Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems

: In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP): c D q 0 + u ( t ) = f ( t , u ( t )) , t ∈ [ 0, T ] , u ( k ) ( 0 ) = ξ k , u ( T ) = m ∑ i = 1 β iRL I p i 0 + u ( η i ) , where n − 1 < q < n , n ≥ 2, m , n ∈ N , ξ k , β i ∈ R , k = 0, 1, . . . , n − 2, i = 1, 2, . . . , m , and c D q 0 + is the Caputo fractional derivatives, f : [ 0, T ] × C ([ 0, T ] , E ) → E , where E is the Banach space. The space E is chosen as an arbitrary Banach space; it can also be R (with the absolute value) or C ([ 0, T ] , R ) with the supremum-norm. RL I p i 0 + is the Riemann–Liouville fractional integral of order p i > 0, η i ∈ ( 0, T ) , and m ∑ i = 1 β i η p i + n − 1 i Γ ( n ) Γ ( n + p i ) (cid:54) = T n − 1 . Via the ﬁxed point theorems of Krasnoselskii and Darbo, the authors study the existence of solutions to this problem. An example is included to illustrate the applicability of their results.


Introduction
The fractional differential equations have an important role in numerous fields of study carried out by mathematicians, physicists, and engineers. They have used it basically to develop mathematical modeling, many physical applications, and engineering disciplines such as viscoelasticity, control, porous media, phenomena in electromagnetics etc. (see [1][2][3]). The major differences between the fractional order differential operator and classical calculus is its nonlocal behavior, that is the feature future state based on the fractional differential operator depends on its current and past states. More details on the fundamental concepts of fractional calculus, fractional differential equations, and fractional integral equations can be found in books like A. A.Kilbas, H. M Srivastava and J. J. Trujillo [1], K. S Miller and B. Ross [2], and J. Banas and K. Goebel [4]. Fractional integro-differential equations involving the Caputo-Fabrizio derivative have been studied by many researchers from different points of view (see, for example, [5][6][7][8], and the references therein). The qualitative theory of differential equations has significant applications, and the existence of solutions and of positive solutions of fractional differential equations, which respect the initial and boundary value, have also received considerable attention. In order to study such a type of problem, different kinds of techniques, such as fixed point theorems [9][10][11], the fixed point index [10,12], the upper and lower solution method [13], coincidence theory [14], etc., are in vogue. For instance, in [15][16][17], the authors investigated the existence of solutions of initial value problems.
In [18], the authors investigated the existence of solutions of the following boundary value problems: In [19], the authors investigated the existence and uniqueness of solutions of the nonlocal fractional integral condition.
where 1 < q ≤ 2, RL D q 0 + is the Riemann-Liouville fractional derivative of order q, H I p i 0 + is the Hadamard fractional integral of order p i > 0, η i ∈ (0, T), f : [0, T] × R → R, and α i ∈ R, i = 1, 2, · · · , n are real constants such that Inspired by the above papers in [15][16][17][18][19], the objective of this paper is to derive the existence solution of the fractional differential equations and nonlocal fractional integral conditions: where n − 1 < q < n, n ≥ 2, m, n ∈ N, ξ k , β i ∈ R, k = 0, 1, . . . , n − 2, i = 1, 2, . . . , m, and c D q 0 + is the Caputo fractional derivatives, f : [0, T] × C([0, T], E) → E, and RL I p i is the Riemann-Liouville fractional integral of order p i > 0, η i ∈ (0, T), and The results obtained in the present paper were based on the fixed point theorems of Krasnoselskii and Darbo. Further, we provide some examples to show the applicability of our results. The next part of the paper is organized in the following order: We recall some notations, definitions, and preliminary facts about fractional differential calculus and Kuratowski's measure of noncompactness (Kuratowski MNC), as well as some known results in Section 2. In Section 3, based on Kransnoselskii's fixed point theorem and Darbo's fixed point theorem together and the idea of the measure of noncompactness, the main result is formulated and proven. We also show an example of the main results.

Background Materials
In this section, we recall some basic notations, definitions, and lemmas regarding fractional differential equations in order to obtain our main results. See [1,3,4,17,20,21], and the references therein. Denote by C([0, T], R) the space of all continuous functions from [0, T] into R. Endowed with the norm: this space is a Banach space. Let (E, · ) be a Banach space. We also denote: Equipped with the norm u C n := provided that the integral exists. The Caputo fractional derivative of order q of u is defined by: provided that the right side is point-wise defined on (0, ∞), where n is the smallest integer greater than or equal to q and Γ denotes the gamma function. If q = n, then c D q 0 + u(t) = u (n) (t).
where n is the smallest integer greater than or equal to q.

For a given set
Next, we provide the definition of the measure of noncompactness and some auxiliary results; see for more details [11,13,15] and the references therein.

Definition 2. Let E be a Banach space and Ω E the collection of subsets of E. The Kuratowski MNC is the map
We also adopt some techniques from the Kuratowski MNC and the theorem of Arzela-Ascoli in Lemma 3. ([4,20,22]). Let E be a Banach space. X and Y are bounded sets, (a) α(X) = 0 ↔X is compact (X is relatively compact), whereX denotes the closure of X, (b) nonsingularity: α is equal to zero on every element set,

E) is uniformly bounded and equicontinuous on [0, T] and for any t
Theorem 1 (Krasnoselskii's fixed point theorem [21]). Let N be a bounded, closed, convex, and nonempty subset of Banach space E. Let A 1 , A 2 : E → E be operators with the following properties: Theorem 2 (Darbo's fixed point theorem [23]). Let E be a Banach space, and let N be a bounded, closed, convex, and nonempty subset of E. Suppose a continuous mapping A : N → N is such that for all closed subsets M of N, where 0 ≤ k < 1. Then, A has a fixed point in N.

Main Result
In this section, we consider the existence of solutions of the nonlocal Riemann-Liouville fractional integral condition and Caputo nonlinear fractional differential Equation (6).
We prove the following lemma to establish the existence of a solution to Problem (6).
where n − 1 < q < n, n ≥ 2, m, n ∈ N, ξ k , β i ∈ R, η ∈ (0, T), k = 0, 1, . . . , n − 2, i = 1, 2, . . . , m. Here, c D q 0 + denotes the Caputo fractional derivatives, and RL I p i 0 + is the Riemann-Liouville non-local fractional integral of order p i > 0. Assume that: Then, the solution of the above BVP has a unique solution given by: Proof. From Lemma (1), we get, for certain constant vectors c 0 , . . . , c n−1 belonging to E, From the first condition in BVP, we see, and so, The substitution T = t yields, and applying the operator RL I p i 0 + results in: By employing the second boundary value condition, we infer: As a consequence, we get, Hence, the result: Let E be the real vector space of all real-valued continuous functions defined on [0, T], that is E = C([0, T], R). Equipped with the supremum norm u ∞ := sup Banach space. Define the non-linear operator A : E → E as follows: Then, the operator A has a fixed point if and only if Problem (6) possesses a solution. In the next theorem, we present the existence of solutions for Problem (6) via the fixed point theorems of Krasnoselskii and Darbo.

Existence Result Via Krasnoselskii's Fixed Point Theorem
We begin with an existence result via the Krasnoselskii's fixed point theorem.
In addition, introduce the operator A 1 and A 2 on E = C([0, T], R) by: For any u, v ∈ B ρ , we get: These inequalities show that A 1 u + A 2 v ∈ B ρ . In order to prove that A 2 is a contraction, we take u, v ∈ E and get, This implies that Hence, A 2 is a contraction. Therefore, the operator A 1 is continuous by the continuity of f . Since for u ∈ E, we have A 1 u ≤ ϕ T q Γ(q+1) , the operator A 1 is uniformly bounded on B ρ . Next, we show that the operator A 1 is compact.
We define sup (t,u)∈[0,T]×B ρ | f (t, u)|= θ < ∞, and for any 0 < τ 1 < τ 2 < T, we get: A consequence of these inequalities is that {A 1 u : u ∈ B ρ } is a uniformly-bounded and equicontinuous set in E. Thus, by the Arzela-Ascoli theorem, the operator A 1 is compact on B ρ . A combination of this property of the operator A 1 with the inclusion property A 1 B ρ + A 2 B ρ ⊂ B ρ implies, by Krasnoselskii's theorem, that the problem (6) has at least one solution on [0, T].

Existence Result via Darbo's Fixed Point Theorem
In order to prove our main result, we assume the following hypotheses are satisfied:

Hypothesis 2 (H2).
There exists a constant L > 0 such that Now, we prove our existence result for the problem (6) by Kuratowski MNC and Darbo's fixed point theorem.

Proof.
A solution to the boundary value problem (6) can be considered as a fixed point of the operator A : E → E, defined by: Step 1: A is continuous. Let {u n } be a sequence such that u n → u in E, when n → ∞. If t ∈ [0, T], we get: Obviously, the set B r is a closed, bounded, convex subset of the Banach space C([0, T], E).
Let u belong to B r . In order to prove that Au ∈ B r , it suffices to show that |Au(t)|≤ r for t ∈ [0, T]. However, for t ∈ [0, T], we have: Thus, we get: and therefore, we see: This implies, where, Thus, we get A(B r ) ⊂ B r .
Step 3: A(B r ) is uniformly bounded and equicontinuous.

From
Step 2, we get A(B r ) = {Au : u ∈ B r } ⊂ B r . Hence, for each u ∈ B r , we get Au ≤ r, which means that A(B r ) is uniformly bounded. Let τ 1 , τ 2 ∈ [0, T], τ 1 < τ 2 , define sup and choose u ∈ B r . Then, we obtain, This implies, As τ 2 → τ 1 , the right-hand side tends to zero. Thus, A(B r ) is equicontinuous and uniformly bounded. Hence, from the Arzela-Ascoli theorem, it follows that the set AB r is relatively compact in B r .