Journal of Differential Equations DE3273 journal of differential equations 138, 188193 (1997) article no. DE973273
Existence Results for Initial Value Problems for Neutral Functional Differential Equations O. Arino Universite de Pau et des Pays de L'Adour, Laboratoire de Mathematiques Appliquees, Pau 64000, France
R. Benkhalti Pacific Lutheran University, Department of Mathematics, Tacoma, Washington 98447
and K. Ezzinbi Faculte des Sciences-Semlalia, Universite Cadi Ayyad, Departement de Mathe matiques, B.P.S. 15, Marrakech, Morocco Received December 9, 1996
1. INTRODUCTION Many works have been devoted to the study of initial value problems for neutral functional differential equations. See, for example Hale [2], Hale and Cruz [3], Ntouyas, Sficas and Tsamatos [5] and the references therein. In this paper, we report and correct an error in the article by Ntouyas, Sficas and Tsamatos [5]. In section 2, we will present some preliminary results. Next, we will provide a counterexample to show that theorem 3.1 in [5] fails. We will also provide the necessary conditions to establish an existence principle for the functional differential equation of neutral type: d [x(t)&g(t, x t )]= f (t, x t ), dt
t # [0, T ] (1)
x 0 =. # C([ &r, 0], R n )=C r , where C r is the Banach space of continuous functions from [&r, 0] into R n, f and g are continuous functions from [0, T ]_C r into R n, r is a fixed positive scalar, and [0, T ] is the interval of existence of a solution. To do this, we will use the same approach as in [5]. In section 4, we will present 188 0022-039697 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
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INITIAL VALUE PROBLEMS
some general comments on how to establish the existence of periodic solutions and solutions in the infinite delay case for equation (1). Finally, for most of the notation that will be used, we refer the reader to [5]. 2. PRELIMINARIES AND COMMENTS The initial value problem associated with the equation (1) is: Given . # C r , find a continuous function x: [&r, T ] R n, such that x 0 =., t x(t)&g(t, x t ) is continuously differentiable and satisfies equation (1) in [0, T ]. Note that x itself may not be differentiable on the interval of existence. If x is a solution of equation (1), then it satisfies the integral equation x(t)=.(0)&g(0, .)+g(t, x t )+
|
t
f (s, x s ) ds,
t # [0, T ].
(2)
0
Conversely, if x satisfies equation (2) and x 0 =., then x is a solution of equation (1) with initial value .. We recall that a function h: [0, T ]_C r R n is completely continuous iff h is continuous and the image of any bounded set in [0, T ]_C r is a bounded set in R n. Now let us recall the invalid result in ([1], theorem 3.1). Theorem 1. Let f, g: [0, T ]_C r R n be a completely continuous function and g is such that the operator G: C r C([0, T ], R n ) defined by (G.)(t)=g(t, .) is compact. Also assume that there exists a constant K such that, &x& [&r, T ] =sup &r t T |x(t)| K, for each solution x of d [x(t)&*g(t, x t )]=*f (t, x t ), dt
t # [0, T ] (3)
x 0 =., and any * # (0, 1). Then the initial value problem (1) has at least one solution on [&r, T ]. For the proof of this theorem, the authors use a result of Schaffer, see for example [1], concerning a homotopy in the Schauder theorem. The argument proposed in the proof is not correct. Indeed, the authors propose that the operator L: C([&r, T ], R n ) C([0, T ], R n ) defined by (Ly)(t)= g(t, y t +.~ t ), where .~(t)=
.(t),
{ .(0),
t # [&r, 0] t # [0, T ]
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ARINO, BENKHALTI, AND EZZINBI
is compact. Their assertion is not correct, as can be shown by the following counterexample. Example 2. Let g(t, .)=.( &1), thus (Ly)(t)=y(t&1)+.~(t&1). It is clear that the operator L is never compact because of the infinite dimension of C([0, T ], R n ). However, G is compact. Hence, the compactness of G is not sufficient to entail the theorem. What is needed for the proof is the following hypothesis. (H 1 ) For any bounded set 4 in C([&r, T ], R n ), the set [t g(t, x t ): x # 4] is equicontinuous in C([0, T ], R n ). Theorem 3. Suppose that f and g are completely continuous and the hypothesis (H 1 ) is satisfied. Moreover, if there exists a constant K such that, &x& [&r, T ] =sup &r t T |x(t)| K, for each solution x of (3) and any * # (0, 1), then the initial value problem (1) has at least one solution on [&r, T ]. Proof. If x is a solution of equation (1), then y(t)=x(t)&.~(t) is a solution of the following equation, y(t)= &g(0, .)+g(t, y t +.~ t )+
|
t
f (s, y s +.~ s ) ds,
t # [0, T ]
0
(4)
y 0 =0.
Let B 0 be the set defined by B 0 =[y # C([&r,T ], R n ), y 0 =0], and define the operator K by
(Ky)(t)=
{
t # [&r, 0]
0, &g(0, .)+g(t, y t +.~ t )+
|
t
f (s, y s +.~ s ) ds,
t # [0,T ].
0
To prove the above theorem, one needs to show the existence of fixed points for the operator K. For, let us recall Scha fer$s result. Lemma 4. Let B 0 be a convex subset of a normed linear space E and assume 0 # B 0 . Let F: B B be a completely continuous operator and let =(F)=[x # B 0 : x=*Fx for some 0
