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.

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