Some Fixed Point Results for Integral Type Mappings in b-Metric Space
Preeti
Department of Applied Sciences and Humanities, PIET, Samalkha, Haryana, 132102, India.
Poonam *
Department of Mathematics, Govt. College for Women, Jassaur Kheri, Jhajjar, India.
*Author to whom correspondence should be addressed.
Abstract
This article explores the fixed point results by generalizing the result proved by (Singh 2016) for Lebesgue integrable mapping satisfying b-(E.A.) property in b- metric spaces. In this manuscript, we establish standard fixed point theorem for the following generalized Lebesgue integrable mapping satisfying b-(E.A.) property in b- metric spaces:
$$
\int_0^{b^{\varepsilon} d(f x, g y)} \varphi(t) d t \leq \int_0^{M_b(x, y)} \varphi(t) d t \text { for all } x, y \in \dot{X} \mid
$$
Where,
$$
M_b(x, y)=\left\{\mathrm{d}(S x, T y), \mathrm{d}(f x, S x), \mathrm{d}(g y, T y), \frac{\mathrm{d}(S x, g y)+\mathrm{d}(f x, T y)}{2 b},\right.
$$
$$
\begin{aligned}
& \mathrm{d}(S x, f x)\left[\frac{1+\mathrm{d}(S x, T y)}{1+\mathrm{d}(T y, f y)}\right], \mathrm{d}(T y, g y)\left[\frac{1+\mathrm{d}(S x, T y)}{1+\mathrm{d}(S x, f x)}\right] \text {, } \\
& \left.\frac{\mathrm{d}^2(S x, f x)}{1+\mathrm{d}(f x, g y)}, \frac{\mathrm{d}^2(T y, g y)}{1+\mathrm{d}(f x, g y)}\right\} .
\end{aligned}
$$
Some examples and corollaries are provided to support our result.
Keywords: Fixed point, generalized b-(E.A) property, weakly compatible mapping, b-metric Space