Oscillation of Second-Order Sublinear Dynamic Equations with Damping on Isolated Time Scales
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Research Article Oscillation of Second-Order Sublinear Dynamic Equations with Damping on Isolated Time Scales Quanwen Lin1, 2 and Baoguo Jia1, 2 1 2
Department of Mathematics, Maoming University, Maoming 525000, China School of Mathematics and Computer Science, Zhongshan University, Guangzhou 510275, China
Correspondence should be addressed to Quanwen Lin, [email protected] Received 8 October 2010; Accepted 27 December 2010 Academic Editor: M. Cecchi Copyright q 2010 Q. Lin and B. Jia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper concerns the oscillation of solutions to the second sublinear dynamic equation with σ damping xΔΔ t qtxΔ t ptxα σt 0, on an isolated time scale Ì which is unbounded above. In 0 < α < 1, α is the quotient of odd positive integers. As an application, we get the difference equation Δ2 xn n−γ Δxn 1 1/nln nβ b−1n /ln nβ xα n 1 0, where γ > 0, β > 0, and b is any real number, is oscillatory.
1. Introduction During the past years, there has been an increasing interest in studying the oscillation of solution of second-order damped dynamic equations on time scale which attempts to harmonize the oscillation theory for continuousness and discreteness, to include them in one comprehensive theory, and to eliminate obscurity from both. We refer the readers to the papers 1–4 and the references cited therein. In 5 , Bohner et al. consider the second-order nonlinear dynamic equation with damping σ xΔΔ t qtxΔ t pt f ◦ xσ t 0,
1.1
where p and q are real-valued, right-dense continuous functions on a time scale ⊂ , with sup ∞. f : → is continuously differentiable and satisfies f x > 0 and xfx > 0 for
2
Advances in Difference Equations
x/ 0. When fx xα , where 0 < α < 1, α > 0 is the quotient of odd positive integers, 1.1 is the second-order sublinear dynamic equation with damping xΔΔ t qtxΔ t ptxα σt 0. σ
1.2
When qt 0, 1.2 is the second-order sublinear dynamic equation xΔΔ t ptxα σt 0. When
1.3
0 , 1.3 is the second-order sublinear difference equation 1.4
Δ2 xn ptxα n 1 0.
In 6 , under the assumption of being ∞ an isolated time scale, we prove that, when pt is allowed to take on negative values, t1 tα ptΔt ∞ is sufficient for the oscillation of the dynamic equation 1.3. As an application, we get that, when pn is allowed to take on α negative values, ∞ n1 n pn ∞ is sufficient for the oscillation of the dynamic equation 1.4, which improves a result of Hooker and Patula 7, Theorem 4.1 and Mingarelli 8 . In this paper, we extend the result of 6 to dynamic equation 1.1. As an application, we get that the difference equation with damping −γ
Δ xn n Δxn 1 2
1 nln nβ
b
−1n ln nβ
xα n 1 0,
1.5
where 0 < α < 1, γ > 0, β > 0,
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