On the Mild Solutions of Second-Order Θ-Caputo Fractional Boundary Value Problems

Abstract

In this paper, we study a class of second-order fractional boundary value problems involving Θ-Caputo derivatives of different orders. By reformulating the problem to an integral equation, we introduce an appropriate notion of a mild solution in the Θ-fractional framework. Existence results are obtained via Krasnoselskii’s fixed point theorem, while uniqueness is established using the Banach contraction principle under suitable Lipschitz-type conditions. The obtained results extend several earlier works on Caputo, Hadamard–Caputo, and Riemann–Liouville fractional derivatives. Two examples are presented to illustrate the applicability of the theoretical results.

1. Introduction

Fractional boundary value problems have become a central topic in the theory of fractional differential equations due to their effectiveness in modeling nonlocal and memory-dependent phenomena. Foundational contributions to fractional calculus and fractional differential equations can be found in the monographs of Miller and Ross [1], Podlubny [2], and Kilbas et al. [3], where the basic theory and applications of fractional operators are systematically developed.

Building on these foundations, extensive research has been devoted to the existence and qualitative analysis of fractional boundary value problems. Classical existence and uniqueness results for nonlinear fractional differential equations were established in works such as [4,5,6], while boundary value problems with singular or nonlocal conditions were investigated in [7,8,9,10,11]. Positive solutions and qualitative properties of fractional boundary value problems were further analyzed in [12,13,14,15,16]. In [17], existence results were established using the measure of noncompactness for fractional problems in Banach spaces. In addition, ref. [18] investigated $(p,q)$-fractional difference equations and provided existence results in Banach spaces. More recent studies have also focused on fractional boundary value problems involving generalized fractional operators and multiple fractional orders. Problems with Riemann–Liouville and Caputo-type derivatives of different orders were examined in [19,20,21]. In contrast, fractional differential equations involving Caputo derivatives with respect to a kernel function were introduced and analyzed in [22]. These developments provide a unified framework encompassing several classical fractional operators and further motivate the study of more general fractional boundary value problems, such as those considered in the present work. Furthermore, numerical methods for fractional differential equations, including finite difference and local discontinuous Galerkin techniques for variable-order time-fractional diffusion equations, were extensively studied in [23].

Niyom et al. [20] investigated the existence and uniqueness of solutions for the following second-order Riemann–Liouville fractional boundary value problem:

$$\left\{\begin{array}{c}\varrho D^{\gamma }\vartheta \left(\varsigma \right)+(1-\varrho )D^{\beta }\vartheta \left(\varsigma \right)=\Phi (\varsigma ,\vartheta \left(\varsigma \right)),\varsigma \in (0,\mathsf{{\rm Y}}),\hfill \\ \vartheta \left(0\right)=0,\lambda D^{{\kappa }_1}\vartheta \left(\mathsf{{\rm Y}}\right)+(1-\lambda )D^{{\kappa }_2}\vartheta \left(\mathsf{{\rm Y}}\right)={\kappa }_3.\hfill \end{array}\right.$$

The authors established existence and uniqueness results by applying Krasnoselskii’s fixed point theorem, the Leray–Schauder nonlinear alternative, and the Banach contraction principle.

The following second-order Caputo fractional boundary value problem was studied in [15]:

$$\left\{\begin{array}{c}{\mathfrak{D}}_{\varsigma }^{\gamma }\vartheta \left(\varsigma \right)=\Phi (\varsigma ,\vartheta \left(\varsigma \right))+{\mathfrak{D}}_{\varsigma }^{\gamma -1}f(\varsigma ,\vartheta \left(\varsigma \right)),0<\varsigma \le \mathsf{{\rm Y}},\hfill \\ \vartheta \left(0\right)={\vartheta }_1>0,{\vartheta }^{\prime }\left(0\right)={\vartheta }_2>0,\hfill \end{array}\right.$$

By employing the method of lower and upper solutions together with the Banach and Schauder fixed point theorems, the authors derived positivity and existence results.

Ardjouni [24] considered the existence and uniqueness of mild solutions for the following second-order Hadamard–Caputo fractional boundary value problem:

$$\left\{\begin{array}{c}\varrho {\mathfrak{D}}_{log\varsigma }^{\gamma }\vartheta \left(\varsigma \right)+(1-\varrho ){\mathfrak{D}}_{log\varsigma }^{\beta }\vartheta \left(\varsigma \right)=\Phi (\varsigma ,\vartheta \left(\varsigma \right)),\varsigma \in (1,\mathsf{{\rm Y}}),\hfill \\ \vartheta \left(1\right)=0,\lambda {\mathfrak{D}}_{log\varsigma }^{{\kappa }_1}\vartheta \left(\mathsf{{\rm Y}}\right)+(1-\lambda ){\mathfrak{D}}_{log\varsigma }^{{\kappa }_2}\vartheta \left(\mathsf{{\rm Y}}\right)={\kappa }_3.\hfill \end{array}\right.$$

Using appropriate fixed point techniques, the author obtained sufficient conditions for the existence and uniqueness of mild solutions.

Motivated by the aforementioned works, we investigate the existence and uniqueness of mild solutions for the following second-order Θ-Caputo fractional boundary value problem:

$$\left\{\begin{array}{c}\varrho {\mathfrak{D}}_{\mathsf{\Theta }}^{\gamma }\vartheta \left(\varsigma \right)+(1-\varrho ){\mathfrak{D}}_{\mathsf{\Theta }}^{\beta }\vartheta \left(\varsigma \right)=\Phi (\varsigma ,\vartheta \left(\varsigma \right)),\varsigma \in ({\tau }_0,\mathsf{{\rm Y}}),\hfill \\ \vartheta \left({\tau }_0\right)=0,\lambda {\mathfrak{D}}_{\mathsf{\Theta }}^{{\kappa }_1}\vartheta \left(\mathsf{{\rm Y}}\right)+(1-\lambda ){\mathfrak{D}}_{\mathsf{\Theta }}^{{\kappa }_2}\vartheta \left(\mathsf{{\rm Y}}\right)={\kappa }_3,\hfill \end{array}\right.$$

(1)

where ${\mathfrak{D}}_{\mathsf{\Theta }}^{\varphi }$ denotes the $\mathsf{\Theta }$-Caputo fractional derivative of order $\varphi \in \{\gamma ,\beta ,{\kappa }_1,{\kappa }_2\}$ such that $1<\gamma ,\beta \le 2$ and $0<{\kappa }_1,{\kappa }_2<\gamma -\beta$. Moreover, ${\kappa }_3\in \mathbb{R}$, $0<\varrho \le 1$, $0\le \lambda \le 1$, and $\Phi :[{\tau }_0,\mathsf{{\rm Y}}]\times \mathbb{R}\to \mathbb{R}$ is a continuous function.

By transforming problem (1) into an integral equation, we employ Krasnoselskii’s fixed point theorem to establish the existence of mild solutions and the Banach contraction principle to ensure uniqueness under appropriate assumptions.

The results obtained in this work extend and unify several earlier contributions for Caputo, Hadamard–Caputo, and Riemann–Liouville fractional boundary value problems. This unification is achieved through the use of the $\mathsf{\Theta }$-Caputo fractional derivative, which recovers several classical fractional operators for appropriate choices of the kernel function $\mathsf{\Theta }$. In particular, if $\mathsf{\Theta }\left(\varsigma \right)=\varsigma$, one obtains the classical Caputo derivative, while $\mathsf{\Theta }\left(\varsigma \right)=ln\varsigma$ yields the Hadamard–Caputo derivative. More generally, different choices of $\mathsf{\Theta }$ generate a broad class of fractional operators with nonlocal kernels. Consequently, problem (1) generalizes a variety of previously studied fractional boundary value problems.

The remainder of this paper is organized as follows. In Section 2, we present the necessary definitions and preliminary results, including Banach’s fixed point theorem and Krasnoselskii’s fixed point theorem. Section 3 is devoted to the existence and uniqueness of mild solutions for problem (1). Finally, Section 4 provides two examples illustrating the applicability of the theoretical results.

2. Preliminaries

We present some of the definitions, theorems and lemmas that are essential to this study.

Definition 1

([22]). Let $\mathsf{\Theta }\in C^1\left([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R}\right)$ with ${\mathsf{\Theta }}^{\prime }\left(\varsigma \right)>0$ for all $\varsigma \in [{\tau }_0,\mathsf{{\rm Y}}]$. The $\mathsf{\Theta }$-Riemann fractional integral of $\vartheta \in C\left([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R}\right)$ of order $\gamma >0$ is defined as

$${\mathfrak{I}}_{\mathsf{\Theta }}^{\gamma }\vartheta \left(\varsigma \right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\gamma \right)}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }(\varsigma )-\mathsf{\Theta }(\iota ){)}^{\gamma -1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota ,\gamma >0.$$

Definition 2

([22]). Let $\mathsf{\Theta }\in C^m\left([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R}\right)$ with ${\mathsf{\Theta }}^{\prime }\left(\varsigma \right)>0$ for all $\varsigma \in [{\tau }_0,\mathsf{{\rm Y}}]$. The $\mathsf{\Theta }$-Caputo fractional derivative of $\vartheta \in C^m\left([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R}\right)$ of order $\gamma >0$ is defined as

$${\mathfrak{D}}_{\mathsf{\Theta }}^{\gamma }\vartheta \left(\varsigma \right)={\displaystyle \frac{1}{\mathsf{\Gamma }(m-\gamma )}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{m-\gamma -1}({\delta }_{\mathsf{\Theta }}^m\vartheta )\left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota ,m-1<\gamma \le m,$$

where ${\delta }_{\mathsf{\Theta }}^m={\left({\displaystyle \frac{d}{{\mathsf{\Theta }}^{\prime }\left(\varsigma \right)d\varsigma }}\right)}^m$, $m\in {\mathbb{N}}^{\ast }$.

Lemma 1

([22]). For $m-1<\gamma \le m$, $m\in {\mathbb{N}}^{\ast }$ and $\vartheta \in C^m\left([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R}\right)$, we have

$$({\mathfrak{I}}_{\mathsf{\Theta }}^{\gamma }{\mathfrak{D}}_{\mathsf{\Theta }}^{\gamma }\vartheta )\left(\varsigma \right)=\vartheta \left(\varsigma \right)-\sum _{k=0}^{m-1}{\displaystyle \frac{\left({\delta }_{\mathsf{\Theta }}^k\vartheta \right)\left({\tau }_0\right)}{\mathsf{\Gamma }(k+1)}}(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right){)}^k.$$

Lemma 2

([22]). For all $\lambda >0$ and $\nu >-1$, we get

$$\begin{array}{cc}\hfill & {\mathfrak{I}}_{\mathsf{\Theta }}^{\lambda }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\nu }\hfill \\ \hfill & ={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\lambda \right)}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\lambda -1}(\mathsf{\Theta }\left(\iota \right){)}^{\nu }{\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & ={\displaystyle \frac{\mathsf{\Gamma }(\nu +1)}{\mathsf{\Gamma }(\lambda +\nu +1)}}(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\lambda +\nu }.\hfill \end{array}$$

Lemma 3.

For $\varkappa \in C\left([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R}\right)$, the $\mathsf{\Theta }$-Caputo fractional boundary value problem

$$\left\{\begin{array}{c}\varrho {\mathfrak{D}}_{\mathsf{\Theta }}^{\gamma }\vartheta \left(\varsigma \right)+(1-\varrho ){\mathfrak{D}}_{\mathsf{\Theta }}^{\beta }\vartheta \left(\varsigma \right)=\varkappa \left(\varsigma \right),\varsigma \in ({\tau }_0,\mathsf{{\rm Y}}),\hfill \\ \vartheta \left({\tau }_0\right)=0,\lambda {\mathfrak{D}}_{\mathsf{\Theta }}^{{\kappa }_1}\vartheta \left(\mathsf{{\rm Y}}\right)+(1-\lambda ){\mathfrak{D}}_{\mathsf{\Theta }}^{{\kappa }_2}\vartheta \left(\mathsf{{\rm Y}}\right)={\kappa }_3,\hfill \end{array}\right.$$

(2)

admits the following mild solution representation

$$\begin{array}{cc}\hfill \vartheta \left(\varsigma \right)& ={\displaystyle \frac{\varrho -1}{\varrho \mathsf{\Gamma }(\gamma -\beta )}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{1}{\varrho \mathsf{\Gamma }\left(\gamma \right)}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -1}\varkappa \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota +{\displaystyle \frac{\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right)}{\rho }}\hfill \\ \hfill & \times \left({\kappa }_3-{\displaystyle \frac{\lambda (\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_1-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right.\hfill \\ \hfill & -{\displaystyle \frac{\lambda }{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_1-1}\varkappa \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & -{\displaystyle \frac{(1-\lambda )(\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_2-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & \left.-{\displaystyle \frac{1-\lambda }{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_2-1}\varkappa \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right),\varsigma \begin{array}{c}\in \end{array}[{\tau }_0,\mathsf{{\rm Y}}],\hfill \end{array}$$

(3)

where $\rho \ne 0$ is defined by

$$\rho ={\displaystyle \frac{\lambda (\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{1-{\kappa }_1}}{\mathsf{\Gamma }(2-{\kappa }_1)}}+{\displaystyle \frac{(1-\lambda )(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{1-{\kappa }_2}}{\mathsf{\Gamma }(2-{\kappa }_2)}}.$$

(4)

Proof. 

According to the first equation of (2), we get

$${\mathfrak{D}}_{\mathsf{\Theta }}^{\gamma }\vartheta \left(\varsigma \right)={\displaystyle \frac{\varrho -1}{\varrho }}{\mathfrak{D}}_{\mathsf{\Theta }}^{\beta }\vartheta \left(\varsigma \right)+{\displaystyle \frac{1}{\varrho }}\varkappa \left(\varsigma \right),\varsigma \in [{\tau }_0,\mathsf{{\rm Y}}].$$

(5)

For both sides of (5), taking the $\mathsf{\Theta }$-Riemann fractional integral of order $\gamma$, we get

$$\begin{array}{cc}\hfill \vartheta \left(\varsigma \right)& ={\displaystyle \frac{\varrho -1}{\varrho \mathsf{\Gamma }(\gamma -\beta )}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{1}{\varrho \mathsf{\Gamma }\left(\gamma \right)}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -1}\varkappa \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota +C_1+C_2\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right),\hfill \end{array}$$

for $C_1,C_2\in \mathbb{R}$. According to the boundary condition of (2), we have $C_1=0$. Hence

$$\begin{array}{cc}\hfill \vartheta \left(\varsigma \right)& ={\displaystyle \frac{\varrho -1}{\varrho \mathsf{\Gamma }(\gamma -\beta )}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{1}{\varrho \mathsf{\Gamma }\left(\gamma \right)}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -1}\varkappa \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota +C_2\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right).\hfill \end{array}$$

(6)

For $\psi \in \{{\kappa }_1,{\kappa }_2\}$ with $0<\psi <\gamma -\beta$, using the $\mathsf{\Theta }$-Caputo fractional derivative of order $\psi$ for (6), we get

$$\begin{array}{cc}\hfill {\mathfrak{D}}_{\mathsf{\Theta }}^{\psi }\vartheta \left(\varsigma \right)& ={\displaystyle \frac{\varrho -1}{\varrho \mathsf{\Gamma }(\gamma -\beta -\psi )}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -\psi -1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{1}{\varrho \mathsf{\Gamma }(\gamma -\psi )}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\psi -1}\varkappa \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +C_2{\displaystyle \frac{1}{\mathsf{\Gamma }(2-\psi )}}(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{1-\psi }.\hfill \end{array}$$

In the above relation, by substituting the values $\psi ={\kappa }_1$ and $\psi ={\kappa }_2$ and applying the second condition of (2), we have

$$\begin{array}{cc}\hfill {\kappa }_3& ={\displaystyle \frac{\lambda (\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_1-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{\lambda }{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_1-1}\varkappa \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{\lambda (\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{1-{\kappa }_1}}{\mathsf{\Gamma }(2-{\kappa }_1)}}{C}_2\hfill \\ \hfill & +{\displaystyle \frac{(1-\lambda )(\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_2-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{1-\lambda }{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_2-1}\varkappa \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{(1-\lambda )(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{1-{\kappa }_2}}{\mathsf{\Gamma }(2-{\kappa }_2)}}{C}_2,\hfill \end{array}$$

which leads to

$$\begin{array}{cc}\hfill C_2& ={\displaystyle \frac{1}{\rho }}\left[{\kappa }_3-{\displaystyle \frac{\lambda (\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_1-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right.\hfill \\ \hfill & -{\displaystyle \frac{\lambda }{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_1-1}\varkappa \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & -{\displaystyle \frac{(1-\lambda )(\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_2-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & \left.-{\displaystyle \frac{1-\lambda }{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_2-1}\varkappa \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right],\hfill \end{array}$$

where

$$\rho ={\displaystyle \frac{\lambda (\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{1-{\kappa }_1}}{\mathsf{\Gamma }(2-{\kappa }_1)}}+{\displaystyle \frac{(1-\lambda )(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{1-{\kappa }_2}}{\mathsf{\Gamma }(2-{\kappa }_2)}},$$

and observe that since ${\mathsf{\Theta }}^{\prime }\left(\mathsf{{\rm Y}}\right)>0$, we have $\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)>\mathsf{\Theta }\left({\tau }_0\right)$. Moreover, for ${\kappa }_1,{\kappa }_2\in \left(0,1\right)$, the Gamma function $\mathsf{\Gamma }(2-{\kappa }_i)$ is finite and positive. Therefore, $\rho >0$, and in particular $\rho \ne 0$.

Substituting the value of $C_2$ in (6), we get the unique mild solution for (3). This completes the proof. □

Definition 3.

$\vartheta \in C\left(\left[{\tau }_0,\mathsf{{\rm Y}}\right],\mathbb{R}\right)$ is said to be a mild solution of (1) if ϑ satisfies the next associated integral equation of (1).

$$\begin{array}{cc}\hfill \vartheta \left(\varsigma \right)& ={\displaystyle \frac{\varrho -1}{\varrho \mathsf{\Gamma }(\gamma -\beta )}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{1}{\varrho \mathsf{\Gamma }\left(\gamma \right)}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -1}\Phi (\iota ,\vartheta \left(\iota \right)){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota +{\displaystyle \frac{\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right)}{\rho }}\hfill \\ \hfill & \times \left({\kappa }_3-{\displaystyle \frac{\lambda (\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_1-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right.\hfill \\ \hfill & -{\displaystyle \frac{\lambda }{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_1-1}\Phi (\iota ,\vartheta \left(\iota \right)){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & -{\displaystyle \frac{(1-\lambda )(\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_2-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & \left.-{\displaystyle \frac{1-\lambda }{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_2-1}\Phi (\iota ,\vartheta \left(\iota \right)){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right),\varsigma \begin{array}{c}\in \end{array}[{\tau }_0,\mathsf{{\rm Y}}].\hfill \end{array}$$

Remark 1.

The notion of a mild solution is particularly appropriate in the present setting, given the infinite-dimensional Banach space framework and the nature of Θ-Caputo fractional derivatives. In general, classical solutions may require stronger regularity assumptions which are not always satisfied.

By reformulating problem (1) as an integral equation, mild solutions allow us to apply fixed point techniques such as Krasnoselskii’s and Banach’s theorems. Moreover, this approach is standard in$\Theta$-Caputo fractional differential equations, where the integral formulation provides a natural extension of classical solutions.

The fixed point theorems that allow us to demonstrate the existence and uniqueness of a mild solution for (1) are stated last in this section.

Definition 4.

Let $(\mathbb{B},∥.∥)$ be a Banach space. The application $\mathcal{P}:\mathbb{B}\to \mathbb{B}$ is a contraction if there is a $\mathsf{\Lambda }\in (0,1)$ such that $\vartheta ,w\in \mathbb{B}$ imply that

$$∥\mathcal{P}\vartheta -\mathcal{P}w∥\le \mathsf{\Lambda }∥\vartheta -w∥.$$

Theorem 1

(Banach [25]). For a convex closed nonempty subset $\mathcal{K}$ of a Banach space $\mathbb{B}$ and a contraction $\mathcal{P}:\mathcal{K}\to \mathcal{K}$, there is a unique $\vartheta \in \mathcal{K}$ with $\mathcal{P}\vartheta =\vartheta$.

Theorem 2

(Krasnoselskii fixed point theorem [25]). Consider a convex closed bounded nonempty subset $\mathcal{K}$ of a Banach space $\mathbb{B}$ and two applications ${\mathcal{P}}_1,{\mathcal{P}}_2:\mathcal{K}\to \mathbb{B}$ with the following conditions:

(i)

${\mathcal{P}}_1\vartheta +{\mathcal{P}}_{2}w\in \mathcal{K}$, for all $\vartheta ,w\in \mathcal{K}$.

(ii)

The application ${\mathcal{P}}_2$ is continuous and compact.

(iii)

The application ${\mathcal{P}}_1$ is a contraction.

Then, there is $z\in \mathcal{K}$ such that $z={\mathcal{P}}_{1}z+{\mathcal{P}}_{2}z$.

3. Existence and Uniqueness Findings

We denote the Banach space of all continuous functions from $[{\tau }_0,\mathsf{{\rm Y}}]$ into $\mathbb{R}$ by $C([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R})$ with the norm $∥\vartheta ∥=sup\left\{\right|\vartheta \left(\varsigma \right)|,$$\varsigma \in [{\tau }_0,\mathsf{{\rm Y}}]\}$.

The Krasnoselskii fixed point theorem serves as the foundation for our first finding.

Theorem 3.

For a continuous function $\Phi :[{\tau }_0,\mathsf{{\rm Y}}]\times \mathbb{R}\to \mathbb{R}$, we suppose the following:

(C1) There exists a function $\sigma \in C([{\tau }_0,\mathsf{{\rm Y}}],{\mathbb{R}}_{+})$ such that

$$\left|\Phi (\varsigma ,\vartheta \left(\varsigma \right))\right|\le \sigma \left(\varsigma \right)fora.e.\varsigma \in [{\tau }_0,\mathsf{{\rm Y}}]andeach\vartheta \in C([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R}).$$

(C2)

$$\begin{array}{cc}\hfill {\eta }_1& ={\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -\beta }|\varrho -1|}{\varrho \mathsf{\Gamma }(\gamma -\beta +1)}}+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -\beta -{\kappa }_1+1}\lambda |\varrho -1|}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_1+1)}}\hfill \\ \hfill & +{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -\beta -{\kappa }_2+1}(1-\lambda )|\varrho -1|}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_2+1)}}\begin{array}{c}<\end{array}1.\hfill \end{array}$$

(7)

Then, (1) admits a mild solution on $[{\tau }_0,\mathsf{{\rm Y}}]$.

Proof. 

For a convex bounded closed subset ${\mathsf{\Omega }}_r=\{\vartheta \in C([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R}):∥\vartheta ∥\le r\}$ of $C([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R})$, where r is a fixed positive constant, we define the application $\mathcal{P}:C([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R})\to C([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R})$ by

$$\begin{array}{cc}\hfill \left(\mathcal{P}\vartheta \right)\left(\varsigma \right)& ={\displaystyle \frac{\varrho -1}{\varrho \mathsf{\Gamma }(\gamma -\beta )}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{1}{\varrho \mathsf{\Gamma }\left(\gamma \right)}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -1}\Phi (\iota ,\vartheta \left(\iota \right)){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota +{\displaystyle \frac{\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right)}{\rho }}\hfill \\ \hfill & \times \left({\kappa }_3-{\displaystyle \frac{\lambda (\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_1-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right.\hfill \\ \hfill & -{\displaystyle \frac{\lambda }{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_1-1}\Phi (\iota ,\vartheta \left(\iota \right)){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & -{\displaystyle \frac{(1-\lambda )(\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_2-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & \left.-{\displaystyle \frac{1-\lambda }{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_2-1}\Phi (\iota ,\vartheta \left(\iota \right)){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right),\varsigma \begin{array}{c}\in \end{array}[{\tau }_0,\mathsf{{\rm Y}}].\hfill \end{array}$$

Let us define ${\mathcal{P}}_1,{\mathcal{P}}_2:C([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R})\to C([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R})$ by

$$\begin{array}{cc}\hfill \left({\mathcal{P}}_1\vartheta \right)\left(\varsigma \right)& ={\displaystyle \frac{(\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta )}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & -{\displaystyle \frac{\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right)}{\rho }}\left({\displaystyle \frac{\lambda (\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_1-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right.\hfill \\ \hfill & \left.+{\displaystyle \frac{(1-\lambda )(\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_2-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left({\mathcal{P}}_2\vartheta \right)\left(\varsigma \right)& ={\displaystyle \frac{1}{\varrho \mathsf{\Gamma }\left(\gamma \right)}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -1}\Phi (\iota ,\vartheta \left(\iota \right)){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota +{\displaystyle \frac{\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right)}{\rho }}\hfill \\ \hfill & \times \left({\kappa }_3-{\displaystyle \frac{\lambda }{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_1-1}\Phi (\iota ,\vartheta \left(\iota \right)){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right.\hfill \\ \hfill & \left.-{\displaystyle \frac{(1-\lambda )}{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_2-1}\Phi (\iota ,\vartheta \left(\iota \right)){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right).\hfill \end{array}$$

Clearly

$$\left(\mathcal{P}\vartheta \right)\left(\varsigma \right)=\left({\mathcal{P}}_1\vartheta \right)\left(\varsigma \right)+\left({\mathcal{P}}_2\vartheta \right)\left(\varsigma \right),\varsigma \in [{\tau }_0,\mathsf{{\rm Y}}].$$

(8)

It is evident that the application $\mathcal{P}$ having a fixed point is equivalent to ${\mathcal{P}}_1+{\mathcal{P}}_2$ having one, so it is sufficient to prove that ${\mathcal{P}}_1+{\mathcal{P}}_2$ has a fixed point. We will demonstrate that the applications ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$ meet every requirement of the Krasnoselskii theorem. This requires several steps.

• Step 1. $\mathcal{P}{\mathsf{\Omega }}_r\subset {\mathsf{\Omega }}_r$. We choose

  $$r\ge {\displaystyle \frac{\parallel \sigma \parallel {\eta }_2+\left|{\kappa }_3\right|\left(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right)\right)/\rho }{1-{\eta }_1}},$$

  where ${\eta }_1$ is defined by (C2) and

  $$\begin{array}{cc}\hfill {\eta }_2& ={\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma }}{\varrho \mathsf{\Gamma }(\gamma +1)}}+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -{\kappa }_1+1}\lambda }{\varrho \rho \mathsf{\Gamma }(\gamma -{\kappa }_1+1)}}\hfill \\ \hfill & +{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -{\kappa }_2+1}(1-\lambda )}{\varrho \rho \mathsf{\Gamma }(\gamma -{\kappa }_2+1)}}.\hfill \end{array}$$

  (9)

  For any $\vartheta \in {\mathsf{\Omega }}_r$, we have

  $$\begin{array}{cc}\hfill \left|\left(\mathcal{P}\vartheta \right)\left(\varsigma \right)\right|& \le \left|{\displaystyle \frac{(\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta )}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right.\hfill \\ \hfill & +{\displaystyle \frac{1}{\varrho \mathsf{\Gamma }\left(\gamma \right)}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -1}\Phi (\iota ,\vartheta \left(\iota \right)){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & -{\displaystyle \frac{\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right)\lambda (\varrho -1)}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_1-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & -{\displaystyle \frac{\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right)(1-\lambda )(\varrho -1)}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_2-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right)}{\rho }}\left({\kappa }_3-{\displaystyle \frac{\lambda }{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_1)}}\right.\hfill \\ \hfill & \times {\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_1-1}\Phi (\iota ,\vartheta \left(\iota \right)){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & \left.\left.-{\displaystyle \frac{(1-\lambda )}{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_2-1}\Phi (\iota ,\vartheta \left(\iota \right)){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right)\right|\hfill \\ \hfill & \le \parallel \vartheta \parallel \left({\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -\beta }|\varrho -1|}{\varrho \mathsf{\Gamma }(\gamma -\beta +1)}}+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -\beta -{\kappa }_1+1}\lambda |\varrho -1|}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_1+1)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -\beta -{\kappa }_2+1}(1-\lambda )|\varrho -1|}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_2+1)}}\right)+{\displaystyle \frac{|{\kappa }_3|\left(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right)\right)}{\rho }}\hfill \\ \hfill & +\parallel \sigma \parallel \left({\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma }}{\varrho \mathsf{\Gamma }(\gamma +1)}}+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -{\kappa }_1+1}\lambda }{\varrho \rho \mathsf{\Gamma }(\gamma -{\kappa }_1+1)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -{\kappa }_2+1}(1-\lambda )}{\varrho \rho \mathsf{\Gamma }(\gamma -{\kappa }_2+1)}}\right)\hfill \\ \hfill & \le r{\eta }_1+\parallel \sigma \parallel {\eta }_2+{\displaystyle \frac{|{\kappa }_3|\left(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right)\right)}{\rho }}\le r,\hfill \end{array}$$

  which implies that $\mathcal{P}{\mathsf{\Omega }}_r\subset {\mathsf{\Omega }}_r$.
• Step 2. The application ${\mathcal{P}}_2$ is compact and continuous. In view of Step 1, the application $P_2$ is uniformly bounded. Let $\vartheta \in {\mathsf{\Omega }}_r$ and ${\varsigma }_1,{\varsigma }_2\in [{\tau }_0,\mathsf{{\rm Y}}]$ with ${\varsigma }_1<{\varsigma }_2$. So, we get

  $$\begin{array}{cc}\hfill & |\left({\mathcal{P}}_2\vartheta \right)\left({\varsigma }_2\right)-\left({\mathcal{P}}_2\vartheta \right)\left({\varsigma }_1\right)|\hfill \\ \hfill & \le {\displaystyle \frac{1}{\varrho \mathsf{\Gamma }\left(\gamma \right)}}{\int }_{{\tau }_0}^{{\varsigma }_1}\left[(\mathsf{\Theta }\left({\varsigma }_2\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -1}-(\mathsf{\Theta }\left({\varsigma }_1\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -1}\right]\sigma \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{1}{\varrho \mathsf{\Gamma }\left(\gamma \right)}}{\int }_{{\varsigma }_1}^{{\varsigma }_2}(\mathsf{\Theta }\left({\varsigma }_2\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -1}\sigma \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{|\mathsf{\Theta }\left({\varsigma }_2\right)-\mathsf{\Theta }\left({\varsigma }_1\right)|}{\rho }}\left(|{\kappa }_3|+{\displaystyle \frac{\lambda }{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_1-1}\sigma \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right.\hfill \\ \hfill & \left.+{\displaystyle \frac{(1-\lambda )}{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_2-1}\sigma \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right)\hfill \\ \hfill & \le {\displaystyle \frac{∥\sigma ∥}{\varrho \mathsf{\Gamma }(\gamma +1)}}\left((\mathsf{\Theta }\left({\varsigma }_2\right){)}^{\gamma }-(\mathsf{\Theta }\left({\varsigma }_1\right){)}^{\gamma }\right)+\left(|{\kappa }_3|+{\displaystyle \frac{\lambda (\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -{\kappa }_1}∥\sigma ∥}{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_1+1)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{(1-\lambda )(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -{\kappa }_2}∥\sigma ∥}{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_2+1)}}\right){\displaystyle \frac{|\mathsf{\Theta }\left({\varsigma }_2\right)-\mathsf{\Theta }\left({\varsigma }_1\right)|}{\rho }},\hfill \end{array}$$

  which tends to zero as ${\varsigma }_2-{\varsigma }_1\to 0$ and is independent of $\vartheta$. Hence, the subset ${\mathcal{P}}_2\left({\mathsf{\Omega }}_r\right)$ is equicontinuous. So, according to the Arzelá–Ascoli theorem, the subset ${\mathcal{P}}_2\left({\mathsf{\Omega }}_r\right)$ is relatively compact. Thus, the application ${\mathcal{P}}_2$ is compact. Moreover, the continuity of $\Phi$ implies that the application ${\mathcal{P}}_2$ is continuous.
• Step 3. The application ${\mathcal{P}}_1$ is a contraction. For $\vartheta ,w\in {\mathsf{\Omega }}_r$, we get

  $$\begin{array}{cc}\hfill & \left|\left({\mathcal{P}}_1\vartheta \right)\left(\varsigma \right)-\left({\mathcal{P}}_{1}w\right)\left(\varsigma \right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{|\varrho -1|}{\varrho \mathsf{\Gamma }(\gamma -\beta )}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -1}|\vartheta \left(\iota \right)-w\left(\iota \right)|{\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{\left(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right)\right)\lambda |\varrho -1|}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_1-1}|\vartheta \left(\iota \right)-w\left(\iota \right)|{\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{\left(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right)\right)(1-\lambda )|\varrho -1|}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_2-1}|\vartheta \left(\iota \right)-w\left(\iota \right)|{\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & \le \left\{{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -\beta }|\varrho -1|}{\varrho \mathsf{\Gamma }(\gamma -\beta +1)}}+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -\beta -{\kappa }_1+1}\lambda |\varrho -1|}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_1+1)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -\beta -{\kappa }_2+1}(1-\lambda )|\varrho -1|}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_2+1)}}\right\}\parallel \vartheta -w\parallel \hfill \\ \hfill & ={\eta }_1\parallel \vartheta -w\parallel .\hfill \end{array}$$

  So, the application ${\mathcal{P}}_1$ is a contraction, since ${\eta }_1<1$.

According to the above steps and the Krasnoselskii theorem, we conclude that the application $\mathcal{P}$ admits a fixed point which is a mild solution of (1). □

The Banach fixed point theorem serves as the foundation for our second finding.

Theorem 4.

For a continuous function $\Phi :[{\tau }_0,\mathsf{{\rm Y}}]\times \mathbb{R}\to \mathbb{R}$, we suppose the following:

(C3) There exists a constant $L_{\Phi }>0$ such that

$$\left|\Phi \left(\varsigma ,\vartheta \left(\varsigma \right)\right)-\Phi \left(\varsigma ,w\left(\varsigma \right)\right)\right|\le L_{\Phi }\left|\vartheta \left(\varsigma \right)-w\left(\varsigma \right)\right|,for\varsigma \in [{\tau }_0,\mathsf{{\rm Y}}],\vartheta ,w\in C([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R}).$$

If

$$L_{\Phi }{\eta }_2+{\eta }_1<1,$$

(10)

where ${\eta }_1$ and ${\eta }_2$ are defined respectively by (7) and (9), then (1) admits a unique mild solution on $[{\tau }_0,\mathsf{{\rm Y}}]$.

Proof. 

We choose

$$R\ge {\displaystyle \frac{N{\eta }_2+\left|{\kappa }_3\right|\left(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right)\right)/\rho }{1-L_{\Phi }{\eta }_2-{\eta }_1}},$$

where $N={sup}_{\varsigma \in [{\tau }_0,\mathsf{{\rm Y}}]}\left|\Phi \left(\varsigma ,0\right)\right|$, and $\rho$ is defined by (4). We demonstrate that $\mathcal{P}{\mathsf{\Omega }}_R\subset {\mathsf{\Omega }}_R$, where

$${\mathsf{\Omega }}_R=\{\vartheta \in C([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R}):∥\vartheta ∥\le R\}.$$

For any $\vartheta \in {\mathsf{\Omega }}_R$, we have

$$\begin{array}{cc}\hfill \left|\left(\mathcal{P}\vartheta \right)\left(\varsigma \right)\right|& \le \left|{\displaystyle \frac{(\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta )}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right.\hfill \\ \hfill & +{\displaystyle \frac{1}{\varrho \mathsf{\Gamma }\left(\gamma \right)}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -1}\Phi (\iota ,\vartheta \left(\iota \right)){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & -{\displaystyle \frac{\left(\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right)\right)\lambda (\varrho -1)}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_1-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & -{\displaystyle \frac{\left(\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right)\right)(1-\lambda )(\varrho -1)}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_2-1}\vartheta \left(\iota \right){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right)}{\rho }}\left({\kappa }_3-{\displaystyle \frac{\lambda }{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_1)}}\right.\hfill \\ \hfill & \times {\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_1-1}\Phi (\iota ,\vartheta \left(\iota \right)){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & \left.\left.-{\displaystyle \frac{(1-\lambda )}{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_2-1}\Phi (\iota ,\vartheta \left(\iota \right)){\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right)\right|\hfill \\ \hfill & \le \parallel \vartheta \parallel \left({\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -\beta }|\varrho -1|}{\varrho \mathsf{\Gamma }(\gamma -\beta +1)}}+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -\beta -{\kappa }_1+1}\lambda |\varrho -1|}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_1+1)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -\beta -{\kappa }_2+1}(1-\lambda )|\varrho -1|}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_2+1)}}\right)+{\displaystyle \frac{|{\kappa }_3|\left(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right)\right)}{\rho }}\hfill \\ \hfill & +\left(L_{\Phi }\parallel \vartheta \parallel +N\right)\left({\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma }}{\varrho \mathsf{\Gamma }(\gamma +1)}}+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -{\kappa }_1+1}\lambda }{\varrho \rho \mathsf{\Gamma }(\gamma -{\kappa }_1+1)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -{\kappa }_2+1}(1-\lambda )}{\varrho \rho \mathsf{\Gamma }(\gamma -{\kappa }_2+1)}}\right)\hfill \\ \hfill & \le \left({\eta }_1+L_{\Phi }{\eta }_2\right)R+N{\eta }_2+{\displaystyle \frac{|{\kappa }_3|\left(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right)\right)}{\rho }}\le R.\hfill \end{array}$$

This means that $∥\mathcal{P}\vartheta ∥\le R$, which leads to $\mathcal{P}{\mathsf{\Omega }}_R\subset {\mathsf{\Omega }}_R$.

Now, for $\vartheta ,w\in C([{\tau }_0,\mathsf{{\rm Y}}],\mathbb{R})$ and $\varsigma \in [{\tau }_0,\mathsf{{\rm Y}}]$, we get

$$\begin{array}{cc}\hfill & \left|\left(\mathcal{P}\vartheta \right)\left(\varsigma \right)-\left(\mathcal{P}w\right)\left(\varsigma \right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{(\varrho -1)}{\varrho \mathsf{\Gamma }(\gamma -\beta )}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -1}\left|\vartheta \left(\iota \right)-w\left(\iota \right)\right|{\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{1}{\varrho \mathsf{\Gamma }\left(\gamma \right)}}{\int }_{{\tau }_0}^{\varsigma }(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -1}\left|\Phi (\iota ,\vartheta (\iota \left)\right)-\Phi (\iota ,w(\iota \left)\right)\right|{\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right)\lambda (\varrho -1)}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_1)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_1-1}\left|\vartheta \left(\iota \right)-w\left(\iota \right)\right|{\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{\left(\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right)\right)(1-\lambda )(\varrho -1)}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_2-1}\left|\vartheta \left(\iota \right)-w\left(\iota \right)\right|{\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & +{\displaystyle \frac{\left(\mathsf{\Theta }\left(\varsigma \right)-\mathsf{\Theta }\left({\tau }_0\right)\right)}{\rho }}\left({\kappa }_3-{\displaystyle \frac{\lambda }{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_1)}}\right.\hfill \\ \hfill & \times {\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -\beta -{\kappa }_1-1}\left|\Phi (\iota ,\vartheta (\iota \left)\right)-\Phi (\iota ,w(\iota \left)\right)\right|{\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \hfill \\ \hfill & \left.+{\displaystyle \frac{(1-\lambda )}{\varrho \mathsf{\Gamma }(\gamma -{\kappa }_2)}}{\int }_{{\tau }_0}^{\mathsf{{\rm Y}}}(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left(\iota \right){)}^{\gamma -{\kappa }_2-1}\left|\Phi (\iota ,\vartheta (\iota \left)\right)-\Phi (\iota ,w(\iota \left)\right)\right|{\mathsf{\Theta }}^{\prime }\left(\iota \right)d\iota \right)\hfill \\ \hfill & \le L_{\Phi }∥\vartheta -w∥\left[{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma }}{\varrho \mathsf{\Gamma }(\gamma +1)}}+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -{\kappa }_1+1}\lambda }{\varrho \rho \mathsf{\Gamma }(\gamma -{\kappa }_1+1)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -{\kappa }_2+1}(1-\lambda )}{\varrho \rho \mathsf{\Gamma }(\gamma -{\kappa }_2+1)}}\right]\hfill \\ \hfill & +∥\vartheta -w∥\left[{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -\beta }|\varrho -1|}{\varrho \mathsf{\Gamma }(\gamma -\beta +1)}}+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -\beta -{\kappa }_1+1}\lambda |\varrho -1|}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_1+1)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{(\mathsf{\Theta }\left(\mathsf{{\rm Y}}\right)-\mathsf{\Theta }\left({\tau }_0\right){)}^{\gamma -\beta -{\kappa }_2+1}(1-\lambda )|\varrho -1|}{\varrho \rho \mathsf{\Gamma }(\gamma -\beta -{\kappa }_2+1)}}\right]\hfill \\ \hfill & \le \left(L_{\Phi }{\eta }_2+{\eta }_1\right)∥\vartheta -w∥,\hfill \end{array}$$

which implies that

$$∥\mathcal{P}\vartheta -\mathcal{P}w∥\le \left(L_{\Phi }{\eta }_2+{\eta }_1\right)∥\vartheta -w∥.$$

According to (10), the application $\mathcal{P}$ is a contraction. So, by applying the Banach theorem, we conclude that the application $\mathcal{P}$ admits a fixed point which is a unique mild solution of (1). □

4. Two Examples

To demonstrate our key findings, we provide two examples in this section.

Example 1.

Consider the following second-order Hadamard–Caputo fractional boundary value problem:

$$\left\{\begin{array}{c}{\displaystyle \frac{32}{45}}{\mathfrak{D}}_{log\varsigma }^{7/4}\vartheta \left(\varsigma \right)+{\displaystyle \frac{13}{45}}{\mathfrak{D}}_{log\varsigma }^{5/4}\vartheta \left(\varsigma \right)={\displaystyle \frac{cos\left(\vartheta \left(\varsigma \right)\right)}{5+13\varsigma }}+{\displaystyle \frac{3}{4}},\varsigma \in [1,e],\hfill \\ \vartheta \left(1\right)=0,{\displaystyle \frac{12}{34}}{\mathfrak{D}}_{log\varsigma }^{1/3}\vartheta \left(e\right)+{\displaystyle \frac{22}{34}}{\mathfrak{D}}_{log\varsigma }^{1/7}\vartheta \left(e\right)={\displaystyle \frac{3}{4}}.\hfill \end{array}\right.$$

(11)

Here $\mathsf{\Theta }\left(\varsigma \right)=log\varsigma$, ${\tau }_0=1$, $\mathsf{{\rm Y}}=e$, $\varrho =32/45$, $\gamma =7/4$, $\beta =5/4$, $\lambda =12/34$, ${\kappa }_1=1/3$, ${\kappa }_2=1/7$, ${\kappa }_3=3/4$, and

$$\Phi (\varsigma ,\vartheta \left(\varsigma \right))={\displaystyle \frac{cos\left(\vartheta \right(\varsigma \left)\right)}{5+13\varsigma }}+{\displaystyle \frac{3}{4}}.$$

Observe that $0<{\kappa }_1,{\kappa }_2<\gamma -\beta =1/2$.

Since $|cos(\vartheta \left)\right|\le 1$, we obtain

$$|\Phi (\varsigma ,\vartheta \left(\varsigma \right))|\le {\displaystyle \frac{1}{5+13\varsigma }}+{\displaystyle \frac{3}{4}}=:\sigma \left(\varsigma \right),$$

which satisfies condition (C1).

Next, substituting the given parameters into (7), we compute

$${\eta }_1={\displaystyle \frac{(\mathsf{\Theta }\left(e\right)-\mathsf{\Theta }\left(1\right){)}^{\gamma -\beta }|\varrho -1|}{\varrho \mathsf{\Gamma }(\gamma -\beta +1)}}+\cdots \simeq 0.877<1,$$

where we used $\mathsf{\Theta }\left(e\right)-\mathsf{\Theta }\left(1\right)=1$.

Thus, all conditions of Theorem 3 are satisfied, and problem (11) admits at least one mild solution.

Furthermore, for $\vartheta ,w\in C\left(\right[1,e],\mathbb{R})$, we have

$$\begin{array}{cc}\hfill \left|\Phi (\varsigma ,\vartheta (\varsigma \left)\right)-\Phi (\varsigma ,w(\varsigma \left)\right)\right|& \le \left|{\displaystyle \frac{cos\vartheta \left(\varsigma \right)-cosw\left(\varsigma \right)}{5+13\varsigma }}\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{5+13\varsigma }}\left|\vartheta \left(\varsigma \right)-w\left(\varsigma \right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{18}}\left|\vartheta \left(\varsigma \right)-w\left(\varsigma \right)\right|,\hfill \end{array}$$

since $5+13\varsigma \ge 18$ for $\varsigma \in [1,e]$. Hence, the Lipschitz constant is $L_{\Phi }=1/18$.

Also, substituting the parameters into (9), we obtain

$${\eta }_2\simeq 1.832.$$

Finally, we compute

$$L_{\Phi }{\eta }_2+{\eta }_1\simeq {\displaystyle \frac{1}{18}}\times 1.832+0.877\simeq 0.979<1.$$

Therefore, by Theorem 4, problem (11) admits a unique mild solution.

Example 2.

Consider the following second-order Caputo fractional boundary value problem

$$\left\{\begin{array}{c}{\displaystyle \frac{47}{50}}{\mathfrak{D}}_{\varsigma }^{8/5}\vartheta \left(\varsigma \right)+{\displaystyle \frac{3}{50}}{\mathfrak{D}}_{\varsigma }^{6/5}\vartheta \left(\varsigma \right)={\displaystyle \frac{\vartheta \left(\varsigma \right)+sin\left(\vartheta \right(\varsigma \left)\right)}{8+\varsigma }}+{\displaystyle \frac{4}{9}},\varsigma \in [0,1],\hfill \\ \vartheta \left(0\right)=0,{\displaystyle \frac{17}{30}}{\mathfrak{D}}_{\varsigma }^{1/4}\vartheta \left(e\right)+{\displaystyle \frac{13}{30}}{\mathfrak{D}}_{\varsigma }^{1/5}\vartheta \left(e\right)={\displaystyle \frac{8}{9}}.\hfill \end{array}\right.$$

(12)

Here $\mathsf{\Theta }\left(\varsigma \right)=\varsigma$, ${\tau }_0=0$, $\mathsf{{\rm Y}}=1$, $\varrho =47/50$, $\gamma =8/5$, $\beta =6/5$, $\lambda =17/30$, ${\kappa }_1=1/4$, ${\kappa }_2=1/5$, ${\kappa }_3=8/9$, and

$$\Phi (\varsigma ,\vartheta \left(\varsigma \right))={\displaystyle \frac{\vartheta \left(\varsigma \right)+sin\left(\vartheta \right(\varsigma \left)\right)}{8+\varsigma }}+{\displaystyle \frac{4}{9}}.$$

Observe that $0<{\kappa }_1,{\kappa }_2<\gamma -\beta =2/5$.

Let $\vartheta ,w\in C\left(\right[0,1],\mathbb{R})$. Then

$$\begin{array}{cc}\hfill |\Phi (\varsigma ,\vartheta \left(\varsigma \right))-\Phi (\varsigma ,w\left(\varsigma \right)\left)\right|& =\left|{\displaystyle \frac{\vartheta \left(\varsigma \right)-w\left(\varsigma \right)+sin\left(\vartheta \right(\varsigma \left)\right)-sin\left(w\right(\varsigma \left)\right)}{8+\varsigma }}\right|\hfill \\ \hfill & \le {\displaystyle \frac{\left|\vartheta \right(\varsigma )-w(\varsigma \left)\right|+|sin(\vartheta \left(\varsigma \right))-sin(w\left(\varsigma \right)\left)\right|}{8+\varsigma }}\hfill \\ \hfill & \le {\displaystyle \frac{1}{4}}\left|\vartheta \left(\varsigma \right)-w\left(\varsigma \right)\right|,\hfill \end{array}$$

since $8+\varsigma \ge 8$, for all $\varsigma \in [0,1]$. Thus, Φ is Lipschitz continuous with constant $L_{\Phi }={\displaystyle \frac{1}{4}}$.

Next, substituting the given parameters into (7) and (9), we compute

$${\eta }_1\simeq 0.1356,{\eta }_2\simeq 1.5504.$$

Finally,

$$L_{\Phi }{\eta }_2+{\eta }_1\simeq 0.5232<1.$$

Therefore, all conditions of Theorem 4 are satisfied, and problem (12) admits a unique mild solution.

5. Conclusions

In this paper, we investigated a class of second-order fractional boundary value problems involving the $\mathsf{\Theta }$-Caputo fractional derivative. By introducing an appropriate notion of a mild solution and reformulating the problem as an integral equation, sufficient conditions for the existence and uniqueness of solutions were established. The analysis is based on the application of Krasnoselskii’s fixed point theorem and the Banach contraction principle.

The obtained results provide new existence conditions by constructing suitable fixed point operators and extend several earlier contributions in the literature. In particular, the $\mathsf{\Theta }$-Caputo framework allows us to recover important classical cases through suitable choices of the kernel function $\mathsf{\Theta }$. For instance, when $\mathsf{\Theta }\left(\varsigma \right)=\varsigma$, the problem is reduced to the classical Caputo fractional case, while choosing $\mathsf{\Theta }\left(\varsigma \right)=log\varsigma$ yields the Hadamard–Caputo fractional case. Thus, our results generalize those reported in [20,24] within a unified setting. Two illustrative examples were presented to demonstrate the applicability and effectiveness of the theoretical results.

The approach developed in this work can be extended in several directions. Future research may consider systems of fractional differential equations, impulsive problems, and boundary value problems with nonlocal or integral conditions. Moreover, it would be of interest to investigate similar problems for fractional partial differential equations and other types of generalized fractional operators. These directions may further enhance the applicability of the proposed framework and contribute to the development of the theory of fractional differential equations.