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

(1)

Available online at www.atnaa.org Research Article

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

Adjimi Naas^a, Maamar Benbachir^b, Mohamed S. Abdo^c, Abdellatif Boutiara^a

aLaboratory of Mathematics And Applied Sciences, University of Ghardaia, Ghardaia 47000. Algeria.

bFaculty of Sciences, Saad Dahlab University, Blida, Algeria.

cDepartment of Mathematics, Hodeidah University, Al-Hodeidah, Yemen.

Abstract

In this paper, we investigate the existence and uniqueness of solutions for a class of fractional dierential 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 xed point theorems are applied to obtain existence results. Some examples are given to illustrate our acquired results.

Keywords: Riesz-Caputo derivative Boundary value problem xed point theorem 2010 MSC: 26A33, 34A07, 93A30, 35R11

1. 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]).

Email addresses: naasadjimi@gmail.com (Adjimi Naas), mbenbachir2001@gmail.com (Maamar Benbachir), msabdo1977@gmail.com (Mohamed S. Abdo), boutiara_a@yahoo.com (Abdellatif Boutiara)

Received April 26, 2021, Accepted September 18, 2021, Online September 21, 2021

(2)

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_Tκ(t) = g(t, κ(t)), 0 < γ ≤ 1, 0 ≤ t ≤ T , κ(0) = κ0, κ(T ) = κT,

where ^RC0 D^ϑ_T is the Riesz-Caputo derivative, g : [0, T ] × R → R is a continuous function, and κ0,κT are constants.

The positive solution of nonlinear FDEs with the Riesz space derivative

_RC

0 D^ϑ₁κ(t) = h(t, κ(t)), 0 < ϑ ≤ 1, t ∈ [0, 1], κ(0) = κ0, κ(1) = κ1, κ0, κ1≥ 0,

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

_RC

0 D_T^ϑκ(t) = g(t, κ(t)), 0 < γ ≤ 1, t ∈ [0, T ], κ(0) + κ(T ) = 0, κ⁰(0) + κ⁰(T ) = 0,

where^RC₀ 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

_RC

0 D_T^ϑκ(t) + F(t, κ(t),^RC0 D_T^ςκ(t)) = 0, t ∈ J := [0, T ],

κ(0) + κ(T ) = 0, µκ⁰(0) + σκ⁰(T ) = 0, (1) where 1 < ϑ ≤ 2 and , 0 < ς ≤ 1 , ^RC₀ 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.

2. 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

E = n

κ : κ ∈ C([0, T ]), ^RCD^δκ ∈ C([0, T ]) o

, endowed with the norm

kκkE = kκk + k^RCD^δκk, and

kκk = sup

t∈J|κ(t)|, k^RCD^δκk = sup

t∈J

|^RCD^δκ(t)|.

We start with denitions.

(3)

Denition 2.1. [9, 36] For 0 ≤ t ≤ T, the classical RieszCaputo fractional derivative is dened by

RC0 D_T^ϑF(t) = 1 Γ(n − ϑ)

Z T 0

|t − ξ|^n−ϑ−1F⁽ⁿ⁾(ξ)dξ

= 1

2(^C₀D^ϑ_t + (−1)^{n C}_t D_T^ϑ)F(t),

where^C0D_t^ϑ and^Ct D_T^ϑ are the left and right Caputo derivative, respectively

C

0D^ϑ_tF(t) = 1 Γ(n − ϑ)

Z t 0

(t − ξ)^n−ϑ−1F⁽ⁿ⁾(ξ)dξ,

C

t D^ϑ_TF(t) = (−1)ⁿ Γ(n − ϑ)

Z T t

(ξ − t)^n−ϑ−1F⁽ⁿ⁾(ξ)dξ.

Remark 2.2. ([27, 31]) In particular if F(t) ∈ C([0, T ]) and 0 < ϑ ≤ 1, then

RC

0 D^ϑ_TF(t) = 1

2(^C₀D^ϑ_t − ^C_t D_T^ϑ)F(t), if F(t) ∈ C²([0, T ]) and 1 < ϑ ≤ 2, then

RC

0 D^ϑ_TF(t) = 1

2(^C₀D^ϑ_t + ^C_t D_T^ϑ)F(t).

Denition 2.3. ([31]) The Riemann-Liouville fractional integrals concepts of order ϑ are dened as

0I_t^ϑF(t) = 1 Γ(ϑ)

Z t 0

(t − ξ)^ϑ−1F(ξ)dξ,

tI_T^ϑF(t) = 1 Γ(ϑ)

Z T t

(ξ − t)^ϑ−1F(ξ)dξ,

0I_T^ϑF(t) = 1 Γ(ϑ)

Z T 0

|ξ − t|^ϑ−1F(ξ)dξ.

Lemma 2.4. ([27]) If F(t) ∈ Cⁿ([0, T ]), then

0I_t^ϑ^C₀D_t^ϑF(t) = F(t) −

n−1

X

k=0

F^(k)(0)

k! (t − 0)^k, and

tI_T^ϑ^C_t D_T^ϑF(t) = (−1)ⁿ

"

F(t) −

n−1

X

k=0

(−1)^kF^(k)(T )

k! (T − t)^k

# . From the above denitions and lemmas, we have

0I_T^ϑ^RC₀ D^ϑ_TF(t) = 1 2

0I_t^ϑ^C₀D_t^ϑ+tI_T^ϑ^C₀D_t^ϑ

F(t) + (−1)ⁿ1

2

0I_t^ϑ^C_t D^ϑ_T +tI_T^ϑ^C_tD^ϑ_T

F(t)

= 1 2

0I_t^ϑ^C₀D_t^ϑ+ (−1)ⁿ_tI_T^ϑ^C_t D_T^ϑ F(t).

In particular, if 1 < ϑ ≤ 2 and F(t) ∈ C²([0, T ]), then

0I_T^ϑ^RC₀ D_T^ϑF(t) = F(t) − 1

2(F(0) + F(T )) −1

2F⁰(0)t + 1

2F⁰(T )(T − t). (2)

(4)

3. Main Results

Lemma 3.1. Assume that g ∈ C(J , R) and κ ∈ C²(J ). Then







RC0 D_T^ϑκ(t) + g(t) = 0, t ∈ [0, T ], 1 < ϑ ≤ 2,

κ(0) + κ(T ) = 0, µκ⁰(0) + σκ⁰(T ) = 0, (3) is equivalent to the integral equation given by

κ(t) = (σ − µ)t + µT (µ + σ)Γ(ϑ − 1)

Z T 0

(T − s)^ϑ−2g(s)ds − 1 Γ(ϑ)

Z t 0

(t − s)^ϑ−1g(s)ds

− 1

Γ(ϑ) Z T

t

(s − t)^ϑ−1g(s)ds. (2)

Proof. Applying Lemma (2.4) on equation (3), we obtain κ(t) = 1

2(κ(0) + κ(T )) +1

2κ⁰(0)t −1

2κ0⁰RCI_T^ϑg(t)

= 1

2(κ(0) + κ(T )) +1

2κ⁰(0)t −1

2κ⁰(T )(T − t) − 1 Γ(ϑ)

Z t 0

(t − ξ)^ϑ−1g(ξ)dξ

− 1

Γ(ϑ) Z T

t

(ξ − t)^ϑ−1g(ξ)dξ. (3)

Then

κ⁰(t) = 1

2(κ⁰(0) + κ⁰(T )) − 1 Γ(ϑ − 1)

Z t 0

(t − ξ)^ϑ−2g(ξ)dξ

+ 1

Γ(ϑ − 1) Z T

t

(ξ − t)^ϑ−2g(ξ)dξ.

By the boundary conditions of (3), we nd that:

κ(0) = −σT

(µ + σ)Γ(ϑ − 1) Z T

0

(T − ξ)^ϑ−2g(ξ)dξ + 1 Γ(ϑ)

Z T 0

(T − ξ)^ϑ−1g(ξ)dξ,

κ(T ) = σT

(µ + σ)Γ(ϑ − 1) Z T

0

(T − ξ)^ϑ−2g(ξ)dξ − 1 Γ(ϑ)

Z T 0

(T − ξ)^ϑ−1g(ξ)dξ,

κ⁰(0) = 2σT (µ + σ)Γ(ϑ − 1)

Z T 0

(T − ξ)^ϑ−2g(ξ)dξ,

κ⁰(T ) = −2µT (µ + σ)Γ(ϑ − 1)

Z T 0

(T − ξ)^ϑ−2g(ξ)dξ.

Substituting the values of κ⁰(0)and κ⁰(T )into (3), we obtain (2).

Let us introduce the following notations:

Ω1= µT^ϑ

(µ + σ)Γ(ϑ) + 2T^ϑ Γ(ϑ + 1), Ω₂= T^ϑ−ς(µ − σ) + 2µT^ϑΓ(2 − ς)

2(µ + σ)Γ(2 − ς)Γ(ϑ) + 2T^ϑ−ς Γ(ϑ − ς + 1), k1 = T^ϑ−ς

Γ(ϑ − ς + 1), k2 = T^ϑ−ς

Γ(ϑ − ς + 1).

(5)

3.1. Uniqueness result via Banach's xed point theorem

Theorem 3.2. Let F : [0, T ] × R × R → R is a continuous function. Assume that

(H₁) there exists nonnegative real numbers L1, L₂ such that for all (ξ, v), (ξ⁰, v⁰) ∈ R², we have

|F(t, ξ, v) − F(t, ξ⁰, v⁰)| ≤ L₁|ξ − ξ⁰| + L₂|v − v⁰|, if

(Ω₁+ Ω₂)(L₁+ L₂) < 1.

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

Hκ(t) = (σ − µ)t + µT (µ + σ)Γ(ϑ − 1)

Z T 0

(T − ξ)^ϑ−2F(ξ, κ(ξ),^RCD^ςκ(ξ))dξ

− 1 (Γ)

Z t 0

(t − ξ)^ϑ−1F(ξ, κ(ξ),^RCD^ςκ(ξ))dξ

− 1 (Γ)

Z T t

(ξ − t)^ϑ−1F(ξ, κ(ξ),^RCD^ςκ(ξ))dξ.

Now, we prove that H is a contraction. For κ, $ ∈ E and for each t ∈ J , we have

|Hκ(t) − H$(t)|

≤ (σ − µ)t + µT | (µ + σ)|Γ(ϑ − 1)

Z T 0

(T − ξ)^ϑ−2|F(ξ, κ(ξ),^RCD^ςκ(ξ)) − F(ξ, $(ξ),^RCD^ς$(ξ))|dξ

+ 1

Γ(ϑ) Z t

0

(t − ξ)^ϑ−1|F(ξ, κ(ξ),^RCD^ςκ(ξ)) − F(ξ, $(ξ),^RCD^ς$(ξ))|dξ

+ 1

Γ(ϑ) Z T

t

(ξ − t)^ϑ−1|F(ξ, κ(ξ),^RCD^ςκ(ξ)) − F(ξ, $(ξ),^RCD^ς$(ξ))|dξ

≤ µT^ϑ

(2µ + σ)Γ(ϑ) L₁kκ − $k + L2k^RCD^ςκ −^RCD^ς$k + 2T^ϑ

Γ(ϑ + 1) L1kκ − $k + L2k^RCD^ςκ −^RCD^ς$k

≤

µT^ϑ

(µ + σ)Γ(ϑ) + 2T^ϑ Γ(ϑ + 1)

L₁+ L₂)(kκ − $k + k^RCD^ςκ −^RCD^ς$k . Consequently, we obtain

kHκ(t) − H$(t)k ≤ Ω1 L1+ L2)(kκ − $k + k^RCD^ςκ −^RCD^ς$k . (12)

(6)

On the other hand, we have

^RC₀ D^ς_THκ(t) −^RC0 D_T^ςHκ(t)

≤ 1

Γ(ϑ − ς) Z t

0

(t − ξ)^ϑ−ς−1|F(ξ, κ(ξ),^RCD^ςκ(ξ)) − F(ξ, $(ξ),^RCD^ς$(ξ))|dξ

+ 1

Γ(ϑ − ς) Z T

t

(ξ − t)^ϑ−ς−1|F(ξ, κ(ξ),^RCD^ςκ(ξ)) − F(ξ, $(ξ),^RCD^ς$(ξ))|dξ +

t^1−ς− (T − t)^1−ς (σ − µ) + 2µΓ(2 − ς)T 2(σ + µ)Γ(2 − ς)Γ(ϑ − 1)

Z T 0

(T − ξ)^ϑ−2|F(ξ, (ξ),^RCD^ςκ(ξ)) − F(ξ, $(ξ),^RCD^ς$(ξ))|dξ

≤ T^ϑ−ς(σ − µ) + 2µT^ϑΓ(2 − ς)

(σ + µ)Γ(2 − ς)Γ(ϑ) L₁kκ − $k + L2k^RCD^ςκ −^RCD^ς$k + 2T^ϑ−ς

Γ(ϑ − ς + 1) L1kκ − $k + L2k^RCD^ςκ −^RCD^ς$k . Thus,

k^RC₀ D^ς_THκ(t) −^RC0 D^ς_THκ(t)k ≤ Ω2 L₁+ L₂)(kκ − $k + k^RCD^ςκ −^RCD^ς$k

(13) From (12) and (13), we get

kHκ(t) − H$(t)kE ≤ (Ω₁+ Ω₂) L₁+ L₂)(kκ − $k + k^RCD^ςκ −^RCD^ς$k .

Hence, H is a contraction. As a consequence of Banach contraction principle, the problem (1) has a unique solution on J .

3.2. Existence result via Shaefer xed point theorem

Lemma 3.3. Let E be a Banach space. Assume that H : E → E be a completely continuous operator, and the set

ω(H) = {$ ∈ E : $ = λH$, λ ∈ (0, 1)} . ω(F)is bounded. Then H has a xed point in E.

Theorem 3.4. Assume that there exists a positive M such that

|F(ξ, κ, $)| < M for t ∈ J , κ, $ ∈ R.

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 ∈ Br= {κ ∈ E : kκkE ≤ r} and t ∈ J , we get

|Hκ(t)| ≤ |(σ − µ)t + µT | (σ + µ)Γ(ϑ − 1)

Z T 0

(T − ξ)^ϑ−2|F(ξ, κ(ξ),^RCD^ςκ(ξ))|dξ

+ 1

Γ(ϑ) Z t

0

(t − ξ)^ϑ−1|F(ξ, κ(ξ),^RCD^ςκ(ξ))|dξ

− 1

Γ(ϑ) Z T

t

(ξ − t)^ϑ−1|F(ξ, κ(ξ),^RCD^ςκ(ξ))|dξ

≤ µT^ϑM

(σ + µ)Γ(ϑ) + 2MT^ϑ Γ(ϑ + 1).

(7)

Which implies that,

kHκ(t)k ≤ MΩ1, (14)

and,

|^RC₀ D^ς_THκ(t)| ≤ 1 Γ(ϑ − ς)

Z t 0

(t − ξ)^ϑ−ς−1|F(ξ, κ(ξ),^RCD^ςκ(ξ))|dξ

+ 1

Γ(ϑ − ς) Z T

t

(t − ξ)^ϑ−ς−1|F(ξ, κ(ξ),^RCD^ςκ(ξ))|dξ +

t^1−ς− (T − t)^1−ς (σ − µ) + 2µΓ(2 − ς)T 2(σ + µ)|Γ(2 − ς)Γ(ϑ − 1)

× Z T

0

≤ T^ϑ−ς(µ + σ) + 2µT^ϑΓ(2 − ς)

2|µ − σ|Γ(2 − ς)Γ(ϑ) M+ 2T^ϑ−ς 2Γ(ϑ − ς + 1)M

≤ T^ϑ−ς(σ − µ)) + 2µT^ϑΓ(2 − ς)

2(σ + µ)Γ(2 − ς)Γ(ϑ) M+ 2T^ϑ−ς Γ(ϑ − ς + 1)M.

Which implies that,

k^RC₀ D_T^ςHκ(t)k ≤ Ω2M. (15)

Adds side of inequality (14)and(15), we get

k^RC₀ D^ς_THκ(t)kE < M (Ω₁+ Ω₂) ∞.

Which implies that H maps bounded sets into bounded sets on E.

step 3. H maps bounded sets into equicontinuous sets in E.

Let Br be a bounded set of E as in step 2, and let κ ∈ Br. For each t1, t2 ∈ J , t₁< t2, we have

|Hκ(t2) − Hκ(t1)| ≤ (σ − µ)(t₂− t₁) (σ + µ)Γ(ϑ − 1)

Z T 0

(T − ξ)^ϑ−2|F(κ(ξ),^RCD^ςκ(ξ))|dξ

+ 1

Γ(ϑ) Z t1

0

(t1− ξ)^ϑ−1− (t₂− ξ)^ϑ−1

|F(κ(ξ),^RCD^ςκ(ξ))|dξ

+ 1

Γ(ϑ) Z t2

t1

(ξ − t1)^ϑ−1− (t₂− ξ)^ϑ−1

+ 1

Γ(ϑ) Z T

t2

(ξ − t₁)^ϑ−1− (ξ − t₂)^ϑ−1

≤

(t^ϑ₁− t^ϑ₂) + (t2− t₁)^ϑ

Γ(ϑ + 1) M+(σ − µ)|t2− t₁|^ϑT^ϑ (σ + µ)Γ(ϑ + 1) M +

(T − t₁)^ϑ− (T − t₂)^ϑ + (t₂− t₁)^ϑ

Γ(ϑ + 1) M

≤

(t^ϑ₁− t^ϑ₂) + (t₂− t₁)^ϑ

Γ(ϑ + 1) M+(σ − µ)|t₂− t₁|^ϑT^ϑ (σ + µ)Γ(ϑ + 1) M +

(T − t1)^ϑ− (T − t₂)^ϑ + (t₂− t₁)^ϑ

Γ(ϑ + 1) M,

(8)

and,

|^RC₀ D_T^ςHκ(t2) −^RC₀ D_T^ςHκ(t1)| ≤ (t^ϑ−ς₁ − t^ϑ−ς₂ ) 2Γ(ϑ − ς + 1)M + (T − t₂)^ϑ−ς− (T − t₁)^ϑ−ς + (t₂− t₁)^ϑ−ς

Γ(ϑ − ς + 1) M

+ (σ − µ)(t^1−ς₂ − t^1−ς₁ ) + ((T − t₂)^1−ς− (T − t₁)^1−ς) T^ϑ−1M

2(σ + µ)Γ(2 − ς)Γ(ϑ) .

Hence,

kHκ(t2) − Hκ(t1)k_κ≤

(t^ϑ₁ − t^ϑ₂) + (t₂− t₁)^ϑ

Γ(ϑ + 1) M+(σ − µ)|t₂− t₁|^ϑT^ϑ (σ + µ)Γ(ϑ + 1) M +

(T − t1)^ϑ− (T − t₂)^ϑ + (t₂− t₁)^ϑ

Γ(ϑ + 1) M

+ (t^ϑ−ς₁ − t^ϑ−ς₂ ) 2Γ(ϑ − ς + 1)M

+ (T − t2)^ϑ−ς− (T − t₁)^ϑ−ς + (t₂− t₁)^ϑ−ς

Γ(ϑ − ς + 1) M

+(σ − µ)(t^1−ς₂ − t^1−ς₁ ) + ((T − t2)^1−ς− (T − t₁)^1−ς) T^ϑ−1M

2(σ + µ)Γ(2 − ς)Γ(ϑ) .

Which implies that kHκ(t2) − Hκ(t1)kE → 0 as t2 → t₁.By Arzela-Ascoli theorem, we conclude that H is completely continuous operator.

step 4. We show that the set ∆ dened by

∆ = {κ ∈ E, κ = ρH(κ), 0 < ρ < 1}

is bounded. Let κ ∈ ∆, for some ρ ∈ (0, 1). For each t ∈ J , we have 1

ρ|κ(t)| ≤ |(σ − µ)t + µT | (σ + µ)Γ(ϑ − 1)

Z T 0

+ 1

Γ(ϑ) Z t

0

(t − ξ)^ϑ|F(ξ, κ(ξ),^RCD^ςκ(ξ))|dξ

+ 1

Γ(ϑ) Z T

t

(ξ − t)^ϑ|F(ξ, κ(ξ),^RCD^ςκ(ξ))|dξ

≤ µT^ϑ

(σ + µ)Γ(ϑ)M+ 2T^ϑ Γ(ϑ + 1)M.

Therefore,

kκk ≤ ρΩ1M. (16)

(9)

and,

1

ρ|^RC₀ D_T^ςκ(t)| ≤ 1 Γ(ϑ − ς)

Z t 0

+ 1

Γ(ϑ − ς) Z T

t

(ξ − t)^ϑ−ς−1|F(ξ, κ(ξ),^RCD^ςκ(ξ))|dξ +

× Z T

0

≤ T^ϑ−ς(µ + σ) + 2µT^ϑΓ(2 − ς)

2|µ − σ|Γ(2 − ς)Γ(ϑ) M+ 2T^ϑ−ς 2Γ(ϑ − ς + 1)M

≤ T^ϑ−ς(σ − µ) + 2µT^ϑΓ(2 − ς)

2(σ + µ)Γ(2 − ς)Γ(ϑ) M+ 2T^ϑ−ς Γ(ϑ − ς + 1)M.

Therefore,

k^RC₀ D^ς_Tκ(t)k ≤ ρΩ2M. (17)

Adds side of inequality (16)and (17), we get

kκkE ≤ ρ(Ω₁+ Ω₂)M.

Hence,

kκkE < ∞.

This shows that ∆ is bounded.

As consequence of Schaefer's xed point theorem, the problem (1) has at least one solution in [0, T ].

3.3. Existence result via 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.

Theorem 3.6. Let F : J × R × R → R be a continuous function, and let the conditions (H1)-(H2) hold. In addition, the function F satisfying the assumptions :

(H3) There exists a nonnegative function Ω ∈ C(J ,R⁺) such that

|F(t, κ, $)| ≤ Ω(t) for any (t, κ, $) ∈ J × R × R, (H₄) (k₁+ k₂)(L₁+ L₂) < 1 ,

Then the problem (1) has a least one solution in J . Proof. We dene two operators H1κ(t) and H2κ(t) as:

(H1κ)(t) = (σ − µ)t) + µT (σ + µ)Γ(ϑ − 1)

Z T 0

(T − ξ)^ϑ−2F(ξ, κ(ξ),^RCD^ςκ(ξ))dξ, (H2κ(t)) = − 1

Γ(ϑ) Z t

0

(t − ξ)^ϑ−1F(ξ, κ(ξ),^RCD^ςκ(ξ))dξ

− 1

Γ(ϑ) Z _T

t

(ξ − t)^ϑ−1F(ξ, κ(ξ),^RCD^ςκ(ξ))dξ.

(10)

Choosing d ≥ (Ω1+ Ω₂)(L₁+ L₂)kΩk, and we consider Bd= {κ ∈ E : kκkE ≤ d}. Step1 We shall prove that H1κ(t) + H2κ(t) ∈ Bd.

For any κ, $ ∈ Bd and for each then t ∈ J , we have

|H₁κ(t) + H2$(t)| ≤ |(σ − µ)t + µT | (σ + µ)Γ(ϑ − 1)

Z T 0

+ 1

Γ(ϑ) Z t

0

(t − ξ)^ϑ−1|F(ξ, κ(ξ),^RCD^ςκ(ξ))|dξ

− 1

Γ(ϑ) Z T

t

(ξ − t)^ϑ−1|F(ξ, κ(ξ),^RCD^ςκ(ξ))|dξ

≤ µT^ϑ

(σ + µ)Γ(ϑ)(L1+ L2) kΩk + 2T^ϑ

Γ(ϑ + 1)(L1+ L2)) kΩk.

Then

kH₁κ(t) + H2$(t)k ≤ Ω₁(L₁+ L₂)kΩk (18) On the other hand,

|^RC₀ D^ς_THκ(t) +^RC0 D^ς_TH$(t)| ≤ 1 Γ(ϑ − ς)

Z t 0

+ 1

Γ(ϑ − ς) Z T

t

(ξ − t)^ϑ−ς−1|F(ξ, κ(ξ),^RCD^ςκ(ξ))|dξ +

Z T 0

≤ T^ϑ−ς(σ − µ) + 2µT^ϑΓ(2 − ς)

2(σ + µ)Γ(2 − ς)Γ(ϑ) (L₁+ L₂) kΩk + 2T^ϑ−ς

Γ(ϑ − ς + 1)(L₁+ L₂) kΩk.

Hence

k^RC₀ D_T^ςHκ(t) +^RC0 D^ς_TH$(t)k ≤ Ω₁(L1+ L2)kΩk (19) It follows from (18) and (19) that

kH₁κ(t) + H2$(t)kE ≤ (Ω₁+ Ω2)(L1+ L2)kΩk ≤ d.

Hence, H1κ(t) + H2κ(t) ∈ Bd .

step2 We shall prove that H1is continuous and compact. The continuity of F implies that the operator H1

is continuous.

Now, we prove that H2 maps bounded sets into bounded sets of E.

For κ ∈ Bd,and for each t ∈ J , we have

|H₁κ(t)| ≤

|(σ − µ) + µT | (σ + µ)Γ(ϑ − 1)

Z T 0

(T − ξ)^ϑ−2|F(ξ, κ(ξ),^RCD^ςκ(ξ))|dξ µT^ϑ

(σ + µ)Γ(ϑ − 1)(L₁+ L₂)kΩk.

Hence

kH₁κ(t)k ≤ µT^ϑ

(σ + µ)Γ(ϑ − 1)(L₁+ L₂)kΩk, (20)

(11)

and

|^RCD^ςH₁κ(t)| ≤

((T − t)^1−ς− t^1−ς)(σ − µ) + 2µT Γ(2 − ς) 2(σ + µ)Γ(2 − ς)Γ(ϑ − 1)

× Z T

0

≤ T^ϑ−ς(σ − µ) + 2µT^ϑΓ(2 − ς)

2(σ + µ)Γ(2 − ς)Γ(ϑ) (L₁+ L₂)kΩk.

Hence

k^RCD^ςH₁κ(t)k ≤ T^ϑ−ς(σ − µ) + 2µT^ϑΓ(2 − ς)

2(σ + µ)Γ(2 − ς)Γ(ϑ) (L₁+ L₂)kΩk. (21) Combining (20) and (21), we get

kH₁κ(t)kE ≤

µT^ϑ

(σ − µ)Γ(ϑ − 1) +T^ϑ−ς(σ − µ) + 2µT^ϑΓ(2 − ς) 2(σ + µ)Γ(2 − ς)Γ(ϑ)

(L₁+ L₂)kΩk, Consequently

kH₁κ(t)kE ≤ ∞.

Thus, it follows the above inequality that operator H1 is uniformly bounded.

The operator H1 maps bounded sets into equicontinuous sets of E. Let t1, t2 ∈ J , t₁ < t2, κ ∈ Bd, then we have :

|H₁κ(t2) − H1κ(t1)| ≤ (σ − µ)(t2− t₁) (σ + µ)Γ(ϑ − 1)

Z T 0

(T − ξ)^ϑ−2|F(κ(ξ),^RCD^ςκ(ξ))|dξ +(σ − µ)|t₂− t₁|^ϑT^ϑ

(σ + µ)Γ(ϑ + 1) (L₁+ L₂)kκk.

On the other hand,

|^RCD^ςH₁κ(t2) −^RCD^ςH₁κ(t1)|

≤ (σ − µ)(t^1−ς₂ − t^1−ς₁ ) + ((T − t2)^1−ς− (T − t₁)^1−ς) T^ϑ−1(L1+ L2)kΩk 2(σ + µ)|Γ(2 − ς)Γ(ϑ)

It follows from (22) and (23) that

kH₁κ(t2) − H₁κ(t1)k_E ≤ (σ − µ)|t₂− t₁|^ϑT^ϑ

(σ + µ)Γ(ϑ + 1) (L₁+ L₂)kκk

+(σ − µ)(t^1−ς₂ − t^1−ς₁ ) + ((T − t₂)^1−ς− (T − t₁)^1−ς) T^ϑ−1(L₁+ L₂)kΩk 2(σ + µ)|Γ(2 − ς)Γ(ϑ).

As t2→ t₁, the right-hand side of this inequality tends to zeros.

Then as a consequence od steps, we can conclude that H1 is continuous and compact.

Step3 Now, we prove that H2 is contraction mapping . Let κ, $ ∈ E. Then, for each t ∈ J , we have

|H₂κ(t) − H2$(t)| ≤ + 1 Γ(ϑ)

Z t 0

(t − ξ)^ϑ−1|F(ξ, κ(ξ),^RCD^ςκ(ξ)) − F(ξ, $(ξ),^RCD^ς$(ξ))|dξ

+ 1

Γ(ϑ) Z T

t

(ξ − t)^ϑ−1|F(ξ, κ(ξ),^RCD^ςκ(ξ)) − F(ξ, $(ξ),^RCD^ς$(ξ))|dξ

≤ 2T^ϑ

Γ(ϑ + 1)(L1+ L2) kκ − $k + k^RCD^ςκ −^RCD^ς$k

(12)

Consequently we obtain

kH₂κ(t) − H2$(t)k ≤ k1(L1+ L2)(kκ − $k + k^RCD^ςκ −^RCD^ς$k) (24) and

|^RC₀ D^ς_TH₂κ(t) −^RC0 D^ς_TH₂κ(t)|

≤ 1

Γ(ϑ − ς) Z t

0

(t − ξ)^ϑ−ς−1|F(ξ, κ(ξ),^RCD^ςκ(ξ)) − F(ξ, $(ξ),^RCD^ς$(ξ))|dξ

+ 1

Γ(ϑ − ς) Z T

t

(ξ − t)^ϑ−ς−1|F(ξ, κ(ξ),^RCD^ςκ(ξ)) − F(ξ, $(ξ),^RCD^ς$(ξ))|dξ

≤ 2T^ϑ−ς

Γ(ϑ − ς + 1)(L1+ L2) kκ − $k + k^RCD^ςκ −^RCD^ς$k , and

k^RCD^ςH₂κ(t) −^RCD^ςH₂$(t)k ≤ k₂(L₁+ L₂)(kκ − $k + k^RCD^ςκ −^RCD^ς$k) (25) It follows from (24) and (25) that

kH₂κ(t) − H2$(t)k_E ≤ (k₁+ k₂)(L₁+ L₂) kκ − $k + k^RCD^ςκ −^RCD^ς$k

Using the condition (H4), we conclude that H2 is a contraction mapping. As consequence of a krasnosselski's

xed point theorem, we deduce that H has a xed point which as solution of (1).

Example 3.7. Consider following nonlinear FDE with Riesz-Caputo derivative:







RC0 D

3 2

Tκ(t) + ⁽

√π+1)|κ(t)|

√

t²+144(1+|κ(t)|)+_(1+e¹2π)²cos(^RCD¹³κ(t)) = 0, t ∈ [0, 1],

$(0) + $(1) = 0, 2$⁰(0) + ¹₂$⁰(1) = 0, (26)

Here, ϑ = ³₂, ς = ¹₃, µ = 2, σ = ¹₂, and,

F(t, κ(t),^RCD^ςκ(t)) = (√

π + 1)|κ|

√

t²+ 144(1 + |κ|) + 1

(1 + e^2π)²cos(^RCD¹³κ(t)).

We have

|F(κ, $) − F(κ⁰, $⁰)| ≤

√π + 1

12 kκ − κ⁰k + 1

(1 + e^2π)²k$ − $⁰k.

Then, the assumption (H1) is satised with L1 =

√π+1

12 , L₂ = _(1+e¹2π)². Using the Matlab program, Ω1 = 2, 1493, Ω2 = 1, 9057.

Therefore, (L1+ L₂)(Ω₁+ Ω₂) = 0, 9369 < 1. By using the theorem (3.2), the problem (26) has a unique solution on [0, 1].

Example 3.8. Consider following nonlinear FDE with Riesz-Caputo derivative:











RC0 D

5 3

Tκ(t) +

e^−2tsin(_1+|κ|^|κ| )

(e^2π+2) + ^|^RC^D

1 2κ(t)|(

√π+1)

(|^RCD¹²κ(t)|+1)(π+2)²

= 0, t ∈ [0, 1],

$(0) + $(1) = 0,³₅$⁰(0) + ²₃$⁰(1) = 0, (27)

Here, ϑ = ⁵₃, ς = ¹₂, µ = ³₅, σ = ²₃, L₁=

√π+1

12 , L₂ = _(1+e¹2π)², and F(t, κ(t),^RCD^ςκ(t)) =

e^−2tsin(_1+|κ|^|κ| )

(e^2π+ 2) + |^RCD¹²κ(t)|(

√π + 1) (|^RCD¹²κ(t)| + 1)(π + 2)²

.

(13)

Moreover,

|F(κ, $) − F(κ⁰, $⁰)| ≤ 1

(e^2π+ 2)kκ − κ⁰k +

√π + 1

(1 + π)²k$ − $⁰k.

Therefor,

|F(t, κ(t),^RCD^ςκ(t))| ≤ e^−2t e^2π+2 +

√π + 1

(π + 2)² = Ω(t).

Ω₁= 2, 1493, Ω₂= 1, 9057, K₁ = 1, 5045, k₂ = 0, 9239 .

Then (k1+ k2)(L1+ L2) = 0, 2406 < 1, (H4) is satised, by using the theorem (3.6), the problem (27) has at least one solution on [0, 1].

4. 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.

References

[1] S. Abbas, M. Benchohra, G.M. N'Guérékata, Topics in Fractional Dierential Equations, Springer, New York, 2012.

[2] S. Abbas, M. Benchohra, G.M. N'Guérékata, Advanced Fractional Dierential and Integral Equations, Nova Science Publishers, New York, 2015.

[3] T. Abdeljawad, On conformable fractional calculus, Journal of computational and Applied Mathematics 279(2015), 57-66.

[4] M.S. Abdo, T. Abdeljawad, K.D. Kucche, M.A. Alqudah , S. M. Ali, M.B. Jeelani, On nonlinear pantograph fractional dierential equations with Atangana- Baleanu-Caputo derivative, Advances in Dierence Equations 1(2021), 1-17.

[5] M.S. Abdo, T. Abdeljawad, S.M. Ali, K. Shah, On fractional boundary value problems involving fractional derivatives with Mittag-Leer kernel and nonlinear integral conditions, Advances in Dierence Equations 1(2021), 1-21.

[6] M.S. Abdo, T. Abdeljawad, S.M. Ali, K. Shah, F. Jarad, Existence of positive solutions for weighted fractional order dierential equations, Chaos Solitons Fractals 141(2020), 110341.

[7] M.S. Abdo, Further results on the existence of solutions for generalized fractional quadratic functional integral equations, Journal of Mathematical Analysis and Modeling 1(1)(2020), 33-46.

[8] M.S. Abdo, T. Abdeljawad, K. Shah, F. Jarad, Study of Impulsive Problems Under Mittag-Leer Power Law, Heliyon 6(10)(2020), e05109.

[9] O.P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, Journal Phys 40(2007), 62876303.

[10] B. Ahmad, S.K. Ntouyas, A. Alsaedi, On fractional dierential inclusions with anti-periodic type integral boundary conditions, Boundary Value Problem 82(2013).

[11] B. Ahmad, S.K. Ntouyas, Existence results for fractional dierential inclusions with Erdelyi-Kober fractional integral conditions, Analele Universitatii" Ovidius" Constanta-Seria Matematica 25(2) (2017), 5-24.

[12] B. Ahmad, V. Otero-Espinar, Existence of solutions for fractional dierential inclusions with antiperiodic boundary conditions, Boundary Value Problem (2009), 625347.

[13] M.A. Almalahi, S.K. Panchal, Some existence and stability results for ψ-Hilfer fractional implicit dierential equation with periodic conditions, Journal of Mathematical Analysis and Modeling, 1(1)(2020), 1-19.

[14] R. Almeida, Fractional variational problems with the Riesz-Caputo derivative, Appl.Math.Lett. 25(2012)142-148.

[15] R. Almeida, Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer.

Simul. 44 (2017), 460481.

[16] A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci. 20(2)(2016), 76369.

[17] M. Benchohra, J.E. Lazreg, Existence and Ulam stability for nonlinear implicit fractional dierential equations with Hadamard derivative, Stud. Univ. Babes-Bolyai Math. 62(1)(2017) 2738.

[18] A. Boutiara, M.S. Abdo, M. Benbachir, Existence results for ψCaputo fractional neutral functional integro-dierential equations with nite delay, Turk J Math (2020) 44: 23802401.

[19] A. Boutiara, Mixed fractional dierential equation with nonlocal conditions in Banach spaces, Journal of Mathematical Modeling 9(3)(2021), 451-463.

[20] A. Boutiara, K. Guerbati, M. Benbachir, Caputo-Hadamard fractional dierential equation with three-point boundary conditions in Banach spaces, AIMS Mathematics 5(1)(2020), 259272.

(14)

[21] A. Boutiara, M. Benbachir, K. Guerbati, Caputo Type Fractional Dierential Equation with Nonlocal Erdélyi-Kober Type Integral Boundary Conditions in Banach Spaces, Surveys in Mathematics and its Applications 15(2020): 399418.

[22] A. Boutiara, M. Benbachir, K. Guerbati, Measure Of Noncompactness for Nonlinear Hilfer Fractional Dierential Equation in Banach Spaces, Ikonion Journal of Mathematics 1(2)(2019).

[23] Y.K. Chang, J.J. Nieto, Some new existence results for fractional dierential inclusions with boundary conditions, Math.

Comput. Modelling 49 (2009), 605-609.

[24] G.Y. Chuan, Z. Jun, W.C. Guo, Positive solution of fractional diferential equations with the Riesz space derivative, Applied Mathematics Letters,Elsevier 95(2019)59-64.

[25] M. Caputo, M. Fabrizio, A new Denition of Fractional Derivative without Singular Kernel, Progress in Fractional Dier- entiation and Applications. 1(2) (2015), 7385.

[26] F.L. Chen, A.P. Chen, X. Wu, Anti-periodic boundary value problems with Riesz-Caputo derivative, Adv. Dier. Equ.

2019(2019), 119.

[27] C. Fulai, C. Anping, W. Xia, Anti-periodic boundary value problems with the Riesz-Caputo derivative, Advanced in Dierence equations.Springer(2019).

[28] M.S. Abdo, T. Abdeljawad, S. M. Ali, K. Shah, F. Jarad, Existence of positive solutions for weighted fractional order dierential equations, Chaos Solitons Fractals 141(2020), 110341.

[29] M.A. Abdulwasaa, M.S. Abdo, K. Shah, T.A. Nofal, S.K. Panchal, S.V. Kawale, A. H. Abdel-Aty, Fractal-fractional mathematical modeling and forecasting of new cases and deaths of COVID-19 epidemic outbreaks in India, Results in Physics, 20(2021), 103702.

[30] M. Iqbal, K. Shah, R.A. Khan, On using coupled xed-point theorems for mild solutions to coupled system of multipoint boundary value problems of nonlinear fractional hybrid pantograph dierential equations, Math. Meth. Appl. Sci. 44(10), (2021), 8113-8124.

[31] C. Fulai, B. Dumitru, W.C. Guo, Existence results of fractional dierential equations with the Riesz-Caputo derivative, Eur. phys. J. special topics 226,4341-4325.

[32] F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-type modication of the Hadamard fractional derivatives, Adv. Dierence Equ. 8(142) (2012).

[33] U.N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6(4) (2014), 1-15.

[34] A.A. Kilbas, M.H. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Dierential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B. V, Amsterdam, 2006.

[35] F. Mainardi, Fractional calculus: Some basic problem in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics. Springer. Vienna (1997).

[36] I. Podlubny, Fractional Dierential Equations, Academic Press, San Diego (1999).

[37] J.V.C. Sousa, E.C.D. Oliveira, On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simula 60 (2018), 72-91.

[38] J.E.N. Valdes, P.M. Guzmán , M.L.M. Bittencurt, A note on the qualitative behavior of some nonlinear local improper conformable dierential equations, Journal of Fractional Calculus and Nonlinear Systems, 1(1)(2020), 13-20.

[39] H.A. Wahash, S.K. Panchal, Positive solutions for generalized Caputo fractional dierential equations using lower and upper solutions method, Journal of Fractional Calculus and Nonlinear Systems 1(1) (2020), 1-12.

[40] F. Xu, Fractional boundary value problems with integral and Anti-periodic boundary conditions, Bull.Malys.Math.Sci.Soc.

39, 571-587.