INITIAL AND BOUNDARY VALUE PROBLEMS FOR PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper we study the existence of solutions to initial and boundary value problems of partial functional differential equations via a fixed-point analysis approach. Using the topological transversality theorem we derive conditions under which an initial or a boundary value problem has a solution.


Introduction
In this paper we study initial and boundary value problems for a class of semilinear, abstract functional differential equations.This class is characterized by the fact that the associated homogeneous, linear differential equation generates a strongly contin- uous linear evolution system of compact operators.
More precisely, we consider the nonlinear Volterra integral equation x(t)-W(t,)x(O)+ / W(t,s)f(S, xs)ds, t [O,b   (1.1) 0 together with an initial condition o (.2) or with a linear boundary condition Lx=a.
(1.3) Our notations follow that of Hale [5]  and Travis and Webb [12].X denotes a Banach space.C C([r, 0], X) is the Banach space of continuous X-valued functions with 158 S.K. NTOUYAS and P. CH.TSAMATOS supremum norm [[ [[, i.e,. I I [[ sup{ (0) [" r _ 0 _ 0}.If u is a continuous function from an interval [-r, b], b > 0 to X, then for t E [0, b], u is the element of C given by ut(O u(t + 0), 0 E r, 0].f" [0, b] x CX is a function.Also {W(t, s): b} is a strongly continuous family of evolution operators on X.Finally, L is a bounded linear operator from C([r, b],X) into X and a X is given.
If the evolution system W(t, s) is generated by a family of densely defined linear operators A(t), then equation (1.1) is an integral form of the abstract functional dif- ferential equation x'(t) A(t)x(t) + f(t, xt) 0 <_ t <_ b (1.4) Initial value problems for partial functional differential equations have been studied by Travis and Webb [12,13], Fitzgibbon [2] and aankin [11].
When r--0, i.e., in the case of ordinary differential equations, the boundary value problem (1.1)-(1.2) has been studied by Ward [15].Also, Rankin [10] consider- ed boundary value problems for partial functional differential equations, with A(t) A, independent of t, the infinitesimal generator of a strongly continuous semigroup T(t) and the boundary condition (1.3) in the case where n has the more specific form Lx Mx 0 + NXb, with M and N bounded linear operators from C into C.
In this paper we prove existence results for initial and boundary value problems by using the "Topological Transversality Method" of Granas [1].This method re- duces the problem of the existence of solutions of an initial or boundary value pro- blem to the establishment of suitable a priori bounds for solutions of these problems.
The advantage of this method for initial value problems (at least in the case of ordinary differential equations) is that this yields simultaneously the existence of solutions and the maximal interval of existence.
Many recent papers deal with this method and the interested reader is referred to [4, 7, 9, 14] and the references given therein.
As a model for the class of partial functional differential equations we study, one may take the equation wt(x t) Wxx(X, t) + f(t, w(x, t r)), 0 x b, t 0 t) (see [2,[11][12][13].) The paper is organized as follows: In Section 2, we present some preliminaries and facts about the evolution system W(t,s) and the linear operator L. In Section 3 we consider initial value problems and we extend the above-mentioned method of Topological Transversality of Granas to the partial functional differential equations.
We prove the basic existence theorem (Theorem 3.1) by assuming a priori bounds on solutions.It is well known (see e.g., [16]) that only the continuity of f is not suffi- cient to assure local existence of solutions, even when X is a Hilbert space.
Therefore, one has to restrict either the function f or the evolution system.In most of the previous works further restrictions on f are imposed.The function f is as- sumed to be locally Lipschitz or monotone or completely continuous.Fitzgibbon [2] assumed that the function f is continuous while the evolution operator compact.
Here we assume, following Ward [14], that the function f satisfies Caratheodory-type conditions, which does not imply that f is completely continuous, and that the evolu- tion operator is compact.In order to apply this basic existence theorem to obtain global existence of solutions, we establish the desired a priori bounds for solutions in Theorem 3.2.
In Section 4 we study boundary value problems.Finally, an application of a par- tial functional differential equation of Sobolev type, is given in Section 5.

Preliminaries
Throughout this paper we shall make the following assumptions on the evolution system W(t, s): (W1) W(t,s)E L(X), the space of bounded linear transformations on X, when- ever 0 _ s _ t _ b and for each x e X the mapping (t,s)W(t,s)x is con- tinuous.
The operar L in (3.1) is a continuous linear operator from C([ r, b], X) into X.We define L L(X) by (2.1) for all x G X. We will assume in the sequel that the operator L has a bounded inverse -1.
Before stating our basic existence theorems we need the following lemma which is referred to as the "Leray-Schauder alternative", [1, p. 61].Lemma 2.1: Let B be a convex subset of a normed linear space E and assume 0 B. Let F'B--,B be a completely continuous operator, i.e., it is continuous and the image of any bounded set is included in a compact set, and let g(F) {x B:x AFx for some O < A < l}.
Then either g(F) is unbounded or F has a fixed point.

Initial Value Problems
In this section we give our basic existence result for the initial value problem (IVP) (1.1)-(1.2).We assume, following Ward [15], that the function f satisfies the follow- ing Caratheodory-type conditions: (C1) For each t E [0, b], the function f(t,.):C-X is continuous, and for each x G C, the function f(., x): [0, b]X is strongly measurable.
(C2) For every positive integer k there exists gk G Ll([0, b]) such that for a.a.
Assume that there exists a constant K such that I I .ll u .(t)< K.
If x(t) y(t) + (t), t [-r, b] it is easy to see that y satisfies Yo--0 f toW(t s)f(s, Ys + s)ds,
It will now be shown that F is a completely continuous operator.
Let Bk-{yeC: IlYll -<k} for some k>_l.We first show that F maps B k N-sup{lW(t,s) l.0 s t b}. (3.3) We remark that N is finite, by the principle of uniform boundedness, since W(t,s) is strongly continuous on the compact triangle 0 < s < t < b.The right-hand side of (3.2) is independent of y E B k and tends to zero as t 2 tl---+0 and c sufficiently small, since the fact of compactness of W(t, s) for t-s > 0, implies the continuity of W(t, s) in the operator norm.
Thus F maps B k into an equicontinuous family of functions.It is easy to see that the family B k is uniform bounded.
Notice that we considered here only the case 0 < t a < t2, since the other cases t 1 < t 2 < 0 or t 1 < 0 < t 2 are very simple.
Since we have shown FB k is an equicontinuous collection, it suffices by the Arzela-Ascoli theorem to show that F maps B k into a precompact set in X.
Therefore, there are precompact sets arbitrary close to the set {(Fy)(t):y E Bk}.
It remains to show that F: 0 0 Cb--*Cb is continuous.Let {Un} 0 C_ C with un+u in C. Then there is an integer M such that u.(t) < M for all n and t G [0, b], so u n B M and u BM.~By (C1) f(t,(t)+s)--.f(t,u(t)+sfor each t G[0, b], and since If(t, Un(t + ts) f(t, u(t) + ts) < 2gM'(t),M' M + ]1 I] we have by dominated convergence f If(s, Un(S) + ts) f(s, u(s) + ) ds-0.0 Thus F is continuous.This completes the proof that F is completely continuous.
Finally, we shall prove that the set $(F)-{y E C'y $Fy,$ (0,1)} is bounded.Indeed, for every solution y in C of (3.1) the function x-y-4-is a solution of (1.1).x.Thus, by hypothesis we have IlYlIb< K+ I1" ConsEquently by Lemma 2.1, the operator F has a fixed point y* in C. Then x* y* -4is a solution of the initial value problem (1.1)-(1.2).The proof of the theorem is now complete. I"1 The applicability of Theorem 3.1 depends upon the existence of a priori bounds for the solutions of the IVP (1.1).x-(1.2),which are independent of $, $ e (0, 1).
In the next theorem we give conditions under which a solution of the IVP (1.1)-(1.2) exists on the interval[-r,b].
Proof: To prove the existence of a solution of the IVP (1.1)-(1.2) we apply Theorem 3.1.In order to apply this theorem we must establish the a priori bounds for the solutions of the IVP (1.1).x-(1.2),e (0, 1).Let x be a solution of the IVP (1.1),x-(1.2).Then we have (t) _< N I I I I + N / m(s)S2( I I xs I I )d, 0 _< t _< b. 0 We consider the function given by e(t) sup{ x(s) r < s _< t}, 0 _< t _< b.
Let t* e[-r, t] be such that /(t).-x(t*)I.If t* [0, t], by the previous inequality Denoting by u(t) the right-hand side of the above inequality we have u(0) N I I II, e(t) _< (t), o _< t _< b, and u'(t) Nm(t)ft(e(t)) < Nm(t)f2(u(t)), 0 <_ t <_ b.This inequality implies that there is a constant K such that u(t)< K, t [0, b] ,and hence e(t) < K, t [0, b].Since for every t [0, b], I I xt I I < e(t), we have where K depends only on b and the functions m and f2.Hence our proof is completer-1 4, Boundary Value Problems Consider in this section, the BVP (1.1)-(1.3).
Then the BVP (1.1)-( 1.3) has at least one solution on [-r,b].f toW(t s)f(s, xs)ds 0 < t < b. (4.3) Since the mapping x--W(., O)x is bounded and linear from X into C b the operator P is continuous and compact.The operator Q is completely continuous as has been proved in Theorem 3.1.Also, the set associated with the operator P is precompact because of the complete continuity of Q, Hence S-P + Q is a completely contin- uous operator.

An Application
As an application, we consider the Cauchy problem for the partial functional differen- tial equation of Sobolev type (Bx(t))' + Ax(t) f(t, xt) 0 < t < b (5.1) (t)-(t),-<_ t <_ 0 (5.2) where f satisfies (C1) (C2) and A and B are closed linear operators with domains contained in a Banach space X and ranges in a Banach space Y, D(B)C D(A), B is bijective and B-1.Y-,D(B) is compact.This problem has been studied by J. Light- bourne III and S. Rankin  where K is a constant, independent of , , and the result follows.
lemark 5.2: In the special case, as in [8], where m L and a(t) t + k we find K BI{MIBI I I I I / MLkw-1}exp[(MLlB-1 + w)b].