Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces

In this paper we deal with fixed point computational problems by strongly convergent methods involving strictly pseudocontractive mappings in smooth Banach spaces. First, we prove that the SS-iteration process recently introduced by Sahu in [14] converges strongly to a unique fixed point of a mapping TT, where TT is κκ-strongly pseudocontractive mapping from a nonempty, closed and convex subset CC of a smooth Banach space into itself. It is also shown that the hybrid steepest descent method converges strongly to a unique solution of a variational inequality problem with respect to a finite family of λiλi-strictly pseudocontractive mappings from CC into itself. Our results extend and improve some very recent theorems in fixed point theory and variational inequality problems. Particularly, the results presented here extend some theorems of Reich (1980) [1] and Yamada (2001) [15] to a general class of λλ-strictly pseudocontractive mappings in uniformly smooth Banach spaces.