In this paper, we obtain the existence and uniqueness of the solution for three self mappings in a complete bipolar metric space under a new Caristi type contraction with an example. We also provide applications to homotopy theory and nonlinear integral equations. MSC: 54H25; 47H10; 54E50 Keywords: Bipolar metric space; Covariant and contravariant map; Compatible mapping and common fixed point 2 Methods/experimental Definition 2.1 ([1]) Let A and B be a two non-empty sets. Suppose that d : A × B → [0, ∞) is a mapping satisfying the following properties: (B 1 ) d(a, b) = 0 if and only if a = b for all (a, b) ∈ A × B, © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

In this paper, we obtain a unique coupled common fixed point theorem for four maps in partial metric spaces. RESUMEN En este artículo obtenemos un teorema del punto fijo clásico acopladoúnico para cuatro aplicaciones en espacios métricos parciales.

In this paper, we introduce the concept of multivalued contraction mappings in partially ordered bipolar metric spaces and establish the existence of unique coupled fixed point results for multivalued contractive mapping by using mixed monotone property in partially ordered bipolar metric spaces. Some interesting consequences of our results are obtained.

In this paper we give some applications to integral equations as well as homotopy theory via fixed point theorems in partially ordered complete S b -metric spaces by using generalized contractive conditions. We also furnish an example which supports our main result.

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