• Bulletin of the Korean Mathematical Society
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• v.41 no.4
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• pp.619-632
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• 2004

Some new oscillation criteria for parabolic neutral delay difference equations corresponding to two sets of boundary conditions are obtained. Our results improve the well known results in the literature.

• Bulletin of the Korean Mathematical Society
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• v.45 no.2
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• pp.299-312
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• 2008

In this paper, we establish some oscillation criteria for nonautonomous second order neutral delay dynamic equations $(x(t){\pm}r(t)x({\tau}(t)))^{{\Delta}{\Delta}}+H(t,\;x(h_1(t)),\;x^{\Delta}(h_2(t)))=0$ on a time scale ${\mathbb{T}}$. Oscillatory behavior of such equations is not studied before. This is a first paper concerning these equations. The results are not only can be applied on neutral differential equations when ${\mathbb{T}}={\mathbb{R}}$, neutral delay difference equations when ${\mathbb{T}}={\mathbb{N}}$ and for neutral delay q-difference equations when ${\mathbb{T}}=q^{\mathbb{N}}$ for q>1, but also improved most previous results. Finally, we give some examples to illustrate our main results. These examples arc [lot discussed before and there is no previous theorems determine the oscillatory behavior of such equations.

• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.125-138
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• 2003

Some Riccati type difference inequalities are established for the second-order nonlinear difference equations with negative neutral term $\Delta$(a(n)$\Delta$(x(n) - px(n-$\tau$))) + f(n, x($\sigma$(n))) = 0 using these inequalities we obtain some oscillation criteria for the above equation.

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.127-144
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• 2005

In this paper, we consider the higher order nonlinear neutral delay difference equation of the form $\Delta^{\gamma}(x_{n}+px_{n-\gamma})+f(n, x_{n-\sigma_1(n)}, x_{n-\sigma_2(n)}, \ldots, x_{n-\sigma{_m}(n)})=0$. We give an integrated classification of nonoscillatory solutions of the above equation according to their asymptotic behaviours. Necessary and sufficient conditions for the existence of nonoscillatory solutions with designated asymptotic properties are also established.

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.173-183
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• 2003

In this paper, we give a classification of nonoscillatory solution of a second-order neutral delay difference equation of the form △²(x/sub n/-c/sub n/x/sub n-r/)=f(n, x/sub g1(n)/, …, x/sub gm(n)/). Some existence results for each kind of nonoscillatory solutions we also established.

• Journal of the Chungcheong Mathematical Society
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• v.12 no.1
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• pp.125-131
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• 1999

We obtain some sufficient conditions for oscillation of the neutral difference equation with positive and negative coefficients $${\Delta}(x_n-cx_{n-m})+px_{n-k}-qx_{n-l}=0$$, where ${\Delta}$ denotes the forward difference operator, m, k, l, are nonnegative integers, and $c{\in}[0,1),p,q{\in}\mathbb{R}^+$.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.355-367
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• 2006

In this paper, we consider the oscillation second order unstable neutral difference equations with continuous arguments $\Delta^2_{/tau}(\chi(t)-p\chi(t-\sigma))=f(t,\chi(g(t)))$ and obtain some criteria for the bounded solutions of this equation to be oscillatory.

• Kyungpook Mathematical Journal
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• v.47 no.2
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• pp.175-190
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• 2007

By employing the Riccati transformation technique some new oscillation criteria for the second-order neutral delay dynamic equation $$(y(t)+r(t)y({\tau}(t)))^{{\Delta}{\Delta}}+p(t)y(\delta(t))=0$$, on a time scale $\mathbb{T}$ are established. Our results as a special case when $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{N}$ improve some well known oscillation criteria for second order neutral delay differential and difference equations, and when $\mathbb{T}=q^{\mathbb{N}}$, i.e., for second-order $q$-neutral difference equations our results are essentially new and can be applied on different types of time scales. Some examples are considered to illustrate the main results.

• The Pure and Applied Mathematics
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• v.14 no.2 s.36
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• pp.49-62
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• 2007

In this paper, the oscillation and existence of nonoscillatory solutions of odd order difference equations with nonlinear neutral terms are studied respectively. Some new criteria are established. Furthermore, some examples are given to illustrate the advance of our results.

• Journal of applied mathematics & informatics
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• v.8 no.1
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• pp.111-128
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• 2001

In this paper, we consider the oscillation of the second-order neutral difference equation Δ²(x/sub n/ - px/sub n-r/) + q/sub n/f(x/sub n/ - σ/sub n/) = 0 as well as the oscillatory behavior of the corresponding ordinary difference equation Δ²z/sub n/ + q/sub n/f(R(n,λ)z/sub n/) = 0