In this paper, we introduce four new types of contractions called in this order Kannan-type cyclic contraction via interpolation, interpolative Ćirić-Reich-Rus type cyclic contraction, and we prove the existence and uniqueness for a fixed point for each situation.

Our main goal of this research is to present the theory of points for relatively cyclic and relatively relatively noncyclic p-contractions in complete locally $\mathbb{K}$ -convex spaces by providing basic conditions to ensure the existence and uniqueness of fixed points and best proximity points of the relatively cyclic and relatively noncyclic p-contractions map in locally $\mathbb{K}$ -convex spaces. The result of this paper is the extension and generalization of the main results of Kirk and A. Abkar.

This paper aimed to obtain common fixed point results by using an interpolative contraction condition given by Karapinar in the setting of complete metric space. Here in this paper, we introduce the notion of interpolative Hardy-Rogers pair-type contraction and prove the corresponding common fixed point theorem by adopting the notion of interpolation. The results are further validated with the application based on them.

Recently, Karapınar introduced a new Hardy-Rogers type contractive mapping using the concept of interpolation and proved a fixed point theorem in complete metric space. This new type of mapping, called "interpolative Hardy-Rogers type contractive mapping" is a generalization of Hardy-Rogers's fixed point theorem. Following this direction of research, in this paper, we will present some fixed point results of Hardy-Rogers-type for cyclic mappings on complete metric spaces. Moreover, an example is given to illustrate the usability of the obtained results.