Abstract
For the equation the existence of positive solutions with non-power asymptotic behavior is proved, namely where is an arbitrary point, h is a positive periodic non-constant function on R. To prove this result, the Hopf bifurcation theorem is used.
Highlights
For the equation y(n) = p x, y, y, . . . , y(n– ) |y|k sgn y, n ≥, k >, ( )Kiguradze posed the problem on the asymptotic behavior of its positive solutions such that lim y(x) = ∞. x→x∗ –with n α(α + ) · · · (α + n – ) k– α=, C= k–p p = const > - is a limit of p(x, y, . . . , yn– ) as x → x∗, y → ∞, . . . , yn– → ∞
The hypothesis of Kiguradze was confirmed in this case
Proof To apply the Hopf bifurcation theorem, we investigate equation ( ) with G(t, V ) ≡ corresponding to the case of the constant function p and the roots of the algebraic equation ( )
Summary
A negative answer to the conjecture of Kiguradze for large n was obtained It was proved [ ] that for any N and K > , there exist an integer n > N and k ∈ R, < k < K , such that equation ( ) has a solution y = x∗ – x –αh log x∗ – x , where α is defined by ( ), h is a positive periodic non-constant function on R. Still, it was not clear how large n should be for the existence of that type of solutions
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