Abstract existence theorems of positive solutions for nonlinear boundary value problems
The present paper deals with the existence of positive solutions to the operator equation \[ Lu=f(x,u),\qquad u\in D(L), \] where \(\Omega\subset {\mathbb R}^n\) is a bounded domain, \(H=L^2(\Omega),\) \(L: L(D)\subset H\to H\) is an unbounded linear operator and \(f: \overline\Omega\times {\mathbb R}\to {\mathbb R}^+\) is a continuous function. The author proposes conditions on \(L\) and \(f\) which ensure the existence of positive solutions to the equation considered. The discussion is based on the Krasnoselskii fixed point theorem of cone mapping and the fixed point index in cones. The abstract theory is applied to some ordinary differential equations (Sturm-Liouville problem, periodic problems for ODEs), extending and improving known results.
scientific article; zbMATH DE number 2102568