Analysis of a fractional boundary value problem involving Riesz-Caputo fractional derivative.

In this paper, we investigate the existence and uniqueness of solutions for a class of fractional di(cid:27)erential equations with boundary conditions in the frame of Riesz-Caputo operators. We apply the methods of functional analysis such that the uniqueness result is established by using Banach’s contraction principle, whereas Schaefer’s and Krasnoslkii’s (cid:28)xed point theorems are applied to obtain existence results. Some examples are given to illustrate our acquired results.


Introduction
Fractional calculus (FC) is a mathematical branch that investigates the properties of derivatives and integrals of non-integer order.The interested readers in the subject should refer to the books [34,35,36].Fractional order models, which provide an excellent description of memory and genetic processes, are more accurate and appropriate than models with integer order.For the development of FC, there are sundry common denitions of fractional derivatives and integrals, such as Rimann-Liouville type, Caputo type, Hadamard type, Hilfer type, ψ-Caputo, ψ-Hilfer type, Caputo-Fabrizio type, Atangana-Baleanu type, conformable type, and Erdelyi-Kober type, etc, (see [11,15,16,25,32,33,37,3]). Some recent contributions have been investigated the existence and uniqueness of solutions for dierent kinds of nonlinear fractional dierential equations (FDEs) and inclusion (FDIs) by using various types of xed point theorems, which can be found in [13,6,17,7,39,38,8,19,20,21,4,5,1,2,18], and the references cited therein.The study of FDEs or FDIs with anti-periodic boundary conditions, that are applied in numerous dierent elds, like chemical engineering, physics, economics, dynamics, etc., has received much attention recently, (see [23,27,40,10,26]) and the papers mentioned therein.On the other hand, the authors in [31] investigated the existence results of the following FDEs RC 0 D ϑ T κ(t) = g(t, κ(t)), 0 < γ ≤ 1, 0 ≤ t ≤ T, κ(0) = κ 0 , κ(T ) = κ T , where RC 0 D ϑ T is the Riesz-Caputo derivative, g : [0, T ] × R → R is a continuous function, and κ 0 ,κ T are constants.The positive solution of nonlinear FDEs with the Riesz space derivative has been studied by Yun Gu et al., in [24].Also, Chen et al., in [27] discussed a class of FDEs with anti-periodic boundary condithions of the form where RC 0 D ϑ T is the Riesz-Caputo derivative and g : [0, 1] × R → R is a continuous function.Motivated by the above cited work, in this paper, we investigat the existence and uniqueness results of the following FDEs with the Riesz-Caputo derivative where 1 < ϑ ≤ 2 and , 0 < ς ≤ 1 , RC 0 D κ T is the Riesz-Caputo fractional derivative of order κ ∈ {ϑ, ς}, F : J × R × R → R, is a continuous function, and µ,σ are nonnegative constants with µ > σ.We refer here to some very recent works that dealt with a similar analysis, see [28,29,30].The paper is marshaled as follows.Section 2 has denitions and some of the most important basic concepts of the FC.In section 3, we prove the existence and uniqueness of solutions for the proposed problem with the Riesz-Caputo derivatives via Banach's, Schaefer's, and Krasnoselskii's xed point theorems.Some illustrative examples associated with our suggested problem are provided in Section 4.

Preliminaries
In this section ,we recall some basic concepts, and preliminary facts.By E = C(J , R) we denote the Banach space of all continuous functions from J into R as follows endowed with the norm We start with denitions.Denition 2.1.[9,36] For 0 ≤ t ≤ T, the classical RieszCaputo fractional derivative is dened by where C 0 D ϑ t and C t D ϑ T are the left and right Caputo derivative, respectively Denition 2.3.( [31]) The Riemann-Liouville fractional integrals concepts of order ϑ are dened as and From the above denitions and lemmas, we have In particular, if

Main Results
is equivalent to the integral equation given by Proof.Applying Lemma (2.4) on equation ( 3), we obtain Then By the boundary conditions of (3), we nd that: Substituting the values of κ (0)and κ (T ) into (3), we obtain (2).Let us introduce the following notations: , there exists nonnegative real numbers L 1 , L 2 such that for all (ξ, v), (ξ , v ) ∈ R 2 , we have Then the problem (1) has a unique solution on J .
Proof.We transform BVP (1) into xed point problem.Then we dene the integral operator H : E → E by Now, we prove that H is a contraction.For κ, ∈ E and for each t ∈ J , we have Consequently, we obtain On the other hand, we have Thus, From ( 12) and ( 13), we get Hence, H is a contraction.As a consequence of Banach contraction principle, the problem (1) has a unique solution on J .Theorem 3.4.Assume that there exists a positive M such that

Existence result via
Then the problem (1) has at least one solution on J .
Proof.We will use the Sheafer's xed point theorem, to prove H has a xed point on E, we subdivided the proof into several steps : Step 1. H is continuous on E: in view of continuity of F, we conclude that operator H is continuous.step 2. H maps bounded sets into bounded sets in E.
For each κ ∈ B r = {κ ∈ E : κ E ≤ r} and t ∈ J , we get and, Which implies that, Adds side of inequality ( 14)and( 15), we get Which implies that H maps bounded sets into bounded sets on E.
step 3. H maps bounded sets into equicontinuous sets in E.
Let B r be a bounded set of E as in step 2, and let κ ∈ B r .For each t 1 , t 2 ∈ J , t 1 < t 2 , we have and, Hence, Which implies that Hκ(t 2 ) − Hκ(t 1 ) E → 0 as t 2 → t 1 .By Arzela-Ascoli theorem, we conclude that H is completely continuous operator.
step 4. We show that the set ∆ dened by is bounded.Let κ ∈ ∆, for some ρ ∈ (0, 1).For each t ∈ J , we have Therefore, and, Therefore, Adds side of inequality ( 16)and ( 17), we get Hence, κ E < ∞.This shows that ∆ is bounded.As consequence of Schaefer's xed point theorem, the problem (1) has at least one solution in [0, T ].Theorem 3.6.Let F : J × R × R → R be a continuous function, and let the conditions (H 1 )-(H 2 ) hold.In addition, the function F satisfying the assumptions :

Existence result via
Then the problem (1) has a least one solution in J .
For any κ, ∈ B d and for each then t ∈ J , we have On the other hand, It follows from ( 18) and ( 19) that step2 We shall prove that H 1 is continuous and compact.The continuity of F implies that the operator H 1 is continuous.Now, we prove that H 2 maps bounded sets into bounded sets of E.
For κ ∈ B d ,and for each t ∈ J , we have Hence and Combining ( 20) and ( 21), we get Thus, it follows the above inequality that operator H 1 is uniformly bounded.
The operator H 1 maps bounded sets into equicontinuous sets of E. Let t 1 , t 2 ∈ J , t 1 < t 2 , κ ∈ B d , then we have : On the other hand, It follows from ( 22) and ( 23) that As t 2 → t 1 , the right-hand side of this inequality tends to zeros.Then as a consequence od steps, we can conclude that H 1 is continuous and compact.
Step3 Now, we prove that H 2 is contraction mapping .

Conclusion
We have eectively achieved several necessary conditions describing the the existence and uniqueness of solutions for a class of fractional dierential equations with boundary conditions involving Riesz-Caputo fractional derivatives.Under some xed point theorems such as Banach, Schaefer, and Krasnoselskii, the necessary results have been investigated.Moreover, by giving appropriate examples, all the main results have been testied.In future such type of analysis can be established for more general type fractional dierential equations involving ψ-Riesz-Caputo fractional derivatives.
Karesnoslskii's xed point theorem Lemma 3.5.(Karasnoselskii's xed point theorem) Let M a closed bounded, convex and nonempty subset of a Banach space E, let A, Bbe operator, such that (a) Aκ + B ∈ M , whenever, κ, ∈ M , (b) A is compact and continuous, (c) B is a contraction mapping, then there exist z ∈ M such that z = Az + Bz.