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In this survey paper, we collected the developmental history of fixed point theory. Some important results from beginning up to now are incorporated in this paper. [ABSTRACT FROM AUTHOR]

In this paper several fixed point theorems for generalized weakly contractive mappings in a metric space setting are proved. The set of generalized weakly contractive mappings considered in this paper contains the family of weakly contractive mappings as a proper subset. Fixed point theorems for single and multi-valued mappings, approximating scheme for common fixed point for some mappings, and fixed point theorems for fuzzy mappings are presented. It extends the work of several authors including Bose and Roychowdhury. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of (α,HLC, fRC)-Caristi type contraction mappings and prove fixed point theorem by using this notion on complete metric space. To illustrate our result, we construct an example. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove a common fixed point theorem, by using the concept of weakly subsequential continuity and compatibility of type (E) for two pairs of self mappings satisfying a linear contractive condition in metric spaces, we give two examples to illustrate our results. [ABSTRACT FROM AUTHOR]

A generalized hypersubstitution of type τ is a mapping which maps any operation symbol to the set of all terms of the same type which does not necessarily preserve the arity. The set of all generalized hypersubstitutions together with a binary operation and the identity element forms a monoid. In this paper we determine all maximal idempotent submonoids of this monoid of type τ = (n). [ABSTRACT FROM AUTHOR]

In this paper, we prove a common fixed point theorem for noncompatible maps in probabilistic metric space using implicit relation. Our result does not require either the completeness of the space or continuity of the maps. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to prove a common fixed point theorem for two pairs of weakly compatible mappings in Menger space using implicit relation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalized iteration scheme with errors for convergence to common fixed points of two nonexpansive mappings. This scheme contains a wide variety of existing iteration schemes as its special cases. The main feature of this scheme is that its special cases can handle both strong convergence like Halpern-type and weak convergence like Ishikawa- type iteration schemes. Our main theorem will in particular generalize a recent result by Kim and Xu [9]. [ABSTRACT FROM AUTHOR]

The concept of occasionally weakly compatible (shortly, owc) mappings introduced by Al-Thagi and Shahzad [2], which is more general than the concept of weakly compatible maps. In this paper, we prove a common fixed point theorem for owc mappings in Menger space using arbitrary continuous t-norm for a nonlinear case. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish a unique common fixed point theorem for two pairs of occasionally weakly compatible single and multi-valued maps in a metric space. This result improves the result of Türkoğlu et al. [6] and references therein. [ABSTRACT FROM AUTHOR]