Mixed Erdélyi–Kober and Caputo Fractional Differential Equations with Nonlocal Fractional Closed Boundary Conditions

Abstract

This work focuses on the analysis of a sequential fractional boundary value problem involving coupled Erdélyi–Kober and Caputo fractional differential operators, together with nonlocal boundary conditions of fractional type. The well-posedness of the problem is addressed by deriving conditions that ensure the existence and uniqueness of solutions. Uniqueness is obtained through an application of Banach’s contraction principle, whereas existence is established by employing Krasnosel’skiĭ’s fixed point approach and the nonlinear alternative of Leray–Schauder. Several numerical examples are presented to demonstrate and support the theoretical findings.

1. Introduction

Fractional calculus constitutes a branch of modern mathematics, focusing on operators of non-integer order. In recent decades, it has attracted considerable attention due to the extensive use of fractional differential and integral operators across numerous scientific disciplines (see, for example, [1,2,3]). Fractional integral operators play a central role in the formulation of many fractional derivatives and provide powerful tools for capturing memory effects and hereditary properties in various physical and material processes. Fractional differential operators have been successfully applied in modeling many physical and engineering phenomena. For instance, fractional models describe viscoelastic materials with memory-dependent stress–strain relations more accurately than classical models [4]. Anomalous diffusion and transport phenomena in heterogeneous or porous media are effectively captured by fractional formulations [5,6]. Self-similar or scaling processes often involve fractional and Erdélyi–Kober operators. Furthermore, boundary value problems with nonlocal fractional operators arise in heat transfer and stochastic processes [7,8]. Systems with nonlocal boundary interactions include feedback-controlled heat transfer mechanisms and certain biological processes (see, for example, [9,10,11]).

A wide variety of fractional derivatives have been introduced and studied, including the Riemann–Liouville, Caputo, Erdélyi–Kober, and Hadamard operators, among others. The Erdélyi–Kober (E–K) fractional integral operator, in particular, extends the classical Riemann–Liouville operator and has proven effective in addressing certain classes of integral equations involving spatially dependent kernels. For further developments and applications of the E–K fractional integral operator, we refer the reader to [12,13,14,15,16,17].

In parallel, boundary value problems (BVPs) subject to various boundary conditions have been extensively investigated, owing to their relevance in applications such as heat transfer, electromagnetic theory, and membrane dynamics in nuclear reactors (see [18,19,20] and the references therein). More recently, BVPs involving Hilfer-type fractional differential equations under different boundary conditions have been explored by several authors [21,22]. These studies typically impose zero initial conditions to ensure the existence of well-defined solutions. In [23], the authors combined the Caputo and Erdélyi–Kober fractional derivatives to investigate a BVP with non-separated boundary conditions of the form

$$\left\{\begin{array}{c}{}^E\mathbb{D}_{\beta }^{\gamma ,\delta }({}^C\mathbb{D}^{\alpha }v)(t)=g(t,v(t)),t\in [0,T],\hfill \\ a_{1}v(0)+a_{2}v(T)=f_1(v),\hfill \\ {\displaystyle b_1{}^C\mathbb{D}^{\alpha -\beta (1+\gamma )}v(0)+b_2{}^C\mathbb{D}^{\alpha -\beta (1+\gamma )}v(T)=f_2(v),}\hfill \end{array}\right.$$

(1)

in which $0<\alpha ,\delta <1$, $\beta >0$, $\gamma >-1$ with $\alpha >\beta (1+\gamma )$, $g\in C([0,T],\mathbb{R})$, $a_i,b_i\in \mathbb{R},i=1,2$ and $f_i:C([0,T],\mathbb{R})⟶\mathbb{R},i=1,2.$

Closed boundary conditions play a pivotal role in fluid dynamics, as they describe physical settings in which mass transfer across the boundary of the domain is prohibited [24]. Such conditions are commonly used to characterize impermeable walls or thermally insulated boundaries and may incorporate free-slip constraints, allowing motion along the boundary while preventing flow in the normal direction. These boundary formulations are indispensable for the realistic representation of a wide range of physical phenomena, including gravitational and radiative effects, elastic wave transmission, and heat transfer governed by radiation mechanisms. Moreover, closed boundary conditions have found extensive applications in areas such as computational fluid dynamics, image reconstruction, and transport processes in structured media, including honeycomb-type lattices [25,26,27,28]. When properly imposed, they ensure physical plausibility, enhance numerical robustness, and improve the precision of analytical and computational models for complex flow systems.

The study of boundary value problems for fractional differential equations and differential inclusions subject to closed boundary conditions was first undertaken in [29]. In that work, the authors considered the following class of problems:

$$\left\{\begin{array}{c}{\displaystyle {}^C\mathcal{D}^{\rho }u(x)=\phi (x,u(x)),x\in J:=[0,b],}\hfill \\ {\displaystyle u(b)={\eta }_{1}u(0)+{\eta }_{2}b{u}^{\prime }(0),}\hfill \\ {\displaystyle b{u}^{\prime }(b)={\xi }_{1}u(0)+{\xi }_{2}b{u}^{\prime }(0),}\hfill \end{array}\right.$$

(2)

where ${}^C\mathcal{D}^{\rho }$ denotes the Caputo fractional derivative of order $\rho ,$ the function $\phi :[0,b]\times \mathbb{R}\to \mathbb{R}$ is continuous and the parameters ${\eta }_1,{\eta }_2,{\xi }_1,{\xi }_2$ are real constants.

For a detailed and up-to-date account of fractional boundary value problems involving closed boundary conditions, the reader is referred to the recent survey presented in [30]. For some recent work on sequential fractional operators, nonlocal fractional boundary conditions, and existence results for fractional differential equations, we refer the reader to [31,32,33,34,35,36,37] and the references cited therein.

Motivated by these developments, the present work investigates a new class of boundary value problems by coupling the Caputo and Erdélyi–Kober fractional derivatives. Specifically, we study a BVP with fractional closed boundary conditions given by

$$\left\{\begin{array}{c}{\displaystyle {}^E\mathcal{D}_{\sigma }^{\vartheta ,\nu }\left({}^C\mathcal{D}^{\rho }u\right)(x)=\phi (x,u(x)),x\in [0,b],}\hfill \\ u(b)={\eta }_{1}u(\zeta )+{\eta }_{2}b{}^C\mathcal{D}^{\rho -\sigma (1+\vartheta )}u(\zeta ),\hfill \\ b{}^C\mathcal{D}^{\rho -\sigma (1+\vartheta )}u(b)={\xi }_{1}u(\zeta )+{\xi }_{2}b{}^C\mathcal{D}^{\rho -\sigma (1+\vartheta )}u(\zeta ),0<\zeta <b,\hfill \end{array}\right.$$

(3)

in which $0<\nu ,\rho <1$, $\sigma >0$, $\vartheta >-1$ with $\rho >\sigma (1+\vartheta )$, ${}^E\mathcal{D}_{\sigma }^{\vartheta ,\nu }$ and ${}^C\mathcal{D}^{\rho }$ are the Erdélyi–Kober and Caputo fractional derivatives operators, respectively, $\phi \in C([0,b],\mathbb{R})$, and ${\eta }_i,{\xi }_i\in \mathbb{R},i=1,2.$

The principal results of this study are established through the application of fixed point methods. In particular, Banach’s contraction principle is employed to guarantee the uniqueness of solutions, whereas the Leray–Schauder nonlinear alternative and Krasnosel’skiĭ’s fixed point theorems are utilized to derive existence results. In addition, several numerical illustrations are provided to validate and complement the theoretical findings.

The originality of the present work stems from coupling the Caputo and Erdélyi–Kober fractional derivatives within a framework of fractional closed boundary conditions. Unlike the standard Caputo or Riemann–Liouville fractional operators, the Erdélyi–Kober operator contains a power-weighted kernel and an intrinsic scaling parameter, and model processes with non-uniform memory effects and scaling invariance. In particular, the presence of the parameter $\rho$ introduces an additional degree of freedom that governs the interaction between memory and dilation effects in contrast to classical fractional operators.

The remainder of the paper is organized as follows. Section 2 introduces the essential preliminaries, including key definitions and auxiliary lemmas required for the subsequent analysis. This section also establishes a useful result that reformulates problem (3) as an equivalent fixed point equation. In Section 3, existence and uniqueness of solutions to problem (3) are demonstrated using appropriate fixed point techniques. Finally, the concluding section presents numerical examples that illustrate and support the obtained theoretical results.

2. Preliminaries

This section introduces several preliminary concepts and auxiliary results that will be used throughout the paper to derive the main findings.

Definition 1

([2]). Let $\rho >0.$ The Riemann–Liouville fractional integral operator of order $\rho >0$ with lower limit a, is defined for $x>a$ by

$$\begin{array}{c}\hfill {}^R\mathcal{I}_{a^{+}}^{\rho }u(x)={\displaystyle \frac{1}{\mathrm{\Gamma }(\rho )}}{\int }_a^x{(x-z)}^{\rho -1}u(z)dz,x>a.\end{array}$$

Based on the above definition, one can introduce the Riemann–Liouville and Caputo fractional derivatives of order $\rho >0$ (see [2]) as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}& {\displaystyle {}^R\mathcal{D}_{a^{+}}^{\rho }u(x)=D^n\left({}^R\mathcal{I}_{a^{+}}^{n-\rho }u\right)(x)=\frac{1}{\mathrm{\Gamma }(n-\rho )}\frac{d^n}{d{x}^n}{\int }_a^x{(x-z)}^{n-\rho -1}u(z)dz,x>a,}\hfill \\ & {\displaystyle {}^C\mathcal{D}_{a^{+}}^{\rho }u(x)={}^R\mathcal{I}_{a^{+}}^{n-\rho }\left(D^{n}u\right)(x)=\frac{1}{\mathrm{\Gamma }(n-\rho )}{\int }_a^x{(x-z)}^{n-\rho -1}{u}^{(n)}(z)dz,x>a,}\hfill \end{array}\end{array}$$

(4)

where $D^n=d^n/d{x}^n$ and n is an integer such that $n-1<\rho <n$.

When the lower terminal is chosen as $a=0$, we adopt the simplified notations ${}^R\mathcal{I}^{\rho }(·)$, ${}^R\mathcal{D}^{\rho }(·)$ and ${}^C\mathcal{D}^{\rho }(·)$. Throughout this work, it is assumed that all integrals and derivatives appearing in the above definitions exist. In the sequel, we present several lemmas that summarize fundamental properties of these fractional operators.

Lemma 1 

([2]). Let $n-1\le \rho \le n$, $\vartheta >0$ be constants and $x>a$. Then, the following properties are satisfied:

$(i)$

${}^R\mathcal{I}_{a^{+}}^{\rho }\left({}^R\mathcal{I}_{a^{+}}^{\vartheta }u\right)(x)={\mathcal{I}}_{a^{+}}^{\rho +\vartheta }u(x)$;

$(ii)$

${\displaystyle {}^R\mathcal{I}_{a^{+}}^{\rho }\left({}^C\mathcal{D}_{a^{+}}^{\rho }u\right)(x)=u(x)-\sum _{j=0}^{n-1}\frac{u^{(j)}(a^{+})}{j!}{(x-a)}^j}$;

$(iii)$

${\displaystyle {}^R\mathcal{I}_{a^{+}}^{\rho }{(x-a)}^{\vartheta -1}=\frac{\mathrm{\Gamma }(\vartheta )}{\mathrm{\Gamma }(\vartheta +\rho )}{(x-a)}^{\vartheta +\rho -1}}$;

$(iv)$

If $\vartheta >\rho +1$ then ${\displaystyle {}^C\mathcal{D}_{a^{+}}^{\rho }{(x-a)}^{\vartheta -1}=\frac{\mathrm{\Gamma }(\vartheta )}{\mathrm{\Gamma }(\vartheta -\rho )}{(x-a)}^{\vartheta -\rho -1};}$

$(u)$

If $\rho <\vartheta$ then ${\displaystyle {}^C\mathcal{D}_{a^{+}}^{\rho }\left({}^R\mathcal{I}_{a^{+}}^{\vartheta }u\right)(x)={}^R\mathcal{I}_{a^{+}}^{\vartheta -\rho }u(x).}$

Definition 2

([2]). Let $\vartheta >0,$$\sigma >0,$ and $\nu \in \mathbb{R}.$ The fractional integral of Erdélyi–Kober type $\vartheta ,$ associated with the parameters σ and ν, acting on a of a continuous function $u:(0,\infty )⟶\mathbb{R},$ is given by

$$\begin{array}{c}\hfill {}^E\mathcal{I}_{\nu ,\vartheta }^{\sigma }u(x)={\displaystyle \frac{\sigma x^{-\sigma (\vartheta +\nu )}}{\mathrm{\Gamma }(\vartheta )}}{\int }_0^x{\displaystyle \frac{z^{\sigma \nu +\sigma -1}u(z)}{{(x^{\sigma }-z^{\sigma })}^{1-\vartheta }}}dz.\end{array}$$

It is worth noting that the above operator reduces to a scaled Riemann–Liouville fractional integral when $\nu =0$ and $\sigma =1$, since in this case ${}^E\mathcal{I}_1^{0,\vartheta }u(x)=x^{-\vartheta }\left({}^R\mathcal{I}^{\vartheta }u(x)\right)$.

Using the Erdélyi–Kober fractional integral, one may define the corresponding Erdélyi–Kober fractional derivative of order $\vartheta$, denoted by ${\displaystyle {}^E\mathcal{D}_{\sigma }^{\nu ,\vartheta }}$, for $n-1<\vartheta \le n$, $\sigma >0$, and $\nu \in \mathbb{R}$, as follows (see [2]):

$${}^E\mathcal{D}_{\sigma }^{\nu ,\vartheta }u(x)=x^{-\sigma \nu }{\left({\displaystyle \frac{1}{\sigma x^{\sigma -1}}}{\displaystyle \frac{d}{dx}}\right)}^n{x}^{\sigma (n+\nu )}\left({}^E\mathcal{I}_{\sigma }^{\nu +\vartheta ,n-\vartheta }u\right)(x).$$

Moreover, by employing the identity

$$\prod _{j=1}^n\left[\nu +j+{\displaystyle \frac{1}{\sigma }}x{\displaystyle \frac{d}{dx}}\right](·)=\left[x^{-\sigma \nu }{\left({\displaystyle \frac{1}{\sigma x^{\sigma -1}}}{\displaystyle \frac{d}{dx}}\right)}^n{t}^{\sigma (n+\nu )}\right](·),$$

an equivalent representation of the Erdélyi–Kober fractional derivative can be obtained as

$$\begin{array}{c}\hfill {\displaystyle \left({}^E\mathcal{D}_{\sigma }^{\nu ,\vartheta }u\right)(x)=\prod _{j=1}^n\left(\nu +j+\frac{1}{\sigma }x\frac{d}{dx}\right)\left({}^E\mathcal{I}_{\sigma }^{\nu +\vartheta ,n-\vartheta }u\right)(x),}\end{array}$$

(5)

(see [16,17]).

Lemma 2

([16]). Let $n-1<\vartheta <n$, $n\in N$, $\vartheta >-\sigma (\nu +1)$ and $u\in C_{\rho }^n[a,b]$. Then we get

$$\begin{array}{c}\hfill {\displaystyle {}^E\mathcal{I}_{\sigma }^{\nu ,\vartheta }\left({}^E\mathcal{D}_{\sigma }^{\nu ,\vartheta }u\right)(x)=u(x)-\sum _{i=0}^{n-1}{c}_i{x}^{-\sigma (1+\nu +i)},}\end{array}$$

where

$$\begin{array}{c}\hfill {\displaystyle c_i=\frac{\mathrm{\Gamma }(n-i)}{\mathrm{\Gamma }(\vartheta -i)}\underset{x⟶0}{lim}{x}^{\sigma (1+\nu +i)}\prod _{j=1+i}^{n-1}\left(1+\nu +j+\frac{1}{\sigma }x\frac{d}{dx}\right)\left({}^E\mathcal{I}_{\sigma }^{\nu +\vartheta ,n-\vartheta }u\right)(x).}\end{array}$$

Lemma 3

([38]). Let $\vartheta ,\sigma ,\mu >0$ and $\nu >-(\mu +\sigma )/\sigma$. Then we have

$$\begin{array}{c}\hfill {}^E\mathcal{I}_{\sigma }^{\nu ,\vartheta }{x}^{\mu }={\displaystyle \frac{x^{\mu }\mathrm{\Gamma }\left(\nu +\frac{\mu }{\sigma }+1\right)}{\mathrm{\Gamma }\left(\nu +\frac{\mu }{\sigma }+\vartheta +1\right)}}.\end{array}$$

Prior to analyzing the boundary value problem involving the sequential Erdélyi–Kober and Caputo fractional differential operators, it is instructive to examine how these operators act in succession on a test function of the form $u(x)=x^{\mu }$.

Example 1.

Let $u(x)=x^{\mu }$, $x\ge 0$. Then we have by using the Lemmas 1 and 3 that

$${\displaystyle {}^E\mathcal{D}_{\sigma }^{\vartheta ,\nu }({}^C\mathcal{D}^{\sigma }u)(x)=\frac{\mathrm{\Gamma }(\mu +1)\mathrm{\Gamma }\left(\vartheta +\nu +\frac{\mu -\rho }{\sigma }+1\right)}{\mathrm{\Gamma }(\mu -\rho +1)\mathrm{\Gamma }\left(\vartheta +\frac{\mu -\rho }{\sigma }+1\right)}{x}^{\mu -\rho }}$$

(6)

and

$${\displaystyle {}^C\mathcal{D}^{\rho }({}^C\mathcal{D}_{\sigma }^{\vartheta ,\nu }u)(x)=\frac{\mathrm{\Gamma }(\mu +1)\mathrm{\Gamma }\left(\vartheta +\nu +\frac{\mu }{\sigma }+1\right)}{\mathrm{\Gamma }(\mu -\rho +1)\mathrm{\Gamma }\left(\vartheta +\frac{\mu }{\sigma }+1\right)}{x}^{\mu -\rho }.}$$

(7)

The lemma stated below plays a central role in rewriting problem (3) as an equivalent integral equation.

Lemma 4.

Let $0<\nu ,\rho <1$ denote the fractional orders, and let $\sigma >0$ and $\vartheta >-1$ be the parameters associated with the Erdélyi–Kober operator, satisfying the condition $\rho >\sigma (1+\vartheta ).$ Assume that $h\in C([0,b],\mathbb{R}),$ and that ${\eta }_i,{\xi }_i\in \mathbb{R}i=1,2$. Under these hypotheses, the linear boundary value problem involving Erdélyi–Kober and Caputo fractional derivatives, together with fractional nonlocal closed boundary conditions of the form

$$\left\{\begin{array}{c}{\displaystyle {}^E\mathcal{D}_{\sigma }^{\vartheta ,\nu }({}^C\mathcal{D}^{\rho }u(x))=h(x),x\in [0,b],}\hfill \\ u(b)={\eta }_{1}u(\zeta )+{\eta }_{2}b{}^C\mathcal{D}^{\rho -\sigma (1+\vartheta )}u(\zeta ),\hfill \\ b{}^C\mathcal{D}^{\rho -\sigma (1+\vartheta )}u(b)={\xi }_{1}u(\zeta )+{\xi }_{2}b{}^C\mathcal{D}^{\rho -\sigma (1+\vartheta )}u(\zeta ),0<\zeta <b,\hfill \end{array}\right.$$

(8)

is equivalent to the integral equation

$$\begin{array}{ccc}\hfill & & u(x)\hfill \\ & =& {\displaystyle {}^R\mathcal{I}^{\rho }({}^R\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(x))}\hfill \\ & & {\displaystyle +\frac{1}{\mathrm{\Lambda }}[\left({\xi }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))+{\xi }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))-b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(b))\right)Q_1}\hfill \\ & & {\displaystyle -\left({\eta }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))+{\eta }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))-{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(b))\right)Q_2]}\hfill \\ & & {\displaystyle +\frac{1}{\mathrm{\Lambda }}[(-{\xi }_1\left({\eta }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))+{\eta }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))-{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(b))\right)}\hfill \\ & & {\displaystyle -(1-{\eta }_1)({\xi }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))+{\xi }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))}\hfill \\ & & {\displaystyle -b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(b)))]\frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}{x}^{\rho -\sigma (1+\vartheta )},}\hfill \end{array}$$

(9)

where

$$\begin{array}{ccc}\hfill Q_1& =& {\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}\left(b^{\rho -\sigma (1+\vartheta )}-{\eta }_1{\zeta }^{\rho -\sigma (1+\vartheta )}\right)-{\eta }_{2}b\mathrm{\Gamma }(1-\sigma (1+\vartheta )),}\hfill \\ \hfill Q_2& =& {\displaystyle b(1-{\xi }_2)\mathrm{\Gamma }(1-\sigma (1+\vartheta ))-{\xi }_1\frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}{\zeta }^{\rho -\sigma (1+\vartheta )},}\hfill \end{array}$$

where it is assumed that $\mathrm{\Lambda }:=-{\xi }_1{Q}_1+({\eta }_1-1)Q_2\ne 0.$

Proof. 

Applying the Erdélyi–Kober fractional integral operator ${\displaystyle {}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }}$ to both sides of the first equation in (8) and invoking Lemma 2, we obtain

$${\displaystyle {}^C\mathcal{D}^{\rho }(u)(x)=c_0{x}^{-\sigma (1+\vartheta )}+{}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(x).}$$

(10)

Next, we apply the Riemann–Liouville fractional integral ${\displaystyle {}^R\mathcal{I}^{\rho }}$ to both sides of (10) and make use Lemma 1 to obtain

$${\displaystyle u(x)=c_1+c_0\frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}{x}^{\rho -\sigma (1+\vartheta )}+{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(x)).}$$

(11)

Finally, in order to compute the fractional derivative appearing in the boundary conditions, we note by Lemma 1 and the fact that the Caputo derivative of a constant vanishes that

$$\begin{array}{ccc}\hfill {\displaystyle {}^C\mathcal{D}^{\rho -\sigma (1+\vartheta )}u(x)}& =& {\displaystyle {}^C\mathcal{D}^{\rho -\sigma (1+\vartheta )}{c}_1+c_0\frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}{}^C\mathcal{D}^{\rho -\sigma (1+\vartheta )}{x}^{\rho -\sigma (1+\vartheta )}}\hfill \\ & & {\displaystyle +{}^C\mathcal{D}^{\rho -\sigma (1+\vartheta )}{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(x))}\hfill \\ & =& {\displaystyle c_0\mathrm{\Gamma }(1-\sigma (1+\vartheta ))+{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(x)).}\hfill \end{array}$$

(12)

By incorporating the prescribed boundary conditions into Equations (11) and (12), we derive a linear system involving the unknown constants $c_0$ and $c_1$ as

$$\left\{\begin{array}{cc}{Q}_1{c}_0+(1-{\eta }_1)c_1\hfill & {\displaystyle ={\eta }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))+{\eta }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))}\hfill \\ \hfill & {\displaystyle -{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(b)),}\hfill \\ Q_2{c}_0-{\xi }_1{c}_1\hfill & {\displaystyle ={\xi }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))+{\xi }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))}\hfill \\ \hfill & {\displaystyle -b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(b)).}\hfill \end{array}\right.$$

(13)

Solving the above system yields explicit expressions for these constants:

$$\begin{array}{ccc}\hfill c_0& =& {\displaystyle \frac{1}{\mathrm{\Lambda }}[-{\xi }_1\left({\eta }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))+{\eta }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))-{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(b))\right)}\hfill \\ & & {\displaystyle -(1-{\eta }_1)({\xi }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))+{\xi }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))}\hfill \\ & & {\displaystyle -b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(b)))],}\hfill \\ \hfill c_1& =& {\displaystyle \frac{1}{\mathrm{\Lambda }}[\left({\xi }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))+{\xi }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))-b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(b))\right)Q_1}\hfill \\ & & {\displaystyle -\left({\eta }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))+{\eta }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(\zeta ))-{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }h(b))\right)Q_2].}\hfill \end{array}$$

Substituting the above values of $c_0$ and $c_1$ into Equation (11), we arrive at the unique solution (9) of the linear boundary value problem (8).

Conversely, the validity of the solution can be verified by directly applying the Caputo and Erdélyi–Kober fractional derivative operators to Equation (9). This computation confirms that the obtained expression satisfies both the governing fractional differential equation and the associated boundary conditions of problem (8). □

Remark 1.

The condition $\mathrm{\Lambda }\ne 0$ ensures that the solution is well defined and non-degenerate, allowing the use of fixed-point theorems to prove existence and uniqueness results for the given problem. The restriction $\rho >\sigma (1+\vartheta )$ guarantees convergence of the Erdélyi–Kober integral and compactness of the associated operator, while balancing the scaling effects of the Erdélyi–Kober operator with the order of the Caputo derivative. From a modeling viewpoint, these assumptions ensure a consistent interplay among memory, scaling and boundary effects, preventing singular or unbounded behavior of solutions.

3. Existence and Uniqueness Results

Let $\mathcal{X}=C([0,b],\mathbb{R})$ denote the Banach space consisting of all real-valued continuous functions defined on the interval $[0,b]$, equipped with the supremum norm $\parallel u\parallel =sup\{|u(x)|:x\in [0,b\left]\right\}$.

Motivated by Lemma 4, we introduce an operator $\mathcal{F}:\mathcal{X}⟶\mathcal{X}$ as

$$\begin{array}{ccc}\hfill \mathcal{F}(u)(x)& =& {\displaystyle {}^R\mathcal{I}^{\rho }\left({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (x,u(x))\right)}\hfill \\ & & {\displaystyle +\frac{1}{\mathrm{\Lambda }}\left[\right({\xi }_1{}^R\mathcal{I}^{\rho }\left({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi \left(\zeta ,u(\zeta )\right)\right)+{\xi }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}\left({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (\zeta ,u(\zeta ))\right)}\hfill \\ & & {\displaystyle -b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (b,u(b))))Q_1-({\eta }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (\zeta ,u(\zeta )))}\hfill \\ & & {\displaystyle +{\eta }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (\zeta ,u(\zeta )))-{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (b,u(b)))\left)Q_2\right]}\hfill \\ & & {\displaystyle +\frac{1}{\mathrm{\Lambda }}[-{\xi }_1({\eta }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (\zeta ,u(\zeta )))+{\eta }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (\zeta ,u(\zeta )))}\hfill \\ & & {\displaystyle -{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (b,u(b))))-(1-{\eta }_1)({\xi }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (\zeta ,u(\zeta )))}\hfill \\ & & {\displaystyle +{\xi }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (\zeta ,u(\zeta )))-b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (b,u(b)))\left)\right]}\hfill \\ & & \times {\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{x}^{\rho -\sigma (1+\vartheta )},x\in [0,b].\hfill \end{array}$$

(14)

For convenience, we set:

$$\begin{array}{ccc}\hfill {\mathcal{M}}_1& =& {\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{1}{\mathrm{\Gamma }(\rho +1)}}\{b^{\rho }\left(1+{\displaystyle \frac{|Q_2|}{|\mathrm{\Lambda }|}}\right)+{\displaystyle \frac{{\zeta }^{\rho }}{|\mathrm{\Lambda }|}}\left(|{\xi }_1||Q_1|+|{\eta }_1||Q_2|\right)\hfill \\ & & +{\displaystyle \frac{|{\xi }_1|}{|\mathrm{\Lambda }|}}\left[{\zeta }^{\rho }(1+2|{\eta }_1|)+b^{\rho }\right]{\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{b}^{\rho -\sigma (1+\vartheta )}\},\hfill \\ \hfill {\mathcal{M}}_2& =& {\displaystyle \frac{1}{|\mathrm{\Lambda }|}}{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{1}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}}\{\left(|{\xi }_2|b{\zeta }^{\sigma (1+\vartheta )}+b^{\sigma (1+\vartheta )+1}\right)|Q_1|\hfill \\ & & +|{\eta }_2|b{\zeta }^{\sigma (1+\vartheta )}|Q_2|+\left[\right((1+|{\eta }_1|)|{\xi }_2|+|{\xi }_1||{\eta }_2|)b{\zeta }^{\sigma (1+\vartheta )}\hfill \\ & & +(1+|{\eta }_1|)b^{\sigma (1+\vartheta )+1}]{\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{b}^{\rho -\sigma (1+\vartheta )}\}.\hfill \end{array}$$

(15)

The theorem below employs Banach’s fixed point principle [39] to establish the existence and uniqueness of a solution to problem (3).

Theorem 1.

Let $\phi :[0,b]\times \mathbb{R}⟶\mathbb{R}$ be a continuous function satisfying the condition:

$(A_1)$

There exists a positive constant $\mathcal{L}$ such that for every $x\in [0,b]$ and all $u_1,u_2\in \mathbb{R},$

$$\begin{array}{c}\hfill |\phi (x,u_1)-\phi (x,u_2)|\le \mathcal{L}|u_1-u_2|.\end{array}$$

Then, problem (3) admits a unique solution on the interval $[0,b],$ provided that

$$\mathcal{L}({\mathcal{M}}_1+{\mathcal{M}}_2)<1,$$

where constants ${\mathcal{M}}_i,i=1,2$ are defined in (15).

Proof. 

Define ${\phi }_0=sup\{|\phi (x,0)|:x\in [0,b]\},$ and consider the closed ball $B_{r_1}=\{u\in \mathcal{X}:\parallel u\parallel \le r_1\}$ where the radius $r_1$ is chosen so that ${\displaystyle r_1\ge \frac{({\mathcal{M}}_1+{\mathcal{M}}_2){\phi }_0}{1-\mathcal{L}({\mathcal{M}}_1+{\mathcal{M}}_2)}}$. We begin by showing that the operator $\mathcal{F}$ maps $B_{r_1}$ into itself, that is, $\mathcal{F}{B}_{r_1}\subseteq B_{r_1}.$ Using condition $(A_1)$, for all $u\in B_{r_1}$ and all $x\in [0,b]$, we obtain

$$\begin{array}{c}\hfill |\phi (x,u(x)|\le |\phi (x,u(x))-\phi (x,0)|+|\phi (x,0)|\le \mathcal{L}\parallel u\parallel +{\phi }_0.\end{array}$$

Consequently, we have

$$\begin{array}{ccc}& & |(\mathcal{F}u)(x)|\hfill \\ & \le & {\displaystyle {}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (x,u(x))|)}\hfill \\ & & {\displaystyle +\frac{1}{|\mathrm{\Lambda }|}[(|{\xi }_1|{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u(\zeta ))|)+|{\xi }_2|b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u(\zeta ))|)}\hfill \\ & & {\displaystyle +b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (b,u(b))|))|Q_1|+(|{\eta }_1|{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u(\zeta ))|)}\hfill \\ & & {\displaystyle +|{\eta }_2|b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u(\zeta ))|)+{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (b,u(b))|))|Q_2|]}\hfill \\ & & {\displaystyle +\frac{1}{|\mathrm{\Lambda }|}[|{\xi }_1|(|{\eta }_1|{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u(\zeta ))|)+|{\eta }_2|b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u(\zeta ))|)}\hfill \\ & & {\displaystyle +{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (b,u(b))|))+(1+|{\eta }_1|)(|{\xi }_1|{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u(\zeta ))|)}\hfill \\ & & {\displaystyle +|{\xi }_2|b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u(\zeta ))|)+b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (b,u(b))|))]}\hfill \\ & & \times {\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{x}^{\rho -\sigma (1+\vartheta )}\hfill \\ & \le & {\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{b^{\rho }}{\mathrm{\Gamma }(\rho +1)}}(\mathcal{L}\parallel u\parallel +{\phi }_0)\hfill \\ & & +{\displaystyle \frac{1}{|\mathrm{\Lambda }|}}[(|{\xi }_1|{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{{\zeta }^{\rho }}{\mathrm{\Gamma }(\rho +1)}}+|{\xi }_2|b{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{{\zeta }^{\sigma (1+\vartheta )}}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}}\hfill \\ & & +b{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{b^{\sigma (1+\vartheta )}}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}})|Q_1|+(|{\eta }_1|{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{{\zeta }^{\rho }}{\mathrm{\Gamma }(\rho +1)}}\hfill \\ & & +|{\eta }_2|b{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{{\zeta }^{\sigma (1+\vartheta )}}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}}+{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{b^{\rho }}{\mathrm{\Gamma }(\rho +1)}})|Q_2|](\mathcal{L}\parallel u\parallel +{\phi }_0)\hfill \\ & & +{\displaystyle \frac{1}{|\mathrm{\Lambda }|}}[|{\xi }_1|(|{\eta }_1|{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{{\zeta }^{\rho }}{\mathrm{\Gamma }(\rho +1)}}+|{\eta }_2|b{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{{\zeta }^{\sigma (1+\vartheta )}}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}}\hfill \\ & & +{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{b^{\rho }}{\mathrm{\Gamma }(\rho +1)}})+(1+|{\eta }_1|)(|{\xi }_1|{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{{\zeta }^{\rho }}{\mathrm{\Gamma }(\rho +1)}}\hfill \\ & & +|{\xi }_2|b{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{{\zeta }^{\sigma (1+\vartheta )}}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}}+b{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{b^{\sigma (1+\vartheta )}}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}})]\hfill \\ & & \times {\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{b}^{\rho -\sigma (1+\vartheta )}(\mathcal{L}\parallel u\parallel +{\phi }_0)\hfill \end{array}$$

$$\begin{array}{ccc}& =& \left[{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{1}{\mathrm{\Gamma }(\rho +1)}}\right\{b^{\rho }\left(1+{\displaystyle \frac{|Q_2|}{|\mathrm{\Lambda }|}}\right)+{\displaystyle \frac{{\zeta }^{\rho }}{|\mathrm{\Lambda }|}}\left(|{\xi }_1||Q_1|+|{\eta }_1||Q_2|\right)\hfill \\ & & +{\displaystyle \frac{|{\xi }_1|}{|\mathrm{\Lambda }|}}\left[{\zeta }^{\rho }(1+2|{\eta }_1|)+b^{\rho }\right]{\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{b}^{\rho -\sigma (1+\vartheta )}\}\hfill \\ & & +{\displaystyle \frac{1}{|\mathrm{\Lambda }|}}{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{1}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}}\{\left(|{\xi }_2|b{\zeta }^{\sigma (1+\vartheta )}+b^{\sigma (1+\vartheta )+1}\right)|Q_1|\hfill \\ & & +|{\eta }_2|b{\zeta }^{\sigma (1+\vartheta )}|Q_2|+[\left((1+|{\eta }_1|)|{\xi }_2|+|{\xi }_1||{\eta }_2|\right)b{\zeta }^{\sigma (1+\vartheta )}\hfill \\ & & +(1+|{\eta }_1|)b^{\sigma (1+\vartheta )+1}]{\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{b}^{\rho -\sigma (1+\vartheta )}\left\}\right](\mathcal{L}\parallel u\parallel +{\phi }_0)\hfill \\ & =& ({\mathcal{M}}_1+{\mathcal{M}}_2)(\mathcal{L}\parallel u\parallel +{\phi }_0)\hfill \\ & \le & \mathcal{L}({\mathcal{M}}_1+{\mathcal{M}}_2)r_1+({\mathcal{M}}_1+{\mathcal{M}}_2){\phi }_0\le r_1.\hfill \end{array}$$

As a consequence, we deduce that $\parallel (\mathcal{F}u)\parallel \le r_1$ which shows that $\mathcal{F}$ leaves the ball $\mathcal{F}{B}_{r_1}$ invariant, i.e., $\mathcal{F}{B}_{r_1}\subseteq B_{r_1}$.

We now proceed to verify that $\mathcal{F}$ is a contraction mapping. Let $u_1,u_2\in B_{r_1}$, and fix $x\in [0,b].$ Then, we obtain

$$\begin{array}{ccc}& & |(\mathcal{F}{u}_1)(x)-(\mathcal{F}{u}_2)(x)|\hfill \\ & \le & {\displaystyle {}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (x,u_1(x))-\phi (x,u_2(x)|)+\frac{1}{\mathrm{\Lambda }}[|{\xi }_1|({}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u_1(\zeta ))-\phi (\zeta ,u_2(\zeta ))|)}\hfill \\ & & {\displaystyle +|{\xi }_2|b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u_1(\zeta ))-\phi (\zeta ,u_2(\zeta ))|)}\hfill \\ & & {\displaystyle +b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (b,u_1(b))-\phi (b,u_2(b))|))|Q_1|}\hfill \\ & & {\displaystyle +(|{\eta }_1|{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u_1(\zeta ))-\phi (\zeta ,u_2(\zeta ))|)}\hfill \\ & & {\displaystyle +|{\eta }_2|b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u_1(\zeta ))-\phi (\zeta ,u_2(\zeta ))|)}\hfill \\ & & {\displaystyle +{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (b,u_1(b))-\phi (b,u_2(b))|))|Q_2|]}\hfill \\ & & {\displaystyle +\frac{1}{\mathrm{\Lambda }}[|{\xi }_1|(|{\eta }_1|{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u_1(\zeta ))-\phi (\zeta ,u_2(\zeta ))|)}\hfill \\ & & {\displaystyle +|{\eta }_2|b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u_1(\zeta ))-\phi (\zeta ,u_2(\zeta ))|)}\hfill \\ & & {\displaystyle +{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (b,u_1(b))-\phi (x,u_2(b))|))}\hfill \\ & & {\displaystyle +(1+|{\eta }_1|)(|{\xi }_1|{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u_1(\zeta ))-\phi (\zeta ,u_2(\zeta ))|)}\hfill \\ & & {\displaystyle +|{\xi }_2|b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^R\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (\zeta ,u_1(\zeta ))-\phi (\zeta ,u_2(\zeta ))|)}\hfill \\ & & {\displaystyle +b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (b,u_1(b))-\phi (b,u_2(b))|))]\frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}{x}^{\rho -\sigma (1+\vartheta )}}\hfill \\ & \le & \left[{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{1}{\mathrm{\Gamma }(\rho +1)}}\right\{b^{\rho }\left(1+{\displaystyle \frac{|Q_2|}{|\mathrm{\Lambda }|}}\right)+{\displaystyle \frac{{\zeta }^{\rho }}{|\mathrm{\Lambda }|}}\left(|{\xi }_1||Q_1|+|{\eta }_1||Q_2|\right)\hfill \end{array}$$

$$\begin{array}{ccc}& & +{\displaystyle \frac{|{\xi }_1|}{|\mathrm{\Lambda }|}}\left[{\zeta }^{\rho }(1+2|{\eta }_1|)+b^{\rho }\right]{\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{b}^{\rho -\sigma (1+\vartheta )}\}\hfill \\ & & +{\displaystyle \frac{1}{|\mathrm{\Lambda }|}}{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{1}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}}\{\left(|{\xi }_2|b{\zeta }^{\sigma (1+\vartheta )}+b^{\sigma (1+\vartheta )+1}\right)|Q_1|\hfill \\ & & +|{\eta }_2|b{\zeta }^{\sigma (1+\vartheta )}|Q_2|+[\left((1+|{\eta }_1|)|{\xi }_2|+|{\xi }_1||{\eta }_2|\right)b{\zeta }^{\sigma (1+\vartheta )}\hfill \\ & & +(1+|{\eta }_1|)b^{\sigma (1+\vartheta )+1}]{\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{b}^{\rho -\sigma (1+\vartheta )}\left\}\right]\mathcal{L}\parallel u_1-u_2\parallel \hfill \\ & =& \mathcal{L}({\mathcal{M}}_1+{\mathcal{M}}_2)\parallel u_1-u_2\parallel .\hfill \end{array}$$

This estimate yields

$$\parallel \mathcal{F}{u}_1-\mathcal{F}{u}_2\parallel \le \mathcal{L}({\mathcal{M}}_1+{\mathcal{M}}_2)\parallel u_1-u_2\parallel .$$

Since $\mathcal{L}({\mathcal{M}}_1+{\mathcal{M}}_2)<1$, it follows that $\mathcal{F}$ is a contraction on $\mathcal{F}{B}_{r_1}.$ Consequently, by Banach’s fixed point theorem, the operator $\mathcal{F}$ admits a unique fixed point, which corresponds to the unique solution to problem (3). This completes the argument. □

The following existence theorem relies on Krasnosel’skiĭ’s fixed point framework [40].

Theorem 2.

Let $\phi :[0,b]\times \mathbb{R}⟶\mathbb{R}$ be a continuous mapping satisfying condition $(A_1).$ In addition, suppose that

$(A_2)$

There exists a function $\varphi \in C([0,b],\mathbb{R})$ such that

$$|\phi (x,u(x))|\le \varphi (x),\forall (x,u)\in [0,b]\times \mathbb{R}.$$

Under these hypotheses, problem (3) admits at least one solution on the interval $[0,b],$ provided that

$$\begin{array}{c}\hfill \mathcal{L}({\mathcal{M}}_1^{\prime }+{\mathcal{M}}_2)<1,\end{array}$$

(16)

where

$$\begin{array}{c}\hfill {\mathcal{M}}_1^{\prime }={\mathcal{M}}_1-{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{b^{\rho }}{\mathrm{\Gamma }(\rho +1)}}.\end{array}$$

(17)

Proof. 

Define the operators ${\mathcal{F}}_1,{\mathcal{F}}_2:\mathcal{X}⟶\mathcal{X}$ by

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{1}u(x)& =& {\displaystyle \frac{1}{\mathrm{\Lambda }}[({\xi }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (\zeta ,u(\zeta )))+{\xi }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (\zeta ,u(\zeta )))}\hfill \\ & & {\displaystyle -b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (b,u(b))))Q_1-({\eta }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (\zeta ,u(\zeta )))}\hfill \\ & & {\displaystyle +{\eta }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (\zeta ,u(\zeta )))-{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (b,u(b))))Q_2]}\hfill \\ & & {\displaystyle +\frac{1}{\mathrm{\Lambda }}[-{\xi }_1({\eta }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (\zeta ,u(\zeta )))+{\eta }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (\zeta ,u(\zeta )))}\hfill \\ & & {\displaystyle -{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (b,u(b))))-(1-{\eta }_1)({\xi }_1{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (\zeta ,u(\zeta )))}\hfill \\ & & {\displaystyle +{\xi }_{2}b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (\zeta ,u(\zeta )))-b{}^R\mathcal{I}^{\sigma (1+\vartheta )}({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (b,u(b))))]}\hfill \\ & & \times {\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{x}^{\rho -\sigma (1+\vartheta )},x\in [0,b].\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{2}u(x)& =& {\displaystyle {}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi (x,u(x)))x\in [0,b].}\hfill \end{array}$$

Denote ${sup}_{x\in [0,b]}|\varphi (x)|=\parallel \varphi \parallel$ and choose a constant $r_2$ such that $r_2\ge ({\mathcal{M}}_1+{\mathcal{M}}_2)\parallel \varphi \parallel$, where the constants ${\mathcal{M}}_i,i=1,2$ are given in (15). Let $B_{r_2}=\{u\in \mathcal{X}:\parallel u\parallel \le r_2\}$ be the corresponding closed ball in $\mathcal{X}$. For any $u\in B_{r_2}$, an argument analogous to that used in the proof of Theorem 1, yields

$$\begin{array}{ccc}& & |{\mathcal{F}}_{1}u(x)+{\mathcal{F}}_{2}u(x)|\hfill \\ & \le & \left[{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{1}{\mathrm{\Gamma }(\rho +1)}}\right\{b^{\rho }\left(1+{\displaystyle \frac{|Q_2|}{|\mathrm{\Lambda }|}}\right)+{\displaystyle \frac{{\zeta }^{\rho }}{|\mathrm{\Lambda }|}}\left(|{\xi }_1||Q_1|+|{\eta }_1||Q_2|\right)\hfill \\ & & +{\displaystyle \frac{|{\xi }_1|}{|\mathrm{\Lambda }|}}\left[{\zeta }^{\rho }(1+2|{\eta }_1|)+b^{\rho }\right]{\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{b}^{\rho -\sigma (1+\vartheta )}\}\hfill \\ & & +{\displaystyle \frac{1}{|\mathrm{\Lambda }|}}{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{1}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}}\{\left(|{\xi }_2|b{\zeta }^{\sigma (1+\vartheta )}+b^{\sigma (1+\vartheta )+1}\right)|Q_1|\hfill \\ & & +|{\eta }_2|b{\zeta }^{\sigma (1+\vartheta )}|Q_2|+[\left((1+|{\eta }_1|)|{\xi }_2|+|{\xi }_1||{\eta }_2|\right)b{\zeta }^{\sigma (1+\vartheta )}\hfill \\ & & +(1+|{\eta }_1|)b^{\sigma (1+\vartheta )+1}]{\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{b}^{\rho -\sigma (1+\vartheta )}\left\}\right]\parallel \varphi \parallel \hfill \\ & =& \parallel \varphi \parallel ({\mathcal{M}}_1+{\mathcal{M}}_2)\le r_2.\hfill \end{array}$$

Consequently, $\parallel {\mathcal{F}}_{1}u+{\mathcal{F}}_{2}u\parallel \le r_2$ and therefore the sum ${\mathcal{F}}_{1}u+{\mathcal{F}}_{2}u$ belongs to the ball $B_{r_2}.$ Moreover, by invoking condition $(A_1)$, we obtain

$$\begin{array}{ccc}& & |{\mathcal{F}}_1{u}_1(x)-{\mathcal{F}}_1{u}_2(x)|\hfill \\ & \le & {\displaystyle \frac{1}{|\mathrm{\Lambda }|}}\left[{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{1}{\mathrm{\Gamma }(\rho +1)}}\right\{b^{\rho }|Q_2|+{\zeta }^{\rho }\left(|{\xi }_1||Q_1|+(1+|{\eta }_1|)|Q_2|\right)\hfill \\ & & +|{\xi }_1|\left[{\zeta }^{\rho }(1+2|{\eta }_1|)+b^{\rho }\right]{\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{b}^{\rho -\sigma (1+\vartheta )}\}\hfill \\ & & +{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{1}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}}\{\left(|{\xi }_2|b{\zeta }^{\sigma (1+\vartheta )}+b^{\sigma (1+\vartheta )+1}\right)|Q_1|\hfill \\ & & +|{\eta }_2|b{\zeta }^{\sigma (1+\vartheta )}|Q_2|+[\left((1+|{\eta }_1|)|{\xi }_2|+|{\xi }_1||{\eta }_2|\right)b{\zeta }^{\sigma (1+\vartheta )}\hfill \\ & & +(1+|{\eta }_1|)b^{\sigma (1+\vartheta )+1}]{\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{b}^{\rho -\sigma (1+\vartheta )}\left\}\right]\mathcal{L}\parallel u_1-u_2\parallel \hfill \\ & =& \mathcal{L}({\mathcal{M}}_1^{\prime }+{\mathcal{M}}_2)\parallel u_1-u_2\parallel .\hfill \end{array}$$

As a result, we obtain $\parallel {\mathcal{F}}_1{u}_1-{\mathcal{F}}_1{u}_2\parallel \le \mathcal{L}({\mathcal{M}}_1^{\prime }+{\mathcal{M}}_2)\parallel u_1-u_2\parallel ,$ which, in view of condition (16), shows that ${\mathcal{F}}_1$ is a contraction operator.

Furthermore, the continuity of the function $\phi$ ensures that the operator ${\mathcal{F}}_2$ is continuous. In addition, for every $u\in B_{r_2}$, we have

$$\begin{array}{c}\hfill \parallel {\mathcal{F}}_{2}u\parallel \le {\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)b^{\rho }}{\mathrm{\Gamma }(\vartheta +\nu +1)\mathrm{\Gamma }(\rho +1)}}\parallel \varphi \parallel ,\end{array}$$

which implies that ${\mathcal{F}}_2$ is uniformly bounded on $B_{r_2}$.

We now turn to the proof of the compactness of the operator ${\mathcal{F}}_2.$ For any $x_1,x_2\in [0,b]$ with $x_1<x_2,$ we have

$$\begin{array}{ccc}\hfill |{\mathcal{F}}_{2}v(x_2)-{\mathcal{F}}_{2}v(x_1)|& =& {\displaystyle |{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi )(x_2)-{}^R\mathcal{I}^{\rho }({}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }\phi )(x_1)|}\hfill \\ & \le & {\displaystyle \frac{1}{\mathrm{\Gamma }(\rho )}}[{\int }_0^{x_1}[{(x_2-z)}^{\rho -1}-{(x_1-z)}^{\rho -1}]{}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (z,u(z))|dz\hfill \\ & & +{\int }_{x_1}^{x_2}{(x_1-z)}^{\rho -1}{}^E\mathcal{I}_{\sigma }^{\vartheta ,\nu }|\phi (z,u(z))|dz]\hfill \\ & \le & {\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +1+\nu )}}{\displaystyle \frac{\parallel \varphi \parallel }{\mathrm{\Gamma }(\rho +1)}}[2{(x_2-x_1)}^{\rho }+|x_2^{\rho }-x_1^{\rho }|].\hfill \end{array}$$

The above expression converges to zero as $x_2-x_1\to 0$ uniformly with respect to u. As a result, the Arzelá–Ascoli theorem guarantees that the operator $F_2$ is compact on $B_{r_2}.$ Consequently, an application of Krasnosel’skiĭ’s fixed point theorem ensures the existence of at least one solution to problem (3) on the interval $[0,b]$. □

Remark 2.

By exchanging the roles of the operators ${\mathcal{F}}_1$ and ${\mathcal{F}}_2,$ one can obtain an alternative existence theorem. In this case, condition (16) takes the form

$$\mathcal{L}{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)b^{\rho }}{\mathrm{\Gamma }(\vartheta +\nu +1)\mathrm{\Gamma }(\rho +1)}}<1.$$

The last existence result relies on the nonlinear alternative of Leray–Schauder [41].

Theorem 3.

Let $\phi :[0,b]\times \mathbb{R}⟶\mathbb{R}$ be a continuous mapping and the following conditions hold:

$(A_3)$

There exist a positive, continuous, and nondecreasing function $\psi :[0,\infty )⟶(0,\infty )$ together with a continuous function $q:[0,b]⟶(0,\infty ),$ such that

$$\begin{array}{c}\hfill |\phi (x,u)|\le q(x)\psi (\parallel u\parallel ),\forall (x,u)\in [0,b]\times \mathbb{R}.\end{array}$$

$(A_4)$

There exists a constant $\mathcal{M}>0$ for which

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathcal{M}}{({\mathcal{M}}_1+{\mathcal{M}}_2)\parallel q\parallel \psi (\mathcal{M})}}>1,\end{array}$$

where the constants ${\mathcal{M}}_i,i=1,2$ are defined in (15).

Under these hypotheses, the boundary value problem (3) admits at least one solution on the interval $[0,b]$.

Proof. 

We begin by verifying that the operator $\mathcal{F}$ sets bounded subsets of $\mathcal{X}$ into bounded subsets of $\mathcal{X}$. Let $B_{r_3}=\{u\in \mathcal{X}:\parallel u\parallel \le r_3\}$ denote an arbitrary bounded set of $\mathcal{X}$. For any $x\in [0,b],$ it then follows that

$$\begin{array}{ccc}\hfill |\mathcal{F}u(x)|& \le & \left[{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{1}{\mathrm{\Gamma }(\rho +1)}}\right\{b^{\rho }\left(1+{\displaystyle \frac{|Q_2|}{|\mathrm{\Lambda }|}}\right)+{\displaystyle \frac{{\zeta }^{\rho }}{|\mathrm{\Lambda }|}}\left(|{\xi }_1||Q_1|+|{\eta }_1||Q_2|\right)\hfill \\ & & +{\displaystyle \frac{|{\xi }_1|}{|\mathrm{\Lambda }|}}\left[{\zeta }^{\rho }(1+2|{\eta }_1|)+b^{\rho }\right]{\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{b}^{\rho -\sigma (1+\vartheta )}\}\hfill \end{array}$$

$$\begin{array}{ccc}& & +{\displaystyle \frac{1}{|\mathrm{\Lambda }|}}{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{1}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}}\{\left(|{\xi }_2|b{\zeta }^{\sigma (1+\vartheta )}+b^{\sigma (1+\vartheta )+1}\right)|Q_1|\hfill \\ & & +|{\eta }_2|b{\zeta }^{\sigma (1+\vartheta )}|Q_2|+[\left((1+|{\eta }_1|)|{\xi }_2|+|{\xi }_1||{\eta }_2|\right)b{\zeta }^{\sigma (1+\vartheta )}\hfill \\ & & +(1+|{\eta }_1|)b^{\sigma (1+\vartheta )+1}]{\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{b}^{\rho -\sigma (1+\vartheta )}\left\}\right]\parallel q\parallel \psi (\parallel u\parallel )\hfill \\ & =& ({\mathcal{M}}_1+{\mathcal{M}}_2)\parallel q\parallel \psi (\parallel u\parallel )\hfill \\ & \le & ({\mathcal{M}}_1+{\mathcal{M}}_2)\parallel q\parallel \psi (r_3),\hfill \end{array}$$

and hence $\parallel \mathcal{F}u\parallel \le ({\mathcal{M}}_1+{\mathcal{M}}_2)\parallel q\parallel \psi (r_3).$

We next establish that the operator $\mathcal{F}$ maps bounded subsets of $\mathcal{X}$ into equicontinuous families. Take $x_1,x_2\in [0,b]$ with $x_1<x_2$ and let $u\in B_{r_3}$. It was shown in Theorem 2 that the operator ${\mathcal{F}}_2$ possesses the equicontinuity property. It therefore remains to verify that the operator ${\mathcal{F}}_1$ enjoys the same property. For arbitrary $x_1,x_2\in [0,b]$ satisfying $x_1<x_2,$ we obtain

$$\begin{array}{ccc}& & |{\mathcal{F}}_{1}u(x_2)-{\mathcal{F}}_{1}u(x_1)|\hfill \\ & \le & |x_2^{\rho -\sigma (1+\vartheta )}-x_1^{\rho -\sigma (1+\vartheta )}|{\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}\hfill \\ & & \times {\displaystyle \frac{1}{|\mathrm{\Lambda }|}}[|{\xi }_1|(|{\eta }_1|{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{{\zeta }^{\rho }}{\mathrm{\Gamma }(\rho +1)}}+|{\eta }_2|b{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{{\zeta }^{\sigma (1+\vartheta )}}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}}\hfill \\ & & +{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{b^{\rho }}{\mathrm{\Gamma }(\rho +1)}})+(1+|{\eta }_1|)(|{\xi }_1|{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{{\zeta }^{\rho }}{\mathrm{\Gamma }(\rho +1)}}\hfill \\ & & +|{\xi }_2|b{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{{\zeta }^{\sigma (1+\vartheta )}}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}}+b{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{b^{\sigma (1+\vartheta )}}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}}\left)\right].\hfill \end{array}$$

The above expression converges to zero as $x_2-x_1\to 0,$ uniformly with respect to u. Hence, by the Arzelá–Ascoli theorem, the operator $\mathcal{F}$ is equicontinuous, which in turn implies that the operator $\mathcal{F}$ itself enjoys the equicontinuity property.

Finally, we show that there exists an open subset $Z\subseteq \mathcal{X}$ such that $u=\lambda \mathcal{F}u$ for $\lambda \in (0,1)$ and every $u\in \partial Z$. Let u be such a solution. Proceeding in a manner analogous to the argument used in the first step, we deduce, for all $x\in [0,b]$, the following estimate holds:

$$\begin{array}{ccc}\hfill |u(x)|& \le & \left[{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{1}{\mathrm{\Gamma }(\rho +1)}}\right\{b^{\rho }\left(1+{\displaystyle \frac{|Q_2|}{|\mathrm{\Lambda }|}}\right)+{\displaystyle \frac{{\zeta }^{\rho }}{|\mathrm{\Lambda }|}}\left(|{\xi }_1||Q_1|+|{\eta }_1||Q_2|\right)\hfill \\ & & +{\displaystyle \frac{|{\xi }_1|}{|\mathrm{\Lambda }|}}\left[{\zeta }^{\rho }(1+2|{\eta }_1|)+b^{\rho }\right]{\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{b}^{\rho -\sigma (1+\vartheta )}\}\hfill \\ & & +{\displaystyle \frac{1}{|\mathrm{\Lambda }|}}{\displaystyle \frac{\mathrm{\Gamma }(\vartheta +1)}{\mathrm{\Gamma }(\vartheta +\nu +1)}}{\displaystyle \frac{1}{\mathrm{\Gamma }(1+\sigma (1+\vartheta ))}}\{\left(|{\xi }_2|b{\zeta }^{\sigma (1+\vartheta )}+b^{\sigma (1+\vartheta )+1}\right)|Q_1|\hfill \\ & & +|{\eta }_2|b{\zeta }^{\sigma (1+\vartheta )}|Q_2|+[\left((1+|{\eta }_1|)|{\xi }_2|+|{\xi }_1||{\eta }_2|\right)b{\zeta }^{\sigma (1+\vartheta )}\hfill \\ & & +(1+|{\eta }_1|)b^{\sigma (1+\vartheta )+1}]{\displaystyle \frac{\mathrm{\Gamma }(1-\sigma (1+\vartheta ))}{\mathrm{\Gamma }(\rho -\sigma (1+\vartheta )+1)}}{b}^{\rho -\sigma (1+\vartheta )}\left\}\right]\parallel q\parallel \psi (\parallel u\parallel )\hfill \\ & =& ({\mathcal{M}}_1+{\mathcal{M}}_2)\parallel q\parallel \psi (\parallel u\parallel ),\hfill \end{array}$$

which leads to

$$\begin{array}{c}\hfill {\displaystyle \frac{\parallel u\parallel }{({\mathcal{M}}_1+{\mathcal{M}}_2)\parallel q\parallel \psi (\parallel u\parallel )}}\le 1.\end{array}$$

Assumption $(A_4)$ guarantees the existence of a constant $\mathcal{M}>0$ such that $\parallel u\parallel \ne \mathcal{M}$. Define

$$\begin{array}{c}\hfill Z=\{u\in \mathcal{X}:\parallel u\parallel \le \mathcal{M}\}.\end{array}$$

It is straightforward to verify that the operator $\mathcal{F}:\overline{Z}⟶C([0,b],\mathbb{R})$ is continuous and completely continuous. Moreover, by the particular choice of the set Z, there exists no element $u\in \partial Z$ satisfying $u=\lambda \mathcal{F}u$ for any $\lambda \in (0,1)$. Therefore, an application of the Leray–Schauder nonlinear alternative yields the existence of a fixed point $u\in Z$ of $\mathcal{F},$ which corresponds to a solution of the boundary value problem (3). This completes the proof. □

4. Examples

In this section, we present several applications of the established results, illustrating that the existence of solutions can be guaranteed for a broad class of mixed Erdélyi–Kober and Caputo fractional differential equations with nonlocal, closed fractional boundary conditions. By appropriately selecting the nonlinear terms and associated functionals, solvability on a prescribed interval is ensured. To this end, we consider the following boundary value problem:

$$\left\{\begin{array}{c}{\displaystyle -{}^E\mathcal{D}_2^{-7/8,1/4}({}^C\mathcal{D}^{1/2}u)(x)=\phi (x,u(x)),x\in [0,3/2],}\hfill \\ u(3/2)=2u(1/2)+3/4{}^C\mathcal{D}^{1/4}u(1/2),\hfill \\ 3/2{}^C\mathcal{D}^{1/4}u(3/2)=1/3u(1/2)+3/10{}^C\mathcal{D}^{1/4}u(1/2).\hfill \end{array}\right.$$

(18)

Setting $\rho =1/2$, $\sigma =2$, $\vartheta =-7/8$, $\nu =1/4$, $b=3/2$, $\zeta =1/2$, ${\eta }_1=2$, ${\eta }_2=1/2$, ${\xi }_1=1/3$, ${\xi }_2=1/5$, $\rho -\sigma (1+\vartheta )=1/4$. Using these constant values, we can find that $Q_1\approx -1.696587$, $Q_2\approx 1.091548$, $\mathrm{\Lambda }\approx 1.657077$, ${\mathcal{M}}_1\approx 16.630105$, ${\mathcal{M}}_2\approx 27.150764$ and ${\mathcal{M}}_1^{\prime }\approx 12.237777$.

Example 2.

Consider the function $\phi :\left[0,3/2\right]\times \mathbb{R}\to \mathbb{R}$ presented by

$$\begin{array}{c}\hfill \phi \left(x,u\right)={\displaystyle \frac{1}{2\sqrt{1933+3(x+1)}}}\left({\displaystyle \frac{u^2+2|u|}{1+|u|}}+{\displaystyle \frac{1}{10}}\right).\end{array}$$

(19)

It can be observed that the function φ defined by (19) satisfies the Lipschitz condition:

$$\begin{array}{c}\hfill \left|\phi (x,u_2)-\phi (x,u_1)\right|\le {\displaystyle \frac{1}{44}}\left|u_2-u_1\right|,\end{array}$$

for all $x\in [0,3/2]$ and $u_1,u_2\in \mathbb{R}$, with Lipschitz constant $\mathcal{L}=1/44$. Using the given data, we obtain $\mathcal{L}({\mathcal{M}}_1+{\mathcal{M}}_2)\approx 0.995019$. Since the conditions of Theorem 1 are satisfied, we conclude that the boundary value problem (18), with the function φ defined in (19), has a unique solution on the interval $x\in [0,3/2]$.

Example 3.

Consider the function $\phi :\left[0,3/2\right]\times \mathbb{R}\to \mathbb{R}$ defined by

$$\begin{array}{c}\hfill \phi \left(x,u\right)={\displaystyle \frac{e^{-{cos}^2\pi x}}{4(11x+10)}}sin|u|+9\sqrt{x^3+7}.\end{array}$$

(20)

It is easy to find that

$$\begin{array}{c}\hfill |\phi \left(x,u\right)|\le {\displaystyle \frac{e^{-{cos}^2\pi x}}{4(11x+10)}}+9\sqrt{x^3+7}:=\varphi (x).\end{array}$$

Furthermore, we can check the Lipschitz condition of the function in (20) by

$$\begin{array}{c}\hfill \left|\phi (x,u_2)-\phi (x,u_1)\right|\le {\displaystyle \frac{1}{40}}\left|u_2-u_1\right|,\end{array}$$

for all $x\in [0,3/2]$ and $u_1,u_2\in \mathbb{R}$, with Lipschitz constant $\mathcal{L}=1/40$. Using the given data, we get $\mathcal{L}({\mathcal{M}}_1^{\prime }+{\mathcal{M}}_2)\approx 0.984713$. Clearly all the assumptions of Theorem 2 are satisfied. In consequence, boundary value problem (18), with the function φ given by (20), has at least one solution on the interval $x\in [0,3/2]$.

Example 4.

Consider the function $\phi :\left[0,3/2\right]\times \mathbb{R}\to \mathbb{R}$ given by

$$\begin{array}{c}\hfill \phi \left(x,u\right)={\displaystyle \frac{1}{7\sqrt{290x+289}}}\left[\left({\displaystyle \frac{u^{2026}}{1+u^{2024}}}\right)+{sin}^2\pi x\right].\end{array}$$

(21)

It is obvious that the above function φ satisfies the inequality

$$\begin{array}{c}\hfill |\phi \left(x,u\right)|\le {\displaystyle \frac{1}{7\sqrt{290x+289}}}(u^2+1):=q(x)\psi (|u|),\end{array}$$

by setting ${\displaystyle q(x)=\frac{1}{7\sqrt{290x+289}}}$ and $\psi \left(|u|\right)=u^2+1$. Then we obtain ${\displaystyle \parallel q\parallel =\frac{1}{119}}$ and there exists a constant $\mathcal{M}\in (0.438719,2.27936)$, which satisfy the condition $(A_4)$ in Theorem 3. Clearly, the conclusion of Theorem 3 leads to the existence of at least one solution for the boundary value problem (18) with the function φ given in (21) on the interval $x\in [0,3/2]$.

5. Conclusions

This work addresses a class of sequential fractional boundary value problems involving a coupling of Erdélyi–Kober and Caputo fractional differential operators, together with nonlocal fractional closed-boundary conditions. Existence and uniqueness properties are derived through the application of fixed point techniques. In particular, uniqueness is obtained by means of Banach’s contraction principle, whereas existence results follow from Krasnosel’skiĭ’s fixed point theorem and the Leray–Schauder nonlinear alternative. The theoretical findings are further supported by several illustrative numerical examples.

We emphasize that these generalizations open promising directions for further theoretical development such as the study of the stability, controllability, and qualitative properties of solutions and their extension to nonlinear, coupled, and variable-order fractional models. They also possess significant potential for applications in multidimensional or multi-component fractional systems arising in complex media, network dynamics, and interacting processes with memory and nonlocal effects.

Finally, it is imperative to mention that the present study will be extended to an associated multivalued problem involving coupling of Erdélyi–Kober and Caputo fractional derivative operators under nonlocal fractional closed boundary conditions in a future work.