Approximating fixed points of weak enriched contractions using Kirk’s iteration scheme of higher order

Abstract

Abstract In this paper, we introduce two types of weak enriched contractions, namely weak enriched F $\mathcal{F}$ -contraction, weak enriched F ′ $\mathcal{F^{\prime}}$ -contraction, and k-fold averaged mapping based on Kirk’s iterative algorithm of order k. The types of contractions introduced herein unify, extend, and generalize several existing classes of enriched and weak enriched contraction mappings. Moreover, K-fold averaged mappings can be viewed as a generalization of the averaged mappings and double averaged mappings. We then prove the existence of a unique fixed point of the k-fold averaged mapping associated with weak enriched contractions introduced herein. We study necessary conditions that guarantee the equality of the sets of fixed points of the k-fold averaged mapping and weak enriched contractions. We show that an appropriate Kirk’s iterative algorithm can be used to approximate a fixed point of a k-fold averaged mapping and of the two weak enriched contractions. We also study the well-posedness, limit shadowing property, and Ulam–Hyers stability of the k-fold averaged mapping. We provide necessary conditions that ensure the periodic point property of each illustrated weak enriched contraction. Some examples are presented to show that our results are a potential generalization of the comparable results in the existing literature.

Keywords

• Fixed point
• Weak enriched F $\mathcal{F}$ -contraction
• Weak enriched F ′ $\mathcal{F^{\prime}}$ -contraction
• Kirk’s iteration
• Well-posedness
• Limit shadowing property