Asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations

2021 ◽  
Author(s):  
Manabu Naito
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonoscillatory Solutions ◽  
Linear Ordinary Differential Equations

scholarly journals Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, I

2021 ◽  
Vol 41 (1) ◽  
pp. 71-94
Author(s):  
Manabu Naito
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Ordinary Differential Equations ◽  
Nonoscillatory Solutions ◽  
Linear Ordinary Differential Equations ◽  
Linear Differential

We consider the half-linear differential equation of the form \[(p(t)|x'|^{\alpha}\mathrm{sgn} x')' + q(t)|x|^{\alpha}\mathrm{sgn} x = 0, \quad t\geq t_{0},\] under the assumption \(\int_{t_{0}}^{\infty}p(s)^{-1/\alpha}ds =\infty\). It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as \(t \to \infty\).

Oscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations

1981 ◽  
Vol 90 (1-2) ◽  
pp. 25-40 ◽  
Author(s):  
Takaŝi Kusano ◽  
Manabu Naito ◽  
Kyoko Tanaka
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Ordinary Differential Equations ◽  
Sufficient Conditions ◽  
Solution Space ◽  
Extreme Situation ◽  
Nonoscillatory Solutions ◽  
Linear Ordinary Differential Equations ◽  
Image Position ◽  
Monotone Solutions

SynopsisThe equation to be considered iswhere pi(t), 0≦i≦n, and q(t) are continuous and positive on some half-line [a, ∞). It is known that (*) always has “strictly monotone” nonoscillatory solutions defined on [a, ∞), so that of particular interest is the extreme situation in which such strictly monotone solutions are the only possible nonoscillatory solutions of (*). In this paper sufficient conditions are given for this situation to hold for (*). The structure of the solution space of (*) is also studied.

scholarly journals The distribution of nonprincipal eigenvalues of singular second-order linear ordinary differential equations

2006 ◽  
Vol 2006 ◽  
pp. 1-7 ◽  
Author(s):  
Juan Pablo Pinasco
Keyword(s):  
Asymptotic Behavior ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Asymptotic Distribution ◽  
Counting Function ◽  
Infinite Interval ◽  
Singular Problem ◽  
Order Linear ◽  
Linear Ordinary Differential Equations

We obtain the asymptotic distribution of the nonprincipal eigenvalues associated with the singular problemx″+λq(t)x=0on an infinite interval[a,+∞). Similar to the regular eigenvalue problem on compact intervals, we can prove a Weyl-type expansion of the eigenvalue counting function, and we derive the asymptotic behavior of the eigenvalues.

Asymptotic behavior of the curves in the Fučík spectrum

2016 ◽  
Vol 19 (04) ◽  
pp. 1650039 ◽  
Author(s):  
Juan Pablo Pinasco ◽  
Ariel Martin Salort
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hyperbolic Type ◽  
Type Curves ◽  
Order Linear ◽  
Linear Ordinary Differential Equations ◽  
Fučík Spectrum ◽  
Fučik Spectrum ◽  
Fucik Spectrum

In this work, we study the asymptotic behavior of the curves of the Fučík spectrum for weighted second-order linear ordinary differential equations. We prove a Weyl type asymptotic behavior of the hyperbolic type curves in the spectrum in terms of some integrals of the weights. We present an algorithm which computes the intersection of the Fučík spectrum with rays through the origin, and we compare their values with the asymptotic ones.

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