AN EXISTENCE THEOREM FOR NONLINEAR DELAY DIFFERENTIAL EQUATIONS *

In this paper we prove a theorem on the existence of solutions of 
nonlinear delay differential equations, with implicit derivatives. The 
result is established using the measure of noncompactness of a set and 
Darbo's fixed point theorem.


INTRODUCTION
The theory of linear and nonlinear delay differential equations has been studied by several researchers [ 1,4].Recently Dacka [5] proved an existence theorem for nonlinear delay differential equations with implicit derivatives using the measure of noncompactness of a set and Darbo's fixed point theorem.By the same method Balachandran and Somasundaram [2] established an existence theorem for nonlinear delay differential equations having implicit derivatives with delay depending on state variable.In [3] Banas proved a theorem about existence of solutions of some nonlinear Volterra integral equations with deviating argument without assuming the Lipschitz condition and using a technique similar to Dacka [5].In this paper we shall prove an existence theorem for nonlinear delay differential equations with delay depending on implicit derivatives.

MATHEMATICAL PRELIMINARIES
Let (X, II.II) be a Banach space and E be bounded set of X.In this paper the following definition of the measure of noncompactness of a set E is used [7].
kt(E) = inf {r > 0: E can be covered by a finite number of balls whose radii are smaller than r} The following version of Darbo's fixed point theorem being a generalization of Schauder's fixed point theorem shows the usefulness of the measure of noncompactness [6]."If S is a nonempty bounded closed convex subset of X and P" S ---> S is a continuous mapping, such that for any set E <2 S we have kt(PE where k is a constant with 0 < k < 1, then P has a fixed point." For the space of continuous functions Cn[t0,t1] with norm Ilxll = max{Ixi(t)I" i = 1,2,...,n, t [t o ,t 1 ] }, the measure of noncompactness of a set E is given by where o)(E,h) is the common modulus of continuity of the functions which belong to the set E, that is, where, as in the space of continuously differentiable functions C [t0,t1] with norm = II x IIc + II k IIc,, we have where DE={ 'x E}.
3. BASIC ASSUMPTIONS Consider the following nonlinear delay differential equation with implicit derivative of the form where x R n and f is an n-vector function.Let r(x(t),(t),t) >_ 0. Set c(t) = t-r(x(t),(t),t), and a =(x,:t,t)inf c(t) and -oo < a < t o Then x(t) (t) on [a,t0].
Definition: The solution of(l) is the function x(t) such that: i) x(t) is defined and continuous on the interval [a,t 1] and is of class C 1 on [t0,t1] such that at the point t o the right side derivative only is taken into account; ii) The function x(t) satisfies (1) on the interval [t0,tl], whereas on the interval [a,t 0] thefunction x(t) (t).
Next we shall prove that the solution of (1) exists in the sense of the above definition.

EXISTENCE THEOREM
Theorem: If the function f(x,y,t) satisfies the conditions (2) and (3), and if r(x(t), :t(t),t) >_ 0 and satisfies the condition (4), then (1) has at least one solution for any For any function x H, x(o(t)) will be the function defined in such a way that if cz(t) < t o for t e [to,tl ] then (6) x((t)) = O(x(t)).
Moreover, consider the bounded closed set B in H as (8) B = {x H" Ilxll _< L, IIkll _< M} where L and M are positive constants such that (9) L = I(to)l+(tl-to)M.
Since f is continuous, T is continuous and maps B into itself.Next let us estimate the modulus of continuity of the function DT(x)(t) for t,s [t0,tl].Since the only functions considered belong to some bounded subset of the space C [t0,tl], and since these all have uniformly bounded derivatives, it follows that they.areequicontinuous.
This follows from the fact that it is formed by the composition of a finite number of functions having a uniformly bounded modulus of continuity.The second and third terms on the fight of inequality (10) have the upper bound Nh+NI31 x(t) = T(x)(t).
Clearly the extension of this function to the interval [a,t 0] by means of the function is a solution of equation (1) having the following form: x(t) = O(to)+ j'f(x(s),x(s-r(x(s),c(s),s)),s)ds, t >_ t o x(t) = (t), a < t < t o Remark: It should be observed that if one assumes that the function f and r satisfy also the Lipschitz condition, then the uniqueness of the solution of (1) can be established by standard techniques used in proving the uniqueness theorem.