Strong Convergence Theorems for Nonexpansive Semigroups and Variational Inequalities in Banach Spaces

Let ð ‘‹ be a uniformly convex Banach space and ð ’® = { 𠑇 ( ð ‘ ) ∶ 0 ≤ ð ‘ < ∞ } be a nonexpansive semigroup such that â‹‚ ð ¹ ( ð ’® ) = ð ‘ > 0 ð ¹ ( 𠑇 ( ð ‘ ) ) ≠∅ . Consider the iterative method that generates the sequence { ð ‘¥ ð ‘› } by the algorithm ð ‘¥ ð ‘› + 1 = ð ›¼ ð ‘› ð ‘“ ( ð ‘¥ ð ‘› ) + ð ›½ ð ‘› ð ‘¥ ð ‘› + ( 1 − ð ›¼ ð ‘› − ð ›½ ð ‘› ) ( 1 / ð ‘ ð ‘› ) ∫ ð ‘ ð ‘› 0 𠑇 ( ð ‘ ) ð ‘¥ ð ‘› ð ‘‘ ð ‘ , ð ‘› ≥ 0 , where { ð ›¼ ð ‘› } , { ð ›½ ð ‘› } , and { ð ‘ ð ‘› } are three sequences satisfying certain conditions, ð ‘“ ∶ ð ¶ â†’ ð ¶ is a contraction mapping. Strong convergence of the algorithm { ð ‘¥ ð ‘› } is proved assuming ð ‘‹ either has a weakly continuous duality map or has a uniformly Gâteaux differentiable norm.