On Coupled Fixed Point Theorems for Nonlinear Contractions in Partially Ordered G‐Metric Spaces

Abstract

Two concepts—one of the coupled fixed point and the other of the generalized metric space—play a very active role in recent research on the fixed point theory. The definition of coupled fixed point was introduced by Bhaskar and Lakshmikantham (2006) while the generalized metric space was introduced by Mustafa and Sims (2006). In this work, we determine some coupled fixed point theorems for mixed monotone mapping satisfying nonlinear contraction in the framework of generalized metric space endowed with partial order. We also prove the uniqueness of the coupled fixed point for such mappings in this setup.

Coupled common fixed point results under new nonlinear contractions in ordered G-metric spaces

“…The mappings considered are commutative. Present study enrich as well as generalize the very recent works present in [15,21,22,23,24]. The work is furnished with suitable illustrations.…”

A Unique Coupled Common Fixed Point Theorem for Symmetric(φ,ψ)-Contractive Mappings in OrderedG-Metric Spaces with Applications

“…Various authors extended and generalized the results of Choudhury and Maity [10] under different contractive conditions in G-metric spaces. For more works, one can see [26][27][28][29][30][31][32][33][34]. Nashine [31] generalized and extended the contractive condition (4) and thereby obtained the coupled coincidence points for a pair of commuting mappings under the following contraction:…”

“…where 𝑘 ∈ [0, 1/2). Mohiuddine and Alotaibi [33] further generalized the contraction (4) by considering the following more general contractive condition:…”

“…Journal of Applied Mathematics 3 where 𝜑, 𝜓 : [0, ∞) → [0, ∞) are functions satisfying some appropriate conditions mentioned in [33]. Interestingly, for 𝜑(𝑡) = 𝑡, 𝜓(𝑡) = ((1 − 𝑘)/2)𝑡, with 0 ≤ 𝑘 < 1, condition (6) reduces to (4).…”

n-Tuplet Coincidence Point Theorems in Intuitionistic Fuzzy Normed Spaces

“…Later on, Lakshmikantham andĆirić [4] introduced the concept of coincidence point which is a generalization of fixed point. By inspiring these works, coupled fixed point theorems have been studied for different type contraction mappings (see [5][6][7][8][9][10][11][12]). The interest on coupled fixed point theorem has motivated the authors to generalize it as tripled fixed point theorem in [13,14], afterwards as quadruple fixed point theorem in [15][16][17], and as -tuplet fixed point theorem in [18][19][20][21].…”