Semilinear Differential Equations with Nonlocal Conditions in Banach Spaces

In this paper we study the existence of mild solutions for the nonlocal Cauchy prob- lem x 0 (t) = Ax(t) + f(t;x(t));0 < tb;x(0) = x0; by using the fixed point techniques,


Introduction and preliminaries
In this paper we discuss the semilinear differential equation with nonlocal condition (1.1) d dt x(t) = Ax(t) + f (t, x(t)), t ∈ (0, b], (1.2) where A is the infinitesimal generator of a strongly continuous semigroup {T (t) : t ≥ 0} of linear operators defined on a Banach space X, f : [0, b] × X → X and g : C([0, b]; X) → X are appropriate given functions.
The theory of differential equations with nonlocal conditions was initiated by Byszewski and it has been extensively studied in the literature.We infer to some of the papers below.Byszewski ([3,4]), Byszewski and Lakshmikantham ( [6]) give the existence and uniqueness of mild solutions and classical solutions when g and f satisfy 0 2000 Mathematics Subject Classification: 35R20, 47D06, 47H09.Lipschitz-type conditions with the special type of g.In [5], Byszewski and Akca give the existence of semilinear functional differential equation when T (t) is compact, and g is convex and compact on a given ball of C([0, b]; X).Dong and Li [8,9] discussed viable domain for semilinear functional differential equation when T (t) is compact.Ntouyas and Tsamatos [14] studied the existence for semilinear evolution equations with nonlocal conditions.Xue [16] proved the existence results for nonlinear nonlocal Cauchy problem.In [17], Xue discussed the semilinear case when f and g are compact and when g is Lipschitz and T (t) is compact.Fan et al. [7], Guedda [10] and Xue [18] studied some semilinear equations under the conditions in respect of the measure of noncompactness.
In this paper, by using the tools involving the measure of noncompactness and fixed point theory, we obtain existence of mild solution of semilinear differential equation with nonlocal conditions (1.1)-(1.2),and the compactness of solution set, without the assumption of compactness or equicontinuity on the associated semigroup.Our results extend and improve the correspondence results in [3,4,5,6,17].We indicate that the method we used in this paper is different from that in [7] or [10].
Throughout this paper X will represent a Banach space with norm • .As usual, C([a, b]; X) denotes the Banach space of all continuous Xvalued functions defined on [a, b] Let A : D(A) ⊂ X → X be the infinitesimal generator of a strongly continuous semigroup {T (t) : t ≥ 0} of linear operators on X.We always assume that For more details of the semigroup theory we refer the readers to [15].

Measure of noncompactness
In this section we recall the concept of the measure of noncompactness in Banach spaces.
where B and convB mean the closure and convex hull of B respectively; ( where Z be a Banach space; EJQTDE, 2009 No. 47, p. 3 In this paper we denote by β the Hausdorf f s measure of noncompactness of X and denote by β c the Hausdorf f s measure of noncompactness of C([a, b]; X).To discuss the existence we need the following lemmas in this paper.
) is measurable and then χ 1 is well defined from the properties of Hausdorff's measure of noncompactness.We also define where mod C (B(t)) is the modulus of equicontinuity of the set of functions B at point t given by the formula Then χ is a monotone and nonsingular measure of noncompactness in the space C([a, b]; X).Further, χ is also regular by the famous Ascoli-Arzela's theorem.Similar definitions with χ 1 , χ 2 and χ can be found in [2].
The following property of regular measure of noncompactness is useful for our results Lemma 2.6.Suppose that Φ is a regular measure of noncompactness in a Banach space Y , and {B n } is a sequence of nonempty, closed and bounded subsets in

The existence of mild solution
In order to define the concept of mild solution for (1.1)-(1.2),by comparison with the abstract Cauchy initial value problem whose properties are well known [15], we associate (1.1)-(1.2) to the integral equation In this section by using the usual techniques of the measure of noncompactnes and its applications in differential equations in Banach spaces (see, e.g.[2], [13]) we give some existence results for the nonlocal problem (1.1)-(1.2).Here we list the following hypotheses.
2) if and only if x is a fixed point of Γ.We shall show that Γ has a fixed point by Schauder's fixed point theorem.
To do this, we first see that Γ is continuous by the usual technique involving (Hf), (Hg) and Lebesgue's dominated convergence theorem.We denote by Define W n+1 = convΓW n for n = 1, 2, • • • .From the above proof we have that {W n } ∞ n=1 is an decreasing sequence of bounded closed, convex and nonempty subsets in C([0, b]; X).Now, for every n ≥ 1 and t ∈ (0, b], W n (t) and ΓW n (t) are bounded subsets of X. Hence for any ε > 0, there is a sequence {x k } ∞ k=1 ⊂ W n such that (see, e.g., [1], pp.125) From the compactness of g, Lemma 2.2, Lemma 2.5 and (Hf)(3), we have Since ε > 0 is arbitrary, it follows from the above inequality that then we obtain from (3.6) that EJQTDE, 2009 No. 47, p. 8 If 0 < t < t then we have It follows that for any t 1 , t 2 ∈ (t 0 − δ 1 , t 0 + δ 1 ), (we may assume that Now, on the basis of the definition of Hausdorff's measure of noncompactness and the fact that β(W n (t 0 − δ 1 )) ≤ ε n , we may find where B(u, r) denote the ball centered at u and radius r.Hence there is an i, 1 ≤ i ≤ k such that (3.11) y(t 0 − δ 1 ) − y i (t 0 − δ 1 ) < 2ε n .
On the other hand, on account of the strong continuity of T (•), there is a δ > 0 (we may choose δ < δ 1 ) such that (3.12) T (τ )y i (t 0 − δ 1 ) − y i (t 0 − δ 1 ) < ε for all τ ∈ (0, δ) and i = 1, 2, • • • , k. From (3.8)-(3.12),we obtain that Coming back to consider f n , since W n is decreasing for n, we know that f (t) = lim n→∞ f n (t) exists for t ∈ [0, b].Taking limit as n → ∞ in (3.7), we have In some of the early related results in references and the two results above, it is supposed that the map g is uniformly bounded.We indicate here that this condition can be released.Indeed, the fact that g is compact implies that g is bounded on bounded subset.And the hypothesis (Hf)(2) may be difficult to be verified sometimes.Here we give an existence result under another growth condition of f when g is not uniformly bounded.Precisely, we replace the hypothesis (Hf)( 2 Remark 3.6.In some previous papers the authors assumed that the space X is a separable Banach space and the semigroup T (t) is equicontinuous (see, e.g., [17,18]).We mention here that these assumptions are not necessary.EJQTDE, 2009 No. 47, p. 11