SOME QUALITATIVE PROPERTIES OF SOLUTIONS OF CERTAIN NONLINEAR THIRD-ORDER STOCHASTIC DIFFERENTIAL EQUATIONS WITH DELAY

Abstract

This study considered certain nonlinear third-order stochastic differential equations with delay. The third-order equation is reduced to an equivalent system of first-order differential equations and used to construct the desired complete Lyapunov-Krasovski\v{\i} functional. Standard conditions guaranteeing stability when the forcing term is zero, boundedness of solutions when the forcing term is non-zero, and lastly the existence and uniqueness of solutions are derived. The obtained results indicated that the adopted technique is effective in studying the qualitative behaviour of solutions. The obtained results are not only new but extend the frontier of knowledge of the qualitative behaviour of solutions of nonlinear stochastic differential with delay. Finally, two special cases are given to illustrate the derived theoretical results.

Keywords

• Existence of solution
• Nonlinear stochastic differential equation
• Uniform stability
• Third-order
• Third-order

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