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

2021 ◽  
pp. 41-60
Author(s):  
Manabu Naito
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonoscillatory Solutions ◽  
Linear Ordinary Differential Equations

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 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 \(t \to \infty\).

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