Iterative Algorithm For Split General Strong Nonlinear Quasi-Variational Inequality

Variational inequality are a very powerful tool of the current mathematical technology,it was widely used optimization,control problems and engineering sciences.Split variational inequalities problem is considered as a kind of variational inequalities,is also a hotspot in the research of mathematics.It is widely used in signal processing,phase retrieval,image reconstruction and itensity-modulated radiation therapy.This paper mainly studied the iterative algorithm in Hilbert space for a class of generalized nonlinear mixed quasi-variational-like inequality problem,split general strong nonlinear quasi-variational inequality problem and system of split general variational inequality problem.In chapter 2,we studied a class of generalized nonlinear mixed quasi-variational-like inequality problem in Hilbert spaces.First,we establish the existence and uniqueness of solution of the corresponding auxiliary generalized nonlinear mixed quasi-variational-like inequality by making use of minimizing sequence of a convex function.Then based on the existence result,we construct an iterative algorithm by using the auxiliary principle technique and proved that the convergence of the iterative sequence generated by the algorithm and the existence of the solution of the generalized nonlinear mixed quasi-variational-like inequality.Our results improve and extend the corresponding results.In chapter 3,the split general strong nonlinear quasi-variational inequality problem and the system of split general variational inequality problem in Hilbert spaces are studied.By using the projection operator as a tool,the problem of splitting general strongly nonlinear quasi variational inequality and the system of split general variational inequality problem is transformed into a fixed point problem.Then based on the existence result,proposed an iterative algorithm,by the condition of strong monotonicity and Lipschitz continuity,we prove the strong convergence of the iterative sequence generated by the iterative algorithm.