Nonlinear quasi-contractions of Ćirić type (Q2873962)

Let \((X,d)\) be a metric space and let \(f,g:X\to X\) be such that \(f(X)\subset g(X)\). Suppose that there exists a nondecreasing function \(\psi:[0,+\infty)\to[0,+\infty)\) such that \(\psi(0)=0\), \(\lim_{t\to+\infty}(t-\psi(t))=+\infty\), \(\lim_{t\to r^+}\psi(t)<r\) for all \(r>0\) and such that NEWLINE\[NEWLINE\begin{aligned} d(fx,fy)\leq\max\biggl\{\psi(d(gx,gy)),\psi(d(fx,gx)),\psi(d(fy,gy)),\\ \psi(d(gx,fy)),\psi(d(fx,gy))\biggr\}\end{aligned}NEWLINE\]NEWLINE for all \(x,y\in X\). If \(f(X)\) or \(g(X)\) is a complete subspace of \(X\) and \(f\) and \(g\) are weakly compatible, the authors prove that \(f\) and \(g\) have a unique common fixed point. Using the scalarization method of \textit{W.-S. Du} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 5, 2259--2261 (2010; Zbl 1205.54040)], they obtain the corresponding result in cone metric spaces 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)].