Halpern–Ishikawa type iterative method for approximating fixed points of non-self pseudocontractive mappings

Abstract

In this paper, we define a Halpern-Ishikawa type iterative method for approximating a fixed point of a Lipschitz pseudocontractive non-self mapping T in a real Hilbert space settings and prove strong convergence result of the iterative method to a fixed point of T under some mild conditions. We give a numerical example to support our results. Our results improve and generalize most of the results that have been proved for this important class of nonlinear mappings. MSC: 37C25; 47H10; 47J05

Halpern–Ishikawa type iterative schemes for approximating fixed points of multi-valued non-self mappings

“…In addition, a Halpern-Ishikawa type iterative method for approximating fixed points of multivalued k-strictly pseudocontractive mappings is introduced and strong convergence results of the scheme are obtained without the end point condition. Our results extend and generalize many of the results in the literature (see, e.g., [6,7,22,23,25,[27][28][29]). More particularly, Theorem 3.2 extends Theorem 3.2 of Zegeye and Tufa [28] from single-valued mapping to multi-valued mapping.…”

“…Our results extend and generalize many of the results in the literature (see, e.g., [6,7,22,23,25,[27][28][29]). More particularly, Theorem 3.2 extends Theorem 3.2 of Zegeye and Tufa [28] from single-valued mapping to multi-valued mapping. Thus, if we assume that T is single-valued mapping in Theorem 3.2, then we get Theorem 3.2 of Zegeye and Tufa [28].…”

“…More particularly, Theorem 3.2 extends Theorem 3.2 of Zegeye and Tufa [28] from single-valued mapping to multi-valued mapping. Thus, if we assume that T is single-valued mapping in Theorem 3.2, then we get Theorem 3.2 of Zegeye and Tufa [28]. Theorem 3.8 extends Theorem 8 of Colao et al [7] from single-valued mapping to multi-valued mapping.…”

“…Recently, Zegeye and Tufa [28] constructed a Halpern-Ishikawa type iterative scheme for single-valued Lipschitz pseudocontractive non-self mappings in Hilbert spaces and obtained strong convergence of the scheme to fixed points of the mappings under some mild conditions. Their result mainly extends the result of Colao et al [7] from k-strictly pseudocontractive to pseudocontractive mapping.…”

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

“…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]…”