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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 '' +g(t,y,y ' )=0fora.e.0<t<+,y(0)=0,yboundedon[0,),

where g satisfies the Carathéodory conditions. Two more conditions are assumed:

(A1): For any H>0, there exists a function ψ H C([0,),[0,)) which does not vanish identically on any subinterval of (0,), and a constant γ[0,1) with g(t,u,v)ψ H (t)v γ ,on [0,)×[0,H] 2 ·

(A2): There exist functions p,q,r:[0,)[0,) such that

Q= 0 q(s)ds<+,Q 1 = 0 sq(s)ds<+,P 1 = 0 sq(s)ds<+,
R= 0 r(s)ds<+,R 1 = 0 sr(s)ds<+,

and |g(t,u,v)|p(t)|u|+q(t)|v|+r(t) for a.e. t[0,) and all (u,v)[0,) 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:
34B18Positive solutions of nonlinear boundary value problems for ODE
34B40Boundary value problems for ODE on infinite intervals