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A new criterion for the existence of multiple solutions in cones. (English) Zbl 1264.47059
The results of the paper extend the one obtained by H. Persson [Appl. Math. Lett. 19, No. 11, 1207–1209 (2006; Zbl 1179.54068)] on the existence of a non-negative fixed point for monotone maps in finite-dimensional spaces to non-decreasing completely continuous operators between infinite-dimensional ordered Banach spaces. One novelty is that the set of supersolutions is not required to be bounded or bounded away from \(0\). Another is that one requires conditions not on two boundaries, but on a point and a boundary. Moreover, the compactness assumption may be relaxed to a condensing condition. At last, the authors illustrate how the results of the paper can be applied to prove the existence of at least two positive solutions for a particular boundary value problem.

47H10 Fixed-point theorems
34B15 Nonlinear boundary value problems for ordinary differential equations
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
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