\displaystyle \frac{d\omega }{\omega }}.\hfill \end{array}$$

(11)

(L4) 

Let $(n+1){\mathfrak{P}}_1>\theta >0,{\mathfrak{P}}_2>0,{\mathfrak{P}}_0\in (0,1],\rho ,x\in \mathbb{R}$ and $\mathrm{\Phi }\in C\left(\right[a,b\left]\right).$ Then,

$$\begin{array}{ccc}\hfill & & {}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{\theta ,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\left[{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }\left(\omega \right){\displaystyle \frac{d\omega }{\omega }}\right]\hfill \\ & & ={\left({\mathfrak{P}}_0\right)}^{\theta }{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1-\theta -1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1-\theta }\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }\left(\omega \right){\displaystyle \frac{d\omega }{\omega }}.\hfill \end{array}$$

(12)

Proof. 

(L1)

Making use of Definition 2 and Lemmas 1 and 2, we have that

$$\begin{array}{ccc}& & {}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\left[{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_4-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_3,{\mathfrak{P}}_4}\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_3}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3}\right)\right]\hfill \\ & & ={}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\left[{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_4-1}\sum _{n=0}^{\infty }{\displaystyle \frac{1}{\Gamma (n{\mathfrak{P}}_3+{\mathfrak{P}}_4)}}{\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_3}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3}\right)}^n\right]\hfill \\ & & =\sum _{n=0}^{\infty }{\displaystyle \frac{{\rho }^n}{{\left({\mathfrak{P}}_0\right)}^{n{\mathfrak{P}}_3}\Gamma (n{\mathfrak{P}}_3+{\mathfrak{P}}_4)}}{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\left[{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{n{\mathfrak{P}}_3+{\mathfrak{P}}_4-1}\right]\hfill \\ & & =\sum _{n=0}^{\infty }{\displaystyle \frac{{\rho }^n}{{\left({\mathfrak{P}}_0\right)}^{n{\mathfrak{P}}_3}\Gamma (n{\mathfrak{P}}_3+{\mathfrak{P}}_4)}}\left[{\displaystyle \frac{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma (n{\mathfrak{P}}_3+{\mathfrak{P}}_4)}{\Gamma (n{\mathfrak{P}}_3+{\mathfrak{P}}_4-{\mathfrak{P}}_1)}}{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{n{\mathfrak{P}}_3+{\mathfrak{P}}_4-{\mathfrak{P}}_1-1}\right]\hfill \\ & & ={\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_4-{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_3,{\mathfrak{P}}_4-{\mathfrak{P}}_1}\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_3}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3}\right).\hfill \end{array}$$

(L2)

Applying Definition 2 and Lemmas 1 and 2, we have that

$$\begin{array}{ccc}& & {}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_1,\ln \left(\sigma \right)}\left[{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_4-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_3,{\mathfrak{P}}_4}\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_3}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3}\right)\right]\hfill \\ & & ={\displaystyle \frac{1}{\Gamma \left({\mathfrak{P}}_1\right)}}{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times \left[{({\displaystyle \frac{\omega }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\omega }{a}})]}^{{\mathfrak{P}}_4-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_3,{\mathfrak{P}}_4}\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_3}{[\ln ({\displaystyle \frac{\omega }{a}})]}^{{\mathfrak{P}}_3}\right)\right]{\displaystyle \frac{d\omega }{\omega }}\hfill \\ & & ={\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1-1}{[\ln ({\displaystyle \frac{\omega }{a}})]}^{{\mathfrak{P}}_4-1}\hfill \\ & & \times \left[\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_3}{[\ln (\frac{\omega }{a})]}^{{\mathfrak{P}}_3}\right)}^n}{\Gamma (n{\mathfrak{P}}_3+{\mathfrak{P}}_4)}}\right]{\displaystyle \frac{d\omega }{\omega }}\hfill \\ & & ={\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_4+{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_3,{\mathfrak{P}}_4+{\mathfrak{P}}_1}\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_3}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3}\right).\hfill \end{array}$$

(L3)

By Definition (2) and (L2), one has

$$\begin{array}{ccc}& & {}_{{\mathfrak{P}}_0}I_{a^{+}}^{\theta ,\ln \left(\sigma \right)}\left[{\int }_a^x{({\displaystyle \frac{x}{\sigma }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{x}{\sigma }})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{x}{\sigma }})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }\left(\sigma \right){\displaystyle \frac{d\omega }{\sigma }}\right]\hfill \\ & & ={\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{\theta }\Gamma \left(\theta \right)}}{\int }_a^x{({\displaystyle \frac{x}{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{x}{\omega }})]}^{\theta -1}\hfill \\ & & \times \left[{\int }_a^{\omega }{({\displaystyle \frac{\omega }{\sigma }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\omega }{\sigma }})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\omega }{\sigma }})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }\left(\sigma \right){\displaystyle \frac{d\sigma }{\sigma }}\right]{\displaystyle \frac{d\omega }{\omega }}\hfill \\ & & ={\int }_a^x{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{\theta }\Gamma \left(\theta \right)}}{\int }_{\sigma }^x{({\displaystyle \frac{x}{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{x}{\omega }})]}^{\theta -1}\hfill \\ & & \times \left[{({\displaystyle \frac{\omega }{\sigma }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\omega }{\sigma }})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\omega }{\sigma }})]}^{{\mathfrak{P}}_1}\right){\displaystyle \frac{d\omega }{\omega }}\right]\mathrm{\Phi }(\sigma ){\displaystyle \frac{d\sigma }{\sigma }}\hfill \\ & & ={\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{\theta }}}{\int }_a^x{({\displaystyle \frac{x}{\sigma }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{x}{\sigma }})]}^{{\mathfrak{P}}_1+\theta -1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1+\theta }\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{x}{\sigma }})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(\sigma ){\displaystyle \frac{d\sigma }{\sigma }}.\hfill \end{array}$$

(L4)

Using (L2), Definition (2), and Lemma (2), we get

$$\begin{array}{ccc}& & {}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{\theta ,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\left[{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }\left(\omega \right){\displaystyle \frac{d\omega }{\omega }}\right]\hfill \\ & & ={}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{\theta ,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\left[\sum _{n=0}^{\infty }{\displaystyle \frac{{\rho }^n}{{\left({\mathfrak{P}}_0\right)}^{n{\mathfrak{P}}_1}\Gamma (n{\mathfrak{P}}_1+{\mathfrak{P}}_1)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{n{\mathfrak{P}}_1+{\mathfrak{P}}_1-1}\mathrm{\Phi }\left(\omega \right){\displaystyle \frac{d\omega }{\omega }}\right]\hfill \\ & & ={\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\sum _{n=0}^{\infty }{\rho }^n\left[{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{\theta ,{\mathfrak{P}}_2,\ln \left(\sigma \right)}{}_{{\mathfrak{P}}_0}I_{a^{+}}^{n{\mathfrak{P}}_1+{\mathfrak{P}}_1,\ln \left(\sigma \right)}\mathrm{\Phi }\left(\sigma \right)\right]\hfill \\ & & ={\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\sum _{n=0}^{\infty }{\rho }^n\left[{}_{{\mathfrak{P}}_0}I_{a^{+}}^{n{\mathfrak{P}}_1+{\mathfrak{P}}_1-\theta ,\ln \left(\sigma \right)}\mathrm{\Phi }\left(\sigma \right)\right]\hfill \\ & & ={\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\sum _{n=0}^{\infty }{\rho }^n[{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{n{\mathfrak{P}}_1+{\mathfrak{P}}_1-\theta }\Gamma (n{\mathfrak{P}}_1+{\mathfrak{P}}_1-\theta )}}\hfill \\ & & \times {\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{n{\mathfrak{P}}_1+{\mathfrak{P}}_1-\theta -1}\mathrm{\Phi }\left(\omega \right){\displaystyle \frac{d\omega }{\omega }}\hfill \\ & & ={\left({\mathfrak{P}}_0\right)}^{\theta }{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1-\theta -1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1-\theta }\left(\rho {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }\left(\omega \right){\displaystyle \frac{d\omega }{\omega }}.\hfill \end{array}$$

□

2.4. Extending the Gronwall Inequality via Proportional Fractional Operators

Theorem 1.

([11,12,23]).

(T1) 

Let ${\mathfrak{P}}_0,{\mathfrak{P}}_1>0,$ and $x,y$ be two non-negative functions locally integrable on $[a,b]$, and z be a non-negative, continuous and non-decreasing function on $[a,b]$. If

$$\begin{array}{c}\hfill x\left(\sigma \right)\le y\left(\sigma \right)+{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)z\left(\sigma \right)\left({}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_1,\ln \left(\sigma \right)}\left[x\left(\sigma \right)\right]\right),\end{array}$$

(13)

then

$$\begin{array}{c}\hfill x\left(\sigma \right)\le y\left(\sigma \right)+{\int }_a^{\sigma }\left[\sum _{m=1}^{\infty }{\displaystyle \frac{{[z\left(\sigma \right)\Gamma \left({\mathfrak{P}}_1\right)]}^m}{\Gamma \left(m{\mathfrak{P}}_1\right)}}{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{m{\mathfrak{P}}_1-1}{\displaystyle \frac{y\left(\omega \right)}{\omega }}\right]d\omega ,\sigma \in [a,b].\end{array}$$

(14)

(T2) 

Let ${\mathfrak{P}}_0,{\mathfrak{P}}_1>0,$ $c>0,$$x,y$ be non-negative functions locally integrable on $[a,b]$ and $z(\sigma )\equiv c$ for $\sigma \in [a,b]$. If

$$\begin{array}{c}\hfill x\left(\sigma \right)\le y\left(\sigma \right)+{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)c({}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_1,\ln \left(\sigma \right)}(x(\sigma ))),\sigma \in [a,b],\end{array}$$

(15)

then

$$\begin{array}{c}\hfill x\left(\sigma \right)\le y\left(\sigma \right)+{\int }_a^{\sigma }\left[\sum _{m=1}^{\infty }{\displaystyle \frac{{(c\Gamma \left({\mathfrak{P}}_1\right))}^m}{\Gamma \left(m{\mathfrak{P}}_1\right)}}{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{m{\mathfrak{P}}_1-1}{\displaystyle \frac{y\left(\omega \right)}{\omega }}\right]d\omega ,\sigma \in [a,b].\end{array}$$

(16)

(T3) 

Let the assumptions in (T1) hold and let y be a non-decreasing function on $[a,b].$ Then,

$$\begin{array}{c}\hfill x\left(\sigma \right)\le y\left(\sigma \right){\mathrm{\Xi }}_{{\mathfrak{P}}_1}\left(z\left(\sigma \right)\Gamma \left({\mathfrak{P}}_1\right){[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right),\sigma \in [a,b].\end{array}$$

(17)

Proof. 

(T1)

Consider for $\sigma \in [a,b]$,

$$L\varphi (\sigma )=z(\sigma ){\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1-1}\varphi \left(\omega \right){\displaystyle \frac{d\omega }{\omega }},$$

(18)

and note that

$$x\left(\sigma \right)\le y\left(\sigma \right)+Lx\left(\sigma \right).$$

(19)

For every $m\in \mathbb{N},$

$$x(\sigma )\le \sum _{k=0}^{m-1}{L}^{k}y(\sigma )+L^{m}x(\sigma ).$$

(20)

We now claim that

$$L^{m}x(\sigma )\le {\int }_a^{\sigma }{\displaystyle \frac{{[z\left(\sigma \right)\Gamma ({\mathfrak{P}}_1)]}^m}{\Gamma \left(m{\mathfrak{P}}_1\right)}}{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{m{\mathfrak{P}}_1-1}x\left(\omega \right){\displaystyle \frac{d\omega }{\omega }},$$

(21)

and $L^{m}x(\sigma )⟶0,$ as $m⟶0,$ for every $\sigma \in [a,b].$ For $m=1,$ the inequality (21) is true. Assume the formula is true for some $m=k\in \mathbb{N},$ i.e.,

$$L^{k}x(\sigma )\le {\int }_a^{\sigma }{\displaystyle \frac{{[z\left(\sigma \right)\Gamma ({\mathfrak{P}}_1)]}^k}{\Gamma \left(k{\mathfrak{P}}_1\right)}}{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{k{\mathfrak{P}}_1-1}x\left(\omega \right){\displaystyle \frac{d\omega }{\omega }}.$$

For $m=k+1,$

$$\begin{array}{ccc}\hfill L^{k+1}x\left(\sigma \right)& =& L\left(L^{k}x\left(\sigma \right)\right)\hfill \\ & \le & L\left({\int }_a^{\sigma }{\displaystyle \frac{{[z\left(\sigma \right)\Gamma \left({\mathfrak{P}}_1\right)]}^k}{\Gamma \left(k{\mathfrak{P}}_1\right)}}{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{k{\mathfrak{P}}_1-1}x\left(\omega \right){\displaystyle \frac{d\omega }{\omega }}\right)\hfill \\ & =& z\left(\sigma \right){\int }_a^{\sigma }{\displaystyle \frac{1}{\omega }}{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times \left[{\int }_a^{\sigma }{\displaystyle \frac{{[z\left(w\right)\Gamma \left({\mathfrak{P}}_1\right)]}^k}{\Gamma \left(k{\mathfrak{P}}_1\right)}}{({\displaystyle \frac{\omega }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\omega }{s}})]}^{k{\mathfrak{P}}_1-1}x\left(s\right){\displaystyle \frac{ds}{s}}\right]d\omega ,\hfill \end{array}$$

(22)

and since $z\left(\omega \right)\le z\left(\sigma \right)$ for $\omega \le \sigma$, we have

$$\begin{array}{ccc}\hfill & & L^{k+1}x\left(\sigma \right)\hfill \\ & & \le {\displaystyle \frac{z^{k+1}\left(\sigma \right){\Gamma }^k\left({\mathfrak{P}}_1\right)}{\Gamma \left(k{\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{\int }_a^{\omega }{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1-1}{({\displaystyle \frac{\omega }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\omega }{s}})]}^{k{\mathfrak{P}}_1-1}x\left(s\right){\displaystyle \frac{dsd\omega }{s\omega }}.\hfill \end{array}$$

(23)

Next, note that

$$\begin{array}{ccc}\hfill & & L^{k+1}x\left(\sigma \right)\hfill \\ & \le & {\displaystyle \frac{z^{k+1}\left(\sigma \right){\Gamma }^k\left({\mathfrak{P}}_1\right)}{\Gamma \left(k{\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }\left[{\int }_s^{\sigma }{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1-1}{({\displaystyle \frac{\omega }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\omega }{s}})]}^{k{\mathfrak{P}}_1-1}{\displaystyle \frac{d\omega }{\omega }}\right]x\left(s\right){\displaystyle \frac{ds}{s}}.\hfill \end{array}$$

(24)

Setting $\ln ({\displaystyle \frac{\omega }{s}})=\eta \ln ({\displaystyle \frac{\sigma }{s}})$ and using the property of the beta function, we have that

$$\begin{array}{ccc}& & {\int }_s^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1-1}{[\ln ({\displaystyle \frac{\omega }{s}})]}^{k{\mathfrak{P}}_1-1}{\displaystyle \frac{d\omega }{\omega }}\hfill \\ & & ={({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{k{\mathfrak{P}}_1+{\mathfrak{P}}_1-1}{\int }_0^1{(1-\eta )}^{{\mathfrak{P}}_1-1}{\eta }^{k{\mathfrak{P}}_1-1}d\eta \hfill \\ & & ={\displaystyle \frac{\Gamma \left({\mathfrak{P}}_1\right)\Gamma \left(k{\mathfrak{P}}_1\right)}{\Gamma (k{\mathfrak{P}}_1+{\mathfrak{P}}_1)}}{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{k{\mathfrak{P}}_1+{\mathfrak{P}}_1-1}.\hfill \end{array}$$

Hence, we get

$$\begin{array}{c}\hfill L^{k+1}x\left(\sigma \right)\le {\displaystyle \frac{z^{k+1}\left(\sigma \right){\Gamma }^{k+1}\left({\mathfrak{P}}_1\right)}{\Gamma \left((k+1){\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{k{\mathfrak{P}}_1+{\mathfrak{P}}_1-1}x\left(s\right){\displaystyle \frac{ds}{s}}.\end{array}$$

(25)

Now, we claim $L^{m}x(\sigma )⟶0$ as $m⟶\infty$. Since $z\in C\left(\right[a,b],\mathbb{R})$, there is a $\delta >0,$ such that $z\left(\sigma \right)\le \delta$, and so, for $\sigma \in [a,b],$ we have

$$\begin{array}{c}\hfill L^{m}x\left(\sigma \right)\le {\displaystyle \frac{{[\delta \Gamma \left({\mathfrak{P}}_1\right)]}^m}{\Gamma \left(m{\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }({\displaystyle \frac{\sigma }{s}}){\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{m{\mathfrak{P}}_1-1}x\left(s\right){\displaystyle \frac{ds}{s}},\end{array}$$

(26)

which immediately (ratio test) guarantees the claim.

Now, we have

$$\begin{array}{ccc}\hfill x\left(\sigma \right)& \le & \sum _{k=0}^{\infty }{L}^{k}y\left(\sigma \right)\hfill \\ & \le & y\left(\sigma \right)+{\int }_a^{\sigma }\sum _{k=1}^{\infty }{\displaystyle \frac{{[z\left(\sigma \right)\Gamma \left({\mathfrak{P}}_1\right)]}^k}{\Gamma \left(k{\mathfrak{P}}_1\right)}}{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{k{\mathfrak{P}}_1-1}y\left(s\right){\displaystyle \frac{ds}{s}}.\hfill \end{array}$$

(27)

(T2)

Set $z\left(\sigma \right)\equiv c$ in (T1) and we obtain the desired result.

(T3)

From (14) and the fact that for $\sigma \in [a,b]$ we have $y\left(\omega \right)\le y\left(\sigma \right)$,

$$\begin{array}{ccc}\hfill x\left(\sigma \right)& \le & y\left(\sigma \right)+{\int }_a^{\sigma }\left[\sum _{m=1}^{\infty }{\displaystyle \frac{{[z\left(\sigma \right)\Gamma \left({\mathfrak{P}}_1\right)]}^m}{\Gamma \left(m{\mathfrak{P}}_1\right)}}{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{m{\mathfrak{P}}_1-1}{\displaystyle \frac{y\left(\omega \right)}{\omega }}\right]d\omega \hfill \\ & \le & y\left(\sigma \right)[1+\sum _{m=1}^{\infty }{\left({\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}z\left(\sigma \right)\Gamma \left({\mathfrak{P}}_1\right)\right)}^m{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{m{\mathfrak{P}}_1}\phi \left(m{\mathfrak{P}}_1\right)}}\hfill \\ & & \times {\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{\omega }})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{m{\mathfrak{P}}_1-1}{\displaystyle \frac{d\omega }{\omega }}].\hfill \end{array}$$

Applying the fact that $0<{\left({\displaystyle \frac{\sigma }{\omega }}\right)}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}\le 1,$ for $a\le \omega \le \sigma \le b,$ we have

$$\begin{array}{ccc}\hfill x\left(\sigma \right)& \le & y\left(\sigma \right)\left[1+\sum _{m=1}^{\infty }{\left({\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}z\left(\sigma \right)\Gamma \left({\mathfrak{P}}_1\right)\right)}^m{\displaystyle \frac{{[\ln (\frac{\sigma }{\omega })]}^{m{\mathfrak{P}}_1}}{{\left({\mathfrak{P}}_0\right)}^{m{\mathfrak{P}}_1}\Gamma (m{\mathfrak{P}}_1+1)}}\right]\hfill \\ & =& y\left(\sigma \right)\sum _{m=0}^{\infty }{\displaystyle \frac{{(z\left(\sigma \right)\Gamma \left({\mathfrak{P}}_1\right))}^m{[\ln (\frac{\sigma }{\omega })]}^{m{\mathfrak{P}}_1}}{\Gamma (m{\mathfrak{P}}_1+1)}}\hfill \\ & =& y\left(\sigma \right){\mathrm{\Xi }}_{{\mathfrak{P}}_1}\left(z\left(\sigma \right)\Gamma \left({\mathfrak{P}}_1\right){[\ln ({\displaystyle \frac{\sigma }{\omega }})]}^{{\mathfrak{P}}_1}\right).\hfill \end{array}$$

□

3. Description of Solutions for Hilfer–Hadamard–Cauchy-Type Problems

The Hilfer–Hadamard fractional derivative holds particular significance compared to other fractional derivatives such as Caputo, Hilfer, and Riemann–Liouville because it offers a versatile framework that interpolates between different types of fractional derivatives. Unlike the Caputo or Riemann–Liouville derivatives, which are often restricted to specific initial conditions, the Hilfer–Hadamard derivative incorporates a parameter that allows for a smooth transition between these traditional derivatives, enabling more flexible modeling of complex phenomena involving memory effects and non-local behaviors. Its logarithmic kernel makes it especially suitable for systems exhibiting multiplicative or scale-invariant properties, which are not as effectively captured by the other derivatives. Consequently, this derivative enhances the ability to describe processes with evolving memory characteristics more accurately, making it a valuable tool in both theoretical analysis and practical applications across various scientific disciplines [12].

Below, we will examine and analyze the solutions of Hilfer–Hadamard–Cauchy- type problems.

3.1. On the Solutions of Proportional Fractional Systems with Constant Coefficients

Making use of Picard’s successive approximation method, we investigate the relationship between the following Hilfer–Hadamard proportional fractional differential equations with constant coefficients and the integral equation in the form of a Mittag–Leffler function. Consider

$$\left\{\begin{array}{c}\begin{array}{cc}{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\varphi \left(\sigma \right)=\eta \varphi \left(\sigma \right)+\mathrm{\Phi }\left(\sigma \right),& \sigma \in [a,b],\\ {}_{{\mathfrak{P}}_0}I_{a^{+}}^{j-{\mathfrak{P}}_3,\ln \left(\sigma \right)}\varphi \left(a\right)=c_j,& j=1,\cdots ,n,\end{array}\hfill \end{array}\right.$$

(28)

where ${\mathfrak{P}}_1\in (n-1,n),{\mathfrak{P}}_2\in [0,1],{\mathfrak{P}}_0\in (0,1],\eta <0,$ ${\mathfrak{P}}_1\le {\mathfrak{P}}_3={\mathfrak{P}}_1+(n-{\mathfrak{P}}_1){\mathfrak{P}}_2,$ $j-{\mathfrak{P}}_3>0,$ and $c_i\in \mathbb{R}.$

Lemma 4.

Let $\mathrm{\Phi }\in C([a,b],\mathbb{R}),\eta \in \mathbb{R},{\mathfrak{P}}_1\in (n-1,n)$ and ${\mathfrak{P}}_0\in (0,1].$ Then, the exact solution of (28) is given by

$$\begin{array}{ccc}\hfill \varphi (\sigma )& =& \sum _{j=1}^n{\displaystyle \frac{c_j}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-j}}}{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3-j}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_3-j+1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & +{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }\left(s\right){\displaystyle \frac{ds}{s}}.\hfill \end{array}$$

(29)

Proof. 

Suppose $\varphi$ is a solution of (28). From Lemma 2, we have that

$$\begin{array}{ccc}\hfill \varphi (\sigma )& =& \sum _{j=1}^n{\displaystyle \frac{c_j{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-k}\Gamma ({\mathfrak{P}}_3-j+1)}}+{\displaystyle \frac{\eta }{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\varphi \left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ & & +{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\mathrm{\Phi }\left(s\right){\displaystyle \frac{ds}{s}},\hfill \end{array}$$

where $c_j={}_{{\mathfrak{P}}_0}I_{a^{+}}^{j-{\mathfrak{P}}_3,\ln \left(\sigma \right)}\varphi \left(a\right),$ for $j=1,\cdots ,n.$ The method of successive approximation is applied to obtain an exact form of the solution. Consider

$$\begin{array}{ccc}& & {\varphi }_0\left(\sigma \right)=\sum _{j=1}^n{\displaystyle \frac{c_j{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-j}\Gamma ({\mathfrak{P}}_3-j+1)}},\hfill \\ & & {\varphi }_k\left(\sigma \right)={\varphi }_0\left(\sigma \right)+{\displaystyle \frac{\eta }{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}{\varphi }_{k-1}\left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ & & +{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\mathrm{\Phi }\left(s\right){\displaystyle \frac{ds}{s}},k=1,2,\cdots .\hfill \end{array}$$

Using Definition 2 and Lemma 2, for $k=1,$ we get

$$\begin{array}{ccc}& & {\varphi }_1\left(\sigma \right)={\varphi }_0\left(\sigma \right)+{\displaystyle \frac{\eta }{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}{\varphi }_0\left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ & & +{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\mathrm{\Phi }\left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ & & =\sum _{j=1}^n{\displaystyle \frac{c_j{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-j}\Gamma ({\mathfrak{P}}_3-j+1)}}\hfill \\ & & +{\displaystyle \frac{\eta }{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\sum _{j=1}^n{\displaystyle \frac{c_j{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-j}\Gamma ({\mathfrak{P}}_3-j+1)}}{\displaystyle \frac{ds}{s}}\hfill \\ & & +{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\mathrm{\Phi }\left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ & & ={\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\mathrm{\Phi }\left(s\right){\displaystyle \frac{ds}{s}}+\sum _{j=1}^n{\displaystyle \frac{c_j{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-j}\Gamma ({\mathfrak{P}}_3-j+1)}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{\eta c_j{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-j}\Gamma ({\mathfrak{P}}_3-j+1)}}\left[{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}{[\ln ({\displaystyle \frac{s}{a}})]}^{{\mathfrak{P}}_3-j}{\displaystyle \frac{ds}{s}}\right]\hfill \\ & & ={\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\mathrm{\Phi }\left(s\right){\displaystyle \frac{ds}{s}}+\sum _{j=1}^n{\displaystyle \frac{c_j{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-j}\Gamma ({\mathfrak{P}}_3-j+1)}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{\eta c_j{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_1+{\mathfrak{P}}_3-j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-j}\Gamma ({\mathfrak{P}}_3-j+1)}}\hfill \\ & & \times \left[{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_0^1{\phi }^{({\mathfrak{P}}_3-j+1)-1}{(1-\phi )}^{{\mathfrak{P}}_1-1}d\phi \right]\hfill \\ & & =\sum _{j=1}^n{\displaystyle \frac{c_j{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-j}}}\left[{\displaystyle \frac{1}{\Gamma ({\mathfrak{P}}_3-j+1)}}+{\displaystyle \frac{\eta {[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_1}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_1+{\mathfrak{P}}_3-j+1)}}\right]\hfill \\ & & +{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\mathrm{\Phi }\left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ & & =\sum _{j=1}^n{\displaystyle \frac{c_j{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-j}}}\left[\sum _{i=1}^2{\displaystyle \frac{{\eta }_{i-1}{[\ln (\frac{\sigma }{a})]}^{(i-1){\mathfrak{P}}_1}}{{\left({\mathfrak{P}}_0\right)}^{(i-1){\mathfrak{P}}_1}\Gamma ((i-1){\mathfrak{P}}_1+{\mathfrak{P}}_3-j+1)}}\right]\hfill \\ & & +{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\mathrm{\Phi }\left(s\right){\displaystyle \frac{ds}{s}}.\hfill \end{array}$$

In a similar way, repeating the same procedure, for $k=1,2,\dots ,m,$ one has

$$\begin{array}{ccc}\hfill {\varphi }_k\left(\sigma \right)& =& \sum _{j=1}^n{\displaystyle \frac{c_j{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-j}}}\left[\sum _{i=1}^{m+1}{\displaystyle \frac{{\eta }^{i-1}{[\ln (\frac{\sigma }{a})]}^{(i-1){\mathfrak{P}}_1}}{{\left({\mathfrak{P}}_0\right)}^{(i-1){\mathfrak{P}}_1}\Gamma ((i-1){\mathfrak{P}}_1+{\mathfrak{P}}_3-j+1)}}\right]\hfill \\ & & +{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}\sum _{i=1}^m{\displaystyle \frac{{\eta }^{i-1}}{{\left({\mathfrak{P}}_0\right)}^{i{\mathfrak{P}}_1}\Gamma \left(i{\mathfrak{P}}_1\right)}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{i{\mathfrak{P}}_1-1}\mathrm{\Phi }\left(s\right){\displaystyle \frac{ds}{s}}.\hfill \end{array}$$

If we proceed inductively and let $m⟶\infty ,$ we have

$$\begin{array}{ccc}\hfill \varphi (\sigma )& =& \sum _{j=1}^n{\displaystyle \frac{c_j{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-j}}}\left[\sum _{i=1}^{\infty }{\displaystyle \frac{{\eta }^{i-1}{[\ln (\frac{\sigma }{a})]}^{(i-1){\mathfrak{P}}_1}}{{\left({\mathfrak{P}}_0\right)}^{(i-1){\mathfrak{P}}_1}\Gamma ((i-1){\mathfrak{P}}_1+{\mathfrak{P}}_3-j+1)}}\right]\hfill \\ & & +{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}\sum _{i=1}^{\infty }{\displaystyle \frac{{\eta }^{i-1}}{{\left({\mathfrak{P}}_0\right)}^{i{\mathfrak{P}}_1}\Gamma \left(i{\mathfrak{P}}_1\right)}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{i{\mathfrak{P}}_1-1}\mathrm{\Phi }\left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ & =& \sum _{j=1}^n{\displaystyle \frac{c_j{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-j}}}\left[\sum _{i=0}^{\infty }{\displaystyle \frac{{\eta }^i{[\ln (\frac{\sigma }{a})]}^{i{\mathfrak{P}}_1}}{{\left({\mathfrak{P}}_0\right)}^{i{\mathfrak{P}}_1}\Gamma (i{\mathfrak{P}}_1+{\mathfrak{P}}_3-j+1)}}\right]\hfill \\ & & +{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\sum _{i=1}^{\infty }{\displaystyle \frac{{\eta }^i}{{\left({\mathfrak{P}}_0\right)}^{(i+1){\mathfrak{P}}_1}\Gamma \left((i+1){\mathfrak{P}}_1\right)}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{i{\mathfrak{P}}_1}\mathrm{\Phi }\left(s\right){\displaystyle \frac{ds}{s}}.\hfill \end{array}$$

□

3.2. On the Solutions of Proportional Fractional Systems with Mixed Boundary Conditions

Here, we investigate the relationship between the following Hilfer–Hadamard proportional fractional differential system with mixed boundary conditions and the integral equation. Consider

$$\left\{\begin{array}{c}\begin{array}{cc}{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\varphi \left(\sigma \right)=\eta \varphi \left(\sigma \right)+\mathrm{\Phi }(\sigma ,\varphi \left(\sigma \right)),\sigma \in (a,b],& \\ \varphi \left(a\right)=0,& \\ \sum _{i=1}^m{\mathrm{\Theta }}_{1i}\varphi \left({\sigma }_{1i}\right)+\sum _{j=1}^n{\mathrm{\Theta }}_{2j}{}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_{4_j},\ln \left(\sigma \right)}\varphi \left({\sigma }_{2j}\right)+\sum _{k=1}^r{\mathrm{\Theta }}_{3k}{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_{5k},{\mathfrak{P}}_2,\ln \left(\sigma \right)}\varphi \left({\sigma }_{3k}\right)=A,& \end{array}\hfill \end{array}\right.$$

(30)

where ${\mathfrak{P}}_0\in (0,1],{\mathfrak{P}}_1\in (1,2),{\mathfrak{P}}_2\in [0,1],{\mathfrak{P}}_3={\mathfrak{P}}_1+{\mathfrak{P}}_2(2-{\mathfrak{P}}_1),{\mathfrak{P}}_{4j}>0,1<{\mathfrak{P}}_{5k}<{\mathfrak{P}}_1\le 2,$ $\eta <0,{\mathrm{\Theta }}_{1i},{\mathrm{\Theta }}_{2j},{\mathrm{\Theta }}_{3k},A\in \mathbb{R},$ ${\sigma }_{1i},{\sigma }_{2j},{\sigma }_{3k}\in [a,b],i=1,2,\cdots ,m,j=1,2,\cdots ,n,$$k=1,2,\cdots ,r$ and $\mathrm{\Phi }\in C\left(\right[a,b]\times \mathbb{R},\mathbb{R}).$

Lemma 5.

Consider the system (30). Then, ϕ is a solution of (30) iff

$$\begin{array}{ccc}\hfill & & \varphi (\sigma )\hfill \\ & =& [A-\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Theta }}_{1i}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{1i}}{({\displaystyle \frac{{\sigma }_{1i}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & \times \mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}-\sum _{j=1}^n{\displaystyle \frac{{\mathrm{\Theta }}_{2j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{2j}}{({\displaystyle \frac{{\sigma }_{2j}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}\hfill \\ & & -\sum _{k=1}^r{\displaystyle \frac{{\mathrm{\Theta }}_{3k}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}}}{\int }_a^{{\sigma }_{3k}}{({\displaystyle \frac{{\sigma }_{3k}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}]\hfill \\ & & \times \left[{\left(X{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\right)}^{-1}{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_3}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right)\right]\hfill \\ & & {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}},\hfill \end{array}$$

(31)

where

$$\begin{array}{ccc}\hfill X& =& \sum _{i=1}^m{\theta }_{1i}{({\displaystyle \frac{{\sigma }_{1i}}{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{a}})]}^{{\mathfrak{P}}_3-1}{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_3+1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_3}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{{\mathrm{\Theta }}_{2j}{(\frac{{\sigma }_{2j}}{a})}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{{\sigma }_{2j}}{a})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_3-1}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_3-1}}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_{4j}+{\mathfrak{P}}_3}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{{\mathrm{\Theta }}_{3k}{(\frac{{\sigma }_{3k}}{a})}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{{\sigma }_{3k}}{a})]}^{{\mathfrak{P}}_3-{\mathfrak{P}}_{5k}-1}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-{\mathfrak{P}}_{5k}-1}}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_3-{\mathfrak{P}}_{5k}}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{a}})]}^{{\mathfrak{P}}_1}\right).\hfill \end{array}$$

(32)

Proof. 

Let $\varphi \in C\left(\right[a,b\left]\right)$ be a solution of (30). Making use of Lemma 4, (30) is equivalent to

$$\begin{array}{ccc}\hfill \varphi (\sigma )& =& {\displaystyle \frac{c_1{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-1}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_3}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & +{\displaystyle \frac{c_2{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-2}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-2}}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_3-1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & +{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}},\hfill \end{array}$$

(33)

where $c_1={}_{{\mathfrak{P}}_0}I_{a^{+}}^{1-{\mathfrak{P}}_3,\ln \left(\sigma \right)}\varphi \left(a\right)\in \mathbb{R}.$ Setting $\sigma =a$ in (33), with ${lim}_{\sigma ⟶a}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3-2}=\infty ,$ we obtain $c_2=0.$ Now, taking ${}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_{4j},\ln \left(\sigma \right)}$ and ${}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_{5k},{\mathfrak{P}}_2,\ln \left(\sigma \right)}$ into (33), we have that

$$\begin{array}{ccc}& & \sum _{i=1}^m{\mathrm{\Theta }}_{1i}\varphi \left({\sigma }_{1i}\right)\hfill \\ & =& c_1\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Theta }}_{1i}{\left(\frac{{\sigma }_{1i}}{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{{\sigma }_{1i}}{a})]}^{{\mathfrak{P}}_3-1}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_3}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & +\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Theta }}_{1i}{\left(\frac{{\sigma }_{1i}}{s}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{{\sigma }_{1i}}{s})]}^{{\mathfrak{P}}_1-1}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{1i}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}},\hfill \end{array}$$

$$\begin{array}{ccc}& & \sum _{j=1}^n{\mathrm{\Theta }}_{2j}{}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_{4j},\ln \left(\sigma \right)}\varphi \left({\sigma }_{2j}\right)\hfill \\ & =& c_1\sum _{j=1}^n{\displaystyle \frac{{\mathrm{\Theta }}_{2j}{(\frac{{\sigma }_{2j}}{a})}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{{\sigma }_{2j}}{a})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_3-1}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_3-1}}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_{4j}+{\mathfrak{P}}_3}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{{\mathrm{\Theta }}_{2j}{(\frac{{\sigma }_{2j}}{s})}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{{\sigma }_{2j}}{s})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}}}\hfill \\ & & \times {\int }_a^{{\sigma }_{2j}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}},\hfill \end{array}$$

and

$$\begin{array}{ccc}& & \sum _{k=1}^r{\mathrm{\Theta }}_{3k}{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_{5k},{\mathfrak{P}}_2,\ln \left(\sigma \right)}\varphi \left({\sigma }_{3k}\right)\hfill \\ & =& c_1\sum _{k=1}^r{\displaystyle \frac{{\mathrm{\Theta }}_{3k}{(\frac{{\sigma }_{3k}}{a})}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{{\sigma }_{3k}}{a})]}^{{\mathfrak{P}}_3-{\mathfrak{P}}_{5k}-1}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-{\mathfrak{P}}_{5k-1}}}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_3-{\mathfrak{P}}_{5k}}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{{\mathrm{\Theta }}_{3k}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}}}{\int }_a^{{\sigma }_{3k}}{({\displaystyle \frac{{\sigma }_{3k}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}.\hfill \end{array}$$

Based on the mixed boundary conditions of (30), one has

$$\begin{array}{ccc}\hfill A& =& c_1[\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Theta }}_{1i}{(\frac{{\sigma }_{1i}}{a})}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{{\sigma }_{1i}}{a})]}^{{\mathfrak{P}}_3-1}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_3}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{{\mathrm{\Theta }}_{2j}{(\frac{{\sigma }_{2j}}{a})}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{{\sigma }_{2j}}{a})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_3-1}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_3-1}}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_{4j}+{\mathfrak{P}}_3}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{{\mathrm{\Theta }}_{3k}{(\frac{{\sigma }_{3k}}{a})}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{{\sigma }_{3k}}{a})]}^{{\mathfrak{P}}_3-{\mathfrak{P}}_{5k}-1}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-{\mathfrak{P}}_{5k-1}}}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_3-{\mathfrak{P}}_{5k}}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{a}})]}^{{\mathfrak{P}}_1}\right)]\hfill \\ & & +\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Theta }}_{1i}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{1i}}{({\displaystyle \frac{{\sigma }_{1i}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{{\mathrm{\Theta }}_{2j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{2j}}{({\displaystyle \frac{{\sigma }_{2j}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{y_j}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{{\mathrm{\Theta }}_{3k}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}}}{\int }_a^{{\sigma }_{3k}}{({\displaystyle \frac{{\sigma }_{3k}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}.\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill c_1& =& {\displaystyle \frac{1}{X}}[A-\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Theta }}_{1i}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{1i}}{({\displaystyle \frac{{\sigma }_{1i}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}\hfill \\ & & -\sum _{j=1}^n{\displaystyle \frac{{\mathrm{\Theta }}_{2j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{2j}}{({\displaystyle \frac{{\sigma }_{2j}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}\hfill \\ & & -\sum _{k=1}^r{\displaystyle \frac{{\mathrm{\Theta }}_{3k}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}}}{\int }_a^{{\sigma }_{3k}}{({\displaystyle \frac{{\sigma }_{3k}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}].\hfill \end{array}$$

Inserting $c_1,c_2$ into (33), we obtain (31). In addition, it is easy to show that $\varphi$ provided by (31) satisfies (30). □

4. The Uniqueness Property

Recall the Banach fixed-point theory [12]: Let U be a Banach space, $C\subset U$ be closed, and $L:C⟶C$ be a strict contraction. Then, L has a fixed point in $C.$

From Lemma 5, consider the operator $L:U⟶U$ (here, $U=C\left(\right[a,b\left]\right)$) defined as

$$\begin{array}{ccc}\hfill \left(L\varphi \right)\left(\sigma \right)& =& {\displaystyle \frac{1}{X}}[A-\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Theta }}_{1i}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{1i}}{\left({\displaystyle \frac{{\sigma }_{1i}}{s}}\right)}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}\hfill \\ & & -\sum _{j=1}^n{\displaystyle \frac{{\mathrm{\Theta }}_{2j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{2j}}{\left({\displaystyle \frac{{\sigma }_{2j}}{s}}\right)}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}\hfill \\ & & -\sum _{k=1}^r{\displaystyle \frac{{\mathrm{\Theta }}_{3k}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}}}{\int }_a^{{\sigma }_{3k}}{({\displaystyle \frac{{\sigma }_{3k}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}]\hfill \\ & & \times \left({\displaystyle \frac{{\left(\frac{\sigma }{a}\right)}^{\frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-1}}{X{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_3}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right)\right)\hfill \\ & & {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}\right).\hfill \end{array}$$

(34)

Let

$$\begin{array}{ccc}\hfill E& :=& {\displaystyle \frac{{[\ln (\frac{b}{a})]}^{{\mathfrak{P}}_1}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_1+1)}}+{\displaystyle \frac{{[\ln (\frac{b}{a})]}^{{\mathfrak{P}}_3-1}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}(\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|{[\ln (\frac{{\sigma }_{1i}}{a})]}^{{\mathfrak{P}}_1}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_1+1)}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{|{\mathrm{\Theta }}_{2j}|{[\ln (\frac{{\sigma }_{2i}}{a})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_{4j}+{\mathfrak{P}}_1+1)}}+\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|{[\ln (\frac{{\sigma }_{3k}}{a})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\Gamma ({\mathfrak{P}}_1-{\mathfrak{P}}_{5k}+1)}}).\hfill \end{array}$$

(35)

Theorem 2.

Let $\mathrm{\Phi }\in C\left(\right[a,b]\times \mathbb{R},\mathbb{R}).$ Assume there is a $J>0,$ such that

$$\begin{array}{c}\hfill \left|\mathrm{\Phi }\right(\sigma ,\varphi )-\mathrm{\Phi }(\sigma ,\psi \left)\right|\le J|\varphi -\psi |,\end{array}$$

(36)

where $\varphi ,\psi \in \mathbb{R}$ and $\sigma \in [a,b].$ If

$$\begin{array}{c}\hfill EJ<1,\end{array}$$

(37)

then (30) has a unique solution on $[a,b].$

Proof. 

Consider $\varphi =L\varphi$, where L is given in (34). Let ${sup}_{\sigma \in [a,b]}\left|\mathrm{\Phi }(\sigma ,0)\right|:={\mathrm{\Phi }}_1^{\ast }<\infty$ and consider $B_{\omega }:=\{\varphi \in U:\parallel \varphi \parallel \le \omega \}$, where

$$\begin{array}{c}\hfill \omega \ge {\displaystyle \frac{E{\mathrm{\Phi }}_1^{\ast }+\frac{\left|A\right|{[\ln (\frac{b}{a})]}^{{\mathfrak{P}}_3-1}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}{1-EJ}}.\end{array}$$

(38)

Note that $B_{\omega }$ is a closed, bounded, and convex subset of $U.$

Step 1.  $L{B}_{\omega }\subseteq B_{\omega }.$

For every $\varphi \in B_{\omega },$ and $\sigma \in [a,b],$ one has

$$\begin{array}{ccc}& & \left|\right(L\varphi \left)\right(\sigma \left)\right|\hfill \\ & \le & \left[\right|A|+\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{1i}}\left|{({\displaystyle \frac{{\sigma }_{1i}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{\left[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})\right]}^{{\mathfrak{P}}_1-1}\right|\hfill \\ & & \times \left|{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1}\right)\right|\left|\mathrm{\Phi }(s,\varphi \left(s\right))\right|{\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{{\mathrm{\Theta }}_{2j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{2j}}\left|{({\displaystyle \frac{{\sigma }_{2j}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}\right|\hfill \\ & & \times \left|{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_1}\right)\right|\left|\mathrm{\Phi }(s,\varphi \left(s\right))\right|{\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}}}{\int }_a^{{\sigma }_{3k}}\left|{({\displaystyle \frac{{\sigma }_{3k}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\right|\hfill \\ & & \times \left|{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1}\right)\right|\left|\mathrm{\Phi }(s,\varphi \left(s\right))\right|{\displaystyle \frac{ds}{s}}]\hfill \\ & & \times \left[{\displaystyle \frac{1}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}}}\left|{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3-1}\right|\left|{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_3}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right)\right|\right]\hfill \\ & & +{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{\int }_a^{\sigma }\left|{\left({\displaystyle \frac{\sigma }{s}}\right)}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\right|\hfill \\ & & \times \left|{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\right|\left|\mathrm{\Phi }(s,\varphi \left(s\right))\right|{\displaystyle \frac{ds}{s}}\hfill \\ & \le & \left[{\displaystyle \frac{1}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}}}\left|{\left({\displaystyle \frac{\sigma }{a}}\right)}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3-1}\right|\left|{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_3}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right)\right|\right]\hfill \\ & & \times \left[\right|A|+\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{1i}}\left|{({\displaystyle \frac{{\sigma }_{1i}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}\right|\hfill \\ & & \times \left|{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1}\right)\right|\left(\left|\mathrm{\Phi }\right(s,\varphi \left(s\right))-\mathrm{\Phi }(s,0\left)\right|+\left|\mathrm{\Phi }\right(s,0\left)\right|\right){\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{{\mathrm{\Theta }}_{2j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{2j}}\left|{({\displaystyle \frac{{\sigma }_{2j}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}\right|\hfill \\ & & \times \left|{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_1}\right)\right|\left(\left|\mathrm{\Phi }\right(s,\varphi \left(s\right))-\mathrm{\Phi }(s,0\left)\right|+\left|\mathrm{\Phi }\right(s,0\left)\right|\right){\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}}}{\int }_a^{{\sigma }_{3k}}\left|{({\displaystyle \frac{{\sigma }_{3k}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\right|\hfill \\ & & \times \left|{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1}\right)\right|\left(\left|\mathrm{\Phi }\right(s,\varphi \left(s\right))-\mathrm{\Phi }(s,0\left)\right|+\left|\mathrm{\Phi }\right(s,0\left)\right|\right){\displaystyle \frac{ds}{s}}]\hfill \\ & & +{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{\int }_a^{\sigma }\left|{\left({\displaystyle \frac{\sigma }{s}}\right)}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\right|\left|{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\right|\hfill \\ & & \times \left(\left|\mathrm{\Phi }\right(s,\varphi \left(s\right))-\mathrm{\Phi }(s,0\left)\right|+\left|\mathrm{\Phi }\right(s,0\left)\right|\right){\displaystyle \frac{ds}{s}}\hfill \\ & \le & \left[{\displaystyle \frac{{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_{3-1}}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}\right]\hfill \\ & & \times \left[\right|A|+(J\parallel \varphi \parallel +{\mathrm{\Phi }}_1^{\ast })\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{{\sigma }_{1i}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}{\displaystyle \frac{ds}{s}}\hfill \\ & & +(J\parallel \varphi \parallel +{\mathrm{\Phi }}_1^{\ast })\sum _{j=1}^n{\displaystyle \frac{|{\mathrm{\Theta }}_{2j}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_{4j}+{\mathfrak{P}}_1)}}{\int }_a^{{\sigma }_{2j}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}{\displaystyle \frac{ds}{s}}\hfill \\ & & +(J\parallel \varphi \parallel +{\mathrm{\Phi }}_1^{\ast })\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\Gamma ({\mathfrak{P}}_1-{\mathfrak{P}}_{5k})}}{\int }_a^{{\sigma }_{3k}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}{\displaystyle \frac{ds}{s}}]\hfill \\ & & (J\parallel \varphi \parallel +{\mathrm{\Phi }}_1^{\ast }){\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}{\displaystyle \frac{ds}{s}}\hfill \\ & \le & \left[{\displaystyle \frac{{[\ln (\frac{b}{a})]}^{{\mathfrak{P}}_{3-1}}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}\right]\hfill \\ & & \times \left[\right|A|+(J\omega +{\mathrm{\Phi }}_1^{\ast })\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_1+1)}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{a}})]}^{{\mathfrak{P}}_1}\hfill \\ & & +(J\omega +{\mathrm{\Phi }}_1^{\ast })\sum _{j=1}^n{\displaystyle \frac{|{\mathrm{\Theta }}_{2j}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_{4j}+{\mathfrak{P}}_1+1)}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{a}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\hfill \\ & & +(J\omega +{\mathrm{\Phi }}_1^{\ast })\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\Gamma ({\mathfrak{P}}_1-{\mathfrak{P}}_{5k}+1)}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{a}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}]\hfill \\ & & +(J\omega +{\mathrm{\Phi }}_1^{\ast }){\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_1+1)}}{\int }_a^{\sigma }{[\ln ({\displaystyle \frac{b}{a}})]}^{{\mathfrak{P}}_1}\hfill \\ & =& EJ\omega +E{\mathrm{\Phi }}_1^{\ast }+{\displaystyle \frac{\left|A\right|{[\ln (\frac{b}{a})]}^{{\mathfrak{P}}_0-1}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}\hfill \\ & \le & \omega .\hfill \end{array}$$

Thus, $L{B}_{\omega }\subseteq B_{\omega }.$

Step 2. $L:U⟶U$ is a contraction.

For every $\varphi ,\psi \in U,$ and $\sigma \in [a,b],$ we have that

$$\begin{array}{ccc}& & \left|\right(L\varphi \left)\right(\sigma )-(L\psi \left)\right(\sigma \left)\right|\hfill \\ & \le & \left[\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{1i}}\right|{({\displaystyle \frac{{\sigma }_{1i}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}\left|\right|{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1}\right)|\hfill \\ & & \times |\mathrm{\Phi }(s,\varphi \left(s\right))-\mathrm{\Phi }(s,\psi \left(s\right))|{\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{|{\mathrm{\Theta }}_{2j}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{2j}}\left|({\displaystyle \frac{{\sigma }_{2j}}{s}}){\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}\right||\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_1}\right)\left|\right|\mathrm{\Phi }(s,\varphi \left(s\right))-\mathrm{\Phi }(s,\psi \left(s\right))|{\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}}}{\int }_a^{{\sigma }_{3k}}\left|{({\displaystyle \frac{{\sigma }_{3k}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\right||\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1}\right)\left|\right|\mathrm{\Phi }(s,\varphi \left(s\right))-\mathrm{\Phi }(s,\psi \left(s\right))\left|{\displaystyle \frac{ds}{s}}\right]\hfill \\ & & \times \left[{\left|X\right|}^{-1}{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_3+1}\left|{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3-1}\right|\left|{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_3}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right)\right|\right]\hfill \\ & & +{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }\left|{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{(\ln ({\displaystyle \frac{\sigma }{s}}))}^{{\mathfrak{P}}_1-1}\right|\left|{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\right|\hfill \\ & & \times |\mathrm{\Phi }(s,\varphi \left(s\right))-\mathrm{\Phi }(s,\psi \left(s\right))|{\displaystyle \frac{ds}{s}}\hfill \\ & \le & EJ\parallel \varphi -\psi \parallel .\hfill \end{array}$$

Hence, $\parallel L\varphi -L\psi \parallel \le EJ\parallel \varphi -\psi \parallel .$ Since $EJ<1,$ then L is a contraction. Now, Banach’s fixed-point theory guarantees that L has a fixed point. □

5. Optimal Designs in Stability Studies

Here, set $\nabla \left(\sigma \right):={min}_{\sigma \in [a,b]}\left\{{\mathrm{\Xi }}_{{\mathfrak{P}}_1}\left(\sigma \right),W_{{\alpha }_1,{\beta }_1}\left(\sigma \right),H_{{\alpha }_2,{\beta }_2,{\gamma }_2}\left(\sigma \right)\right\},$ where ${\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2,{\gamma }_2\in \mathbb{C},$ and $\Re \left({\alpha }_1\right),\Re \left({\alpha }_2\right),\Re \left({\beta }_1\right),\Re \left({\beta }_2\right),\Re \left({\gamma }_2\right)>0.$

Definition 3.

We consider the following concepts:

(D1) 

Let $\mathrm{\Omega }<0,$ and $\mathrm{\Phi }\in C([a,b]\times {\mathbb{R}}^2,\mathbb{R}).$ The proportional fractional system (30) is called maximal UH-stable, if there exists a $R_{\mathrm{\Phi }}>0,$ such that, for every $\epsilon >0,$ and for every $\psi \in U$ of

$$\begin{array}{c}\hfill |{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}-\mathrm{\Omega }\varphi \left(\sigma \right)-\mathrm{\Phi }(\sigma ,\varphi \left(\sigma \right))|\le \epsilon ,\sigma \in [a,b],\end{array}$$

(39)

there is a $\varphi \in U$ of (30) with

$$\begin{array}{c}\hfill |\psi \left(\sigma \right)-\varphi \left(\sigma \right)|\le R_{\mathrm{\Phi }}\epsilon \nabla \left(T_{\mathrm{\Phi }}{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right),\end{array}$$

for every $\sigma \in [a,b]$ and $T_{\mathrm{\Phi }}\ge 0.$

(D2) 

The proportional fractional system (30) is called maximal GUH-stable if there exists $G_{\mathrm{\Phi }}\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+}),$ such that, for every $\epsilon >0$ and every $\psi \in U$ of (39), there is a $\varphi \in U$ of (30) with

$$\begin{array}{c}\hfill |\psi \left(\sigma \right)-\varphi \left(\sigma \right)|\le G_{\mathrm{\Phi }}\left(\epsilon \right)\nabla \left(T_{\mathrm{\Phi }}{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right)\end{array}$$

(40)

where $\sigma \in [a,b]$ and $T_{\mathrm{\Phi }}\ge 0.$

(D3) 

The proportional fractional system (30) is called maximal UHR-stable with respect to φ if there exists a $R_{\mathrm{\Phi },\phi }>0,$ such that, for every $\epsilon >0$ and for every $\psi \in U$ of

$$\begin{array}{c}\hfill |{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\varphi \left(\sigma \right)-\mathrm{\Omega }\varphi \left(\sigma \right)-\mathrm{\Phi }(\sigma ,\varphi \left(\sigma \right))|\le \epsilon \phi \left(\sigma \right),\sigma \in [a,b],\end{array}$$

(41)

there exists a $\varphi \in U$ of (30) with

$$\begin{array}{c}\hfill |\psi \left(\sigma \right)-\varphi \left(\sigma \right)|\le R_{\mathrm{\Phi },\phi }\epsilon \phi \left(\sigma \right)\nabla \left(T_{\mathrm{\Phi }}{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right),\end{array}$$

for every $\sigma \in [a,b]$ and $T_{\mathrm{\Phi }}\ge 0.$

(D4) 

The proportional fractional system (30) is called maximal GUHR-stable with respect to φ if there exists a $R_{\mathrm{\Phi },\phi }>0,$ such that, for every $\epsilon >0$ and for every $\psi \in U$ of

$$\begin{array}{c}\hfill |{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\varphi \left(\sigma \right)-\mathrm{\Omega }\psi \left(\tau \right)-\mathrm{\Phi }(\sigma ,\varphi \left(\sigma \right))|\le \phi \left(\sigma \right),\sigma \in [a,b],\end{array}$$

(42)

there exists a $\varphi \in U$ of (30) with

$$\begin{array}{c}\hfill |\psi \left(\sigma \right)-\varphi \left(\sigma \right)|\le R_{\mathrm{\Phi },\phi }\phi \left(\sigma \right)\nabla \left(T_{\mathrm{\Phi }}{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right),\end{array}$$

where $\sigma \in [a,b]$ and $T_{\mathrm{\Phi }}\ge 0.$

Remark 1.

We have that

(N1) 

$\psi \in U$ is a solution of (39) iff there exists a ${\rm Y}\in U,$ such that,

$$|{\rm Y}(\sigma )|\le \epsilon \nabla \left({\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right),$$

for every $\sigma \in [a,b],$ and

$$\begin{array}{c}\hfill {}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\psi \left(\sigma \right)=\mathrm{\Omega }\psi \left(\sigma \right)+\mathrm{\Phi }(\sigma ,\psi \left(\sigma \right))+{\rm Y}\left(\sigma \right),\end{array}$$

where $\sigma \in [a,b].$

(N2) 

$\psi \in U$ is a solution of (41) iff there exists a $\mathrm{\Lambda }\in U,$ such that,

$$|\mathrm{\Lambda }(\sigma )|\le \epsilon \phi \left(\sigma \right)\epsilon \nabla \left({\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right),$$

for every $\sigma \in [a,b],$ and

$$\begin{array}{c}\hfill {}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\psi \left(\sigma \right)=\mathrm{\Omega }\psi \left(\sigma \right)+\mathrm{\Phi }(\sigma ,\psi \left(\sigma \right))+\mathrm{\Lambda }\left(\sigma \right),\end{array}$$

where $\sigma \in [a,b].$

5.1. The Maximal UH and GUH-Stability

Let

$$\begin{array}{ccc}\hfill E_1& :=& {\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{[\ln ({\displaystyle \frac{b}{a}})]}^{{\mathfrak{P}}_1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1+1}\left({\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{b}{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & +{\displaystyle \frac{{[\ln (\frac{b}{a})]}^{{\mathfrak{P}}_3-1}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}{\left[\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}\ln ({\displaystyle \frac{{\sigma }_{1i}}{a}})\right]}^{{\mathfrak{P}}_1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1+1}\left({\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{|{\mathrm{\Theta }}_{2j}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{a}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1+1}\left({\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{a}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}+1}\left({\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{a}})]}^{{\mathfrak{P}}_1}\right),\hfill \end{array}$$

(43)

and

$$\begin{array}{ccc}\hfill E_2& :=& {\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}+{\displaystyle \frac{{[\ln (\frac{b}{a})]}^{{\mathfrak{P}}_3-1}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}[\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{|{\mathrm{\Theta }}_{2j}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_{4j}+{\mathfrak{P}}_1)}}+\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\Gamma ({\mathfrak{P}}_1-{\mathfrak{P}}_{5k})}}].\hfill \end{array}$$

(44)

Lemma 6.

Let ${\mathfrak{P}}_1\in (1,2),{\mathfrak{P}}_2\in [0,1]$ and ${\mathfrak{P}}_0\in (0,1].$ If $\psi \in U$ satisfies (39), then, one has

$$\begin{array}{ccc}& & |\psi \left(\sigma \right)-W_{\psi }\left(\sigma \right)-{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }({\displaystyle \frac{\sigma }{s}}){\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & \times \mathrm{\Phi }(s,\psi \left(s\right)){\displaystyle \frac{ds}{s}}|\le E_1\epsilon ,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill & & W_{\psi }\left(\sigma \right)\hfill \\ & =& [A-\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Theta }}_{1i}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{1i}}{({\displaystyle \frac{{\sigma }_{1i}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\psi \left(s\right)){\displaystyle \frac{ds}{s}}\hfill \\ & & -\sum _{j=1}^n{\displaystyle \frac{{\mathrm{\Theta }}_{2j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{2j}}{({\displaystyle \frac{{\sigma }_{2j}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\psi \left(s\right)){\displaystyle \frac{ds}{s}}\hfill \\ & & -\sum _{k=1}^r{\displaystyle \frac{{\mathrm{\Theta }}_{3k}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}}}\hfill \\ & & \times {\int }_a^{{\sigma }_{3k}}{({\displaystyle \frac{{\sigma }_{3k}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\psi \left(s\right)){\displaystyle \frac{ds}{s}}]\hfill \\ & & \times \left[{\left(X{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\right)}^{-1}{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_3}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right)\right].\hfill \end{array}$$

(45)

Proof. 

Let $\psi$ be a solution of (39). Applying Remark 1, we have that

$$\left\{\begin{array}{c}\begin{array}{cc}{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\psi \left(\sigma \right)=\eta \psi \left(\sigma \right)+\Psi (\sigma ,\psi \left(\sigma \right))+{\rm Y}\left(\sigma \right),\sigma \in (a,b],& \\ \psi \left(a\right)=0,& \\ \sum _{i=1}^m{\mathrm{\Theta }}_{1i}\psi \left({\sigma }_{1i}\right)+\sum _{j=1}^n{\mathrm{\Theta }}_{2j}{}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_{4_j},\ln \left(\sigma \right)}\psi \left({\sigma }_{2j}\right)+\sum _{k=1}^r{\mathrm{\Theta }}_{3k}{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_{5k},{\mathfrak{P}}_2,\ln \left(\sigma \right)}\psi \left({\sigma }_{3k}\right)=A.& \end{array}\hfill \end{array}\right.$$

(46)

Making use of Lemma 29, the solution of (46) can be written as

$$\begin{array}{ccc}\hfill & & \psi (\sigma )=\hfill \\ & & W_{\psi }\left(\sigma \right)+{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}\hfill \\ & & +{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right){\rm Y}\left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ & & -[\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Theta }}_{1i}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{1i}}{({\displaystyle \frac{{\sigma }_{1i}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1}\right){\rm Y}\left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{{\mathrm{\Theta }}_{2j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{2j}}{({\displaystyle \frac{{\sigma }_{2j}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right){\rm Y}\left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{{\mathrm{\Theta }}_{3k}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}}}{\int }_a^{{\sigma }_{3k}}{({\displaystyle \frac{{\sigma }_{3k}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1}\right){\rm Y}\left(s\right){\displaystyle \frac{ds}{s}}]\hfill \\ & & \times \left[{\displaystyle \frac{1}{X{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}}}{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right)\right].\hfill \end{array}$$

(47)

Using Note 44 and Lemma 1, we get

$$\begin{array}{ccc}& & |\psi \left(\sigma \right)-W_{\psi }\left(\sigma \right)-{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\psi \left(s\right)){\displaystyle \frac{ds}{s}}|\hfill \\ & \le & {\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\left|{\rm Y}\left(s\right)\right|{\displaystyle \frac{ds}{s}}\hfill \\ & & +{\displaystyle \frac{{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-1}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}[\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{{\sigma }_{1i}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}\left|{\rm Y}\left(s\right)\right|{\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{\left|{\rm Y}\right|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_{4j}+{\mathfrak{P}}_1)}}{\int }_a^{{\sigma }_{2j}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}\left|{\rm Y}\left(s\right)\right|{\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\Gamma ({\mathfrak{P}}_1-{\mathfrak{P}}_{5k})}}{\int }_a^{{\sigma }_{3k}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\left|{\rm Y}\left(s\right)\right|{\displaystyle \frac{ds}{s}}]\hfill \\ & \le & \epsilon \{{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\nabla \left({\eta }^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{s}{a}})]}^{{\mathfrak{P}}_1}\right){\displaystyle \frac{ds}{s}}+{\displaystyle \frac{{[\ln (\frac{b}{a})]}^{{\mathfrak{P}}_3-1}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}\hfill \\ & & \times [\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{{\sigma }_{1i}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}\nabla \left({\eta }^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{s}{a}})]}^{{\mathfrak{P}}_1}\right){\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{|{\mathrm{\Theta }}_{2j}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_{4j}+{\mathfrak{P}}_1)}}{\int }_a^{{\sigma }_{2j}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}\nabla \left({\eta }^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{s}{a}})]}^{{\mathfrak{P}}_1}\right){\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\Gamma ({\mathfrak{P}}_1-{\mathfrak{P}}_{5k})}}\hfill \\ & & \times {\int }_a^{{\sigma }_{3k}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\nabla \left({\eta }^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{s}{a}})]}^{{\mathfrak{P}}_1}{\displaystyle \frac{ds}{s}}\right)\}.\hfill \end{array}$$

We now have

$$\begin{array}{ccc}& & |\psi \left(\sigma \right)-W_{\psi }\left(\sigma \right)-{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\psi \left(s\right)){\displaystyle \frac{ds}{s}}|\hfill \\ & \le & \epsilon \{{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\sum _{n=0}^{\infty }{\displaystyle \frac{{[\ln (\frac{s}{a})]}^{n{\mathfrak{P}}_1}}{{\left({\mathfrak{P}}_0\right)}^{n{\mathfrak{P}}_1}\Gamma (n{\mathfrak{P}}_1+1)}}{\displaystyle \frac{ds}{s}}+{\displaystyle \frac{{[\ln (\frac{b}{a})]}^{{\mathfrak{P}}_3-1}}{{\left|X\right|}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}\hfill \\ & & \times [\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{{\sigma }_{1i}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}\sum _{n=0}^{\infty }{\displaystyle \frac{{[\ln (\frac{s}{a})]}^{n{\mathfrak{P}}_1}}{{\left({\mathfrak{P}}_0\right)}^{n{\mathfrak{P}}_1}\Gamma (n{\mathfrak{P}}_1+1)}}{\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{|{\mathrm{\Theta }}_{2j}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_{4j}+{\mathfrak{P}}_1)}}{\int }_a^{{\sigma }_{2j}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}\sum _{n=0}^{\infty }{\displaystyle \frac{{[\ln (\frac{s}{a})]}^{n{\mathfrak{P}}_1}}{{\left({\mathfrak{P}}_0\right)}^{n{\mathfrak{P}}_1}\Gamma (n{\mathfrak{P}}_1+1)}}{\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\Gamma ({\mathfrak{P}}_1-{\mathfrak{P}}_{5k})}}\hfill \\ & & \times {\int }_a^{{\sigma }_{3k}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\sum _{n=0}^{\infty }{\displaystyle \frac{{[\ln (\frac{s}{a})]}^{n{\mathfrak{P}}_1}}{{\left({\mathfrak{P}}_0\right)}^{n{\mathfrak{P}}_1}\Gamma (n{\mathfrak{P}}_1+1)}}{\displaystyle \frac{ds}{s}}\left]\right\}\hfill \\ & =& \epsilon \{{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}\sum _{n=0}^{\infty }{\displaystyle \frac{{[\ln (\frac{b}{a})]}^{n{\mathfrak{P}}_1+{\mathfrak{P}}_1}}{{\left({\mathfrak{P}}_0\right)}^{n{\mathfrak{P}}_1}\Gamma (n{\mathfrak{P}}_1+{\mathfrak{P}}_1+1)}}\hfill \\ & & +{\displaystyle \frac{{[\ln (\frac{b}{a})]}^{{\mathfrak{P}}_3-1}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}[\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}\sum _{n=0}^{\infty }{\displaystyle \frac{{[\ln (\frac{{\sigma }_{1i}}{a})]}^{n{\mathfrak{P}}_1+{\mathfrak{P}}_1}}{{\left({\mathfrak{P}}_0\right)}^{n{\mathfrak{P}}_1}\Gamma (n{\mathfrak{P}}_1+{\mathfrak{P}}_1+1)}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{|{\mathrm{\Theta }}_{2j}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}}}\sum _{n=0}^{\infty }{\displaystyle \frac{{[\ln (\frac{{\sigma }_{2j}}{a})]}^{n{\mathfrak{P}}_1+{\mathfrak{P}}_1{+}^{{\mathfrak{P}}_{4j}}}}{{\left({\mathfrak{P}}_0\right)}^{n{\mathfrak{P}}_1}\Gamma (n{\mathfrak{P}}_1+1+{\mathfrak{P}}_{4j})+{\mathfrak{P}}_1}}\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}-{\mathfrak{P}}_{5k}}}\sum _{n=0}^{\infty }{\displaystyle \frac{{[\ln (\frac{{\sigma }_{3k}}{a})]}^{n{\mathfrak{P}}_1+{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}}{{\left({\mathfrak{P}}_0\right)}^{n{\mathfrak{P}}_1}\Gamma (n{\mathfrak{P}}_1+{\mathfrak{P}}_1+1-{\mathfrak{P}}_{5k})}}\left]\right\}\hfill \\ & & =\{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{b}{a}})]}^{{\mathfrak{P}}_1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1+1}\left({\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{b}{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & +{\displaystyle \frac{{[\ln (\frac{b}{a})]}^{{\mathfrak{P}}_3-1}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}[\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{a}})]}^{{\mathfrak{P}}_1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1+1}\left({\eta }^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{|{\mathrm{\Theta }}_{2j}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{a}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1+1}\left({\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{a}})]}^{{\mathfrak{P}}_1}\right)\hfill \\ & & \sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{a}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}+1}\left({\eta }^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{a}})]}^{{\mathfrak{P}}_1}\right)\left]\right\}\epsilon \hfill \\ & \le & E_1\epsilon .\hfill \end{array}$$

This implies (46). □

Theorem 3.

Let the assumptions of Theorem 2 hold. Then, (30) is maximal UH-stable and consequently maximal GUH-stable on $[a,b].$

Proof. 

Let $\epsilon >0,$ and $\psi \in U$ be a function satisfying (39). Let $\varphi \in U$ be the unique solution of

$$\left\{\begin{array}{c}\begin{array}{cc}{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\varphi \left(\sigma \right)=\eta \varphi \left(\sigma \right)+\mathrm{\Phi }(\sigma ,\varphi \left(\sigma \right)),\sigma \in (a,b],& \\ \varphi \left(a\right)=0,& \\ \sum _{i=1}^m{\mathrm{\Theta }}_{1i}\varphi \left({\sigma }_{1i}\right)+\sum _{j=1}^n{\mathrm{\Theta }}_{2j}{}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_{4_j},\ln \left(\sigma \right)}\varphi \left({\sigma }_{2j}\right)+\sum _{k=1}^r{\mathrm{\Theta }}_{3k}{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_{5k},{\mathfrak{P}}_2,\ln \left(\sigma \right)}\varphi \left({\sigma }_{3k}\right)=A.& \end{array}\hfill \end{array}\right.$$

(48)

According to Lemma 5, one has

$$\begin{array}{ccc}& & \varphi \left(\sigma \right)=W_{\varphi }\left(\sigma \right)\hfill \\ & & +{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{[{\displaystyle \frac{\sigma }{s}}]}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}},\hfill \end{array}$$

where $\sigma \in (a,b].$

Note that

${}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_{4j},\ln \left(\sigma \right)}\varphi \left({\sigma }_{2j}\right)={}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_{4j},\ln \left(\sigma \right)}\psi \left({\sigma }_{2j}\right)$,

${}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_{5k},{\mathfrak{P}}_2,\ln \left(\sigma \right)}\varphi \left({\sigma }_{3k}\right)={}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_{5k},{\mathfrak{P}}_2,\ln \left(\sigma \right)}\psi \left({\sigma }_{3k}\right),$$\varphi \left(a\right)=\psi \left(a\right),$$\varphi \left({\sigma }_{1i}\right)=\psi \left({\sigma }_{1i}\right),$ and

$$\begin{array}{ccc}& & |W_{\varphi }\left(\sigma \right)-W_{\psi }\left(\sigma \right)|\hfill \\ & \le & {\displaystyle \frac{{[\ln (\frac{b}{a})]}^{{\mathfrak{P}}_3-1}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}\hfill \\ & & \times [\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{{\mathfrak{P}}_0{\sigma }_{1i}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}|\mathrm{\Phi }(s,\varphi \left(s\right))-\mathrm{\Phi }(s,\psi \left(s\right))|{\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{|{\mathrm{\Theta }}_{2j}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}}+{\mathfrak{P}}_1\Gamma ({\mathfrak{P}}_{4j}+{\mathfrak{P}}_1)}}\hfill \\ & & \times {\int }_a^{{\sigma }_{2j}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}|\mathrm{\Phi }(s,\varphi \left(s\right))-\mathrm{\Phi }(s,\psi \left(s\right))|{\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\Gamma ({\mathfrak{P}}_1-{\mathfrak{P}}_{5k})}}\hfill \\ & & \times {\int }_a^{{\mathfrak{P}}_{3k}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}|\mathrm{\Phi }(s,\varphi \left(s\right))-\mathrm{\Phi }(s,\psi \left(s\right))|{\displaystyle \frac{ds}{s}}]\hfill \\ & \le & {\displaystyle \frac{J{[\ln (\frac{b}{a})]}^{{\mathfrak{P}}_3-1}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}[\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{{\sigma }_{1i}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}|\varphi \left(s\right)-\psi \left(s\right)|{\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{|{\mathrm{\Theta }}_{2j}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_{4j}+{\mathfrak{P}}_1)}}{\int }_a^{{\sigma }_{2j}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}|\varphi \left(s\right)-\psi \left(s\right)|{\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\Gamma ({\mathfrak{P}}_1-{\mathfrak{P}}_{5k})}}{\int }_a^{{\sigma }_{3k}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}|\varphi \left(s\right)-\psi \left(s\right)|{\displaystyle \frac{ds}{s}}]\hfill \\ & =& 0,\hfill \end{array}$$

so, $W_{\varphi }\left(\sigma \right)=W_{\psi }\left(\sigma \right).$

Using Lemma 6 with $|\varphi -\psi |\le \left|\varphi \right|+\left|\psi \right|,$ for every $\sigma \in [a,b],$ we get

$$\begin{array}{ccc}& & |\psi (\sigma )-\varphi (\sigma )|\hfill \\ & \le & |\psi \left(\sigma \right)-W_{\varphi }\left(\sigma \right)-{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}|\hfill \\ & \le & |\psi \left(\sigma \right)-W_{\psi }\left(\sigma \right)-{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\psi \left(s\right)){\displaystyle \frac{ds}{s}}|\hfill \\ & & +|{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\psi \left(s\right))-\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}|\hfill \\ & & +|W_{\psi }\left(\sigma \right)-W_{\varphi }\left(\sigma \right)|\hfill \\ & \le & E_1\epsilon +{\displaystyle \frac{J}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}|\psi \left(s\right)-\varphi \left(s\right)|{\displaystyle \frac{ds}{s}}.\hfill \end{array}$$

Making use of Theorem 1, we obtain $|\varphi \left(\sigma \right)-\psi \left(\sigma \right)|\le E_1\epsilon \nabla \left(J{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right).$ By setting $R_{\mathrm{\Phi }}=E_1$ and $T_{\mathrm{\Phi }}=J,$ one has $|\varphi \left(\sigma \right)-\psi \left(\sigma \right)|\le R_{\mathrm{\Phi }}\epsilon \nabla \left(T_{\mathrm{\Phi }}{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right).$ Thus, (30) is maximal UH-stable. In addition, by setting $G_{\mathrm{\Phi }}\left(\epsilon \right)=R_{\mathrm{\Phi }}\epsilon$ with $G_{\mathrm{\Phi }}\left(0\right)=0,$ $|\varphi \left(\sigma \right)-\psi \left(\sigma \right)|\le G_{\mathrm{\Phi }}\left(\epsilon \right)\nabla \left(T_{\mathrm{\Phi }}{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right).$ Thus, (30) is maximal GUH-stable. □

5.2. The Maximal UHR- and GUHR-Stability

Consider the assumption:

$\left(A\right):$ Suppose $\phi \in C\left(\right[a,b],\mathbb{R})$ is a non-decreasing function and there exists ${\eta }_{\phi }\in {\mathbb{R}}^{+},$ such that, for $\sigma \in [a,b],$

$$\begin{array}{c}\hfill {\int }_a^{\sigma }{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\nabla \left({\eta }^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{s}{a}})]}^{{\mathfrak{P}}_1}\right)\phi \left(s\right){\displaystyle \frac{ds}{s}}\le {\eta }_{\phi }\phi (\sigma ).\end{array}$$

(49)

Lemma 7.

Consider ${\mathfrak{P}}_1\in (1,2),{\mathfrak{P}}_2\in [0,1]$ and ${\mathfrak{P}}_0\in (0,1].$ If $\psi \in C\left(\right[a,b],\mathbb{R})$ satisfies (41), then, ψ satisfies

$$\begin{array}{ccc}& & |\psi \left(\sigma \right)-W_{\psi }\left(\sigma \right)-{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\psi \left(s\right)){\displaystyle \frac{ds}{s}}|\hfill \\ & & \le E_2\epsilon {\eta }_{\phi }\phi \left(\sigma \right).\hfill \end{array}$$

Proof. 

Let $\psi$ be a solution of (41). Based on Remark 1, we have that

$$\left\{\begin{array}{c}\begin{array}{cc}{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\psi \left(\sigma \right)=\eta \psi \left(\sigma \right)+\mathrm{\Phi }(\sigma ,\psi \left(\sigma \right))+\mathrm{\Lambda }\left(\sigma \right),\sigma \in (a,b],& \\ \psi \left(a\right)=0,& \\ \sum _{i=1}^m{\mathrm{\Theta }}_{1i}\psi \left({\sigma }_{1i}\right)+\sum _{j=1}^n{\mathrm{\Theta }}_{2j}{}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_{4_j},\ln \left(\sigma \right)}\psi \left({\sigma }_{2j}\right)+\sum _{k=1}^r{\mathrm{\Theta }}_{3k}{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_{5k},{\mathfrak{P}}_2,\ln \left(\sigma \right)}\psi \left({\sigma }_{3k}\right)=A.& \end{array}\hfill \end{array}\right.$$

(50)

Making use of Lemma 5, the solution of (50) can be written as

$$\begin{array}{ccc}\hfill & & \psi (\sigma )=\hfill \\ & & W_{\psi }\left(\sigma \right)+{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}\hfill \\ & & +{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Lambda }\left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ & & -[\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Theta }}_{1i}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{1i}}{({\displaystyle \frac{{\sigma }_{1i}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Lambda }\left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{{\mathrm{\Theta }}_{2j}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}}}{\int }_a^{{\sigma }_{2j}}{({\displaystyle \frac{{\sigma }_{2j}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Lambda }\left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{{\mathrm{\Theta }}_{3k}}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}}}{\int }_a^{{\sigma }_{3k}}{({\displaystyle \frac{{\sigma }_{3k}}{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Lambda }\left(s\right){\displaystyle \frac{ds}{s}}]\hfill \\ & & \times \left[{\displaystyle \frac{1}{X{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}}}{({\displaystyle \frac{\sigma }{a}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_3-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right)\right].\hfill \end{array}$$

(51)

Using Remark 1 with Lemma 1, one has

$$\begin{array}{ccc}& & |\psi \left(\sigma \right)-W_{\psi }\left(\sigma \right)-{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\psi \left(s\right)){\displaystyle \frac{ds}{s}}|\hfill \\ & \le & {\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\left|\mathrm{\Lambda }\left(s\right)\right|{\displaystyle \frac{ds}{s}}\hfill \\ & & +{\displaystyle \frac{{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-1}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}(\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{{\sigma }_{1i}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}\left|\mathrm{\Lambda }\left(s\right)\right|{\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{|{\mathrm{\Theta }}_{2j}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_{4j}+{\mathfrak{P}}_1)}}{\int }_a^{{\sigma }_{2j}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}\left|\mathrm{\Lambda }\left(s\right)\right|{\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\Gamma ({\mathfrak{P}}_1-{\mathfrak{P}}_{5k})}}{\int }_a^{{\sigma }_{3k}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\left|\mathrm{\Lambda }\left(s\right)\right|{\displaystyle \frac{ds}{s}})\hfill \\ & \le & \{{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\nabla \left({\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{s}{a}})]}^{{\mathfrak{P}}_1}\right)\phi \left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ & & +{\displaystyle \frac{{[\ln (\frac{\sigma }{a})]}^{{\mathfrak{P}}_3-1}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}(\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{{\sigma }_{1i}}{[\ln ({\displaystyle \frac{{\sigma }_{1i}}{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times \nabla \left({\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{s}{a}})]}^{{\mathfrak{P}}_1}\right)\phi \left(s\right){\displaystyle \frac{ds}{s}}+\sum _{j=1}^n{\displaystyle \frac{|{\mathrm{\Theta }}_{2j}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_0{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_{4j}+{\mathfrak{P}}_1)}}\hfill \\ & & \times {\int }_a^{{\sigma }_{2j}}{[\ln ({\displaystyle \frac{{\sigma }_{2j}}{s}})]}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1-1}\nabla \left({\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{s}{a}})]}^{{\mathfrak{P}}_1}\right)\phi \left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ & & +\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\Gamma ({\mathfrak{P}}_1-{\mathfrak{P}}_{5k})}}{\int }_a^{{\sigma }_{3k}}{[\ln ({\displaystyle \frac{{\sigma }_{3k}}{s}})]}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}-1}\hfill \\ & & \times \nabla \left({\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{s}{a}})]}^{{\mathfrak{P}}_1}\right)\phi \left(s\right){\displaystyle \frac{ds}{s}}\left)\right\}\epsilon .\hfill \end{array}$$

Based on the assumption (A), we obtain

$$\begin{array}{ccc}& & |\psi \left(\sigma \right)-W_{\psi }\left(\sigma \right)-{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\psi \left(s\right)){\displaystyle \frac{ds}{s}}|\hfill \\ & \le & \{{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}+{\displaystyle \frac{{[\ln (\frac{b}{a})]}^{{\mathfrak{P}}_3-1}}{\left|X\right|{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_3-1}\Gamma \left({\mathfrak{P}}_3\right)}}(\sum _{i=1}^m{\displaystyle \frac{|{\mathrm{\Theta }}_{1i}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}\hfill \\ & & +\sum _{j=1}^n{\displaystyle \frac{|{\mathrm{\Theta }}_{2j}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_{4j}+{\mathfrak{P}}_1}\Gamma ({\mathfrak{P}}_{4j}+{\mathfrak{P}}_1)}}+\sum _{k=1}^r{\displaystyle \frac{|{\mathrm{\Theta }}_{3k}|}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1-{\mathfrak{P}}_{5k}}\Gamma ({\mathfrak{P}}_1-{\mathfrak{P}}_{5k})}}\left)\right\}\epsilon {\eta }_{\phi }\phi \left(\sigma \right)\hfill \\ & \le & E_2\epsilon {\eta }_{\phi }\phi \left(\sigma \right).\hfill \end{array}$$

□

Theorem 4.

Suppose $\mathrm{\Phi }\in C\left(\right[a,b]\times \mathbb{R},\mathbb{R})$ and let the assumptions of Theorem 2 hold. Then, (30) is maximal UHR- and GUHR-stable on $[a,b].$

Proof. 

Let $\epsilon >0,$ and $\psi \in \mho$ be a function satisfying (41). Suppose $\varphi \in \mho$ is the unique solution of

$$\left\{\begin{array}{c}\begin{array}{cc}{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_1,{\mathfrak{P}}_2,\ln \left(\sigma \right)}\psi \left(\sigma \right)=\eta \psi \left(\sigma \right)+\mathrm{\Phi }(\sigma ,\psi \left(\sigma \right)),\sigma \in (a,b],& \\ \psi \left(a\right)=0,& \\ \sum _{i=1}^m{\mathrm{\Theta }}_{1i}\psi \left({\sigma }_{1i}\right)+\sum _{j=1}^n{\mathrm{\Theta }}_{2j}{}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_{4_j},\ln \left(\sigma \right)}\psi \left({\sigma }_{2j}\right)+\sum _{k=1}^r{\mathrm{\Theta }}_{3k}{}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_{5k},{\mathfrak{P}}_2,\ln \left(\sigma \right)}\psi \left({\sigma }_{3k}\right)=A.& \end{array}\hfill \end{array}\right.$$

(52)

Making use of Lemma 5, one has

$$\begin{array}{ccc}& & \varphi \left(\sigma \right)=W_{\varphi }\left(\sigma \right)\hfill \\ & & +{\displaystyle \frac{1}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}{\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}},\hfill \end{array}$$

for every $\sigma \in (a,b].$

In addition, $\varphi \left(a\right)=\psi \left(a\right),\varphi \left({\sigma }_{1i}\right)=\psi \left({\sigma }_{1i}\right),$ ${}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_{4j},\ln \left(\sigma \right)}\varphi \left({\sigma }_{2j}\right)={}_{{\mathfrak{P}}_0}I_{a^{+}}^{{\mathfrak{P}}_{4j},\ln \left(\sigma \right)}\psi \left({\sigma }_{2j}\right)$ and ${}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_{5k},{\mathfrak{P}}_2,\ln \left(\sigma \right)}\varphi \left({\sigma }_{3k}\right)={}_{{\mathfrak{P}}_0}{}^{HH}D_{a^{+}}^{{\mathfrak{P}}_{5k},{\mathfrak{P}}_2,\ln \left(\sigma \right)}\psi \left({\sigma }_{3k}\right)$ yields $W_{\varphi }\left(\sigma \right)=W_{\psi }\left(\sigma \right).$

From Lemma 6, for every $\sigma \in [a,b],$ we have that

$$\begin{array}{ccc}& & |\psi (\sigma )-\varphi (\sigma )|\hfill \\ & \le & |\psi \left(\sigma \right)-W_{\varphi }\left(\sigma \right)-{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}|\hfill \\ & \le & |\psi \left(\sigma \right)-W_{\psi }\left(\sigma \right)-{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\psi \left(s\right)){\displaystyle \frac{ds}{s}}|\hfill \\ & & +|{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}\hfill \\ & & \times {\mathrm{\Xi }}_{{\mathfrak{P}}_1,{\mathfrak{P}}_1}\left(\eta {\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1}\right)\mathrm{\Phi }(s,\psi \left(s\right))-\mathrm{\Phi }(s,\varphi \left(s\right)){\displaystyle \frac{ds}{s}}|\hfill \\ & & +|W_{\varphi }\left(\sigma \right)-W_{\psi }\left(\sigma \right)|\hfill \\ & \le & E_2\epsilon {\eta }_{\phi }\phi \left(\sigma \right)+{\displaystyle \frac{J}{{\left({\mathfrak{P}}_0\right)}^{{\mathfrak{P}}_1}\Gamma \left({\mathfrak{P}}_1\right)}}{\int }_a^{\sigma }{({\displaystyle \frac{\sigma }{s}})}^{{\displaystyle \frac{{\mathfrak{P}}_0-1}{{\mathfrak{P}}_0}}}{[\ln ({\displaystyle \frac{\sigma }{s}})]}^{{\mathfrak{P}}_1-1}|\psi \left(s\right)-\varphi \left(s\right)|{\displaystyle \frac{ds}{s}}.\hfill \end{array}$$

Using Theorem 1, we have

$$\begin{array}{c}\hfill |\varphi \left(\sigma \right)-\psi \left(\sigma \right)|\le E_2{\eta }_{\phi }\epsilon \phi \left(\sigma \right)\nabla \left(J{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right).\end{array}$$

Setting $R_{\mathrm{\Phi },\phi }=E_2{\eta }_{\phi }$ and $T_{\mathrm{\Phi }}=J,$ we get

$$\begin{array}{c}\hfill |\varphi \left(\sigma \right)-\psi \left(\sigma \right)|\le R_{\mathrm{\Phi },\phi }\epsilon \phi \left(\sigma \right)\nabla \left(T_{\mathrm{\Phi }}{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right).\end{array}$$

Also, we have

$$\begin{array}{c}\hfill |\varphi \left(\sigma \right)-\psi \left(\sigma \right)|\le R_{\mathrm{\Phi },\phi }\phi \left(\sigma \right)\nabla \left(T_{\mathrm{\Phi }}{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right).\end{array}$$

Hence, the proof is complete. □

6. Application

Consider the Hilfer–Hadamard proportional fractional differential system (30) as follows:

$$\left\{\begin{array}{c}\begin{array}{cc}{}_{0.70}{}^{HH}D_{a^{+}}^{1.80,0.80,\ln \left(\sigma \right)}\varphi \left(\sigma \right)=-2\varphi \left(\sigma \right)+\mathrm{\Phi }(\sigma ,\varphi \left(\sigma \right)),& \sigma \in (a,b],\\ \varphi \left(a\right)=0,& \\ \sum _{i=1}^3(4-i){(5-i)}^{-1}\varphi ({\displaystyle \frac{2i}{5}}-{\displaystyle \frac{1}{5}})+{\displaystyle \frac{2}{9}}{}_{0.70}I_{a^{+}}^{2.50,\ln \left(\sigma \right)}\varphi \left(0.90\right)\\ +\sum _{k=1}^2({\displaystyle \frac{k}{18-3k}}+{\displaystyle \frac{2}{18-3k}}){}_{0.70}{}^{HH}D_{a^{+}}^{1.90-0.20k,0.80,\ln \left(\sigma \right)}\varphi (0.50k-0.30)=4,\end{array}\hfill \end{array}\right.$$

(53)

where $$\begin{gathered} {\mathfrak{P}}_0=0.70,{\mathfrak{P}}_1=1.80,{\mathfrak{P}}_2=0.80,a=0.15,b=3,\eta =-2,{\mathrm{\Theta }}_{1i}={\displaystyle \frac{4}{5-i}}-{\displaystyle \frac{i}{5-i}}, \\ {\sigma }_{1i}={\displaystyle \frac{2i}{5}}-{\displaystyle \frac{1}{5}},{\mathrm{\Theta }}_{2j}=2.90,{\sigma }_{2j}=0.90,{\mathrm{\Theta }}_{3k}={\displaystyle \frac{k}{18-3k}}+{\displaystyle \frac{2}{18-3k}},{\sigma }_{3k}=0.50k-0.30, \\ {\mathfrak{P}}_{4j}=2.50,{\mathfrak{P}}_{5k}=1.90-0.20k,A=4,i,j,k=1,2,3,\mathrm{and} \mathrm{thus},X\approx 1.20. \end{gathered}$$

Consider $\mathrm{\Phi }(\sigma ,\varphi \left(\sigma \right))={\displaystyle \frac{cos({\sigma }^2-2)}{8\sigma sin(4-2\sigma )+6}}+{\displaystyle \frac{3sin\left(4\sigma \pi \right)}{4\left[\ln \right(2\sigma +2)+\exp (\sigma \left)\right]}}\times {\displaystyle \frac{\left|\varphi \right(\sigma \left)\right|}{2+\left|\varphi \right(\sigma \left)\right|}}.$

For every $\sigma \in [a,b]$ and $\alpha ,\beta \in \mathbb{R},$ we have $\left|\mathrm{\Phi }\right(\sigma ,\alpha )-\mathrm{\Phi }(\sigma ,\beta \left)\right|\le J|\alpha -\beta |,$ where $J\in (0,1).$ Hence, by (37), we obtain $EJ\approx 0.4567$ in which $E\approx 0.2698.$ Thus, the assumption of Theorem 2 is satisfied. We now can conclude that (53) has a unique solution on $[a,b].$ Making use of (43), we obtain $E_1\approx 0.2750.$ If we set $R_{\mathrm{\Phi }}:=E_1$ and $T_{\mathrm{\Phi }}:=J,$ then, via Theorem 3, (53) is maximal UH-stable. Plus, if we set $G_{\mathrm{\Phi }}\left(\epsilon \right)=R_{\mathrm{\Phi }}\epsilon ,$ then, (53) is also maximal GUH-stable on $[a,b].$ Finally, by letting $\phi \left(\sigma \right)={[\ln ({\displaystyle \frac{\sigma }{a}})]}^{0.50}$ in (49), the assumption (A) is thus satisfied under ${\eta }_{\mathrm{\Phi }}=0.0245.$

From (44), we get $E_2\approx 2.6491,$ and $E_2{\eta }_{\mathrm{\Phi }}\approx 0.1054.$ If we set $R_{\mathrm{\Phi },\phi }:=E_2{\eta }_{\phi }$ and $T_{\mathrm{\Phi }}=J,$ then, via Theorem 4, (53) is maximal UHR stable on $[a,b].$ In addition, if we consider $R_{\mathrm{\Phi },\phi }=R_{\mathrm{\Phi },\phi }\epsilon ,$ then, (53) is also maximal GUHR stable on $[a,b].$

Here, set

$$\begin{array}{ccc}& & \mathcal{E}\left(\sigma \right):=R_{\mathrm{\Phi }}\epsilon \nabla \left(T_{\mathrm{\Phi }}{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right),\hfill \\ & & {\mathcal{E}}_1\left(\sigma \right):=R_{\mathrm{\Phi }}\epsilon H_{{\alpha }_2,{\beta }_2,{\gamma }_2}\left(T_{\mathrm{\Phi }}{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right),\hfill \\ & & {\mathcal{E}}_2\left(\sigma \right):=R_{\mathrm{\Phi }}\epsilon W_{{\alpha }_1,{\beta }_1}\left(T_{\mathrm{\Phi }}{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right),\hfill \\ & & {\mathcal{E}}_3\left(\sigma \right):=R_{\mathrm{\Phi }}\epsilon {\mathrm{\Xi }}_{{\mathfrak{P}}_1}\left(T_{\mathrm{\Phi }}{\left({\mathfrak{P}}_0\right)}^{-{\mathfrak{P}}_1}{[\ln ({\displaystyle \frac{\sigma }{a}})]}^{{\mathfrak{P}}_1}\right),\hfill \end{array}$$

where all parameters have been introduced in the previous sections.

Contour plots of $\mathcal{E}$ for different $\mathfrak{P}$ values are shown in Figure 1. As can be seen, with increasing $\mathfrak{P}$ from $1.1$ to $1.9$, the resulting error generally increases. However, at certain points, the error values tend to converge and become nearly zero. Similarly, the error plots of ${\mathcal{E}}_i$ for $(i=1,2,3)$, obtained by applying the hypergeometric, Wright, and Mittag–Leffler control functions, respectively, are presented in Figure 2. As observed, the resulting error magnitude is notably lower when the hypergeometric controller is employed compared to the other configurations. This highlights that the choice of special control function plays a crucial role in optimizing stability problems, underscoring the significant impact of controller selection on the overall performance.

Figure 1. The contour plots of $\mathcal{E}\left(\sigma \right),$ for two different domains $\sigma \in (0.15,1.15)$ and $\sigma \in (1.24,1.29),$ and different values of ${\mathfrak{P}}_1=1.1$ (purple), ${\mathfrak{P}}_1=1.3$ (cyan), ${\mathfrak{P}}_1=1.5$ (green), ${\mathfrak{P}}_1=1.7$ (yellow), and ${\mathfrak{P}}_1=1.9$ (red).

Figure 2. The plots of ${\mathcal{E}}_3$ (yellow), ${\mathcal{E}}_2$ (green), and ${\mathcal{E}}_1$ (blue), for $\sigma \in (2.80,2.85)$ and ${\mathfrak{P}}_1=1.5$.

Special functions play a fundamental role in fractional calculus and the precise modeling of complex systems. Their unique properties enable an accurate representation of fractional-order dynamics, which are increasingly prevalent in stability analysis and control design. In optimization tasks, these functions facilitate the development of more effective controllers by capturing intricate system behaviors that conventional methods may overlook. Incorporating special functions allows for enhanced computational efficiency and higher solution accuracy in fractional calculus applications. Consequently, their utilization becomes essential for advancing stability and optimization strategies, ultimately leading to more robust and reliable control systems. Their significance is underscored by their ability to bridge theoretical models with practical, real-world system behaviors.

The numerical results presented in Table 1 provide a comprehensive validation of the aforementioned insights. Based on the data, it can be concluded that the previous claims are indeed accurate and thoroughly substantiated, confirming the effectiveness of the approaches discussed.

Table 1. The obtained errors ${\mathcal{E}}_i,i=1,2,3,$ for different values of ${\mathfrak{P}}_1$ in different domains.

Considered Domains	$\sigma \in (1.25,1.50)$			$\sigma \in (1.50,1.75)$
The Obtained Errors	${\mathcal{E}}_1$	${\mathcal{E}}_2$	${\mathcal{E}}_3$	${\mathcal{E}}_1$	${\mathcal{E}}_2$	${\mathcal{E}}_3$
${\mathfrak{P}}_1=1.20$	0.012577	0.028937	0.029436	0.029103	0.032205	0.033622
${\mathfrak{P}}_1=1.25$	0.013902	0.033012	0.037011	0.034678	0.039110	0.041003
${\mathfrak{P}}_1=1.30$	0.014113	0.039278	0.042126	0.040118	0.044391	0.050863
${\mathfrak{P}}_1=1.35$	0.014997	0.043105	0.049649	0.047325	0.048890	0.057118

7. Conclusions

In this study, we introduced a novel version of the Gronwall inequality tailored to the Hilfer–Hadamard proportional fractional derivative, establishing new theoretical foundations for such equations. Through the application of Picard’s successive approximations and properties of Mittag–Leffler functions, we derived a robust representation formula expressed in terms of the Mittag–Leffler kernel. Additionally, leveraging Banach’s fixed-point theorem, we demonstrated the uniqueness of the solution under specific conditions. Furthermore, we explored optimal stability results associated with special functions. Our exploration of stability results further solidifies the mathematical framework, offering deeper insights into the behavior of fractional systems governed by this class of equations. These findings contribute meaningfully to the growing body of research in fractional differential equations and their applications. For future research, we recommend exploring other well-known special functions, including the Fox H-function, the G-function, and the Meijer G-function, as control functions [12,24,25]. The results obtained from these functions will be compared with the findings of this study to thoroughly evaluate their effectiveness and relevance in the context of our proposed methodology.