The purpose of this article is to study the fixed point and weak convergence problem for the new defined class of point-dependent l-hybrid mappings relative to a Bregman distance D f in a Banach space. We at first extend the Aoyama-Iemoto-Kohsaka-Takahashi fixed point theorem for l-hybrid mappings in Hilbert spaces in 2010 to this much wider class of nonlinear mappings in Banach spaces. Secondly, we derive an Opial-like inequality for the Bregman distance and apply it to establish a weak convergence theorem for this new class of nonlinear mappings. Some concrete examples in a Hilbert space showing that our extension is proper are also given. 2010 MSC: 47H09; 47H10.
Ž. We introduce the class S-KKM X, Y, Z consisting of all multifunctions T : Y ª 2 Z that have the S-KKM property and establish a fixed-point theorem for Ž . compact and closed T in S-KKM X, Y, Z that generalizes a large number of classical fixed-point theorems. We also deduce some KKM-type theorems with application to the generalizations of the Ky Fan matching theorem and the Fan᎐Browder fixed-point theorem. ᮊ 1999 Academic Press
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