Recall in the Exercises of Chapter 1 (Ampadu [1]), we introduced the higher-order Reich contraction. Alternatively, let  be a metric space, one says  is a higher-order Reich contraction, if the following hold for all x,  and any

for all  and  If  in the above definition, then we obtain the notion of Reich contraction (Reich [2]). On the other hand if  then we obtain the notion of convex type 2 contraction (Istratescu [3]). In this paper, if  then in the above definition, we say Tis a convex Reich contraction mapping of order 2 instead of convex type 2 contractions as originally coined in Istratescu [3]. This paper investigates fixed point theory for the convex Reich contraction mapping of order 2 in various settings.

fixed point theory, Reich contraction, convexity condition.