Solvability of a fractional boundary value problem with fractional derivative condition

In this paper, we investigate a boundary value problem for fractional differential equations with fractional derivative condition. Some new existence results are obtained using Banach contraction principle and Leray–Schauder nonlinear alternative.

Gorenflo et al. in [8] presented some general results for the fractional boundary value problems. They dealt with boundary value problems for pseudo-differential equations with the operator: where A(D) is an elliptic pseudo-differential operator and with boundary operators depending on a positive real parameter α.
Bai [2] considered the existence of positive solutions of the fractional boundary value problem: where D α 0 + denotes the Riemann-Liouville differentiation. Goodrich studied in [6] a similar problem for fractional differential equation where the nonlinear term depends only on u and t, he considered the following problem: where 0 ≤ i ≤ n − 2, 1 ≤ α ≤ n − 2, v > 3 satisfying n − 1 < v ≤ n, n ∈ N, is given, and D v 0 + is the standard Riemann-Liouville fractional derivative of order v. The author established the existence of positive solution using cone theoretic techniques, then in [7] he extended this study to systems of differential equations of fractional order.
Motivated by the results mentioned above, we are concerned with the existence and uniqueness of solutions of the fractional boundary value problem generated by a fractional differential equation and fractional derivative condition (FBVP)(P1): where f : [0, 1] × R × R → R is a given function, 2 < q < 3, 0 < σ < 1 and c D q 0 + represents the standard Caputo fractional derivative of order q. The case q = 2 is studied in [9], where the second-order equations are used to model various phenomena in physics, chemistry and epidemiology. It is shown that by introducing fractional derivatives and fractional integrals, we get an adequate mathematical modelling of real objects and processes. Moreover, the introduction of the Caputo's fractional derivative, allows the utilization of physically interpretable boundary conditions. For more details on the geometric and physical interpretation for fractional derivatives Caputo types, see [18].
By the use of nonlinear alternative of Leray-Schauder and the Banach fixed-point theorem, we show the existence and uniqueness of solutions for the above problem. Our results allow the derivative condition to depend on the fractional derivative c D σ 0 + u, which leads to extra difficulties. No contributions exist, as far as we know, concerning the existence of solutions of the fractional differential Eq. (1.1) jointly with fractional derivative condition (1.2).
The rest of this paper is organized as follows. First, we list some notations, definitions and lemmas to be used later. In Sect. 3, we present and prove our main results which consist of uniqueness and existence theorems. Finally, we give some examples to illustrate our work.

Preliminaries and lemmas
In this section, we cite definitions and some fundamental facts from fractional calculus which can be found in [13]. b]) and α > 0, then the Riemann-Liouville fractional integral is defined by The following lemmas give some properties of Riemann-Liouville fractional integrals and Caputo fractional derivative.

Lemma 2.5 ([13])
Let p, q ≥ 0, and f ∈ L 1 ([a, b]). Then The following lemma is fundamental in the proof of the existence theorem. We start by solving an auxiliary problem and we give the Green function.
is given by Proof Assume that u is a solution of the fractional boundary value problem (P0). Then using Lemma 2.5, we have gives so u(t) can be written as where G is defined by (2.2). The proof is complete.
Define the integral operator T : Proof Let u be a solution of (P1). Then using the same method as used in Lemma 2.8, we can prove that Conversely u satisfies and denotes the right-hand side of the equation by v(t). Then by Lemma 2.5, we obtain Hence, v(t) is a solution of the fractional differential Eq. (1.1). Also it is easy to verify that v satisfies conditions (1.2), then it is a solution for the problem (P1). This achieves the proof.

Existence and uniqueness results
In this section, we prove the existence and uniqueness of solutions in the Banach space E.
Then the FBVP (P1) has a unique solution u in E.
Proof We shall use Banach fixed-point theorem. For this, we need to verify that T is a contraction. Let u, v ∈ E. Applying (2.4) we get Let us estimate the term 1 0 |G (t, s)| g (s) ds: On the other hand, we have From the above, we deduce ∂s h (r ) dr ds, (3.4) and using (3.2), we obtain max 0≤t≤1

Taking (3.3) and (3.5) into account, we acquire
then, T is a contraction. As a consequence of Banach fixed-point theorem, we deduce that T has a fixed point which is the unique solution of the FBVP (P1). The proof is complete. Now, we give an existence result for the fractional boundary value problem (P1).

7)
where Then the FBVP (P1) has at least one nontrivial solution u * ∈ E.
Proof In view of the continuity of f and G, the operator T is continuous. Let B r = [u ∈ E, ||u|| ≤ r } be a bounded subset in E.
(i) For u ∈ B r , using (3.6) and the fact that φ and ψ are nondecreasing, we obtain Thus, we have In addition, Hence, it follows that therefore, The following estimate holds As t 1 → t 2 , the right-hand sides of the above inequalities (3.12) and (3.13) tend to 0, consequently T (B r ) is equicontinuous. By means of the Arzela-Ascoli Theorem, we conclude that T is completely continuous.
In what follows, we establish an existence result using the nonlinear alternative of Leray-Schauder. Setting = {u ∈ E : u < r } then for u ∈ ∂ , such that u = λT u, 0 < λ < 1. Using (3.8) we get (3.14) In addition, From (3.7), (3.14) and (3.15) we deduce that this contradicts the fact that u ∈ ∂ . By Lemma 2.7, we conclude that T has a fixed point u * ∈ and then the FBVP (P1) has a nontrivial solution u * in E. This achieves the proof We illustrate our work with two examples.
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