Existence results of ψ-Hilfer integro-differential equations with fractional order in Banach space

In this article we present the existence and uniqueness results for fractional integro-differential equations with ψ-Hilfer fractional derivative. The reasoning is mainly based upon different types of classical fixed point theory such as the Mönch fixed point theorem and the Banach fixed point theorem. Furthermore, we discuss Eα-Ulam-Hyers stability of the presented problem. Also, we use the generalized Gronwall inequality with singularity to establish continuous dependence and uniqueness of the δ-approximate solution.


[172]
Mohammed A. Almalahi, Satish K. Panchal p, q ∈ R, c ∈ E, p + q = 0 and t a k(t, s)y(s)ds is a linear integral operator with η = max{ t a |k(t, s)|ds : (t, s) ∈ J × J}, k : J × J → R. The fractional calculus has been given proper attention by many researchers in the last few decades. This branch of mathematics was founded by Leibniz and Newton in seventeenth century. In the eighteenth century some notable definitions about fractional derivatives were given by some famous mathematicians like Riemann, Liouville, Grönwal, Letnikove, Hadamard and many others, for more detail see [10,12,16,17]. In the last few decades significant work has been done on various aspects of fractional calculus due to the fact that, the modelling of various phenomenons in the fields of science and engineering is done more precisely using fractional differential equations as compared to ordinary differential equations. Since a boundary value problem of differential equations represent an important class of applied analysis, the area mentioned was given more importance, see [1,2,8,16,23] and the references therein. An important characteristic is that engineers and scientists have developed some new models that involve fractional differential equations. These models have been applied successfully, for instance in theory of viscoelasticity and viscoplasticity, modelling of polymers and proteins, transmission of of ultrasound waves, modelling of human tissue under mechanical loads, etc. There have been extensive consideration in the last decades of the existence theory of boundary value problems including fractional differential equation, see [5,4,19,7,3,18]. This paper is organized as follows. In Section 2, we introduce some notations, definitions, and preliminary facts, which are use throughout this paper. By using measure of noncompactness and Mönch fixed point theorem we present the existence result of our problem in Section 3. We discuss E α -Ulam-Hyers stability of problem (1) in Section 4. Finally, in Section 5, by using generalized Gronwall inequality with singularity we establish continuous dependence and uniqueness of δ-approximate solution of problem (1).

Preliminaries
Let J := (a, b], (−∞ < a < b < ∞) be a finite interval and let C[J, E] be the Banach space of continuous functions on J into E with the norm y C[J,E] = sup{ y(t) : t ∈ J}, ψ : J → R be an increasing function such that ψ (t) = 0 for all t ∈ J. For 0 ≤ γ < 1 and n ∈ N, the weighted spaces are the Banach spaces with the norms Existence results of ψ-Hilfer integro-differential equations respectively. For n = 0, we have C 0 Definition 2.1 ( [20]) Let α > 0 and ψ be a positive and increasing function, having a continuous derivative ψ on the interval (a, b). Then the left-sided ψ-Riemann-Liouville fractional integral of a function f : [a, ∞) → R of order α is defined by be two functions such that ψ is increasing and ψ (t) = 0, for all t ∈ [a, b]. The left-sided ψ-Hilfer fractional derivative of function f of order α and type 0 ≤ β ≤ 1 is defined by Lemma 2.1 ( [12]) Let α, γ > 0, then Now, we give definitions of fundamental spaces. For γ = α + β − αβ and 0 < α, β, γ < 1, 0 ≤ µ < 1, we define [174] Next, we introduce the Hausdorff measure of noncompactness Φ(·) on each bounded subset K ⊂ E by Φ(K) = inf{r > 0 : for which K has a finite r-net in E}.
In the following Lemmas, we recall some basic properties of Φ(·). (2) for all bounded subsets for every x ∈ E and every nonempty subset A ⊆ E; is equivalent to the integral equation Then the problem (1) is equivalent to the following integral equation Proof. In view of lemma 2.7, a solution of the first equation of (1) can be expressed by Now, by using condition For more details, see [4,19].

Existence and uniqueness of solution
To obtain our results, the following conditions must be satisfied.
(H 3 ) There exist constant numbers L, M > 0 such that for each t ∈ J and x 1 , x 2 , y 1 , y 2 ∈ E and where y 1 , y 2 are bounded subsets of E.
Now, by using the Mönch fixed point theorem, we present the existence result for the problem (1).
Then the problem (1) has at least one solution in C γ Clearly, G is well defined and the fixed point of the operator G is a solution of the problem (1). Define a bounded, closed and convex set Existence results of ψ-Hilfer integro-differential equations .
Claim (1). The operator G maps the set k ξ in to itself (Gk ξ ⊂ k ξ ).
For any y ∈ k ξ , t ∈ J, we have It follows that Gy C 1−γ,ψ ≤ ξ. Thus Gk ξ ⊂ k ξ . [178] Mohammed A. Almalahi, Satish K. Panchal Claim (2). The operator G is continuous on k ξ . Let {y n } ∞ n=1 be a sequence such that y n → y in k ξ as n → ∞, then for each y ∈ k ξ , t ∈ J, we have where f yn = f (s, y n (s), (By n )(s)) and f y = f (s, y(s), (By)(s)). By Lebesgue convergence theorem, we conclude that Gy n − Gy → 0 as n → ∞, and hence the operator G is continuous on k ξ .
Claim (3). The operator G is equicontinuous on k ξ . For any t 1 , t 2 ∈ J such that a < t 1 < t 2 < b, y ∈ k ξ , we have Existence results of ψ-Hilfer integro-differential equations
Claim (4). The Mönch condition is satisfied. For brevity, let K be a bounded subset of a Banach space C [J, E] and Ω be the measure of noncompactness in the Banach space C[J, E] which is defined by where (K) is the collection of all countable subsets of K, and σ is the real measure of noncompactness defined by such that Z(t) = {y(t) : y ∈ Z}, t ∈ J, L is the suitably constant and mod c (Z) is the modulus of equicontinuity of Z given by Observe that Ω is well defined [19,11] and is a monotone, nonsingular and regular measure of noncompactness. Let U ⊂ k ξ be a countable set such that U ⊂ conv(G(U)∪{0}). Now we need to show that U is precompact. Let {x n } ∞ n=1 ⊆ G(U) be a countable set. Then there exists a set {y n } ∞ n=1 such that x n (t) = (Gy n )(t) for all t ∈ J, n ≥ 1. Using (H 4 ) and Lemmas 2.3, 2.4, we get It follows where R ∈ (0, 1) is the suitable constant, such that Notice that which implies that σ({y n } ∞ n=1 ) = 0 and hence σ({x n } ∞ n=1 ) = 0. Now, by step 3, we have an equicontinuous set {x n } ∞ n=1 on J.
Hence Ω(U) ≤ Ω(conv(G(U) ∪ {0})) ≤ Ω(G(U)), where Ω(G(U)) = Ω({x n } ∞ n=1 ) = 0. Thus U is precompact. Hence, by Lemma 2.6, the operator G has a fixed point y * , which is a solution of the problem (1) in C 1−γ,ψ [J, E]. Finally, we need to show that such a fixed point Since y * is a fixed point of operator G in C 1−γ,ψ [J, E], then, for each t ∈ J, we have Existence results of ψ-Hilfer integro-differential equations Applying D γ a + on both sides and using Lemma 2.1, we get . As a consequence of the above steps, we conclude that the problem (1) has at least one solution in C γ In the forthcoming theorem, by using Banach contraction principle, we present the uniqeness of solution for the problem (1).

Theorem 3.2
Assume that (H 1 ) and (H 3 ) hold. If then the problem (1) has a unique solution in C γ Proof. By using the Banach contraction principle we shall show that the operator G, defined by (2), has a unique fixed point, which is a unique solution of the problem (1) in and t ∈ J, then, by our hypotheses, we have By (3), the operator G : is a contracting mapping. According to the Banach contraction principle we conclude that the operator G has a unique fixed point y * in C 1−γ,ψ [J, E] which is a unique solution of (1).

E α -Ulam-Hyers stability
In this part, we discuss the E α -Ulam-Hyers stability of problem (1). The following observations are taken from [13,15,22]. Then If y be a nondecreasing function on [a, b]. Then we have Existence results of ψ-Hilfer integro-differential equations where E α (·) is the Mittag-Leffler function defined by , y ∈ C, (α) > 0.
if and only if there exists a function g ∈ C γ if there exists a real number C E > 0 such that, for each ε > 0 and each z ∈ Lemma 4.2 Let γ = α+β −αβ be such that α ∈ (0, 1), β ∈ [0, 1]. If a function z ∈ C γ 1−γ,ψ [J, E] satisfies (4), then z satisfies the following integral inequality Proof. According to Theorem 2.2 and Remark 4.1, the following equation The above implies that Now, in the following theorem we prove the E α -Ulam-Hyers stability result for ψ-Hilfer problem (1).
Proof. Let ε > 0 and let z ∈ C γ 1−γ,ψ [J, E] be a solution of the following inequality where , we can easily find that T u = T z . Hence using (5), we get for each t ∈ J, Using Lemma 4.1, we obtain where Thus (6) is E α -Ulam-Hyers stable.