Authors’ abstract: Let be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. The first purpose of this paper is to introduce a modified viscosity iterative process with perturbation for a continuous pseudocontractive self-mapping and prove that this iterative process converges strongly to , where is the unique solution in to the following variational inequality:
The second aim of the paper is to propose two modified implicit iterative schemes with perturbation for a continuous pseudocontractive self-mapping and prove that these iterative schemes strongly converge to the same point . Basically, we show that if the perturbation mapping is nonexpansive, then the convergence property of the iterative process holds. In this respect, the results presented here extend, improve and unify some very recent theorems in the literature [L. C. Zeng and J. C. Yao, Nonlinear Anal., Theory Methods Appl. 64, No. 11 (A), 2507–2515 (2006; Zbl 1105.47061); H. K. Xu, J. Math. Anal. Appl. 298, No. 1, 279–291 (2004; Zbl 1061.47060); Y. S. Song and R. D. Chen, Nonlinear Anal., Theory Methods Appl. 67, No. 2 (A), 486–497 (2007; Zbl 1126.47054)].