  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. 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.
