scholarly journals Oscillatory behavior of second order unstable type neutral difference equations

2005 ◽  
Vol 36 (1) ◽  
pp. 57-68
Author(s):  
E. Thandapani ◽  
S. Pandian ◽  
R. K. Balasubramanian
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Difference Equations ◽  
Neutral Type ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Neutral Difference Equations ◽  
Unbounded Solutions ◽  
Unstable Type ◽  
Positive Integers

This paper deals with the oscillatory behavior of all bounded/ unbounded solutions of second order neutral type difference equation of the form$$ \Delta (a_n(\Delta_c y_n+py_{n-k}))^\alpha)-g_nf(y_{\sigma(n)})=0, $$where $ p $ is real, $ \alpha $ is a ratio of odd positive integers, $ k $ is a positive integer and $ \{\sigma(n)\} $ is a sequence of integers. Examples are provided to illustrate 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.

scholarly journals Kamenev-Type Oscillation Criteria for Generalized Second Order Sublinear -Difference Equations

2018 ◽  
Vol 7 (4.10) ◽  
pp. 1046
Author(s):  
A. Benevatho Jaison ◽  
Sk. Khadar Babu ◽  
V. Chandrasekar
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Second Order ◽  
Nonlinear Difference Equation ◽  
Riccati Transformation ◽  
Transformation Techniques ◽  
Oscillation Criteria ◽  
Positive Integers ◽  
Type Oscillation

By means of Riccati transformation techniques, authors establish some new oscillation criteria for generalized second order nonlinear -difference equation when  and  are quotient of odd positive integers.   

scholarly journals Results on the solutions of several second order mixed type partial differential difference equations

2021 ◽  
Vol 7 (2) ◽  
pp. 1907-1924
Author(s):  
Wenju Tang ◽  
◽  
Keyu Zhang ◽  
Hongyan Xu ◽  
◽  
...  
Keyword(s):  
Difference Equation ◽  
Finite Order ◽  
Difference Equations ◽  
Mixed Type ◽  
Existence Of Solutions ◽  
Second Order ◽  
Entire Solutions ◽  
Partial Differential ◽  
Differential Difference Equation ◽  
Positive Integers

<abstract><p>This article is concerned with the existence of entire solutions for the following complex second order partial differential-difference equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left(\frac{\partial^2 f(z_1, z_2)}{\partial z_1^2}+\frac{\partial^2 f(z_1, z_2)}{\partial z_2^2}\right)^{l}+f(z_1+c_1, z_2+c_2)^{k} = 1, $\end{document} </tex-math></disp-formula></p> <p>where $ c_1, c_2 $ are constants in $ \mathbb{C} $ and $ k, l $ are positive integers. In addition, we also investigate the forms of finite order transcendental entire solutions for several complex second order partial differential-difference equations of Fermat type, and obtain some theorems about the existence and the forms of solutions for the above equations. Meantime, we give some examples to explain the existence of solutions for some theorems in some cases. Our results are some generalizations of the previous theorems given by Qi <sup>[<xref ref-type="bibr" rid="b23">23</xref>]</sup>, Xu and Cao <sup>[<xref ref-type="bibr" rid="b35">35</xref>]</sup>, Liu, Cao and Cao <sup>[<xref ref-type="bibr" rid="b17">17</xref>]</sup>.</p></abstract>

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 for Super Linear/Linear Second Order Neutral Difference Equations with Variable Several Delays

2020 ◽  
Vol 5 (4) ◽  
pp. 663-681
Author(s):  
Chittaranjan Behera ◽  
Radhanath Rath ◽  
Prayag Prasad Mishra
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Second Order ◽  
Multiple Delays ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Forward Difference ◽  
Several Delays

This article, is concerned with finding sufficient conditions for the oscillation and non oscillation of the solutions of a second order neutral difference equation with multiple delays under the forward difference operator, which generalize and extend some existing results.This could be possible by extending an important lemma from the literature.

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.

Impact of different types of non linearity on the oscillatory behavior of higher order neutral difference equations

2021 ◽  
Vol 71 (4) ◽  
pp. 941-960
Author(s):  
Ajit Kumar Bhuyan ◽  
Laxmi Narayan Padhy ◽  
Radhanath Rath
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Unbounded Solution ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Forward Difference ◽  
Different Types

Abstract In this article, sufficient conditions are obtained so that every solution of the neutral difference equation Δ m ( y n − p n L ( y n − s ) ) + q n G ( y n − k ) = 0 , $$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})=0, \end{equation*}$$ or every unbounded solution of Δ m ( y n − p n L ( y n − s ) ) + q n G ( y n − k ) − u n H ( y α ( n ) ) = 0 , n ≥ n 0 , $$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})-u_nH(y_{\alpha(n)})=0,\quad n\geq n_0, \end{equation*}$$ oscillates, where m=2 is any integer, Δ is the forward difference operator given by Δy n = y n+1 − y n ; Δ m y n = Δ(Δ m−1 y n ) and other parameters have their usual meaning. The non linear function L ∈ C (ℝ, ℝ) inside the operator Δ m includes the case L(x) = x. Different types of super linear and sub linear conditions are imposed on G to prevent the solution approaching zero or ±∞. Further, all the three possible cases, p n ≥ 0, p n ≤ 0 and p n changing sign, are considered. The results of this paper generalize and extend some known results.

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