(k,ψ)-Hilfer Nonlocal Integro-Multi-Point Boundary Value Problems for Fractional Differential Equations and Inclusions

Abstract

In this paper, we establish existence and uniqueness results for single-valued as well as multi-valued $(k,\psi )$-Hilfer boundary value problems of order in $(1,2],$ subject to nonlocal integro-multi-point boundary conditions. In the single-valued case, we use Banach and Krasnosel’skiĭ fixed point theorems as well as a Leray–Schauder nonlinear alternative to derive the existence and uniqueness results. For the multi-valued problem, we prove two existence results for the convex and non-convex nature of the multi-valued map involved in a problem by applying a Leray–Schauder nonlinear alternative for multi-valued maps, and a Covitz–Nadler fixed point theorem for multi-valued contractions, respectively. Numerical examples are presented for illustration of all the obtained results.

1. Introduction

Fractional calculus is concerned with integral and derivative operators of non-integer order. The tools of this branch of mathematical analysis are found to be of great help in improving the mathematical models associated with many engineering and scientific phenomena arising in a variety of fields such as physics [1], continuum mechanics [2], bioengineering [3], blood flow in small-lumen arterial vessels [4], financial economics [5], fractals [6], COVID-19 infection of epithelial cells [7], HIV/AIDS [8], chaotic synchronization [9], immune systems [10], ecology [11], vaccination for COVID-19 with the fear factor [12], etc. For an updated account of fractional differential equations, we refer the reader to the books [13,14,15,16,17,18,19,20,21]. Usually, fractional integral operators with different kernel functions appear in definitions of fractional derivatives. Many kinds of fractional derivative operators such as Riemann–Liouville, Caputo, Hadamard, Katugampola and Hilfer, etc., are proposed in the literature. It has been observed that certain fractional derivatives include the other types of fractional derivatives. For example, the generalized fractional derivative introduced by Katugampola in [22,23] includes Riemann–Liouville and Hadamard fractional derivatives, while the Hilfer fractional derivative [1] specializes to Riemann–Liouville and Caputo fractional derivatives. Another fractional derivative operator, known as the $\psi$-fractional derivative operator [24], unifies Caputo, Caputo–Hadamard and Caputo–Erdélyi–Kober fractional derivative operators. A wide class of fractional operators is covered by the $(k,\psi )$-Hilfer fractional derivative operator introduced in [25]; for details, see Remark 1.

In [25], the authors proved an existence and uniqueness result via Banach’s fixed point theorem, for the following nonlinear $(k,\psi )$-Hilfer initial value problem:

$$\left\{\begin{array}{c}{}^{k,H}{D}_{a+}^{\alpha ,\beta ;\psi }x\left(t\right)=f(t,x\left(t\right)),t\in (a,b],0<\alpha <k,0\le \beta \le 1,\hfill \\ {}^k{I}^{k-{\theta }_k;\psi }x\left(a\right)=x_a\in \mathbb{R},{\theta }_k=\alpha +\beta (k-\alpha ),\hfill \end{array}\right.$$

(1)

where ${}^{k,H}{D}^{\alpha ,\beta ;\psi }$ is the $(k,\psi )$-Hilfer type fractional derivative of order $\alpha$, $0<\alpha \le 1$ and parameter $\beta$, $0\le \beta \le 1$, and $f:[a,b]\times \mathbb{R}\to \mathbb{R}$ is a continuous function.

Ahmad et al. in [26] applied the fixed-point approach to study Hilfer type single-valued and multi-valued fractional sequential boundary value problems with integro-multistrip-multipoint boundary conditions given by

$$\left\{\begin{array}{c}\left({}^H{D}^{\alpha ,\beta }+k{}^H{D}^{\alpha -1,\beta }\right)x\left(t\right)=f(t,x\left(t\right)),\mathrm{and}\hfill \\ {\displaystyle \left({}^H{D}^{\alpha ,\beta }+k{}^H{D}^{\alpha -1,\beta }\right)x\left(t\right)\in F(t,x\left(t\right)),t\in J:=[a,b],}\hfill \\ {\displaystyle x^{\left(i\right)}\left(a\right)=0,i=0,1,2,\dots ,n-2,}\hfill \\ {\displaystyle {\int }_a^{b}x\left(s\right)ds=\sum _{i=2}^p{\lambda }_{i-1}{\int }_{{\eta }_{i-1}}^{{\eta }_i}x\left(s\right)ds+\sum _{j=1}^q{\mu }_{j}x\left({\rho }_j\right),}\hfill \end{array}\right.$$

(2)

where ${}^H{D}^{\alpha ,\beta }$ denotes the Hilfer type fractional derivative of order $\alpha ,$$n-1<\alpha \le n$ with $n\ge 3,$ and parameter $\beta ,0\le \beta \le 1,$$f\in C(J\times \mathbb{R},\mathbb{R})$, $a<{\eta }_1<{\eta }_2<\dots <{\eta }_p<{\rho }_1<{\rho }_2<\dots <{\rho }_q<b,$ and $k,{\lambda }_i,{\mu }_j>0,i=2,3,\dots ,p,$$j=1,2,\dots ,n$ with $p,q\in N$ and $F:J\times \mathbb{R}\to \mathcal{P}\left(\mathbb{R}\right)$ is a multi-valued map ($\mathcal{P}\left(\mathbb{R}\right)$ represents a family of all nonempty subsets of $\mathbb{R}$).

In [27], Sitho et al. proved the existence and uniqueness of solutions for a new class of $\psi$-Hilfer type single-valued and multi-valued fractional boundary value problems involving integral multi-point boundary conditions of the form

$$\left\{\begin{array}{c}{\displaystyle \left({}^H{D}^{\alpha ,\beta ;\psi }+k{}^H{D}^{\alpha -1,\beta ;\psi }\right)x\left(t\right)=f(t,x\left(t\right)),\mathrm{and}}\hfill \\ {\displaystyle \left({}^H{D}^{\alpha ,\beta ;\psi }+k{}^H{D}^{\alpha -1,\beta ;\psi }\right)x\left(t\right)\in F(t,x\left(t\right),t\in [a,b],}\hfill \\ {\displaystyle x\left(a\right)=0,x\left(b\right)=\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right)x\left(s\right)ds+\sum _{j=1}^m{\theta }_{j}x\left({\xi }_j\right).}\hfill \end{array}\right.$$

(3)

Here, ${}^H{D}^{\alpha ,\beta ;\psi }$ is the $\psi$-Hilfer type fractional derivative of order $\alpha$, $1<\alpha <2,$$0\le \beta \le 1$, $k\in \mathbb{R}$, $f\in C\left(\right[a,b]\times \mathbb{R},\mathbb{R})$, $a\ge 0,$${\mu }_i,{\theta }_j\in \mathbb{R}$, ${\eta }_i,{\xi }_j\in (a,b]$, $i=1,2,\dots ,n$, $j=1,2,\dots m$ and $\psi$ is an increasing positive function on $(a,b]$, which has on $(a,b)$ a continuous derivative ${\psi }^{\prime }\left(t\right)$, and $F:J\times \mathbb{R}\to \mathcal{P}\left(\mathbb{R}\right)$ is a multi-valued map.

Recently, Tariboon et al. initiated in [28] the study of $(k,\psi )$-Hilfer fractional boundary value problems for single and multi-valued differential equations of the form:

$$\left\{\begin{array}{c}{\displaystyle {}^{k,H}{D}^{\alpha ,\beta ;\psi }x\left(t\right)=f(t,x\left(t\right)),t\in (a,b],\mathrm{and}}\hfill \\ {\displaystyle {}^{k,H}{D}^{\alpha ,\beta ;\psi }x\left(t\right)\in F(t,x\left(t\right)),t\in (a,b],}\hfill \\ {\displaystyle x\left(a\right)=0,x\left(b\right)=\sum _{i=1}^m{\lambda }_{i}x\left({\xi }_i\right),}\hfill \end{array}\right.$$

(4)

where ${}^{k,H}{D}^{\alpha ,\beta ;\psi }$ is the $(k,\psi )$-Hilfer type fractional derivative of order $\alpha$, $1<\alpha <2,$$0\le \beta \le 1$, $k>0$, $f\in C\left(\right[a,b]\times \mathbb{R},\mathbb{R})$, ${\lambda }_i\in \mathbb{R},$ and $a<{\xi }_i<b,i=1,2,\dots ,m$ and $F:[a,b]\times \mathbb{R}\to \mathcal{P}\left(\mathbb{R}\right)$ is a multi-valued map. Here $\mathcal{P}\left(\mathbb{R}\right)$ denotes the family of all nonempty subsets of $\mathbb{R}$. Existence and uniqueness was established by using standard fixed point theorems. However, the literature on $(k,\psi )$-Hilfer type fractional boundary value problems is very limited at the moment.

Motivated by the aforementioned work (especially Remark 1 on page 5) and a recent article dealing with a $(k,\psi )$-Hilfer variational problem [29], our objective of the present work is to study the following $(k,\psi )$-Hilfer nonlocal integro-multi-point fractional boundary value problem:

$$\left\{\begin{array}{c}{\displaystyle {}^{k,H}{D}^{\alpha ,\beta ;\psi }x\left(t\right)=f(t,x\left(t\right)),t\in (a,b],}\hfill \\ {\displaystyle x\left(a\right)=0,x\left(b\right)=\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right)x\left(s\right)ds+\sum _{j=1}^m{\zeta }_j{}^k{I}^{{\varphi }_j;\psi }x\left(z_j\right),}\hfill \end{array}\right.$$

(5)

where ${}^{k,H}{D}^{\alpha ,\beta ;\psi }$ denotes the $(k,\psi )$-Hilfer type fractional derivative of order $\alpha$, $1<\alpha <2,$$0\le \beta \le 1$, $k>0$, $f\in C\left(\right[a,b]\times \mathbb{R},\mathbb{R})$, ${}^k{I}^{{\varphi }_j;\psi }$ is the $(k,\psi )$-Riemann–Liouville fractional integral of order ${\varphi }_j>0,$${\mu }_i,{\zeta }_j\in \mathbb{R},$ and $a<{\eta }_i,z_j<b,$$i=1,2,\dots ,n,j=1,2,\dots ,m.$ We establish existence and uniqueness results by using Banach’s and Krasnosel’skiĭ’s fixed point theorems, as well as the Laray–Schauder nonlinear alternative.

As a companion problem, we also investigate the multi-valued problem associated with (5) given by

$$\left\{\begin{array}{c}{\displaystyle {}^{k,H}{D}^{\alpha ,\beta ;\psi }x\left(t\right)\in F(t,x\left(t\right)),t\in (a,b],}\hfill \\ {\displaystyle x\left(a\right)=0,x\left(b\right)=\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right)x\left(s\right)ds+\sum _{j=1}^m{\zeta }_j{}^k{I}^{{\varphi }_j;\psi }x\left(z_j\right),}\hfill \end{array}\right.$$

(6)

where $F:[a,b]\times \mathbb{R}\to \mathcal{P}\left(\mathbb{R}\right)$ is a multi-valued map and the other parameters are the same as explained in problem (5), and $\mathcal{P}\left(\mathbb{R}\right)$ denotes the family of all nonempty subsets of $\mathbb{R}$. For the multi-valued problem (6), we establish the existence results for the cases of convex and non-convex multi-valued map $F(t,x(t\left)\right)$ by using Laray–Schauder nonlinear alternative for multi-valued maps, and the Covitz–Nadler fixed point theorem for multi-valued contractions, respectively. Here we mention that differential inclusions are found to be of great help in studying the dynamical systems and stochastic processes ([30,31,32]). Further applications of differential inclusions concerning climate control, queuing networks, etc., can be found in [33]. Now we enlist some applications of fractional order inclusions. In [34], the authors studied oscillation and nonoscillation for Caputo–Hadamard impulsive fractional differential inclusions. The problem of time fractional Navier–Stokes delay differential inclusions was discussed in [35]. The authors in [36] investigated a fractional differential inclusion with oscillatory potential.

The remainder of our paper is organized as follows. In Section 2, some definitions and lemmas related to our study are recalled. In Section 3, an auxiliary lemma is proved, which is used to transform the $(k,\psi )$-Hilfer type fractional nonlocal integro-multi-point boundary value problem (6) into an equivalent fixed point problem. The main existence and uniqueness results for the single-valued $(k,\psi )$-Hilfer type fractional nonlocal integro-multi-point boundary value problem (5) are presented in Section 4. In Section 5, we study the existence results for the multi-valued $(k,\psi )$-Hilfer type fractional nonlocal integro-multi-point boundary value problem (6). Section 6 contains illustrative examples for all the theoretical results obtained in the paper. Finally, Section 7 includes the conclusions of the paper.

2. Preliminaries

In this section we recall some definitions and lemmas that will be used throughout the paper.

Definition 1

([14]). Suppose that $f\in L^1([a,b],\mathbb{R})$, $a<b<+\infty .$ Then the Riemann–Liouville fractional integral is defined by

$$I_{a+}^{\alpha }f\left(t\right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_a^t{(t-u)}^{\alpha -1}f\left(u\right)du,t>a,$$

(7)

where $\mathsf{\Gamma }(·)$ denotes the classical Euler gamma function.

Definition 2

([14]). Let $f\in C\left(\right[a,b],\mathbb{R})$, $a<b<+\infty .$ Then the Riemann–Liouville fractional derivative operator of order $\alpha >0$ is defined by

$${}^{RL}{D}_{a+}^{\alpha }f\left(t\right)=D^n{I}_{a+}^{n-a}f\left(t\right)=\frac{1}{\mathsf{\Gamma }(n-\alpha )}\frac{d^n}{d{t}^n}{\int }_a^t{(t-u)}^{n-\alpha -1}f\left(u\right)du,t>a,$$

(8)

where $n-1<\alpha \le n$ and $n\in \mathbb{N}.$

Definition 3

([14]). Let $f\in C^n([a,b],\mathbb{R})$, $a<b<+\infty .$ Then the Caputo fractional derivative operator of order $\alpha >0$ is defined by

$${}^C{D}_{a+}^{\alpha }f\left(t\right)=I_{a+}^{n-a}{D}^{n}f\left(t\right)=\frac{1}{\mathsf{\Gamma }(n-\alpha )}{\int }_a^t{(t-u)}^{n-\alpha -1}{f}^{\left(n\right)}\left(u\right)du,t>a,$$

(9)

where $n-1<\alpha \le n$ and $n\in \mathbb{N}.$

Definition 4

([37]). Let $\mathfrak{h}\in L^1([a,b],\mathbb{R})$ and $k,\alpha \in {\mathbb{R}}^{+}.$ Then the k-Riemann–Liouville fractional derivative of order α of the function $\mathfrak{h}$ is given by

$${}^k{I}_{a+}^{\alpha }\mathfrak{h}\left(t\right)=\frac{1}{k{\mathsf{\Gamma }}_k\left(\alpha \right)}{\int }_a^t{(t-u)}^{\frac{\alpha }{k}-1}\mathfrak{h}\left(u\right)du,$$

(10)

where ${\mathsf{\Gamma }}_k$ is the k-Gamma function for $z\in \mathbb{C}$ with $\Re \left(z\right)>0$ and $k\in \mathbb{R},k>0$ which is defined in [38] by

$${\mathsf{\Gamma }}_k\left(z\right)={\int }_0^{\infty }{s}^{z-1}{e}^{-\frac{s^k}{k}}ds.$$

The following relations are well known.

$$\mathsf{\Gamma }\left(\theta \right)=\underset{k\to 1}{lim}{\mathsf{\Gamma }}_k\left(\theta \right),{\mathsf{\Gamma }}_k\left(\theta \right)=k^{\frac{\theta }{k}-1}\mathsf{\Gamma }\left(\frac{\theta }{k}\right)\mathrm{and}{\mathsf{\Gamma }}_k(\theta +k)=\theta {\mathsf{\Gamma }}_k\left(\theta \right).$$

Definition 5

([39]). Let $\mathfrak{h}\in L^1([a,b],\mathbb{R})$ and $k,\alpha \in {\mathbb{R}}^{+}.$ Then the k-Riemann–Liouville fractional derivative of order α of the function $\mathfrak{h}$ is given by

$${}^{k,RL}{D}_{a+}^{\alpha }\mathfrak{h}\left(t\right)={\left(k\frac{d}{dt}\right)}^n{}^k{I}^{nk-{\alpha }_{a+}}\mathfrak{h}\left(t\right),n=⌈\frac{\alpha }{k}⌉,$$

(11)

where $⌈\frac{\alpha }{k}⌉$ is the ceiling function of $\frac{\alpha }{k}.$

Definition 6

([14]). Let $\mathfrak{h}\in L^1([a,b],\mathbb{R})$ and an increasing function $\psi :[a,b]\to \mathbb{R}$ with ${\psi }^{\prime }\left(\theta \right)\ne 0$ for all $\theta \in [a,b].$ Then the ψ-Riemann–Liouville fractional integral of the function $\mathfrak{h}$ is given by

$$I^{\alpha ;\psi }\mathfrak{h}\left(t\right)=\frac{1}{{\mathsf{\Gamma }}_k\left(\alpha \right)}{\int }_a^t{\psi }^{\prime }\left(u\right){(\psi \left(t\right)-\psi \left(u\right))}^{\alpha -1}\mathfrak{h}\left(u\right)du.$$

(12)

Definition 7

([14]). Let $n-1<\alpha \le n,$$\psi \in C^n([a,b],\mathbb{R}),$${\psi }^{\prime }\left(\theta \right)\ne 0,\theta \in [a,b],$ and $\mathfrak{h}\in C\left(\right[a,b],\mathbb{R}).$ Then the ψ-Riemann–Liouville fractional derivative of the function $\mathfrak{h}$ of order α is given by

$${}^{RL}{D}^{\alpha ;\psi }\mathfrak{h}\left(t\right)={\left(\frac{1}{{\psi }^{\prime }\left(t\right)}\frac{d}{dt}\right)}^n{I}_{a+}^{n-\alpha ;\psi }\mathfrak{h}\left(t\right),$$

(13)

Definition 8

([24]). Let $n-1<\alpha \le n,\psi \in C^n([a,b],\mathbb{R}),$${\psi }^{\prime }\left(\theta \right)\ne 0,\theta \in [a,b],$ and $\mathfrak{h}\in C\left(\right[a,b],\mathbb{R}).$ Then the ψ-Caputo fractional derivative of the function $\mathfrak{h}$ of order α is given by

$${}^C{D}^{\alpha ;\psi }\mathfrak{h}\left(t\right)=I_{a+}^{n-\alpha ;\psi }{\left(\frac{1}{{\psi }^{\prime }\left(t\right)}\frac{d}{dt}\right)}^n\mathfrak{h}\left(t\right).$$

(14)

Definition 9

([40]). Let $n-1<\alpha \le n,$$\psi \in C^n([a,b],\mathbb{R}),$${\psi }^{\prime }\left(\theta \right)\ne 0,\theta \in [a,b],$ and $\mathfrak{h}\in C\left(\right[a,b],\mathbb{R}).$ Then the ψ-Hilfer fractional derivative of the function $\mathfrak{h}\in C\left(\right[a,b],\mathbb{R})$ of order $\alpha \in (n-1,n]$ and type $\beta \in [0,1]$ is defined by

$${}^H{D}^{\alpha ,\beta ;\psi }\mathfrak{h}\left(t\right)=I_{a+}^{\beta (n-\alpha );\psi }{\left(\frac{1}{{\psi }^{\prime }\left(t\right)}\frac{d}{dt}\right)}^n{I}_{a+}^{(1-\beta )(n-\alpha );\psi }\mathfrak{h}\left(t\right).$$

(15)

Definition 10

([41]). Let $\mathfrak{h}\in L^1([a,b],\mathbb{R})$ and $k>0.$ Then the $(k,\psi )$-Riemann–Liouville fractional integral of order $\alpha >0$ ($\alpha \in \mathbb{R}$) of the function $\mathfrak{h}$ is given by

$${}^k{I}_{a+}^{\alpha ;\psi }\mathfrak{h}\left(t\right)=\frac{1}{k{\mathsf{\Gamma }}_k\left(\alpha \right)}{\int }_a^t{\psi }^{\prime }\left(u\right){(\psi \left(t\right)-\psi \left(u\right))}^{\frac{\alpha }{k}-1}\mathfrak{h}\left(u\right)du.$$

(16)

Definition 11

([25]). Let $\alpha ,k\in {\mathbb{R}}^{+}=(0,\infty ),$$\beta \in [0,1],$$\psi \in C^n([a,b],\mathbb{R}),$${\psi }^{\prime }\left(\theta \right)\ne 0,\theta \in [a,b]$ and $\mathfrak{h}\in C^n([a,b],\mathbb{R}).$ Then the $(k,\psi )$-Hilfer fractional derivative of the function $\mathfrak{h}$ of order α and type β is defined by

$${}^{k,H}{D}^{\alpha ,\beta ;\psi }\mathfrak{h}\left(t\right)={}^k{I}_{a+}^{\beta (nk-\alpha );\psi }{\left(\frac{k}{{\psi }^{\prime }\left(t\right)}\frac{d}{dt}\right)}^n{}^k{I}_{a+}^{(1-\beta )(nk-\alpha );\psi }\mathfrak{h}\left(t\right),n=⌈\frac{\alpha }{k}⌉.$$

(17)

Remark 1.

Note that:

• For $\beta =0,$ (17) reduces to $(k,\psi )$-Riemann–Liouville fractional derivative operator, while for $\beta =1,$ (17) reduces to $(k,\psi )$-Caputo fractional derivative operator [25].
• If $\psi \left(t\right)=t^{\rho }$, then (17) reduces to k-Hilfer–Katugampola fractional derivative operator, while if $\psi \left(t\right)=logt,$ then (17) reduces to k-Hilfer–Hadamard fractional derivative operator.

Remark 2.

The $(k,\psi )$-Hilfer fractional derivative can be expressed in terms of a $(k,\psi )$-Riemann–Liouville fractional derivative as

$$\begin{array}{ccc}\hfill {}^{k,H}{D}^{\alpha ,\beta ;\psi }\mathfrak{h}\left(t\right)& =& {}^k{I}_{a+}^{{\theta }_k-\alpha ;\psi }{\left(\frac{k}{{\psi }^{\prime }\left(t\right)}\frac{d}{dt}\right)}^n{}^k{I}_{a+}^{nk-{\theta }_k;\psi }\mathfrak{h}\left(t\right)\hfill \\ & =& {}^k{I}_{a+}^{{\theta }_k-\alpha ;\psi }\left({}^{k,RL}{D}^{{\theta }_k;\psi }\mathfrak{h}\right)\left(t\right),\hfill \end{array}$$

where $(1-\beta )(nk-\alpha )=nk-{\theta }_k.$ Note for $\beta \in [0,1]$ and $n-1<\frac{\alpha }{k}\le n,$ we have $n-1<\frac{{\theta }_k}{k}\le n.$

Now we recall some useful lemmas, which are used in this paper.

Lemma 1

([25]). Let $\mu ,k\in {\mathbb{R}}^{+}=(0,\infty )$ and $n=⌈\frac{\mu }{k}⌉.$ Assume that $\mathfrak{h}\in C^n([a,b],\mathbb{R})$ and ${}^k{I}_{a+}^{nk-\mu ;\psi }\mathfrak{h}\in C^n([a,b],\mathbb{R}).$ Then

$${}^k{I}^{\mu ;\psi }\left({}^{k,RL}{D}^{\mu ;\psi }\mathfrak{h}\left(t\right)\right)=\mathfrak{h}\left(t\right)-\sum _{j=1}^n\frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{\mu }{k}-j}}{{\mathsf{\Gamma }}_k(\mu -jk+k)}{\left[{\left(\frac{k}{{\psi }^{\prime }\left(t\right)}\frac{d}{dt}\right)}^{n-j}{}^k{I}_{a+}^{nk-\mu ;\psi }\mathfrak{h}\left(t\right)\right]}_{z=a}.$$

Lemma 2

([25]). Let $\alpha ,k\in {\mathbb{R}}^{+}=(0,\infty )$ with $\alpha <k,$$\beta \in [0,1]$ and ${\theta }_k=\alpha +\beta (k-\alpha ).$ Then

$${}^k{I}^{{\theta }_k;\psi }\left({}^{k,RL}{D}^{{\theta }_k;\psi }\mathfrak{h}\right)\left(t\right)={}^k{I}^{\alpha ;\psi }\left({}^{k,H}{D}^{\alpha ,\beta ;\psi }\mathfrak{h}\right)\left(t\right),\mathfrak{h}\in C^n([a,b],\mathbb{R}).$$

Lemma 3

([25]). Let $\zeta ,k\in {\mathbb{R}}^{+}=(0,\infty )$ and $\eta \in \mathbb{R}$ such that $\frac{\eta }{k}>-1.$ Then

$$\left(i\right){}^k{\mathcal{I}}^{\zeta ,\psi }{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{\eta }{k}}=\frac{{\mathsf{\Gamma }}_k(\eta +k)}{{\mathsf{\Gamma }}_k(\eta +k+\zeta )}{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{\eta +\zeta }{k}}.$$

$$\left(ii\right){}^k{\mathcal{D}}^{\zeta ,\psi }{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{\eta }{k}}=\frac{{\mathsf{\Gamma }}_k(\eta +k)}{{\mathsf{\Gamma }}_k(\eta +k-\zeta )}{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{\eta -\zeta }{k}}.$$

3. An Auxiliary Result

We prove the following lemma, which concerns a linear variant of the $(k,\psi )$-Hilfer nonlocal integral fractional boundary value problem (5) and is used in transforming the nonlinear $(k,\psi )$-Hilfer type nonlocal integral multi-point fractional boundary value problem (5) into a fixed point problem.

Lemma 4.

Assume that $a<b,$$k>0,$$1<\alpha \le 2,$$\beta \in [0,1],$${\theta }_k=\alpha +\beta (2k-\alpha ),$$\mathfrak{g}\in C^2([a,b],\mathbb{R})$ and

$$\begin{array}{cc}\mathsf{\Lambda }:\hfill & {\displaystyle =\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{{\mathsf{\Gamma }}_k\left({\theta }_k\right)}-\frac{k}{{\mathsf{\Gamma }}_k({\theta }_k+k)}\sum _{i=1}^n{\mu }_i{(\psi \left({\eta }_i\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}}}\hfill \\ \hfill & {\displaystyle -\sum _{j=1}^m\frac{{\zeta }_j}{{\mathsf{\Gamma }}_k({\theta }_k+{\varphi }_j)}{(\psi \left(z_j\right)-\psi \left(a\right))}^{\frac{{\theta }_k+{\varphi }_j}{k}-1}\ne 0.}\hfill \end{array}$$

(18)

Then the function $x\in C\left(\right[a,b],\mathbb{R})$ is a solution of the $(k,\psi )$-Hilfer type fractional nonlocal integro-multi-point boundary value problem

$$\left\{\begin{array}{c}{\displaystyle {}^{k,H}{D}^{\alpha ,\beta ;\psi }x\left(t\right)=\mathfrak{g}\left(t\right),t\in (a,b],}\hfill \\ {\displaystyle x\left(a\right)=0,x\left(b\right)=\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right)x\left(s\right)ds+\sum _{j=1}^m{\zeta }_j{}^k{I}^{{\varphi }_j;\psi }x\left(z_j\right),}\hfill \end{array}\right.$$

(19)

if and only if

$$\begin{array}{cc}x\left(t\right)\hfill & {\displaystyle ={}^k{I}^{\alpha ;\psi }\mathfrak{g}\left(t\right)}\hfill \\ \hfill & {\displaystyle +\frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\mathsf{\Lambda }{\mathsf{\Gamma }}_k\left({\theta }_k\right)}\left[\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }\mathfrak{g}\left(s\right)ds+\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }\mathfrak{g}\left(z_j\right)-{}^k{I}^{\alpha ;\psi }\mathfrak{g}\left(b\right)\right].}\hfill \end{array}$$

(20)

Proof. 

In the first stage, we proceed to assume that x is a solution of the $(k,\psi )$-Hilfer type boundary value problem (19). We take fractional integral ${}^k{I}^{\alpha ;\psi }$ on both sides of equation in (19). We have, by using Lemmas 1 and 2,

$$\begin{array}{ccc}\hfill {}^k{I}^{\alpha ;\psi }\left({}^{k,H}{D}^{\alpha ,\beta ;\psi }x\right)\left(t\right)& =& {}^k{I}^{{\theta }_k;\psi }\left({}^{k,RL}{D}^{{\theta }_k;\psi }x\right)\left(t\right)\hfill \\ & =& x\left(t\right)-\frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{{\mathsf{\Gamma }}_k\left({\theta }_k\right)}{\left[\left(\frac{k}{{\psi }^{\prime }\left(t\right)}\frac{d}{dt}\right){}^k{I}^{2k-{\theta }_k;\psi }x\left(t\right)\right]}_{w=a}\hfill \\ & & -\frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-2}}{{\mathsf{\Gamma }}_k({\theta }_k-k)}{\left[{}^k{I}^{2k-{\theta }_k;\psi }x\left(t\right)\right]}_{w=a}.\hfill \end{array}$$

Consequently,

$$x\left(t\right)={}^k{I}^{\alpha ;\psi }\mathfrak{g}\left(t\right)+c_0\frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{{\mathsf{\Gamma }}_k\left({\theta }_k\right)}+c_1\frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-2}}{{\mathsf{\Gamma }}_k({\theta }_k-k)},$$

(21)

where

$$c_0={\left[\left(\frac{k}{{\psi }^{\prime }\left(t\right)}\frac{d}{dt}\right){}^k{I}^{2k-{\theta }_k;\psi }x\left(t\right)\right]}_{w=a},c_1={\left[{}^k{I}^{2k-{\theta }_k;\psi }x\left(t\right)\right]}_{w=a}.$$

From the first boundary condition $x\left(a\right)=0$, we get $c_1=0,$ since $\frac{{\theta }_k}{k}-2<0$ by Remark 2.

By using Lemma 3 we get

$$\begin{array}{ccc}\hfill & & {\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right)[{}^k{I}^{\alpha ;\psi }\mathfrak{g}\left(s\right)+c_0\frac{{(\psi \left(s\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{{\mathsf{\Gamma }}_k\left({\theta }_k\right)}]ds\hfill \\ & =& {\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }\mathfrak{g}\left(s\right)ds+\frac{k{c}_0}{{\mathsf{\Gamma }}_k({\theta }_k+k)}{(\psi \left({\eta }_i\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}},\hfill \end{array}$$

(22)

and

$$\begin{array}{ccc}\hfill & & {\displaystyle {}^k{I}^{{\varphi }_j;\psi }[{}^k{I}^{\alpha ;\psi }\mathfrak{g}\left(t\right)+c_0\frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{{\mathsf{\Gamma }}_k\left({\theta }_k\right)}]}\hfill \\ & =& {\displaystyle {}^k{I}^{{\varphi }_j+\alpha ;\psi }g\left(t\right)+c_0\frac{1}{{\mathsf{\Gamma }}_k({\theta }_k+{\varphi }_j)}{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k+{\varphi }_j}{k}-1}.}\hfill \end{array}$$

(23)

From (22) and (23), and the second boundary condition $x\left(b\right)={\displaystyle \sum _{i=1}^n}{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right)x\left(s\right)ds+{\displaystyle {\sum }_{j=1}^m}{\zeta }_j{}^k{I}^{{\varphi }_j;\psi }x\left(z_j\right),$ we found

$$c_0=\frac{1}{\mathsf{\Lambda }}\left[\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }\mathfrak{g}\left(s\right)ds+\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }g\left(z_j\right)-{}^k{I}^{\alpha ;\psi }\mathfrak{g}\left(b\right)\right].$$

Inserting the values of $c_0$ and $c_1$ in (21), we get the solution (20). The converse follows easily by direct computation. This finishes the proof. □

4. The Single-Valued Problem

We denote by $C\left(\right[a,b],\mathbb{R})$ the Banach space of all continuous functions from $[a,b]$ to $\mathbb{R}$ endowed with the sup-norm $\parallel x\parallel ={sup}_{t\in [a,b]}\left|x\left(t\right)\right|.$ By Lemma 4, we define an operator $\mathcal{A}:C\left(\right[a,b],\mathbb{R})\to C\left(\right[a,b],\mathbb{R})$ by

$$\begin{array}{cc}\left(\mathcal{A}x\right)\left(t\right)\hfill & {\displaystyle ={}^k{I}^{\alpha ;\psi }f(t,x\left(t\right))+\frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\mathsf{\Lambda }{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }f(s,x\left(s\right))ds}\hfill \\ \hfill & {\displaystyle +\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }f(z_j,x\left(z_j\right))-{}^k{I}^{\alpha ;\psi }f(b,x\left(b\right))].}\hfill \end{array}$$

(24)

We notice that the fixed points of $\mathcal{A}$ are solutions of the $(k,\psi )$-Hilfer type fractional nonlocal integro-multi-point boundary value problem (6).

For convenience, we put:

$$\begin{array}{ccc}\hfill \mathfrak{G}& =& \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n\left|{\mu }_i\right|\frac{k{(\psi \left({\eta }_i\right)-\psi \left(a\right))}^{\frac{\alpha }{k}+1}}{{\mathsf{\Gamma }}_k(\alpha +2k)}\hfill \\ & & +\sum _{j=1}^m\left|{\zeta }_j\right|\frac{{(\psi \left(z_j\right)-\psi \left(a\right))}^{\frac{\alpha +{\varphi }_j}{k}}}{{\mathsf{\Gamma }}_k(\alpha +{\varphi }_j+k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}].\hfill \end{array}$$

(25)

4.1. Existence of a Unique Solution

By using Banach’s contraction mapping principle [42], we prove the existence and uniqueness result for the $(k,\psi )$-Hilfer nonlocal integro-multi-point fractional boundary value problem (5).

Theorem 1.

Assume that:

$\left(H_1\right)$

$\left|f\right(t,x)-f(t,y\left)\right|\le \mathfrak{L}|x-y|,\mathfrak{L}>0for eacht\in [a,b]andx,y\in \mathbb{R}.$

Then, if

$$\mathfrak{L}\mathfrak{G}<1,$$

(26)

where $\mathfrak{G}$ is given in $\left(25\right),$ the $(k,\psi )$-Hilfer type nonlocal integro-multi-point fractional boundary value problem (5) has a unique solution on $[a,b].$

Proof. 

Using the operator $\mathcal{A}$ defined in $\left(24\right)$, we transform the $(k,\psi )$-Hilfer type nonlocal integro-multi-point fractional boundary value problem (5) into a fixed point problem, and we will show that it has a unique fixed point.

We let ${sup}_{t\in [a,b]}\left|f(t,0)\right|=\mathfrak{M}<\infty$, and choose

$$r\ge \frac{{\displaystyle \mathfrak{M}\mathfrak{G}}}{1-\mathfrak{L}\mathfrak{G}}.$$

(27)

Let $B_r=\{x\in C([a,b],\mathbb{R}):\parallel x\parallel \le r\}$. In the first step, we will show that $\mathcal{A}{B}_r\subset B_r.$ We have, using $\left(H_1\right),$ that

$$\begin{array}{ccc}\hfill \left|f\right(t,x\left(t\right)\left)\right|& \le & \left|f\right(t,x\left(t\right))-f(t,0\left)\right|+\left|f\right(t,0\left)\right|\hfill \\ & \le & \mathfrak{L}\left|x\right(t\left)\right|+\mathfrak{M}\le \mathfrak{L}\parallel x\parallel +\mathfrak{M}\le \mathfrak{L}r+\mathfrak{M}.\hfill \end{array}$$

For any $x\in B_r$, we have

$$\begin{array}{ccc}\hfill \left|\right(\mathcal{A}x\left)\right(t\left)\right|& \le & \underset{t\in [a,b]}{sup}\{\frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n|{\mu }_i|{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }\left|f(s,x\left(s\right))\right|ds\hfill \\ & & +\sum _{j=1}^m|{\zeta }_j|{}^k{I}^{\alpha +{\varphi }_j;\psi }|f(z_j,x\left(z_j\right))|+{}^k{I}^{\alpha ;\psi }|f(b,x\left(b\right))\left|\right]+{}^k{I}^{\alpha ;\psi }\left|f(t,x\left(t\right))\right|\}\hfill \\ & \le & {}^k{I}^{\alpha ;\psi }\left(\right|f(t,x\left(t\right))-f(t,0)|+|f(t,0)\left|\right)\hfill \\ & & +\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n|{\mu }_i|{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }\left(\right|f(s,x\left(s\right))-f(s,0)|+|f(s,0)\left|\right)ds\hfill \\ & & +\sum _{j=1}^m|{\zeta }_j|{}^k{I}^{\alpha +{\varphi }_j;\psi }\left(\right|f(z_j,x\left(z_j\right))-f(z_j,0)|+|f(z_j,0)\left|\right)\hfill \\ & & {+}^k{I}^{\alpha ;\psi }\left(\right|f(b,x\left(b\right))-f(b,0)|+|f(b,0)\left|\right))\hfill \\ & \le & \{\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n\left|{\mu }_i\right|\frac{k{(\psi \left({\eta }_i\right)-\psi \left(a\right))}^{\frac{\alpha }{k}+1}}{{\mathsf{\Gamma }}_k(\alpha +2k)}\hfill \\ & & +\sum _{j=1}^m|{\zeta }_j|\frac{{(\psi \left(z_j\right)-\psi \left(a\right))}^{\frac{\alpha +{\varphi }_j}{k}}}{{\mathsf{\Gamma }}_k(\alpha +{\varphi }_j+k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}\left]\right\}(\mathfrak{L}\parallel x\parallel +\mathfrak{M})\hfill \\ & \le & (\mathfrak{L}r+\mathfrak{M})\mathfrak{G}\le r.\hfill \end{array}$$

Consequently, $\parallel \mathcal{A}x\parallel \le r$ and thus $\mathcal{A}{B}_r\subset B_r$.

Now we will show that $\mathcal{A}$ is a contraction. For $t\in [a,b]$ and $x,y\in C\left(\right[a,b],\mathbb{R}),$ we have

$$\begin{array}{ccc}& & \left|\right(\mathcal{A}x\left)\right(t)-(\mathcal{A}y\left)\right(t\left)\right|\hfill \\ & \le & {}^k{I}^{\alpha ;\psi }|f(t,x\left(t\right))-f(t,y\left(t\right))|\hfill \\ & & +\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{|\mathsf{\Lambda }|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n|{\mu }_i|{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }|f(s,x\left(s\right))-f(s,y\left(s\right)|ds\hfill \\ & & +\sum _{j=1}^m|{\zeta }_j|{}^k{I}^{\alpha +{\varphi }_j;\psi }|f(z_j,x\left(z_j\right))-f(z_j,y\left(z_j\right))|{+}^k{I}^{\alpha ;\psi }|f(b,x\left(b\right))-f(b,y\left(b\right))|)\hfill \\ & \le & \{\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n|{\mu }_i|\frac{k{(\psi \left({\eta }_i\right)-\psi \left(a\right))}^{\frac{\alpha }{k}+1}}{{\mathsf{\Gamma }}_k(\alpha +2k)}\hfill \\ & & +\sum _{j=1}^m|{\zeta }_j|\frac{{(\psi \left(z_j\right)-\psi \left(a\right))}^{\frac{\alpha +{\varphi }_j}{k}}}{{\mathsf{\Gamma }}_k(\alpha +{\varphi }_j+k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}\left]\right\}\mathfrak{L}\parallel x-y\parallel \hfill \\ & =& \mathfrak{L}\mathfrak{G}\parallel x-y\parallel .\hfill \end{array}$$

Hence $\parallel \mathcal{A}x-\mathcal{A}y\parallel \le \mathfrak{L}\mathfrak{G}\parallel x-y\parallel$ and we deduce, by $\mathfrak{L}\mathfrak{G}<1,$ that $\mathcal{A}$ is a contraction mapping. Consequently, the operator $\mathcal{A}$ has a unique fixed point, by the Banach’s contraction mapping principle, which is the unique solution of $(k,\psi )$-Hilfer type nonlocal integral multi-point fractional boundary value problem (5). The proof is completed. □

4.2. Existence Results

Now we will prove existence results for the $(k,\psi )$-Hilfer type nonlocal integral multi-point fractional boundary value problem (5) via Krasnosel’skiĭ’s fixed point theorem [43] and nonlinear alternative of Leray–Schauder type [44].

Theorem 2.

Assume that $f:[a,b]\times \mathbb{R}\to \mathbb{R}$ is a continuous function satisfying $\left(H_1\right).$ In addition, we suppose that:

$\left(H_2\right)$

$\left|f\right(t,x\left)\right|\le \varpi \left(t\right),\forall (t,x)\in [a,b]\times \mathbb{R}$, and $\varpi \in C([a,b],{\mathbb{R}}^{+})$.

Then, if ${\mathfrak{G}}_1\mathfrak{L}<1$, where

$${\mathfrak{G}}_1:=\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}\left[\sum _{i=1}^n|{\mu }_i|\frac{k{(\psi \left({\eta }_i\right)-\psi \left(a\right))}^{\frac{\alpha }{k}+1}}{{\mathsf{\Gamma }}_k(\alpha +2k)}+\sum _{j=1}^m\left|{\zeta }_j\right|\frac{{(\psi \left(z_j\right)-\psi \left(a\right))}^{\frac{\alpha +{\varphi }_j}{k}}}{{\mathsf{\Gamma }}_k(\alpha +{\varphi }_j+k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}\right],$$

(28)

the $(k,\psi )$-Hilfer type nonlocal integro-multi-point fractional boundary value problem (5) has at least one solution on $[a,b].$

Proof. 

Set ${sup}_{t\in [a,b]}\varpi \left(t\right)=\parallel \varpi \parallel$ and $B_{\rho }=\{x\in C([a,b],\mathbb{R}):\parallel x\parallel \le \rho \},$ with $\rho \ge \parallel \varpi \parallel \mathfrak{G}.$ We define on $B_{\rho }$ two operators ${\mathcal{A}}_1$, ${\mathcal{A}}_2$ by

$$\begin{array}{ccc}\hfill {\mathcal{A}}_{1}x\left(t\right)& =& {}^k{I}^{\alpha ;\psi }f(t,x\left(t\right)),t\in [a,b],\hfill \\ \hfill \left({\mathcal{A}}_{2}x\right)\left(t\right)& =& \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\mathsf{\Lambda }{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }f(s,x\left(s\right))ds\hfill \\ & & +\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }f(z_j,x\left(z_j\right))-{}^k{I}^{\alpha ;\psi }f(b,x\left(b\right))],t\in [a,b].\hfill \end{array}$$

For any $x,y\in B_{\rho }$, we have

$$\begin{array}{ccc}& & |\left({\mathcal{A}}_{1}x\right)\left(t\right)+\left({\mathcal{A}}_{2}y\right)\left(t\right)|\hfill \\ & \le & \underset{t\in [a,b]}{sup}\left\{\frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}\right[\sum _{i=1}^n|{\mu }_i|{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }\left|f(s,y\left(s\right))\right|ds\hfill \\ & & +\sum _{j=1}^m|{\zeta }_j|{}^k{I}^{\alpha +{\varphi }_j;\psi }|f(z_j,y\left(z_j\right))|+{}^k{I}^{\alpha ;\psi }|f(b,y\left(b\right))\left|\right]+{}^k{I}^{\alpha ;\psi }\left|f(t,x\left(t\right))\right|\}\hfill \\ & \le & \{\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n\left|{\mu }_i\right|\frac{k{(\psi \left({\eta }_i\right)-\psi \left(a\right))}^{\frac{\alpha }{k}+1}}{{\mathsf{\Gamma }}_k(\alpha +2k)}\hfill \\ & & +\sum _{j=1}^m|{\zeta }_j|\frac{{(\psi \left(z_j\right)-\psi \left(a\right))}^{\frac{\alpha +{\varphi }_j}{k}}}{{\mathsf{\Gamma }}_k(\alpha +{\varphi }_j+k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}\left]\right\}\parallel \varpi \parallel \hfill \\ & =& \mathfrak{G}\parallel \varpi \parallel \le \rho .\hfill \end{array}$$

Consequently, $\parallel \left({\mathcal{A}}_{1}x\right)+\left({\mathcal{A}}_{2}y\right)\parallel \le \rho ,$ which shows that ${\mathcal{A}}_{1}x+{\mathcal{A}}_{2}y\in B_{\rho }.$

We can prove easily that ${\mathcal{A}}_2$ is a contraction mapping by using (28).

The operator ${\mathcal{A}}_1$ is continuous, since f is continuous. Moreover, ${\mathcal{A}}_1$ is uniformly bounded on $B_{\rho }$ as

$$\parallel {\mathcal{A}}_{1}x\parallel \le \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}\parallel \varpi \parallel .$$

Now we will prove the compactness of the operator ${\mathcal{A}}_1.$ For $w_1,w_2\in [a,b]$ with $w_1<w_2$ we have

$$\begin{array}{ccc}& & |\left({\mathcal{A}}_{1}x\right)\left(t_2\right)-\left({\mathcal{A}}_{1}x\right)\left(t_1\right)|\hfill \\ & \le & \frac{1}{{\mathsf{\Gamma }}_k\left(\alpha \right)}|{\int }_a^{w_1}{\psi }^{\prime }\left(s\right)[{(\psi \left(t_2\right)-\psi \left(s\right))}^{\frac{\alpha }{k}-1}-{(\psi \left(t_1\right)-\psi \left(s\right))}^{\frac{\alpha }{k}-1}]f(s,x\left(s\right))ds\hfill \\ & & +{\int }_{w_1}^{w_2}{\psi }^{\prime }\left(s\right){(\psi \left(t_2\right)-\psi \left(s\right))}^{\frac{\alpha }{k}-1}f(s,x\left(s\right))ds|\hfill \\ & \le & \frac{\parallel \varpi \parallel }{{\mathsf{\Gamma }}_k(\alpha +k)}[2{(\psi \left(t_2\right)-\psi \left(t_1\right))}^{\frac{\alpha }{k}}+|{(\psi \left(t_2\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}-{(\psi \left(t_1\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}\left|\right],\hfill \end{array}$$

which tends to zero as $w_2-w_1\to 0,$ independently of $x.$ Thus, ${\mathcal{A}}_1$ is equicontinuous. So ${\mathcal{A}}_1$ is completely continuous by the Arzelá–Ascoli theorem. Hence, the $(k,\psi )$-Hilfer type nonlocal intego-multi-point fractional boundary value problem (5) has at least one solution on $[a,b],$ by Krasnosel’skiĭ’s fixed point theorem. The proof is completed. □

Theorem 3.

For the continuous function $f:[a,b]\times \mathbb{R}\to \mathbb{R}$, suppose that:

($H_3$)

there exists a continuous and nondecreasing function $\chi :[0,\infty )\to (0,\infty )$ and a continuous positive function σ such that

$$\left|f\right(t,u\left)\right|\le \sigma \left(t\right)\chi \left(\right|u\left|\right)\mathrm{for}\mathrm{each}(t,u)\in [a,b]\times \mathbb{R};$$

($H_4$)

there exists a constant $\mathfrak{K}>0$ such that

$$\frac{\mathfrak{K}}{\chi \left(\mathfrak{K}\right)\parallel \sigma \parallel \mathfrak{G}}>1.$$

Then, there exists at least one solution on $[a,b]$ of the $(k,\psi )$-Hilfer type nonlocal integro-multi-point fractional boundary value problem (5).

Proof. 

Consider the operator $\mathcal{A}$ defined by (24). For $r>0$, let $B_r=\{x\in C([a,b],\mathbb{R}):\parallel x\parallel \le r\}.$ We will show that $\mathcal{A}$ maps bounded sets into bounded set in $C\left(\right[a,b],\mathbb{R}).$ For $t\in [a,b]$, we have

$$\begin{array}{ccc}\hfill \left|\right(\mathcal{A}x\left)\right(t\left)\right|& \le & \underset{t\in [a,b]}{sup}\{\frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n|{\mu }_i|{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }\left|f(s,x\left(s\right))\right|ds\hfill \\ & & +\sum _{j=1}^m|{\zeta }_j|{}^k{I}^{\alpha +{\varphi }_j;\psi }|f(z_j,x\left(z_j\right))|+{}^k{I}^{\alpha ;\psi }|f(b,x\left(b\right))\left|\right]+{}^k{I}^{\alpha ;\psi }\left|f(t,x\left(t\right))\right|\}\hfill \\ & \le & \{\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n\left|{\mu }_i\right|\frac{k{(\psi \left({\eta }_i\right)-\psi \left(a\right))}^{\frac{\alpha }{k}+1}}{{\mathsf{\Gamma }}_k(\alpha +2k)}\hfill \\ & & +\sum _{j=1}^m|{\zeta }_j|\frac{{(\psi \left(z_j\right)-\psi \left(a\right))}^{\frac{\alpha +{\varphi }_j}{k}}}{{\mathsf{\Gamma }}_k(\alpha +{\varphi }_j+k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}\left]\right\}\parallel \sigma \parallel \chi (\parallel x\parallel ),\hfill \end{array}$$

and consequently,

$$\parallel \mathcal{A}x\parallel \le \chi \left(r\right)\parallel \sigma \parallel \mathfrak{G}.$$

In the next step, we will show that $\mathcal{A}$ maps bounded sets into equicontinuous sets of $C\left(\right[a,b],$$\mathbb{R}).$ Let $t_1,t_2\in [a,b]$ with $t_1<t_2$ and $x\in B_r.$ Then we have

$$\begin{array}{ccc}& & |\left(\mathcal{A}x\right)\left(t_2\right)-\left(\mathcal{A}x\right)\left(t_1\right)|\hfill \\ & \le & \frac{1}{{\mathsf{\Gamma }}_k\left(\alpha \right)}|{\int }_a^{w_1}{\psi }^{\prime }\left(s\right)[{(\psi \left(t_2\right)-\psi \left(s\right))}^{\frac{\alpha }{k}-1}-{(\psi \left(t_1\right)-\psi \left(s\right))}^{\frac{\alpha }{k}-1}]f(s,x\left(s\right))ds\hfill \\ & & +{\int }_{w_1}^{w_2}{\psi }^{\prime }\left(s\right){(\psi \left(t_2\right)-\psi \left(s\right))}^{\frac{\alpha }{k}-1}f(s,x\left(s\right))ds|\hfill \\ & & +\frac{{(\psi \left(t_2\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}-{(\psi \left(t_1\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n\left|{\mu }_i\right|\frac{k{(\psi \left({\eta }_i\right)-\psi \left(a\right))}^{\frac{\alpha }{k}+1}}{{\mathsf{\Gamma }}_k(\alpha +2k)}\hfill \\ & & +\sum _{j=1}^m\left|{\zeta }_j\right|\frac{{(\psi \left(z_j\right)-\psi \left(a\right))}^{\frac{\alpha +{\varphi }_j}{k}}}{{\mathsf{\Gamma }}_k(\alpha +{\varphi }_j+k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}]\hfill \\ & \le & \frac{\parallel \sigma \parallel \chi \left(r\right)}{{\mathsf{\Gamma }}_k(\alpha +k)}[2{(\psi \left(t_2\right)-\psi \left(t_1\right))}^{\frac{\alpha }{k}}+|{(\psi \left(t_2\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}-{(\psi \left(t_1\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}\left|\right],\hfill \\ & & +\frac{{(\psi \left(t_2\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}-{(\psi \left(t_1\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n\left|{\mu }_i\right|\frac{k{(\psi \left({\eta }_i\right)-\psi \left(a\right))}^{\frac{\alpha }{k}+1}}{{\mathsf{\Gamma }}_k(\alpha +2k)}\hfill \\ & & +\sum _{j=1}^m|{\zeta }_j|\frac{{(\psi \left(z_j\right)-\psi \left(a\right))}^{\frac{\alpha +{\varphi }_j}{k}}}{{\mathsf{\Gamma }}_k(\alpha +{\varphi }_j+k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}]\parallel \sigma \parallel \chi \left(r\right),\hfill \end{array}$$

which tends to zero independently of $x\in B_r,$ as $t_2-t_1\to 0.$ Therefore, the operator $\mathcal{A}:C\left(\right[a,b],\mathbb{R})\to C\left(\right[a,b],\mathbb{R}),$ by the Arzelá–Ascoli theorem, is completely continuous.

Finally, we will show that the set of all solutions to equations $x=\lambda \mathcal{A}x$ for $\lambda \in (0,1)$ is bounded.

Let x be a solution. Then, for $t\in [a,b],$ and working as in the first step, we have

$$\left|x\right(t\left)\right|\le \chi (\parallel x\parallel )\parallel \sigma \parallel \mathfrak{G},$$

or

$$\frac{\parallel x\parallel }{\chi (\parallel x\parallel )\parallel \sigma \parallel \mathfrak{G}}\le 1.$$

By $\left(H_4\right)$, there exists $\mathfrak{K}$ such that $\parallel x\parallel \ne \mathfrak{K}.$ Note that the operator $\mathcal{A}:\overline{U}\to C([a,b],\mathbb{R})$ is continuous and completely continuous, where $U=\{x\in C([a,b],\mathbb{R}):\parallel x\parallel <\mathfrak{K}\}.$ From the choice of $U,$ there is no $x\in \partial U$ such that $x=\lambda \mathcal{A}x$ for some $\lambda \in (0,1).$ We deduce that $\mathcal{A}$ has a fixed point $x\in \overline{U},$ by the nonlinear alternative of Leray–Schauder type, which is a solution of the $(k,\psi )$-Hilfer type nonlocal integral multi-point fractional boundary value problem (5). The proof is finished. □

5. The Multi-Valued Problem

For a normed space $(\mathfrak{X},\parallel ·\parallel )$, we define:

• ${\mathcal{P}}_{cl}\left(\mathfrak{X}\right)=\{\mathfrak{R}\in \mathcal{P}\left(\mathfrak{X}\right):\mathfrak{R}$ is closed},
• ${\mathcal{P}}_{cp}\left(\mathfrak{X}\right)=\{\mathfrak{R}\in \mathcal{P}\left(\mathfrak{X}\right):\mathfrak{R}$ is compact}, and
• ${\mathcal{P}}_{cp,c}\left(\mathfrak{X}\right)=\{\mathfrak{R}\in \mathcal{P}\left(\mathfrak{X}\right):\mathfrak{R}$ is compact and convex$\}.$

We refer the interested reader for more details of multi-valued analysis to [45,46]. For a brief summary, see also [19].

As usual, we defined by

$$S_{F,x}:=\{f\in L^1([a,b],\mathbb{R}):f\left(t\right)\in F(t,x\left(t\right))on[a,b]\},$$

the set of selections of $f,$ for each $x\in C\left(\right[a,b],\mathbb{R}).$

Definition 12.

A function $x\in C\left(\right[a,b],\mathbb{R})$ is said to be a solution of the $(k,\psi )$-Hilfer type nonlocal integro-multi-point fractional boundary value problem (6) if there exists a function $f\in L^1([a,b],\mathbb{R})$ with $f\left(t\right)\in F(t,x)$ for almost every $t\in [a,b]$, such that x satisfies the differential equation ${}^{k,H}{D}^{\alpha ,\beta ;\psi }x\left(t\right)=f\left(t\right)$ on $[a,b]$ and the boundary conditions $x\left(a\right)=0,x\left(b\right)={\displaystyle \sum _{i=1}^n}{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right)x\left(s\right)ds+{\displaystyle \sum _{j=1}^m}{\zeta }_j{}^k{I}^{{\varphi }_j;\psi }x\left(z_j\right).$

We define the graph of G to be the set $\mathit{Gr}\left(G\right)=\left\{\right(x,y)\in X\times Y,y\in G(x\left)\right\}$ and recall two useful results regarding closed graphs and upper-semicontinuity.

Lemma 5

([42], Proposition 1.2). If $G:X\to {\mathcal{P}}_{cl}\left(Y\right)$ is u.s.c., then $\mathit{Gr}\left(G\right)$ is a closed subset of $X\times Y$; i.e., for every sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}\subset X$ and ${\left\{y_n\right\}}_{n\in \mathbb{N}}\subset Y$, if when $n\to \infty$, $x_n\to x_{\ast }$, $y_n\to y_{\ast }$ and $y_n\in G\left(x_n\right)$, then $y_{\ast }\in G\left(x_{\ast }\right)$. Conversely, if G is completely continuous and has a closed graph, then it is upper semi-continuous.

Lemma 6

([47]). Let $F:[a,b]\times \mathbb{R}\to {\mathcal{P}}_{cp,c}\left(\mathbb{R}\right)$ be an $L^1-$ Carathéodory multi-valued map and let Θ be a linear continuous mapping from $L^1([a,b],\mathbb{R})$ to $C\left(\right[a,b],\mathbb{R})$. Then the operator

$$\Theta \circ S_F:C([a,b],\mathbb{R})\to {\mathcal{P}}_{cp,c}\left(C([a,b],\mathbb{R})\right),x\mapsto (\Theta \circ S_F)\left(x\right)=\Theta \left(S_{F,x}\right)$$

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

Remark 3.

We recall that a multi-valued map $F:[a,b]\times \mathbb{R}\to \mathcal{P}\left(\mathbb{R}\right)$ is said to be $L^1-$ Carathéodory if $\left(i\right)$ $t⟼F(t,x)$ is measurable for each $x\in \mathbb{R}$; $\left(ii\right)$ $x⟼F(t,x)$ is upper semicontinuous for almost all $t\in [a,b];$ $\left(iii\right)$ for each $\alpha >0$, there exists ${\phi }_{\alpha }\in L^1([a,b],{\mathbb{R}}^{+})$ such that $\parallel F(t,x)\parallel =sup\left\{\right|v|:v\in F(t,x)\}\le {\phi }_{\alpha }\left(t\right)$ for all $x\in \mathbb{R}$ with $\parallel x\parallel \le \alpha$ and for a.e. $t\in [a,b].$

Now we establish our first existence result for the $(k,\psi )$-Hilfer nonlocal integro-multi-point fractional boundary value problem (6), via a nonlinear alternative of Leray–Schauder type for multi-valued maps [44].

Lemma 7

(Nonlinear alternative for Kakutani maps [44]). Let E be a Banach space, C a closed convex subset of $E,$ U an open subset of C and $0\in U.$ Suppose that $F:\overline{U}\to {\mathcal{P}}_{cp,c}\left(C\right)$ is an upper semicontinuous compact map. Then either

(i)

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

(ii)

there is a $u\in \partial U$ and $\mu \in (0,1)$ with $u\in \mu F\left(u\right).$

Theorem 4.

Suppose that:

$\left(G_1\right)$

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

$\left(G_2\right)$

There exists a continuous function $z:[0,\infty )\to (0,\infty )$ which is nondecreasing, and a positive continuous function q such that

$${\parallel F(t,x)\parallel }_{\mathcal{P}}:=sup\left\{\right|f|:f\in F(t,x)\}\le q\left(t\right)z(\parallel x\parallel )fore ach(t,x)\in [a,b]\times \mathbb{R};$$

$\left(G_3\right)$

there exists a constant $\mathfrak{K}>0$ such that

$$\frac{{\displaystyle \mathfrak{K}}}{{\displaystyle \parallel q\parallel z\left(\mathfrak{K}\right)\mathfrak{G}}}>1.$$

Then, there exists at least one solution on $[a,b]$ of the $(k,\psi )$-Hilfer type nonlocal integro-multi-point fractional boundary value problem (6).

Proof. 

We define an operator $\mathcal{F}:C\left(\right[a,b],\mathbb{R})⟶\mathcal{P}\left(C\right([a,b],\mathbb{R}\left)\right)$ by

$$\mathcal{F}\left(x\right)=\left\{\begin{array}{cc}& h\in C\left(\right[a,b],\mathbb{R}):\hfill \\ & h\left(t\right)=\left\{\begin{array}{c}{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\mathsf{\Lambda }{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }f\left(s\right)ds}\hfill \\ {\displaystyle +\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }f\left(z_j\right)-{}^k{I}^{\alpha ;\psi }f\left(b\right)]+{}^k{I}^{\alpha ;\psi }f\left(t\right),}\hfill \end{array}\right.\hfill \end{array}\right\}$$

for $t\in [a,b]$ and $f\in S_{F,x}.$ It is obvious that the fixed points of $\mathcal{F}$ are the solutions of the $(k,\psi )$-Hilfer nonlocal integro-multi-point fractional boundary value problem (6).

For making the proof readable, we split it in several steps.

Step 1. $\mathcal{F}\left(x\right)$ is convex, for each $x\in C\left(\right[a,b],\mathbb{R}).$

We omit the proof, because this step is obvious, since $S_{F,x}$ is convex.

Step 2. Bounded sets are mapped by $\mathcal{F}$ into bounded sets in $C\left(\right[a,b],\mathbb{R}).$

Let $B_r=\{x\in C([a,b],\mathbb{R}):\parallel x\parallel \le r\},r>0.$ Then, for each $h\in \mathcal{F}\left(x\right),x\in B_r$, there exists $f\in S_{F,x}$ such that

$$\begin{array}{ccc}\hfill h\left(t\right)& =& {\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\mathsf{\Lambda }{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }f\left(s\right)ds+\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }f\left(z_j\right)}\hfill \\ & & -{}^k{I}^{\alpha ;\psi }f\left(b\right)]+{}^k{I}^{\alpha ;\psi }f\left(t\right).\hfill \end{array}$$

Then, for $t\in [a,b],$ we have

$$\begin{array}{ccc}\hfill \left|h\right(t\left)\right|& \le & \underset{t\in [a,b]}{sup}\left\{\frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}\right[\sum _{i=1}^n|{\mu }_i|{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }\left|f\left(s\right)\right|ds\hfill \\ & & +\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }|f\left(z_j\right)|+{}^k{I}^{\alpha ;\psi }|f\left(b\right)\left|\right]+{}^k{I}^{\alpha ;\psi }\left|f\left(t\right)\right|\}\hfill \\ & \le & \{\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n\left|{\mu }_i\right|\frac{k{(\psi \left({\eta }_i\right)-\psi \left(a\right))}^{\frac{\alpha }{k}+1}}{{\mathsf{\Gamma }}_k(\alpha +2k)}\hfill \\ & & +\sum _{j=1}^m|{\zeta }_j|\frac{{(\psi \left(z_j\right)-\psi \left(a\right))}^{\frac{\alpha +{\varphi }_j}{k}}}{{\mathsf{\Gamma }}_k(\alpha +{\varphi }_j+k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}\left]\right\}\parallel q\parallel z(\parallel x\parallel ),\hfill \end{array}$$

and consequently,

$$\parallel h\parallel \le z\left(r\right)\parallel q\parallel \mathfrak{G}.$$

Step 3. Bounded sets are mapped by $\mathcal{F}$ into equicontinuous sets of $C\left(\right[a,b],\mathbb{R}).$

Let $t_1,t_2\in [a,b]$ with $t_1<t_2$ and $x\in B_r.$ Then, for each $h\in \mathcal{F}\left(x\right),$ we obtain

$$\begin{array}{ccc}& & |h\left(t_2\right)-h\left(t_1\right)|\hfill \\ & \le & \frac{1}{{\mathsf{\Gamma }}_k\left(\alpha \right)}|{\int }_a^{w_1}{\psi }^{\prime }\left(s\right)[{(\psi \left(t_2\right)-\psi \left(s\right))}^{\frac{\alpha }{k}-1}-{(\psi \left(t_1\right)-\psi \left(s\right))}^{\frac{\alpha }{k}-1}]f\left(s\right)ds\hfill \\ & & +{\int }_{w_1}^{w_2}{\psi }^{\prime }\left(s\right){(\psi \left(t_2\right)-\psi \left(s\right))}^{\frac{\alpha }{k}-1}f\left(s\right)ds|\hfill \\ & & +\frac{{(\psi \left(t_2\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}-{(\psi \left(t_1\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n|{\mu }_i|{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }\left|f\left(s\right)\right|ds\hfill \\ & & +\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }|f\left(z_j\right)|+{}^k{I}^{\alpha ;\psi }\left|f\left(b\right)\right|]\hfill \\ & \le & \frac{\parallel q\parallel z\left(r\right)}{{\mathsf{\Gamma }}_k(\alpha +k)}[2{(\psi \left(t_2\right)-\psi \left(t_1\right))}^{\frac{\alpha }{k}}+|{(\psi \left(t_2\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}-{(\psi \left(t_1\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}\left|\right],\hfill \\ & & +\frac{{(\psi \left(t_2\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}-{(\psi \left(t_1\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n\left|{\mu }_i\right|\frac{k{(\psi \left({\eta }_i\right)-\psi \left(a\right))}^{\frac{\alpha }{k}+1}}{{\mathsf{\Gamma }}_k(\alpha +2k)}\hfill \\ & & +\sum _{j=1}^m|{\zeta }_j|\frac{{(\psi \left(z_j\right)-\psi \left(a\right))}^{\frac{\alpha +{\varphi }_j}{k}}}{{\mathsf{\Gamma }}_k(\alpha +{\varphi }_j+k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}]\parallel q\parallel z\left(r\right).\hfill \end{array}$$

Hence, we have $|h\left(t_2\right)-h\left(t_1\right)|\to 0$ as $w_2-w_1\to 0,$ independently of $x\in B_r.$ Thus $\mathcal{F}:C\left(\right[a,b],\mathbb{R})\to \mathcal{P}\left(C\right([a,b],\mathbb{R}\left)\right)$ is completely continuous, by the Arzelá–Ascoli theorem.

In order to prove that $\mathcal{F}$ is upper semi-continuous multi-valued mapping, by virtue of the Lemma 5 it is enough to prove that the $\mathcal{F}$ has a closed graph.

Step 4. $\mathcal{F}$ has a closed graph.

Let $x_n\to x_{\ast },h_n\in \mathcal{F}\left(x_n\right)$ and $h_n\to h_{\ast }.$ Then we must show that $h_{\ast }\in \mathcal{F}\left(x_{\ast }\right).$ Since $h_n\in \mathcal{F}\left(x_n\right)$, therefore, there exists $f_n\in S_{F,x_n}$ such that, for each $t\in [a,b],$

$$\begin{array}{ccc}\hfill h_n\left(t\right)& =& {\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\mathsf{\Lambda }{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }{f}_n\left(s\right)ds+\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }{f}_n\left(z_j\right)}\hfill \\ & & -{}^k{I}^{\alpha ;\psi }{f}_n\left(b\right)]+{}^k{I}^{\alpha ;\psi }{f}_n\left(t\right).\hfill \end{array}$$

Hence it suffices to show that there exists $f_{\ast }\in S_{F,x_{\ast }}$ such that, for each $t\in [a,b],$

$$\begin{array}{ccc}\hfill h_{\ast }\left(t\right)& =& {\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\mathsf{\Lambda }{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }{f}_{\ast }\left(s\right)ds+\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }{f}_{\ast }\left(z_j\right)}\hfill \\ & & -{}^k{I}^{\alpha ;\psi }{f}_{\ast }\left(b\right)]+{}^k{I}^{\alpha ;\psi }{f}_{\ast }\left(t\right).\hfill \end{array}$$

Consider the linear operator $\Theta :L^1([a,b],\mathbb{R})\to C([a,b],\mathbb{R})$ given by

$$\begin{array}{ccc}\hfill f\mapsto \Theta \left(f\right)\left(t\right)& =& {\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\mathsf{\Lambda }{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }f\left(s\right)ds+\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }f\left(z_j\right)}\hfill \\ & & -{}^k{I}^{\alpha ;\psi }f\left(b\right)]+{}^k{I}^{\alpha ;\psi }f\left(t\right).\hfill \end{array}$$

Obviously, $\parallel h_n\left(t\right)-h_{\ast }\left(t\right)\parallel \to 0,$ as $n\to \infty .$ Therefore, $\Theta \circ S_f$ is a closed graph operator by Lemma 6. Also, $h_n\left(t\right)\in \Theta \left(S_{f,x_n}\right).$ Since $x_n\to x_{\ast },$ therefore we deduce by Lemma 6 that

$$\begin{array}{ccc}\hfill h_{\ast }\left(t\right)& =& {\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\mathsf{\Lambda }{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }{f}_{\ast }\left(s\right)ds+\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }{f}_{\ast }\left(z_j\right)}\hfill \\ & & -{}^k{I}^{\alpha ;\psi }{f}_{\ast }\left(b\right)]+{}^k{I}^{\alpha ;\psi }{f}_{\ast }\left(t\right).\hfill \end{array}$$

for some $f_{\ast }\in S_{F,x_{\ast }}$.

Step 5. We give now an a priori bound of solutions.

Let $\nu \in (0,1)$ and $x\in \nu \mathcal{F}\left(x\right).$ Then there exists $f\in L^1([a,b],\mathbb{R})$ with $f\in S_{F,x}$ such that, for $t\in [a,b]$, we have

$$\begin{array}{ccc}\hfill x\left(t\right)& =& {\displaystyle \nu \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\mathsf{\Lambda }{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }f\left(s\right)ds+\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }f\left(z_j\right)}\hfill \\ & & -{}^k{I}^{\alpha ;\psi }f\left(b\right)]+\nu {}^k{I}^{\alpha ;\psi }f\left(t\right).\hfill \end{array}$$

Following the procedure, as in the second step, we get

$$\begin{array}{c}\hfill \left|x\right(t\left)\right|\le \parallel p\parallel z(\parallel x\parallel )\mathfrak{G},\end{array}$$

from which we have

$$\begin{array}{c}\hfill \parallel x\parallel \le \parallel p\parallel z(\parallel x\parallel )\mathfrak{G},\end{array}$$

or

$$\frac{{\displaystyle \parallel x\parallel }}{{\displaystyle \parallel q\parallel z(\parallel x\parallel )\mathfrak{G}}}\le 1.$$

Then, there exists $\mathfrak{K},$ by $\left(H_3\right)$, such that $\parallel x\parallel \ne \mathfrak{K}$. Let us set

$$\mathcal{U}=\{x\in C([a,b],\mathbb{R}):\parallel x\parallel <\mathfrak{K}\}.$$

Note that the operator $\mathcal{F}:\overline{U}\to \mathcal{P}\left(C([a,b],\mathbb{R})\right)$ is an upper semi-continuous, compact multi-valued map with convex closed values. From the choice of $\mathcal{U},$ there is no $x\in \partial \mathcal{U}$ such that $x\in \nu \mathcal{F}\left(x\right)$ for some $\nu \in (0,1).$ Hence, $\mathcal{F}$ has a fixed point $x\in \overline{\mathcal{U}},$ by the nonlinear alternative of Leray–Schauder type, which is a solution of the $(k,\psi )$-Hilfer type nonlocal integro-multi-point fractional boundary value problem (6). The proof is finished. □

Next we study the case when F is not necessarily non-convex valued. Our approach here is based on a Covitz and Nadler fixed point theorem for multi-valued contractive maps [48].

Lemma 8

(Covitz and Nadler fixed point theorem [48]). Let $(X,d)$ be a complete metric space. If $N:X\to {\mathcal{P}}_{cl}\left(X\right)$ is a contraction, then $FixN\ne \varnothing$.

Theorem 5.

Suppose that:

$\left(A_1\right)$

$F:[a,b]\times \mathbb{R}\to {\mathcal{P}}_{cp}\left(\mathbb{R}\right)$ is such that $F(·,x):[a,b]\to {\mathcal{P}}_{cp}\left(\mathbb{R}\right)$ is measurable for each $x\in \mathbb{R}$.

$\left(A_2\right)$

$H_d(F(t,x),F(t,\overline{x}))\le m\left(t\right)|x-\overline{x}|$ for almost all $t\in [a,b]$ and $x,\overline{x}\in \mathbb{R}$ with $m\in C([a,b],{\mathbb{R}}^{+})$ and $d(0,F(t,0\left)\right)\le m\left(t\right)$ for almost all $t\in [a,b]$.

Then, if

$$\delta :=\mathfrak{G}\parallel m\parallel <1,$$

(29)

the $(k,\psi )$-Hilfer type nonlocal integro-multi-point fractional boundary value problem (6) has at least one solution on $[a,b].$

Proof. 

Transform the $(k,\psi )$-Hilfer type nonlocal integro-multi-point fractional boundary value problem (6) into a fixed point problem. Consider the operator $\mathcal{F}$ defined by (24). For each $x\in C\left(\right[a,b],\mathbb{R}),$ by $\left(A_1\right),$ the set $S_{F,x}$ is nonempty. Therefore, F has a measurable selection (see Theorem III.6 [49]). Now, for each $x\in C\left(\right[a,b],\mathbb{R}),$ we will show that $\mathcal{F}\left(x\right)\in {\mathcal{P}}_{cl}\left(C([a,b],\mathbb{R})\right).$ Let ${\left\{u_n\right\}}_{n\ge 0}\in \mathcal{F}\left(x\right)$ be such that $u_n\to u(n\to \infty )$ in $C\left(\right[a,b],\mathbb{R}).$ Then $u\in C\left(\right[a,b],\mathbb{R})$, and there exists $f_n\in S_{f,x_n}$, such that, for each $t\in [a,b]$,

$$\begin{array}{ccc}\hfill u_n\left(t\right)& =& {\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\mathsf{\Lambda }{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }{f}_n\left(s\right)ds+\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }{f}_n\left(z_j\right)}\hfill \\ & & -{}^k{I}^{\alpha ;\psi }{f}_n\left(b\right)]+{}^k{I}^{\alpha ;\psi }{f}_n\left(t\right).\hfill \end{array}$$

We have that $f_n$ converges to f in $L^1([a,b],\mathbb{R}),$ as F has compact values. Thus, $f\in S_{F,x}$ and for each $t\in [a,b]$, we have

$$\begin{array}{ccc}\hfill u_n\left(t\right)\to u\left(t\right)& =& {\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\mathsf{\Lambda }{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }f\left(s\right)ds+\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }f\left(z_j\right)}\hfill \\ & & -{}^k{I}^{\alpha ;\psi }f\left(b\right)]+{}^k{I}^{\alpha ;\psi }f\left(t\right).\hfill \end{array}$$

Consequently, $u\in \mathcal{F}\left(x\right).$

In the next step, we will show that there exists $\delta <1$ (defined by (29)) such that

$$H_d(\mathcal{F}\left(x\right),\mathcal{F}\left(\overline{x}\right))\le \delta \parallel x-\overline{x}\parallel ,\delta <1,\mathrm{for}\mathrm{each}x,\overline{x}\in C^2([a,b],\mathbb{R}).$$

Let $x,\overline{x}\in C^2([a,b],\mathbb{R})$ and $h_1\in \mathcal{F}\left(x\right)$. Then there exists $f_1\left(t\right)\in F(t,x\left(t\right))$ such that, for each $t\in [a,b]$,

$$\begin{array}{ccc}\hfill h_1\left(t\right)& =& {\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\mathsf{\Lambda }{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }{f}_1\left(s\right)ds+\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }{f}_1\left(z_j\right)}\hfill \\ & & -{}^k{I}^{\alpha ;\psi }{f}_1\left(b\right)]+{}^k{I}^{\alpha ;\psi }{f}_1\left(t\right).\hfill \end{array}$$

By $\left(A_2\right)$, we have

$$H_d(F(t,x),F(t,\overline{x}))\le m\left(t\right)|x\left(t\right)-\overline{x}\left(t\right)|.$$

So, there exists $\omega \in F(t,\overline{x}\left(t\right))$ such that

$$|f_1\left(t\right)-\omega |\le m\left(t\right)|x\left(t\right)-\overline{x}\left(t\right)|,t\in [a,b].$$

Define $U:[a,b]\to \mathcal{P}\left(\mathbb{R}\right)$ by

$$U\left(t\right)=\{w\in \mathbb{R}:|f_1\left(t\right)-\omega |\le m\left(t\right)|x\left(t\right)-\overline{x}\left(t\right)\left|\right\}.$$

By Proposition III.4 [49], the multi-valued operator $U\left(t\right)\cap F(t,\overline{x}\left(t\right))$ is measurable. Hence there exists a function $f_2\left(t\right)$ which is a measurable selection for U. So $f_2\left(t\right)\in F(t,\overline{x}\left(t\right))$ and for each $t\in [a,b]$, we have $|f_1\left(t\right)-f_2\left(t\right)|\le m\left(t\right)|x\left(t\right)-\overline{x}\left(t\right)|$.

For each $t\in [a,b]$, let us define

$$\begin{array}{ccc}\hfill h_2\left(t\right)& =& {\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\mathsf{\Lambda }{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n{\mu }_i{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }{f}_2\left(s\right)ds+\sum _{j=1}^m{\zeta }_j{}^k{I}^{\alpha +{\varphi }_j;\psi }{f}_2\left(z_j\right)}\hfill \\ & & -{}^k{I}^{\alpha ;\psi }{f}_2\left(b\right)]+{}^k{I}^{\alpha ;\psi }{f}_2\left(t\right).\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}& & |h_1\left(t\right)-h_2\left(t\right)|\hfill \\ & \le & \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n|{\mu }_i|{\int }_a^{{\eta }_i}{\psi }^{\prime }\left(s\right){}^k{I}^{\alpha ;\psi }|f_1\left(s\right)-f_2\left(s\right)|ds\hfill \\ & & +\sum _{j=1}^m|{\zeta }_j|{}^k{I}^{\alpha +{\varphi }_j;\psi }|f_1\left(z_j\right)-f_2\left(z_j\right)|+{}^k{I}^{\alpha ;\psi }f\left(b\right)]+{}^k{I}^{\alpha ;\psi }|f_1\left(b\right)-f_2\left(b\right)|\hfill \\ & \le & \{\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{{\theta }_k}{k}-1}}{\left|\mathsf{\Lambda }\right|{\mathsf{\Gamma }}_k\left({\theta }_k\right)}[\sum _{i=1}^n\left|{\mu }_i\right|\frac{k{(\psi \left({\eta }_i\right)-\psi \left(a\right))}^{\frac{\alpha }{k}+1}}{{\mathsf{\Gamma }}_k(\alpha +2k)}\hfill \\ & & +\sum _{j=1}^m|{\zeta }_j|\frac{{(\psi \left(z_j\right)-\psi \left(a\right))}^{\frac{\alpha +{\varphi }_j}{k}}}{{\mathsf{\Gamma }}_k(\alpha +{\varphi }_j+k)}+\frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\frac{\alpha }{k}}}{{\mathsf{\Gamma }}_k(\alpha +k)}\left]\right\}\parallel m\parallel \parallel x-\overline{x}\parallel \hfill \\ & =& \mathfrak{G}\parallel m\parallel \parallel x-\overline{x}\parallel ,\hfill \end{array}$$

which implies that

$$\parallel h_1-h_2\parallel \le \mathfrak{G}\parallel m\parallel \parallel x-\overline{x}\parallel .$$

Interchanging the roles of x and $\overline{x}$, we obtain

$$\begin{array}{c}\hfill H_d(\mathcal{F}\left(x\right),\mathcal{F}\left(\overline{x}\right))\le \mathfrak{G}\parallel m\parallel \parallel x-\overline{v}\parallel .\end{array}$$

Hence $\mathcal{F}$ is a contraction and therefore, by Covitz and Nadler theorem, $\mathcal{F}$ has a fixed point x which is a solution of the $(k,\psi )$-Hilfer type nonlocal integro-multi-point fractional boundary value problem (6). The proof is completed. □

6. Examples

Let us present some examples to show the applicability of our theorems.

Consider the following $(k,\psi )$-Hilfer type nonlocal integro-multi-point fractional boundary value problem of the form

$$\left\{\begin{array}{c}{\displaystyle {}^{\frac{6}{5},H}{D}^{\frac{3}{2},\frac{1}{2};ln(t^2+1)}x\left(t\right)=f(t,x\left(t\right)),\frac{1}{4}<w<\frac{9}{4},}\hfill \\ {\displaystyle x\left(\frac{1}{4}\right)=0,x\left(\frac{9}{4}\right)=\frac{1}{6}{\int }_{\frac{1}{4}}^{\frac{1}{2}}\left(\frac{s}{s^2+1}\right)x\left(s\right)ds+\frac{3}{16}{\int }_{\frac{1}{4}}^{\frac{3}{4}}\left(\frac{s}{s^2+1}\right)x\left(s\right)ds}\hfill \\ {\displaystyle +\frac{5}{52}{}^{\frac{6}{5}}{I}^{\frac{3}{4};ln(t^2+1)}x\left(\frac{5}{4}\right)+\frac{7}{72}{}^{\frac{6}{5}}{I}^{\frac{5}{4};ln(t^2+1)}x\left(\frac{3}{2}\right)}\hfill \\ {\displaystyle +\frac{9}{92}{}^{\frac{6}{5}}{I}^{\frac{7}{4};ln(t^2+1)}x\left(\frac{7}{4}\right).}\hfill \end{array}\right.$$

(30)

Here $k=6/5$, $\alpha =3/2$, $\beta =1/2$, $\psi \left(t\right)=ln(t^2+1)$, $a=1/4$, $b=9/4$, $n=2$, ${\mu }_1=1/12$, ${\eta }_1=1/2$, ${\mu }_2=3/32$, ${\eta }_2=3/4$, $m=3$, ${\zeta }_1=5/52$, ${\varphi }_1=3/4$, $z_1=5/4$, ${\zeta }_2=7/72$, ${\varphi }_2=5/4$, $z_2=3/2$, ${\zeta }_3=9/92$, ${\varphi }_3=7/4$, and $z_3=7/4$. From the data, computation yields ${\theta }_{\frac{6}{5}}=39/20$, $\mathsf{\Lambda }\approx 1.220347192$, $\mathfrak{G}\approx 3.138780815$, and ${\mathfrak{G}}_1\approx 1.732908250$.

(i) Let f be a nonlinear unbounded Lipschitzian function given by

$$f(t,x)=\left(\frac{x^2+16\left|x\right|}{8(8+|x\left|\right)}\right)e^{-{(4w-1)}^2}+\frac{1}{2}{sin}^{2}w+\frac{2}{3}.$$

(31)

It is easy to check that

$$|f(t,x_1)-f(t,x_2)|\le \frac{1}{4}|x_1-x_2|,$$

for all $t\in [1/4,9/4]$ and $x_1,x_2\in \mathbb{R}$, which implies that $\mathfrak{L}=1/4$. Moreover, $\mathfrak{L}\mathfrak{G}\approx 0.7846952038<1$. By Theorem 1, the $(k,\psi )$-Hilfer nonlocal integro-multi-point fractional boundary value problem (30) with f given in (31) has a unique solution on the interval $[1/4,9/4]$.

(ii)Consider a nonlinear bounded Lipschitzian function defined by

$$f(t,x)=\left(\frac{5\left|x\right|}{9+\left|x\right|}\right){cos}^2\pi w+\frac{1}{3}{w}^2+\frac{1}{2}w+1.$$

(32)

Clearly f is bounded, since

$$\left|f(t,x)\right|\le 5{cos}^2\pi w+\frac{1}{3}{w}^2+\frac{1}{2}w+1:=\varpi \left(t\right),$$

for all $t\in [1/4,9/4]$. In addition, the function f is Lipschitzian with Lipschitz constant $\mathfrak{L}=5/9$. The $(k,\psi )$-Hilfer nonlocal integro-multi-point fractional boundary value problem (30), with f given by (32), has at least one solution on $[1/4,9/4]$ since $\mathfrak{L}{\mathfrak{G}}_{\mathfrak{1}}\approx 0.9627268056<1$, a condition in Theorem 2. However, the uniqueness of solutions for (30)–(32) does not follow from Theorem 1 because $\mathfrak{L}\mathfrak{G}\approx 1.743767120>1$.

(iii) To apply the result in Theorem 3, we choose

$$f(t,x)=\sigma \left(t\right)\left(\frac{{A\left|x\right|}^{n+1}}{(1+|x{|}^n)}+B\right),$$

(33)

where $n\in {\mathcal{I}}_0$, $\sigma :[1/4,9/4]\to {\mathcal{R}}^{+}$, $0\le A<1/(\parallel \sigma \parallel \mathfrak{G})$ and $B>0$. Then we have

$$\left|f(t,x)\right|\le \parallel \sigma \parallel \left(A\left|x\right|+B\right).$$

By setting $\chi \left(u\right)=A\left|u\right|+B$, there exists a constant $\mathfrak{K}$ satisfying condition $\left(H_4\right)$ in Theorem 3 as

$$\mathfrak{K}>\frac{B\parallel \sigma \parallel \mathfrak{G}}{1-A\parallel \sigma \parallel \mathfrak{G}}.$$

By Theorem 3, the $(k,\psi )$-Hilfer nonlocal integro-multi-point fractional boundary value problem (30) with f given in (33) has at least one solution on $[1/4,9/4].$

(iv) For another one application of Theorem 3, we choose

$$f(t,x)=\sigma \left(t\right)\left(\frac{{A\left|x\right|}^{n+2}}{(1+|x{|}^n)}+B\right),$$

(34)

where $n\in {\mathcal{I}}_0$, $\sigma :[1/4,9/4]\to {\mathcal{R}}^{+}$, constants $A,B>0$ with $AB<1/(4\parallel \sigma {\parallel }^2{\mathfrak{G}}^2)$. Obviously,

$$\left|f(t,x)\right|\le \parallel \sigma \parallel \left(A{x}^2+B\right).$$

For $\chi \left(u\right)=A{u}^2+B$, the condition $\left(H_4\right)$ in Theorem 3 is satisfied for

$$\mathfrak{K}\in \left(\frac{1-\sqrt{1-{4AB\parallel \sigma \parallel }^2{\mathfrak{G}}^2}}{2A\parallel \sigma \parallel \mathfrak{G}},\frac{1+\sqrt{1-{4AB\parallel \sigma \parallel }^2{\mathfrak{G}}^2}}{2A\parallel \sigma \parallel \mathfrak{G}}\right).$$

By Theorem 3, the $(k,\psi )$-Hilfer nonlocal integro-multi-point fractional boundary value problem (30) with f given in (34) has at least one solution on $[1/4,9/4].$

(v) Suppose that the first equation of (30) is replaced by

$${}^{\frac{6}{5},H}{D}^{\frac{3}{2},\frac{1}{2};ln(t^2+1)}x\left(t\right)\in F(t,x\left(t\right)),\frac{1}{4}<w<\frac{9}{4},$$

(35)

where

$$F(t,x)=\left[0,\frac{1}{2(2w+3)}\left(\frac{x^2+2\left|x\right|}{1+\left|x\right|}+1\right)\right].$$

It is obvious that $F(t,x)$ is a measurable set. Moreover,

$$H_d\left(F(t,x),F(t,\overline{x})\right)\le \frac{1}{(2w+3)}|x-\overline{x}|.$$

Setting $m\left(t\right)=1/(2w+3)$, we obtain $d(0,F(t,0\left)\right)\le 1/\left(2\right(2w+3\left)\right)<1/(2w+3)=m\left(t\right)$ for almost all $t\in [1/4,9/4]$. Since $\delta =\mathfrak{G}\parallel m\parallel \approx 0.896794518<1$, the $(k,\psi )$-Hilfer fractional inclusion (35), with nonlocal integro-multi-point boundary conditions presented in (30), has at least one solution on $[1/4,9/4]$.

7. Conclusions

In this paper, we have investigated $(k,\psi )$-Hilfer type fractional boundary value problems with nonlocal integro-multi-point fractional boundary conditions. The single and multi-valued cases are considered. First the given problem was transformed into a fixed point problem, by using a linear variant of of the single-valued problem. Then we studied the existence and uniqueness of solutions for the single-valued problem by using Banach contraction mapping principle, Krasnosel’skiĭ fixed point theorem and the Leray–Schauder nonlinear alternative. Afterward, the multi-valued problem was discussed for both convex and non-convex values of the multi-valued map involved in the problem. By applying the Leray–Schauder nonlinear alternative for multi-valued maps, an existence result was established for the convex case, while the existence result for the non-convex case was based on the Covitz–Nadler fixed point theorem for contractive multi-valued maps. Numerical examples were constructed to demonstrate the application of the obtained theoretical results. The exposition of the chosen tools of the fixed point-theory to the given $(k,\psi )$-Hilfer type nonlocal integro-multi-point fractional boundary value problems is new. Our results are novel and contribute to the existing material on $(k,\psi )$-Hilfer type fractional differential equations and inclusions of order in $(1,2]$ supplemented with nonlocal integro-multi-point fractional boundary conditions.