On the asymptotic behavior of solutions of x ′′+ a ( t ) f ( x ) = e ( t )
1972; Mathematical Sciences Publishers; Volume: 41; Issue: 1 Linguagem: Inglês
10.2140/pjm.1972.41.43
ISSN1945-5844
AutoresTheodore Burton, Ronald Grimmer,
Tópico(s)Nonlinear Differential Equations Analysis
ResumoAssume first that (iii) holds.Then, for s ^ s 0 ^ 0, Fas a function of s along a solution of (6) satisfiesNow, if V(s) is eventually bounded away from zero, say V(s) ^ k > for s ^ s 0 , we haveand by Gronwall's inequality, S s ΓsX^u) du ^ V(s 0 ) exp 1 (X 2 (u)/k) du .From (i), (ii), and ( 4), however, it follows that X 2 (u) du < oo and I λ x (u) du --ĵ and so it must be that V(s) tends to zero ass->oo, This contradiction implies that there must be a sequence {s n } tending to infinity with n with the property that V(s n ) -• 0 as ^ -> oo.Now, for s ^ s n , we have, F(s) ^ F(s n ) exp 1 \(u) du + 1 X 2 (u) exp \ λ^t?) di; du
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