On some enriched contractions in Banach spaces and an application

Abstract

In this paper, we introduce two new types of enriched contractions, viz., enriched $\mathcal{A}$-contraction and enriched $\mathcal{A}'$-contraction. Then we obtain fixed points of mappings satisfying such contractions using the fixed point property of the average operator of the mappings. Further, we study the well-posedness and limit shadowing property of the fixed point problem involving the contractions, and give some examples to validate the results proved. Our work improves Berinde and P\u{a}curar's recent results on different kinds enriched contractions and some well known classical fixed point results. As an application, we prove an existence and uniqueness theorem for an integro-differential equation.