EXISTENCE AND UNIQUENESS OF SOLUTIONS OF FRACTIONAL QUASILINEAR MIXED INTEGRODIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITION IN BANACH SPACES

In this paper, we discuss the existence and uniqueness of mild and classical solutions of quasilinear mixed integrodifferential equations of frac- tional orders with nonlocal condition in Banach spaces. Furthermore, we study continuous dependence of mild solutions. Our analysis is based on fractional calculus, resolvent operators and Banach's fixed point theorem.


Introduction
In recent years a considerable interest has been shown in the so-called fractional calculus, which allows us to consider integration and differentiation of any order, not necessarily integer.To a large extent this is due to the applications of the fractional calculus to problems in different areas of physics and engineering.The fractional calculus can be considered an old and yet novel topic.Starting from some speculations of Leibniz and Euler, followed by the works of other eminent mathematicians including Laplace, Fourier, Abel, Liouville and Riemann, it has undergone a rapid development especially during the past two decades.One of the emerging branches of this study is the theory of fractional quasilinear equations, i.e. quasilinear equations where the integer derivative with respect to time is replaced by a derivative of fractional order.The increasing interest in this class of equations is motivated both by their application to problems from viscoelasticity, heat conduction in materials with memory, electrodynamics with memory, and also because they can be employed to approach nonlinear conservation laws [1,6,7,8,9,10,27].
Recently, the existence of solutions of fractional abstract differential equations with nonlocal initial condition was investigated by [30].Much attention has been paid to existence results for the nonlinear mixed integrodifferential equations with nonlocal condition in Banach spaces, see Dhakne et al. [20].Several authors have studied the existence of solutions of abstract nonlocal problems by using different techniques, see [3,12,21,25,26,36,37] and the references given therein.
In this paper our aims is to study the existence, uniqueness and other properties of solutions of the problem (1.1)-(1.2).The main tool employed in our analysis is based on the Banach fixed point theorem, resolvent operators and fractional calculus.Our results generalizes the correspondence results in [20] to nonlocal quasilinear mixed integrodifferential equations of arbitrary orders.We indicate that the definition of resolvent operators used in this paper is different from that in [16].
The rest of this article is organized as follows: In section 2 we recall briefly some basic definitions and preliminary facts which are used throughout this paper.The existence and uniqueness theorems for the problem (1.1)-(1.2) and their proofs are arranged in section 3. Finally in section 4 we give example to illustrate the application of our results.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.
Here we recall the following known definition, for more details see [23,29,33].
Definition 2.2.The Caputo derivative of order α, for a function x : [0, ∞) → R can be written as where x ′ (s) = dx(s) ds .If x is an abstract function with values in X, then the integrals and derivatives which appear in (2. Here R z (t, s) can be extracted from the evolution operator of the generator −A(t, z).
Next we introduce the so-called "Mild Solution" and "Classical Solution" for (1.1)-(1.2).Definition 2.4 (Compare [35] with [16]).A continuous solution x of the integral equation 3) with t ∈ J, is said to be a mild solution of (1.1)-(1.2) on J. Definition 2.5 ( [16,18]).By a classical solution of (1.1)-(1.2) on J, we mean a function x with values in X such that: (i) x is continuous function on J and x(t) ∈ D(A), (ii) d α x dt α exists and is continuous on (0, T ), and satisfying (1.1)-(1.2) on J. Also, we need the following lemma Lemma 2.6.[16,Lemma 3.1] Let Ω ⊂ X, Y be a densely and continuously imbedded Banach space in X and let R z (t, s) be the resolvent operator for the problem for every z 1 , z 2 ∈ E with values in Ω and every ω ∈ Y .Now, we list the following hypotheses for our convenience.For the rest of paper, let Z be taken as both X and Y .Also, we denote by E the Banach space C(J; X) of X-valued continuous functions on J equipped with the sup-norm.
(H1) There exists a constant G > 0 such that (H3) The constants x 0 , M, G 1 , L, K, K 1 , H, H 1 , T and r satisfy the following two inequalities: With these preparations we are now in a position to state our main results to be proved in the present paper.

Main Results
Theorem 3.1.Assume that (i) hypotheses (H1)-(H3) hold, (ii) f : J × X × X × X → Z is continuous in t on J and there exists a constant L > 0 such that 2) has a unique mild solution on J.
Proof of Theorem 3.1.We shall use the notions and notations introduced in the preceding section.We define an operator for t ∈ J.It follows from assumption on the functions f , h and k that F : E → E and for every z ∈ E, F z(0) = x 0 − g(t 1 , t 2 , . . ., t p , z(•)).
EJQTDE, 2012 No. 51, p. 4 Let S be the nonempty closed and bounded set given by Then for z ∈ S we have Thus, we have F : S → S. Now, for every z 1 , z 2 ∈ S and t ∈ J, we have where and Using Lemma (2.6) and hypotheses (H1), (H2), we obtain Applying Lemma (2.6), hypotheses (H2), and assumptions (ii), (iii), we get Again by using Lemma (2.6), hypotheses (H2), and assumptions (ii), (iii), we obtain Hence from (3.3)-(3.5),we have Γ(α+1) [1 + KT + HT ], with 0 < q < 1.Thus F is a strict contraction map from S into S and therefore by Banach contraction principle there exists unique fixed point x of F in S and this point is the mild solution of problem (1.1)-(1.2) on J.This completes the proof of the Theorem 3.1.
To establish the existence of unique classical solution to (1.1)-(1.2),we shall require the following lemma.
Proof.It follows from (b) of Definition 2.3 that R z (t, s)x is continuously differentiable in t ∈ J. Using mean value theorem for derivatives, we obtain where | t2 − t1 | ≤ 1, 0 < α ≤ 1 and sup t∈J ∂Rz ∂t (t, s)x ≤ N 0 x for some N 0 > 0.
Theorem 3.3.Assume that EJQTDE, 2012 No. 51, p. 8 (i) hypotheses (H1)-(H3) hold, (ii) X is a reflexive Banach space with norm • and x 0 ∈ D(A), the domain of A(t, •), (iii) g(t 1 , t 2 , . . ., t p , x(•)) ∈ D(A), (iv) There exists a constant L > 0 such that Proof of Theorem 3.3.All the assumptions of Theorem 3.1 are being satisfied, then problem (1.1)-(1.2) has a unique mild solution belonging to S and given by (3.8) Since J is compact it is easy to check that x is Hölder continuous on J if it is locally Hölder continuous.Now we will show that x is locally Hölder continuous.
For simplification, set Then (3.8) can be written as (3.10) Since x is continuous on J and the map f satisfy the assumptions (iv) and (v), it fellows that f is continuous, and therefore bounded on J, set N 1 := sup t∈J f (t) .EJQTDE, 2012 No. 51, p. 9 Next, let t ∈ J be fixed and let t1 , t2 be in (t − δ, t + δ) with t1 ≤ t2 and δ > 0, we have Using Lemma (3.2) for a small enough δ > 0, we get ) for Ĩ2 , we have and for Ĩ3 , we have with µ = 1 − α.Here we can use the calculation presented in [31, Theorem 3.2] to find the upper bound of integral and thus we get where c = (1 − µ) 1 µ and 0 < δ 1 ≤ 1.Using again (3.6), we may calculate the bound of Ĩ4 as EJQTDE, 2012 No. 51, p. 10 Hence from (3.13)-(3.17),locally Hölder continuity of x(t) follows.
As pointed out earlier in this proof, we may deduce that x(t) is Hölder continuous on J.The Hölder continuity of x(t) on J combined with (iv) and (v) of Theorem (3.3) implies f (t) is Hölder continuous on J.According to [16,Theorem 3.4], we observe that the equation has a unique classical solution y(t) on J satisfying the equation Consequently, x(t) is the classical solution of initial value problem (1.1)-(1.2) on J.This completes the proof of Theorem 3.3.

Application
In this section we present an example to illustrate the applications of some of our main results, we consider the fractional mixed Volterra-Fredholm partial integrodifferential equation = P (t, w(u, t), Let X = L 2 [0, 1] be the space of square integrable functions.Define the operator A(t, •) : X → X by (A(t, •)z)(u) = a(u, t, •)z ′′ with dense domain D(A(t, •)) = {z ∈ X : z, z ′ are absolutely continuous, z ′′ ∈ X and z(0) = z(1) = 0}, generates an evolution system and R x (t, s) can be extracted from evolution system, such that R x (t, s) ≤ M 0 , M 0 > 0 for s < t and x ∈ Ω ⊂ X (see [16,17,34]).