A generalization of Banach’s contraction principle

Abstract

Let T : M → M T:M \to M be a mapping of a metric space ( M , d ) (M,d) into itself. A mapping T T will be called a quasi-contraction iff d ( T x , T y ) ⩽ q max { d ( x , y ) ; d ( x , T x ) ; d ( y , T y ) ; d ( x , T y ) ; d ( y , T x ) } d(Tx,Ty) \leqslant q\max \{ d(x,y);d(x,Tx);d(y,Ty);d(x,Ty);d(y,Tx)\} for some q > 1 q > 1 and all x , y ∈ M x,y \in M . In the present paper the mappings of this kind are investigated. The results presented here show that the condition of quasi-contractivity implies all conclusions of Banach’s contraction principle. Multi-valued quasi-contractions are also discussed.