Algorithms for General System of Generalized Resolvent Equations with Corresponding System of Variational Inclusions

Very recently, Ahmad and Yao (2009) introduced and considered a system of generalized resolvent equations with corresponding system of variational inclusions in uniformly smooth Banach spaces. In this paper we introduce and study a general system of generalized resolvent equations with corresponding general system of variational inclusions in uniformly smooth Banach spaces. We establish an equivalence relation between general system of generalized resolvent equations and general system of variational inclusions. The iterative algorithms for finding the approximate solutions of general system of generalized resolvent equations are proposed. The convergence criteria of approximate solutions of general system of generalized resolvent equations obtained by the proposed iterative algorithm are also presented. Our results represent the generalization, improvement, supplement, and development of Ahmad and Yao corresponding ones.

Generalized Resolvent Equations and Unsymmetric Dirichlet Spaces

Let (X, , μ) and (X, , μ′) be measure spaces with the measures μ and μ′ totally finite. Suppose {Uλ: λ > 0} is a family of positive (i.e., ϕ ≧ 0 ⇒ Uλϕ ≧ 0) continuous linear operators from L2(X, dμ′) to L2(X,dμ) with the following additional properties: if ϕ ≧ 0 then Uλϕ is non-decreasing as λ increases, while λ−1Uλϕ is nonincreasing.A family {Mλ:λ > 0} of continuous linear operators from L2(X, dμ) to L2(X, dμ′) satisfies the “generalized resolvent equation” relative to {Uλ} if(0.1)for positive λ and v. If Uλ = λI, then (0.1) is just the well-known resolvent equation. The family {Mλ} is called submarkov if Mλ is a positive operator and(0.2)it is conservative if(0.3)