Asymptotic Behavior of Non-oscillatory Solutions of Second order Integro-Dynamic Equations on Time Scales

1 1 2 2 ( , ) ( ) ( , ) ( ) 0 0, β γ ≥ ≤ < ≥ f t x p t x and f t x p t x for x and t By solutions of equation (1.1) we mean a deltadifferentiable function defined on T that is nontrivial in a neighborhood of ∞ . A solution x of equation (1.1) is said to be oscillatory if for every 1 0 ≥ t 0 0 ≥ t we have inf t ≥ t1 x(t)<0<sup t ≥ t1 x(t) and non-oscillatory otherwise. With respect to dynamic equations on time-scales, it is a fairly new topic and for general basic ideas and background, we refer the reader to [1]. Oscillation and non-oscillation results for integral equations of Volterra-type are scant and only a few references exist on this subject. Related studies can be found in [2-5]. To the best of our knowledge, there appear to be no such results on asymptotic behavior of nonoscillatory solutions of equations (1.1). Therefore; the main goal of this paper is to establish some new criteria for the asymptotic behavior of non-oscillatory solutions of equation (1.1) and other related equations. Also we provide some numerical examples to illustrate the obtained results when T=R. Main Results We shall employ the following lemma. Lemma 4.1: If X and Y are nonnegative [6], then


Introduction
In this paper we are concerned with the asymptotic behavior of the non-oscillatory solutions of second order integro-dynamic equation on time-scale of the form ii) : × → F T R R is continuous and assume that there exist 1 2 , : × → f f T R R are continuous such that we have inf t ≥ t 1 x(t)<0<sup t ≥ t 1 x(t) and non-oscillatory otherwise.
With respect to dynamic equations on time-scales, it is a fairly new topic and for general basic ideas and background, we refer the reader to [1].
Oscillation and non-oscillation results for integral equations of Volterra-type are scant and only a few references exist on this subject. Related studies can be found in [2][3][4][5]. To the best of our knowledge, there appear to be no such results on asymptotic behavior of nonoscillatory solutions of equations (1.1). Therefore; the main goal of this paper is to establish some new criteria for the asymptotic behavior of non-oscillatory solutions of equation (1.1) and other related equations. Also we provide some numerical examples to illustrate the obtained results when T=R.

Main Results
We shall employ the following lemma. Lemma 4.1: If X and Y are nonnegative [6], then where equality hold if and only if X = Y.
We define

for all t a s
Here is our first result. First assume x is eventually positive. Fix Assume x t for t t for some t t  By assumption (i), we have Hence from (2.5) and (2.6), we get ( ) If we apply (2.1) with Integrating this equality from t 1 to t, we have Or, Integrating this equality from t 1 to t we get, Now assume x is eventually negative, say x(t)<0 for t ≥ t 1 for some t ≥ t 0 .
Dividing by both sides of this inequality This completes the proof.
Next, we give the following simple result.  The rest of the proof is similar to that of Theorem 2.1 and hence is omitted The following corollary is immediate.
for all 0 0.  x and Y p p We have  The rest of the proof is similar to that of Theorem 4.1 and hence is omitted.
Finally, we present the following results with different nonlinearities i.e. with β>1 and 1 γ < .
for all

t s p s x s s x s s
As in the proof of Theorems 2.1 and 2.3, we can easily find ( ) The rest of the proof is similar to that of Theorem 4.1 and hence is omitted. Theorem 4.4 can be re-stated as follows: for all 0 0. For the case of forced integro-differential equation

x t a t s F s x s s e t
Where : → e T R . Now, if in addition to the hypotheses of all the results presented above, we assume that is rd-contiuous function.

Numerical Examples
As we already mentioned that the results of the present paper are new even for the cases when T = R i.e., the continuous case or when T=Z i.e., the discrete case.
As a numerical illustration of Theorems 4.3, 4.4 and 4.1 respectively, let us consider the following equation x s x s ds t t s (3.1) with initial conditions x t x and x t x Equation (3.1) can be converted to two simultaneous first order ordinary differential equations by substituting ' . = tx y This will lead to the following system:

t x s x s ds y t t x t s
Many numerical techniques can be used to solve (3.2). In the current work, the second order, accurate modified Euler technique is considered. The time interval [t 0 ,T] will be divided into N equal subdivisions with width for each one. The prediction and correction steps of the modified Euler technique will be: Where ( , , ) = y f t x y t (3.4) and x s x s ds t s will have negligible effect on the solution behavior, as shown in Figure  3. In this case the solution asymptotes, approximately, to the same straight line in Figure 2.

General Remarks
We conclude by presenting several remarks and extensions of the results given above i) The results presented in this paper are new for T=R and T=Z.
ii) The results of this paper are presented in a form which is essentially new for equation (1.1) with different nonlinearities. Corollaries similar to Corollary 2.1 can be obtained. Here we omit the details.
iii) The results of this paper will remain the same if we replace (1.2) of assumption (i) by