EQUATIONS OF VOLTERRA TYPE AND EXTENSION OF LYAPUNOV ’ S METHOD

In this paper, the stability properties of nonlinear difference equations of Volterra type is discussed. For this purpose some comparison Theorems are developed, then using these results the stability of the nonlinear difference equations of Volterra type is investigated.

Stability results of (1.1) are discussed recently by linearization method in [3].In this paper, we are interested in extending the Lyapunov's method to discuss stability properties of the system (1.1).
When we employ a Lyapunov function, we are faced with two questions.One is to estimate variation of the Lyapunov function relative to the system (1.1) in terms of a function, in which case a basic question is to select a minimal class of functions for which this can be done.Thus by using the theory of difference inequalities and choosing minimal classes of functions suitably, it is possible to establish stability properties of difference equations of Volterra type (1.1) by reducing the study of (1.1) to a simple difference equation.The second approach is to estimate the vairation of Lyapunov function by means of a functional so that the study of (1.1) is reduced to the study of a rlatively simple differenece equation of Volterra type.This approach makes the choice of minimal classes unnecessary but it requires developing the theory of difference inequalities of Volterra type.Therefore, finding the stability properties of even simple diffenence equation of Volterra type is comparatively more difficult than simple difference equations.Both methods offer a unified approach.
Extension of Lyapunov method for integro-differential equations is discussed in [1].For stability results using Lyapunov method for difference equations see [2]. 2.
Comparison results.
It is well known that comparison principle is one of the most efficient methods for studying the qualitative behavior of solutions of nonlinear systems.Let us begin by proving the following comparison results relative to the scalar difference equation of Volterra type./ Theorem 2. 1. Assume that g N n o nondecreasing in u, v for fixed n E + N n o X + X + , and g(n, u, v) is Suppose further that + U'Nn0 + and n > no, u(n0)=u 0 _> 0, Then u(n) _< r(n), n >_ no, where r(n) = r(n, n o u0) is the solution of the scalar difference equation of Volterra type.n-1 Proof.
Hence, using the monotone character of g, we get

S-I10
This contradiction proves the theorem.
The next comparison result is more general but requires the functional to be nonnegative.
Theorem 2.2.Assume that g.N + + no X + X + +, H'Nn0 and g(n, u, v), H(n, s, u) are nondecreasing in u, u(no)-U 0 >_0.n >_ no, where r(n) r(n, no, u0) is the solution of the scalar n-1 where Ais the antidifference operator and W(n) is an arbitrary function of period 1.
If Z(n) = &'lP(n) + W(n), then we see that XZ(n)= P(n) > 0 since g _> 0. Hence Z(n) is nondecreasing and therefore we have u(n) <_ Z(n), for every n >_ n o Consequently using the monotone character of g and H, we get Hence by comparison theorem for difference equation, we have Z(n) <r(n), n > n o where r(n) is the solution of (2.2).Since u(n) <_ Z(n), the proof is complete.
Next we shall discuss a comparison result in terms of Lyapunov function.For this purpose, we need to define the variation of a Lyapunov function.
If V. N + X Rd N+, n o system (1.1) by then we define the variation of V relative to the AV(n, x(n)) V(n+l, x(n+l)) V(n, x(n)), Also let us define the minimal set f2 given by [x(n).Nn 0+ d.V(s, x(s)) _< V(n, x(n)), n o _< s _< n].
Suppose the assertion is false.Then there exists a k > n o such thar V(k, x(k)) _< u(k), and V(k+l, x(k/l)) > u(k/l).
This leads to the contradiction and hence the proof is complete.
Another comparison theorem which is sometimes useful, is the following.
.N + Theorem 2.4.Assume that g no X + +, g(n, u) is nondecreasing in u where u(n) is the solution of (2.4).

Stability Results.
Having the necessary comparison results, it is now easy to investigate stability properties of solutions of the system (1.1).For this purpose, we slaall assume that f(n, 0, 0) = 0, and G(n, s, 0) = 0, so that we have the trivial solution x(n) = 0 for the system (1.1).
Let us recall that a function (u) is said to be of class n if it is continuous in [0, p), strictly increasing in u and (0) = 0. Using the comparison Theorem 2. 3, we can now prove stability porperties of the null solution of (1.1) in a unified way.
Theorem 3. 1. Suppose that there exists two functions V(n, x) and g(n, u) satisfying the conditions: .N + g no X+ , g(n, 0)=0, g(n,u) is nondecreasing in u for Then the stability properties of the trivial solution of (2.4) imply the corresponding stability properties of the Volterra system (1.1).
Choose 6 3(n0, ) > 0 such that a(6) < 61.Then we claim that the null solution of (1.1) is stable with this 6.If this is false, then there would exist a solution x(n) of (1.1)   such that II x(n 0) I I < and an n > n o with (3.2) I I x(nl)II-e and I I x(s)II < < ;, n o < s < n 1.