NEW STABILITY AND BOUNDEDNESS RESULTS FOR SOLUTIONS OF A CERTAIN THIRD-ORDER NONLINEAR STOCHASTIC DIFFERENTIAL EQUATION

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Published: 2015-06-12

Page: 60-70


A. M. A. ABOU-EL-ELA *

Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt.

A. I. SADEK

Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt

A. M. MAHMOUD

Department of Mathematics, Faculty of Science, New Valley Branch, Assiut University, New Valley, El-Khargah 72111, Egypt

E. S. FARGHALY *

Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt

*Author to whom correspondence should be addressed.


Abstract

In this paper we consider the nonlinear third-order stochastic differential equation (1.1) and use Lyapunov functions to prove the stochastic asymptotic stability of the zero solution when p ≡ 0. When p ̸≡ 0, two results are studied:
(1) The uniform stochastic boundedness of all solutions,
(2) The exponential asymptotic stability in probability of the zero solution.
We obtained the results in a very simple form. In the last section, we give two examples to illustrate our main results of stability and boundedness.

Keywords: Asymptotic stability, boundedness, exponential asymptotic stability, Lyapunov function;, nonlinear stochastic dierential equations of third-order


How to Cite

ABOU-EL-ELA, A. M. A., SADEK, A. I., MAHMOUD, A. M., & FARGHALY, E. S. (2015). NEW STABILITY AND BOUNDEDNESS RESULTS FOR SOLUTIONS OF A CERTAIN THIRD-ORDER NONLINEAR STOCHASTIC DIFFERENTIAL EQUATION. Asian Journal of Mathematics and Computer Research, 5(1), 60–70. Retrieved from https://ikprress.org/index.php/AJOMCOR/article/view/145

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