On asymptotic relationships between two higher order dynamic equations on time scales

článek v časopise ve Web of Science, Jimp

en

We consider the $n$-th order dynamic equations $x^{\Delta^n}\!+p_1(t)x^{\Delta^{n-1}}+\cdots+p_n(t)x=0$ and $y^{\Delta^n}+p_1(t)y^{\Delta^{n-1}}+\cdots+p_n(t)y=f(t,y(\tau(t)))$ on a time scale $\mathbb{T}$, where $\tau$ is a composition of the forward jump operators, $p_i$ are real rd-continuous functions and $f$ is a continuous function; $\mathbb{T}$ is assumed to be unbounded above. We establish conditions that guarantee asymptotic equivalence between some solutions of these equations. No restriction is placed on whether the solutions are oscillatory or nonoscillatory. Applications to second order Emden-Fowler type dynamic equations and Euler type dynamic equations are shown.

We consider the $n$-th order dynamic equations $x^{\Delta^n}\!+p_1(t)x^{\Delta^{n-1}}+\cdots+p_n(t)x=0$ and $y^{\Delta^n}+p_1(t)y^{\Delta^{n-1}}+\cdots+p_n(t)y=f(t,y(\tau(t)))$ on a time scale $\mathbb{T}$, where $\tau$ is a composition of the forward jump operators, $p_i$ are real rd-continuous functions and $f$ is a continuous function; $\mathbb{T}$ is assumed to be unbounded above. We establish conditions that guarantee asymptotic equivalence between some solutions of these equations. No restriction is placed on whether the solutions are oscillatory or nonoscillatory. Applications to second order Emden-Fowler type dynamic equations and Euler type dynamic equations are shown.

higher order dynamic equation; time scale; asymptotic equivalence

23.04.2017

Elsevier

0893-9659

2017

73

84–90

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