Fixed-point results for multi-valued operators in quasi-ordered metric spaces (Q712618)

Let \((X,d,\leq)\) be a sequentially complete quasi-ordered metric space, and \(M:X\to {\mathcal N}(X)\) be a multivalued map fulfilling: \({\mathcal H}_d(Mx,My)\leq\) \(\alpha\) \([D(x,My)+D(y,Mx)]-\varphi(D(x,My),D(y,Mx))\), for all comparable \(x,y\in X\), where \(\alpha> 0\) is a constant and \(\varphi:\mathbb R_+^2\to \mathbb R_+\) is a continuous function. Sufficient conditions are given upon \((X,d,\leq)\), \(\alpha\) and \(\varphi\) such that i) \(M\) has a fixed point \(x^*\in X\), ii) for each \(x_0\in X\), some iterated sequence \((x_n)\) with \((x_n\in Mx_{n-1}, n\geq 1)\) converges to \(x^*\). Reviewer's remark: The Ran-Reurings and Nieto-Lopez fixed point results in ordered metric spaces were obtained two decades ago by the reviewer, in [J. Math. Anal. Appl. 117, 100--127 (1986; Zbl 0613.47037)] and [Demonstr. Math. 19, 171--180 (1986; Zbl 0651.54020)].