The Existence Of The Solution And Algortithms For A New System Of Generalized Co-Compliementarity Problems

In a recent paper[36], J.Y Chen, N-C Wong and J-C Yao introduced a class of co-complementariy Problems, and constructed a news iterative algorithm which included some known algorithm as special cases to solve variational inequalities and complementarity problems, and the author also proves the convergence of the iteratives sequences generated by this algorithm. And Nan-Jing Huang [15] studied a new class of generalized nonlinear implicit quasivariational inequalities for set-valued mappings, and proved the existence of solutions for the nonlinear implicit quasivariational inequalities for set-valued mappings without compactness and the convergence of iterative sequences generated by the algorithms.Motivated and inspied by these works, in this paper, we introduce and study a new systems of generalized co-complementarity problems for set-valued mappings and construct some new iterative algortithms. we prove the exsitence of the solutions for this class of generalized co-complementarity problems for set-valued mappings and the convergence of iterative sequences generated by the algorithms.Let X be a Hilbert space endowed with a norm || · || and inner product (., .). Let CB(X) be the familiy of all nonempty subsets of X. Let K1, K2 be the convex and closed subset of X, g1, g2, m1, m2 : X → X, F, G : X ×X ×X → X,U ,V :X→CB{X). We consider the following problem :Find x, y ∈ X, u∈ U(x), v ∈ V(y), such that gi(x) ∈ Ki(x), i = 1, 2where (K1(x) - g1(x))*, (K2(y) - g2(y))* are the dual cone, Ki : X → 2X, K1(x) = m1(x)+ K1, K2(y) = m2(y) + K2, i = 1,2.In this paper, we prove, under the conditon that gi : X → X,i = 1,2 is strongly monotone and Lipschitz continuous, m, : X → X, i = 1,2 is Lipschitz continuous, F: X× X× X→X is Lipschitz continuous with respect to the first, the second , the third argument, and strongly monotone with respect to the first argument, G : X × X ×X →X is Lipschitz continuous with respect to the first, the second, the third argument and strongly monotone with respect to the third argument, U and V are Lipschitz continuous, and all Lipschitz continuous and strongly monotone parameters satisfy with some conditions, then (*) has a unique solution. And x, y ∈ X, u ∈ U(x), v ∈ V(y) is the solution of (*) if and only if x, y ∈ X, u∈ U(x), v ∈ V(y), satisfy with :x = x - g1(x) + m1(x) + PK1(gx(x) - pF(x,u,y) -m1(x)), y = y- g2(y) + m2(y) + Pk2(g2(x) - pG{x,v,y) -m2(y)).Under the condition that, U = V = I, K1 = K2 = K, We also construct the algorithm: for any given x0,y0 ∈ X, satisfy withAnd we prove the algorithm converge to the solution of () under some conditions.