A variant of the Meir-Keeler-type theorem in ordered Banach spaces
The author shows a fixed point theorem in ordered Banach spaces. Let \(B\) be a real Banach space partially ordered by a cone \(P\), i.e., \(x\leq y\) iff \(y-x\in P\). In this paper it is assumed that the norm is monotone, i.e., \(0\leq x\leq y\) implies that \(\|x\|\leq\|y\|\). A normal cone \(P\) will guarantee that \(B\) has an equivalent norm that is monotone. \(\mathring P\) stands for the interior of \(P\). Let \(D\subset B\). An operator \(A: D\to B\) is said to be increasing if \(Ax\leq Ay\) for \(x\leq y\). \(x,y\in P-\{0\}\) are said to be comparable if there exist positive numbers \(\lambda\) and \(\mu\) such that \(\mu y\leq x\leq\lambda y\). This equivalent relation divides \(P\setminus\{0\}\) into disjoint components of \(P\). If \(\mathring P\neq\varnothing\), then it is a component of \(P\). In this section, \(C\) will always stand for a component of \(P\). For \(x,y\in C\), if \(y- x\in C\), then we denote \(x\ll y\). Since \(\mathring P\) is a component of \(P\), \(x\ll y\) means \(y-x\in\mathring P\) if \(x,y\in\mathring P\). The author proves the following theorem: Let \(A: C\to C\) be increasing and satisfy: there exists \(\varepsilon\in (0,1)\) such that for any \(\alpha\in (\varepsilon, 1)\), we have \(\beta\in (\varepsilon, \alpha)\) so that \[ t\in (\beta,\alpha]\quad\text{implies}\quad A(tx)\gg\alpha Ax. \] Then there exists a unique fixed-point \(x^*\) of \(A\) with \(\lim_{n\to\infty} A^nx= x^*\) for any \(x\in C\).
scientific article; zbMATH DE number 1340189
scientific article