Fixed point theory with partially ordered metric spaces and homotopy

The importance of fixed point theory and for that matter metric spaces have significantly manifested in sciences and in many branches of mathematical analysis over the years. The paper presents fixed point theory on a complete metric space with contractive mappings. Provision of some fixed point results of contraction mappings on a partially metric spaces with example. However, generalization of findings using the contraction mappings on partially metric spaces with application to Homotopy theory is provided.