Electronic Journal of Qualitative Theory of Differential Equations

In this paper, we prove the existence, uniqueness, and continuous dependence of the mild solutions for a class of fractional abstract differential equations with infinite delay. The results are obtained by using the Krasnoselskii’s fixed point theorem and the theory of resolvent operators for integral equations.


Introduction
Motivated by the fact that abstract functional differential equations with infinite delay arise in many areas of applied mathematics, this type of equations has received much attention in recent years [1].The works of Bahuguna, Pandey and Ujlayan [2], Liang, Xiao and Casteren [3], Liang and Xiao [4], Henríquez and Lizama [1], Baghli and Benchohra [5], and references therein, provide a basic theory for abstract functional differential or integrodifferential equations with infinite delay.They established theorems about the existence, uniqueness, or existence of periodic solutions.
It is worth pointing out that all of the aforementioned problems have been restricted to integer-order differential equations.Recent investigations in physics, engineering, biological sciences and other fields have demonstrated that the dynamics of many systems are described more accurately using fractional differential equations [6][7][8][9].There have been some attempts on fractional evolution equations and their optimal control or controllability [10][11][12][13][14][15].Also, the problem of the existence of a mild solution for abstract differential equations with fractional derivatives has started to draw some initial research attention leading to inspiring results, see e.g.[16][17][18].In [16][17], Zhou and Jiao introduced an appropriate concept for mild solution and established the criteria on existence and uniqueness of mild solutions to nonlocal Cauchy problem of fractional evolution equations by considering an integral equation which was given in terms of probability density and semigroup.Furthermore, Wang, Wei and Zhou [19] studied the fractional finite time delay evolution systems and optimal controls in infinite-dimensional spaces.They obtained some sufficient conditions for the existence, uniqueness and continuous dependence of mild solutions of these control systems.Then, Wang, Zhou and Medved [20] derived the solvability and optimal controls of a class of fractional integrodifferential evolution systems with infinite delay in Banach spaces.In [20], existence and continuous dependence of mild solutions were investigated by utilizing the techniques of a priori estimation and extension of step by steps.
Particularly, Hernández, O'Regan and Balachandran in [18] considered a different approach to treating a general class of abstract fractional differential equations where 0 < α < 1.The authors of [18] gave the definition of the mild solution to the problem (1.1) by using the well developed theory of resolvent operators for integral equations and studied the existence of this class of solutions.Motivated by the approach of [18], we consider the following fractional abstract differential equations with infinite delay where a > 0, 0 < α < 1, A is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators T (t) t≥0 , on a Banach space E with norm .; the functions f, g, γ will be given later.Let x t (.) denote First, we apply the Krasnoselskii's fixed point theorem and the theory of resolvent operators for integral equations to establish our existence result.Then, we derive the uniqueness of mild solutions by using the Banach contraction principle.Furthermore, the continuous dependence of the mild solutions on the initial condition is investigated.

Preliminaries
First, we give some notations needed to establish our results.We assume that E is a Banach space with norm .; A is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators.D(A) is the domain of A endowed with the graph norm x D(A) = x + Ax .We denote by L(Z, W ) the space of bounded linear operators from Z to W endowed with the operator norm denoted by .L(Z,W ) .Especially when Z = W , we write simply L(Z) and .L(Z) .Let J ⊂ R and denote C(J, E) to be the Banach space of continuous functions form J into E with the norm x C(J,E) = sup t∈J x(t) , where x ∈ C(J, E).
We introduce some basic definitions, properties and lemmas which are used throughout this paper.Definition 2.1.[18,21] Definition 2.2.[18,21] A one parameter family of bounded linear operators S(t) t≥0 on E is called a resolvent operator for (2.1) if the following conditions are satisfied: (S1) S(.)x ∈ C([0, ∞), E) and S(0)x = x for all x ∈ E; Lemma 2.1.[18,21] If the resolvent S(t) t≥0 for (2.1) is analytic, then i) for each Lemma 2.2.[18] Assume S(t) is compact for all t > 0. Then S ′ (t) is compact for all t > 0 and the inclusion map i c : Throughout the study, we assume that the phase space (B, .B ) is a seminormed linear space consisting of functions from (−∞, 0] into E satisfying the following assumptions (cf.Hale and Kato in [22]).(A1) If x : (−∞, a] → E is continuous on J = [0, a] and x 0 ∈ B, then for every t ∈ J the following conditions hold: (i) Before giving the definition of mild solution of (1.2) , we rewrite (1.2) in the equivalent integral equation: Let Ω be set defined by The concept of a mild solution of (1.2) can now be stated. where The key tool in our approach is the following fixed-point theorem.
x has a solution on Q.

Main results
In this section, we present and prove our main results.First, we list some assumptions: (H1) The resolvent operator S(t) t≥0 is compact.(H2) g : [0, a] × B −→ D(A) is a continuous function and there exist a constant K g and a function for all t ∈ [0, a], and every x, y ∈ Ω. (H3) γ : for all t ∈ [0, a], and every x, y ∈ Ω. (H5) There exists a function for all t ∈ [0, a], x, y, w, u ∈ Ω.

Existence of solutions
We now establish our existence result.
Theorem 3.1.Assume α ∈ (0, 1) and the following conditions are satisfied: Then the Cauchy problem (1.2) has a mild solution on (−∞, b] for some 0 < b ≤ a. For any positive constant k, x ∈ Ω and x C(J,E) ≤ k, we introduce the map Γ : Now we prove that Γ is well defined.
To bring out the results, we define the operator P : Ω 0 → Ω 0 by That the operator Γ has a fixed point is equivalent to P has a fixed point, and so we turn to proving that P has a fixed point.
For any positive constant k, let z ∈ B k = {z ∈ Ω 0 , z Ω 0 ≤ k}.We can decompose P as P = P 1 + P 2 , where We will show the operator equation z = P 1 z + P 2 z has a solution in B k 0 .Our proof will be divided into three steps.
For any z 1 , z 2 ∈ B k 0 and t ∈ [0, b], we get by using the assumptions (H2)and (A1) That is According to (3.2) we see P 1 is a contraction map on B k 0 .
Step 3. P 2 is a completely continuous operator.At first, we prove that P 2 is continuous on We have by (H 3 ) and (A1) From the assumption (H 4 ), we obtain f (s, ẑn s +y s , Let fz (s) = f (s, ẑs + y s , s 0 γ(s, τ, ẑτ + y τ )dτ ).By the assumptions (H 3 ) ∼ (H 4 ), we have We infer that there exists 0 < δ < b such that ≤ ε for all z ∈ B k 0 and every 0 < h < δ.Then we have which implies {P 2 z, z ∈ B k 0 } is right equicontinuous at t ∈ (0, b).Using the same argument, we can get {P 2 z, z ∈ B k 0 } is left equicontinuous at t ∈ (0, b] and the right equicontinuous at zero.Thus Let t ∈ (0, b] and 0 < ε < min{t, 1}.From Bochner integral (see [23,Lemma 2.1.3]),for x ∈ B k 0 , we have where co(S) denote the convex hull of a set S.

Uniqueness of solutions
We now discuss the uniqueness of mild solutions by using the Banach contraction principle.
EJQTDE, 2012 No. 56, p. 13 For k > 0, x ∈ Ω and x C(J,E) ≤ k, we define an operator Φ : Ω → Ω by where G(t) and F (t) are as Definition 2.3.From Theorem 3.1, we see that Φ is well defined.Similar as Theorem 3.1, we also define the operator P : Ω 0 → Ω 0 by where Ω 0 , G z (t) and F z (t) are as Theorem 3.1.
It is clear that the operator Φ is a contraction on Ω is equivalent to P is a contraction on Ω 0 .In fact, for z 1 , z 2 ∈ Ω 0 and t ∈ [0, b], From the assumption (H 2 ) and (A1), we obtain By (H 3 ) and (H 5 ), we see that Thus we have According to (3.4), we see that P is a contraction on Ω 0 .By the Banach contraction principle, we know the Cauchy problem (1.2) has a unique mild solution.

Continuous dependence of solutions
In the next result we investigate the continuous dependence of the mild solutions on the initial condition.Theorem 3.3.For each φ 1 , φ 2 ∈ B, suppose that there exists a constant K * such that φ 1 (0) − φ 2 (0) D(A) ≤ K * φ 1 − φ 2 B and suppose the conditions in Theorem 3.2 are satisfied.Then for the corresponding mild solutions x 1 (t), x 2 (t) of the problem ) we have the inequality From Theorem 3.2, we know that C 6 < 1.
and the derivative D α is understood here in the Caputo sense.