Abstract

This manuscript consists a common fixed point result for four weakly compatible self-maps $\hat{P},\hat{Q}, \hat{S},\hat{T}$ on a metric space $(M, d^*)$ satisfying the following contractive inequality of integral type:

$$\int_{0}^{d^*(\hat{T}\mu, \hat{S}\nu)}\xi(t) dt\leq\beta(d^*(\mu, \nu)) \int_{0}^{\Delta_1(\mu, \nu)}\xi(t) dt,$$

where ($\xi, \beta$) $\in$ $\xi_1\times\xi_3$ and for all $\mu$, $\nu$ in $M$.

\begin{equation*}

\begin{split}

\Delta_1(\mu, \nu) &= \textit{max}\lbrace d^*(\hat{T}\mu, \hat{S}\nu),d^*(\hat{T}\mu, \hat{P}\mu),d^*(\hat{S}\nu, \hat{Q}\nu),\\

&\quad \frac{1}{2}[d^*(\hat{P}\mu, \hat{S}\nu)+d^*(\hat{Q}\nu, \hat{T}\mu)],

\frac{d^*(\hat{P}\mu, \hat{T}\mu).d^*(\hat{Q}\nu, \hat{S}\nu)}{1+d^*(\hat{T}\mu, \hat{S}\nu)},\\

&\quad \frac{d^*(\hat{P}\mu, \hat{S}\nu).d^*(\hat{Q}\nu, \hat{T}\mu)}{1+d^*(\hat{T}\mu, \hat{S}\nu)},

d^*(\hat{T}\mu, \hat{P}\mu)[\frac{1+d^*(\hat{T}\mu, \hat{Q}\nu)+d^*(\hat{S}\nu, \hat{P}\mu)}{1+d^*(\hat{T}\mu, \hat{P}\mu)+d^*(\hat{S}\nu, \hat{Q}\nu)}]\rbrace.

\end{split}

\end{equation*}

Also, some common fixed point results for the above mentioned weakly compatible self - maps along with E.A. property and (CLR) property are proved. A suitable illustrative example is also provided to support our result.

Keywords and Phrases

Fixed Point, Coincidence Point, Weakly Compatible Maps, E.A. Property, (CLR) Property.

A.M.S. subject classification

47H10, 54H25.

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