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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.

MSC:
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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