The omega limit sets of Ray-contractive operators (Q5949583)

In this paper the author deals with omega limit sets of so-called ray-contractive mappings introduced by \textit{P. Takač} [Nonlinear Anal., Theory Methods Appl. 26, No. 11, 1753-1777 (1996; Zbl 0864.47031)]. The main result of the paper states that if \(K\) is a cone in a Banach space, \(\text{Int} K\) is the interior of \(K\), \(T: K\to K\) is a ray-contractive mapping such that \(T(K\setminus\{0\})\subset\text{Int} K\) and there exists a nonempty compact subset \(\Omega\) of \(\text{Int} K\) such that \(T(\Omega)=\Omega\), then \(\omega(x)\) either is a single point, which is a fixed point of \(T\), or forms a \(2\)-cycle of \(T\). In the last part of the paper the author indicates classes of mappings, which appear in the literature and are ray-contractive.