INEQUALITIES OF SOLUTIONS OF VOLTERRA INTEGRAL AND DIFFERENTIAL EQUATIONS

In this paper, we study solutions of Volterra integral and differential equations , x ′ (t) = −a(t)x(t) + t t−h b(s)x(s)ds + f (t, x t), x ∈ R, or X(t) = a(t) + t t−α g(t, s)X(s)ds, X ∈ R n. With Lyapunov functionals, we obtain inequalities for the solutions of these equations. As a corollary, we also obtain a result on asymptotic stability which is simpler and better than some existing results.


Introduction
Before proceeding, we shall set forth notation and terminology that will be used throughout this paper.Let A = (a ij ) be an n × n matrix.A T denotes the transpose of A, A T = (a ji ), and |A| = Σ n i,j=1 a 2 ij .Let (C, || • ||) be the Banach space of continuous functions φ : [−h, 0]→ R n with the norm ||φ|| = max −h≤s≤0 |φ(s)| and | • | is any convenient norm in R n .In this paper, we will use the norm defined by |X| = Σ n i=1 x 2 i for X = (x 1 , x 2 , ..., x n ) T ∈ R n .Given H > 0, by C H we denote the subset of C for which ||φ|| < H. X ′ (t) denotes the right-hand derivative at t if it exists and is finite.Definitions of stability and boundedness can be found in [1].

T. Wang 2 Some Results on Inequalities of Solutions of Functional Differential Equations
There have been a lot of discussions on estimating solutions of differential equations.
For the system of ordinary differential equations where A is an n × n real matrix of continuous functions defined on R + = [0, ∞), solutions are estimated by Ważewski's inequality, which is stated as Theorem 1.1 below and its proof can be found in [3,12].
For the linear Volterra integro-differential system where h > 0 is a constant, A is an n × n real matrix of continuous functions defined on For the linear scalar functional differential equation where a, b : R + → R continuous, and h > 0 is a constant, we obtained the following three inequalities [9]. where where where For the general abstract functional differential system with finite delay we obtained the following results [10].
Theorem 2.7 Let V : R + × C XH → R + be continuous and D : R + × C XH → R + be continuous along the solutions of ( 4), and η, L, and P : R + → R + be integrable.Suppose the following conditions hold: Then the solutions of (4), u(t) = u(t, t 0 , φ), satisfy the following inequality: where Theorem 2.8 Let M and c be positive constants, and let u(t) = u(t, t 0 , φ) be a solution of (4).Let V : R + ×C XH → R + be continuous and D : R + ×C XH → R + be continuous along the solutions of ( 4), and assume the following conditions hold: Then there is a constant ε > 0 such that the solutions of (4) satisfy the following inequality: where K = V (t 0 , φ) + M(e ch − 1) Applying Theorem 2.8 to the following partial functional differential equation, with ω a real constant and f : R → R continuous, we obtained the following estimate on its solutions.

More Estimates on Volterra Integral and Differential Equations
In this part, we investigate more Volterra integral and differential equations.Our results are new and improve some former results.
Theorem 3.1 Suppose that the following conditions hold.
i) There exists a continuous function, P (t) : ii) There is a constant θ > 0 with 0 < θh < where η(t) In Theorem 2.7, take η(t) = (1 − θh)a(t), L(t) = θha(t).By Theorem 2.7, we obtain (10).Many authors have studied (8).Wang [11] gave results on uniform boundedness and ultimately uniform boundedness.Here we give an estimate for solutions with simpler conditions.For f (t, x t ) = 0, Burton, Casal and Somolinos [2] and Wang [5,6] studied asymptotic stability, uniform stability and uniformly asymptotic stability.In the following theorem, we obtain asymptotic stability with weaker and simpler conditions and an estimate for the solutions.Its proof is a direct corollary of Theorem 3.1.(8).Suppose that there is a constant θ > 0 with where Let us consider the following integral equation: Wang [8] obtained uniform boundedness and ultimate uniform boundedness.We will give an estimate of its solutions.Theorem 3.3 Let a(t) ∈ R n be continuous on R and g(t, s) be an n × n real matrix of continuous functions on where D r is the derivative from the right with respect to t.