Mann and Ishikawa-Type Iterative Schemes for Approximating Fixed Points of Multi-valued Non-Self Mappings

Abstract

A Mann-type iterative scheme which converges strongly to a fixed point of a multi-valued nonexpansive non-self mapping T is constructed in a real Hilbert space H. We also constructed a Manntype sequence which converges to a fixed point of a multi-valued quasinonexpansive non-self mapping under appropriate conditions. In addition, an Ishikawa-type iterative scheme which approximates the fixed points of multi-valued Lipschitz pseudocontractive non-self mappings is constructed in Banach spaces. The results obtained in this paper improve and extend the known results in the literature.Mathematics Subject Classification. 47H04, 47H09, 47J05, 47J25.

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.…”

“…Then they proved that {x n } converges strongly to a fixed point of T under some mild conditions. In 2016, Tufa and Zegeye [27] pointed out that the above results hold for approximating fixed points of self-mappings which are not always the cases in practical applications. Motivated by the result of Colao and Marino obtained in [6], Tufa and Zegeye introduced and studied Mann-type iterative scheme for multi-valued nonexpansive non-self mappings in a real Hilbert space.…”

A new iterative method for approximating common fixed points of two non-self mappings in a CAT(0) space

“…This method of Colao and Marino [5] have been used by many authors to construct and study iterative processes for approximating fixed points of non-self mappings (see, e.g., [6,7,12,29,[32][33][34]). In particular, Tufa and Zegeye [33] introduced an iterative scheme for approximating fixed points of nonexpansive non-self mappings in the setting of CAT(0) spaces (see Corollary 3.5 of [33]).…”

Approximating common fixed points of a family of non-self mappings in CAT(0) spaces

“…Recently, in [36] and [37], Tufa and Zegeye have extended the result of Colao and Marino [9] to multi-valued mappings in Hilbert spaces and CAT(0) spaces, respectively.…”