Stability Behavior of the Zero Solution for Nonlinear Damped Vectorial Second Order Differential Equation

In this paper, a theoretical treatment of the stability behavior of the zero solution of nonlinear damped oscillator in the vectorial case is investigated. We study the sufficient conditions for the boundedness of solution of the nonlinear damped vectorial oscillator and the conditions for the stability of the zero solution to be uniformly stable as well as asymptotically stable.


Introduction
We consider the nonlinear second order vectorial differential equation of the form x g x 0  . R (1) where; , , Stability problems for the second order ordinary differential equation has been intensively and widely studied [1][2][3][4][5].Based on Schauder fixed point theorem T. A. Burton and T. Furumochi [2] introduced a new method to study the stability of the zero solution for Equation (1).This problem is considered also by Gheorghe Morosanu and Cristian Vladimirescu [5,6].In [6], they used relatively classical arguments to prove the stability of the zero solution of Equation (1).While in [5], they obtained new stability results for this ordinary differential equation under more general assumptions.Their approach allows extensions to both the vector case and the case of the whole real line.In [7] the dynamics of various oscillators had been studied.

The Main Results
In the next theorem we state sufficient conations for the boundedness of the solution of Equation (1) are given.

Theorem 1
If the following hypotheses are hold: , where t R     denotes some norm in .then the solution of Equation ( 1) is bounded.

Proof
For the n-dimensional system, we have , , , , , , , , where m = 2n.Applying the transformation, 1) can be converted into a first order system of differential equations of the form: where , and are n × n matrices.Note that,  and I are the zero and the identity matrices, respectively.
 , let is an arbitrary fixed and let be a fundamental matrix solution to the linear system:

z z
This gives us the following integral inequality: where .3) and ( 4) give us the following differential equations: is a decreasing function, so Equations ( 6) and ( 7) lead to: By the same way we can obtain from Equations ( 8) and ( 9) the following: For , , , , , , , , , , , , , , satisfies (4), then we get: Using Shwartz inequality, Minkowski inequality [8] and suitable assumptions lead to: We have also:   As mentioned the system satisfies the integral inequality (5), then inequalities (12) and (13) give For all t we can replace g by another function say g defined as follows.By hypothesis (4) it follows that there exists a We defined the function by: : and we h , , , x x x  ginal function .So, we w g satisfies

 
ill admit from now that the all the properties of the ori  g .Since , with positive constant D [9,10].This gives s the followin , , t t z z can be extended to the right of l.This contradicts the maximality of l.This means that the solution of Equation ( 1) is bo nded.The proof of theorem 1 is complete.

Theorem 2
If the hypotheses of theorem 1 are hold and lowing assumption is satisfied: 1) There exist two constants such that: then the zero solution of Equation ( 1 Assume h l    .We are going to find an estimate for on the interval   , h l .From hypothesis (1) of theorem 2, we have for t h l   : By integration we get: From (15) it follows that Therefore, with the same  16), ( 18) and ( 19) it follows tha zero solution to (1) is asymptotically stable.The proof of theorem 2 is complete.

Examples
We confirm the results of the introduced theorems by considering two numerical examples for which the functions t the

Conclusion
ifo ly stable as well as