• 대한수학회보
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• 제40권3호
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• pp.489-501
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• 2003

In this paper we shall consider the nonlinear delay difference equation $\Delta({p_n}{\Deltax_N})\;+\;q_nf(x_{n-\sigma})\;=\;0,\;n\;=\;0,\;1,\;2,\;...$ when (equation omitted). We will establish some sufficient conditions which guarantee that every solution is oscillatory or converges to zero.

• 대한수학회보
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• 제51권1호
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• pp.55-66
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• 2014

In this paper, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive p-Laplace equation $u_t=div({\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)+a{\int}_{\Omega}u^q(y,t)dy$, 1 < p < 2, in a bounded domain ${\Omega}{\subset}R^N$ with $N{\geq}1$. More precisely, it is shown that if q > p-1, any solution vanishes in finite time when the initial datum or the coefficient a or the Lebesgue measure of the domain is small, and if 0 < q < p-1, there exists a solution which is positive in ${\Omega}$ for all t > 0. For the critical case q = p-1, whether the solutions vanish in finite time or not depends crucially on the value of $a{\mu}$, where ${\mu}{\int}_{\Omega}{\phi}^{p-1}(x)dx$ and ${\phi}$ is the unique positive solution of the elliptic problem -div(${\mid}{\nabla}{\phi}{\mid}^{p-2}{\nabla}{\phi}$) = 1, $x{\in}{\Omega}$; ${\phi}(x)$=0, $x{\in}{\partial}{\Omega}$. This is a main difference between equations with local and nonlocal sources.

• Journal of applied mathematics & informatics
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• 제16권1_2호
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• pp.79-90
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• 2004

In this paper the authors present sufficient conditions for all bounded solutions of the second order neutral difference equation ${\Delta}^2(y_n\;-\;py_{n-{\kappa}})\;-\;q_nf(y_{n-e})\;=\;0,\;n\;{\in}\;N$ to be oscillatory. Examples are provided to illustrate the results.

• Journal of applied mathematics & informatics
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• 제9권1호
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• pp.87-99
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• 2002

We consider the problem of oscillation and nonoscillation solutions for unstable type second-order neutral difference equation : $\Delta^2(x(n))-p(n)x(n-\tau))=q(n)x(g(n))$. (1) In this paper, we obtain some conditions for the bounded solutions of Eq(1) to be oscillatory and for the existence of the nonoscillatory solutions.

• Kyungpook Mathematical Journal
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• 제57권2호
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• pp.251-263
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• 2017

The aim of this paper is to investigate the dynamical behavior of the difference equation $$x_{n+1}={\frac{{\alpha}x_n}{{\beta}+{\gamma}x^p_{n-1}x^q_{n-2}}},\;n=0,1,{\ldots}$$, where the parameters ${\alpha}$, ${\beta}$, ${\gamma}$, p, q are non-negative numbers and the initial values $x_{-2}$, $x_{-1}$, $x_0$ are positive numbers. Also, some numerical examples are given to verify our theoretical results.

• Journal of applied mathematics & informatics
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• 제35권3_4호
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• pp.267-276
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• 2017

The purpose of this paper is to investigate the local asymptotic stability of equilibria, the periodic nature of solutions, the existence of unbounded solutions and the global behavior of solutions of the fractional difference equation $$x_{n+1}=\frac{^{{\alpha}x}n-1(k+1)}{{\beta}+{\gamma}x^p_{n-k}x^q_{n-(k+2)}}$$, $$n=0,1,{\dots}$$ where the parameters ${\alpha}$, ${\beta}$, ${\gamma}$, p, q are non-negative numbers and the initial values $x_{-(k+2)}$,$x_{-(k+1)}$, ${\dots}$, $x_{-1}$, $x_0{\in}\mathb{R}^+$.

• Journal of applied mathematics & informatics
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• 제28권5_6호
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• pp.1285-1304
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• 2010

The purpose of this paper is to establish some sufficient conditions for oscillation of solutions of the second-order functional dynamic equation of Emden-Fowler type $\[a(t)x^{\Delta}(t)\]^{\Delta}+p(t)|x^{\gamma}(\tau(t))|\|x^{\Delta}(t)\|^{1-\gamma}$ $sgnx(\tau(t))=0$, $t\;{\geq}\;t_0$, on a time scale $\mathbb{T}$, where ${\gamma}\;{\in}\;(0,\;1]$, a, p and $\tau$ are positive rd-continuous functions defined on $\mathbb{T}$, and $lim_{t{\rightarrow}{\infty}}\;{\tau}(t)\;=\;\infty$. Our results include some previously obtained results for differential equations when $\mathbb{T}=\mathbb{R}$. When $\mathbb{T}=\mathbb{N}$ and $\mathbb{T}=q^{\mathbb{N}_0}=\{q^t\;:\;t\;{\in}\;\mathbb{N}_0\}$ where q > 1, the results are essentially new for difference and q-difference equations and can be applied on different types of time scales. Some examples are worked out to demonstrate the main results.

• Journal of applied mathematics & informatics
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• 제7권3호
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• pp.833-842
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• 2000

Consider the even-order neutral difference equation (*) ${\delta}^m(x_n{-}p_ng(x_{n-k}))-q_nh(x_{n-1})=0$, n=0,1,2,... where $\Delta$ is the forward difference operator, m is even, ${-p_n},{q_n}$ are sequences of nonnegative real numbers, k, l are nonnegative integers, g(x), h(x) ${\in}$ C(R, R) with xg(x) > 0 for $x\;{\neq}\;0$. In this paper, we obtain some linearized oscillation theorems of (*) for $p_n\;{\in}\;(-{\infty},0)$ which are discrete results of the open problem by Gyori and Ladas.

• Journal of applied mathematics & informatics
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• 제27권3_4호
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• pp.749-755
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• 2009

In this paper, we investigate the invariant interval, periodic character, semicycle and global attractivity of all positive solutions of the equation $Y_{n+1}\;=\;\frac{p+qy_{n-k}}{1+y_n+ry_{n-k}}$, n = 0, 1, ..., where the parameters p, q, r and the initial conditions $y_{-k}$, ..., $y_{-1}$, $y_0$ are positive real numbers, k $\in$ {1, 2, 3, ...}. It is worth to mention that our results solve the open problem proposed by Kulenvic and Ladas in their monograph [Dynamics of Second Order Rational Difference Equations: with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002]

• 대한수학회논문집
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• 제12권2호
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• pp.287-291
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• 1997

Assume $H_i : R_+ \times R_+ \to R_+ (i = 1, 2)$ are monotonically increasing (in both variables), homogeneous mapping for which $H_1(tu, tv) = t^p(H_1(u, v) (p > 0)$ and $H_2(u, v)^{t^q} (q \leq 1)$ hold for $t, u, v \geq 0$. Using an idea from the paper of Baker, Lawrence and Zorzitto [2], the superstability problems of the functional inequalities $\Vert f(x+y) - f(x)f(y) \Vert \leq H_i (\Vert x \Vert, \Vert y \Vert)$ shall be investigated.