Oscillation and nonoscillation for second order neutral dynamic equations with positive and negative coefficients on tim
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RESEARCH
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Oscillation and nonoscillation for second order neutral dynamic equations with positive and negative coefficients on time scales Xun-Huan Deng1 , Qi-Ru Wang1 and Ravi P Agarwal2,3* * Correspondence: [email protected] 2 Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363-8202, USA 3 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article
Abstract We investigate oscillation and nonoscillation of certain second order neutral dynamic equations with positive and negative coefficients. We apply the results from the theory of lower and upper solutions for related dynamic equations along with some additional estimates on positive solutions and use different techniques to obtain some oscillatory theorems. Also, we apply Kranoselskii’s fixed point theorem to obtain nonoscillatory results and then give two sufficient and necessary conditions for the equations to be oscillatory. Some interesting examples are given to illustrate the versatility of our results. Keywords: neutral dynamic equations; positive and negative coefficients; oscillation and nonoscillation; lower and upper solutions; Kranoselskii’s fixed point theorem
1 Introduction In this paper, we investigate oscillation and nonoscillation of second order neutral functional dynamic equations with positive and negative coefficients of the form
p(t) x(t) + r(t)x g(t) + h t, xσ (t), x τ (t) , x τ (t) , x ξ (t) , x ξ (t) = ,
(.)
where t ∈ T, h ∈ C(T × R++ , R), γ (u) = |u|γ – u, γ > , h(t, x , x , x , x , x ) = q (t)γ (x ) + q (t)γ (x ) – q (t)γ (x ) + s (t)γ (x ) – s (t)γ (x ).
(.)
Throughout this paper, we shall assume that T is a time scale satisfying inf T = t and sup T = ∞, and ∞ (B) p ∈ Crd ([t , ∞)T , (, ∞)) satisfies t p(s) s = ∞; (B) there exists a constant r , ≤ r < such that r ∈ Crd (T, [, r ]); (B) g ∈ Crd (T, T), g(t) ≤ t, limt→∞ g(t) = ∞; (B) τ , τ ∈ Crd (T, T) are injective, τ (t) ≤ τ (t) ≤ t, limt→∞ τ (t) = ∞ and Im τ (T) ⊇ Im τ (T); ©2014 Deng et al.; 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.
Deng et al. Advances in Difference Equations 2014, 2014:115 http://www.advancesindifferenceequations.com/content/2014/1/115
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(B) ξ , ξ ∈ Crd (T, T) are injective, ξ (t) ≥ ξ (t) ≥ σ (t), and for sufficiently large T ∈ T, there exists T ∈ T such that ξ (T ) = ξ (T ) and A = {Im ξ (t) : t ≥ T } ⊇ A = {Im ξ (t) : t ≥ T }; (B) q (t), q (t), q (t), s (t), s (t) ∈ Crd (T, R) are eventually positive, and satisfy q τ– τ (t) ≥ q (t), q σ – ξ (t) ≥ s (t),
s ξ– ξ (t) ≥ s (t), q σ – τ
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