Solvability And Iterative Algorithms For Nonlinear Variational Inclusions And System Of Inclusions

Variational inequality theory has been prominent in mathematics.Among many aspects of it,the most important and interesting is to develop effective numerical methods to generate approximate solutions.This paper will discuss it in the following way:Firstly,the background and current state of variational inequality theory will be discussed.Secondly,it will introduce a new class of G-f-η-monotone mappings and the generalized resolvent operators elaborated in Hilbert space.These operators can be used to testify the existence of the solvability of a class of completely generalized implicit nonlinear variational-like inclusions.Thirdly,it will prove the solvability of a class of mixed nonlinear variationallike inclusion with fuzzy mappings involving G-f-η-monotone mappings in Hilbert space and discuss the convergence criteria and stability of the Mann-type iterative algorithm and p-step Ishikawa-type iterative algorithm for this class of inclusions.Finally,the(P,f,η)-proximal-point mapping,which is derived from a new (P,f,η)-accretive mapping is revealed to be single-valued and Lipschitz continuous within real q-uniformly smooth Banach spaces.And based on this,the solvability and the convergence criteria and stability of the Mann-type iterative algorithm and m-step Ishikawa-type iterative algorithm for this system of mixed quasi-variational inclusions can be proved as well.