Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications

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

“…Our iteration scheme seems simpler and better than those studied in [31,24,8,16]. Our theorems improve and generalize the results of Chang et al [8], Kang, et al [16], Takahashi and Zambayeshi [24], Zegeye [31] and those of host of other authors. Application of our theorem to iterative approximation of solution of equation of Hammerstein-type is of independent interest.…”

“…It is easy to see that the iteration process studied in this paper seems simpler than the schemes studied by [8,16,24,31]. Moreover, Theorem 13 generalizes the corresponding results of Chang et al [8], Kang, et al [16], Takahashi and Zambayeshi [24], Zegeye [31] and those of host of other authors to approximation of common element of null spaces of finite family of c inverse strongly monotone mappings, set of common fixed point of finite family of weak relatively nonexpansive mappings and solution set of generalized mixed equilibrium problem.…”

“…In the results obtained in [8,16,24,31] and many other results published in this direction, the assumptions VI(C, A) -; and kAxk 6 kAx À Auk "x 2 C, u 2 VI(C, A) were made. It is of interest to mention here that these two assumptions are equivalent to the single assumption A À1 (0) -;.…”

“…Recently, for finding a common element of F 0 = F(S) \ VI(A, C) \ EP(f), Kang et al [16] introduced the following scheme in a 2-uniformly convex and uniformly smooth Banach space E : x 0 2 C, y n ¼ P C J À1 ðJx n À b n Ax n Þ; z n ¼ J À1 ða n Jx n þ ð1 À a n ÞJSy n Þ;…”

Parallel and sequential hybrid methods for a finite family of asymptotically quasi $$\phi $$ ϕ -nonexpansive mappings

“…x 0 ∈ C, y n = J −1 (α n Jx n + (1 − α n )JT x n ), C n = {v ∈ C : φ (v, y n ) ≤ φ (v, x n )} , Q n = {v ∈ C : Jx 0 − Jx n , x n − v ≥ 0} , x n+1 = Π C n Q n x 0 , n ≥ 0. This algorithm has been modified and generalized for finding a common fixed point of a finite or infinite family of relatively nonexpansive mappings by several authors, such as Takahashi et al [29], Takahashi and Zembayashi [30], Wang and Xuan [32], Reich and Sabach [24,25], Kang, Su, and Zhang [13], Plubtieng and Ungchittrakool [22], etc... In 2011, Liu [20] introduced the following cyclic method for a finite family of relatively nonexpansive mappings:…”