Fixed points of \((\psi, \varphi )\)-weak contractions on cone metric spaces
Let \((X,d)\) be a cone metric space in the sense of [\textit{L-G. Huang} and \textit{X. Zhang}, J. Math. Anal. Appl. 332, No.~2, 1468--1476 (2007; Zbl 1118.54022)] over a regular cone \(P\), such that \(d(x,y)\in\text{int}\,P\) for \(x\neq y\). Let \(T:X\to X\) be a mapping satisfying \(\psi(d(Tx,Ty))\preceq\psi(d(x,y))-\phi(d(x,y))\) for all \(x,y\in X\), where \(\psi,\phi:\text{int}\,P\cup\{0\}\to\text{int}\,P\cup\{0\}\) are continuous and increasing, satisfying: (a)~\(\psi(t)=\phi(t)=0\) iff \(t=0\); (b)~\(t-\psi(t)\in \text{int}\,P\cup\{0\}\), \(\phi(t)\in\text{int}\,P\) for \(t\in \text{int}\,P\); (c)~\(\psi(t_1+t_2)\preceq\psi(t_1)+\psi(t_2)\) for \(t_1,t_2\in\text{int}\,P\) and (d)~either \(\psi(t),\phi(t)\preceq d(x,y)\) or \(d(x,y)\preceq\psi(t),\phi(t)\) for \(t\in \text{int}\,P\) and \(x,y\in X\). Under these assumptions, the authors prove that \(T\) has a unique fixed point in~\(X\). A similar result is obtained for the existence of a common fixed point for two self-mappings.
scientific article; zbMATH DE number 5935721