# STABILITY AND BOUNDEDNESS IN VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH DELAY.

1. INTRODUCTION

During the last years, many good results have been obtained on the qualitative behaviors in Volterra integro-differential equations without delay. In particular, for some works on the stability and boundedness in certain Volterra integro- differential equations without delay, we referee the interested reader to the papers of Becker ([1], [2], [3]), Burton ([4], [5], [6]), Burton et al. [7], Burton and Mahfoud [8], Corduneanu [9], Furumochi and Matsuoka [11], Gripenberg et al. [13], Hara et al.[14], Miller [16], Staffans [19], Tunc [20], Tunc and Ayhan [22], Vanualailai and Nakagiri [24] and theirs references.

Besides, concerning stability and boundedness in Volterra integro-differential equations with delay, we can find a few interesting results in the papers by Graef and Tunc [12], Raffoul [17], Raffoul and Unal [18], Tunc [21] and in theirs references.

In 1982, Burton [5] considered the following non-linear homogeneous scalar Volterra integro-differential equation without delay

(1.1) x'(t) = A(t)f (x(t)) + [[integral].sup.t.sub.0] B(t, s)g(x(s))ds,

where t [greater than or equal to] 0, x [member of] R, A(t) : R [right arrow] [0, [infinity]) and f, g : R [right arrow] R are continuous functions with f (0) = g(0) = 0, and B(t, s) is a continuous function for all 0 [less than or equal to] s [less than or equal to] t [less than or equal to] [infinity]. The author studied the stability, boundedness, convergence of bounded solutions of equation (1.1) by using the Lyapunov functionals.

Next, in the same paper, Burton [5] considered the following non-linear homogeneous vector Volterra integro-differential equation without delay of the form

(1.2) x'(t) = Ax(t) + [[integral].sup.t.sub.0] B(t, s)E(x(s))x(s)ds,

where t [greater than or equal to] 0, x is an n-vector, n [greater than or equal to] 1, A is an n x n-constant matrix, B(t, s) is an n x n-continuous matrix function for 0 [less than or equal to] s [less than or equal to] t < [infinity], E(x) is n x n- -matrix valued continuous function for x [member of] [R.sup.n]. Burton [5] discussed the stability, boundedness, convergence of bounded solutions and square integrability of solutions of equation (1.2) by defining a Lyapunov functional.

Later, in 1999, Furumochi and Matsuoka [11] discussed the stability and boundedness of solutions of vector Volterra integro-differential equations without delay of the form

(1.3) x'(t)= a(x(t)) + [[integral].sup.t.sub.0] C(t, s)f(x(s))ds + g(t, x(t)),

when g(t, x(t)) [equivalent to] 0 and g(t, x(t)) [not equal to] 0, respectively. In the proofs, the Lyapunov functionals are applied by Furumochi and Matsuoka [11].

Recently, the author in [21] considered the non-linear scalar Volterra integro- differential equation with delay

(1.4) x'(t) = -a(t)f (x(t)) + [[integral].sup.t.sub.t-[tau]] B(t, s)g(x(s))ds,

where t [greater than or equal to] 0, [tau] is a positive constant, fixed delay, x [member of] R, a(t) : [0, [infinity]) [right arrow] (0, [infinity]), f, g : R [right arrow] R are continuous functions with f (0) = g(0) = 0, B(t, s) is a continuous function for 0 [less than or equal to] s [less than or equal to] t < [infinity]. The author discussed the stability, boundedness and convergence of bounded solutions of equation (1.4) when t [right arrow] [infinity] by defining suitable Lyapunov functionals.

In this paper, we consider the vector Volterra integro-differential equations with delay of the form:

(1.5) x' = -Ax + H (x) + [[integral].sup.t.sub.t-[tau]] C(t, s)G(s, x(s))ds + E (t, x),

where t [greater than or equal to] 0, [tau] is a positive constant fixed delay, x is an n-vector, n [greater than or equal to] 1, A is an n x n-symmetric matrix, H : [R.sup.n] [right arrow] [R.sup.n] is a continuous function with H(0) = 0, C(t, s) is an n x n-continuous symmetric matrix function for 0 [less than or equal to] s [less than or equal to] t < [infinity], G, E : [R.sup.+] x [R.sup.n] [right arrow] [R.sup.n] are continuous functions with G(t, 0) = 0, and [K.sup.+] = [0, [infinity]).

The objective of this paper is to investigate sufficient conditions for the stability of zero solution and boundedness of solutions of equation (1.5) by employing Lyapunov functionals, when E(t, x) = 0 and E(t, x) = 0, respectively. It is clear that equation (1.1)-(1.3) and equation (1.4) are special cases of equation (1.1) when [tau] = 0 and [tau] [not equal to] 0, respectively.

It should be noted any investigation of the stability and boundedness in a Volterra integro-differential equation, using the Lyapunov functional method, first requires the definition or construction of a suitable Laypunov functional. In fact, this case can be an arduous task. The situation becomes more difficult when we replace the ordinary differential equation with a functional integro-differential equation. However, once a viable Lyapunov functional has been defined or constructed, researchers may end up with working with it for a long time, deriving more more information about stability. To arrive at the objective of this paper, we define two new suitable Lyapunov functionals.

In view of the mentioned information, it follows that the Volterra integro- differential equations discussed by Burton [5] and Furumochi and Matsuoka [11] are without delay. However, in this paper, the Volterra integro-differential equations to be studied are with delay. This is a novelty and improvement for the case without delay to the case with delay. That is from the ordinary case to the functional case. Besides, our equation, equation (1.5) includes and extends the equations discussed by Burton [4], and Furumochi and Matsuoka [11], when [tau] = 0. In addition, our equation includes and improves equation (1.4) in [21] from the scalar case to the system form.

Our results will also be different from that obtained in the literature (see, Becker ([1], [2], [3]), Burton ([4], [5], [6]), Burton et al. [7], Burton and Mahfoud [8], Corduneanu [9], Furumochi and Matsuoka [11], Gripenberg et al. [13], Hara et al. [14], Miller [16], Rafffoul [17], Raffoul and Unal [18], Staffans [19], Tunc ([20], [21]), Vanualailai and Nakagiri [24] and the references thereof). By this way, we mean that the Volterra integro-differential equations considered and the assumptions to be established here are different from that in the mentioned papers above. This paper has also a contribution to the subject in the literature, and it may be useful for researchers working on the qualitative behaviors of solutions to Volterra integro- differential equations. In view of all the mentioned information, it can be checked the novelty and originality of the present paper.

We give some basic information related equation (1.5).

We use the following notation throughout this paper.

For any [t.sub.0] [greater than or equal to] 0 and initial function [phi] [member of] [[t.sub.0] - [tau], [t.sub.0]], let x(t) = x(t, [t.sub.0], [phi]) denote the solution of equation (1.5) on [[t.sub.0] - [tau], [infinity]) such that x(t) = [phi](t) on [phi] [member of] [[t.sub.0] - [tau], [t.sub.0]].

Let C[[t.sub.0], [t.sub.1]] and C[[t.sub.0], [infinity]) denote the set of all continuous real-valued functions on [[t.sub.0], [t.sub.1]], [[t.sub.0], [infinity]), respectively.

For [mathematical expression not reproducible].

Definition 1.1. The zero solution of equation (1.5) with E(t, x) [equivalent to] 0 is stable if for each [epsilon] > 0 and each [t.sub.0] [greater than or equal to] 0, there exists [delta] = [delta]([epsilon], [t.sub.0]) > 0 such that [phi] [member of] C[0, [t.sub.0]] with [mathematical expression not reproducible] to imply [absolute value of (x(t, [t.sub.0], [phi]))] < [epsilon].

Definition 1.2. The solutions of equation (1.5) are bounded with E(t, x) [not equal to] 0 if for each T > 0, there exists D > 0 such that [mathematical expression not reproducible] imply [absolute value of (x(t))] < D.

The following lemma plays a key role in proving our main results.

Lemma 1.3 (Horn and Johnson [15]). Let A be a real symmetric n x n-matrix. Then, for any X [member of] [R.sub.n],

[[alpha].sub.1] [[parallel]X[parallel].sup.2] [less than or equal to] <AX, X> [greater than or equal to] [a.sub.0] [[parallel]X[parallel].sup.2],

where [a.sub.0] and [a.sub.1] are the least and greatest eigenvalues, respectively, of A.

The stability result in this paper is based in the following theorem.

Theorem 1.4 (Driver [10]). If there exists a functional V(t, [phi](*)), defined whenever t [greater than or equal to] [t.sub.0] [greater than or equal to] 0 and [phi] [member of] C([0, t], [R.sup.n]), such that

(i) V(t, 0) [equivalent to] 0, V is continuous in t and locally Lipschitz in [phi],

(ii) V(t, [phi](*)) [greater than or equal to] W([absolute value of ([phi](t))]), W : [0, [infinity]) [right arrow] [0, [infinity]) is a continuous function with W(0) = 0, W(r) > 0 if r > 0, and W strictly increasing (positive definiteness), and

(iii) V'(t, f(*)) [less than or equal to] 0,

then the zero solution of equation (1.5) is stable, and

V(t, [phi](*)) = V(t, [phi](s) : 0 [less than or equal to] s [less than or equal to] t)

is called a Lyapunov functional for system (1.5).

2. STABILITY

In this section we use a Lyapunov functional and establish sufficient conditions to obtain a stability result on zero solution of equation(1.5).

Let

E (t, x) [equivalent to] 0

and

[mathematical expression not reproducible].

A. Assumptions. We assume the following holds:

(A1) There exist a symmetric matrix B and positive constants [delta] and [gamma] such that

[A.sup.T]B + BA = I, [x.sup.T]Bx > 0 for x [member of] [R.sup.n], x [not equal to] 0,

(A2) [mathematical expression not reproducible].

Theorem 2.1. Let assumptions (A1) and (A2) hold, K is a positive constant. If [beta](t) [less than or equal to] K < 1 holds for t [greater than or equal to] [t.sub.0] - [tau] [greater than or equal to] 0, then the zero solution of equation (1.5) is stable.

Proof. We define a functional [W.sub.0] = [W.sub.0](t) = [W.sub.0](t, x(t)) defined by

[mathematical expression not reproducible].

If the assumptions of Theorem 2.1 hold, then it is clear that the functional [W.sub.0] is positive definite.

Differentiating the functional [W.sub.0] with respect to t, we obtain

[mathematical expression not reproducible].

Thus, in view of the discussion made and Theorem 1.4, we can arrive at that the zero solution of equation (1.5) is stable. This completes the proof of Theorem 2.1.

3. BOUNDEDNESS

Let E(t, x) [not equal to] 0 and

[beta](t) = -K [[parallel]x[parallel].sup.2] + 2[theta](t) [parallel]B[parallel] [[parallel]x[parallel].sup.2] + 2[theta](t) [parallel]B[parallel] [parallel]x[parallel] - L[theta](t).

B. Assumptions.

(B1) [parallel]E(t, x)[parallel] [less than or equal to] [theta](t)([parallel]x[parallel] + 1), [theta] : [R.sup.+] [right arrow] [R.sup.+], [R.sup.+] = [0, [infinity]), [theta] is a continuous function such that

[[integral].sup.[infinity].sub.0] [theta](s)ds < [infinity] and [theta](t) [right arrow] 0 as t [right arrow] [infinity].

(B2) There exists a positive constant [K.sub.1] such that [[beta].sub.1](t) [less than or equal to] -[K.sub.1] [[parallel]x[parallel].sup.2].

Theorem 3.1. We suppose that assumptions (A1), (A2), (B1) and (B2) hold. Then all solutions of equation (1.5) are bounded.

Proof. We define a functional [W.sub.1] = [W.sub.1](t) = [W.sub.1](t, x(t)) by

[mathematical expression not reproducible],

where L is a positive constant.

In the light of the assumptions of Theorem 3.1, it follows that the functional [W.sub.1] is positive definite.

Differentiating the functional [W.sub.1] with respect to t, using the assumptions of Theorem 3.1 and the inequality [absolute value of (mn)] [less than or equal to] [2.sup.-1]([m.sup.2] + [n.sup.2]) we have

[mathematical expression not reproducible].

Integrating the estimate [W'.sub.1](t) [less than or equal to] 0 from zero [t.sub.0] to t, we get

[mathematical expression not reproducible]

Then, the boundedness of solutions can be readily followed.

Example 3.2. For the case n = 1, as a special case of equation (1.5), we consider the nonlinear Volterra integro-differential equation with delay, [tau] [greater than or equal to] 0,

[mathematical expression not reproducible],

for t - [tau] [greater than or equal to] 0, x [member of] R.

When we compare this equation with equation (1.5) and consider the assumption of Theorem 2.1 and Theorem 3.1, it follows that

[mathematical expression not reproducible].

Hence, all the assumptions of Theorem 2.1 and Theorem 3.1 hold if [beta](t) [less than or equal to] K < 1 and [beta](t) [less than or equal to] - [K.sub.1][x.sup.2].

4. CONCLUSION

A class of vector non-linear Volterra integro-differential equations of first order is considered. The stability and boundedness of solutions are studied by means of the Lyapunov's functional approach. The obtained results improve some results in the literature.

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CEMIL TUNC

Department of Mathematics, Faculty of Sciences, Yuzuncu Yil University 65080,

Van--Turkey