A generalization of the Banach contraction principle with high order of convergence of successive approximations

In this paper, we present the following generalization of the Banach contraction principle. Let T:D⊂X→XT:D⊂X→X be a continuous operator on a complete metric space (X,d)(X,d) satisfying d(Tx,T2x)≤φ(d(x,Tx))for all x∈D,Tx∈D with d(x,Tx)∈J, where J⊂R+J⊂R+ is an interval containing 0, φ:J→Jφ:J→J is a gauge function of order r≥1r≥1 on JJ in the sense that φ(0)=0,ϕ(t)/trφ(0)=0,ϕ(t)/tr is nondecreasing on J∖{0}J∖{0} and ϕ(t)