Modified Forward-Backward Splitting Method Without Cocoercivity for the sum of two Monotone Operators in Banach Spaces

Abstract

In this thesis, we present an algorithm for solving a variation inclusion problem of sum of two monotone operators in real Banach spaces which uses variable step sizes that are updated over each iteration by some cheap computations. These step sizes are found without using Linesearch Procedure or prior knowledge of the Lipschitz constant. More precisely, we provide the following theorem. Theorem. Let E be a real 2-uniformly convex and uniformly smooth Banach space. Let A : E → 2E∗ be a maximal monotone operator and B : E → E∗ be monotone and Lipschitz, and assume that (A + B)−1 (0) 6= ∅ and J is weakly sequentially continuous. For x0, x−1 ∈ E define the sequence xn iteratively by xn+1 = J Aλn◦ J−1 (Jxn − λnBxn − λn−1(Bxn − Bxn−1)), n ≥ 0;λn+1 := min λn, θkxn+1 − xnk kBxn+1 − Bxnk , θ ∈ (0,12µ), µ ≥ 1. Then, the sequence {xn} converges weakly to s an element of (A + B)−1(0). Finally, we give some real life physical applications to show how results can be ap plied