New Existence Results for Nonlinear Fractional Integrodifferential Equations

This paper discusses a boundary value problem of nonlinear fractional integrodi ﬀ erential equations of order 1 < α ≤ 2 and 1 < β ≤ 2 and boundary conditions of the form x ð 0 Þ = x ð 1 Þ = c D β x ð 1 Þ = c D β x ð 0 Þ = 0 . Some new existence and uniqueness results are proposed by using the ﬁ xed point theory. In particular, we make use of the Banach contraction mapping principle and Krasnoselskii ’ s ﬁ xed point theorem under some weak conditions. Moreover, two illustrative examples are studied to support the results.

Some physical applications of fractional differential equations include viscoelasticity, Schrodinger equation, fractional diffusion equation, and fractional relaxation equation; for more details, we refer the readers to [17].
In addition, fractional integrodifferential equations are used as an important tool to gain insight into some emerging problems from several science areas, for more details, we give the following references [18][19][20][21][22][23].
More recently, in [24], the existence and uniqueness of positive solutions for the fractional integrodifferential equation were proved.
In [25], the authors discussed the existence and uniqueness of solutions for nonlinear integrodifferential equations of fractional order with three-point nonlocal fractional boundary conditions. The existence of solutions for nonlinear fractional integrodifferential equations has been studied in [26].
Motivated by all these works and by the fact that there are no papers dealing with the new existence results for nonlinear fractional integrodifferential equations, in this work, we consider the existence and uniqueness of solutions for the following problem: where λ, δ : ½0, 1 × ½0, 1 − ⟶½0,+∞Þ, with ϕ * = sup t∈½0,1 This paper is organized as follows. In Section 2, we present some preliminaries and notations that will be required for the later sections. After that, in Section 3, we establish the main results by using the fixed point theory. And, in the last section, we give two examples to illustrate the results.

Preliminaries and Notations
In this section, we give some notations, definitions, and lemmas which will be required for the rest of the paper.
Definition 1 [5]. The fractional integral of order α > 0 with the lower limit zero for a function f can be defined as Definition 2 [5]. The Caputo derivative of order α > 0 with the lower limit zero for a function f can be defined as where n ∈ ℕ, 0 ≤ n − 1 < α < n, t > 0.
Theorem 3 [27]. Let M be a bounded, closed, convex, and nonempty subset of a Banach space X. Let A and B be two operators such that (i) Ax + By ∈ M whenever x, y ∈ M (ii) A is compact and continuous (iii) B is a contraction mapping Then, there exists z ∈ M such that z = Az + Bz.
Lemma 4 [5]. Let α, β ≥ 0; then, the following relation hold: Lemma 5 [5]. Let n ∈ ℕ and n − 1 < α < n. If f is a continuous function, then we have Lemma 6. Let h ∈ Cð½0, 1, ℝÞ. Then, the unique solution of the problem is given by Proof. By applying Lemma 5, we have where a 0 , a 1 , a 2 , a 3 ∈ ℝ. So And by using c D β xð0Þ = xð0Þ = 0, we obtain a 0 = a 2 = 0. As a result of c D β xð1Þ = 0, we have that Now, we use xð1Þ = 0 to get By substituting the value of a 0 , a 1 , a 2 , a 3 , we obtain the following Conversely, by direct computations, we obtain the desired result.
For x, y ∈ B r , we have je υs e υs ds Therefore, Then, Now, we prove that F 1 is a contraction. For x, y ∈ B r , we have By using the condition of the new norm, we conclude that F 1 is a contraction. Next, we will prove that F 2 is compact and continuous. Continuity of f implies that the operator F 2 is continuous. Also, F 2 is uniformly bounded on B r as Suppose that 0 ≤ t 1 < t 2 ≤ 1. We have Then, jF 2 yðt 2 Þ − F 2 yðt 1 Þj ⟶ 0, as t 1 ⟶ t 2 independently from y ∈ B r . This shows that the operator F 2 is relatively compact on B r . Thus, by the Arzela Ascoli theorem, we obtain that F 2 is compact on B r .
By the Krasnoselskii fixed point theorem, the problem (1) has at least one solution on B r .
We consider the set B r = fx ∈ X : kxk ≤ rg, where r ≥ ðr 2 /ð1 − r 1 ÞÞ, with For each t ∈ ½0, 1 and x ∈ B r , we have Then, kFxk ≤ r. Therefore, FB r ⊆ B r . Next, we show that F is a contraction mapping. For x, y ∈ B r , we have