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On positive solutions of boundary value problems on the half-line. (English) Zbl 1003.34024
The boundary value problem $x''(t)- k^2 x(t)+ f(t,x(t), x'(t))= 0,\quad x(0)= 0,\quad \lim_{t\to\infty} x(t)= 0,$ is studied. The author considers separately the case when $$f$$ is independent of $$x'$$. It is assumed that the function $$f\geq 0$$ is continuous and $$k> 0$$. Sufficient conditions are obtained for the existence of at least one positive solution in an appropriate functional space equipped with Bielecki’s norm. The proofs are based on the fixed-point theorem in cones of Krasnoselskij.

MSC:
 34B40 Boundary value problems on infinite intervals for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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References:
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