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

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 View the MathML sourced(Tx,T2x)≤φ(d(x,Tx))for all x∈D,Tx∈D with d(x,Tx)∈J, Turn MathJax on 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)