Set-Valued Variational Inclusions And A Series Of Strong Convergence Theorems

In this paper, a series of strong convergence theorems are discussed, which concern with generarized set-valued variational inclusions, a class of strongly accretive operator equation and a class of nonexpansive mappings.In the second chapter, we study the concepts of resolvent operator associated with (H,η )-monotone operators in Hilbert space. Using the resolvent operator technique, we construct a new algorithm for approximating the solution of this class of variational inclusions, and discuss the convergence of iterative sequences generated by the algorithm.In the third chapter, we introduce a class of if-accretive operators in Banach space, and give the concept of a new class of (H,η)-accretive operators. Using the new resolvent operator technique, we obtain the approximate solution for a system of set-valued quasi-variational inclusions.In the fourth chapter, we consider the generalized duality map J_φ with gauge φ, and some basic inequalities about J_φ. By using a class of uniformly continuous and strongly accretive operator, we discuss the strong convergence problem of the Mann iterative process with errors for computing solutions of the equation Tx= f.In the fifth chapter, we discuss the follow convergence problem in a real reflexive Banach space: Let C be a closed convex subset of Banach space X, which has a weakly sequentially continuous duality map J_φ; f be a contractive mapping, T be a nonexpansive mapping. For n ≥ 1, there exists z_n ∈ C,λ_n≥ 0, we haveAgain, for t ∈ (0,1). we define {x_t} aswhere P is a sunny nonexpansive retraction of X onto C. If T satisfies condition(*), then we prove that {x_t} strongly converges to a fixed point of T as t → 0.