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Abstract existence theorems of positive solutions for nonlinear boundary value problems. (English) Zbl 1064.47058

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.

MSC:

47J05 Equations involving nonlinear operators (general)
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
34L30 Nonlinear ordinary differential operators
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