More precisely in Section 3 we consider first-order impulsive semilinear neutral functional differential inclusions of the form d dt[x(t)−f(t, xt)]∈Ax(t) +G(t, xt) a.e

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EXISTENCE RESULTS FOR IMPULSIVE PARTIAL NEUTRAL FUNCTIONAL DIFFERENTIAL INCLUSIONS

SOTIRIS K. NTOUYAS

Abstract. In this paper we prove existence results for first order semilinear impulsive neutral functional differential inclusions under the mixed Lipschitz and Carath´eodory conditions.

1. Introduction

The theory of impulsive differential equations is emerging as an important area of investigation since it is much richer that the corresponding theory of differential equations; see the monograph of Lakshmikantham et al [3]. In this paper, we study the existence of solutions for initial value problems for first order impulsive semilinear neutral functional differential inclusions. More precisely in Section 3 we consider first-order impulsive semilinear neutral functional differential inclusions of the form

d

dt[x(t)−f(t, xt)]∈Ax(t) +G(t, xt) a.e. t∈J := [0, T], t6=t_k k= 1, . . . , m,

(1.1) x(t⁺_k)−x(t⁻_k) =I_k(x(t⁻_k)), k= 1, . . . , m, (1.2)

x(t) =φ(t), t∈[−r,0], (1.3)

where Ais the infinitesimal generator of an analytic semigroup of bounded linear operators,S(t), t≥0 on a Banach spaceX,f :J× D →X andG:J× D → P(X);

Dconsists of functionsψ: [−r,0]→Xsuch thatψis continuous everywhere except for a finite number of points s at whichψ(s) and the right limit ψ(s⁺) exist and ψ(s⁻) = ψ(s); φ ∈ D, (0 < r < ∞), 0 = t0 < t1 < · · · < tm < tm+1 = T, Ik :X →X (k= 1,2, . . . , m),x(t⁺_k) and x(t⁻_k) are respectively the right and the left limit ofxat t=tk, andP(X) denotes the class of all nonempty subsets ofX.

For any continuous functionxdefined on the interval [−r, T]\ {t1, . . .,tm} and anyt∈J, we denote byx_tthe element ofDdefined by

x_t(θ) =x(t+θ), θ∈[−r,0].

2000Mathematics Subject Classification. 34A60, 34K05, 34K45.

Key words and phrases. Impulsive neutral functional differential inclusions;

fixed point theorem; existence theorem.

c

2005 Texas State University - San Marcos.

Submitted February 9, 2005. Published March 14, 2005.

1

Forψ∈ Dthe norm ofψis defined by

kψk_D = sup{|ψ(θ)|, θ∈[−r,0]}.

The main tools used in the study is a fixed point theorem proved by Dhage [1].

In the following section, we give some auxiliary results needed in the subsequent part of the paper.

2. Auxiliary results

Throughout this paper,X will be a separable Banach space provided with norm k · kandA:D(A)→Xwill be the infinitesimal generator of an analytic semigroup, S(t), t≥0, of bounded linear operators onX. For the theory of strongly continuous semigroup, refer to Pazy [5]. IfS(t), t ≥0, is a uniformly bounded and analytic semigroup such that 0∈ρ(A), then it is possible to define the fraction power (−A)^α, for 0< α≤1, as closed linear operator on its domainD(−A)^α. Furthermore, the subspaceD(−A)^αis dense inX, and the expression

kxkα=k(−A)^αxk, x∈D(−A)^α

defines a norm onD(−A)^α. Hereafter we denote byX_αthe Banach spaceD(−A)^α normed withk · k_α. Then for each 0< α≤1,X_αis a Banach space, andX_α,→X_β for 0< β≤α≤1 and the imbedding is compact whenever the resolvent operator ofAis compact. Also for every 0< α≤1 there existsC_α>0 such that

k(−A)^αS(t)k ≤ C_α

t^α, 0< t≤T. (2.1) Let P(X) denote the class of all nonempty subsets of X. Let Pbd,cl(X) and Pcp,cv(X) denote respectively the classes of all bounded-closed and compact-convex subsets ofX. Forx∈X andY, Z∈ Pbd,cl(X) we denote byD(x, Y) = inf{kx−yk: y∈Y}, andρ(Y, Z) = sup_a∈YD(a, Z).

Define the functionH :Pbd,cl(X)× Pbd,cl(X)→R⁺ by H(A, B) = max{ρ(A, B), ρ(B, A)}.

The functionH is called a Hausdorff metric onX. Note thatkYk=H(Y,{0}).

A correspondence G : X → P(X) is called a multi-valued mapping on X. A pointx0∈X is called a fixed point of the multi-valued operatorG:X → P(X) if x0∈G(x0). The fixed points set ofGwill be denoted by Fix(G).

Definition 2.1. Let G : X → Pbd,cl(X) be a multi-valued operator. Then G is called a multi-valued contraction if there exists a constantk∈(0,1) such that for eachx, y∈X we have

H(G(x), G(y))≤kkx−yk.

The constantkis called a contraction constant ofG.

A multi-valued mappingG:X → P(X) is calledlower semi-continuous(shortly l.s.c.) (resp. upper semi-continuous (shortly u.s.c.)) if B is any open subset of X then {x ∈ X : Gx∩B 6= ∅}(resp. {x ∈ X : Gx ⊂ B}) is an open subset of X. The multi-valued operatorGis calledcompact ifG(X) is a compact subset of X. Again G is called totally bounded if for any bounded subset S of X, G(S) is a totally bounded subset of X. A multi-valued operatorG:X → P(X) is called completely continuousif it is upper semi-continuous and totally bounded onX, for each boundedB∈ P(X). Every compact multi-valued operator is totally bounded

but the converse may not be true. However the two notions are equivalent on a bounded subset ofX.

We apply the following form of the fixed point theorem by Dhage [1] in the sequel.

Theorem 2.2. Let X be a Banach space, A : X → Pcl,cv,bd(X) and B : X → Pcp,cv(X)two multi-valued operators satisfying

(a) A is contraction with a contraction constantk, and (b) B is completely continuous.

Then either

(i) The operator inclusionλx∈Ax+Bxhas a solution forλ= 1, or (ii) The set E={u∈X :λu∈Au+Bu, λ >1} is unbounded.

3. Existence results

Let us state what we mean by a solution of problem (1.1)–(1.3). For this purpose, we consider the space P C([−r, T], X) consisting of functionsx: [−r, T]→X such that x(t) is continuous almost everywhere except for some tk at which x(t⁻_k) and x(t⁺_k),k= 1, . . . , m exist andx(t⁻_k) =x(tk).

Obviously, for any t ∈ [0, T] we have xt ∈ D and P C([−r, T], X) is a Banach space with the norm

kxk= sup{|x(t)|:t∈[−r, T]}.

In the following we set for convenience

Ω =P C([−r, T], X).

Also we denote by AC(J, X) the space of all absolutely continuous functions x : J →X.

A function x∈Ω∩AC((t_k, t_k+1), X), k= 1, . . . , m, is said to be a solution of (1.1)–(1.3) ifx(t)−f(t, x_t) is absolutely continuous onJ\ {t₁, . . . , t_m} and (1.1)–

(1.3) are satisfied.

A multi-valued map G : J → Pcp,cv(Rⁿ) is said to be measurable if for every y∈Rⁿ, the functiont→d(y, G(t)) = inf{ky−xk:x∈G(t)}is measurable.

A multi-valued mapG:J× D → Pcl(X) is said to beL¹-Carath´eodory if (i) t7→G(t, x) is measurable for eachx∈ D,

(ii) x7→G(t, x) is upper semi-continuous for almost allt∈J, and

(iii) for each real number ρ > 0, there exists a function h_ρ ∈ L¹(J,R⁺) such that

kG(t, u)k:= sup{kvk:v∈G(t, u)} ≤h_ρ(t), a.e. t∈J for allu∈ DwithkukD≤ρ.

Then we have the following lemmas due to Lasota and Opial [4].

Lemma 3.1. If dim(X)<∞ and F : J×X → P(X) isL¹-Carath´eodory, then S_G¹(x)6=∅ for each x∈X.

Lemma 3.2. Let X be a Banach space, Gan L¹-Carath´eodory multi-valued map withS¹_G6=∅where

S_G¹(x) :={v∈L¹(I,Rⁿ) :v(t)∈G(t, x_t)a.e. t∈J},

andK:L¹(J, X)→C(J, X)be a linear continuous mapping. Then the operator K ◦S_G¹ :C(J, X)→ Pcp,cv(C(J, X))

is a closed graph operator inC(J, X)×C(J, X).

We need also the following result from [2].

Lemma 3.3. Let v(·), w(·) : [0, T] → [0,∞) be continuous functions. If w(·) is nondecreasing and there are constantsθ >0,0< α <1 such that

v(t)≤w(t) +θ Z t

0

v(s)

(t−s)^1−αds, t∈[0, T], then

v(t)≤e^{θnΓ(α)ntnα/Γ(nα)}

n−1

X

J=0

θT^α α

^j w(t),

for every t ∈ [0, T] and every n ∈ N such that nα > 1, and Γ(·) is the Gamma function.

We consider the following set of assumptions in the sequel.

(H1) There exist constants 0 < β < 1, c₁, c₂, L_f such that f is X_β-valued, (−A)^βf is continuous, and

(i) k(−A)^βf(t, x)k ≤c1kxkD+c2, (t, x)∈J× D

(ii) k(−A)^βf(t, x₁)−(−A)^βf(t, x₂)k ≤ L_fkx1−x₂kD, (t, x_i) ∈ J × D, i= 1,2, with

L_f

k(−A)^−βk+C_1−βT^β β <1.

(H2) The multivalued map G(t, x) has compact and convex values for each (t, x)∈J× D.

(H3) The semigroupS(t) is compact fort >0, and there existsM ≥1 such that kS(t)k ≤M, for allt≥0.

(H4) GisL¹-Carath´eodory.

(H5) There exists a function q ∈ L¹(I,R) with q(t) > 0 for a.e. t ∈ J and a nondecreasing functionψ:R⁺ →(0,∞) such that

kG(t, x)k:= sup{kvk:v∈G(t, x)} ≤q(t)ψ(kxkD) a.e. t∈J, for allx∈ D.

(H6) The impulsive functionsI_k are continuous and there exist constantsc_ksuch thatkI_k(x)k ≤c_k,k= 1, . . . , mfor eachx∈X.

Theorem 3.4. Assume that (H1)–(H6) hold. Suppose that

bK2

Z T

0

q(s)ds <

Z ∞

K0

ds s+ψ(s), where

K₀= F

1−c1k(−A)^−βk, K₂= M

1−c1k(−A)^−βk, b=e^{K1n(Γ(β))nTnβ/Γ(nβ)}

n−1

X

j=0

K₁T^β β

^j ,

and

F =MkφkD{1 +c₁k(−A)^−βk}+c₂k(−A)^−βk{M+ 1}+M

m

X

k=1

c_k+C_1−βc₂T^β

β .

Then the initial-value problem (1.1)–(1.3) has at least one solution on[−r, T].

Proof. Transform the problem (1.1)–(1.3) into a fixed point problem. Consider the operatorN : Ω→ P(Ω) defined by

N x(t) =n

h∈Ω :h(t) =φ(t) fort∈[−r,0], andh(t) =S(t)[φ(0)−f(0, φ(0))]

+f(t, xt) + Z t

0

AS(t−s)f(s, xs)ds+ Z t

0

S(t−s)v(s)ds

+ X

0<t_k<t

S(t−tk)Ik(x(t⁻_k)) fort∈Jo ,

wherev∈S_G¹(x).

Now, we define two operators as follows. A: Ω→Ω by Ax(t) =

(0, ift∈[−r,0], −S(t)f(0, φ) +f(t, xt) +Rt

0AS(t−s)f(s, xs)ds , ift∈J,

(3.1) and the multi-valued operatorB : Ω→ P(Ω) by

Bx(t) =n

h∈Ω :h(t) =φ(t) fort∈[−r,0], andh(t) =S(t)φ(0) +

Z t

0

S(t−s)v(s)ds+ X

0<t_k<t

S(t−tk)Ik(x(t⁻_k)) fort∈Jo .

(3.2)

Then N = A+B. We shall show that the operators A and B satisfy all the conditions of Theorem 2.2 on [−r, T]. For better readability, we break the proof into a sequence of steps.

Step I. First we remark thatA for each x ∈Ω, has closed, convex values on Ω.

Next we show thatAhas bounded values for bounded sets inX. To show this, let S be a bounded subset of Ω, with boundρ. Then, for any x∈S one has

kAx(t)k ≤Mkf(0, φ)k+k(−A)^−βk[c₁kx_tk_D+c₂] +

Z t

0

k(−A)^1−βS(t−s)kk(−A)^βf(s, x_s)kds

≤Mkf(0, φ)k+k(−A)^−βk[c1kxtkD+c₂] +

Z t

0

C1−βc1

(t−s)^1−βkxskDds+C1−βc2T^β β

≤Mkf(0, φ)k+k(−A)^−βk[c1ρ+c₂] +C_1−βT^β

β [ρc₁+c₂], and consequently

kAxk ≤Mkf(0, φ)k+k(−A)^−βk[c1ρ+c2] +C1−βT^β

β [ρc1+c2].

HenceA is bounded on bounded subsets of Ω.

Step II. Next we prove that Bx is a convex subset of Ω for each x ∈ Ω. Let u1, u2∈Bx. Then there existsv1andv2in S_G¹(x) such that

u_j(t) =S(t)φ(0) + X

0<t_k<t

S(t−s_k)I_k(x(t⁻_k)) + Z t

0

S(t−s)v_j(s)ds, j= 1,2.

SinceG(t, x) has convex values, one has for 0≤µ≤1, [µv₁+ (1−µ)v₂](t)∈S_G¹(x)(t), ∀t∈J.

As a result we have [µu1+ (1−µ)u2](t)

=S(t)φ(0) + X

0<t_k<t

S(t−tk)Ik(x(t⁻_k)) + Z t

0

S(t−s)[µv1(s) + (1−µ)v2(s)]ds.

Therefore, [µu₁+ (1−µ)u₂] ∈Bx and consequentlyBx has convex values in Ω.

Thus we haveB : Ω→ P_cv(Ω).

Step III.We show thatAis a contraction on Ω. Letx, y∈X. By hypothesis (H1) kAx(t)−Ay(t)k ≤ kf(t, x_t)−f(t, y_t)k+

Z t

0

AS(t−s)[f(s, x_s)−f(s, y_s)]ds

≤ k(−A)^−βkLfkxt−ytk_D+ Z t

0

C_1−β

(t−s)^1−β ds Lfkxt−ytk_D

≤Lf

k(−A)^−βk+C1−βT^β

β kxt−ytkD. Taking supremum overt,

kAx−Ayk ≤L₀kx−ykD, L₀:=L_f

k(−A)^−βk+C_1−βT^β

β .

This shows thatA is a multi-valued contraction, sinceL0<1.

Step IV.Now we show that the multi-valued operatorB is completely continuous on Ω. First we show that B maps bounded sets into bounded sets in Ω. To see this, letQbe a bounded set in Ω. Then there exists a real numberρ >0 such that kxk ≤ρ,∀x∈Q.

Now for eachu∈Bx, there exists av∈S_G¹(x) such that u(t) =S(t)φ(0) + X

0<t_k<t

S(t−tk)Ik(x(t⁻_k)) + Z t

0

S(t−s)v(s)ds, t∈J.

Then for eacht∈J,

ku(t)k ≤Mkφ(0)k+M

m

X

k=1

ck+M Z t

0

|v(s)|ds

≤Mkφk_D+M

m

X

k=1

ck+M Z t

0

hρ(s)ds

≤Mkφk_D+M

m

X

k=1

ck+Mkhρk_L1.

This implies

kuk ≤Mkφk_D+M

m

X

k=1

ck+Mkhρk_L1

for allu∈Bx⊂B(Q) =S

x∈QB(x). HenceB(Q) is bounded.

Next we show that B maps bounded sets into equi-continuous sets. Let Qbe, as above, a bounded set andh∈Bxfor somex∈Q. Then there existsv∈S_G¹(x) such that

h(t) =S(t)φ(0) + X

0<tk<t

S(t−tk)Ik(x(t⁻_k)) + Z t

0

S(t−s)v(s)ds, t∈J.

Letτ1, τ2∈J\{t1, . . . , tm}, τ1< τ2. Then we have kh(τ2)−h(τ₁)k

≤ k[S(τ2)−S(τ1)]φ(0)k+ Z τ₁−

0

kS(τ2−s)−S(τ1−s)kϕq(s)ds +

Z τ1

τ₁−

kS(τ2−s)−S(τ₁−s)kϕq(s)ds+ Z τ2

τ₁

kS(τ2−s)kϕq(s)ds

+ X

0<tk<τ2−τ1

M ck+ X

0<tk<τ2

kS(τ2−tk)−S(τ1−tk)kck.

Asτ2→τ1andbecomes sufficiently small the right-hand side of the above inequal- ity tends to zero, sinceS(t) is a strongly continuous operator and the compactness ofS(t) fort >0 implies the continuity in the uniform operator topology.

This proves the equicontinuity for the case where t 6=ti, i = 1, . . . , m+ 1. It remains to examine the equicontinuity att=ti. Set

h1(t) =S(t)φ(0) + X

0<t_k<t

S(t−tk)Ik(y(t⁻_k)) and

h₂(t) = Z t

0

S(t−s)v(s)ds.

First we prove equicontinuity att=t⁻_i . Fixδ1 >0 such that{tk : k6=i} ∩[ti− δ1, ti+δ1] =∅,

h₁(t_i) =S(t_i)φ(0) + X

0<tk<ti

S(t−t_k)I_k(y(t⁻_k))

=S(ti)φ(0) +

i−1

X

k=1

T(ti−tk)Ik(y(t⁻_k)).

For 0< h < δ1 we have kh₁(t_i−h)−h₁(t_i)k

≤ k(S(ti−h)−S(t_i))φ(0) +

i−1

X

k=1

|[S(ti−h−t_k)−S(t_i−t_k)]I(y(t⁻_k))k.

The right-hand side tends to zero ash→0. Moreover kh₂(t_i−h)−h₂(t_i)k ≤

Z ti−h

0

k[S(t_i−h−s)−S(t_i−s)]v(s)kds+ Z ti

t_i−h

M φ_q(s)ds,

which tends to zero ash→0. Define

ˆh0(t) =h(t), t∈[0, t1] and

ˆh_i(t) =

(h(t), ift∈(ti, ti+1], h(t⁺_i ), ift=ti

Next we prove equicontinuity att =t⁺_i . Fixδ2>0 such that{tk :k 6=i} ∩[ti− δ2, ti+δ2] =∅. Then

ˆh(t_i) =S(t_i)φ(0) + Z ti

0

S(t_i−s)v(s) +

i

X

k=1

S(t_i−t_k)I_k(y(t_k)).

For 0< h < δ₂ we have kˆh(ti+h)−h(tˆ i)k

≤ k(S(ti+h)−S(ti))φ(0)k+ Z t_i

0

k[S(ti+h−s)−S(ti−s)]v(s)kds +

Z t_i+h

t_i

M ϕq(s)ds+

i

X

k=1

k[S(ti+h−tk)−S(ti−tk)]I(y(t⁻_k))k.

The right-hand side tends to zero ash→0.

The equicontinuity for the casesτ₁ < τ₂ ≤0 andτ₁ ≤0 ≤τ₂ follows from the uniform continuity of φon the interval [−r,0]. As a consequence of Steps 1 to 3, together with the Arzel´a-Ascoli theorem it suffices to show that B maps Qinto a precompact set inX.

Let 0< t≤bbe fixed and letbe a real number satisfying 0< < t. Forx∈Q we define

h(t)

=S(t)φ(0) +S() Z t−

0

S(t−s−)v1(s)ds+S() X

0<t_k<t−

S(t−tk−)Ik(y(t⁻_k)), where v₁ ∈ S_F¹. Since S(t) is a compact operator, the set H(t) = {h(t) : h ∈ N(y)} is precompact inX for every, 0< < t. Moreover, for everyh∈N(y) we have

|h(t)−h(t)| ≤ Z t

t−

kS(t−s)kϕq(s)ds+ X

t−<tk<t

kS(t−tk)kck.

Therefore, there are precompact sets arbitrarily close to the setH(t) ={h(t) :h∈ N(y)}. Hence the set H(t) ={h(t) :h∈B(Q)} is precompact in X. Hence, the operatorB : Ω→ P(Ω) is completely continuous.

Step V. Next we prove thatB has a closed graph. Let {xn} ⊂Ω be a sequence such thatxn→x_∗ and let{yn}be a sequence defined byyn∈Bxn for eachn∈N such thatyn →y_∗. We will show that y_∗ ∈Bx_∗. Sinceyn ∈Bxn, there exists a vn ∈S_G¹(xn) such that

yn(t) =φ(0) + X

0<t_k<t

S(t−tk)Ik(yn(t⁻_k)) + Z t

0

vn(s)ds.

Consider the linear and continuous operatorK:L¹(J,Rⁿ)→C(J,Rⁿ) defined by

Kv(t) = Z t

0

vn(s)ds.

Now

y_n(t)−φ(0)− X

0<t_k<t

S(t−t_k)I_k(y_n(t⁻_k))

−

y_∗(t)−φ(0)− X

0<tk<t

S(t−tk)Ik(y_∗(t⁻_k)) →0,

as n → ∞. From Lemma 3.2 it follows that (K ◦S_G¹) is a closed graph operator and from the definition ofKone has

yn(t)−φ(0)− X

0<tk<t

S(t−tk)Ik(yn(t⁻_k))∈(K ◦S_F¹(yn)).

Asx_n→x_∗andy_n→y_∗, there is av∈S_G¹(x_∗) such that

y_∗(t) =φ(0) + X

0<tk<t

S(t−t_k)I_k(y_∗(t⁻_k)) + Z t

0

v_∗(s)ds.

Hence the multi-valued operatorB is an upper semi-continuous operator on Ω.

Step VI.Finally we show that the set

E={u∈Ω :λu∈Au+Bufor someλ >1}

is bounded. Letu∈ E be any element. Then there existsv∈S_G¹(u) such that

u(t) =λ⁻¹S(t)[φ(0)−f(0, φ(0))] +λ⁻¹f(t, xt) +λ⁻¹

Z t

0

AS(t−s)f(s, xs)ds+λ⁻¹ Z t

0

S(t−s)v(s)ds +λ⁻¹ X

0<t_k<t

S(t−tk)Ik(x(t⁻_k)).

Then

ku(t)k ≤MkφkD+Mk(−A)^−βk[c1kφkD+c2] +k(−A)^−βk[c1kutkD+c2] +

Z t

0

k(−A)^1−βS(t−s)kk(−A)^βf(s, xs)kds +M

Z t

0

q(s)ψ(kusk_D)ds+M

m

X

k=1

ck

≤Mkφk_D+Mk(−A)^−βk[c1kφk_D+c2] +k(−A)^−βk[c1kutk_D+c2]

+ Z t

0

C_1−βc1

(t−s)^1−βkusk_Dds+C_1−βc2T^β β

+M Z t

0

q(s)ψ(kuskD)ds+M

m

X

k=1

c_k

≤F+c₁k(−A)^−βkku_tk_D

+ Z t

0

C_1−βc1

(t−s)^1−βku_sk_Dds+M Z t

0

q(s)ψ(ku_sk_D)ds, t∈J, where

F =MkφkD{1 +c₁k(−A)^−βk}+c₂k(−A)^−βk{M+ 1}+M

m

X

k=1

c_k+C_1−βc₂T^β

β .

Putw(t) = max{ku(s)k:−r≤s≤t}, t∈J. ThenkutkD ≤w(t) for allt∈J and there is a pointt^∗∈[−r, t] such that w(t) =ku(t^∗)k. Hence we have

w(t) =ku(t^∗)k

≤F+c₁k(−A)^−βkku_t^∗k_D+C_1−βc₁ Z t^∗

0

ku_sk_D (t−s)^1−βds +M

Z t^∗

0

q(s)ψ(kusk_D)ds

≤F+c1k(−A)^−βkw(t) +C1−βc1

Z t

0

w(s)

(t−s)^1−β ds+M Z t

0

q(s)ψ(w(s))ds,

or

w(t)≤ F

1−c1k(−A)^−βk

+ 1

1−c1k(−A)^−βk n

C1−βc1

Z t

0

w(s)

(t−s)^1−β ds+M Z t

0

q(s)ψ(w(s))dso

≤K0+K1

Z t

0

w(s)

(t−s)^1−βds+K2

Z t

0

q(s)ψ(w(s))ds, t∈I, where

K₀= F

1−c1k(−A)^−βk, K₁= C_1−βc₁

1−c1k(−A)^−βk and K₂= M

1−c1k(−A)^−βk.

From Lemma 3.3 we have

w(t)≤b K₀+K₂ Z t

0

q(s)ψ(w(s))ds ,

where

b=e^{K1n(Γ(β))nTnβ/Γ(nβ)}

n−1

X

j=0

K1T^β β

^j .

Let

m(t) =b

K₀+K₂ Z t

0

q(s)ψ(w(s))ds

, t∈J.

Then we havew(t)≤m(t) for allt∈J. Differentiating with respect tot, we obtain m⁰(t) =bK2q(t)ψ(w(t)), a.e. t∈J, m(0) =K0.

This impliesm⁰(t)≤bK2q(t)ψ(m(t)) a.e. t∈J; that is, m⁰(t)

ψ(m(t)) ≤bK2q(t), a.e. t∈J.

Integrating from 0 tot, we obtain Z t

0

m⁰(s)

ψ(m(s))ds≤bK2

Z t

0

q(s)ds.

By the change of variable, Z m(t)

K0

ds

ψ(s) ≤bK₂ Z T

0

q(t)ds <

Z ∞

K0

ds ψ(s).

Hence there exists a constantM such thatm(t)≤M for allt∈J, and therefore w(t)≤m(t)≤M for allt∈J.

Now from the definition ofwit follows that kuk= sup

t∈[−r,T]

ku(t)k=w(T)≤m(T)≤M,

for allu∈ E. This shows that the setEis bounded in Ω. As a result the conclusion (ii) of Theorem 2.2 does not hold. Hence the conclusion (i) holds and consequently the initial value problem (1.1)–(1.3) has a solution x on [−r, T]. This completes

the proof.

References

[1] B. C. Dhage; Multi-valued mappings and fixed points I,Nonlinear Func. Anal. & Appl.(to appear).

[2] E. Hernandez; Existence results for partial neutral functional integrodifferential equations with unbounded delay,J. Math. Anal. Appl.292(2004), 194–210.

[3] V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov; Theory of Impulsive Differential Equations, World Scientific Pub. Co., Singapore, 1989.

[4] A. Lasota and Z. Opial; An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys.13 (1965), 781-786.

[5] A. Pazy;Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Methematical Sciences, vol. 44, Springer Verlag, New York, 1983.

Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece E-mail address:[email protected]