Abstract
This paper is concerned with the study of a class of forward–backward splitting methods based on Lyapunov distance for variational inequalities and convex minimization problem in Banach spaces.
Highlights
Let X be a reflexive, strictly convex and smooth Banach space with the dual space X∗, A : X ⇒ X∗ a general maximal monotone operator, and C a closed convex set in X
In [3], convergence results have been obtained for the backward–backward splitting method (2) under the key Fenchel conjugate assumption that λnβn Ψ ∗
In [18], the authors prove that every sequence generated by a projection iterative method converges strongly to a common minimum norm solution of a variational inequality problem for an inverse strongly monotone mapping in Banach spaces
Summary
Let X be a reflexive, strictly convex and smooth Banach space with the dual space X∗, A : X ⇒ X∗ a general maximal monotone operator, and C a closed convex set in X. In [4], the authors prove that every sequence generated by the forward–backward splitting method converges weakly to a solution of the minimization problem if either the penalization function or the objective function is inf-compact. In [18], the authors prove that every sequence generated by a projection iterative method converges strongly to a common minimum norm solution of a variational inequality problem for an inverse strongly monotone mapping in Banach spaces. It is obvious from the definition of W that x – y 2 ≤ W (x, y) ≤ x + y 2, ∀x, y ∈ X
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