NONLINEAR SEQUENTIAL RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS WITH NONLOCAL AND INTEGRAL BOUNDARY CONDITIONS

In this paper, we discuss the existence and uniqueness of solutions for a new class of sequential fractional differential equations of Riemann-Liouville and Caputo types with nonlocal integral boundary conditions, by using standard fixed point theorems. We also demonstrate the application of the obtained results with the aid of examples. Received 2018-09-10; accepted 2018-10-24; published 2019-01-04. 2010 Mathematics Subject Classification. 26A33, 34A08, 34B15.

Very recently in [19], the authors discussed existence and uniqueness of solutions for two sequential Caputo-Hadamard and Hadamard-Caputo fractional differential equations subject to separated boundary conditions as and where C D p and H D q are the Caputo and Hadamard fractional derivatives of orders p and q, respectively, Motivated by the above papers, we consider in the present paper the following boundary value problem where RL D q , C D r are Riemann-Liouville and Caputo fractional derivatives of orders q, r ∈ (0, 1), respectively, I p is the Riemann-Liouville fractional integral of order p > 0, f, g : J × R → R are given continuous functions and φ, h : C(J, R) → R are two given functionals.
The rest of the paper is arranged as follows.In Section 2, we establish basic results that lays the foundation for defining a fixed point problem equivalent to the given problem (1.5).The main results, based on Banach's contraction mapping principle, Krasnoselskii's fixed point theorem and nonlinear alternative of Leray-Schauder type, are obtained in Section 3. Examples illustrating the obtained results are also included.

Preliminaries
In this section, we recall some basic concepts of fractional calculus [12,16] and present known results needed in our forthcoming analysis.
Definition 2.1.The Riemann-Liouville fractional derivative of order q for a function f : (0, ∞) → R is defined by where [q] denotes the integer part of the real number q, provided the right-hand side is pointwise defined on (0, ∞).
Definition 2.2.The Riemann-Liouville fractional integral of order q for a function f : (0, ∞) → R is defined by provided the right-hand side is pointwise defined on (0, ∞).
In the following, for simplicity, we use the notation I q for RL I q .
Lemma 2.6.Let p > 0, 0 < q, r ≤ 1, with q + r > 1, Λ = Γ(q) Γ(q + r) and ĝ, y ∈ C(J, R) and two functionals φ, h : C(J, R) → R. The unique solution of the linear problem is given by x(t) = I r ĝ(t) + I q+r y(t) Proof.Firstly, we apply the Riemann-Liouville fractional integral of order q to both sides of equation (2.2), and using Lemma 2.4, we have where a constant c 1 ∈ R.After that, using Riemann-Liouville fractional integral of order r to both sides the above equation and applying Lemma 2.5, we get where a constant c 2 ∈ R. Observe that the equation (2.4) is well defined as q + r > 1.
Using nonlocal boundary condition of problem (2.2) to the above equation, we obtain the linear system c 1 Γ(q) Γ(q + 1) Note that the two functionals φ(x) and h(x) are constants.Solving the system of linear equations for constants c 1 , c 2 , we have T p+q+r−1 φ(x) − I r ĝ(η) − I q+r y(η) .
Substituting two constants c 1 and c 2 into equation (2.4), we obtain the required solution.The converse follows by direct computation.The proof is completed.

Main results
Let J = [0, T ] and C = C(J, R) denotes the Banach space of all continuous functions from J to R endowed with the norm defined by x = sup t∈J |x(t)|.By Lemma 2.6, we define an operator with Λ = 0.It should be noticed that problem (1.5) can be transformed into a fixed point equation x = Ax.
To accomplish of the study, we will use fixed point theorems to prove that the operator A has fixed points.
For the sake of convenience, we define four constants by T p+q+r Γ(p + q + r + 1) The first existence and uniqueness result is obtained by using Banach contraction mapping principle.
Theorem 3.1.Let g, f : J × R → R, be continuous functions and φ, h : C(J, R) → R be two functionals satisfying the assumption: (H 1 ) there exist positive constants L i , i = 1, 2, 3, 4 such that: If the inequality holds, then the boundary value problem (1.5) has a unique solution on J.
Proof.By using the Banach's contraction mapping principle, we shall show that A of a fixed point problem, x = Ax, has a unique fixed point which is the unique solution of problem (1.5).
To prove the embedding property, we set and choose Now, we show that AB r ⊂ B r , where B r = {x ∈ C : x ≤ r}.For any x ∈ B r , and taking into account assumption (H 1 ), we obtain T p+q+r Γ(p + q + r + 1) T p+q+r Γ(p + q + r + 1) This mean that Ax ≤ r which yields AB r ⊂ B r .For all t ∈ [0, T ] and for each x, y ∈ C, we have T p+q+r Γ(p + q + r + 1) η q+r−1 + Γ(q) Γ(p + q + 1) The above result implies that Ax − Ay ≤ Ω 1 x − y .As Ω 1 < 1, therefore A is a contraction operator.
Hence, by the Banach contraction mapping principle, we obtain that A has a unique fixed point which is the unique solution of the problem (1.5).The proof is completed.The second existence result will be proved by using the following Krasnoselskii's fixed point theorem.
Lemma 3.1.(Krasnoselskii's fixed point theorem) [13].Let M be a closed, bounded, convex and nonempty subset of a Banach space X.Let A, B be the operators such that (a) Ax+By ∈ M whenever x, y ∈ M ; (b) A is compact and continuous; (c) B is a contraction mapping.Then there exists z ∈ M such that z = Az + Bz.
Theorem 3.2.Assume that g, f : J ×R → R, are continuous functions and two functionals φ, h : C(J ×R) → R satisfying the assumption (H 1 ).In addition we suppose that: If the inequality then the boundary value problem (1.5) has at least one solution on J.
Proof.To applied Lemma 3.1, we let sup t∈J |δ 1 (t)| = δ 1 , sup t∈J |δ 2 (t)| = δ 2 , and a positive constant r as Define a ball B r by B r = {x ∈ C : x ≤ r} which is closed, bounded, convex and nonempty subset of a Banach space C. In addition, we define the operators P and Q on B r as Obvious that Ax = Px + Qx.To prove that P and Q satisfy (a) of Lemma 3.1, for x, y ∈ B r , we have T p+r Γ(p + r + 1) η q+r−1 + Γ(q) Γ(p + q + 1) T p+q+r Γ(p + q + r + 1) η q+r−1 + + Γ(q) Γ(p + q + 1) This shows that Px + Qy ∈ B r .
The operator Q satisfies the condition (c) of Lemma 3.1 from assumption (H 1 ) together with (3.4).The final step is to show that the operator P is satisfied condition (b) of Lemma 3.1.Since the functions f, g are continuous, we get that the operator P is continuous.Now we will show that the operator P is compact.
Therefore, the set P(B r ) is uniformly bounded.Let us let sup (t,x)∈J×Br |g(t, x)| = g < ∞ and sup Then we have which is independent of x and tends to zero as t 1 → t 2 .Thus, the set P(B r ) is equicontinuous.Hence, by the Arzelá-Ascoli theorem, the set P(B r ) is relatively compact.Therefore, the operator P is compact which is satisfied condition (b) of Lemma 3.1.Thus all the assumptions of Lemma 3.1 are satisfied.So the boundary value problem (1.5) has at least one solution on J.The proof is completed.
Remark 3.1.In the above theorem we can interchange the roles of the operators P and Q to obtain a second result replacing (3.4) by the following condition: Remark 3.2.Since Ω 2 < Ω 1 and Ω 3 < Ω 1 , the condition (3.2) can be relaxed by (3.4) and (3.5).However, the conclusion of both theorems has different mentions between uniqueness and multiplicity of solutions.
Example 3.2.Consider the following nonlinear sequential Riemann-Liouville and Caputo fractional differential equation with nonlocal integral boundary conditions Setting constants q = 1/2, r = 2/3, p = 2/3, η = 2, T = 4, then we can fine that Φ  [11].Let E be a Banach space, C be a closed, convex subset of E, U be an open subset of C and 0 ∈ U .Suppose that A : U → C is a continuous, compact (that is, A(U ) is a relatively compact subset of C) map.Then either (i) A has a fixed point in U , or (ii) there is a u ∈ ∂U (the boundary of U in C) and λ ∈ (0, 1) with u = λA(u).
T p+q+r Γ(p + q + r + 1) Therefore, from the above result, we conclude that Then the set A(B R ) is uniformly bounded.Next, we show that the operator A maps bounded sets into equicontinuous sets of C. Let ν 1 , ν 2 ∈ J with ν 1 < ν 2 and for any x ∈ B R , then we have Obviously the right hand side of the above inequality tends to zero independently of x ∈ B R as ν 1 → ν 2 , which implies that the set A(B R ) is equicontinuous.Therefore it follows by the Arzelá-Ascoli theorem that the set A(B R ) is relative compact.Then the operator A is compact.
Let x(t) be a solution of problem (1.5).Then, for t ∈ J and x ∈ B R , we have Consequently, we have Let us define a subset of B R as where N is satisfied the condition (H 4 ).Note that the operator A : U → C is continuous and completely continuous.From the choice of U , there is no x ∈ ∂U such that x = θAx for some θ ∈ (0, 1).Then, by nonlinear alternative of Leray-Schauder type, Lemma 3.2, we get that the operator A has a fixed point in U , which is a solution of the boundary value problem (1.5).This completes the proof.Setting constants q = 4/5, r = 2/5, p = 3/5, η = 2, T = 5, then we get Φ

Example 3 . 3 .
Consider the following nonlinear sequential Riemann-Liouville and Caputo fractional differential equation with nonlocal integral boundary conditions
The problem(3.6)cannot be applied by Theorem 3.1 since Ω 1 = 1.000204 > 1.Now, third existence result is based on Leray-Schauder's Nonlinear Alternative.