Some Results on Fixed Points--II

Abstract

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Generalization of Rakotch's fixed Point Theorem

“…Finally, by the use of the concept of ω-distance they proved a fixed point theorem in a complete metric space. This theorem generalized the fixed theorems of Subrahmanyan [14], Kannan [7] and Ciric [3]. In the same year T. Suzuki & W. Takahashi [15] gave another property of the ω-distance and using this notion they proved a fixed point theorem for set-valued mappings on complete metric spaces j.r. morales which are related with Nadler's fixed point theorem [9] and Edelstein theorem [4].…”

On the weakly(α, ψ, ξ)-contractive condition for multi-valued operators in metric spaces and related fixed point results

“…[4], the main results of Kannan [19], the main results Chaterjea [20], Nadler's contraction principle [3] and many others.…”

A Generalized Common Fixed Point Theorem for Two Families of Self-Maps

“…Though the fixed point theory for a single map without continuity is traced back to Kannan [17], the following notion was introduced in [26] in the study of common fixed points for noncompatible and discontinuous maps: Definition 2.5. Self-maps f and r on X are reciprocally continuous at z ∈ X if for any sequence x n ∞ n=1 ⊂ X with the choice (2.6), we have lim n→∞ f rx n = f z and lim n→∞ rf x n = rz.…”

“…In 1968, Kannan [17] analyzed a substantially new contractive type condition to ensure the existence of fixed point for maps that have discontinuity in its domain. Kannan's result is effective in characterizing metric completeness [45], though it is independent of Banach's theorem.…”