Stochastic differential equations and fixed point technique.

Abstract

Fixed point theory has a long history of being used in nonlinear di§erential equations, in order to prove existence, uniqueness, or other qualitative properties of solutions. However, using the contraction mapping principle for stability and asymptotic stability of solutions is of more recent appearance. Lyapunovís direct method has been very e§ective in establishing stability results for a wide variety of general nonlinear systems without solving the systems themselves. Nevertheless, the application of this method to problems of stability in di§erential equations with delay has encountered serious di¢ culties if the delay is unbounded or if the equation has unbounded terms. Applying Lyapunov techniques can be challenging, and the Banach Öxed point method has been shown to yield less restrictive criteria for stability of delayed FDEs. The Öxed point theory does not only solve the problem on stability but has a signiÖcant advantage over Lyapunovís direct method. The conditions of the former are often averages but those of the latter are usually pointwise. In this Thesis, we will extend a contraction mapping stability result that gives mean square asymptotic stability of a nonlinear stochastic di§erential equations with variables delays.