New oscillation criteria for third order nonlinear neutral difference equations

2011 ◽  
Vol 61 (4) ◽  
Author(s):  
S. Saker
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Nonnegative Integer ◽  
Real Numbers ◽  
Third Order ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Oscillation Criteria ◽  
The Third ◽  
Positive Integers

AbstractIn this paper, we are concerned with oscillation of the third-order nonlinear neutral difference equation $\Delta (c_n [\Delta (d_n \Delta (x_n + p_n x_{n - \tau } ))]^\gamma ) + q_n f(x_{g(n)} ) = 0,n \geqslant n_0 ,$ where γ > 0 is the quotient of odd positive integers, c n, d n, p n and q n are positive sequences of real numbers, τ is a nonnegative integer, g(n) is a sequence of nonnegative integers and f ∈ C(ℝ,ℝ) such that uf(u) > 0 for u ≠ 0. Our results extend and improve some previously obtained ones. Some examples are considered to illustrate the main results.

scholarly journals New Oscillation Criteria for Third-Order Nonlinear Mixed Neutral Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Elmetwally Mohammed Elabbasy ◽  
Magdy Yoseph Barsom ◽  
Faisal Saleh AL-dheleai
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Third Order ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Oscillation Criteria

Some new oscillation criteria are established for a third-order nonlinear mixed neutral difference equation. Our results improve and extend some known results in the literature. Several examples are given to illustrate the importance of the results.

scholarly journals ON EXISTENCE OF POSITIVE SOLUTIONS OF NEUTRAL DIFFERENCE EQUATIONS

1994 ◽  
Vol 25 (3) ◽  
pp. 257-265
Author(s):  
J. H. SHEN ◽  
Z. C. WANG ◽  
X. Z. QIAN
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Difference Equations ◽  
Real Numbers ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Existence Of Positive Solutions

Consider the neutral difference equation \[\Delta(x_n- cx_{n-m})+p_nx_{n-k}=0, n\ge N\qquad (*) \] where $c$ and $p_n$ are real numbers, $k$ and $N$ are nonnegative integers, and $m$ is positive integer. We show that if \[\sum_{n=N}^\infty |p_n|<\infty \qquad (**) \] then Eq.(*) has a positive solution when $c \neq 1$. However, an interesting example is also given which shows that (**) does not imply that (*) has a positive solution when $c =1$.

scholarly journals Oscillation theorems for second order difference equations woth negative neutral term

2015 ◽  
Vol 46 (4) ◽  
pp. 441-451 ◽  
Author(s):  
Ethiraju Thandapani ◽  
Devarajulu Seghar ◽  
Sandra Pinelas
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Second Order ◽  
Neutral Difference Equation ◽  
Order Difference ◽  
Oscillation Criteria ◽  
Neutral Term ◽  
Oscillation Theorems ◽  
Positive Integers

In this paper we obtain some new oscillation criteria for the neutral difference equation \begin{equation*} \Delta \Big(a_n (\Delta (x_n-p_n x_{n-k}))\Big)+q_n f(x_{n-l})=0 \end{equation*} where $0\leq p_n\leq p0$ and $l$ and $k$ are positive integers. Examples are presented to illustrate the main results. The results obtained in this paper improve and complement to the existing results.

Asymptotic behavior in neutral difference equations with negative coefficients

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
G. Chatzarakis ◽  
G. Miliaras
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Difference Equations ◽  
Difference Operator ◽  
Retarded Argument ◽  
Real Numbers ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Forward Difference ◽  
Deviated Argument

AbstractIn this paper, we study the asymptotic behavior of the solutions of a neutral difference equation of the form $\Delta [x(n) + cx(\tau (n))] - p(n)x)(\sigma (n)) = 0,$, where τ(n) is a general retarded argument, σ(n) is a general deviated argument, c ∈ ℝ, (−p(n))n≥0 is a sequence of negative real numbers such that p(n) ≥ p, p ∈ ℝ+, and Δ denotes the forward difference operator Δx(n) = x(n+1)−x(n).

scholarly journals Oscillatory Behavior of Solutions of Fourth-order Mixed Neutral Difference Equations with Asynchronous Non Linearities

2019 ◽  
Vol 8 (10S) ◽  
pp. 63-66
Keyword(s):  
Difference Equation ◽  
Fourth Order ◽  
Difference Equations ◽  
Mixed Type ◽  
Oscillatory Behavior ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Oscillation Criteria

In this article, oscillation criteria for solutions of fourth order mixed type neutral difference equation with asynchronous non linearities of the form where{an}, {bn}, {cn}, {qn} and {pn} are established. Examples are provided to illustrate the results

scholarly journals Oscillation and comparison theorems for certain neutral difference equations

1992 ◽  
Vol 34 (2) ◽  
pp. 245-256 ◽  
Author(s):  
B. S. Lalli ◽  
B. G. Zhang
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Comparison Theorems ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Oscillation Criteria ◽  
Arbitrary Sign ◽  
Image Position ◽  
Mixed Arguments

AbstractSome comparison theorems and oscillation criteria are established for the neutral difference equationas well as for certain neutral difference equations with coefficients of arbitrary sign. Neutral difference equations with mixed arguments are also considered.

scholarly journals Oscillation criteria for even order neutral difference equations

2019 ◽  
Vol 39 (1) ◽  
pp. 91-108 ◽  
Author(s):  
S. Selvarangam ◽  
S. A. Rupadevi ◽  
E. Thandapani ◽  
S. Pinelas
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Geometric Mean ◽  
Sufficient Conditions ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Oscillation Criteria ◽  
Even Order

In this paper, we present some new sufficient conditions for oscillation of even order nonlinear neutral difference equation of the form \[\Delta^m(x_n+ax_{n-\tau_1}+bx_{n+\tau_2})+p_nx_{n-\sigma_1}^{\alpha}+q_nx_{n+\sigma_2}^{\beta}=0,\quad n\geq n_0\gt0,\] where \(m\geq 2\) is an even integer, using arithmetic-geometric mean inequality. Examples are provided to illustrate the main results.

scholarly journals Oscillation criteria of third-order nonlinear neutral delay difference equations with noncanonical operators

2021 ◽  
pp. 11-11
Author(s):  
G. Ayyappan ◽  
G.E. Chatzarakis ◽  
T. Gopal ◽  
E. Thandapani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Neutral Delay ◽  
Third Order ◽  
Neutral Difference Equations ◽  
Oscillation Criteria ◽  
All Solutions ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
New Criteria

In this paper, we present some new oscillation criteria for nonlinear neutral difference equations of the form ?(b(n)?(a(n)?z(n))) + q(n)x?(?(n)) = 0 where z(n) = x(n) + p(n)x(?(n)),? > 0, b(n) > 0, a(n) > 0, q(n) ? 0 and p(n) > 1. By summation averaging technique, we establish new criteria for the oscillation of all solutions of the studied difference equation above. We present four examples to show the strength of the new obtained results.

scholarly journals A Comparison Theorem for Oscillation of the Even-Order Nonlinear Neutral Difference Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Quanxin Zhang
Keyword(s):  
Difference Equation ◽  
Comparison Theorem ◽  
Difference Equations ◽  
Neutral Delay ◽  
Neutral Difference Equation ◽  
Oscillation Criteria ◽  
Even Order ◽  
Delay Difference Equations ◽  
Delay Difference

A comparison theorem on oscillation behavior is firstly established for a class of even-order nonlinear neutral delay difference equations. By using the obtained comparison theorem, two oscillation criteria are derived for the class of even-order nonlinear neutral delay difference equations. Two examples are given to show the effectiveness of the obtained results.

scholarly journals OSCILLATION AND COMPARISON THEOREMS FOR NEUTRAL DIFFERENCE EQUATIONS

1994 ◽  
Vol 25 (4) ◽  
pp. 343-352
Author(s):  
B. G. ZHANG ◽  
PENGXIANG YAN
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Comparison Theorems ◽  
Qualitative Properties ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Riccati Techniques ◽  
Qualitative Properties Of Solutions

In this paper we study qualitative properties of solutions of the neutral difference equation $$ \Delta(y_n-py_{n-k})+\sum_{i=1}^m q_n^i y_{n-k_i} =0 $$ $$ y_n=A_n \quad \text{ for } n=-M, \cdots, -1, 0$$ where $p \ge 1$, $M =\max\{k, k_1, \cdots, k_m\}$, and $k$, $k_i$, $i =1, \cdots, m$, are nonnegative integers. Riccati techniques are used.  

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