Fixed point results for modified weak and rational α-ψ-contractions in ordered 2-metric spaces

Abstract

We first introduce the concept of a triangular 2-α-η-admissible mapping which extends the notion of α-admissible mapping with respect to η to 2-metric spaces.Next, we introduce the concepts of modified weak and modified rational α-ψ-contractions and establish the existence and uniqueness of fixed points for such mappings in complete 2-metric spaces. As an application of the obtained results, we prove some fixed point results in partially ordered 2-metric spaces. The presented theorems generalize and improve certain existing results in the literature and provide main results in Dung and Hang (Fixed Point Theory Appl. 2013:161, 2013 as corollaries. Moreover, some examples and an application to integral equations are provided to illustrate the usability of the obtained results. MSC: 46N40; 47H10; 54H25; 46T99

Common fixed point results of generalized almost rational contraction mappings with an application

“…Following this trend, Samet et al [34] first introduced α-admissible mappings and then α-ψ-contractive type mappings to obtain some interesting generalizations of Banach contraction principle. For more results in this direction, we refer to [6,13,14,16,17,20,21,22,24,28,30,32] and references mentioned therein. Recently, Alizadeh et al [5] defined the concept of cyclic (α, β)-admissible mapping as follows: Definition 1.1 ( [5]).…”

“…Due to its applications in mathematics and other related disciplines, Banach contraction principle has been generalized in many directions. Extensions of Banach contraction principle have been obtained either by generalizing the domain of the mapping or by extending the contractive condition on the mappings (see, [1,2,3,4,5,6,7,10,11,13,14,15,16,18,19,22,23,24,26,27,28,29,30,31,32,34,35] and references therein).…”

Cone $$b_2$$ b 2 -metric spaces over Banach algebra with applications

“…In this Section, inspired by the work of [7] we shall discuss the existence of solutions to the integral equation.…”

On Suzuki-Wardowski type fixed point theorems

“…(F 2 ) inf F = −∞ or, also, by (F 2 ) there exists a sequence {α n } ∞ n=1 of positive real numbers such that lim n→∞ F (α n ) = −∞. Very recently Piri et al [18] utilized the following condition (F 3 ) instead of (F 3 ) in Definition (6).…”