EXISTENCE OF SOLUTIONS OF ABSTRACT FRACTIONAL IMPULSIVE SEMILINEAR EVOLUTION EQUATIONS

In this paper we prove the existence of solutions of fractional impulsive semilinear evolution equations in Banach spaces. A nonlocal Cauchy problem is discussed for the evolution equations. The results are obtained using fractional calculus and fixed point theorems. An example is provided to illustrate the theory.


Introduction
Fractional differential equations are increasingly used for many mathematical models in science and engineering.In fact fractional differential equations are considered as an alternative model to nonlinear differential equations [8].The theory of fractional differential equations has been extensively studied by several authors [11,[16][17][18][19].In [12,14] the authors have proved the existence of solutions of abstract differential equations by using semigroup theory and fixed point theorem.Many partial fractional differential equations can be expressed as fractional differential equations in some Banach spaces [13].
The nonlocal Cauchy problem for abstract evolution differential equation was first studied by Byszewski [9].Subsequently several authors have investigated the problem for different types of nonlinear differential equations and integrodifferential equations including functional differential equations in Banach spaces [2-4, 10, 20].Mophou and N'Guérékata [22,23,24] and Balachandran and Park [5] discussed the existence of solutions of abstract fractional differential equations with nonlocal initial conditions.
Impulsive differential equations have become important in recent years as mathematical models of phenomena in both physical and social sciences.There has been a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments.Recently Benchohra and Slimani [6,7] discussed the existence and uniqueness of solutions of impulsive fractional differential equations and Ahmad and Sivasundaram [1] discussed the existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations.Motivated by this work we study in this paper the existence of solutions of fractional impulsive semilinear evolution equations in Banach spaces by using fractional calculus and fixed point theorems.

Preliminaries
We need some basic definitions and properties of fractional calculus which are used in this paper.By C(J, X) we denote the Banach space of continuous functions x(t) with x(t) ∈ X for t ∈ J, a compact interval in R and x C(J,X) = max t∈J x(t) .
Definition2.1.A real function f (t) is said to be in the space C α , α ∈ R if there exists a real number p > α, such that f (t) = t p g(t), where g ∈ C[0, ∞) and it is said to be in the space Definition2.2.The Riemann-Liouville fractional integral operator of order β > 0 of function f ∈ C α , α ≥ −1 is defined as where Γ(.) is the Euler gamma function.

Definition2.3. If the function f ∈ C m
−1 and m is a positive integer then we define the fractional derivative of f (t) in the Caputo sense as where f ′ (s) = df (s) ds and f is an abstract function with values in X.For basic facts about fractional derivatives and fractional calculus one can refer to the books [15,21,25,26].Consider the Banach space Consider the linear fractional impulsive evolution equation where 0 < q ≤ 1 and A(t) is a bounded linear operator on a Banach space X, ) represent the right and left limits of u(t) at t = t k .Our Eq.(2.1) is equivalent to the integral equation (2.2) Definition 2.4.By a solution of the abstract Cauchy problem (2.1), we mean an abstract function u such that the following conditions are satisfied: (i) u ∈ P C(J, X) and u ∈ D(A(t)) for all t ∈ J ′ ; (ii) d q u dt q exists on J ′ where 0 < q < 1; (iii) u satisfies Eq.(2.1) on J ′ , and satisfy the conditions Now, we assume the following conditions to prove the existence of a solution of the evolution Eq.(2.1).(HA) A(t) is a bounded linear operator on X for each t ∈ J and the function t → A(t) is continuous in the uniform operator topology.(HI) The functions I k : X → X are continuous and there exists a constant L 1 > 0 such that For brevity let us take T q Γ(q + 1) = γ.Proof: The proof is based on the application of Picard's iteration method.Let M = max 0≤t≤T A(t) and define a mapping Let u, v ∈ P C(J, X).Then from Eq.(2.3), we have for each Then by induction we have Since 1 n! Mγ(m + 1) + mL 1 n < 1 for large n, then by the well-known generalization of the Banach contraction principle, F has a unique fixed point u ∈ P C([0, T ] : X).This fixed point is the solution of Eq.(2.1).

Semilinear Evolution Equation
Now consider the semilinear fractional impulsive evolution equation ∆u where A(t) is a bounded linear operator and f : J × X → X is continuous.This Eq.(3.1) is equivalent to the integral equation We need the following additional assumptions to prove the existence of solution of the Eq.(3.1).
(Hf) f : J × X → X is continuous and there exist constants L 2 > 0, N > 0 such that and we have to show that Φ has a fixed point.This fixed point is the solution of Eq.(3.1).Choose r ≥ 2 u 0 + (m + 1)Nγ .Then we can show that ΦB r ⊂ B r , where B r := {u ∈ Z : u ≤ r}.From the assumptions we have EJQTDE, 2010 No. 4, p. 5 Thus, Φ maps B r into itself.Now, for u 1 , u 2 ∈ Z, we have Φu Hence Φ is a contraction mapping and therefore there exists a unique fixed point u ∈ B r such that Φu(t) = u(t).Any fixed point of Φ is the solution of Eq.(3.1).

Nonlocal Cauchy Problem
In this section we discuss the existence of solution of the impulsive evolution equation (3.1) with nonlocal condition of the form where g : P C(J, X) → X is a given function which satisfies the following condition.(Hg) g : P C(J, X) → X is continuous and there exists a constant G > 0 such that EJQTDE, 2010 No. 4, p. 6 has a fixed point.This fixed point is then a solution of Eq.(3.1) and Eq.(4.1).Choose r ≥ 2 u 0 + g(0) + (m + 1)Nγ .Then it is easy to see that ΨB r ⊂ B r .Further, for u 1 , u 2 ∈ Z, we have Ψu The result follows by the application of contraction mapping principle.
Our next result is based on the following well-known fixed point theorem.
Krasnoselskii Theorem.Let S be a closed convex nonempty subset of a Banach space X.Let P, Q be two operators such that (i) P x + Qy ∈ S whenever x, y ∈ S; (ii) P is a contraction mapping; (iii) Q is compact and continuous.Then there exists z ∈ S such that z = P z + Qz.Now, we assume the following condition instead of (Hf) and apply the above fixed point theorem.
(Hf) ′ f : J × X → X is continuous and there exists a continuous function µ ∈ L 1 (J) such that f (t, u) ≤ µ(t), for all (t, u) ∈ J × X.where µ 0 = sup t∈J µ(t) and define the operators P and Q on B r as and EJQTDE, 2010 No. 4, p. 7 For any u, v ∈ B r , we have Hence, we deduce that P u + Qv ≤ r.Next, for any t ∈ I, u, v ∈ X we have And since G + γ(m + 1)M + mL 1 < 1, then P is a contraction mapping.Now, let us prove that Q is continuous and compact.
Let {u n } be a sequence in B r , such that u n → u in B r .Then because the function f is continuous on I × X.Now, for each t ∈ I, we have

Consequently, lim n→∞
Qu n (t) − Qu(t) = 0.In other words, Q is continuous.Let's now note that Q is uniformly bounded on B r .This follows from the inequality Now, let's prove that Qu, u ∈ B r is equicontinuous.Let t 1 , t 2 ∈ J, t 1 < t 2 , and let B r be a bounded set in X.Let u ∈ B r , we have As t 1 → t 2 , the right hand side of the above inequality tends to zero.In short we have proven that Q(B r ) is relatively compact for t ∈ I.By Arzela Ascoli's theorem, Q is compact.Hence by the Krasnoselskii theorem there exists a solution of the problem (3.1) with nonlocal condition (4.1).