Existence of positive solutions for second-order boundary value problems on infinity intervals. (English) Zbl 1046.34045

The author considers the second-order boundary value problem \[ y^{\prime \prime} +g(t,y,y^{\prime })=0\text{ for a.e. }0<t<+\infty,\; y(0)=0,\; y \text{ bounded on }[0,\infty), \] where \(g\) satisfies the Carathéodory conditions. Two more conditions are assumed:
(A1): For any \(H>0,\) there exists a function \(\psi _{H}\in C([ 0,\infty ) ,[ 0,\infty ) )\) which does not vanish identically on any subinterval of \(( 0,\infty ) ,\) and a constant \(\gamma \in [ 0,1) \) with \(g( t,u,v) \geq \psi _{H}( t) v^{\gamma },\;\)on \([ 0,\infty ) \times [ 0,H] ^{2}.\)
(A2): There exist functions \(p,\;q,\;r:[0,\infty) \to [0,\infty)\) such that \[ Q=\int_{0}^{\infty }q(s)\,ds<+\infty ,\;Q_{1}=\int_{0}^{\infty }sq(s)\,ds<+\infty ,\;P_{1}=\int_{0}^{\infty }sq(s)\,ds<+\infty , \]
\[ \;R=\int_{0}^{\infty }r(s)\,ds<+\infty ,\;R_{1}=\int_{0}^{\infty }sr(s)\,ds<+\infty ,\; \] and \(| g( t,u,v) | \leq p(t)| u| +q(t)| v| +r(t)\) for a.e. \(t\in [0,\infty)\) and all \(( u,v) \in [ 0,\infty ) ^{2}.\) The main result asserts that the problem above has at least one solution provided \(P_{1}+Q<1.\)
The Leray-Schauder continuation theorem is used for establishing the result. However, the method of obtaining the needed a priori estimates is different from that of either the pionneer related works [R. P. Agarwal and D. O’Regan, Tohoku Math. J., II. Ser. 51, No. 3, 391–397 (1999; Zbl 0942.34026)] or [A. Constantin, Ann. Mat. Pura Appl., IV. Ser. 176, 379–394 (1999; Zbl 0969.34024)]. The author gives an interesting example to which the main result of his article applies but not those of Agarwal and O’Regan nor Constantin cited above.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
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[1] Gupta, C.P.; Trofimchuk, S.I., A sharper condition for the solvability of a three-point second order boundary value problem, J. math. anal. appl., 205, 586-597, (1997) · Zbl 0874.34014
[2] Agarwal, R.P.; O’Regan, D., Boundary value problems of nonsingular type on the semi-infinity interval, Tohoku math. J., 51, 391-397, (1999) · Zbl 0942.34026
[3] Constantin, A., On an infinite interval boundary value problem, Annali di Mathematica ed applicata (IV), CLXXVI, 379-394, (1999) · Zbl 0969.34024
[4] Mawhin, J., Topological degree methods in nonlinear boundary value problems, () · Zbl 0414.34025
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