Existence and Stability Analysis for Fractional Differential Equations with Mixed Nonlocal Conditions

In this paper, we study the existence and uniqueness of solution for fractional differential equations with mixed fractional derivatives, integrals and multi-point conditions. After that, we also establish different kinds of Ulam stability for the problem at hand. Examples illustrating our results are also presented.

One important and interesting area of research of fractional differential equations is devoted to the stability analysis.The notion of Ulam stability, which can be considered as a special type of data dependence was initiated by Ulam [19,20].Hyers, Aoki, Rassias and Obloza contributed in the development of this field (see [21][22][23][24][25] and the references therein).Meanwhile, there have been few works considering the Ulam stability of variety of classes of fractional differential equations [26][27][28][29][30].
Fixed point theorems are used to investigate existence results.After that we also study different types of Ulam stability for the proposed problem.The rest of this paper is organized as follows.Some definitions from fractional calculus theory are recalled in Section 2. In Section 3, we will prove the existence and uniqueness of solutions for Problem (1).In Section 4, we discuss the Ulam stability results.Finally, examples are given in Section 5 to illustrate the usefulness of our main results.

Preliminaries and Background Materials
Now we recall in this section some preliminary concepts of fractional calculus [1].Let C([0, T], R) and L([0, T], R) denote the spaces of continuous real-valued and integrable real-valued functions respectively.Definition 1.The Riemann-Liouville fractional integral of order ρ > 0 starting at a point 0 of the function f ∈ L([0, T], R) is defined by where the right hand side exists and Γ(•) is the classical gamma function defined by Definition 2. The Caputo fractional derivative of order ρ > 0 starting at a point 0 for the n-times differentiable function f is defined by where n = [ρ] + 1 and [ρ] denotes the integer part of the real number ρ.
where the function f is defined in (1).Let x ∈ C 1 ([0, T], R) be a solution of the Problem (1).If there is a non zero positive constant κ such that Then the Problem (1) is said to be Ulam-Hyers (UH) stable.
Then the Problem ( 1) is said to be generalized Ulam-Hyers (GUH) stable.
Definition 5.For every ε > 0, the function z where R) be a solution of the Problem (1).If there exists a non zero positive constant κ Φ, f such that Then the problem ( 1) is said to be Ulam-Hyers-Rassias (UHR) stable.
Lemma 2 ((Schaefer fixed point theorem) [32]).Suppose that E is a Banach space.Let T : E → E be a completely continuous operator and be a bounded set.Then T has a fixed point in E.

Existence Results for the Problem
The following lemma concern a linear variant of Problem (1).
Then, the unique solution of the linear problem is given by the integral equation Proof.By Lemma 1 and the first Equation ( 4), we obtain where c 0 an arbitrary constant.Using condition the boundary condition, we get from which we have Substituting the constant c 0 into (6), we get the integral Equation ( 5).The converse can be proven by direct computation.The proof is completed.
In the following, we set an abbreviated notation for the Riemann-Liouville fractional integral of order ρ > 0, for a function with two variables as Moreover, for computational convenience we put Using Lemma 3 we define the operator A : E → E by Theorem 1.Let f : [0, T] × R → R be a continuous function.Suppose that: (H1) f (t, x) is Lipschtiz continuous (in x) i.e. there exists a constant L > 0 such that Proof.Let a ball B r be defined as B r = {x ∈ E : x ≤ r}, where r is a positive constant with which, by taking the norm on [0, T], yields Ax ≤ r.This shows that AB r ⊂ B r .In order to show that A is a contraction, we put x, y ∈ B r , we obtain which implies that A is a contraction, since K * L/|Ω| < 1.By Banach contraction mapping principle the operator A defined in (8), has the unique fixed point, which implies that Problem (1) has the unique solution on [0, T].The proof is completed.
Proof.Now, we need to show that the operator A is compact by applying the well known Arzelá-Ascoli theorem.So we will show that the operator AB r is a uniformly bounded set, where B r = {x ∈ E : x ≤ r, r > 0} and equicontinuous set.Let ϕ * = sup{ϕ(t) : t ∈ [0, T]}.For x ∈ B r , it follows that and consequently which implies that the set AB r is uniformly bounded.Next, we are going to prove that AB r is equicontinuous set.For The right-hand side of the above inequality tends to zero as τ 1 → τ 2 independently of x which implies that AB r is equicontinuous set.By using Arzelá-Ascoli theorem, the set AB r is relative compact, that is, the operator A is completely continuous.Fianlly we will show that W = {x ∈ which yields x ≤ (|A| + ϕ * K * )/|Ω|.Therefore W is bounded and the proof is completed by using Schaefer fixed point theorem (Lemma 2).
Proof.Form Remark 1 (I I) and Lemma 3, we have Then, by Remark 1 (I), we obtain which is satisfied inequality in (9).This completes the proof.
R) be a solution of the inequality in (3) and assume that Proof.Form Remark 1 (I I) and Lemma 3 we have Therefore, which leads to inequality in (10).Proof.Let z ∈ C 1 ([0, T], R) be a solution of the inequality in (3) and let x(t) be the unique solution of Problem (1).Next we consider Therefore, the Problem (1) is UHR stable.Next by putting Φ * f (ε) = εΦ f (t) with Φ * f (0) = 0, we deduce that the Problem (1) is GUHR stable.This completes the proof.

Examples
In this section, we would like to show the applicability of our theoretical results to specific numerical examples.

Conclusions
We have proved the existence and uniqueness of solutions for fractional differential equations with mixed fractional derivatives, integrals and multi-point conditions.We applied Banach and Schaefer fixed point theorems.Different kinds of Ulam stability, such as, Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability are also investigated.The obtained results are illustrated by numerical examples.It seems that the results of this paper can be extended to cover the case 1 < α ≤ 2.