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Solvability of impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators.

1. Introduction

In this paper we study the existence of mild solutions for the following system in general Banach space X

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

t [member of] J - {[t.sub.1], [t.sub.2], ..., [t.sub.m]} where J = [0,a], (1.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.3)

x(0) = g(x) [member of] X (1.4)

where A is the infinitesimal generator of a compact, analytic resolvent operator R(t),t > 0 in a Banach space X, [x.sub.0] [member of] X, 0 = [t.sub.0] < [t.sub.1] < ... < [t.sub.m] < [t.sub.m+1] = a, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represent the right and left limits of x(t) at t = [t.sub.k]. [h.sub.i]: J [right arrow] J, i = 1, 2, f(t), t [member of] J is a bounded linear operator, and F, G, g are appropriate functions specified later.

The theory of impulsive differential equations appears as a natural description of several real processes subject to certain perturbations whose duration is negligible in comparison with the duration of the process. It has seen considerable development in the past decade, see the monographs of Benchohra et al.[2], and the papers [6, 7, 8, 18, 19, 22, 23, 24, 27].

The nonlocal Cauchy problem was considered by Byszewski [4] and the importance of nonlocal conditions in different fields has been discussed in [4] and [10] and references therein. For example, in [10] the author described the diffusion phenomenonof a small amount of gas in a transparent tube by using the formula

g(x) = [p.summation over (i=0)][c.sub.i]x([t.sub.i]),

where [c.sub.i], i = 0, 1, ..., p are given constants and 0 < [t.sub.0] < t1 < ... < [t.sub.p] < a. In this case the above equation allows the additional measurement at [t.sub.i], i = 0, 1, ..., p. In the past several years theorems about existence, uniqueness and stability of differential and functional differential abstract evolution Cauchy problem with nonlocal conditions have been studied by Byszewski and Lakshmikantham [3], by Byszewski [5], by Fu [14], by Fu and Ezzinbi [15] studied the existence results for impulsive neutral functional differential equations with nonlocal conditions.

In the present paper, we discuss the existence of mild solutions for the problem (1.1)-(1.3) with [alpha]-norm as in [12]. The rest of this paper is organized as follows: In section 2 we recall briefly some basic definitions and preliminary facts which will be used throughout this paper. The existence theorems for the problem (1.1)-(1.3) and their proofs and arranged in section 3. Finally, in section 4 an example is presented to illustrate the applications of the obtained result.

2. Preliminaries

In this section, we introduce some basic definitions, notations, and lemmas that are used throughout this paper.

Let (X, [parallel]x[parallel]) be a Banach space. C(J, X) is the Banach space of continuous functions from J to X with the norm [[parallel]x[parallel].sub.J] = sup{[parallel]x(t)[parallel]: t [member of] J}. B(x) denotes the Banach space of bounded linear operators from X to X, with the norm [[parallel]L[parallel].sub.B(X)] = sup{[parallel]L(y)[parallel]:[parallel]y[parallel] = 1}.

A measurable function x : J [right arrow] X is Bochnerintegrable ifand only if [parallel]x[parallel] is Lebegue integrable (For properties of the Bochner integral see Yosida [26]). [L.sup.1] (J, X) denotes the Banach space of Bochner integrable functions x : J [right arrow] X with norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let (X, d) be a metric space. We use the notations P(X) = {Y [member of] X : Y [not equal to] [empty set]}, [P.sub.cl](X) = {Y [member of] P(X): Y closed}, [P.sub.b](X) = {Y [member of] P(X): Y bounded}, [P.sub.c](X) = {Y [member of] P(X): Y convex}, and [P.sub.cp](X) = {Y [member of] P(X): Y compact}.

A multi-valued map N : X [right arrow] P(X) is said to be convex (closed) valued if N(x) is convex (closed) for all x [member of] X. N is said to be bounded on bounded sets if N(B) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is bounded in X for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

N is called upper semicontinuous (u.s.c.) on X if for each [bar.x] [member of] X, the set N([bar.x]) is a nonempty, closed subset of X, and if for each open set U of X containing N([bar.x]), there exists an open neighborhood V of [bar.x] such that N(V) [subset or equal to] U.

N is said to be completely continuous if N(B) is relatively compact for every B [member of] [P.sub.b](X). If the multivalued map N is completely continuous with nonempty compact values, then N is u.s.c. if and only if N has a closed graph (i.e., [x.sub.n] [right arrow] [x.sub.*], [y.sub.n] [right arrow] [y.sup.*], [y.sub.n] [member of] N([x.sub.n]) imply [y.sub.*] [member of] N([x.sub.*])). N has a fixed point if there is x [member of] X such that x [member of] N(x).

An upper semicontinuous multivalued map G : X [right arrow] P(X) is said to be condensing [9] if for any subset B [member of] X with o(B) [not equal to] 0, we have o(G(B)) < o(B), where o denotes the Kuratowski measure of noncompactness [1]. For more details on multivalued maps, we refer to the book by Deimling [9] and Hu et al. [20].

Definition 2.1. [16] A family of bounded linear operators R(t) [member of] B(X) for t [member of] J is called a resolvent operator for

dx/dt = A[x(t) + [[integral].sup.t.sub.0] f(t - s)x(s)ds],

(i) R(0) = I, the identity operator on X.

(ii) For all u [member of] X, R(t)u is continuous for t [member of] J.

(iii) R(t) [member of] B(Y), t [member of] J, where Y is the Banach space formed from D(A) endowed with the graph norm. For y [member of] Y, R(x) y [member of] [C.sup.1] (J, X) [intersection] C(J, Y) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Assume that [parallel]R(t)[parallel] [less than or equal to] M[16]. Let 0 [member of] [rho](A), then it is possible to define the fractional power [A.sup.[alpha]], for 0 < [alpha] [less than or equal to] 1, as a closed linear operator on its domain D([A.sup.[alpha]]). Furthermore, the subspace D([A.sup.[alpha]]) is dense in X, and the expression

[[parallel]x[parallel].sub.[alpha]] = [parallel][A.sup.[alpha]]x[parallel], x [member of] D([A.sup.[alpha]])

defines a norm on D([A.sup.[alpha]]). Hereafter, we denote by [X.sub.[alpha]] the Banach space D([A.sup.[alpha]]) with norm [[parallel]x[parallel].sub.[alpha]]. The following properties are well known [17].

Lemma 2.2. Under the above conditions we have

(i) [A.sup.[alpha]]: [X.sub.[alpha]] [right arrow] [X.sub.[alpha]], then [X.sub.[alpha]] is a Banach space for 0 [less than or equal to] [alpha] [less than or equal to] 1.

(ii) If the resolvent operator of A is compact, then [X.sub.[alpha]] [right arrow] [X.sub.[beta]] is continuous and compact for 0 < [beta] [less than or equal to] [alpha].

(iii) For each [alpha] > 0, there exists a positive constant [C.sub.[alpha]] such that

[parallel][A.sup.[alpha]]R(t)[parallel] [less than or equal to] [[C.sub.[alpha]]/[t.sup.[alpha]]], t [member of] J.

Lemma 2.3. (see [11]) Let B (0,r) and B [0,r] denote respectively the open and closed balls in a Banach space X centered at the origin and of radius r and let A: X [right arrow] [P.sub.bcc](X) and B: B[0,r] [right arrow] [P.sub.cp,c](X) be two multi-valued operators satisfying:

(I) A is a multi-valued contraction, and

(II) B is upper semicontinuous and completely continuous. Then either

(a) the operator inclusion x [member of] Ax + Bx has a solution in B[0,r], or

(b) there exists a u [member of] X with [parallel]u[parallel] = r such that [delta]u [member of] Au + Bu for some [delta ] > 1.

In order to define mild solutions of the problem (1.1)-(1.3), we introduce the following space [OMEGA] = {x: J [right arrow] X: [x.sub.k] [member of] C([J.sub.k], X), k = 0, 1, ..., m and there exist x([t.sup.-.sub.k]) and x([t.sup.+.sub.k]), k = 1, 2, ..., m with x([t.sup.-.sub.k]) = x([t.sub.k])}, which is a Banach space with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [x.sub.k] is the restriction of x to [J.sub.k] = ([t.sub.k], [t.sub.k+1]], k = 0, ..., m, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now we define the mild solution for the system (1.1)-(1.3).

Definition 2.4. A function x [member of] [OMEGA] is said to be a mild solution of the problem (1.1)-(1.3) if the following holds: x(0) + g(x) = [x.sub.0] for each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the restriction of x(x) to the interval [0,a) - {[t.sub.1], [t.sub.2], ..., [t.sub.m]} is continuous and for each s [member of] [0,t) the function AR(t - s) F(s, x([h.sub.1](s))) is integrable and the impulsive integral equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is satisfied.

3. Existence Results

In this section, we state and prove the existence theorem for the problem (1.1)-(1.3). Let us list the following hypothesis: for some [alpha] [member of] (0, 1),

(H1) There exist constants [M.sub.1], [M.sub.2], [M.sub.3] such that

[parallel]R(t)[parallel] [less than or equal to] [M.sub.1], t [member of] J, [parallel][A.sup.-[beta]][parallel] [less than or equal to] [M.sub.2], [parallel]f(t)[parallel] [less than or equal to] [M.sub.3].

(H2) There exists a constant [beta] [member of] (0, 1) such that F: J x [X.sub.[alpha]] [right arrow] [X.sub.[beta]] is a continuous function, and [A.sup.[beta]] F: J x [X.sub.[alpha]] [right arrow] [X.sub.[alpha]] satisfies the Lipschitz condition, i.e., there exists a constant [L.sub.*] > 0 such that

[parallel][A.sup.[beta]] F(s, [x.sub.1]) - [A.sub.[beta]]F(t, [x.sub.2]) [less than or equal to] [L.sub.*]([absolute value of s - t] + [[parallel][x.sub.1] - [x.sub.2][parallel].sub.[alpha]]).

for any s, t [member of] J, [x.sub.1], [x.sub.2] [member of] [X.sub.[alpha]]. Moreover, there exists a constant L1 > 0 such that for any x [member of] [X.sub.[alpha]]

[[parallel][A.sup.[beta]]F(t, x)[parallel].sub.[alpha]] [less than or equal to] [L.sub.1]([[parallel]x[parallel].sub.[alpha]] + 1) holds for any x [member of] [X.sub.[alpha]],

with

[L.sub.0] = [L.sub.*][(1 + [M.sub.1]) [M.sub.2] + [[C.sub.1-[beta]]/[beta]][a.sup.[beta]] + [[.sub.C1-[beta]]] [a.sup.1+[beta]][M.sub.3] < 1. (3.1)

(H3) The multi-valued map G: J x [X.sub.[alpha]] [right arrow] [P.sub.c,cp](X) satisfies the following conditions:

(i) For each t [member of] J, the function G(t, *): [X.sub.[alpha]] [right arrow] [P.sub.c,cp](X) is u.s.c.; and for each x [member of] [X.sub.[alpha]], the function G(*, x) is measurable. And for each fixed u [member of] Q the set [S.sub.G,u] = {v [member of] [L.sup.1](J, X): v(t) [member of] G(t, u) for a.e t [member of] J} is nonempty.

(ii) for each positive number l > 0, there exists a positive function w(l) dependent on l such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(H4) There exist positive constants c, d such that [[parallel]g(x)[parallel].sub.[alpha]] [less than or equal to] c[[parallel]x[parallel].sub.[OMEGA]] + d for x [member of] [OMEGA] and

g: [OMEGA] [right arrow] [X.sub.[alpha]] is completely continuous.

(H5) For each t [member of] J, K(t, s) is measurable on J and

K(t) = ess sup{[absolute value of K(t, s)], 0 [less than or equal to] s [less than or equal to] t}

is bounded on J. The map t [member of] [K.sub.t] is continuous from to [L.sup.[infinity]](J, R), here [K.sub.t](s) = K(t, s).

(H6) [I.sub.k] [member of] C([X.sub.[alpha]], [X.sub.[alpha]]), k = 1, 2, ..., m, and there exist nondecreasing functions [[PSI].sub.k] [member of] (J, [R.sub.+]), k = 1, 2, ..., m such that

[[parallel][I.sub.k](x)[parallel].sub.[alpha]] [less than or equal to] [[PSI].sub.k]([[parallel]x[parallel].sub.[alpha]]), for x [member of] [X.sub.[alpha]].

(H7) There exists a real number r > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

where

[([M.sub.1][M.sub.2] + [M.sub.2] + [[C.sub.1-[beta]]/[beta]] [a.sup.[beta]] + a[M.sub.3] [[C.sub.1- [beta]]/[beta]][a.sup.[beta]]) [L.sub.1] + [M.sub.1]c] < 1 (3.3)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 3.1. Let [x.sub.0] [member of] [X.sub.[alpha]]. If the hypotheses (H1)-(H7) are satisfied, then the system (1.1)-(1.3) admits at least one mild solution on J.

Proof. Consider the operator N: [OMEGA] [right arrow] P([OMEGA]) defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Clearly the fixed points of N are mild solutions of (1.1)-(1.3).

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In order to apply lemma (2.2), we give the proof in several steps

Step 1. A is a contraction. Take [x.sub.1], [x.sub.2] [member of] [B.sub.r] arbitrarily. Then for each t [member of] J and by conditions (H1)-(H2), we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus,

[parallel]A([x.sub.1]) - A([x.sub.2])[parallel] [less than or equal to] [L.sub.0][[parallel]x - y[parallel].sub.[alpha]].

Therefore, by assumption 0 < [L.sub.0] < 1 (see(3.1)), we can see that A is a contraction.

Step 2. B(x) is convex for each x [member of] Q. Indeed, if [[phi].sub.1], [[phi].sub.2] [member of] N(x), then there exist [v.sub.1], [v.sub.2] [member of] [S.sub.G,x] such that for each t [member of] J, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let 0 [less than or equal to] [lambda] [less than or equal to] 1. Then for each t [member of] J we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [S.sub.G,x] is convex(because G has convex values), then [lambda][[phi].sub.1] + (1 - [lambda])[[phi].sub.2] [member of] N(x).

Step 3. B is bounded on bounded sets of [OMEGA].

Let [B.sub.r] = {x [member of] Q: [[parallel]x[parallel].sub.[alpha]] [less than or equal to] r} be a bounded set in [OMEGA]. Now for each x [member of] [B.sub.r], [phi] [member of] B(x), there exists a function v [member of] [S.sub.G,x] such that for each t [member of] J. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence B is bounded on bounded sets of [OMEGA] for each [phi] e[member of] B([B.sub.r]).

Step 4. B sends bounded sets into equicontinuous sets on [OMEGA].

Let [[tau].sub.1], [[tau].sub.2] [member of] J, [[tau].sub.1] < [[tau].sub.2] and [epsilon] > 0 be arbitrarily small. Let x [member of] [B.sub.r] and [phi] [member of] B(x), then there exists v [member of] [S.sub.S,x] such that, for each t [member of] J, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As [[tau].sub.2] - [[tau].sub.1] [right arrow] 0 with [epsilon] sufficiently small, the right-hand side tends to zero; as R(t) is compact and analytic, the functions R(t), [A.sup.[alpha]]R(t) are continuous in the uniform operator topology on (0,a].

Step 5. (B[B.sub.l])(t) is relatively compact for each t [member of] J, where (B[B.sub.l])(t) = {[phi](t): [phi] [member of] B([B.sub.l])}, t [member of] J.

Obviuosly, by condition(), (B[B.sub.l])(t) is relatively compact in [OMEGA] for t = 0. Let 0 < t [less than or equal to] a be fixed and let e be a real number satisfying 0 < [epsilon] < t. For x [member of] [B.sub.l] and [phi] [member of] B(x), there exists a function v [member of] [S.sub.G,x] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because R(t) is compact, the set [V.sub.[epsilon]] = {[phi](t): [phi] [member of] B([B.sub.l])} is relatively compact in [OMEGA] for every [epsilon], 0 < [epsilon] < t. Moreover, for every [phi] [member of] B([B.sub.l]).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, letting [epsilon] [right arrow] 0, we see that, there are relatively compact sets arbitrarily close to the set {[phi](t): [phi] [member of] B([B.sub.l])[parallel]. Hence the set {[phi](t): [phi] [member of] [B.sub.Bl])[parallel] is relatively compact in [OMEGA].

As a consequence of steps 3-5 together with the Arzela-Ascoli theorem, we can conclude that B is a compact multi-valued map.

Step 6. B has a closed graph.

Let [x.sub.n] [right arrow] [x.sub.*], [x.sub.n] [member of] [B.sub.l], [[phi].sub.n] [member of] B([x.sub.n]), and [[phi].sub.n] [right arrow] [[phi].sub.*]. We shall prove that [[phi].sub.*] [member of] B([x.sub.*]). [[phi].sub.n] [member of] B([x.sub.n]) means that there exists [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that, for each t [member of] J,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We must prove that there exists [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that, for each t [member of] J,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because [I.sub.k], k = 1, 2, ..., m and g are continuous we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consider the linear continuous operator defined by

[GAMMA] : [L.sup.1](J, X) [right arrow] C(J, X) v [right arrow] [GAMMA](v)(t) = [[integral].sup.t.sub.0] R(t - s) [[integral].sup.S.sub.0] K(s, [tau])v(x[tau])d[tau]ds.

From lemma (2.1), it follows that [GAMMA] x [S.sub.G,x] is a closed graph operator. Moreover we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because [x.sub.n] [right arrow] [x.sub.*], it follows from lemma (2.1) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that is, there must exist a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, B has a closed graph and so B is u.s.c.

Step 7. The operator inclusion x [member of] A(x) + B(x) has a solution in B[0,r].

Define an open ball B(0,r) in [OMEGA]. Where the real number r satisfies the inequality given in condition (3.2). As a consequence of steps 1 - 6. We can see that the operators A and B satisfy all conditions of lemma (2.2). Now we show that the second assertion of lemma (2.2) is not true. Let u [member of] [OMEGA] be a possible solution for [lambda]u e Au + Bu for some [lambda] > 1 with [[parallel]u[parallel].sub.[alpha]] = r. Then we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus by (H1) - (H6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Taking the supremum over t we obtain,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting [parallel]u[parallel] = r in the above inequality and noting that (3.3) holds. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Which is contradiction to (3.2). Hence the operator inclusions x [member of] A(x) + B(x) has a solution in B[0, r]. It further implies that the problem (1.1)-(1.3) has at least one mild solution x in [OMEGA]. The proof is complete.

4. Example

Consider the following neutral partial integrodifferential inclusion with nonlocal initial condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4.2)

where p is a positive integer, 0 < [s.sub.0] < [s.sub.1] < ... < [s.sub.p] < 1, and 0 < [t.sub.1] < [t.sub.2] < ... < [t.sub.m] < 1. [z.sub.0](x) [member of] X = [L.sup.2]([0, [pi]]) and A is defined by Af = f" with domain

D(A) = [H.sup.2.sub.0]([0, [pi]]) = {f(x) [member of] X: f, f" [member of] X, f(0) = f([pi]) = 0}.

Then A generates a strongly continuous semigroup that is analytic, and resolvent operator R(t) can be extracted from this analytic semigroup(see [17]).

On the other hand, A has a discrete spectrum, the eigenvalues are -[n.sup.2], n [member of] N, with the corresponding normalized eigenvectors [e.sub.n](x) = [square root of 2/[pi]] sin(nx) (see [12]). Moreover (i) If f [member of] D(A), then

Af = [[infinity].summation (n=1)][n.sup.2](f, [e.sub.n])[e.sub.n].

(ii) The operator [A.sup.1/2] is given by

[A.sup.1/2] f = [[infinity].summation over (n=1)] n(f, [e.sub.n])[e.sub.n].

with the domain D([A.sup.1/2]) = {f [member of] X: [[infinity].summation over (n=1)] n(w, [e.sub.n])[e.sub.n]}.

Here we choose [alpha] = [beta] = 1/2. According to [25], if z [member of] D([A.sup.1/2]) then z is absolutely continuous, z' [member of] X and [parallel]z'[parallel] = [parallel][A.sup.1/2]z[parallel].

For (t, z) [member of] [0, 1] x [X.sub.1/2], w [member of] [OMEGA] ([OMEGA] is defined as in section 3), we can define respectively that

F(t,z)(x) = [[integral].sup.[pi].sub.0] a(t, y, x)[z(y) + z'(y)]dy',

G(t,z)(x) = h(t,z(x),z'(x))

and

g(w(t)) = [p.summation over (i=0)] [k.sub.i]w([t.sub.i]), w [member of] [OMEGA].

Let [h.sub.1](t) = [h.sub.2](t) = sin t. Then (4.1) takes the following abstract form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Moreover,

F: J x [X.sub.1/2] [right arrow] [X.sub.1/2],

[A.sub.1/2]F: J x [X.sub.1/2] [right arrow] [X.sub.1/2]

and

G: J x [X.sub.1/2] [right arrow] X (see [13]).

On the other hand, if we impose some suitable conditions on the above defined functions to verify assumptions on theorem (3.1), we can conclude (4.1) admits at least one mild solution on [0,1].

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A. Anguraj (1)

Department of Mathematics, PSG College of Arts and Science, Coimbatore-641 014.

E-mail: angurajpsg@yahoo.com

K. Karthikeyan

Department of Mathematics, KSR College of Technology, Tiruchengode-637 215

E-mail: karthi_phd2010@yahoo.co.in

(1) Corresponding author.
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Author:Anguraj, A.; Karthikeyan, K.
Publication:International Journal of Computational and Applied Mathematics
Date:Sep 1, 2010
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