In this paper, we prove that coupled and tripled coincidence point theorems under ðF; gÞ-invariant sets for weakly contractive mappings defined on a G-metric space are immediate consequences of corresponding results via rectangular G-a-admissible mappings. This idea can also be applied to obtain coupled and tripled fixed point theorems in other spaces under various contractive conditions which reduces the proof considerably. Keywords G b-metric space Á Coupled coincidence point Á Common coupled fixed point Á Tripled coincidence point Á Common tripled fixed point Á F-invariant set Á Admissible mapping Mathematics Subject Classification Primary 47H10 Á Secondary 54H25 Introduction and mathematical preliminaries The concept of generalized metric space, or a G-metric space, was introduced by Mustafa and Sims. Definition 1.1 (G-Metric Space, [14]) Let X be a nonempty set and G: X Â X Â X ! R þ be a function satisfying the following properties: (G1) Gðx; y; zÞ ¼ 0 iff x ¼ y ¼ z; (G2) 0\Gðx; x; yÞ, for all x; y 2 X with x 6 ¼ y; (G3) Gðx; x; yÞ Gðx; y; zÞ, for all x; y; z 2 X with y 6 ¼ z; (G4) Gðx; y; zÞ ¼ Gðx; z; yÞ ¼ Gðy; z; xÞ ¼. . ., (symmetry in all three variables); (G5) Gðx; y; zÞ Gðx; a; aÞ þ Gða; y; zÞ, for all x; y; z; a 2 X (rectangle inequality). Then, the function G is called a G-metric on X and the pair ðX; GÞ is called a G-metric space. Recently, Aghajani et al. [1] motivated by the concept of b-metric [27] introduced the concept of generalized bmetric spaces (G b-metric spaces) and then they presented some basic properties of G b-metric spaces. The following is their definition of G b-metric spaces. Definition 1.2 [1] Let X be a nonempty set and s ! 1 be a given real number. Suppose that a mapping G: X Â X Â X ! R þ satisfies: (G b 1) Gðx; y; zÞ ¼ 0 if x ¼ y ¼ z; (G b 2) 0\Gðx; x; yÞ for all x; y 2 X with x 6 ¼ y; (G b 3) Gðx; x; yÞ Gðx; y; zÞ for all x; y; z 2 X with y 6 ¼ z; (G b 4) Gðx; y; zÞ ¼ Gðpfx; y; zgÞ; where p is a permutation of x; y; z (symmetry); (G b 5) Gðx; y; zÞ s½Gðx; a; aÞ þ Gða; y; zÞ for all x; y; z; a 2 X (rectangle inequality). Then, G is called a generalized b-metric and the pair ðX; GÞ is called a generalized b-metric space or a G b-metric space. Each G-metric space is a G b-metric space with s ¼ 1.
In this paper, we extend Darbo’s fixed point theorem via weak JS-contractions in a Banach space. Our results generalize and extend several well-known comparable results in the literature. The technique of measure of non-compactness is the main tool in carrying out our proof. As an application, we study the existence of solutions for a system of integral equations. Finally, we present a concrete example to support the effectiveness of our results.
We establish the existence of fixed point for F-contractive multivalued mappings of Hardy–Rogers type in metric-like spaces. Then, we introduce a new notion called bilateral approximate fixed points for multivalued mappings and establish a common bilateral approximate fixed point result for two [Formula: see text]-dominated multivalued contractive mappings in the sense of [M. Arshad, Z. Kadelburgb, S. Radenovic, A. Shoaibe and S. Shukla, Filomat. 31(11) (2017) 3041–3056] on a closed ball of a K-sequentially complete quasi metric-like space. Finally, one can observe that the common bilateral approximate fixed point results coincide with common fixed point results, whenever we reduce the quasi metric-like spaces to metric-like spaces.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.