Convergence theorems of iterative algorithms for continuous pseudocontractive mappings

In a real reflexive and strictly convex Banach space EE with a uniformly Gâteaux differentiable norm, we introduced the viscosity iterative process {xt}{xt} given by H.-K. Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291] to continuous pseudocontractive self-mapping, and proved that the {xt}{xt} strongly converges x∗∈F(T)x∗∈F(T) as t→0t→0 and x∗x∗ is a unique solution in F(T)F(T) to the following variational inequality: 〈f(x∗)−x∗,j(z−x∗)〉≤0for all z∈F(T). We also proposed the modified implicit iteration process {xn}{xn} for continuous pseudocontractive self-mapping, and show that {xn}{xn} strongly converges x∗∈F(T)x∗∈F(T). The results presented extended and improved the corresponding results of H.-K. Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291].