Existence results for operator equations in partially ordered sets and applications
In this paper, the author improves a previous result from [\textit{S. Heikkilä}, J. Math. Anal. Appl. 292, 262-273 (2004; Zbl 1072.47057)] regarding the solution existence of abstract equations of the form \(Lu=Nu\), where \(L,N\) are operators between partially ordered sets satisfying some conditions. In the author's previous work [loc. cit.] \(L\) is a bijection with \(L^{-1}\) increasing, while here \(L\) is not necessarily injective. The author proves the main result (Theorem 2.1) by reducing it to the case from [op. cit.]. As the author explains in the next sections, the result in abstract form covers a large class of practical problems, for example, operator equations in ordered Banach spaces, discontinuous functional parabolic problems, or elliptic boundary value problems. Moreover, these nice examples show that the result is applicable in some cases when standard theories such as the theory of monotone operators, or the method of upper and lower solutions cannot be used under the hypotheses given in Theorem 2.1.
scientific article; zbMATH DE number 2137245