A Strong Convergence for the sum of three Monotone Operators in a Real Banach Space

Abstract

Let E be a real 2-uniformly convex Banach space with topological dual E∗. We established strong convergence for the class of variational inclusion for the sum of three monotone operators. More over, we give a variant of this algorithm in which the stepsizes which are diminishing and non summable. More precisely, we provide the following theorem: Theorem. Let E be a real 2-uniformly convex Banach space. Let A : E → 2E∗ be a maximal monotone operator and B : E → E∗ be γ− strongly monotone and L-Lipschitz and C : E → E be monotone and c-cocoercive . For x−1, x0 ∈ E de ne the sequence {xk} iteratively by xk+1 = JAαk◦ J−1 (Jxk − αkBxk − αk−1(Bxk − Bxk−1) − αkCxk, where αn ⊆ (0, ∞) and X∞ n=0 αn = +∞, limn→∞ αn = 0 converges strongly to x ∗ an element of (A + B + C)−(0). Finally, few applications were also provided to illustrate the relevance of our proposed scheme. Our results extend and complement several existing results in the literature.