Oscillation criteria for second-order nonlinear neutral dynamic equations with distributed deviating arguments on time s
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RESEARCH
Open Access
Oscillation criteria for second-order nonlinear neutral dynamic equations with distributed deviating arguments on time scales Tuncay Candan* *
Correspondence: [email protected] Department of Mathematics, ˘ Faculty of Art and Science, Nigde ˘ University, Nigde, 51200, Turkey
Abstract In this article, we establish some new oscillation criteria and give sufficient conditions to ensure that all solutions of nonlinear neutral dynamic equation of the form
(r(t)((y(t) + p(t)y(τ (t))) )γ ) +
b
a
f (t, y(δ (t, ξ )))ξ = 0
are oscillatory on a time scale T, where γ ≥ 1 is a quotient of odd positive integers. Keywords: oscillation; dynamic equations; time scales; distributed deviating arguments
1 Introduction The aim of this article is to develop some oscillation theorems for a second-order nonlinear neutral dynamic equation
γ r(t) y(t) + p(t)y τ (t) +
b
f t, y δ(t, ξ ) ξ =
()
a
on a time scale T. Throughout this paper, it is assumed that γ ≥ is a quotient of odd positive integers, < a < b, τ (t) : T → T, is rd-continuous function such that τ (t) ≤ t and τ (t) → ∞ as t → ∞, δ(t, ξ ) : T × [a, b] → T is rd-continuous function such that decreasing with respect to ξ , δ(t, ξ ) ≤ t for ξ ∈ [a, b], δ(t, ξ ) → ∞ as t → ∞, r(t) > and ≤ p(t) < are real valued rd-continuous functions defined on T, p(t) is increasing and (H )
∞ t
γ ( r(t) ) t = ∞,
(H ) f : T × R → R is a continuous function such that uf (t, u) > for all u = and there exists a positive function q(t) defined on T such that |f (t, u)| ≥ q(t)|uγ |. [ty , ∞] and A nontrivial function y(t) is said to be a solution of () if y(t) + p(t)y(τ (t)) ∈ Crd r(t)((y(t) + p(t)y(τ (t))) )γ ∈ Crd [ty , ∞] for ty ≥ t and y(t) satisfies equation () for ty ≥ t .
A solution of (), which is nontrivial for all large t, is called oscillatory if it has no last zero. Otherwise, a solution is called nonoscillatory. © 2013 Candan; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Candan Advances in Difference Equations 2013, 2013:112 http://www.advancesindifferenceequations.com/content/2013/1/112
Page 2 of 8
We note that if T = R, we have σ (t) = t, μ(t) = , y (t) = y (t) and, therefore, () becomes a second-order neutral differential equation with distributed deviating arguments
γ r(t) y(t) + p(t)y τ (t) +
b
f t, y δ(t, ξ ) dξ = .
a
If T = N, we have σ (t) = t + , μ(t) = , y (t) = y(t) = y(t + ) – y(t) and therefore () becomes a second-order neutral difference equation with distributed deviating arguments b– γ r(t) y(t) + p(t)y τ (t) + f t, y δ(t, ξ ) = ξ =a
and if T = hN, h > , we have σ (t) = t + h, μ(t) = h, y (t) = h y(t) = y(t+h)–y(t) and, thereh fore, () becomes a second-order neu
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