# IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH CAUSAL OPERATORS.

1. Introduction and preliminaries

The study of functional equations with causal operators has recently been developed and some results on existence, stability and control are found in the monographs [7, 14, 23]. The term causal operators or Volterra abstract operator was introduced by Tonelli  (see also Tikhonov , ). The theory of these operators has the advantage of unifying some classes of differential equations as: ordinary differential equations, integrodifferential equations, differential equations with finite or infinite delay, Volterra integral equations, and neutral functional equations, and so on. Many papers in the literature address various aspects of the theory of causal operators. Control problems involving causal operators were studied in [4, 8, 18, 36]. A new class of abstract integral equations has been introduced in . We note that differential equations with causal operators were studied by several authors, see , , -, , - and the references therein. The properties of the solutions of the differential equations with causal operators were studied in [2, 21, 34, 35, 40]. The existence of solutions for impulsive differential equations with causal operators in finite dimensional spaces were studied in [19, 26].

Let E be a real separable Banach space endowed with the norm [parallel]*[parallel]. For x [member of] E and r > 0 let [B.sub.r] (x) := {y [member of] E; [parallel]y - x[parallel] < r} be the open ball centered at x with radius r, and let [B.sub.r] [x] be its closure. The space of all (classes of) strongly measurable functions u(*) : [0, b] [right arrow] E such that

[[parallel]u(*)[parallel].sub.p] := [([[integral].sup.b.sub.0] [[parallel]u(t)[parallel].sup.p]).sup.1/b] < [infinity]

for 1 [less than or equal to] p < [infinity] and [[parallel]u(*)[parallel].sub.[infinity]] := ess [sup.sub.t[member of][0b]] [parallel]u(t)[parallel] < [infinity], will be denoted by [L.sup.p] ([0, b], E). This is a Banach space with respect to the norm [[parallel]u(*)[parallel].sub.p]. We denote by PC ([0, b], E) the set of all functions u : [0, b] [right arrow] E such that u is continuous at t [not equal to] [t.sub.k], left continuous at t = [t.sub.k] and the right limit u([t.sup.+.sub.k]) exists for k = 1, 2, ..., m. Then PC([0, b], E) is a Banach space with respect to the norm [parallel]u(*)[parallel] = [sup.sub.0[less than or equal to]t[less than or equal to]b] [parallel]u (t)[parallel].

The following definition of causal operator was given by Tonelli .

An operator Q : PC([0, b], E) [right arrow] [L.sup.p.sub.loc] ([0, b], E) is a causal operator or a Volterra operator if, for each [tau] [member of] [0, b) and for all u(*), v(*) [member of] PC([0, b], E) with u(t) = v(t) for every t [member of] [0, [tau]], we have Qu(t) = Qv(t) for a.e. t [member of] [0, [tau]].

In this paper we study the following functional impulsive differential equation:

(1.1) [mathematical expression not reproducible]

where Q : PC([0, b], E) [right arrow] [L.sup.p] ([0, b], E), 1 [less than or equal to] p [less than or equal to] [infinity], is a continuous causal operator, [xi] [member of] E, m [member of] N, 0 = [t.sub.0] < [t.sub.1] < [t.sub.2] < ... < [t.sub.m] < [t.sub.m+1] = b and [I.sub.k] : E [right arrow] E is a continuous operator for each k = 1, 2, ..., m. Now we provide some examples of impulsive differential equations that can be included in impulsive differential equations with causal operators of the form (1.1). The impulsive differential equation

[mathematical expression not reproducible],

can be considered as a causal impulsive differential equations by identifying F(t, u(t)) with (Qu)(t). Another example is the general integro-differential equation

(1.2) [mathematical expression not reproducible].

Also, the differential equation with "maxima":

[mathematical expression not reproducible],

is another example of a causal impulsive differential equation. Finally, we remark that the Fredholm operator, given by

(Qu)(t) = [[integral].sup.a.sub.0] K(t, s, u(s)) ds,

is a causal operator if and only if K(t, s, u) [equivalent to] 0 for t < s < a.

We denote by [chi](A) the Hausdorff measure of non-compactness of a nonempty bounded set A [subset] E, and it is defined by (, ):

[chi] (A) = inf{[epsilon] > 0; A admits a finite cover by balls of radius [less than or equal to] [epsilon]}.

This is equivalent to the measure of non-compactness introduced by Kuratowski (see , ).

If dim(A) = sup{[parallel]x - y[parallel]; x, y [member of] A} is the diameter of the bounded set A, then we have that [chi](A) [less than or equal to] dim(A) and [chi](A) [less than or equal to] 2d if [sup.sub.x[member of]A] [parallel]x[parallel] [less than or equal to] d. We recall some properties of [chi] (see , ). If A, B are bounded subsets of E and [bar.A] denotes the closure of A, then

(1) [chi](A) = 0 if and only if [bar.A] is compact;

(2) [chi](A) = [chi]([bar.A]) = [chi]([bar.co](A));

(3) [chi]([lambda]A) = [absolute value of ([lambda])][chi](A) for every [lambda] [member of] R;

(4) [chi](A) [less than or equal to] [chi] (B) if A [subset] B;

(5) [chi] (A + B) = [chi] (A) + [chi] (B);

(6) If T : E [right arrow] E is a bounded linear operator, then [gamma] (TA) [less than or equal to] [parallel]T[parallel] [gamma](A).

If V [subset] PC([0, b], E) is equicontinuous, then

[mathematical expression not reproducible],

where V(t) := {u(t) : u(*) [member of] V}, is the Hausdorff measure of non- compactness in the space PC([0, b], E) (see ).

We recall the following lemma due to Kisielewicz [22, Lemma 2.2].

Lemma 1.1. Let {[u.sub.n] (*); n [greater than or equal to] 1} be a subset in [L.sup.1] ([0, b], E) for which there exists m(*) [member of] [L.sup.1] ([0, b], [R.sup.+]) such that [parallel][u.sub.n](t)[parallel] [less than or equal to] m(t) for each n [greater than or equal to] 1 and for a.e. t [member of] [0, b]. Then the function t [??] [chi](t) := [chi]({[u.sub.n] (t); n [greater than or equal to] 1}) is integrable on [0, b] and, for each t [member of] [0, b], we have

[chi] ({[[integral].sup.t.sub.0] [u.sub.n] (t) dt; n [greater than or equal to] 1}) [less than or equal to] [[integral].sup.t.sub.0] [chi] (t) dt.

2. An existence result

Consider the following functional impulsive differential equation:

(2.1) [mathematical expression not reproducible],

where Q : PC([0, b], E) [right arrow] [L.sup.p] ([0, b], E), 1 [less than or equal to] p [less than or equal to] [infinity], is a continuous causal operator, [xi] [member of] E, m [member of] N, 0 = [t.sub.0] < [t.sub.1] < [t.sub.2] < ... < [t.sub.m] < [t.sub.m+1] = b, [I.sub.k] : E [right arrow] E is continuous for each k = 1, 2, ..., m.

We consider the following assumptions:

([H.sub.1]) Q : PC([0, b], E) [right arrow] [L.sup.p] ([0, b], E), 1 [less than or equal to] p [less than or equal to] [infinity], is a continuous causal operator, and [I.sub.k] : E [right arrow] E is continuous for each k = 1, 2, ..., m.

([H.sub.2]) For each r > 0 there exist [psi], [eta] [member of] [L.sup.p] ([0, b], [R.sub.+]) such that, for each u(*) [member of] PC([0, b], E) with [mathematical expression not reproducible], we have

[mathematical expression not reproducible].

([H.sub.3]) For each bounded subsets A [subset] PC([0, b], E) and B [subset] E there exist constants [[gamma].sub.A], [[delta].sup.k.sub.B] > 0 (k = 1, 2, ..., m) such that

(2.2) [chi] ((QA) (t)) [less than or equal to] [[gamma].sub.A] [chi] (A (t)),

and

[chi] ([I.sub.k] (B)) [less than or equal to] [[delta].sup.k.sub.B] [chi] (B), k = 1, 2, ..., m,

for all t [member of] [0, b], where (QA)(t) := {(Qu)(t) : u(*) [member of] A}.

By solution of (2.1) we mean a function u(*) : [0, b] [right arrow] E such that u(0) = [xi], u(*) is continuous on ([t.sub.k], [t.sub.k+1]) for k =1, 2, ..., m, u'(t) = (Qu)(t) for a.e. t [member of] [0, b] \ {[t.sub.1], [t.sub.2], ..., [t.sub.m]}, and u([t.sup.+.sub.k]) = u([t.sup.- .sub.k]) + [I.sub.k] (u([t.sup.-.sub.k])), k = 1, 2, ..., m.

It is easy to show that (see ) a function u(*) [member of] PC([0, b], E) is a solution for (2.1) on [0, b], if and only if

(2.3) u(t) = u(0) + [[integral].sup.t.sub.0] (Qu)(s)ds + [summation over (0<[t.sub.k]<t)] [I.sub.k] (u([t.sup.- .sub.k])) for t [member of] [0, b].

For a fixed [xi] [member of] E, by [S.sub.T] ([xi]) we denote the set of solutions u(*) of Cauchy problem (2.1) on an interval [0, T] with T [member of] (0, b].

Theorem 2.1. Let Q : PC([0, b], E) [right arrow] [L.sup.p] ([0, b], E) be a causal operator such that conditions ([H.sub.1])-([H.sub.3]) hold. Then, for every [xi] [member of] E, there exists T [member of] (0, 6] such that the set [S.sub.T] ([xi]) is nonempty and compact set in PC([0, T], E).

Proof. First we shall show that there exists T [member of] (0, b] such that the set [S.sub.T] ([xi]) is nonempty. Let [delta] > 0 be any number and let r := [parallel][xi][parallel] + [delta]. We choose T [member of] (0, b] such that

[[integral].sup.T.sub.0] [psi] (s) ds [less than or equal to] [delta]/4 and [[integral].sup.T.sub.0] [eta] (s) ds [less than or equal to] [delta]/4

and we consider the set B defined as follows

B = {u [member of] PC([0, T], E); [parallel]u(t) - [xi][parallel] [less than or equal to] [delta]}.

Further on, we consider the integral operator [LAMBDA] : B [right arrow] PC([0, T], E) given by

([LAMBDA]u)(t) = [xi] + [[integral].sup.t.sub.0] (Cu)(s)ds + [summation over (0<[t.sub.k]<1)] [I.sub.k] (u([t.sup.-.sub.k])), for t [member of] [0, T]

and we prove that this is a continuous operator from B into B. First, we observe that if u(*) G B, then [sup.sub.0[less than or equal to]t[less than or equal to]b] [parallel]u(t)[parallel] < r, and so [parallel](Cu)(t)[parallel] [less than or equal to] [psi](t) for a.e. t [member of] [0, T]. Hence, for each u(*) [member of] B, we have

[mathematical expression not reproducible]

and thus, [LAMBDA](B) [subset] B. Further on, let [u.sub.n] (*) [right arrow] u(*) in B. Then we have

[mathematical expression not reproducible]

if 1 [less than or equal to] p < [infinity] and 1/p +1/q = 1, and

[mathematical expression not reproducible]

if p = [infinity]. Using Lemma 1.15 from , by ([H.sub.1]) and ([H.sub.2]) it follows that

[mathematical expression not reproducible],

so that [LAMBDA] : B [right arrow] B is a continuous operator. Moreover, it follows that [LAMBDA](B) is bounded. Further on, we show that [LAMBDA](B) is equicontinuous on [0, T]. Let [epsilon] > 0. On the closed set [0, T], the functions t [??] [[integral].sup.t.sub.0] [psi] (s)ds and t [??] [[integral].sup.t.sub.0] [eta](s)ds, are uniformly continuous, and so there exist [eta] > 0 such that

[absolute value of ([[integral].sup.t.sub.s][psi]([tau])d[tau])] [less than or equal to] [epsilon]/4 and [absolute value of ([[integral].sup.t.sub.s][eta]([tau])d[tau])] [less than or equal to] [epsilon]/4 for every t, s [member of] [0, T] with [absolute value of (t - s)] < [eta]. Let t, s [member of] [0, T] are such that [absolute value of (t - s)] [less than or equal to] [eta]. If we suppose that 0 [less than or equal to] s [less than or equal to] t [less than or equal to] T then, for each u(*) [member of] B, we have

[mathematical expression not reproducible].

Therefore, we conclude that [LAMBDA](B) is uniformly equicontinuous on [0, T]. Next, we construct a sequence [{[u.sub.n](*)}.sub.n[greater than or equal to]1] of continuous functions [u.sub.n] (*) : [0, T] [right arrow] E as follows. Let n [member of] N. For i =1, 2, ..., n, we define [u.sup.1.sub.n] (t) = [xi], t [member of] [0, T] and

[mathematical expression not reproducible],

for i > 1. It is easy to see that if i [member of] {1, 2, ..., n - 1} and [parallel][u.sup.i.sub.n](t)[parallel] [less than or equal to] r for t [member of] [0, iT/n], then [parallel][u.sup.i+1.sub.n] (t)[parallel] [less than or equal to] r for t [member of] [0, iT/n] and, by ([H.sub.2]), [parallel] (C[u.sup.i.sub.n])(t) [parallel] [less than or equal to] [psi](t) for a.e. t [member of] [0, iT/n] and

[summation over (0<[t.sub.k]<t-T/n)] [parallel][I.sub.k] ([u.sup.i-1.sub.n] ([t.sup.-.sub.k]))[parallel] [less than or equal to] [[integral].sup.t-T/n.sub.0] [eta] (s) ds

for t [member of] [0, iT/n]. It follows that

[mathematical expression not reproducible],

for all t [member of] [0, (i + 1)T/n]. Since [parallel][u.sup.1.sub.n](t)[parallel] [less than or equal to] r for t [member of] [0, T/n], then by induction on k we have that [parallel][u.sup.i.sub.n](t)[parallel] [less than or equal to] r for all k = 1, 2, ..., n, t [member of] [0, iT/n]. In the following, to simplify the notation, we put [u.sub.n] (*) = [u.sup.n.sub.n] (*), n [member of] N. Since [u.sup.n.sub.n] (s) = [u.sup.n-1.sub.n] (s) for all s [member of] [0, (n - 1)T/n] and C is a causal operator, then

(C[u.sup.n.sub.n]) (s) = (C[u.sup.n-1.sub.n]) (s) for all s [member of] [0, (n - 1) T/n].

Moreover, we have that [u.sup.n.sub.n] ([t.sup.-.sub.k]) = [u.sup.n- 1.sub.n]([t.sup.-.sub.k]) and so [I.sub.k] ([u.sup.n.sub.n] ([t.sup.-.sub.k])) = [I.sub.k] ([u.sup.n-1.sub.n] ([t.sup.- .sub.k])) for 0 < [t.sub.k] < t - T/n with t [member of] [T/n, (n - 1)T/n]. Next, if t [member of] [(n - 1)T/n, T], then t - T/n [less than or equal to] (n - 1)T/n and consequently

[[integral].sup.t-T/n.sub.0] (C[u.sup.n.sub.n]) (s) ds = [[integral].sup.t- T/n.sub.0] (C[u.sup.n-1.sub.n]) (s) ds

for t [member of] [(n - 1)T/n, T]. It follows that the sequence [{[u.sub.n] (*)}.sub.n[greater than or equal to]1] can be written as

[mathematical expression not reproducible],

for every n [member of] N. Moreover, it is easy to see that [u.sub.n] (*) [member of] PC([0, T], E) for all n [greater than or equal to] 1. Further, if 0 [less than or equal to] t [less than or equal to] T/n, then we have

[mathematical expression not reproducible].

If T/n [less than or equal to] t [less than or equal to] T, then we have

[mathematical expression not reproducible].

Therefore, it follows that

(2.4) [mathematical expression not reproducible].

Let A = {[u.sub.n] (*); n [greater than or equal to] 1}. Denote by I the identity mapping on B. From (2.4) it follows that (I - [LAMBDA])(A) is a equicontinuous subset of B. Since A [subset] (I - [LAMBDA])(A) + [LAMBDA](A) and the set [LAMBDA](A) is equicontinuous, then we infer that the set A is also equicontinuous on [0, T]. Set A(t) = {[u.sub.n] (t); n [greater than or equal to] 1} for t [member of] [0, T]. Then, by Lemma 1.1 and the properties of the measure of non-compactness we have

[mathematical expression not reproducible].

Note that, given [epsilon] > 0, we can find n([epsilon]) > 0 such that [[integral].sup.t.sub.t-T/n] [psi] (s) ds < [epsilon]/2 for t [member of] [0, T] and n [greater than or equal to] n([epsilon]). Hence we have that

[mathematical expression not reproducible].

Using the last inequality, we obtain that

[chi] (A (t)) [less than or equal to] [chi] ([[integral].sup.t.sub.0] (CA) (s) ds) + [summation over (0<[t.sub.k]<t- T/n)] [chi] (([I.sub.k]A) ([t.sup.-.sub.k])).

Since for every t [member of] [0, T], A(t) is bounded then, by Lemma 1.1, ([H.sub.3]) and the and properties of the measure of non-compactness we have that

[mathematical expression not reproducible]

for every t [member of] [0, T]. Therefore, if we put m(t) := [chi](A(t)), t [member of] [0, T], then we infer that

m(t) [less than or equal to] [[integral].sup.t.sub.0] [[gamma].sub.r] m (s) ds + [m.summation over (k=1)] [[delta].sup.k.sub.r] m ([t.sup.-.sub.k]),

for every t [member of] [0, T]. Then, by Gronwall's lemma for impulsive integral inequalities (see [13, Theorem 1.5.1]), we must have that m(t) = [chi](A(t)) = 0 for every t [member of] [0, T]. Moreover, since (see ) [[chi].sub.PC] (A) = [sup.sub.0[less than or equal to]t[less than or equal to]T] [chi](A(t)) we deduce that [[chi].sub.PC] (A) = 0. Therefore, A is relatively compact subset of PC([0, T], E). Then, by Arzela- Ascoli theorem (see [13, Theorem 1.1.5]), and extracting a subsequence if necessary, we may assume that the sequence [{[u.sub.n] (*)}.sub.m[greater than or equal to]1] converges on [0, T] to a function u(*) [member of] B. Therefore, since

[mathematical expression not reproducible]

then, by (2.4) and by the fact that [LAMBDA] is a continuous operator, we obtain that sup)0 [less than or equal to] t [less than or equal to] T[parallel]([LAMBDA]u)(t) - u(t)[parallel] = 0. It follows that

u(t) = ([LAMBDA]u)(t) = [xi] + [[integral].sup.t.sub.0] (Cu)(s))ds + [summation over (0<[t.sub.k]<t)] [I.sub.k] (u([t.sup.-.sub.k]))

for every t [member of] [0, T], that is u(*) = [LAMBDA]u(*). Hence

u(t) = [xi] + [[integral].sup.t.sub.0] (Cu)(s)ds + [summation over (0<[t.sub.k]<t)] [I.sub.k] (u([t.sup.-.sub.k])), for t [member of] [0, T]

solve the Cauchy problem (2.1), that is, u(*) [member of] [S.sub.T] ([xi]) and so [S.sub.T] ([xi]) is a nonempty set. Since [LAMBDA] is continuous, then [S.sub.T] ([xi]) is a closed subset in PC([0, T], E). Moreover, since [S.sub.T] ([xi]) = [LAMBDA] ([S.sub.T] ([xi])) it follows that [chi] (([S.sub.T] ([xi])) (t)) = [chi] ([LAMBDA] ([S.sub.T] ([xi])) (t)) for every t [member of] [0, T], where ([S.sub.T] ([xi])) (t) := {u(t); u [member of] [S.sub.T] ([xi])}. Therefore, following the same argument as above, we obtain that relatively compact subset of PC([0, T], E). Since [S.sub.T] ([xi]) is a closed subset in PC ([0, T], E) it follows that [S.sub.T] ([xi]) is a compact subset in PC([0, T\, E).

Remark 2.2. The conclusion of Theorem 2.1 is also true if we replace the condition (2.2) with the condition:

([H'.sub.3]) For each bounded subsets A [subset] PC([0, b], E) there exists [[gamma].sub.A] > 0 such that

(2.5) [mathematical expression not reproducible].

3. An optimal control problem

In the following, we shall establish necessary conditions for the existence of an optimal solution for the control problem:

(3.1) [mathematical expression not reproducible],

where g(*) : E [right arrow] R is a given function. For this aim, it will need to establish some preliminary results. For a fixed [xi] [member of] E we denote by [A.sub.T] ([xi]) the attainable set of Cauchy problem (2.1); that is, [A.sub.T] ([xi]) = {u(T); u(*) [member of] [S.sub.T] ([xi])}.

Lemma 3.1. Assume that Q : PC([0, b], E) [right arrow] [L.sup.p] ([0, b], E) is a causal operator such that the condition ([H.sub.1])-([H.sub.3]) hold. Then the multifunction [S.sub.T] : E [right arrow] PC([0, T], E) is upper semicontinuous.

Proof. Let K be a closed set in PC ([0, T ], E) and G = ([xi] [member of] E; [S.sub.T] ([xi]) [intersection] K [not equal to] 0}. We must show that G is closed in E. For this, let [{[[xi].sub.n]}.sub.n[greater than or equal to]1] be a sequence in G such that [[xi].sub.n] [right arrow] [xi]. Further on, for each n [greater than or equal to] 1, let un(*) [member of] [S.sub.T] ([[xi].sub.n]) [intersection] K. Then

[u.sub.n] (t) = [[xi].sub.n] + [[integral].sup.t.sub.0] (Q[u.sub.n]) (s)ds + [summation over (0<[t.sub.k]<t)] [I.sub.k] ([u.sub.n] ([t.sup.-.sub.k]))

for every t [member of] (0, T]. As in proof of Theorem 1.1 we can show that [{[u.sub.n] (*)}.sub.n[greater than or equal to]1] converges uniformly on [0, T] to a continuous function u(*) [member of] K. Since

[mathematical expression not reproducible]

for every t [member of] [0, T], we deduce that u(*) [member of] [S.sub.T] ([xi]) [intersection] K. This prove that G is closed and so [xi] [??] [S.sub.T] ([xi]) is upper semi continuous.

Corollary 3.2. Assume that Q : PC([0, b], E) [right arrow] [L.sup.p] ([0, b], E) is a causal operator such that conditions ([H.sub.1])-([H.sub.3]) hold. Then, for any [xi] [member of] E and any t [member of] [0, T] the attainable set [A.sub.t] ([xi]) is compact in C([0, t], E) and the multifunction (t, [xi]) [right arrow] [A.sub.t] ([xi]) is jointly upper semicontinuous.

Theorem 3.3. Let [K.sub.0] be a compact set in E and let g(*) : E [right arrow] R be a lower semicontinuous function. If Q : PC([0, b], E) [right arrow] [L.sup.p] ([0, b], E) is a causal operator such that the condition ([H.sub.1])-([H.sub.3]) hold, then the control problem (3.1) has an optimal solution; that is, there exists [[xi].sub.0] [member of] [K.sub.0] and [u.sub.0] (*) [member of] [S.sub.T] ([[xi].sub.0]) such that g([u.sub.0] (T)) = inf (g(u(T)); u(*) [member of] [S.sub.T] ([[xi].sub.0]), [[xi].sub.0] [member of] [K.sub.0]}.

Proof. From Corollary 3.2 we deduce that the attainable set [A.sub.T] ([xi]) is upper semicontinuous. Then the set [mathematical expression not reproducible] is compact in E and so, since g(/) is lower semicontinuous, there exists [[xi].sub.0] [member of] [K.sub.0] such that g([u.sub.0] (T)) = inf {g(u(T)); u(*) [member of] [S.sub.T] ([[xi].sub.0]), [[xi].sub.0] [member of] [K.sub.0]}.

4. An example

Consider the following impulsive differential equation:

(4.1) [mathematical expression not reproducible],

where g (*), [lambda] (*) [member of] [L.sup.p] ([0, b], E), p [greater than or equal to] 1, and K : [0, b] x [0, b] [right arrow] L(E) is strongly continuous. Let M := [sup.sub.s,t[member of][0,b]] [parallel]K(t, s)[parallel]. Assume that

(f1) f : [0, b] x E [right arrow] E is a Caratheodory function; that is, t [??] f (t, u) is strongly measurable for all u [member of] E, u [??] f (t, u) is continuous for a.e. t [member of] [0, b], and there c (*) [member of] [L.sup.P] ([0, b], [R.sub.+])

[parallel]f (t, u)[parallel] [less than or equal to] c (t), t [member of] [0, b], u [member of] E,

(f2) For each bounded set A [subset] E there exist [l.sub.A] > 0 such that

[chi](f (s, A)) [less than or equal to] [l.sub.A] [chi] (A) for every t [member of] [0, b].

If we put

(Cu)(t) := g(t) + [[integral].sup.t.sub.0] K(t, s)f(s, u(s))ds, t [member of] [0, b],

and

[mathematical expression not reproducible],

for all u [member of] CP([0, b], E), then equations (4.1) can be written in abstract form (2.1). It is easy to see that C : PC([0, b], E) [right arrow] [L.sup.p] ([0, b], E), 1 [less than or equal to] p [less than or equal to] [infinity], is a continuous causal operator, and [I.sub.k] : E [right arrow] E is continuous for each k = 1, 2, ..., m. Next, by (f1) we have that

[mathematical expression not reproducible],

so that [psi] (*) := [parallel]g (*)[parallel] + M[b.sup.1/p'] [[parallel]c[parallel].sub.p] [member of] [L.sup.p] ([0, b], [R.sub.+]) and

[parallel](Cu)(t)[parallel] [less than or equal to] [psi] (t) for a.e. t [member of] [0, b].

Now, let s, t [member of] [0, b] be such that s < t and let {[t.sub.v], [t.sub.v+1], ..., [t.sub.r]} [subset] {[t.sub.1], [t.sub.2], ..., [t.sub.m]} be such that s < [t.sub.v] < [t.sub.v+1] < ... < [t.sub.r] < t. Then,

[mathematical expression not reproducible],

that is,

[r.summation over (k=v)] [parallel][I.sub.k](u([t.sup.-.sub.k]))[parallel] [less than or equal to] [[integral].sup.t.sub.s] [eta] ([tau]) d[tau],

where n (*) := [[parallel]u[parallel].sub.CP] [lambda](*) [member of] [L.sup.p] ([0, b], [R.sub.+]). Therefore, ([H.sub.2]) is verified as true. Next, if A [subset] PC([0, b], E) is a bounded set, then using the properties of noncompactness measure, Mean Value Theorem (see ) and (f2), we have

[mathematical expression not reproducible],

that is,

[mathematical expression not reproducible].

Also, it is easy to see that

[chi] ([I.sub.k] (B)) [less than or equal to] [[delta].sup.k.sub.B] [chi] (B), k = 1, 2, ..., m,

for each bounded set B [subset] E, where [mathematical expression not reproducible]. Consequently, all the hypothesis of Theorem 2.1 are satisfied (see also Remark 2.2), so that (4.1) has a solution on [0, b].

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TAHIRA JABEEN, RAVI P. AGARWAL, DONAL O'REGAN, AND VASILE LUPULESCU

Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan Department of Mathematics, Texas A&M University-Kingville, Kingsville, TX, USA

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland Constantin Brancusi University, Republicii 1, 210152 Targu-Jiu, Romania