On weakly nonlinear contractions
The author generalizes some known fixed point theorems to cone-valued metric spaces. The main theorem reads as follows: Theorem. Let \((X,d)\) be a complete, convex, \(K\)-metric space (where \(K\) denotes a strongly normed cone in a reflexive Banach space \(E\)) and \(S\) a non-empty closed subset of \(X\) such that for each \(x\in S\), there exists an element \(U\) of \(S\) with \(d(x,u)+d(u,Tx)=d(x,Tx)\). Suppose that \(T: S\to X\) satisfies (i) \(d(Tx,Ty)\leq \phi(d(x,y)),\) \(x\neq y\in S\), (ii) \(\phi (t)<t\) for any \(t\in cl\{d(x,y):x,y\in S\}=P_ 1(S),\) where \(\phi:P_ 1(S)\to K\) is upper semicontinuous, (iii) \(Tx\in S\) for every \(x\) of the boundary of \(S\). Then \(T\) has a unique fixed point in \(S\). Some important corollaries of this theorem are shown to be themain results of various authors.
scientific article; zbMATH DE number 3877954
scientific article