Common Fixed Point Theorems for Weakly Compatible Maps Satisfying Integral Type Contraction in G-Metric Spaces - Kumar, Narinder; Kumar, Manoj; Ashish, (Communications in Mathematics and Applications, 2023)
The main purpose of this manuscript is to prove a common fixed point theorem for two weakly compatible maps satisfying the following integral type contraction in \(G\)-metric space:\[\int^{G(\mathcal{F}x,\mathcal{F}y,\mathcal{F}z)}_0\varphi (t)dt\le \alpha \int^{L(x,y,z)}_0\varphi (t)dt,\] for all \(x,y, z\in X\), where\begin{align*}L(x,y,z)&=\max\{G(gx, gy, gz), G(gx, \mathcal{F}x, \mathcal{F}x), G(gx, \mathcal{F}y, \mathcal{F}y),\\ &\qquad\quad \ \ G(gy, \mathcal{F}y, \mathcal{F}y),  G(gy, \mathcal{F}x, \mathcal{F}x), G(gz, \mathcal{F}z, \mathcal{F}z),\\ &\qquad\quad \ \ G(gz, \mathcal{F}x, \mathcal{F}x), G(gz, \mathcal{F}y, \mathcal{F}y)\}.\end{align*} Also, we have proved common fixed point theorems for the above mentioned weakly compatible self-mappings along with E.A property and (\(\mathit{CLR}_g\)) property.
Keywords: Fixed point; G-metric space; Coincidence point; Weakly compatible maps; E.A property; (CLRg) property