Abstract
In this paper, we prove that coupled and tripled coincidence point theorems under $$(F, g)$$ -invariant sets for weakly contractive mappings defined on a $$G$$ -metric space are immediate consequences of corresponding results via rectangular $$G$$ - $$\alpha $$ -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.
Highlights
Introduction and mathematical preliminariesThe 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: XÂXÂX ! Rþ be a function satisfying the following properties:(G1) Gðx; y; zÞ 1⁄4 0 iff x 1⁄4 y 1⁄4 z; (G2) 0\Gðx; x; yÞ, for all x; y 2 X with x 61⁄4 y; (G3) Gðx; x; yÞ Gðx; y; zÞ, for all x; y; z 2 X with y 61⁄4 z; (G4) Gðx; y; zÞ 1⁄4 Gðx; z; yÞ 1⁄4 Gðy; z; xÞ 1⁄4 . . ., (symmetry in all three variables); (G5) Gðx; y; zÞ Gðx; a; aÞ þ Gða; y; zÞ, for all x; y; z; a 2X (rectangle inequality).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 (Gb-metric spaces) and they presented some basic properties of Gb-metric spaces.The following is their definition of Gb-metric spaces.Definition 1.2 [1] Let X be a nonempty set and s ! 1 be a given real number
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
Hold whenever fxng and fyng are sequences in X such that limn!1 f ðxn; ynÞ 1⁄4 limn!1 gxn and limn!1 f ðyn; xnÞ 1⁄4 limn!1 gyn: On the other hand, Berinde and Borcut [24] introduced the concept of tripled fixed point and obtained some tripled fixed point theorems for contractive type mappings in partially ordered metric spaces
Summary
Introduction and mathematical preliminariesThe 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: XÂXÂX ! Rþ be a function satisfying the following properties:(G1) Gðx; y; zÞ 1⁄4 0 iff x 1⁄4 y 1⁄4 z; (G2) 0\Gðx; x; yÞ, for all x; y 2 X with x 61⁄4 y; (G3) Gðx; x; yÞ Gðx; y; zÞ, for all x; y; z 2 X with y 61⁄4 z; (G4) Gðx; y; zÞ 1⁄4 Gðx; z; yÞ 1⁄4 Gðy; z; xÞ 1⁄4 . . ., (symmetry in all three variables); (G5) Gðx; y; zÞ Gðx; a; aÞ þ Gða; y; zÞ, for all x; y; z; a 2X (rectangle inequality).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 (Gb-metric spaces) and they presented some basic properties of Gb-metric spaces.The following is their definition of Gb-metric spaces.Definition 1.2 [1] Let X be a nonempty set and s ! 1 be a given real number. Choudhury and maity [6] have established some coupled fixed point results for mappings with mixed monotone property in partially ordered G-metric spaces.
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