# Existence of Solutions for Anti-Periodic Fractional Differential Inclusions Involving ψ-Riesz-Caputo Fractional Derivative

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**The left Caputo fractional derivative order α ($1<\alpha \le 2$) of a function $f\in {C}^{2}([a,b])$ is given by

**Definition**

**2**

**.**The fractional left, right and Riemann-Liouville integrals of order $\beta >0$ are defined as

**Definition**

**3**

**.**The fractional left, right integral of order $\alpha >0$ of a function $f\in C([a,b])$ with respect to ψ are defined by

**Definition**

**4**

**.**The left, right ψ-Caputo fractional derivative of order α ($1<\alpha \le 2$) of a function $f\in {C}^{2}([a,b])$ are defined as

**Remark**

**1.**

**Definition**

**5.**

**Definition**

**6**

- (a)
- denote the set $Gr(G)=\{(x,y)\in X\times Y,y\in G(x)\}$ as the graph of G,$$t\mapsto d(y,G(t))=inf\{|y-z|:z\in G(t)\}$$
- (b)
- if $G:X\to {\mathcal{P}}_{cl}(X)$ is called $\gamma -$Lipschitz if and only if there exists $\gamma >0$ such that$${H}_{d}(N(x),N(y))\le \gamma d(x,y),foreachx,y\in X.$$
- (c)
- if $G:X\to {\mathcal{P}}_{cl}(X)$ is called contraction if and only if it is $\gamma -$Lipschitz with $\gamma <1.$
- (d)
- G is said to be measurable if for every $y\in R,$ the function

**Definition**

**7**

**.**Assume that $F:J\times R\to \mathcal{P}(R)$ is a multivalued map with nonempty compact values. Denote a multivalued operator $\mathcal{F}:C(J\times R)\to \mathcal{P}({L}^{1}(J,R)$ associated with F as

**Definition**

**8**

**.**Assume that Y is a separable metric space and $N:Y\to \mathcal{P}({L}^{1}(J,R))$ is a multivalued operator. If N is lower semi-continuous(l.s.c.) and has nonempty closed and decomposable values, we say N has a property (BC).

**Definition**

**9**

**.**For each $u\in C(J,R),$ $t\in J=[a,b],$ denote the selection set of F as

**Definition**

**10**

**.**Let $A,B\in {\mathcal{P}}_{cl}(X.)$ The Pompeiu-Hausdorff distance of $A,B$ is defined by

**Property**

**1**

**.**Let G be a completely continuous multi-valued map with nonempty compact values, then T is u.s.c. ⟺ G has a closed graph.

**Lemma**

**1**

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

- (i)
- F has a fixed point in $\overline{U}$, or
- (ii)
- there exist a $u\in \partial U$ and $\lambda \in (0,1)$ satisfying $u\in \lambda F(u)$.

**Lemma**

**2**

**.**Let $(X,d)$ be a complete metric space. If $N:X\to {\mathcal{P}}_{cl}(X)$ is a contraction, then $FixN\ne \varnothing .$

**Lemma**

**3**

**.**Let X be a Banach space, and $F:J\times X\to (P)(X)$ be a ${L}^{1}-$Carathédory set-valued map with ${S}_{F}\ne \varnothing $ and let $\Theta :{L}^{1}(J,X)\to C(J,X)$ be a linear continuous mapping. Then the set-valued map $\Gamma \circ {S}_{F}:C(J,X)\to \mathcal{P}(C(J,X))$ defined by

**Lemma**

**4**

**.**Assume that Y is a separable metric space and $N:Y\to \mathcal{P}({L}^{1}(J,R))$ is a multivalued operator with the property (BC). Then there exists a continuous single-valued function $g:Y\to {L}^{1}(J,R)$ satisfying $g(x)\in N(x)$ for every $x\in Y,$ i.e., N has a continuous selection.

**Lemma**

**5.**

**Lemma**

**6.**

**Lemma**

**7.**

**Lemma**

**8.**

**Proof.**

## 3. Main Results

- (a)
- $(x,u)\mapsto F(x,u)$ is $\mathcal{L}\u2a02\mathcal{B}$ is measurable.
- (b)
- $u\mapsto F(x,u)$ is lower semicontinuous for each $x\in [a,b],$

**Theorem**

**1.**

**Proof.**

**Part (i).**T maps the bounded sets into bounded sets of $C([a,b],R)$. Set ${B}_{r}=\{v\in C([a,b],R):\parallel v\parallel \le r,\phantom{\rule{1.em}{0ex}}r>0\},$ which is a bounded ball in $C([a,b],R)$, then for $h\in T(u),$$u\in {B}_{r},$ there exists $f\in {S}_{F,u}$ such that

**Part (ii).**T maps bounded set into equicontinuous sets. Let $u\in {B}_{r},$${t}_{1},{t}_{2}\in [a,b],$${t}_{1}<{t}_{2},$ where ${B}_{r}$ is a bounded set in $C([a,b],R)$, for $u\in T(u),$ we have

**Part (iii).**T has a closed graph. Set ${u}_{n}\to {u}_{*},$${h}_{n}\in T({u}_{n})$ and ${h}_{n}\to {h}_{*}.$ Then, we shall show that ${h}_{*}\in T({u}_{*}).$ For ${h}_{n}\in T({u}_{n}),$ there exist ${f}_{n}\in {S}_{F,{u}_{n}}$ such that

**Part (iv).**T is convex for each $x\in C([a,b],R).$ Since ${S}_{F,u}$ is convex, it is obviously true.

**Part (v).**We show that there exists a open set $U\subset C([a,b],R)$, with $u\notin T(u)$ for any $\eta \in (0,1)$ and all $u\in \partial U.$ Let $\eta \in (0,1),$$u\in \eta T(u).$ Then for $t\in [a,b],$ there exists $f\in {S}_{F,u}$ such that

**Theorem**

**2.**

**Proof.**

**Claim 1.**For each $h\in C([a,b],\mathbb{R})$ the operator T is closed. Let ${\left\{{h}_{n}\right\}}_{n\ge 0}\in T(u)$ be such that ${h}_{n}\to h(n\to \infty )$ in $C([a,b],R).$ Then $h\in C([a,b],\mathbb{R})$, and there exists ${v}_{n}\in {S}_{F,u}$ such that for each $t\in [a,b],$

**Claim 2.**We shall show that there exists $\gamma <1$ such that

**Theorem**

**3.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

## 4. Applications

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Yang, D.; Bai, C.
Existence of Solutions for Anti-Periodic Fractional Differential Inclusions Involving *ψ*-Riesz-Caputo Fractional Derivative. *Mathematics* **2019**, *7*, 630.
https://doi.org/10.3390/math7070630

**AMA Style**

Yang D, Bai C.
Existence of Solutions for Anti-Periodic Fractional Differential Inclusions Involving *ψ*-Riesz-Caputo Fractional Derivative. *Mathematics*. 2019; 7(7):630.
https://doi.org/10.3390/math7070630

**Chicago/Turabian Style**

Yang, Dandan, and Chuanzhi Bai.
2019. "Existence of Solutions for Anti-Periodic Fractional Differential Inclusions Involving *ψ*-Riesz-Caputo Fractional Derivative" *Mathematics* 7, no. 7: 630.
https://doi.org/10.3390/math7070630