Oscillation of higher order neutral functional difference equations with positive and negative coefficients

2010 ◽  
Vol 60 (3) ◽  
Author(s):  
R. Rath ◽  
B. Barik ◽  
S. Rath
Keyword(s):  
Positive Integer ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Functional Difference ◽  
Real Numbers ◽  
Positive And Negative Coefficients ◽  
Forward Difference ◽  
Functional Difference Equation ◽  
Functional Difference Equations ◽  
Infinite Sequences

AbstractSufficient conditions are obtained so that every solution of the neutral functional difference equation $$ \Delta ^m (y_n - p_n y_{\tau (n)} ) + q_n G(y_{\sigma (n)} ) - u_n H(y_{\alpha (n)} ) = f_n , $$ oscillates or tends to zero or ±∞ as n → ∞, where Δ is the forward difference operator given by Δx n = x n+1 − x n, p n, q n, u n, f n are infinite sequences of real numbers with q n > 0, u n ≥ 0, G,H ∈ C(ℝ,ℝ) and m ≥ 2 is any positive integer. Various ranges of {p n} are considered. The results hold for G(u) ≡ u, and f n ≡ 0. This paper corrects, improves and generalizes some recent results.

scholarly journals Oscillation of nonlinear third order perturbed functional difference equations

2019 ◽  
Vol 6 (1) ◽  
pp. 57-64 ◽  
Author(s):  
P. Dinakar ◽  
S. Selvarangam ◽  
E. Thandapani
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Functional Difference ◽  
Third Order ◽  
All Solutions ◽  
Functional Difference Equation ◽  
Functional Difference Equations

AbstractThis paper deals with oscillatory and asymptotic behavior of all solutions of perturbed nonlinear third order functional difference equation\Delta {\left( {{b_n}\Delta ({a_n}(\Delta {x_n}} \right)^\alpha })) + {p_n}f\left( {{x_{\sigma \left( n \right)}}} \right) = g\left( {n,{x_n},{x_{\sigma (n)}},\Delta {x_n}} \right),\,\,\,n \ge {n_0}.By using comparison techniques we present some new sufficient conditions for the oscillation of all solutions of the studied equation. Examples illustrating the main results are included.

scholarly journals Positive solutions of boundary value problems for $ p $-Laplacian functional difference equations

2007 ◽  
Vol 38 (4) ◽  
pp. 291-299
Author(s):  
Changxiu Song
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Difference Equation ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Functional Difference ◽  
Functional Difference Equation ◽  
Functional Difference Equations

In this paper, the author studies the boundary value problems of $ p $-Laplacian functional difference equation. By using a fixed point theorem in cones, sufficient conditions are established for the existence of the positive solutions.

scholarly journals Oscillation of a Class of Third Order Generalized Functional Difference Equation

2018 ◽  
Author(s):  
P.Venkata Mohan Reddy ◽  
Adem Kilicman ◽  
Maria Susai Manuel
Keyword(s):  
Difference Equation ◽  
Sufficient Conditions ◽  
Functional Difference ◽  
Third Order ◽  
Oscillation Criteria ◽  
Functional Difference Equation

The authors intend to establish new oscillation criteria for a class of generalized third order functional difference equation of the form \begin{equation}{\label{eq01}} \Delta_{\ell}\left(a_2(n)\left[\Delta_{\ell}\left(a_1(n)\left[\Delta_{\ell}z(n)\right]^{\beta_1}\right)\right]^{\beta_2}\right)+q(n)f(x(g(n)))=0, ~~n\geq n_0, \end{equation} where $z(n)=x(n)+p(n)x(\tau(n))$. We also present sufficient conditions for the solutions to converges to zero. Suitable examples are presented to validate our main results.

scholarly journals Oscillatory and Asymptotic Behavior of a First-Order Neutral Equation of Discrete Type with Variable Several Delay under Δ Sign

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Radhanath Rath ◽  
Chittaranjan Behera
Keyword(s):  
Difference Operator ◽  
Sufficient Conditions ◽  
Neutral Equation ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
First Order ◽  
Forward Difference ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Delay Difference

We obtain necessary and sufficient conditions so that every solution of neutral delay difference equation Δyn-∑j=1kpnjyn-mj+qnG(yσ(n))=fn oscillates or tends to zero as n→∞, where {qn} and {fn} are real sequences and G∈C(R,R), xG(x)>0, and m1,m2,…,mk are positive integers. Here Δ is the forward difference operator given by Δxn=xn+1-xn, and {σn} is an increasing unbounded sequences with σn≤n. This paper complements, improves, and generalizes some past and recent results.

scholarly journals Convergence and Divergence of the Solutions of a Neutral Difference Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
G. E. Chatzarakis ◽  
G. N. Miliaras
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Difference Operator ◽  
Neutral Type ◽  
Retarded Argument ◽  
Real Numbers ◽  
Positive Real ◽  
Neutral Difference Equation ◽  
Forward Difference ◽  
Deviated Argument

We investigate the asymptotic behavior of the solutions of a neutral type difference equation of the form , where is a general retarded argument, is a general deviated argument (retarded or advanced), , is a sequence of positive real numbers such that , , and denotes the forward difference operator . Also, we examine the asymptotic behavior of the solutions in case they are continuous and differentiable with respect to .

scholarly journals On Nonlinear Boundary Value Problems for Functional Difference Equations withp-Laplacian

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Yong Wan ◽  
Yuji Liu
Keyword(s):  
Boundary Value Problems ◽  
Difference Equations ◽  
Nonlinear Boundary ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Functional Difference ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value ◽  
Functional Difference Equations

Sufficient conditions for the existence of solutions of nonlinear boundary value problems for higher-order functional difference equations withp-Laplacian are established by making of continuation theorems. We allowfto be at most linear, superlinear, or sublinear in obtained results.

A note on the existence of three positive periodic solutions of functional difference equation

2011 ◽  
Vol 18 (1) ◽  
pp. 39-52
Author(s):  
Shengping Chen
Keyword(s):  
Difference Equation ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Functional Difference ◽  
Positive Periodic Solutions ◽  
Functional Difference Equation ◽  
Functional Difference Equations

Abstract It is shown that, under certain assumptions, the functional difference equations have at least three positive periodic solutions. Applications are given to illustrate the main 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 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.

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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