On some enriched contractions in Banach spaces

In this paper, we introduce two new types of enriched contractions, viz., enriched $\mathcal{A}$-contraction and enriched ${\mathcal{A}}^{\mathrm{\prime }}$-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. We frame an open question related to the existence of a fixed point of such contractions. We also show that Berinde and P\u{a}curar’s recent results on different kinds enriched contractions and some well known classical fixed point results are particular cases of our results.