Existence Of Solutions And Perturbed Iterative Algorithms For A Class Of Nonlinear Projection Equations

It is well-known that the variational inequality problem on the nonempty closed subset of Hilbert spaces can be formulated into the projection equation problem. Recently, Zhao and Sun introduced and discussed a class of nonlinear projection equations. In this paper, we introduce and study a new class of nonlinear projection equations, which include many kinds of variational inequalities, quasi-variational inequalities and implicit quasi-variational inequalities as special cases. We discuss the existence of the solution for this class of nonlinear projection equations in Hilbert space and the convergence of sequences generated by algorithms.We obtained the following results.Theorem A. Let H be a real Hilbert space endowed with a norm and inner product <.,.>. Let K:H2H be a set-valued mapping such that K(x) is a closed convex subset of H for each H be Lipschitz continuous with Lipschitz constants , respectively, h strongly monotone with constant , and T strongly monotone with respect to g with constant . If there exists a constant such thatandthen the equationhas a unique solution.Algorithm B. Let H be a real Hilbert space and K(x) be a closedconvex subset of H for each. For given mappings T,g,h:H H and point H , define the iterative sequences as follows:where and are two sequences in two sequences in [0,1] satisfying following conditions:Theorem C. Let H be a real Hilbert space and Kn,K:H be set-valued mappings such that are closed convex subsets of H and for each . Let h,g,T:H-H be Lipschitz continuous with Lipschitz constants respectively, h strongly monotone with constant , and T strongly monotone with respect to g with constant . Suppose that there exists a constant 6 > 0 such thatthen converges strongly to the unique solution of the projection equation...