Solutions of the Systems of Differential Equations with random hand side
by
Ali Mahmud Ateiwi
Department of Mathematics and Statistics, Faculty of Science, Al-Hussein Bin Talal University , P.O. Box (20), Ma'an-Jordan . E:ateiwi@hotmail.com.

Solutions of the Systems of Differential Equations with random hand side

Ali Mahmud Ateiwi

Department of Mathematics and Statistics, Faculty of Science, Al-Hussein Bin Talal University , P.O. Box (20), Ma'an-Jordan . E:ateiwi@hotmail.com.

Consider the system of differential equations with random right hand side and random impulse action at the fixed moments of time:

(1)

Where t≥0,x∈Rⁿ , (x,ω) is the sequence of random variables in Rⁿ, describing the magnitudes of the impulses, F(t,x,ξ(t)) is a random process for all x∈Rⁿ , is the sequence of the moments of impulse action, such that .

Definition 1 The system (1) is called dissipative, if all solutions of this system can be unboundedly extended to the right and the random variables | | are bounded in probability uniformly in and in for any .

We will study the conditions for dissipativity in probability of the solutions of the system (1).

Theorem 1 Let in the domain x∈Rⁿ , t≥0 there exists a non-negative Lyapunov's function, V(t,x)∈C₀, satisfying the condition

= V(t,x)→∞ , R→∞

and the conditions :

where , C₂,C₃ are positive constants.

Besides, let F and σ satisfy the local Lipshitz condition in x , and ‖σ(t,x)‖≤C₄ , C₄›0 .

Then the system (1) is dissipative for any measurable, separable random process ξ(t) and any sequence of random variables { }

such that

, .

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Date received: November 25, 2007