Forward-Backward splitting methods for accretive operators in Banach spaces

Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two (possibly simpler) nonlinear operators. Most of the investigation on splitting methods is however carried out in the framework of Hilbert spaces. In this paper, we consider these methods in the setting of Banach spaces. We shall introduce two iterative forward-backward splitting methods with relaxations and errors to find zeros of the sum of two accretive operators in the Banach spaces. We shall prove the weak and strong convergence of these methods under mild conditions. We also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem

Alternative iterative methods for nonexpansive mappings, rates of convergence and application

Alternative iterative methods for a nonexpansive mapping in a Banach space are proposed and proved to be convergent to a common solution to a fixed point problem and a variational inequality. We give rates of asymptotic regularity for such iterations using proof-theoretic techniques. Some applications of the convergence results are presented

Strong convergence theorem for accretive mapping in Banach spaces

Suppose K is a closed convex subset of a real reflexive Banach space E which has a uniformly Gâteaux differentiable norm and every nonempty closed convex bounded subset of E has the fixed point property for nonexpansive mappings. We prove a strong convergence theorem for an m-accretive mapping from K to E. The results in this paper are different from the corresponding results in [] and they improve the corresponding results in [6,14