On some contractive mappings and a new version of implicit function theorem in topological spaces

Abstract

Study on existence of fixed points of contraction and contractive (type) mappings in topological spaces is a challenging task. The main goal of this article is to deal with this challenging task. To achieve our goal, we define two new contractive type mappings, namely, $h$-$\mathcal{A}$-contractive and $h$-$\mathcal{A}_1$-contractive mappings on a topological space $X$, where $h:X\times X\to \mathbb{R}_+$ is a function and $\mathcal{A}$, $\mathcal{A}_1$ are two collections of implicit functions. Then, we obtain some fixed point results concerning such contractive type mappings. Finally, as an application of one of the above mentioned fixed point result, we obtain a newer version of the implicit function theorem on topological spaces.