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.


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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