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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.

##### MSC:
 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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