Strong convergence of asymptotically pseudocontractive semigroup by viscosity iteration Rajshree Dewangan a , Balwant Singh Thakur a , Mihai Postolache b,⇑ a School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur 492010, C.G., India b University Politehnica of Bucharest, Faculty of Applied Sciences, 313 Splaiul Independent ßei, 060042 Bucharest, Romania article info Keywords: Viscosity iteration process Semigroup of asymptotically pseudocontractive mappings Strong convergence Common fixed point abstract In this paper, we study the strong convergence of viscosity iteration and modified viscosity iteration process for strongly continuous semigroup of uniformly Lipschitzian asymptoti- cally pseudocontractive mappings. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction Analytical and numerical construction of fixed points of nonexpansive mappings, and of common fixed points of nonexpansive semigroups, became in recent years important topics in Optimization Theory; please, see [1] Agarwal et al., Takahashi [2,3] Aoyama et al., Li et al. [4], Chang [5]. That is why they found various utilizations in a large number of applied areas. We have in mind image recovery and signal processing; Byrne [6], Podilchuk and Mammone [7], Sezan and Stark [8], Youla [9,10]. The most straightforward way to study nonexpansive mappings is to use contraction mapping to approximate fixed point of nonexpansive mapping; Browder [11], Browder and Petryshyn [12], Deimling [13], Reich [14,15], Shou [16], Suzuki [17], Xu [18]. Viscosity method provides an efficient approach to a large number of problems coming from different branches of Math- ematical Analysis. A major feature of these methods is to provide as a limit of the solution of the approximate problems, a particular solution of the original problem, called a viscosity solution. It has been successfully applied to various problems coming from calculus of variations, minimal surface problems, plasticity theory and phase transition; Kohn and Sternberg [19], Ladyzenskaya and Uralceva [20], Lions [21]. Various applications of the viscosity methods can be found in optimal con- trol theory, singular perturbations, minimal cost problem; Attouch [22], Lions [23,24], and in stochastic control theory; Flem- ing and McEneaney [25]. First abstract formulation of the properties of the viscosity approximation have been given by Tykhonov [26] in 1963 when studying ill-posed problems; see Dontchev and Zolezzi [27] for details. The concept of viscosity solution for Hamilton–Jacobi equations, which plays a crucial role in control theory, game theory and partial differential equations has been introduced by Crandall and Lions [28]; also, see Cho and Kang [29]. In 2000, Moudafi [30] introduced a viscosity approximation method to compute fixed points of nonexpansive mappings. Xu [31] studied further the viscosity approximation method for nonexpansive mapping in uniformly smooth Banach spaces, while Song and Xu [33] studied the convergence of their implicit viscosity iterative scheme for nonexpansive semigroup. Song and Chen [34] proposed implicit viscosity iterative scheme for a fixed Lipschitz strongly pseudocontractive mapping and a continuous pseudocontractive mapping. http://dx.doi.org/10.1016/j.amc.2014.09.115 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail addresses: rajshreedewangan1@gmail.com (R. Dewangan), balwantst@gmail.com (B.S. Thakur), emscolar@yahoo.com (M. Postolache). Applied Mathematics and Computation 248 (2014) 160–168 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc