Fixed points for Geraghty-Contractions in partial metric spaces

Abstract

We establish some fixed point theorems for mappings satisfying Geraghty-type contractive conditions in the setting of partial metric spaces and ordered partial metric spaces. Presented theorems extend and generalize many existing results in the literature. Examples are given showing that these results are proper extensions of the existing ones.

Some fixed point theorems for (α, β)-admissible Geraghty type contractive mappings and related results

“…It is well known that the contractive-type conditions are very indispensable in the study of fixed point theory and Banach's fixed point theorem [1] for contraction mappings is one of the pivotal results in analysis. This theorem that has been extended and generalized by various authors (see, e.g., [2][3][4][5][6][7][8][9][10][11][12][13][14][15]) and has many applications in mathematics and other related disciplines as well.…”

“…Thereafter, the existence of fixed points of a-admissible contractive-type mappings in complete metric spaces has been studied by several researchers (see [2,11,13,14] and references cited therein). In this paper, we introduced the concept of (a, b)-admissible Geraghty type contractive mappings.…”

Fixed points of α-Θ-Geraghty type and Θ-Geraghty graphic type contractions

“…Suppose that there exists β ∈ F such that, for all x, y ∈ X, Then T has a unique fixed point z ∈ X and {T n x} converges to the point z for each x ∈ X. Since Geraghty's fixed point theorem, some authors have studied this theorem in several ways (see [11,23,21,8,25,7]). On the other hand, in 2012 and 2013, Samet et al [27] and Hussain et al [13] introduced the concept of α-admissible mappings in metric spaces and proved some fixed point theorems for these mappings.…”

A New Class of Contraction inb-Metric Spaces and Applications

“…They established some fixed point theorems for such mappings in complete metric spaces and showed some examples and applications to ordinary differential equations. Since then, many researchers extended the idea and generalized fixed point results for single-valued and multivalued -admissible mappings in various abstract spaces (see, e.g., [2,3,11,19,20] and more references in the literature). The following definitions and examples reveal the basics of -admissible mappings.…”

“…In 2014, a new notion of h--admissible mapping was introduced by Rosa and Vetro [20]. The definition of this notion is as follows.…”