Common Fixed Point Theorems for Weakly Compatible Maps Satisfying Integral Type Contraction in G-Metric Spaces

Abstract

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.