Lipschitz mapping appears inevitably in many branches of mathematics, especially in functional analysis, and leads to the study of new results in metric fixed point theory. Goebel and Sims (resp. Goebel and Japon-Pineda) introduced a class of the Lipschitz mappings termed as (Formula presented.) -Liptschitz mappings and studied not only the modified form of the Lipschitz condition, but also the behavior of a finite number of their iterates. The purpose of this paper is to discuss the various types of (Formula presented.) -contractions with cyclic representation that extend the results due to Banach, Kannan, and Chatterjea. Moreover, based on such types of contractions and the property of symmetry, we obtain some related fixed-point results in the setting of metric spaces. Some examples are studied to illustrate the validity of our obtained results. As an application of our results, we establish the existence of the solution to a class of Fredholm integral equations.
- (α,p)-Chatterjea-type contraction
- (α,p)-Kannan-type contraction
- (α,p)-cyclic contraction
- fixed point