Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type

Abstract

Keywords:Fixed point problems Generalized mixed equilibrium problem Hammerstein equations Monotone operators Uniformly convex and uniformly smooth Banach spaces Weak relatively nonexpansive mappings a b s t r a c tIn this paper we study hybrid iterative scheme for finding a common element of set of solutions of generalized mixed equilibrium problem, set of common fixed points of finite family of weak relatively nonexpansive mapping and null spaces of finite family of c-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. Our results extend, improve and generalize the results of several authors which are announced recently. Application of our theorem to solution of equations of Hammersteintype is of independent interest.

Strong convergence theorems for nonlinear operator equations with total quasi-ϕ-asymptotically nonexpansive mappings and applications

“…As an application, we shall utilize our results to study the iterative solutions of the nonlinear Hammerstian type equation. The results presented in the article improve and extend the corresponding results in [1][2][3][4][5][7][8][9][10][11][12][13][14][15][16].…”

“…In 2005, Matsushita and Takahashi [7] obtained some weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Recently, Ofoedu and Malonza [4], Zhang [5], Su et al [8], Zhang and Su [9], Zegeye and Shahzad [10], Wattanawitoon and Kumam [11], Qin et al [12], Takahashi and Zembayashi [13] extend the notions from relatively nonexpansive mappings, weakly relatively nonexpansive mappings or quasi-j-nonexpansive mappings to quasi-j-asymptotically nonexpansive mappings and also proved some strong convergence theorems to approximate a common fixed point of quasi-j-nonexpansive mappings or quasi-jasymptotically nonexpansive mappings.…”

Strong convergence theorems of nonlinear operator equations for countable family of multi-valued total quasi-ϕ-asymptotically nonexpansive mappings with applications

“…The results presented in the article not only generalize the corresponding results of [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] from single-valued mappings to multi-valued mappings, but also improve and extend the main results of Homaeipour and Razani [3]. The method given in this article is quite different from that one adopted in [3].…”

“…Concerning the weak and strong convergence of iterative sequences to approximate a common element of the set of solutions for a generalized MEP, the set of solutions for variational inequality problems, and the set of common fixed points for single-valued relatively non-expansive mappings, single-valued quasi-j-nonexpansive mappings, single-valued quasi-j-asymptotically nonexpansive mappings and single-valued total quasij-asymptotically non-expansive mappings have been studied by many authors in the setting of Hilbert or Banach spaces (see, for example, [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and the references therein). Very recently, in 2011, Homaeipour and Razani [3] introduced the concept of multivalued relatively nonexpansive mappings and proved some weak and strong convergence theorems to approximation a fixed point for a single relatively nonexpansive multivalued mapping in a uniformly convex and uniformly smooth Banach space X which improve and extend the corresponding results of Matsushita and Takahashi [5].…”

“…Let C be a unit ball in a real Hilbert space l 2 and let T : C C be a mapping defined by 11) where {a i } is a sequence in (0, 1) such that…”