Existence and Uniqueness Analysis for (k, ψ)-Hilfer and (k, ψ)-Caputo Sequential Fractional Differential Equations and Inclusions with Non-Separated Boundary Conditions

Abstract

This paper investigates the existence and uniqueness of solutions to a class of sequential fractional differential equations and inclusions involving the $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo derivatives under non-separated boundary conditions. By reformulating the problems into equivalent fixed-point systems, several classical fixed-point theorems, including those of Banach, Krasnosel’ski$\breve{\mathrm{i}}$’s, Schaefer, and the Leray–Schauder alternative, are employed to derive rigorous results. The study is further extended to the multi-valued setting, where existence results are established for both convex- and nonconvex-valued multi-functions using appropriate fixed-point techniques. Numerical examples are provided to illustrate the applicability and effectiveness of the theoretical findings.

1. Introduction

Recently, fractional differential equations have garnered substantial interest due to their superior capacity to capture memory and hereditary properties in complex systems, surpassing the limitations of classical integer-order models. Although the Riemann–Liouville and Caputo derivatives remain the most commonly employed, various problems in science and engineering exhibit structural complexities that are not effectively addressed by these traditional forms. To overcome such limitations, mathematicians have introduced a range of generalized fractional derivatives—including those of Hadamard, Erdélyi–Kober, Katugampola, and others—each tailored to better suit the specific nature of the underlying problems. The Hilfer fractional derivative, introduced by R. Hilfer in [1], provides a unified framework that generalizes both the Riemann–Liouville and Caputo derivatives. It is defined by an order $\alpha$ and a parameter $\beta \in [0,1]$. Notably, the Hilfer derivative reduces to the Riemann–Liouville derivative when $\beta =0$ and to the Caputo derivative when $\beta =1$, thereby serving as an interpolation between the two classical formulations. Fractional differential equations involving the Hilfer derivative have numerous applications; see [2,3] and the references therein for further details.

In the last decades, the interest in fractional differential equations and, in particular, in boundary value problems for fractional differential equations has increased. Some properties of solutions, like the existence and uniqueness of solutions for fractional boundary value problems, have been widely investigated, for example, for boundary value problems with multiple orders of fractional derivatives and integrals [4], for generalized Hilfer fractional integro-differential equations [5], for sequential fractional differential equations [6], for sequential $\psi$-Hilfer fractional differential equations [7,8], for ($k,$$\psi$)-Hilfer fractional differential equations [9] and $\psi$-Hilfer generalized proportional fractional differential equations and inclusions [10]. The Hilfer fractional derivative, which frequently appears in the study of boundary value problems, typically requires the initial condition to be zero. This restriction significantly limits its applicability in problems involving more general boundary conditions. To address this constraint, a sequential application of Hilfer and Caputo fractional derivatives can be utilized. This method facilitates the examination of boundary value problems involving nonzero initial conditions.

Souza and Oliveira [11] introduced a new fractional derivative, the $\psi$-Hilfer derivative, to unify different types of fractional derivatives into a single operator. Diaz and Pariguan defined the k-gamma function and k-beta function in [12]. Mubeen and Habibullah, in [13], introduced the k-Riemann–Liouville fractional integral operator based on the definition of the k-gamma function. In [14], Romero et al. introduced the k-Riemann–Liouville fractional derivative. In [15], the authors introduced the $(k,\psi )$-Riemann–Liouville integral operator. In [16], Kucche and Mali introduced the $(k,\psi )$-Riemann–Liouville, $(k,\psi )$-Caputo and $(k,\psi )$-Hilfer fractional derivative operators.

Researchers have shown great interest in studying these new fractional derivatives. Studying boundary value problems for fractional differential equations with non-separated boundary conditions is very important for theoretical mathematics and practical applications. By considering non-separated boundary conditions, we take more realistic behavior into account and improve accuracy in modeling real-world phenomena.

Samadi et al. [17] examined the following type of sequential fractional boundary value problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {}^H{D}_{0^{+}}^{\vartheta ,\beta ;\psi }\left({}^C{D}_{0^{+}}^{\gamma ;\psi }\varkappa \left(\delta \right)\right)=\phi \left(\delta ,\varkappa \left(\delta \right),I_{0^{+}}^{\sigma ,\psi }\varkappa \left(\delta \right),{\int }_0^b\varkappa \left(\varsigma \right)d\tau (\varsigma )\right),\delta \in [0,b],}\hfill \\ \\ \varkappa (0)+{\lambda }_1\varkappa (b)=0,{}^C{D}_{0^{+}}^{\zeta +\gamma -1;\psi }\varkappa (0)+{\lambda }_2{}^C{D}_{0^{+}}^{\zeta +\gamma -1;\psi }\varkappa (b)=0,\hfill \end{array}\right.\end{array}$$

where ${}^H{D}_{0^{+}}^{\vartheta ,\beta ;\psi }$ and ${}^C{D}_{0^{+}}^{\gamma ;\psi }$, ${}^C{D}_{0^{+}}^{\zeta +\gamma -1;\psi }$, $0<\vartheta ,\beta ,\gamma <1$, $\zeta =\vartheta +\beta (1-\vartheta )$, $\zeta +\gamma >1$ denotes the $\psi$-Hilfer and $\psi$-Caputo fractional derivative operators, respectively. Additionally, ${\lambda }_1,{\lambda }_2\in \mathbb{R}$, $I_{0^{+}}^{\sigma ,\psi }$ represents the Riemann–Liouville fractional integral operator of order $\sigma >0$ with respect to the function $\psi .$ The function $\phi :[0,b]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is a nonlinear continuous mapping, ${\int }_0^b\varkappa \left(\varsigma \right)d\tau (\varsigma )$ denotes the Riemann–Stieltjes integral with respect to a function of bounded variation $\tau :[0,b]\to \mathbb{R}$.

Ahmed et al. [18] investigated a class of separated boundary value problems of the following form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^H{D}_q^{\vartheta ,\beta }\left({}^C{D}_q^{\gamma }\varkappa (\delta )\right)=\phi (\delta ,\varkappa (\delta )),\delta \in [0,b],\hfill \\ \varkappa (0)+{\lambda }_1{}^C{D}_q^{\zeta +\gamma -1}\varkappa (0)=0,\varkappa (b)+{\lambda }_2{}^C{D}_q^{\zeta +\gamma -1}\varkappa (b)=0,\hfill \\ 0<\vartheta ,\gamma ,q<1,0\le \beta \le 1,{\lambda }_1,{\lambda }_2\in \mathbb{R},b>0,\hfill \end{array}\right.\end{array}$$

where ${}^C{D}_q^{\gamma }(·)$ and ${}^H{D}_q^{\vartheta ,\beta }(·)$, respectively, are the q-Caputo and q-Hilfer fractional quantum derivatives of orders $\gamma$, $\vartheta$, and of type $\beta$, such that $\zeta =\vartheta +\beta (1-\vartheta )$ with $\zeta +\gamma >1$, and $\phi :[0,b]\times \mathbb{R}\to \mathbb{R}$ is a continuous function.

Focusing on the aforementioned works, this paper investigates both the k-Hilfer and the k-Caputo fractional derivatives with respect to a function, $\psi$, supplemented with non-separated boundary conditions, which makes the problem under study more interesting. More precisely, we study the following problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{k,H}{D}_{0^{+}}^{\vartheta ,\beta ;\psi }\left({}^{k,C}{D}_{0^{+}}^{\gamma ;\psi }\varkappa \left(\delta \right)\right)=\phi \left(\delta ,\varkappa \left(\delta \right)\right),\delta \in [0,b],\hfill \\ {\lambda }_1\varkappa \left(0\right)+{\lambda }_2\varkappa \left(b\right)=0,\hfill \\ {\lambda }_3{}^{k,C}{D}_{0^{+}}^{{\zeta }_k+\gamma -k;\psi }\varkappa \left(0\right)+{\lambda }_4{}^{k,C}{D}_{0^{+}}^{{\zeta }_k+\gamma -k;\psi }\varkappa \left(b\right)=0,\hfill \end{array}\right.\end{array}$$

(1)

where the differential operator ${}^{k,H}{D}_{0^{+}}^{\vartheta ,\beta ;\psi }$ is the $(k,\psi )$-Hilfer fractional differential operator of order $0<\vartheta <1$ with the parameters $0\le \beta \le 1$. ${}^{k,C}{D}_{0^{+}}^{\gamma ;\psi }$ and ${}^{k,C}{D}_{0^{+}}^{{\zeta }_k+\gamma -k;\psi }$ are the $(k,\psi )$-Caputo fractional differential operators of orders $0<\gamma <1$ and ${\zeta }_k+\gamma -k>0$, respectively, where ${\zeta }_k=\vartheta +\beta (k-\vartheta )$. Moreover, $k>0$, ${\lambda }_{\iota }\in \mathbb{R}$$(\iota =1,2,3,4)$ and $\phi :[0,b]\times \mathbb{R}\to \mathbb{R}$ is a continuous function. This paper seeks to demonstrate the existence and uniqueness of solutions through the application of fixed point theorems by Banach, Schaefer, and Krasnosel’ski$\breve{\mathrm{i}}$, along with the Leray–Schauder nonlinear alternative. To showcase the effectiveness of the theoretical findings, we provide several numerical examples.

Subsequently, we extend our analysis to the multi-valued case of the nonlinear sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo fractional boundary value problem (1), given by the following:

$$\left\{\begin{array}{cc}\hfill & {}^{k,H}{D}_{0^{+}}^{\vartheta ,\beta ;\psi }\left({}^{k,C}{D}_{0^{+}}^{\gamma ;\psi }\varkappa \left(\delta \right)\right)\in F\left(\delta ,\varkappa \left(\delta \right)\right),\delta \in [0,b],\hfill \\ \hfill & {\displaystyle {\lambda }_1\varkappa \left(0\right)+{\lambda }_2\varkappa \left(b\right)=0,}\hfill \\ \hfill & {\lambda }_3{}^{k,C}{D}_{0^{+}}^{{\zeta }_k+\gamma -k;\psi }\varkappa \left(0\right)+{\lambda }_4{}^{k,C}{D}_{0^{+}}^{{\zeta }_k+\gamma -k;\psi }\varkappa \left(b\right)=0,\hfill \end{array}\right.$$

(2)

where $F:[0,b]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ is a multi-valued map ($\mathcal{P}(\mathbb{R})$ denotes the family of all nonempty subsets of $\mathbb{R}$) and the other symbols are the same as defined in the problem (1). We address two distinct cases—when the right-hand side of the inclusion is convex-valued and when it is nonconvex-valued. Existence results are established using the Leray–Schauder nonlinear alternative for the former, and the Covitz–Nadler fixed point theorem for multi-valued contractions in the latter.

The structure of this paper is as follows. Section 2 provides the necessary definitions and preliminary lemmas that form the foundation of the study. In Section 3, we establish the main results by applying the Banach contraction principle to demonstrate the existence and uniqueness of solutions. Further existence results are obtained using the Leray–Schauder nonlinear alternative, Schaefer’s fixed point theorem, and Krasnosel’ski$\breve{\mathrm{i}}$’s fixed point theorem. Section 4 addresses the multi-valued case of the sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo fractional boundary value problems, presenting corresponding existence results. Finally, several numerical examples are provided to illustrate the applicability of the theoretical results.

2. Preliminaries

In this section, we provide several fundamental definitions, lemmas, and a remark that will be employed in the subsequent analysis. For a comprehensive overview of the basic concepts and definitions in fractional calculus, the reader is referred to [19,20,21,22].

In the following, we suppose that $\psi \in C^1([a,b],\mathbb{R})$ and $\psi$ is a positively continuous and increasing function satisfying the condition ${\psi }^{\prime }(\delta )\ne 0$ for each $\delta \in [a,b]$.

Definition 1

(Definition 4, p. 3, [15]). Let $\vartheta ,k\in {\mathbb{R}}^{+}$ and $\phi \in L^1([a,b],\mathbb{R})$. Then the $(k,\psi )$-Riemann–Liouville fractional integral of order ϑ for a function φ is defined by

$$\begin{array}{c}\hfill {{}^{k}I}_{a^{+}}^{\vartheta ;\psi }\phi (\delta )={\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta )}}{\int }_a^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta }{k}}-1}\phi \left(\varsigma \right)d\varsigma .\end{array}$$

Definition 2

(Definition 3.2, p. 3, [16]). Let $\vartheta ,k\in {\mathbb{R}}^{+}$ and $\phi \in C([a,b],\mathbb{R})$. Then the $(k,\psi )$-Caputo fractional derivative of order ϑ for a function φ is defined by

$$\begin{array}{cc}\hfill {}^{k,C}{D}_{a^{+}}^{\vartheta ;\psi }\phi (\delta )& ={}^k{I}_{a^{+}}^{nk-\vartheta ;\psi }{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\right)}^n\phi (\delta )\hfill \\ & ={\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(nk-\vartheta )}}{\int }_a^{\delta }{\psi }^{\prime }(\varsigma ){(\psi (\delta )-\psi (\varsigma ))}^{n-{\displaystyle \frac{\vartheta }{k}}-1}{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^n\phi (\varsigma )d\varsigma .\hfill \end{array}$$

where $n=⌈{\displaystyle \frac{\vartheta }{k}}⌉$ is the ceiling function of ${\displaystyle \frac{\vartheta }{k}}$.

Definition 3

(Definition 3.2, p. 3, [16]). Let $\vartheta ,k\in {\mathbb{R}}^{+}$, $0\le \beta \le 1$ and $\phi \in C([a,b],\mathbb{R})$. Then the $(k,\psi )$-Hilfer fractional derivative of order ϑ and type β for a function φ is defined by the following:

$$\begin{array}{c}\hfill {}^{k,H}{D}_{a^{+}}^{\vartheta ,\beta ;\psi }\phi (\delta )={}^k{I}_{a^{+}}^{\beta (nk-\vartheta );\psi }{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\right)}^n{}^k{I}_{a^{+}}^{(1-\beta )(nk-\vartheta );\psi }\phi (\delta ),n=⌈{\displaystyle \frac{\vartheta }{k}}⌉.\end{array}$$

Lemma 1

(Lemma 4.10, p. 5, [16]). Let $m\in \mathbb{N},k>0,(k\in \mathbb{R})$ and $h\in C([a,b],\mathbb{R}).$ Then,

$$\begin{array}{cc}\hfill (a)& {\left({\displaystyle \frac{1}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\right)}^m{I}_{a+}^{m,\psi }h(\delta )=h(\delta ),\\ \hfill (b)& {\left({\displaystyle \frac{k}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\right)}^m{}^k{I}_{a+}^{mk,\psi }h(\delta )=h(\delta ).\end{array}$$

Lemma 2.

Let $\vartheta ,k\in {\mathbb{R}}^{+}$ and $n=⌈{\displaystyle \frac{\vartheta }{k}}⌉$. Suppose that $\phi \in C^n([a,b],\mathbb{R})$. Then, we have the following:

$$\begin{array}{c}\hfill {}^k{I}_{a^{+}}^{\vartheta ;\psi }\left({}^{k,C}{D}_{a^{+}}^{\vartheta ;\psi }\phi (\delta )\right)=\phi (\delta )-\sum _{\dot{𝚥}=0}^{n-1}{\displaystyle \frac{{\left(\psi (\delta )-\psi (a)\right)}^{\dot{𝚥}}}{{\mathsf{\Gamma }}_k(\dot{𝚥}k+k)}}{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\right)}^{\dot{𝚥}}\phi (\delta )\right]}_{\delta =a}.\end{array}$$

Proof. 

Using Lemma 1 (b), for $n=1$, we obtain the following:

$$\begin{array}{cc}\hfill & {}^k{I}_{a^{+}}^{\vartheta ;\psi }\left({}^{k,C}{D}_{a^{+}}^{\vartheta ;\psi }\phi (\delta )\right)\hfill \\ \hfill & =\left({\displaystyle \frac{k}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\right){}^k{I}_{a^{+}}^{k;\psi }\left[{}^k{I}_{a^{+}}^{\vartheta ;\psi }\left({}^k{I}_{a^{+}}^{nk-\vartheta ;\psi }{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^n\phi (\delta )\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{k}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\left[{}^k{I}_{a^{+}}^{nk+k;\psi }{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^n\phi (\delta )\right]\hfill \\ \hfill & ={\displaystyle \frac{k}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\left[{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(nk+k)}}{\int }_a^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^n{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^n\phi (\varsigma )d\varsigma \right]\hfill \\ \hfill & ={\displaystyle \frac{1}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\left[{\displaystyle \frac{1}{k^n\mathsf{\Gamma }(n+1)}}{\int }_a^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^n{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^n\phi (\varsigma )d\varsigma \right].\hfill \end{array}$$

(3)

Now, using the definition of the $(k,\psi )$-Caputo fractional derivative and integration by parts, we obtain the following:

$$\begin{array}{ccc}\hfill & & \left[{\displaystyle \frac{1}{k^n\mathsf{\Gamma }(n+1)}}{\int }_a^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^n{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^n\phi (\varsigma )d\varsigma \right]\hfill \\ & & ={\displaystyle \frac{k}{k^n\mathsf{\Gamma }(n+1)}}{\int }_a^{\delta }{\left(\psi (\delta )-\psi (\varsigma )\right)}^n{\displaystyle \frac{d}{d\varsigma }}\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{n-1}\phi (\varsigma )\right]d\varsigma \hfill \\ & & =-{\displaystyle \frac{k^{1-n}}{\mathsf{\Gamma }(n+1)}}{\left(\psi (\delta )-\psi (a)\right)}^n{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{n-1}\phi (\varsigma )\right]}_{\varsigma =a}\hfill \\ & & +{\displaystyle \frac{k^{2-n}}{\mathsf{\Gamma }(n)}}{\int }_a^{\delta }{\left(\psi (\delta )-\psi (\varsigma )\right)}^{n-1}{\displaystyle \frac{d}{d\varsigma }}\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{n-2}\phi (\varsigma )\right]d\varsigma \hfill \\ & & =-{\displaystyle \frac{k^{1-n}}{\mathsf{\Gamma }(n+1)}}{\left(\psi (\delta )-\psi (a)\right)}^n{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{n-1}\phi (\varsigma )\right]}_{\varsigma =a}\hfill \\ & & -{\displaystyle \frac{k^{2-n}}{\mathsf{\Gamma }(n)}}{\left(\psi (\delta )-\psi (a)\right)}^{n-1}{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{n-2}\phi (\varsigma )\right]}_{\varsigma =a}\hfill \\ & & +{\displaystyle \frac{k^{3-n}}{\mathsf{\Gamma }(n-1)}}{\int }_a^{\delta }{\left(\psi (\delta )-\psi (\varsigma )\right)}^{n-2}{\displaystyle \frac{d}{d\varsigma }}\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{n-3}\phi (\varsigma )\right]d\varsigma \hfill \\ & & =-{\displaystyle \frac{k^{1-n}}{\mathsf{\Gamma }(n+1)}}{\left(\psi (\delta )-\psi (a)\right)}^n{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{n-1}\phi (\varsigma )\right]}_{\varsigma =a}\hfill \\ & & -{\displaystyle \frac{k^{2-n}}{\mathsf{\Gamma }(n)}}{\left(\psi (\delta )-\psi (a)\right)}^{n-1}{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{n-2}\phi (\varsigma )\right]}_{\varsigma =a}\hfill \\ & & -{\displaystyle \frac{k^{3-n}}{\mathsf{\Gamma }(n-1)}}{\left(\psi (\delta )-\psi (a)\right)}^{n-2}{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{n-3}\phi (\varsigma )\right]}_{\varsigma =a}\hfill \\ & & +{\displaystyle \frac{k^{4-n}}{\mathsf{\Gamma }(n-2)}}{\int }_a^{\delta }{\left(\psi (\delta )-\psi (\varsigma )\right)}^{n-3}{\displaystyle \frac{d}{d\varsigma }}\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{n-4}\phi (\varsigma )\right]d\varsigma \hfill \\ & & =-\sum _{\dot{𝚥}=n-3}^{n-1}{\displaystyle \frac{k^{-\dot{𝚥}}}{\mathsf{\Gamma }(\dot{𝚥}+2)}}{\left(\psi (\delta )-\psi (a)\right)}^{\dot{𝚥}+1}{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{\dot{𝚥}}\phi (\varsigma )\right]}_{\varsigma =a}\hfill \\ & & +{\displaystyle \frac{k^{4-n}}{\mathsf{\Gamma }(n-2)}}{\int }_a^{\delta }{\left(\psi (\delta )-\psi (\varsigma )\right)}^{n-3}{\displaystyle \frac{d}{d\varsigma }}\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{n-4}\phi (\varsigma )\right]d\varsigma \hfill \\ & & =-\sum _{\dot{𝚥}=0}^{n-1}{\displaystyle \frac{k^{-\dot{𝚥}}}{\mathsf{\Gamma }(\dot{𝚥}+2)}}{\left(\psi (\delta )-\psi (a)\right)}^{\dot{𝚥}+1}{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{\dot{𝚥}}\phi (\varsigma )\right]}_{\varsigma =a}\hfill \\ & & +{\displaystyle \frac{k}{\mathsf{\Gamma }(1)}}{\int }_a^{\delta }{\left(\psi (\delta )-\psi (\varsigma )\right)}^0{\displaystyle \frac{d}{d\varsigma }}\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{-1}\phi (\varsigma )\right]d\varsigma \hfill \\ & & ={\int }_a^{\delta }{\psi }^{\prime }(\varsigma )\phi (\varsigma )d\varsigma -\sum _{\dot{𝚥}=0}^{n-1}{\displaystyle \frac{k^{-\dot{𝚥}}}{\mathsf{\Gamma }(\dot{𝚥}+2)}}{\left(\psi (\delta )-\psi (a)\right)}^{\dot{𝚥}+1}{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{\dot{𝚥}}\phi (\varsigma )\right]}_{\varsigma =a}.\hfill \end{array}$$

(4)

Using (3) in (4), and the definition of the k-gamma function, we obtain the following:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\left\{{\int }_a^{\delta }{\psi }^{\prime }(\varsigma )\phi (\varsigma )d\varsigma -\sum _{\dot{𝚥}=0}^{n-1}{\displaystyle \frac{k^{-\dot{𝚥}}}{\mathsf{\Gamma }(\dot{𝚥}+2)}}{\left(\psi (\delta )-\psi (a)\right)}^{\dot{𝚥}+1}{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{\dot{𝚥}}\phi (\varsigma )\right]}_{\varsigma =a}\right\}\hfill \\ \hfill & =\phi (\delta )-\sum _{\dot{𝚥}=0}^{n-1}{\displaystyle \frac{k^{-\dot{𝚥}}}{\mathsf{\Gamma }(\dot{𝚥}+1)}}{\left(\psi (\delta )-\psi (a)\right)}^{\dot{𝚥}}{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{\dot{𝚥}}\phi (\varsigma )\right]}_{\varsigma =a}\hfill \\ \hfill & =\phi (\delta )-\sum _{\dot{𝚥}=0}^{n-1}{\displaystyle \frac{{\left(\psi (\delta )-\psi (a)\right)}^{\dot{𝚥}}}{{\mathsf{\Gamma }}_k(\dot{𝚥}k+k)}}{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^{\dot{𝚥}}\phi (\varsigma )\right]}_{\varsigma =a}.\hfill \end{array}$$

□

Lemma 3

(Theorems 6.3, 5,5, p. 6, [16]). Let $\vartheta ,k\in {\mathbb{R}}^{+}$, $\beta \in [0,1]$, ${\zeta }_k=\vartheta +\beta (nk-\vartheta )$ and $n=⌈{\displaystyle \frac{\vartheta }{k}}⌉$. Suppose that $\phi \in C^n([a,b],\mathbb{R})$ and ${{}^{k}I}_{a^{+}}^{nk-{\zeta }_k;\psi }\phi \in C^n([a,b],\mathbb{R})$. Then, we have the following:

$$\begin{array}{c}\hfill {{}^{k}I}_{a^{+}}^{\vartheta ;\psi }\left({}^{k,H}{D}_{a^{+}}^{\vartheta ,\beta ;\psi }\phi (\delta )\right)=\phi (\delta )-\sum _{\dot{𝚥}=1}^n{\displaystyle \frac{{\left(\psi (\delta )-\psi (a)\right)}^{\frac{{\zeta }_k}{k}-\dot{𝚥}}}{{\mathsf{\Gamma }}_k({\zeta }_k-\dot{𝚥}k+k)}}{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\right)}^{n-\dot{𝚥}}{{}^{k}I}_{a^{+}}^{nk-{\zeta }_k;\psi }\phi (\delta )\right]}_{\delta =a}.\end{array}$$

Remark 1

(Definition 3.2, p. 3, [16]). The $(k,\psi )$-Hilfer fractional derivative can be expressed in terms of the $(k,\psi )$-Riemann–Liouville fractional integral as follows:

$$\begin{array}{cc}\hfill {}^{k,H}{D}_{a^{+}}^{\vartheta ,\beta ;\psi }\phi \left(\delta \right)& ={{}^{k}I}_{a^{+}}^{{\zeta }_k-\vartheta ;\psi }{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\right)}^n{{}^{k}I}_{a^{+}}^{(nk-{\zeta }_k);\psi }\phi (\delta ),\hfill \\ \hfill & ={{}^{k}I}_{a^{+}}^{{\zeta }_k-\vartheta ;\psi }\left({}^{k,RL}{D}_{a^{+}}^{{\zeta }_k;\psi }\right)\phi \left(\delta \right).\hfill \end{array}$$

If we put ${\zeta }_k=\vartheta +\beta (nk-\vartheta )$, then we obtain $(1-\beta )(nk-\vartheta )=nk-{\zeta }_k$. Moreover, we have $n-1<{\displaystyle \frac{{\zeta }_k}{k}}\le n$, for $n-1<{\displaystyle \frac{\vartheta }{k}}\le n$ and $0\le \beta \le 1$.

Lemma 4

(Theorem 4.1, p. 3, [16]). Let ${\vartheta }_1,{\vartheta }_2,k\in {\mathbb{R}}^{+}$. Then,

$$\begin{array}{c}\hfill {{}^{k}I}_{a^{+}}^{{\vartheta }_1;\psi }{{}^{k}I}_{a^{+}}^{{\vartheta }_2;\psi }\phi (\delta )={{}^{k}I}_{a^{+}}^{{\vartheta }_1+{\vartheta }_2;\psi }\phi (\delta ).\end{array}$$

Lemma 5.

Let ${\vartheta }_1,{\vartheta }_2,k\in {\mathbb{R}}^{+}$ with ${\vartheta }_2>{\vartheta }_1$. Then, we have the following:

$$\begin{array}{c}\hfill {}^{k,C}{D}_{a^{+}}^{{\vartheta }_1;\psi }{{}^{k}I}_{a^{+}}^{{\vartheta }_2;\psi }\phi (\delta )={{}^{k}I}_{a^{+}}^{{\vartheta }_2-{\vartheta }_1;\psi }\phi (\delta ).\end{array}$$

Proof. 

Using Definition 2 and utilizing the definition of the k-Gamma function, we have the following:

$$\begin{array}{cc}\hfill & {}^{k,C}{D}_{a^{+}}^{{\vartheta }_1;\psi }{{}^{k}I}_{a^{+}}^{{\vartheta }_2;\psi }\phi (\delta )\hfill \\ \hfill & ={}^k{I}_{a^{+}}^{nk-{\vartheta }_1;\psi }{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\right)}^n{{}^{k}I}_{a^{+}}^{{\vartheta }_2;\psi }\phi (\delta )\hfill \\ \hfill & ={\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(nk-{\vartheta }_1)}}{\int }_a^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^{n-{\displaystyle \frac{{\vartheta }_1}{k}}-1}{\left({\displaystyle \frac{k}{{\psi }^{\prime }(\varsigma )}}{\displaystyle \frac{d}{d\varsigma }}\right)}^n{{}^{k}I}_{a^{+}}^{{\vartheta }_2;\psi }\phi (\varsigma )d\varsigma ,\hfill \end{array}$$

(5)

where

$$\begin{array}{cc}\hfill & {\left({\displaystyle \frac{k}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\right)}^n{{}^{k}I}_{a^{+}}^{{\vartheta }_2;\psi }\phi (\delta )\hfill \\ & ={\left({\displaystyle \frac{k}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\right)}^n{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k({\vartheta }_2)}}{\int }_a^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^{{\displaystyle \frac{{\vartheta }_2}{k}}-1}\phi \left(\varsigma \right)d\varsigma \hfill \\ & ={\left({\displaystyle \frac{k}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\right)}^{n-1}{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k({\vartheta }_2-k)}}{\int }_a^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^{{\displaystyle \frac{{\vartheta }_2}{k}}-2}\phi \left(\varsigma \right)d\varsigma \hfill \\ & ={\left({\displaystyle \frac{k}{{\psi }^{\prime }(\delta )}}{\displaystyle \frac{d}{d\delta }}\right)}^{n-2}{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k({\vartheta }_2-2k)}}{\int }_a^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^{{\displaystyle \frac{{\vartheta }_2}{k}}-3}\phi \left(\varsigma \right)d\varsigma \hfill \\ \hfill & .\hfill \\ \hfill & .\hfill \\ \hfill & .\hfill \\ \hfill & ={\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k({\vartheta }_2-nk)}}{\int }_a^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^{{\displaystyle \frac{{\vartheta }_2}{k}}-n-1}\phi \left(\varsigma \right)d\varsigma \hfill \\ \hfill & ={{}^{k}I}_{a^{+}}^{{\vartheta }_2-nk;\psi }\phi (\delta ).\hfill \end{array}$$

Substituting into Equation (5) and applying Lemma 4, we have the following:

$$\begin{array}{c}\hfill {{}^{k}I}_{a^{+}}^{{\vartheta }_2-nk;\psi }{}^k{I}_{a^{+}}^{nk-{\vartheta }_1;\psi }\phi (\delta )={{}^{k}I}_{a^{+}}^{{\vartheta }_2-{\vartheta }_1;\psi }\phi (\delta ).\end{array}$$

□

Lemma 6

(Theorem 4.3, p. 3, [16]). Let $\vartheta ,k\in {\mathbb{R}}^{+}$ and $\mu \in \mathbb{R}$ such that ${\displaystyle \frac{\mu }{k}}>-1$. Then, we have the following:

(i) 

${\displaystyle {}^k{I}_{a^{+}}^{\vartheta ;\psi }{\left(\psi (\delta )-\psi (a)\right)}^{\frac{\mu }{k}}=\frac{{\mathsf{\Gamma }}_k(\mu +k)}{{\mathsf{\Gamma }}_k(\mu +k+\vartheta )}{\left(\psi (\delta )-\psi (a)\right)}^{\frac{\mu +\vartheta }{k}}}$

(ii) 

${\displaystyle {}^{k,C}{D}_{a^{+}}^{\vartheta ;\psi }{\left(\psi (\delta )-\psi (a)\right)}^{\frac{\mu }{k}}=\frac{{\mathsf{\Gamma }}_k(\mu +k)}{{\mathsf{\Gamma }}_k(\mu +k-\vartheta )}{\left(\psi (\delta )-\psi (a)\right)}^{\frac{\mu -\vartheta }{k}}}$.

The following lemma addresses a linear variant of the sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo fractional differential equations subject to non-separated boundary conditions as specified in (1). This result is instrumental in reformulating the nonlinear problem (1) into an equivalent fixed-point problem.

Lemma 7.

Assume that $h\in C([0,b],\mathbb{R})$, $0<\vartheta ,\gamma <1$, $0\le \beta \le 1$, $k>0$, ${\zeta }_k+\gamma -k>0$, ${\zeta }_k=\vartheta +\beta (k-\vartheta )$, ${\lambda }_1+{\lambda }_2\ne 0$ and ${\lambda }_3+{\lambda }_4\ne 0$. Then the integral expression of the solution for the following linear boundary value problem

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{k,H}{D}_{0^{+}}^{\vartheta ,\beta ;\psi }\left({}^{k,C}{D}_{0^{+}}^{\gamma ;\psi }\varkappa \left(\delta \right)\right)=h\left(\delta \right),\delta \in [0,b],\hfill \\ {\lambda }_1\varkappa \left(0\right)+{\lambda }_2\varkappa \left(b\right)=0,\hfill \\ {\lambda }_3{}^{k,C}{D}_{0^{+}}^{{\zeta }_k+\gamma -k;\psi }\varkappa \left(0\right)+{\lambda }_4^{k,C}{D}_{0^{+}}^{{\zeta }_k+\gamma -k;\psi }\varkappa \left(b\right)=0,\hfill \end{array}\right.\end{array}$$

(6)

is given by

$$\begin{array}{c}\hfill \varkappa (\delta )=\left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }h\left(b\right)-{\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }h\left(b\right)+{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }h\left(\delta \right),\end{array}$$

(7)

where

$$\mathsf{\Lambda }(\delta )={\displaystyle \frac{{\lambda }_4}{{\lambda }_3+{\lambda }_4}}{\displaystyle \frac{{(\psi (\delta )-\psi (0))}^{\frac{{\zeta }_k+\gamma }{k}-1}}{{\mathsf{\Gamma }}_k({\zeta }_k+\gamma )}}.$$

(8)

Proof. 

By applying the fractional integral operator ${}^k{I}_{0^{+}}^{\vartheta ;\psi }$ to both sides of the first equation in (6) and utilizing Lemma 3, we obtain the following result:

$$\begin{array}{c}\hfill {}^{k,C}{D}_{0^{+}}^{\gamma ;\psi }\varkappa \left(\delta \right)={}^k{I}_{0^{+}}^{\vartheta ;\psi }h\left(\delta \right)+{\displaystyle \frac{{(\psi (\delta )-\psi (0))}^{\frac{{\zeta }_k}{k}-1}}{{\mathsf{\Gamma }}_k({\zeta }_k)}}{c}_1,\end{array}$$

(9)

where ${\zeta }_k=\vartheta +\beta (k-\vartheta )$ and $c_1={\left[{{}^{k}I}_{0^{+}}^{k-{\zeta }_k;\psi }h(\delta )\right]}_{\delta =0}$. Now, by taking the fractional integral ${}^k{I}_{0^{+}}^{\gamma ;\psi }$ on both sides of Equation (9) and applying Lemma 2, we obtain the following:

$$\begin{array}{c}\hfill \varkappa \left(\delta \right)={}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }h\left(\delta \right)+{\displaystyle \frac{{(\psi (\delta )-\psi (0))}^{\frac{{\zeta }_k+\gamma }{k}-1}}{{\mathsf{\Gamma }}_k({\zeta }_k+\gamma )}}{c}_1+d_1.\end{array}$$

(10)

By Lemma 5, we have the following:

$$\begin{array}{c}\hfill {}^{k,C}{D}_{0^{+}}^{{\zeta }_k+\gamma -k;\psi }\varkappa \left(\delta \right)={}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }h\left(\delta \right)+c_1.\end{array}$$

(11)

From (11) and by applying the boundary conditions in the second equation of (6), we obtain the following:

$$\begin{array}{cc}\hfill {\lambda }_1{d}_1+{\lambda }_2{d}_1+{\lambda }_2{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }h\left(b\right)+{\lambda }_2{\displaystyle \frac{{(\psi (b)-\psi (0))}^{\frac{{\zeta }_k+\gamma }{k}-1}}{{\mathsf{\Gamma }}_k({\zeta }_k+\gamma )}}{c}_1& =0,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\lambda }_3{c}_1+{\lambda }_4{c}_1+{\lambda }_4{}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }h\left(b\right)& =0.\hfill \end{array}$$

(13)

From (12) and (13), we have the following:

$$\begin{array}{c}\hfill c_1=-{\displaystyle \frac{{\lambda }_4}{{\lambda }_3+{\lambda }_4}}{}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }h\left(b\right)\end{array}$$

and

$$\begin{array}{c}\hfill d_1={\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\left[{\displaystyle \frac{{\lambda }_4}{{\lambda }_3+{\lambda }_4}}{\displaystyle \frac{{(\psi (b)-\psi (0))}^{\frac{{\zeta }_k+\gamma }{k}-1}}{{\mathsf{\Gamma }}_k({\zeta }_k+\gamma )}}{}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }h\left(b\right)-{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }h\left(b\right)\right].\end{array}$$

Replacing the above constants $c_1$ and $d_1$ in (10), we obtain the following:

$$\begin{array}{cc}\hfill \varkappa \left(\delta \right)& ={\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\left[{\displaystyle \frac{{\lambda }_4}{{\lambda }_3+{\lambda }_4}}{\displaystyle \frac{{(\psi (b)-\psi (0))}^{\frac{{\zeta }_k+\gamma }{k}-1}}{{\mathsf{\Gamma }}_k({\zeta }_k+\gamma )}}{}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }h\left(b\right)-{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }h\left(b\right)\right]\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_4}{{\lambda }_3+{\lambda }_4}}{\displaystyle \frac{{(\psi (\delta )-\psi (0))}^{\frac{{\zeta }_k+\gamma }{k}-1}}{{\mathsf{\Gamma }}_k({\zeta }_k+\gamma )}}{}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }h\left(b\right)+{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }h\left(\delta \right)\hfill \\ \hfill & =\left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }h\left(b\right)-{\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }h\left(b\right)+{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }h\left(\delta \right).\hfill \end{array}$$

The converse of the lemma can be readily verified through direct computation, thereby completing the proof. □

3. The Single-Valued Case

Let $X=C([0,b],\mathbb{R})$ be the Banach space of all continuous functions from $[0,b]$ to $\mathbb{R}$ endowed with the norm $∥\varkappa ∥=\sup \{\varkappa (\delta ):\delta \in [0,b]\}$.

In view of Lemma 7, we define an operator $T:X\to X$ by the following:

$$\begin{array}{cc}\hfill T(\varkappa )(\delta )& =\left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }\phi \left(b,\varkappa (b)\right)\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\phi \left(b,\varkappa (b)\right)+{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\phi \left(\delta ,\varkappa (\delta )\right),\delta \in [0,b].\hfill \end{array}$$

(14)

For computational convenience, let us set the following:

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_1& ={\left(\psi (b)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}\left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right)[{\displaystyle \frac{\left|{\lambda }_4\right|}{\left|{\lambda }_3+{\lambda }_4\right|{\mathsf{\Gamma }}_k({\zeta }_k+\gamma ){\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+2k)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}],\hfill \end{array}$$

(15)

and

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_2& ={\mathsf{\Omega }}_1{-(\psi (b)-\psi (0)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}{\displaystyle \frac{1}{{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}.\hfill \end{array}$$

(16)

3.1. Existence and Uniqueness Result via Banach Contraction Mapping Principle

Applying Banach’s contraction principle (Theorem 17.1, p. 187, [23]), we establish existence and uniqueness results for the $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo sequential fractional boundary value problems with non-separated boundary conditions (1).

Theorem 1.

Let $\phi :[0,b]\times \mathbb{R}\to \mathbb{R}$ be a continuous function. Assume that the following condition is satisfied:

(${\mathcal{H}}_1$) 

There exists a positive real constant $\mathcal{L}$ such that

$$\begin{array}{c}\hfill \left|\phi (\delta ,\varkappa )-\phi (\delta ,y)\right|\le \mathcal{L}\left|\varkappa -y\right|,\end{array}$$

for all $\delta \in [0,b]$ and $\varkappa ,y\in \mathbb{R}$.

If

$$\begin{array}{c}\hfill \mathcal{L}{\mathsf{\Omega }}_1<1,\end{array}$$

(17)

where ${\mathsf{\Omega }}_1$ is given by (15), then the $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo sequential fractional boundary value problem (1) admits a unique solution on the interval $[0,b]$.

Proof. 

We define ${\displaystyle \mathcal{M}=\underset{0\le \delta \le b}{sup}\left\{\left|\phi (\delta ,0)\right|:\delta \in [0,b]\right\}<\infty }$ and consider the set

$$B_r=\left\{\varkappa \in X:∥\varkappa ∥\le r\right\}$$

with ${\displaystyle r\ge \frac{\mathcal{M}{\mathsf{\Omega }}_1}{1-\mathcal{L}{\mathsf{\Omega }}_1}}$.

By assumption $({\mathcal{H}}_1)$, it follows that

$$\begin{array}{c}\hfill \left|\phi (\delta ,\varkappa (\delta ))\right|\le \left|\phi (\delta ,\varkappa (\delta ))-\phi (\delta ,0)\right|+\left|\phi (\delta ,0)\right|\le \mathcal{L}\left|\varkappa (\delta )\right|+\mathcal{M}\le \mathcal{L}∥\varkappa ∥+\mathcal{M}\le \mathcal{L}r+\mathcal{M}.\end{array}$$

Let us first show that $T(B_r)\subset B_r$. For each $\varkappa \in B_r$ and $\delta \in [0,b],$ we have the following:

$$\begin{array}{cc}\hfill & \left|T(\varkappa )(\delta )\right|\hfill \\ \hfill & \le \left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right)\left|\mathsf{\Lambda }(b)\right|{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}\left|\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & \le \left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right)\left|\mathsf{\Lambda }(b)\right|{\displaystyle \frac{(\mathcal{L}r+\mathcal{M})}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}{\displaystyle \frac{(\mathcal{L}r+\mathcal{M})}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\mathcal{L}r+\mathcal{M}}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}d\varsigma \hfill \\ \hfill & \le \left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right)(\mathcal{L}r+\mathcal{M}){\displaystyle \frac{\left|{\lambda }_4\right|}{\left|{\lambda }_3+{\lambda }_4\right|}}{\displaystyle \frac{(\psi (b)-\psi {(0)}^{\frac{{\zeta }_k+\gamma }{k}-1}}{{\mathsf{\Gamma }}_k({\zeta }_k+\gamma )}}{\displaystyle \frac{{\left(\psi (b)-\psi (0)\right)}^{\frac{\vartheta -{\zeta }_k}{k}+1}}{{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+2k)}}\hfill \\ \hfill & +{\displaystyle \frac{\left|{\lambda }_2\right|(\mathcal{L}r+\mathcal{M})}{\left|{\lambda }_1+{\lambda }_2\right|{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}{\left(\psi (b)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}+{\displaystyle \frac{\mathcal{L}r+\mathcal{M}}{{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}{\left(\psi (b)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}\hfill \\ \hfill & =(\mathcal{L}r+\mathcal{M}){\left(\psi (b)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}\left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right)[{\displaystyle \frac{\left|{\lambda }_4\right|}{\left|{\lambda }_3+{\lambda }_4\right|{\mathsf{\Gamma }}_k({\zeta }_k+\gamma ){\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+2k)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}]\hfill \\ \hfill & =(\mathcal{L}r+\mathcal{M}){\mathsf{\Omega }}_1\le r.\hfill \end{array}$$

Therefore, we have $|T(\varkappa )|\le r$, which implies that $T(B_r)\subset B_r$.

We now proceed to show that T is a contraction. For $\delta \in [0,b]$ and $\varkappa ,y\in X$, we have the following:

$$\begin{array}{cc}\hfill & \left|T(\varkappa )(\delta )-T(y)(\delta )\right|\hfill \\ \hfill & \le \left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right){\displaystyle \frac{|\mathsf{\Lambda }(b)|}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}\left|\phi (\varsigma ,\varkappa (\varsigma ))-\phi (\varsigma ,y(\varsigma ))\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\left|{\lambda }_2\right|}{k\left|{\lambda }_1+{\lambda }_2\right|{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\phi (\varsigma ,\varkappa (\varsigma ))-\phi (\varsigma ,y(\varsigma ))\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\phi (\varsigma ,\varkappa (\varsigma ))-\phi (\varsigma ,y(\varsigma ))\right|d\varsigma \hfill \\ \hfill & \le \mathcal{L}\parallel \varkappa -y\parallel {\left(\psi (b)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}\left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right)[{\displaystyle \frac{\left|{\lambda }_4\right|}{\left|{\lambda }_3+{\lambda }_4\right|{\mathsf{\Gamma }}_k({\zeta }_k+\gamma ){\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+2k)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}]\hfill \\ \hfill & =\mathcal{L}{\mathsf{\Omega }}_1\parallel \varkappa -y\parallel .\hfill \end{array}$$

Hence, $\parallel T(\varkappa )-T(y)\parallel \le \mathcal{L}{\mathsf{\Omega }}_1\parallel \varkappa -y\parallel ,$ which, in view of condition (17), implies that the operator T is a contraction. Therefore, by the Banach contraction mapping principle, the operator T admits a unique fixed point. Consequently, the sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo fractional boundary value problem given in (1) has a unique solution on the interval $[0,b]$. The proof is finished. □

3.2. Existence Result via Schaefer’s Fixed Point Theorem

Theorem 2

(Schaefer’s fixed point theorem, (Theorem 4.3.2, p. 29, [24])). Let X be a Banach space. Assume that $T:X\to X$ is a completely continuous operator and the set $V=\{\varkappa \in X|\varkappa =\mathcal{A}T(\varkappa ),0<\mathcal{A}<1\}$ is bounded. Then T has a fixed point in $X.$

Theorem 3.

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

(${\mathcal{H}}_2$) 

There exists a real constant $\mathcal{N}>0$ such that for all $\delta \in [0,b]$, $\varkappa \in \mathbb{R}$,

$$\begin{array}{c}\hfill \left|\phi (\delta ,\varkappa )\right|\le \mathcal{N}.\end{array}$$

   Then, there exists at least one solution for the $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo sequential fractional boundary value problem (1) on $[0,b]$.

Proof. 

First, we demonstrate that the operator T is a completely continuous operator. To prove the continuity of T, let $\left\{{\varkappa }_n\right\}$ be a sequence set that ${\varkappa }_n\to \varkappa$ in X. Then, for each $\delta \in [0,b]$, we have

$$\begin{array}{cc}\hfill & \left|T({\varkappa }_n)(\delta )-T(\varkappa )(\delta )\right|\hfill \\ \hfill & \le \left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right){\displaystyle \frac{|\mathsf{\Lambda }(b)|}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}\left|\phi (\varsigma ,{\varkappa }_n(\varsigma ))-\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\left|{\lambda }_2\right|}{k\left|{\lambda }_1+{\lambda }_2\right|{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\phi (\varsigma ,{\varkappa }_n(\varsigma ))-\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\phi (\varsigma ,{\varkappa }_n(\varsigma ))-\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma ,\hfill \end{array}$$

from which we conclude that

$$\begin{array}{c}\hfill \left|T({\varkappa }_n)-T(\varkappa )\right|\to 0\mathrm{as}{\varkappa }_n\to \varkappa .\end{array}$$

This means that T is continuous.

Now, we show that the operator T maps bounded sets into bounded sets in X. Let us define

$$B_R=\left\{\varkappa \in X:∥\varkappa ∥\le R\right\}.$$

For any $\varkappa \in B_R$, we have the following:

$$\begin{array}{cc}\hfill & \left|T(\varkappa )(\delta )\right|\hfill \\ \hfill & \le \left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right)\left|\mathsf{\Lambda }(b)\right|{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}\left|\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\left|{\lambda }_2\right|}{k\left|{\lambda }_1+{\lambda }_2\right|{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & \le \mathcal{N}\left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right){\displaystyle \frac{\left|{\lambda }_4\right|}{\left|{\lambda }_3+{\lambda }_4\right|}}{\displaystyle \frac{(\psi (b)-\psi {(0)}^{\frac{{\zeta }_k+\gamma }{k}-1}}{{\mathsf{\Gamma }}_k({\zeta }_k+\gamma )}}{\displaystyle \frac{{\left(\psi (b)-\psi (0)\right)}^{\frac{\vartheta -{\zeta }_k}{k}+1}}{{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+2k)}}\hfill \\ \hfill & +{\displaystyle \frac{\mathcal{N}\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}{\left(\psi (b)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}+{\displaystyle \frac{\mathcal{N}}{{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}{\left(\psi (b)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}\hfill \\ \hfill & =\mathcal{N}{\mathsf{\Omega }}_1,\hfill \end{array}$$

and consequently,

$$∥T(\varkappa )∥\le \mathcal{N}{\mathsf{\Omega }}_1.$$

Thus, T is bounded.

In order to establish the equicontinuity of $T(B_R)$, we proceed as follows: let ${\delta }_1,{\delta }_2\in [0,b]$ such that ${\delta }_1<{\delta }_2$. For any $\varkappa \in B_R$, we have

$$\begin{array}{cc}\hfill & \left|T(\varkappa )({\delta }_2)-T(\varkappa )({\delta }_1)\right|\hfill \\ \hfill & \le {\displaystyle \frac{(\mathsf{\Lambda }\left({\delta }_2\right)-\mathsf{\Lambda }\left({\delta }_1\right))}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}\left|\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{{\delta }_1}{\psi }^{\prime }(\varsigma )\left({\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}-{\left(\psi ({\delta }_1)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\right)\left|\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_{{\delta }_1}^{{\delta }_2}{\psi }^{\prime }(\varsigma ){\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & \le {\displaystyle \frac{\mathcal{N}(\mathsf{\Lambda }\left({\delta }_2\right)-\mathsf{\Lambda }\left({\delta }_1\right))}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\mathcal{N}}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{{\delta }_1}{\psi }^{\prime }(\varsigma )\left({\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}-{\left(\psi ({\delta }_1)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\right)d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\mathcal{N}}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_{{\delta }_1}^{{\delta }_2}{\psi }^{\prime }(\varsigma ){\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}d\varsigma \hfill \\ \hfill & \le {\displaystyle \frac{\mathcal{N}(\mathsf{\Lambda }\left({\delta }_2\right)-\mathsf{\Lambda }\left({\delta }_1\right))}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\mathcal{N}}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{{\delta }_1}{\psi }^{\prime }(\varsigma )\left({\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}-{\left(\psi ({\delta }_1)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\right)d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\mathcal{N}}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_{{\delta }_1}^{{\delta }_2}{\psi }^{\prime }(\varsigma ){\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}d\varsigma ,\hfill \\ \hfill & ={\displaystyle \frac{\mathcal{N}(\mathsf{\Lambda }\left({\delta }_2\right)-\mathsf{\Lambda }\left({\delta }_1\right))}{{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+2k)}}{\left(\psi (b)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}+1}\hfill \\ \hfill & +{\displaystyle \frac{\mathcal{N}}{{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}\left[{\left(\psi ({\delta }_2)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}-{\left(\psi ({\delta }_1)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}\right],\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill \left|T(\varkappa )({\delta }_2)-T(\varkappa )({\delta }_1)\right|\to 0,\mathrm{as}{\delta }_1\to {\delta }_2,\end{array}$$

independently of $\varkappa \in B_{\rho }.$ Consequently, the operator T is equicontinuous. Therefore, we can conclude, by the Arzelá–Ascoli theorem (Theorem 11.18, p. 181, [25]), that the operator T is completely continuous.

It remains to prove that the set

$$\begin{array}{c}\hfill V=\{\varkappa \in X:\varkappa =\mathcal{A}T(\varkappa ),\mathcal{A}\in (0,1)\}\end{array}$$

is bounded. In view of the hypothesis $({\mathcal{H}}_2)$, we have the following:

$$\begin{array}{cc}\hfill \left|\varkappa (\delta )\right|& =\left|\mathcal{A}T(\varkappa )(\delta )\right|\hfill \\ \hfill & \le \left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right)\left|\mathsf{\Lambda }(b)\right|{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}\left|\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\left|{\lambda }_2\right|}{k\left|{\lambda }_1+{\lambda }_2\right|{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & \le \mathcal{N}{\mathsf{\Omega }}_1.\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill \parallel \varkappa \parallel \le \mathcal{N}{\mathsf{\Omega }}_1.\end{array}$$

This confirms that the set V is bounded. Consequently, by Schaefer’s fixed point theorem, the operator T possesses at least one fixed point, which corresponds to a solution of problem (1), thereby completing the proof. □

3.3. Existence Result via Leray–Schauder Nonlinear Alternative

Lemma 8

(Leray–Schauder nonlinear alternative, (Theorem (4.1), p. 14, [26])). Let X be a Banach space, C a closed, convex subset of X, U an open subset of C, and $0\in U$. Suppose that $T:\overline{U}\to C$ is a continuous, compact map (that is, $T(U)$ is a relatively compact subset of C). Then either

(i) 

T has a fixed point in $\overline{U}$, or

(ii) 

there is a $\varkappa \in \partial U$ (the boundary of U in C) and $\widehat{\mathcal{A}}\in (0,1)$ with $\varkappa =\widehat{\mathcal{A}}T(\varkappa )$.

Theorem 4.

Let $\phi :[0,b]\times \mathbb{R}\to \mathbb{R}$ be a continuous function. Assume that the following conditions are satisfied:

(${\mathcal{H}}_3$) 

There exists a continuous, non-decreasing function $\sigma :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ and a positive continuous function $\omega \in C([0,b],{\mathbb{R}}^{+})$ such that

$$\begin{array}{c}\hfill \left|\phi (\delta ,\varkappa )\right|\le \omega (\delta )\sigma (\left|\varkappa \right|)\end{array}$$

(18)

for all $\delta \in [0,b]$ and $\varkappa \in \mathbb{R}$.

(${\mathcal{H}}_4$) 

There exists a positive constant $\mathcal{K}$ such that

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathcal{K}}{\parallel \omega \parallel \sigma (\mathcal{K}){\mathsf{\Omega }}_1}}>1,\end{array}$$

(19)

where ${\mathsf{\Omega }}_1$ is defined by (15).

    Then, the $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo sequential fractional boundary value problem (1) has at least one solution on $[0,b]$.

Proof. 

We demonstrate only the existence of an open set $U\subseteq C([0,b],\mathbb{R})$ such that $\varkappa \ne \widehat{\mathcal{A}}T(\varkappa )$ for all $\widehat{\mathcal{A}}\in (0,1)$ and $\varkappa \in \partial U$, as the complete continuity of the operator T has already been established in Theorem 3.

Let $\varkappa \in C([0,b],\mathbb{R})$ be such that $\varkappa =\widehat{\mathcal{A}}T(\varkappa )$ for some $0<\widehat{\mathcal{A}}<1$. Then, for each $\delta \in [0,b]$, we have the following:

$$\begin{array}{c}\hfill \left|\varkappa (\delta )\right|=\widehat{\mathcal{A}}\left|T(\varkappa )(\delta )\right|\le \parallel \omega \parallel \sigma (\parallel \varkappa \parallel ){\mathsf{\Omega }}_1,\end{array}$$

which implies that

$$\begin{array}{c}\hfill {\displaystyle \frac{\parallel \varkappa \parallel }{\parallel \omega \parallel \sigma (\parallel \varkappa \parallel ){\mathsf{\Omega }}_1}}\le 1.\end{array}$$

In view of $({\mathcal{H}}_4)$, there is no solution $\varkappa$ such that $\parallel \varkappa \parallel \ne \mathcal{K}$. Let us set

$$\begin{array}{c}\hfill U=\left\{\varkappa \in C([0,b],\mathbb{R}):∥\varkappa ∥<\mathcal{K}\right\}.\end{array}$$

Note that the operator $T:\overline{U}\to C([0,b],\mathbb{R})$ is both continuous and completely continuous. Furthermore, due to the construction of the set U, there is no $\varkappa \in \partial U$ such that $\varkappa =\widehat{\mathcal{A}}T(\varkappa )$ for any $\widehat{\mathcal{A}}\in (0,1)$. Consequently, by applying the Leray–Schauder nonlinear alternative, we conclude that the operator T admits at least one fixed point $\varkappa$. It follows that the $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo sequential fractional boundary value problem (1) has at least one solution on the interval $[0,b]$. This completes the proof. □

3.4. Existence Result via Krasnosel’ski$\breve{i}$’s Fixed Point Theorem

Theorem 5

(Theorem 4.4.1, p. 31, [24]). Let B be a bounded, closed, convex, and nonempty subset of a Banach space X. Let $T_1$ and $T_2$ be two operators such that:

(i) 

$T_1(\varkappa )+T_2(y)\in B$ whenever $\varkappa ,y\in B.$

(ii) 

$T_1(\varkappa )$ is compact and continuous.

(iii) 

$T_2(y)$ is a contraction mapping.

Then, there exists $z\in B$ such that $z=T_1(z)+T_2(z)$.

Theorem 6.

Let $\phi :[0,b]\times \mathbb{R}\to \mathbb{R}$ be a continuous function satisfying the assumption $({\mathcal{H}}_1)$. Moreover, we suppose the following:

(${\mathcal{H}}_5$) 

A continuous function $\varphi \in C([0,b],{\mathbb{R}}^{+})$ exists such that

$$\begin{array}{c}\hfill \left|\phi (\delta ,\varkappa )\right|\le \varphi (\delta )\end{array}$$

for each $(\delta ,\varkappa )\in [0,b]\times \mathbb{R}$.

If

$$\begin{array}{c}\hfill \mathcal{L}{\mathsf{\Omega }}_2<1,\end{array}$$

(20)

where ${\mathsf{\Omega }}_2$ is given by (16), then the $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo sequential fractional boundary value problem (1) has at least one solution on $[0,b]$.

Proof. 

We can fix $\varrho \ge \parallel \varphi \parallel {\mathsf{\Omega }}_1$ and consider $B_{\varrho }=\{\varkappa \in X:\parallel \varkappa \parallel \le \varrho \}$. We define the operators, $T_1$ and $T_2$ on $B_{\varrho }$ as

$$\begin{array}{cc}\hfill T_1(\varkappa )(\delta )& ={}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\phi \left(\delta ,\varkappa (\delta )\right),\delta \in [0,b],\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill T_2(\varkappa )(\delta )& =\left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }\phi \left(b,\varkappa (b)\right)\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\phi \left(b,\varkappa (b)\right),\delta \in [0,b].\hfill \end{array}$$

For any $\varkappa ,y\in B_{\varrho }$, we find that

$$\begin{array}{cc}\hfill & \left|T_1(\varkappa )(\delta )+T_2(y)(\delta )\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & +\left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right)\left|\mathsf{\Lambda }(b)\right|{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}\left|\phi (\varsigma ,y(\varsigma ))\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\phi (\varsigma ,y(\varsigma ))\right|d\varsigma \hfill \\ \hfill & \le \parallel \varphi \parallel {\left(\psi (b)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}\left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right)[{\displaystyle \frac{\left|{\lambda }_4\right|}{\left|{\lambda }_3+{\lambda }_4\right|{\mathsf{\Gamma }}_k({\zeta }_k+\gamma ){\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+2k)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}]\hfill \\ \hfill & =\parallel \varphi \parallel {\mathsf{\Omega }}_1\le \varrho .\hfill \end{array}$$

Thus, $\parallel T_1(\varkappa )+T_2(y)\parallel \le \varrho ,$ which implies that $T_1(\varkappa )+T_2(y)\in B_{\varrho }$.

Now, we show that $T_2$ is a contraction mapping. From the assumption $({\mathcal{H}}_1)$ together with (20), we obtain the following:

$$\begin{array}{cc}\hfill & \left|T_2(\varkappa )(\delta )-T_2(y)(\delta )\right|\hfill \\ \hfill & \le \left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right){\displaystyle \frac{|\mathsf{\Lambda }(b)|}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}\left|\phi (\varsigma ,\varkappa (\varsigma ))-\phi (\varsigma ,y(\varsigma ))\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\left|{\lambda }_2\right|}{k\left|{\lambda }_1+{\lambda }_2\right|{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\phi (\varsigma ,\varkappa (\varsigma ))-\phi (\varsigma ,y(\varsigma ))\right|d\varsigma \hfill \\ \hfill & \le \mathcal{L}\parallel \varkappa -y\parallel \left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right){\displaystyle \frac{\left|{\lambda }_4\right|}{\left|{\lambda }_3+{\lambda }_4\right|}}{\displaystyle \frac{(\psi (b)-\psi {(0)}^{\frac{{\zeta }_k+\gamma }{k}-1}}{{\mathsf{\Gamma }}_k({\zeta }_k+\gamma )}}{\displaystyle \frac{{\left(\psi (b)-\psi (0)\right)}^{\frac{\vartheta -{\zeta }_k}{k}+1}}{{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+2k)}}\hfill \\ \hfill & +\mathcal{L}\parallel \varkappa -y\parallel {\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}{\left(\psi (b)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}\hfill \\ \hfill & \le \mathcal{L}\parallel \varkappa -y\parallel (\psi (b)-\psi {(0)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}[\left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right){\displaystyle \frac{\left|{\lambda }_4\right|}{\left|{\lambda }_3+{\lambda }_4\right|{\mathsf{\Gamma }}_k({\zeta }_k+\gamma ){\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+2k)}}\hfill \\ \hfill & +{\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}]\hfill \\ \hfill & =\mathcal{L}{\mathsf{\Omega }}_2\parallel \varkappa -y\parallel .\hfill \end{array}$$

Hence $\parallel T_2(\varkappa )-T_2(y)\parallel \le \mathcal{L}{\mathsf{\Omega }}_2\parallel \varkappa -y\parallel ,$ which implies that the operator $T_2$ is a contraction.

Continuity of $\phi$ implies that the operator $T_1$ is continuous. Also, $T_1$ is uniformly bounded on $B_{\varrho }$ as

$$\parallel T_1(\varkappa )\parallel \le {\displaystyle \frac{{(\psi (b)-\psi (0))}^{\frac{\vartheta +\gamma }{k}}}{{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}\parallel \varphi \parallel .$$

Now, we prove the compactness of the operator $T_1$. In view of $({\mathcal{H}}_5)$, we have the following:

$$\begin{array}{cc}\hfill & \left|T_1(\varkappa )({\delta }_2)-T_1(\varkappa )({\delta }_1)\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{{\delta }_1}{\psi }^{\prime }(\varsigma )\left({\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}-{\left(\psi ({\delta }_1)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\right)\left|\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_{{\delta }_1}^{{\delta }_2}{\psi }^{\prime }(\varsigma ){\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\phi (\varsigma ,\varkappa (\varsigma ))\right|d\varsigma \hfill \\ \hfill & \le {\displaystyle \frac{\parallel \varphi \parallel }{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{{\delta }_1}{\psi }^{\prime }(\varsigma )\left({\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}-{\left(\psi ({\delta }_1)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\right)d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\parallel \varphi \parallel }{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_{{\delta }_1}^{{\delta }_2}{\psi }^{\prime }(\varsigma ){\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}d\varsigma \hfill \\ \hfill & ={\displaystyle \frac{\parallel \varphi \parallel }{{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}\left[{\left(\psi ({\delta }_2)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}-{\left(\psi ({\delta }_1)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}\right],\hfill \end{array}$$

which is independent of $\varkappa$ and tends to zero as ${\delta }_2\to {\delta }_1$. Thus, $T_1$ is relatively compact on $B_{\varrho }$. Hence, by the Arzelá–Ascoli Theorem (Theorem 11.18, p. 181, [25]), $T_1$ is compact on $B_{\varrho }$. Hence, according to the conclusion of Krasnosel’ski$\breve{\mathrm{i}}$’s fixed point theorem, problem (1) admits at least one solution on the interval $[0,b]$, which concludes the proof. □

3.5. Illustrative Examples for Single-Valued Case

In this section, we present several illustrative examples to demonstrate the applicability of the main results obtained in the previous section.

Example 1.

Consider the following sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo fractional boundary value problem with non-separated boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {}^{\frac{13}{19},H}{D}_{0^{+}}^{\frac{11}{19},\frac{3}{5};\psi }\left({}^{\frac{13}{19},C}{D}_{0^{+}}^{\frac{8}{19};\psi }\varkappa \left(\delta \right)\right)=\phi (\delta ,\varkappa (\delta )),\delta \in \left[0,\frac{25}{23}\right],}\hfill \\ \\ {\displaystyle \frac{3}{31}\varkappa \left(0\right)+\frac{5}{51}\varkappa \left(\frac{25}{23}\right)=0,\frac{7}{71}{}^{\frac{13}{19},C}{D}_{0^{+}}^{\frac{36}{95};\psi }\varkappa \left(0\right)+\frac{9}{91}{}^{\frac{13}{19},C}{D}_{0^{+}}^{\frac{36}{95};\psi }\varkappa \left(\frac{25}{23}\right)=0.}\hfill \end{array}\right.\end{array}$$

(21)

Here, $k=13/19$, $\vartheta =11/19$, $\beta =3/5$, $\gamma =8/19$, and $b=25/23$. We can verify that $\vartheta /k=11/13$ and $\gamma /k=8/13$ both belong to the interval $(0,1)$, which implies that $n=1$. In addition, we have ${\zeta }_k=61/95$ and ${\zeta }_k+\gamma -k=36/95>0$. Therefore, the problem (21) is well-defined. The well-known relation

$${\mathsf{\Gamma }}_k(y)=k^{{\displaystyle \frac{y}{k}}-1}\mathsf{\Gamma }\left({\displaystyle \frac{y}{k}}\right),$$

can be used to compute certain constants, such as ${\mathsf{\Gamma }}_k({\zeta }_k+\gamma )\approx 0.7206064300$, ${\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+2k)\approx 0.6833938837$, ${\mathsf{\Gamma }}_k(\vartheta +\gamma +k)\approx 0.7433150185$. Naturally, the boundary value problem in (18) should be accompanied by fully specified functions. However, in order to explore the implications of the main results and demonstrate the applicability of various theorems presented in this work, we allow modifications of the involved functions across different cases. This enables a multifaceted illustration of the theorems’ utility. As for the boundary conditions, the constants can be explicitly assigned as ${\lambda }_1=3/31$, ${\lambda }_2=5/51$, ${\lambda }_3=7/71$, and ${\lambda }_4=9/91$, since the boundary conditions have been clearly stated.

$(i)$ Let the nonlinear unbounded Lipschitzian function $\phi (\delta ,\varkappa )$ be defined by

$$\phi (\delta ,\varkappa )={\displaystyle \frac{1}{{\delta }^2+10}}\left({\displaystyle \frac{{\varkappa }^2+2\left|\varkappa \right|}{1+\left|\varkappa \right|}}\right)+{\displaystyle \frac{1}{4}},$$

(22)

and $\psi (\delta )={\delta }^2+1$.

Based on the parameters and conditions described above, we now compute the corresponding constant, yielding ${\mathsf{\Omega }}_1\approx 4.531099536$. Next, we can show that

$$|\phi (\delta ,\varkappa )-\phi (\delta ,y)|\le {\displaystyle \frac{1}{5}}\parallel \varkappa -y\parallel ,$$

and thus $\mathcal{L}=1/5$. Therefore, we have $\mathcal{L}{\mathsf{\Omega }}_1\approx 0.9062199072<1$. By applying Theorem 1, we can conclude that the boundary value problem (18), involving the nonlinear function $\phi$ defined by (22) and $\psi (\delta )={\delta }^2+1$, has a unique solution on the interval $[0,25/23]$.

$(ii)$ The function $\psi$ can be generalized by setting $\psi (\delta )={\delta }^n+1$ where $n>0$. The nonlinear function $\phi$ is taken as given in Equation (22). By virtue of Theorem 1, and due to relation (17), we conclude that the boundary value problems (21) and (22), with $\psi (\delta )={\delta }^n+1$, admit a unique solution, provided that $n<11.55411928$.

$(iii)$ Observe that the nonlinear Lipschitzian function $\phi$ in case (i) is unbounded. However, if the function is modified to be bounded, for example, as

$$\phi (\delta ,\varkappa )={\displaystyle \frac{1}{{\delta }^2+3}}\left({\displaystyle \frac{\left|\varkappa \right|}{1+\left|\varkappa \right|}}\right)+{\displaystyle \frac{1}{4}},$$

(23)

then the analysis can proceed as follows.

The Lipschitz constant is determined to be $\mathcal{L}=1/3$, as for Equation (23) it holds that

$$|\phi (\delta ,\varkappa )-\phi (\delta ,y)|\le {\displaystyle \frac{1}{3}}\parallel \varkappa -y\parallel .$$

However, Theorem 1 is not applicable in this case because the inequality $\mathcal{L}{\mathsf{\Omega }}_1\approx 1.510366512>1$ contradicts the required assumption. On the other hand, we observe that ${\mathsf{\Omega }}_2\approx 2.814464159$, which leads to $\mathcal{L}{\mathsf{\Omega }}_2\approx 0.9381547197<1$, satisfying inequality (20) of Theorem 6. In addition, the nonlinear function is bounded by

$$\left|\phi (\delta ,\varkappa )\right|\le {\displaystyle \frac{1}{{\delta }^2+3}}+{\displaystyle \frac{1}{4}}:=\omega (\delta ).$$

Therefore, the conclusion of Theorem 6 can be applied to ensure that the boundary value problem (21) with (23) has at least one solution on the interval $[0,25/23]$.

$(iv)$ A non-Lipschitzian, but bounded nonlinear function is illustrated in this case. For example, let $\phi$ be defined as

$$\phi (\delta ,\varkappa )={\displaystyle \frac{1}{{(\sqrt{\delta }+2)}^2}}\left({\displaystyle \frac{{\varkappa }^{2568}}{5(1+{\varkappa }^{2566})}}+{\displaystyle \frac{1}{\sqrt[3]{\delta }+6}}{e}^{-\left|\varkappa \right|}\right),$$

(24)

and $\psi (\delta )=\delta +\sqrt{\delta }.$

Here, we can find the constant ${\mathsf{\Omega }}_1\approx 10.71893315$ and the nonlinear bound of $\phi$ as

$$\left|\phi (\delta ,\varkappa )\right|\le {\displaystyle \frac{1}{{(\sqrt{\delta }+2)}^2}}\left({\displaystyle \frac{1}{5}}{\varkappa }^2+{\displaystyle \frac{1}{6}}\right):=\upsilon (\delta )\sigma (\left|\varkappa \right|).$$

Then $\parallel \upsilon \parallel =1/4$ and $\sigma (u)=(1/5)u^2+1/6$. Hence, by taking advantage of Theorem 4, and since there exists a constant $\mathcal{K}\in (0.7405161881,1.125341143)$ satisfying inequality (19), we deduce that the boundary value problem (21) with (24) admits at least one solution on $[0,25/23]$.

$(v)$ This case allows for the analysis of a nonlinear function that is bounded by a constant but does not satisfy the Lipschitz condition. For instance, by modifying case $(iv)$, the function can be expressed as

$$\phi (\delta ,\varkappa )={\displaystyle \frac{1}{{(\sqrt{\delta }+2)}^2}}\left({\displaystyle \frac{{\varkappa }^{2568}}{5(1+{\varkappa }^{2568})}}+{\displaystyle \frac{1}{\sqrt[3]{\delta }+6}}{e}^{-\left|\varkappa \right|}\right).$$

(25)

Clearly, the function presented in (25) is bounded by a constant, as demonstrated below

$$\left|\phi (\delta ,\varkappa )\right|\le {\displaystyle \frac{1}{4}}\left({\displaystyle \frac{1}{5}}+{\displaystyle \frac{1}{6}}\right)={\displaystyle \frac{11}{120}}.$$

Consequently, assumption $({\mathcal{H}}_2)$ of Theorem 3 is fulfilled. It follows that the boundary value problem (21), involving the function defined in (25), admits at least one solution on the interval $[0,25/23]$.

4. Multi-Valued Case

Let $(X,|·|)$ be a normed space. We denote by ${\mathcal{P}}_{cl},$${\mathcal{P}}_b,$${\mathcal{P}}_{cp},$ and ${\mathcal{P}}_{cp,c}$ the collections of all closed, bounded, compact, and compact convex subsets of X, respectively.

The set of selections of F for each $\varkappa \in C([0,b],\mathbb{R})$ is defined as

$$S_{F,\varkappa }:=\{z\in L^1([0,b],\mathbb{R}):z(\delta )\in F(\delta ,\varkappa (\delta ))fora.e.\delta \in [0,b]\}.$$

For details on multi-valued analysis, see [27,28,29].

Definition 4.

A function $\varkappa \in C([0,b],\mathbb{R})$ is called a solution of the sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo inclusion fractional boundary value problem (2), if there exists a function $\nu \in L^1([0,b],\mathbb{R})$ with $\nu (\delta )\in F(\delta ,\varkappa )$ almost everywhere (a.e.) on $[0,b]$ such that ϰ satisfies the differential equation ${}^{k,H}{D}_{0^{+}}^{\vartheta ,\beta ;\psi }\left({}^{k,C}{D}_{0^{+}}^{\gamma ;\psi }\varkappa \left(\delta \right)\right)=\nu (\delta )$ on $[0,b]$ and the boundary conditions ${\displaystyle {\lambda }_1\varkappa \left(0\right)+{\lambda }_2\varkappa \left(b\right)=0,{\lambda }_3{}^{k,C}{D}_{0^{+}}^{{\zeta }_k+\gamma -k;\psi }\varkappa \left(0\right)+{\lambda }_4{}^{k,C}{D}_{0^{+}}^{{\zeta }_k+\gamma -k;\psi }\varkappa \left(b\right)=0.}$

In the proof of the following result, we make use of the nonlinear alternative for Kakutani maps [26] and the closed graph operator theorem [30].

Theorem 7

(Nonlinear alternative for Kakutani maps (Theorem (8.5), p. 169, [26])). Assume that $\mathcal{C}$ is a Banach space; ${\mathcal{C}}_1$ is a convex closed subset of $\mathcal{C};$ U is an open subset of ${\mathcal{C}}_1$; and $0\in U.$ If $F:\overline{U}\to {\mathcal{P}}_{cp,c}({\mathcal{C}}_1)$ is an upper semi-continuous compact map, then either F has a fixed point in $\overline{U},$ or there is a $\varphi \in \partial U$ and $k\in (0,1)$ with $\varphi \in kF(\varphi ).$

The following lemma is used in the forthcoming result.

Lemma 9

(Theorem 2, p. 783, [30]). Assume that $\mathsf{\Phi }:[0,b]\times \mathbb{R}\to {\mathcal{P}}_{cp,c}(\mathbb{R})$ is an $L^1-$ Carathéodory multi-valued map. If Q is a linear continuous mapping from $L^1([0,b],\mathbb{R})$ to $C([0,b],\mathbb{R}),$ then the operator

$$Q\circ S_{\mathsf{\Phi }}:C([0,b],\mathbb{R})\to {\mathcal{P}}_{cp,c}(C([0,b],\mathbb{R})),\varkappa \mapsto (Q\circ S_{\mathsf{\Phi }})(\varkappa )=Q(S_{\mathsf{\Phi },\varkappa })$$

is a closed graph operator in $C([0,b],\mathbb{R})\times C([0,b],\mathbb{R}).$

4.1. Case 1: Convex-Valued Multi-Functions

We address the case where the multi-function F takes convex values and establish an existence result for the sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo fractional inclusion boundary value problem (2). The proof relies on the nonlinear alternative for Kakutani maps [26] and the closed graph operator theorem [30], assuming that F satisfies the $L^1$-Carathéodory conditions.

Definition 5

(Definition 1.5, p. 142, [30]). A set-valued function $F:[a,b]\times \mathbb{R}\to {\mathcal{P}}_{cp,c}(\mathbb{R})$ is defined as Carathéodory if

(i) 

the mapping $\delta \to F(\delta ,\varkappa )$ is measurable for every $\varkappa \in \mathbb{R}$ and

(ii) 

the function $\varkappa \to F(\delta ,\varkappa )$ is upper semi-continuous for almost every $\delta \in [a,b]$.

Moreover, a Carathéodory function F is referred to as an $L^1$-Carathéodory if

(iii) 

for each $\rho >0$, there exists a function ${{\rm Y}}_{\rho }\in L^1([a,b],{\mathbb{R}}^{+})$ such that $\parallel F(\delta ,\varkappa )\parallel =sup\left\{\right|u|:u\in F(\delta ,\varkappa )\}\le {{\rm Y}}_{\rho }(\delta )$ for all $\varkappa \in \mathbb{R}$ with $\parallel \varkappa \parallel \le \rho$ and for almost every $\delta \in [a,b]$.

Theorem 8.

Suppose that:

$(A_1)$

The multi-function $F:[0,b]\times \mathbb{R}\to {\mathcal{P}}_{cp,c}(\mathbb{R})$ is $L^1$-Carathéodory;

$(A_2)$

There exists a nondecreasing function $\chi \in C([0,b],{\mathbb{R}}^{+}),$ and a continuous function q: $[0,b]\to {\mathbb{R}}^{+}$ such that

${\parallel F(\delta ,\varkappa )\parallel }_{\mathcal{P}}:=sup\left\{\right|z|:z\in F(\delta ,\varkappa )\}\le q(\delta )\chi (\parallel \varkappa \parallel )foreach(\delta ,\varkappa )$$\in [0,b]\times \mathbb{R};$

$(A_3)$

There exists a positive number M such that

$${\displaystyle \frac{M}{{\displaystyle \chi (M)\parallel q\parallel {\mathsf{\Omega }}_1}}>1,}$$

where ${\mathsf{\Omega }}_1$ is given by (15).

   Then, the sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo inclusion fractional boundary value problem (2) has at least one solution on $[0,b].$

Proof. 

We introduce a multi-valued operator: $N:C([0,b],\mathbb{R})\to \mathcal{P}(C([0,b],\mathbb{R}))$ as

$$N(\varkappa )=\left\{\begin{array}{cc}\hfill & h\in C([0,b],\mathbb{R}):\hfill \\ \hfill & h(\delta )=\left\{\begin{array}{c}{\displaystyle \left(\frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }\nu \left(b\right)}\hfill \\ {\displaystyle -\frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\nu \left(b\right)+{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\nu \left(\delta \right),\nu \in S_{F,\varkappa }.}\hfill \end{array}\right.\hfill \end{array}\right\}$$

(26)

We aim to show that the operator N meets the conditions of the Leray–Schauder nonlinear alternative for Kakutani maps (Theorem 7), and we do so through several steps.

Step 1. The operator N is bounded on bounded subsets of $C([0,b],\mathbb{R})$.

Let $B_r=\varkappa \in C([0,b],\mathbb{R}):\left|\varkappa \right|\le r$ for some $r>0$, representing a bounded subset of $C([0,b],\mathbb{R})$. For each $h\in N(\varkappa )$ with $\varkappa \in B_r$, there exists $\nu \in S_{F,\varkappa }$ such that

$$\begin{array}{ccc}\hfill h(\delta )& =& \left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }\nu \left(b\right)-{\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\nu \left(b\right)+{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\nu \left(\delta \right).\hfill \end{array}$$

For $\delta \in [0,b],$ using the assumption $(A_2)$, we obtain

$$\begin{array}{ccc}\hfill |h(\delta )|& \le & \left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right)\left|\mathsf{\Lambda }(b)\right|{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}\left|\nu (\varsigma )\right|d\varsigma \hfill \\ & & +{\displaystyle \frac{\left|{\lambda }_2\right|}{k\left|{\lambda }_1+{\lambda }_2\right|{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\nu (\varsigma )\right|d\varsigma \hfill \\ & & +{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\nu (\varsigma )\right|d\varsigma \hfill \\ & \le & \left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right)\parallel q\parallel \chi (\parallel \varkappa \parallel ){\displaystyle \frac{\left|{\lambda }_4\right|}{\left|{\lambda }_3+{\lambda }_4\right|}}{\displaystyle \frac{(\psi (b)-\psi {(0)}^{\frac{{\zeta }_k+\gamma }{k}-1}}{{\mathsf{\Gamma }}_k({\zeta }_k+\gamma )}}{\displaystyle \frac{{\left(\psi (b)-\psi (0)\right)}^{\frac{\vartheta -{\zeta }_k}{k}+1}}{{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+2k)}}\hfill \\ & & +{\displaystyle \frac{\left|{\lambda }_2\right|\parallel q\parallel \chi (\parallel \varkappa \parallel )}{\left|{\lambda }_1+{\lambda }_2\right|{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}{\left(\psi (b)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}+{\displaystyle \frac{\parallel q\parallel \chi (\parallel \varkappa \parallel )}{{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}{\left(\psi (b)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}\hfill \\ & =& \parallel q\parallel \chi (\parallel \varkappa \parallel ){\mathsf{\Omega }}_1,\hfill \end{array}$$

and consequently

$$\parallel h\parallel \le \parallel q\parallel \chi (r){\mathsf{\Omega }}_1.$$

Step 2. Bounded sets are mapped by N into equicontinuous sets of $C([0,b],\mathbb{R}).$

Let $\varkappa \in B_r$ and $h\in N(\varkappa ).$ Then there exists $\nu \in S_{F,\varkappa }$ such that

$$\begin{array}{ccc}\hfill h(\delta )& =& \left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }\nu \left(b\right)-{\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\nu \left(b\right)+{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\nu \left(\delta \right).\hfill \end{array}$$

Let ${\delta }_1,{\delta }_2\in [0,b],{\delta }_1<{\delta }_2$. Then,

$$\begin{array}{cc}\hfill & |h({\delta }_2)-h({\delta }_1)|\hfill \\ \hfill & \le {\displaystyle \frac{(\mathsf{\Lambda }\left({\delta }_2\right)-\mathsf{\Lambda }\left({\delta }_1\right))}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}\left|\nu (\varsigma )\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{{\delta }_1}{\psi }^{\prime }(\varsigma )\left({\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}-{\left(\psi ({\delta }_1)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\right)\left|\nu (\varsigma )\right|d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_{{\delta }_1}^{{\delta }_2}{\psi }^{\prime }(\varsigma ){\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|\nu (\varsigma )\right|d\varsigma \hfill \\ \hfill & \le {\displaystyle \frac{(\mathsf{\Lambda }\left({\delta }_2\right)-\mathsf{\Lambda }\left({\delta }_1\right))\parallel q\parallel \chi (r)}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\parallel q\parallel \chi (r)}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{{\delta }_1}{\psi }^{\prime }(\varsigma )\left({\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}-{\left(\psi ({\delta }_1)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\right)d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\parallel q\parallel \chi (r)}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_{{\delta }_1}^{{\delta }_2}{\psi }^{\prime }(\varsigma ){\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}d\varsigma \hfill \\ \hfill & \le {\displaystyle \frac{(\mathsf{\Lambda }\left({\delta }_2\right)-\mathsf{\Lambda }\left({\delta }_1\right))\parallel q\parallel \chi (r)}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\parallel q\parallel \chi (r)}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{{\delta }_1}{\psi }^{\prime }(\varsigma )\left({\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}-{\left(\psi ({\delta }_1)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\right)d\varsigma \hfill \\ \hfill & +{\displaystyle \frac{\parallel q\parallel \chi (r)}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_{{\delta }_1}^{{\delta }_2}{\psi }^{\prime }(\varsigma ){\left(\psi ({\delta }_2)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}d\varsigma \to 0,\hfill \end{array}$$

as ${\delta }_1\to {\delta }_2$, independently of $\varkappa \in B_r$. Therefore, by the Arzelà–Ascoli theorem (Theorem 11.18, p. 181, [25]), the operator $N:C([0,b],\mathbb{R})\to \mathcal{P}(C([0,b],\mathbb{R}))$ is completely continuous.

Step 3. The set $N(\varkappa )$ is convex for every $\varkappa \in C([0,b],\mathbb{R})$.

This follows directly from the fact that $S_{F,\varkappa }$ is convex, which holds by assumption, since the multi-function F takes convex values.

Step 4. The graph of N is closed.

Suppose ${\varkappa }_n\to {\varkappa }_{\ast }$, with $h_n\in N({\varkappa }_n)$ and $h_n\to h_{\ast }$. We aim to demonstrate that $h_{\ast }\in N({\varkappa }_{\ast })$. Note that $h_n\in N({\varkappa }_n)$ implies the existence of ${\nu }_n\in S_{F,{\varkappa }_n}$ such that, for every $\delta \in [0,b]$, the following holds:

$$\begin{array}{ccc}\hfill h_n(\delta )& =& \left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }{\nu }_n\left(b\right)-{\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }{\nu }_n\left(b\right)+{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }{\nu }_n\left(\delta \right).\hfill \end{array}$$

For each $\delta \in [0,b]$, we must have ${\nu }_{\ast }\in S_{F,{\varkappa }_{\ast }}$ such that

$$\begin{array}{ccc}\hfill h_{\ast }(\delta )& =& \left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }{\nu }_{\ast }\left(b\right)-{\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }{\nu }_{\ast }\left(b\right)+{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }{\nu }_{\ast }\left(\delta \right).\hfill \end{array}$$

Introduce a continuous linear operator $\mathsf{\Phi }:L^1([0,b],\mathbb{R})\to C([0,b],\mathbb{R})$ as

$$\begin{array}{ccc}& & \nu \to \mathsf{\Phi }(\nu )(\delta )\hfill \\ & =& \left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }\nu \left(b\right)-{\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\nu \left(b\right)+{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\nu \left(\delta \right).\hfill \end{array}$$

It is evident that $\parallel h_n-h_{\ast }\parallel \to 0$ as $n\to \infty .$ Therefore, by applying the closed graph theorem for operators (Lemma 9), it follows that $\mathsf{\Phi }\circ S_{F,\varkappa }$ is a closed graph operator. Moreover, since $h_n\in \mathsf{\Phi }(S_{F,{\varkappa }_n})$ we have

$$\begin{array}{ccc}\hfill h_{\ast }(\delta )& =& \left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }{\nu }_{\ast }\left(b\right)-{\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }{\nu }_{\ast }\left(b\right)+{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }{\nu }_{\ast }\left(\delta \right),\hfill \end{array}$$

for some ${\nu }_{\ast }\in S_{F,{\varkappa }_{\ast }}.$ Hence, the operator N possesses a closed graph. This, in turn, implies that N is upper semi-continuous, since—according to (Proposition 1.2, p. 8, [27])—any completely continuous operator with a closed graph is necessarily upper semi-continuous.

Step 5. There exists an open set $U\subseteq C([0,b],\mathbb{R}),$ such that, for every $\kappa \in (0,1)$ and all $\varkappa \in \partial U,$ we have $\varkappa \notin \kappa N(\varkappa ).$

Now, suppose $\varkappa \in \kappa N(\varkappa ),$ for some $\kappa \in (0,1).$ Then, there exists a function $\nu \in L^1([0,b],\mathbb{R})$ with $\nu \in S_{F,\varkappa }$ such that for $\delta \in [0,b]$, the following holds:

$$\begin{array}{ccc}\hfill \varkappa (\delta )& =& \kappa \left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }\nu \left(b\right)-\kappa {\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\nu \left(b\right)+\kappa {}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\nu \left(\delta \right).\hfill \end{array}$$

Using the same reasoning as in Step 1, we find that for every $\delta \in [0,b],$

$$\begin{array}{c}\hfill \left|\varkappa (\delta )\right|\le \parallel q\parallel \chi (\parallel \varkappa \parallel ){\mathsf{\Omega }}_1.\end{array}$$

This leads to the inequality

$${\displaystyle \frac{\parallel \varkappa \parallel }{{\displaystyle \chi (\parallel \varkappa \parallel )\parallel q\parallel {\mathsf{\Omega }}_1}}\le 1.}$$

By assumption $(A_3)$, there exists a positive constant M such that $\parallel \varkappa \parallel \ne M$. We now define the set

$$\mathsf{\Theta }=\{x\in C([0,b],\mathbb{R}):\parallel \varkappa \parallel <M\}.$$

It is clear that $N:\overline{\mathsf{\Theta }}\to \mathcal{P}(C([0,b],$$\mathbb{R}))$ is a multi-valued operator with compact, convex values and is upper semi-continuous. Moreover, by construction of the set $\mathsf{\Theta },$ no element $\varkappa \in \partial \mathsf{\Theta }$ satisfies $\varkappa \in \kappa N(\varkappa )$ for any $\kappa \in (0,1).$

As a result, we can apply the Leray–Schauder nonlinear alternative for Kakutani-type maps (Theorem 7), which guarantees that N has a fixed point $\varkappa \in \overline{\mathsf{\Theta }}.$ Therefore, the boundary value problem given by Equation (2) admits at least one solution on the interval $[0,b],$ completing the proof. □

4.2. Case 2: Nonconvex Valued Multi-Functions

In this section, we establish the existence of a solution to the sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo fractional boundary value inclusion problem (2) involving a non-convex valued multi-valued mapping, using the fixed point theorem for contractive multi-valued mappings proposed by Covitz and Nadler [26,31].

Lemma 10

(Covitz–Nadler fixed point theorem (Theorem (3.1), p. 28, [26])). Let $(X,d)$ be a complete metric space. If $G:X\to P_{cl}(\varkappa )$ is a contraction, then $FixG\ne \varnothing .$

Definition 6

(p. 28, [26]). Let $(X,d)$ be a metric space induced from the normed space $(X,\parallel ·\parallel )$ and $H_{\overline{d}}:\mathcal{P}(\varkappa )\times \mathcal{P}(\varkappa )\to \mathbb{R}\cup \{\infty \}$ be defined by

$$H_{\overline{d}}(A,B)=\max \{\underset{c\in A}{\sup }\overline{d}(c,d),\underset{d\in B}{\sup }\overline{d}(c,d)\},$$

where $\overline{d}(c,d)={inf}_{c\in A}\overline{d}(c,d)$ and $\overline{d}(c,d)={inf}_{d\in B}\overline{d}(c,d)$.

Definition 7

(p. 28, [26]). A multi-valued mapping $N:X\to {\mathcal{P}}_{cl}(X)$ is said to be

(a) 

$\gamma$–Lipschitz if there exists a constant $\gamma >0$ such that

$$H_d(N(\varkappa ),N(y))\le \gamma d(\varkappa ,y)forall\varkappa ,y\in X;$$

(b) 

a contraction, if it is $\gamma -\mathit{Lipschitz}\mathit{with}$$\gamma <1$.

Theorem 9.

Assume that:

$(B_1)$

$F:[0,b]\times \mathbb{R}\to {\mathcal{P}}_{cp}(\mathbb{R})$ is such that $F(·,\varkappa ):[0,b]\to {\mathcal{P}}_{cp}(\mathbb{R})$ is measurable for each $\varkappa \in \mathbb{R}$;

$(B_2)$

$H_{\overline{d}}(F(\delta ,\varkappa ),F(\delta ,\overline{\varkappa }))\le \varrho (\delta )|\varkappa -\overline{\varkappa }|$ for almost all $\delta \in [0,b]$ and $\varkappa ,\overline{\varkappa }\in \mathbb{R}$ with $\varrho \in C([0,b],{\mathbb{R}}^{+})$ and $\overline{d}(0,F(t,0))\le \varrho (\delta )$ for almost all $\delta \in [0,b]$.

   Then, the sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo inclusion fractional boundary value problem (2) has at least one solution on $[0,b],$ if

$${\mathsf{\Omega }}_1\parallel \varrho \parallel <1,$$

where ${\mathsf{\Omega }}_1$ is given by (15).

Proof. 

We aim to show that the operator $N:C([0,b],\mathbb{R})\to \mathcal{P}(C([0,b],\mathbb{R}))$, defined by (26), fulfills the conditions of Covitz–Nadler’s fixed point theorem for multi-valued mappings (Lemma 10)

Step I. N is nonempty and closed for every $\nu \in S_{F,\varkappa }.$

The set-valued function $F(·,\varkappa (·))$ is measurable due to the measurable selection theorem (Theorem III.6, p. 65, [32]). Consequently, there exists a measurable function $\nu :[0,b]\to \mathbb{R}$ that selects from $F.$ According to assumption $(B_2),$ we have the bound $|\nu (\delta )|\le \varrho (\delta )(1+|\varkappa (\delta )|),$ which implies that $\nu \in L^1([0,b],\mathbb{R}).$ As a result, F is integrably bounded, leading to the conclusion that $S_{F,\varkappa }\ne \varnothing$.

Next, we establish that $N(\varkappa )\in {\mathcal{P}}_{cl}(C([0,b],$$\mathbb{R}))$ for every $\varkappa \in C([0,b],\mathbb{R})$. To do so, suppose a sequence ${\left\{u_n\right\}}_{n\ge 0}\in N(\varkappa )$ satisfies $u_n\to u$ in $C([0,b],\mathbb{R})$ as $n\to \infty .$ Then $u\in C([0,b],\mathbb{R})$ and for each $\delta \in [0,b],$ there exists corresponding selection ${\nu }_n\in S_{F,{\varkappa }_n}$ such that

$$\begin{array}{ccc}\hfill u_n(\delta )& =& \left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }{\nu }_n\left(b\right)-{\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }{\nu }_n\left(b\right)+{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }{\nu }_n\left(\delta \right).\hfill \end{array}$$

Hence, due to the compactness of the values of $F,$ we can extract a subsequence (if needed) such that ${\nu }_n\to v$ in $L^1([0,b],\mathbb{R}).$ This implies that $\nu \in S_{F,\varkappa },$ and for every $\delta \in [0,b]$, we have

$$\begin{array}{ccc}& & u_n(\delta )\to \nu (\delta )\hfill \\ & =& \left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }\nu \left(b\right)-{\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\nu \left(b\right)+{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }\nu \left(\delta \right).\hfill \end{array}$$

Thus, $u\in N(\varkappa ).$

Step II. Here we establish that there exists $0<\overline{m_0}<1$$(\overline{m_0}={\mathsf{\Omega }}_1\parallel \varrho \parallel )$ such that

$$H_{\overline{d}}(N(\varkappa ),N(\overline{\varkappa }))\le \overline{m_0}\parallel x-\overline{\varkappa }\parallel for eachx,\overline{\varkappa }\in C([0,b],\mathbb{R}).$$

Let $x,\overline{\varkappa }\in C([0,b],\mathbb{R})$ and $h_1\in N(\varkappa )$. Then there exists ${\nu }_1(\delta )\in F(\delta ,w(\delta ))$ such that, for each $\delta \in [0,b]$,

$$\begin{array}{ccc}\hfill h_1(\delta )& =& \left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }{\nu }_1\left(b\right)-{\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }{\nu }_1\left(b\right)+{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }{\nu }_1\left(\delta \right).\hfill \end{array}$$

Using assumption $(B_2)$, we obtain

$$H_{\overline{d}}(F(\delta ,\varkappa ),F(\delta ,\overline{\varkappa }))\le \varrho (\delta )|\varkappa (\delta )-\overline{\varkappa }(\delta )|.$$

Therefore, there exists an element $z\in F(\delta ,\overline{\varkappa }(\delta ))$ such that

$$|{\nu }_1(\delta )-z|\le \varrho (\delta )|\varkappa (\delta )-\overline{\varkappa }(\delta )|,\forall \delta \in [0,b].$$

Let us define the set-valued mapping $\mathcal{V}:[0,b]\to \mathcal{P}(\mathbb{R})$ by

$$\mathcal{V}(\delta )=\{z\in \mathbb{R}:|{\nu }_1(\delta )-z|\le \varrho (\delta )|\varkappa (\delta )-\overline{\varkappa }(\delta )\left|\right\}.$$

Since the intersection $\mathcal{V}(\delta )\cap F(\delta ,\overline{\varkappa }(\delta ))$ forms a measurable multi-function (as ensured by (Proposition III.4, p. 63, [32])), it admits a measurable selection ${\nu }_2(\delta ).$ This means ${\nu }_2(\delta )\in F(\delta ,\overline{\varkappa }(\delta ))$ and for all $\delta \in [0,b]$, we have $|{\nu }_1(\delta )-{\nu }_2(\delta )|\le \varrho (\delta )|\varkappa (\delta )-\overline{\varkappa }(\delta )|$. Accordingly, for every $\delta \in [0,b]$, it follows that

$$\begin{array}{ccc}\hfill h_2(\delta )& =& \left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}\mathsf{\Lambda }(b)-\mathsf{\Lambda }(\delta )\right){}^k{I}_{0^{+}}^{\vartheta -{\zeta }_k+k;\psi }{\nu }_2\left(b\right)-{\displaystyle \frac{{\lambda }_2}{{\lambda }_1+{\lambda }_2}}{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }{\nu }_2\left(b\right)+{}^k{I}_{0^{+}}^{\vartheta +\gamma ;\psi }{\nu }_2\left(\delta \right).\hfill \end{array}$$

In consequence, we obtain

$$\begin{array}{ccc}& & |h_1(\delta )-h_2(\delta )|\hfill \\ & \le & \left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right){\displaystyle \frac{|\mathsf{\Lambda }(b)|}{k{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+k)}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta -{\zeta }_k}{k}}}\left|{\nu }_2(b)-{\nu }_1(b)\right|d\varsigma \hfill \\ & & +{\displaystyle \frac{\left|{\lambda }_2\right|}{k\left|{\lambda }_1+{\lambda }_2\right|{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^b{\psi }^{\prime }(\varsigma ){\left(\psi (b)-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|{\nu }_2(b)-{\nu }_1(b)\right|d\varsigma \hfill \\ & & +{\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(\vartheta +\gamma )}}{\int }_0^{\delta }{\psi }^{\prime }(\varsigma ){\left(\psi (\delta )-\psi (\varsigma )\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}-1}\left|{\nu }_2(s)-{\nu }_1(s)\right|d\varsigma \hfill \\ & \le & {\left(\psi (b)-\psi (0)\right)}^{{\displaystyle \frac{\vartheta +\gamma }{k}}}\left({\displaystyle \frac{\left|{\lambda }_2\right|}{\left|{\lambda }_1+{\lambda }_2\right|}}+1\right)[{\displaystyle \frac{\left|{\lambda }_4\right|}{\left|{\lambda }_3+{\lambda }_4\right|}}{\displaystyle \frac{1}{{\mathsf{\Gamma }}_k({\zeta }_k+\gamma )}}{\displaystyle \frac{1}{{\mathsf{\Gamma }}_k(\vartheta -{\zeta }_k+2k)}}\hfill \\ & & +{\displaystyle \frac{1}{{\mathsf{\Gamma }}_k(\vartheta +\gamma +k)}}]\parallel \varrho \parallel \parallel x-\overline{\varkappa }\parallel \hfill \\ & =& \parallel \varrho \parallel \parallel x-\overline{\varkappa }\parallel {\mathsf{\Omega }}_1,\hfill \end{array}$$

which leads to

$$\parallel h_1-h_2\parallel \le {\mathsf{\Omega }}_1\parallel \varrho \parallel \parallel x-\overline{\varkappa }\parallel .$$

By interchanging the roles of $\varkappa$ and $\overline{\varkappa }$, we obtain

$$H_{\overline{d}}(N(\varkappa ),N(\overline{\varkappa }))\le {\mathsf{\Omega }}_1\parallel \varrho \parallel \parallel x-\overline{\varkappa }\parallel ,$$

demonstrating that the operator N is a contraction. As a result, the Covitz–Nadler fixed point theorem (Lemma 10) guarantees the existence of a fixed point $\varkappa$ for $N.$ This fixed point represents a solution to the sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo inclusion fractional boundary value problem given in equation (2). This concludes the proof. □

4.3. Illustrative Examples for Multi-Valued Case

Example 2.

Consider the following sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo fractional differential inclusion with non-separated boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {}^{\frac{13}{19},H}{D}_{0^{+}}^{\frac{11}{19},\frac{3}{5};\delta e^{\delta }}\left({}^{\frac{13}{19},C}{D}_{0^{+}}^{\frac{8}{19};\delta e^{\delta }}\varkappa \left(\delta \right)\right)\in F(\delta ,\varkappa (\delta )),\delta \in \left[0,\frac{25}{23}\right],}\hfill \\ \\ {\displaystyle \frac{3}{31}\varkappa \left(0\right)+\frac{5}{51}\varkappa \left(\frac{25}{23}\right)=0,\frac{7}{71}{}^{\frac{13}{19},C}{D}_{0^{+}}^{\frac{36}{95};\delta e^{\delta }}\varkappa \left(0\right)+\frac{9}{91}{}^{\frac{13}{19},C}{D}_{0^{+}}^{\frac{36}{95};\delta e^{\delta }}\varkappa \left(\frac{25}{23}\right)=0.}\hfill \end{array}\right.\end{array}$$

(27)

It is worth noting that several constants in the boundary value problem (27) are inherited from problem (21), with the exception of the function $\psi (\delta )=\delta e^{\delta }$. As a result, we obtain ${\mathsf{\Omega }}_1\approx 19.64313607$.

$(i)$ Assume that the given multi-function is presented as follows,

$$F(\delta ,\varkappa )=\left[{\displaystyle \frac{1}{\delta +12}}\left({\displaystyle \frac{{\varkappa }^{2568}}{8(1+|\varkappa {|}^{2567})}}+{\displaystyle \frac{1}{9}}{e}^{-2{\varkappa }^2}\right),{\displaystyle \frac{1}{\delta +5}}\left({\displaystyle \frac{{\varkappa }^{2568}}{4(1+|\varkappa {|}^{2567})}}+{\displaystyle \frac{1}{6}}{e}^{-\left|\varkappa \right|}\right)\right].$$

(28)

From the above, we can see that

$${\parallel F(\delta ,\varkappa )\parallel }_{\mathcal{P}}\le \left({\displaystyle \frac{1}{\delta +5}}\right)\left({\displaystyle \frac{1}{4}}\left|\varkappa \right|+{\displaystyle \frac{1}{6}}\right).$$

Selecting $q(\delta )=1/(\delta +5)$ (we have $\parallel q\parallel =1/5$) and $\chi (|\varkappa |)=(1/4)\left|\varkappa \right|+(1/6)$, it can be shown that there exists a constant $M>36.69584664$ such that the inequality in hypothesis $(A_3)$ in Theorem 8 holds. Therefore, by invoking Theorem 8, we conclude that the sequential fractional inclusion boundary value problem (27), involving the multi-function defined in (28), admits at least one solution on the interval $[0,25/23]$.

$(ii)$ If the multi-function $F(\delta ,\varkappa )$ is defined as follows,

$$F(\delta ,\varkappa )=\left[0,{\displaystyle \frac{1}{10{(\delta +2)}^2}}\left({\displaystyle \frac{2\left|\varkappa \right|+{\varkappa }^2}{1+\left|\varkappa \right|}}+{\displaystyle \frac{1}{2}}\right)\right],$$

(29)

then it can be demonstrated that

$$H_{\overline{d}}(F(\delta ,\varkappa ),F(\delta ,\overline{\varkappa }))\le {\displaystyle \frac{1}{5{(\delta +2)}^2}}|\varkappa -\overline{\varkappa }|,$$

for almost all $\delta \in [0,25/23]$ and $\varkappa ,\overline{\varkappa }\in \mathbb{R}.$ In addition, we can show the following relation

$$\overline{d}(0,F(\delta ,0))\le {\displaystyle \frac{1}{20{(\delta +2)}^2}}<{\displaystyle \frac{1}{5{(\delta +2)}^2}}:=\varrho (\delta ).$$

Then we have $\parallel \varrho \parallel =1/20$. Since the inequality ${\mathsf{\Omega }}_1\parallel \varrho \parallel \approx 0.9821568035<1$ holds, fulfilling condition $(B_2)$ of Theorem 9, and all other assumptions of the theorem are satisfied, it follows that the sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo fractional inclusion boundary value problem (27), involving the multi-function defined in (29), admits at least one solution on $[0,25/23]$.

5. Conclusions

In this study, we explored fractional boundary value problems involving the $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo types of fractional differential equations and inclusions, accompanied by non-separated boundary conditions. Initially, we addressed the case of single-valued mappings. By reformulating the original problem as a fixed-point problem, we employed several classical tools—namely, the Banach contraction principle, Schaefer’s fixed-point theorem, Krasnoselski$\breve{\mathrm{i}}$’s fixed-point theorem, and the Leray–Schauder alternative—to derive results on existence and uniqueness.

Subsequently, we turned our attention to the multi-valued setting, analyzing both convex- and nonconvex-valued multi-valued mappings. For the convex case, we utilized the Leray–Schauder nonlinear alternative tailored for multi-valued operators to prove an existence result. In the nonconvex scenario, we applied the Covitz–Nadler fixed-point theorem suited to contractive multi-valued mappings.

We also provided numerical examples to demonstrate the validity of the theoretical findings. Although the techniques applied are well-established, their implementation within the framework of $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo fractional differential problems with non-separated boundary conditions is novel. As far as we are aware, this work presents the first results specifically addressing boundary value problems of this nature. Therefore, our contributions are expected to advance the ongoing development in this emerging area of research.

In future studies, we intend to utilize this innovative approach to explore various types of boundary value problems involving nonzero initial conditions, as well as coupled systems of fractional differential equations that incorporate both Hilfer and Caputo fractional derivative operators.