Search

In this paper, we propose a new splitting algorithm to find the zero of a monotone inclusion problem that features the sum of three maximal monotone operators and a Lipschitz continuous monotone operator in Hilbert spaces. We prove that the sequence of iterates generated by our proposed splitting algorithm converges weakly to the zero of the considered inclusion problem under mild conditions on the iterative parameters. Several splitting algorithms in the literature are recovered as special cases of our proposed algorithm. Another interesting feature of our algorithm is that one forward evaluation of the Lipschitz continuous monotone operator is utilized at each iteration. Numerical results are given to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

Representation formulas for faces and support functions of the values of maximal monotone operators are established in two cases: either the operators are defined on reflexive and locally uniformly convex real Banach spaces with locally uniformly convex duals, or their domains have nonempty interiors on real Banach spaces. Faces and support functions are characterized by the limit values of the minimal-norm selections of maximal monotone operators in the first case while in the second case they are represented by the limit values of any selection of maximal monotone operators. These obtained formulas are applied to study the structure of maximal monotone operators: the local unique determination from their minimal-norm selections, the local and global decompositions, and the unique determination on dense subsets of their domains. [ABSTRACT FROM AUTHOR]

This paper deals with a method for approximating a solution of the following problem: find hierarchically a common fixed point of a finite family of nonself nonexpansive mappings with respect to a nonexpansive self mapping on a closed convex subset of a smooth and reflexive Banach space X, which admits a weakly sequentially continuous duality mapping. First, we prove a weak convergence theorem which extends and improves one recent result proved by Yao and Liou (see Inverse Problems 24 (), doi:). Secondly, when the self mapping is a contraction, we prove, under different restrictions on parameters, a strong convergence result which generalize some recent results in the literature. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and study two hybrid proximal-point algorithms for finding a common element of the set of solutions of an equilibrium problem and the set of solutions to the equation 0∈ Tx for a maximal monotone operator T in a uniformly smooth and uniformly convex Banach space X. Strong and weak convergence results of these two hybrid proximal-point algorithms are established, respectively. [ABSTRACT FROM AUTHOR]