Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 47, 1-13; http://www.math.u-szeged.hu/ejqtde/

EXISTENCE OF SOLUTIONS FOR SEMILINEAR DIFFERENTIAL EQUATIONS WITH NONLOCAL

CONDITIONS IN BANACH SPACES

Qixiang Dong and Gang Li

School of Mathematical Science, Yangzhou University Yangzhou 225002, P. R. China e-mail: qxdongyz@yahoo.com.cn gli@yzu.edu.cn

Abstract. This paper is concerned with semilinear differential equations with nonlocal conditions in Banach spaces. Using the tools involving the measure of noncompactness and fixed point theory, existence of mild solutions is obtained without the assumption of compactness or equicontinuity on the associated linear semigroup.

1. Introduction and preliminaries

In this paper we discuss the semilinear differential equation with nonlocal condition

(1.1)  \frac{d}{dt}x(t) = Ax(t) + f(t, x(t)), t ∈ (0, b],
(1.2)  x(0) = x_{0} + g(x)

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

EJQTDE, 2009 No. 47, p. 1 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.

02000 Mathematics Subject Classification: 35R20, 47D06, 47H09.

0Keywords: Differential equation, nonlocal condition, measure of noncompactness, C_{0}-semigroup, mild solution.

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] with norm ∥x∥_{[a,b]} = sup_{s∈[a,b]} ∥x(s)∥ for x ∈ C([a, b]; X).

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 ∥T (t)∥ ≤ M (M ≥ 1) for every t ∈ [0, b].

For more details of the semigroup theory we refer the readers to [15].

2. Measure of noncompactness

In this section we recall the concept of the measure of noncompactness in Banach spaces.

EJQTDE, 2009 No. 47, p. 2 Definition 2.1. Let E^{+} be a positive cone of an ordered Banach space (E, ≤). A function Φ on the collection of all bounded subsets of a Banach space X with values in E^{+} is called a measure of noncompactness if Φ(\overline{co}B) = Φ(B) for all bounded subset B ⊂ X, where \overline{co}B stands for the closed convex hull of B.

A measure of noncompactness Φ is said to be:

(i) monotone if for all bounded subset B_{1}, B_{2} of X, B_{1} ⊂ B_{2} implies Φ(B_{1}) ≤ Φ(B_{2});

(ii) nonsingular if Φ({a} ∪ B) = Φ(B) for every a ∈ X and every nonempty subset B ⊂ X;

(iii) regular if Φ(B) = 0 if and only if B is relatively compact in X. One of the most important examples of measure of noncompactness

is the Hausdorff′s measure of noncompactness β_{Y}, which is defined by

β_{Y} (B) = inf{r > 0; B can be covered with a finite

number of balls of radii smaller then r}

for bounded set B in a Banach space Y.

The following properties of Hausdorff’s measure of noncompactness are well known:

Lemma 2.2. ([2]): Let Y be a real Banach space and B, C ⊆ Y be bounded,the following properties are satisfied :

(1). B is pre-compact if and only if β_{Y} (B) = 0 ;

(2). β_{Y} (B) = β_{Y} (\overline{B}) = β_{Y} (convB) where \overline{B} and convB mean the closure and convex hull of B respectively;

(3). β_{Y} (B) ≤ β_{Y} (C) when B ⊆ C;

(4). β_{Y} (B + C) ≤ β_{Y} (B) + β_{Y} (C) where B + C = {x + y; x ∈ B, y ∈ C};

(5). β_{Y} (B ∪ C) ≤ max{β_{Y} (B), β_{Y} (C)};

(6). β_{Y} (λB) = |λ|β_{Y} (B) for any λ ∈ R;

(7). If the map Q : D(Q) ⊆ Y → Z is Lipschitz continuous with constant k then β_{Z}(QB) ≤ kβ_{Y} (B) for any bounded subset B ⊆ D(Q),where Z be a Banach space;

EJQTDE, 2009 No. 47, p. 3

(8). β_{Y} (B) = inf{d_{Y} (B, C); C ⊆ Y be precompact} = inf{d_{Y} (B, C); C ⊆ Y be finite valued}, where d_{Y} (B, C) means the nonsymmetric (or symmetric) Hausdorff distance between B and C in Y.

(9). If {W_{n}}_{n=1}^{+∞} is a decreasing sequence of bounded closed nonempty subsets of Y and lim_{n→+∞} β_{Y} (W_{n}) = 0, then ⋂_{n=1}^{+∞} W_{n} is nonempty and compact in Y.

The map Q : W ⊆ Y → Y is said to be a β_{Y} − contraction if there exists a positive constant k < 1 such that β_{Y} (Q(B)) ≤ kβ_{Y} (B) for any bounded closed subset B ⊆ W, where Y is a Banach space.

Lemma 2.3. ([12]): Let W ⊂ Y is bounded closed and convex and Q : W → W is a continuous β_{Y} −contraction. If the fixed point set of Q is bounded, then it is compact.

In this paper we denote by β the Hausdorff′s measure of noncompactness of X and denote by β_{c} the Hausdorff′s measure of noncompactness of C([a, b]; X). To discuss the existence we need the following lemmas in this paper.

Lemma 2.4. ([2]): If W ⊆ C([a, b]; X) is bounded, then

β(W (t)) ≤ β_{c}(W)

for all t ∈ [a, b], where W (t) = {u(t); u ∈ W } ⊆ X.Furthermore if W is equicontinuous on [a,b], then β(W (t)) is continuous on [a, b] and

β_{c}(W) = sup{β(W (t)), t ∈ [a, b]}.

Lemma 2.5. ([11, 12]): If {u_{n}}_{n=1}^{∞} ⊂ L^{1}(a, b; X) is uniformly integrable, then β({u_{n}(t)}_{n=1}^{∞}) is measurable and

(2.1)  β({∫_{a}^{t} u_{n}(s)ds}_{n=1}^{∞}) ≤ 2 ∫_{a}^{t} β({u_{n}(s)}_{n=1}^{∞})ds.

Now we consider another measure of noncompactness in the Banach space C([a, b]; X). For a bounded subset B ∈ C([a, b]; X), we define

χ_{1}(B) = sup_{t∈[a,b]} β(B(t)),

EJQTDE, 2009 No. 47, p. 4 then χ_{1} is well defined from the properties of Hausdorff’s measure of noncompactness. We also define

χ_{2}(B) = sup_{t∈[a,b]} mod_{C}(B(t)),

where mod_{C}(B(t)) is the modulus of equicontinuity of the set of functions B at point t given by the formula

mod_{C}(B(t)) = lim sup_{δ→0}{∥x(t_{1}) − x(t_{2})∥ : t_{1}, t_{2} ∈ (t − δ, t + δ), x ∈ B}.

Define

χ(B) = χ_{1}(B) + χ_{2}(B).

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 Y satisfying B_{n+1} ⊂ B_{n} for n = 1, 2, · · ·. If lim_{n→∞} Φ(B_{n}) = 0, B = ∩_{n≥1}B_{n} ≠ ∅ and B is a compact subset in Y.

3. 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

\frac{d}{dt}x(t) = Ax(t) + f(t), x(0) = \overline{x} ∈ X,

whose properties are well known [15], we associate (1.1)-(1.2) to the integral equation

(3.1) x(t) = T (t)(x_{0} + g(x)) + ∫_{0}^{t} T (t − s)f(s, x(s))ds, t ∈ [0, b].

Definition 3.1. A continuous function x : [0, b] → X is said to be a mild solution to the nonlocal problem (1.1)-(1.2) if x(0) = x_{0} + g(x) and (3.1) is satisfied.

EJQTDE, 2009 No. 47, p. 5

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.

(Hf):(1) f : [0, b] × X → X satisfies the Carathéodory-type condition, i.e.,f(·, x) : [0, b] → X is measurable for all x ∈ X and f(t, ·) : X → X is continuous for a.e. t ∈ [0, b];

(2): There exists a function h : [0, b] × ℝ^{+} → ℝ^{+} such that h(·, s) ∈ L(0, b; ℝ^{+}) for every s ≥ 0, h(t, ·) is continuous and increasing for a.e. t ∈ [0, b], and ∥f(t, x)∥ ≤ h(t, ∥x∥) for a.e. t ∈ [0, b] and all x ∈ X, and for each positive constant K, the following scalar equation

(3.2)  m(t) = MK + M ∫_{0}^{t} h(s, m(s))ds, t ∈ [0, b]

has at least one solution; (3):There exists η ∈ L(0, b; ℝ^{+}) such that:

(3.3)  β(T (s)f(t, D)) ≤ η(t)β(D)

for a.e. t, s ∈ [0, b] and any bounded subset D ⊂ X.

(Hg): (1) g : C([0, b]; X) → X is continuous and compact;

(2) There exists a constant N > 0 such that

(3.4)  ∥g(x)∥ ≤ N

for all x ∈ X.

Remark 3.2. If the semigroup {T (t) : t ≥ 0} or the function f is compact (see, e.g.,[5, 17]), or f satisfies Lipschitz-type condition (see, e.g., [3, 4]), then (Hf)(3) is automatically satisfied.

Now, we give an existence result under the above hypotheses.

Theorem 3.3. Assume the hypotheses (Hf) and (Hg) are satisfied. Then for each x_{0} ∈ X, the solution set of the problem (1.1)-(1.2) is a nonempty compact subset of the space C([0, b]; X).

Proof. Let m : [0, b] → ℝ^{+} be a solution of the scalar equation:

(3.5)  m(t) = M(∥x_{0}∥ + N) + M ∫_{0}^{t} h(s, m(s))ds, t ∈ [0, b].

EJQTDE, 2009 No. 47, p. 6 Define a map Γ : C([0, b]; X) → C([0, b]; X) by

Γx(t) = T (t)(x_{0} + g(x)) + ∫_{0}^{t} T (t − s)f(s, x(s))ds, t ∈ [0, b]

for all x ∈ C([0, b] : X). It is easily seen that x ∈ C([0, b]; X) is a mild solution of problem (1.1)-(1.2) 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 W_{0} = {x ∈ C([0, b]; X) : ∥x(t)∥ ≤ m(t), ∀t ∈ [0, b]}. Then W_{0} ⊂ C([0, b]; X) is bounded and convex.

Define W_{1} = \overline{conv}ΓW_{0}, where \overline{conv} means the closure of the convex hull in C([0, b]; X). Then it is easily seen that W_{1} ⊂ C([0, b]; X) is closed and convex. Furthermore, for every x ∈ C([0, b]; X), we have

∥Γx(t)∥ ≤ M(∥x_{0}∥ + N) + M ∫_{0}^{t} h(s, m(s))ds = m(t)

for t ∈ [0, b]. Hence Γx ∈ W_{0} for all x ∈ W_{0}. It then follows that W_{1} is bounded and W_{1} ⊂ W_{0}.

Define W_{n+1} = \overline{conv}ΓW_{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)

β(ΓW_{n}(t)) ≤ 2β({Γx_{k}(t)}_{k=1}^{∞}) + ε
≤ 2β(T (t)(x_{0} + g({x_{k}}_{k=1}^{∞})
+2β(∫_{0}^{t} T (t − s)f(s, {x_{k}(s)}_{k=1}^{∞})ds) + ε.

EJQTDE, 2009 No. 47, p. 7 From the compactness of g, Lemma 2.2, Lemma 2.5 and (Hf)(3), we have

β(W_{n+1}(t)) = β(ΓW_{n}(t))
≤ 2β(∫_{0}^{t} T (t − s)f(s, {x_{k}(s)}_{k=1}^{∞})ds) + ε
≤ 2 ∫_{0}^{t} β(T (t − s)f(s, {x_{k}(s)}_{k=1}^{∞}))ds + ε
≤ 2 ∫_{0}^{t} η(s)β({x_{k}(s)}_{k=1}^{∞})ds + ε
≤ 2 ∫_{0}^{t} η(s)β(W_{n}(s))ds + ε.

Since ε > 0 is arbitrary, it follows from the above inequality that

(3.6)  β(W_{n+1}(t)) ≤ 2 ∫_{0}^{t} η(s)β(W_{n}(s))ds

for all t ∈ (0, b]. Define functions f_{n} : [0, b] → [0, +∞) by

f_{n}(t) = β(W_{n}(t)),

then we obtain from (3.6) that

f_{n+1}(t) ≤ 2 ∫_{0}^{t} η(s)f_{n}(s)ds (3.7)

for all t ∈ [0, b].

Notice that χ_{1}(W_{n}) = sup_{0≤t≤b} β(W_{n}(t)) = sup_{0≤t≤b} f_{n}(t). Denote ε_{n} = χ_{1}(W_{n}), then β(W_{n}(t)) ≤ ε_{n} for all t ∈ [0, b]. Now we consider χ_{2}(W_{n}). We claim that there is a constant K_{1} such that χ_{2}(W_{n}) ≤ K_{1}ε_{n}. To do this, it is sufficient to prove that for every t_{0} ∈ [0, b], mod_{C}(W_{n}(t_{0})) ≤ K_{1}ε_{n}.

Take ε > 0 arbitrary. First note that, since f(s, (W_{n})(s)) is uniformly integrable on [0, b], there is δ_{1} > 0 such that

(3.8)  ∥ ∫_{τ_{1}}^{τ_{2}} T (t − s)f(s, x(s))ds∥ < ε

for 0 < τ_{2} − τ_{1} < 2δ_{1}, 0 ≤ s ≤ t ≤ b and every x ∈ W_{n}. For every y ∈ Γ(W_{n−1}), there is an x ∈ W_{n−1} such that for all t ∈ [0, b],

y(t) = T (t)(x_{0} + g(x)) + ∫_{0}^{t} T (t − s)f(s, x(s))ds.

EJQTDE, 2009 No. 47, p. 8 If 0 < t′ < t then we have

y(t) = T (t − t′)T (t′)(x_{0} + g(x)) + T (t − t′) ∫_{0}^{t′} T (t′ − s)f(s, x(s))ds
+ ∫_{t′}^{t} T (t − s)f(s, x(s))ds
= T (t − t′)y(t′) + ∫_{t′}^{t} T (t − s)f(s, x(s))ds.

It follows that for any t_{1}, t_{2} ∈ (t_{0} − δ_{1}, t_{0} + δ_{1}), (we may assume that t_{0} − δ_{1} > 0),

(3.9) y(t_{1}) = T (t_{1} − t_{0} + δ_{1})y(t_{0} − δ_{1}) + ∫_{t_{0}−δ_{1}}^{t_{1}} T (t_{1} − s)f(s, x(s))ds,

(3.10) y(t_{2}) = T (t_{2} − t_{0} + δ_{1})y(t_{0} − δ_{1}) + ∫_{t_{0}−δ_{1}}^{t_{2}} T (t_{2} − s)f(s, x(s))ds.

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 y_{1}, y_{2}, ⋯, y_{k}, such that

W_{n}(t_{0} − δ_{1}) ⊂ ⋃_{i=1}^{k} B(y_{i}(t_{0} − δ_{1}), 2ε_{n}),

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

∥y(t_{1}) − y(t_{2})∥ ≤ ∥T (t_{1} − t_{0} + δ_{1})y(t_{0} − δ_{1}) − y(t_{0} − δ_{1})∥
+2∥y(t_{0} − δ_{1}) − y_{i}(t_{0} − δ_{1})∥
+∥T (t_{2} − t_{0} + δ_{1})y(t_{0} − δ_{1}) − y(t_{0} − δ_{1})∥

+∥ ∫_{t_{0}−δ_{1}}^{t_{1}} T (t_{1} − s)f(s, x_{s})ds − ∫_{t_{0}−δ_{1}}^{t_{2}} T (t_{2} − s)f(s, x_{s})ds∥

≤ 4ε_{n} + 4ε

EJQTDE, 2009 No. 47, p. 9 for t_{1}, t_{2} ∈ (t_{0} − δ, t_{0} + δ). Since ε > 0 is taken arbitrary, we get that

∥y(t_{1}) − y(t_{2})∥ ≤ 4ε_{n}

for t_{1}, t_{2} ∈ (t_{0} − δ, t_{0} + δ), which implies that

mod_{C}(W_{n}(t_{0})) ≤ 4ε_{n}

for every t_{0} ∈ [0, b]. Accordingly,

(3.13)  χ_{2}(W_{n}) ≤ 4ε_{n} = 4χ_{1}(W_{n}).

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

(3.14)  f(t) ≤ 2 ∫_{0}^{t} η(s)f(s)ds,

for t ∈ [0, b]. It then follows that f(t) = 0 for all t ∈ [0, b]. This means that lim_{n→∞} χ_{1}(W_{n}) = 0. The inequality (3.13) implies that lim_{n→∞} χ_{2}(W_{n}) = 0, and hence lim_{n→∞} χ(W_{n}) = 0. Using Lemma 2.6 we know that W = ∩_{n=1}^{∞}W_{n} is convex compact and nonempty in C([0, b]; X) and ΓW ⊂ W. By the famous Schauder’s fixed point theorem, there exists at least one fixed point x ∈ W of Γ, which is the mild solution of (1.1)-(1.2). From the proof we can also see that all the fixed points of Γ are in W which is compact in C([0, b]; X). The continuity of the map Γ implies the closeness of the fixed point set. Hence we conclude that the solution set of problem (1.1)-(1.2) is compact in the space C([0, b]; X). □

Since {T (t)} is a C_{0}−semigroup, condition (Hf)(3) can be replaced by

(Hf)(3′): There exists η̃ ∈ L(0, b; ℝ^{+}) such that:

(3.15)  β(f(t, D)) ≤ η̃(t)/M · β(D)

for a.e. t, s ∈ [0, b] and any bounded subset D ⊂ X.

From Theorem 3.3, we can get the following obvious result:

Theorem 3.4. Assume the hypotheses (Hf)(1)(2)(3′) and (Hg) are satisfied. Then for each x_{0} ∈ X, the solution set of problem (1.1)-(1.2) is a nonempty compact subset of the space C([0, b]; X).

EJQTDE, 2009 No. 47, p. 10

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) by

(Hf)(2′): There exists a function α ∈ L(0, b; ℝ^{+}) and an increasing function Ω : ℝ^{+} → ℝ^{+} such that

∥f(t, v)∥ ≤ α(t)Ω(∥v∥)

for a.e. t ∈ [0, b] and all v ∈ X.

Theorem 3.5. Suppose that the hypotheses (Hf)(1)(2′)(3) and (Hg)(1) are satisfied. If

(3.16)  lim sup_{k→∞} \frac{M}{k} (γ(k) + Ω(k) ∫_{0}^{b} α(s)ds < 1,

where γ(k) = sup{∥g(x)∥ : ∥x∥ ≤ k}, then for each x_{0} ∈ X, the solution set of problem (1.1)-(1.2) is a nonempty compact subset of the space C([0, b]; X).

Proof. The inequality (3.16) implies that there exists a constant k > 0 such that

M(∥x∥ + γ(k) + Ω(k) ∫_{0}^{b} α(s)ds) < k.

As in the proof of Theorem 3.3, let W_{0} = {x ∈ X : ∥x∥ ≤ k} and W_{1} = \overline{conv}ΓW_{0}. Then for any x ∈ W_{1}, we have

∥x(t)∥ ≤ M(∥x∥ + γ(k)) + MΩ(k) ∫_{0}^{b} α(s)ds < k

for t ∈ [0, b]. It follows that W_{1} ⊂ W_{0}. So we can complete the proof similarly to Theorem 3.3. □

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

References

[1] D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Isreal J. Math. 108(1998) 109-138.

[2] J. Banas, K. Goebel, Measure of Noncompactness in Banach spaces, Lecture Notes in Pure and Applied Math. Vol 60, Dekker, New York, 1980.

[3] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162(1991) 497-505.

[4] L. Byszewski, Existence and uniqueness of solutions of semilinear evolution nonlocal Cauchy problem, Zesz. Nauk. Pol. Rzes. Mat. Fiz., 18(1993) 109-112.

[5] L. Byszewski, H. Akca, Existence of solutions of a semilinear functionaldifferential evolution nonlocal problem, Nonlinear Anal. 34(1998) 65-72.

[6] L. Byszewski, V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal. 40(1990) 11-19.

[7] Q. Dong, Z. Fan, G. Li, Existence of solutions to nonlocal neutral functional differential and integrodifferential equations, Intern J. Nonlinear Sci., 5(2008) No.2, 140-151.

[8] Q. Dong, G. Li, Viability for Semilinear Differential Equations of Retarded Type, Bull. Korean Math. Soc. 44(2007), No.4, 731-742.

[9] Q. Dong, G. Li, Viability for a class of semilinear differential equations of retarded type, Appl. Math. J. Chinese Univ., 24(2009), No.1, 36-44.

[10] L. Guedda, On the existence of mild solutions for neutral functional differential inclusions in Banach spaces, Electronic J. Qualitative Theory of Differential Equations, 2007(2007) No.2 1-15.

[11] H. P. Heinz, On the behavior of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal. TMA. 7(1983) 1351-1371.

[12] M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing multivalued maps and semilinear differential inclusions in Banach spaces, De Gruyter Ser. Nonlinear Anal. Appl., Vol.7, de Gruyter, Berlin, 2001.

[13] M. Kisielewicz, Multivalued differential equations in separable Banach spaces, J. Optim. Theory Appl. 37(1982) 231-249.

[14] S. K. Ntouyas, P. Ch. Tsamatos, Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl. 210(1997) 679-687.

[15] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.

[16] X. Xue, Nonlinear differential equations with nonlocal conditions in Banach spaces, Nonlinear Anal. 63(2005) 575-586.

EJQTDE, 2009 No. 47, p. 12

[17] X. Xue, Existence of solutions for semilinear nonlocal Cauchy problems in Banach spaces, Elec. J. D. E. 2005(2005) No.64, 1-7.

[18] X. Xue, Semilinear nonlocal differential equations with measure of noncompactness in Banach spaces, Journal of Nanjing University Mathematical Biquarterly, 24(2007), No.2, 264-275.

(Received April 12, 2009)

EJQTDE, 2009 No. 47, p. 13