The Application Of The Fixed Point Theorem In Nonlinear Differential Equation

This thesis mainly investigate the existence, stability and exponential stability of solutionsof some classes of stochastic differential equations. On the other hand, theorems of strongconvergence of the viscosity approximation processes for a nonexpansive semigroup areestablished in reflexive Banach spaces, meanwhile, the solvable problems of variationalinequalities contained the above operator is investigated. We will divide into the followingfour chapters.Chapter1presents the preliminaries of this thesis. Basic definitions, formulas andtheorems are stated.Chapter2is concerned with the p th exponential stability and almost surely stability forstochastic Volterra-Levin equations. Using nonlinear operator fixed point theorem, combinedwith differential integral theory, the related problems are solved. Compared with the existingresults, our results extend and improve the exponential stability of solutions of stochasticVolterra-Levin equation from second to p th (p≥2).Chapter3is studies the existence, uniqueness and stability of solutions of stochastic integro-differential equation with variable delay. By the use of fixed point theorem, the authors givesufficient and necessary conditions to ensure the solution of stochastic integro-differential equationbe mean square asymptotic stable.In chapter4, using the techniques of analysis and variational inequality theory, theoremsof strong convergence of viscosity approximation processes for a nonexpansive semigroup areestablished in reflexive Banach spaces. Meanwhile, this chapter investigates the solvableproblems of variational inequalities contained the above operator.