An iterative Algorithm for Hemicontractive Mappings in Banach Spaces

Abstract

We First introduce a three-step iterative algorithm for approximating the fixed points of the hemicontractive mappings in Banach spaces. Consequently, we prove the strong convergence of the proposed algorithm under some assumptions. Since three-step iterations include Ishikawa iterations as special cases, our result continue to hold for these problems. Our main results can be viewed as an important refinement of the previously known results.

Convergence and stability of modified multi-step Noor iterative procedure with errors for strictly hemicontractive-type mappings in Banach spaces

“…The obtained results of this paper provided easy and straightforward techniques for proving the convergence, almost common-stability and common-stability criteria of the proposed MMNIPE given by (1.5). Furthermore, the results of this paper extended the corresponding results of Hussain et al [1,[7][8][9], Zegeye et al [2], Meche et al [3], Chidume and Osilike [4], Chidume [5], Liu et al [12], Zeng [13], Yu et al [11], Yang [25], Chidume [36], Deng [37,38] and Liu [39]. According to the Remark 3.6, our results generalized and unify the corresponding results of Hussain et al [1], Mann [28], Ishikawa [29], Xu and Noor [30], Liu [31] and Xu [32] and Cho et al [33] in the case of establishing the fixed-point theorem-based iterative procedures for two Lipschitz strictly hemicontractive-type mappings.…”

“…Application of strictly hemicontractive-type mapping was initiated by Chidume and Osilike [4] for improving the consequence of Chidume [5]. After Chidume and Osilike [4], several researchers studied strictly hemicontractive-type mapping in many directions; see for instance [1][2][3][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and the references cited therein. Among the articles cited in [1][2][3][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21], Hussain et al [1] studied Lipschitz strictly hemicontractive-type mapping in arbitrary Banach spaces to extend and improve the equivalent consequences of the monographs [4,5,[12][13][14]…”

“…Among the above-mentioned articles, Hussain et al [1] studied the following special type of Ishikawa iterative procedure with errors (STIIPE) for two Lipschitz strictly hemicontractive-type mappings in arbitrary Banach spaces: For a given u 0 ∈ B, the iterative sequences {u n } ∞ n=0 defined by u n+1 = u (2) n = a (2) n u n + b (2) n Tu (1) n + c (2) n v (2) n , u (1) n = a (1) n u n + b (1) n Su n + c (1) n v (1) n , n ≥ 0, (1.4) where {v (1) n }, {v (2) n } are bounded sequences in B and {a (i) n }, {b (i) n }, {c (i) n } are appropriate real sequences in [0, 1] satisfying a (i) n + b (i) n + c (i) n = 1 for all i ∈ {1, 2}. Stimulated by the work of Hussain et al [1,9], Plubtieng and Wangkeeree [24], Yu et al [11], Agwu and Igbokwe [17] and Zegeye and Tufa [19] in this paper, we propose and study the following modified multi-step Noor iterative procedure with errors (MMNIPE) for two Lipschitz strictly hemicontractive-type mappings in arbitrary Banach spaces: For a given u 0 ∈ B, and a fixed m ∈ N, we compute the iterative sequences {u n } ∞ n=0 by (2) n = a (2) n u n + b (2) n Tu (1) n + c (2) n v (2) n , u (1) n = a (1) n u n + b (1) n Su n + c (1) n v (1) n , n ≥ 0,…”

Algorithms of common solutions for a fixed point of hemicontractive-type mapping and a generalized equilibrium problem

“…Several authors have studied iterative algorithms for approximating fixed points of various classes of nonlinear mappings (including hemicontractive mapping); see, for example, [5,8,14,15,21] and the references cited therein. In 2007, Rafiq [13] introduced the following Mann-type implicit iteration process and showed that the process converges strongly to a fixed point of a continuous hemicontractive mapping T from a compact and convex subset C of a real Hilbert space into itself:…”