In this paper, we introduce two new types of enriched contractions, viz., enriched A-contraction and enriched 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 Păcurar'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