ON THE STABILITY OF DELAY INTEGRO-DIFFERENTIAL EQUATIONS

Some new stability results are given for a delay integro-differential equation. A basis theorem on the behavior of solutions of delay integro-differential equations is established. As a consequence of this theorem, a stability criterion is obtained.


INTRODUCTION AND PRELIMINARIES
This paper deals with the stability of the trivial solution for a delay integrodifferential equation.An estimate of the solutions is established and sufficient conditions for the stability and the asymptotic stability of the trivial solution are given.Since the first systematic study was carried out by Volterra [10], this type of equations have been investigated in various fields, such as mathematical biology and control theory (see, e.g., [4,5,8]).
In this paper, we will give a basic theorem on the behavior of solutions of scaler delay integro-differential equations.As a consequence of this theorem, we will establish a criterion for the stability of the trivial solution; see [3,6,7,9] for related stability results.A root of the associated characteristic equation is used in obtaining our results.For the basis theory of integral equations, we choose to refer to the books by Burton [1] and Corduneanu [2].
Note that linear neutral delay differential equations with periodic coefficients have been studied by Philos and Purnaras [9] and the behavior of solutions of linear differential equations with unbounded delay have been obtained by Kordonis and Philos [6].
Let us consider initial value problems for delay integro-differential equations with a delay It is known (see, for example, [1]) that, for any given initial function φ , there exists a unique solution x of the delay integro-differential equation (1), where φ is defined in (2), x will be called the solution of the initial problem (1)-( 2) or, more briefly, the solution of ( 1)-( 2).
If we look for a solution of (1) of the form Under this hypothesis, the characteristic equation ( 3) has a unique root 0 Indeed, let ) (λ F denote the characteristic function of (1), i.e., .Moreover, by the secod inequality of (4), we obtain for , and consequently, the function F is strictly increasing on the interval ) , ( ∞ γ .Furthermore, for every γ λ ≥ , we have i.e., 0 λ satisfies (5).Note that (5) implies in particular that Before closing this section, we will give two well-known definitions (see, for example, [1]).The trivial solution of ( 1) is said to be "stable" (at 0) if for every 0 > ε , there exists a number 0 Moreover, The trivial solution of ( 1) is called "asymptotically stable" (at 0) if it is stable in the above sense and in addition there exists a number 0 0 > δ such that, for any initial fuction φ with 0 δ φ < , the solution x of (1)-(2) satisfies

MAIN RESULTS
The main results of the paper are the following theorem and a corollary of this theorem.
Theorem.Assume that (4) holds and let 0 λ be the unique root of (3) in the interval Proof.Let now x be the solution of ( 1) Then it is easy to see that the fact that x satisfies (1) for all Moreover, the initial condition (2) can be equivalently written Furthermore, by using the fact that 0 λ is a root of (3) and taking into account ( 8) and (11), we can verify that ( 10) is equivalent to Then we can see that (12) reduces to the following equivalent equation: On the other hand, (11) takes the equivalent form Because of the definitions of y and z , ( 7) is equivalent to The proof will be accomplished by proving (15).Now, in view of ( 9) and ( 14), we have We will show that To this end, let us consider an arbitrary number 0 > ε .We claim that Otherwise, by ( 16), there exists a 0 * > t such that Then from (13), we obtain which, in view of ( 5), leads to a contradiction.So, our claim is true.Since (18) holds for every 0 > ε , it follows that (17) is always satisfied.By using (17), from (13), we derive for all 0  Proof.First of all, we observe that, because of ( 5) and ( 6), formula (20) defines a real number Θ with 1 > Θ .By our theorem, ( 7) is satisfied, where ) (φ L and ) (φ M are defined by ( 8) and (9), respectively.From (7) Thus, (19) gives 2)where a , b and c are real number, solution" of the delay integro-differential equation (1), we mean a continuous real-valued function x defined on the real line IR , which is continuously differentiable on the interval ) , 0 [ ∞ and satisffies (1) for all 0 ≥ t .
we see that λ is a root of the "characteristic equation"