scholarly journals Oscillation results for nonlinear second order difference equations with mixed neutral terms

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Said R. Grace ◽  
Jehad Alzabut
Keyword(s):  
Difference Equations ◽  
Wide Class ◽  
Second Order ◽  
Order Difference ◽  
Oscillation Criteria ◽  
First Order ◽  
Oscillatory Behaviors

AbstractIn this paper, we establish new oscillation criteria for nonlinear second order difference equations with mixed neutral terms. The key idea of our approach is to compare with first order equations whose oscillatory behaviors are already known. The obtained results not only improve and extend existing results reported in the literature but also provide a new platform for the investigation of a wide class of nonlinear second order difference equations. The results are supported by examples to demonstrate the validity of the theoretical findings.

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 TRANSFORMATION OF THE LINEAR DIFFERENCE EQUATION INTO A SYSTEM OF THE FIRST ORDER DIFFERENCE EQUATIONS

2019 ◽  
pp. 76-80
Author(s):  
M.I. Ayzatsky
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Linear Difference Equation ◽  
Second Order ◽  
Dimensional System ◽  
Order Difference ◽  
Order Linear ◽  
First Order ◽  
Second Order Difference Equation ◽  
Order Difference Equation

The transformation of the N-th-order linear difference equation into a system of the first order difference equations is presented. The proposed transformation opens possibility to obtain new forms of the N-dimensional system of the first order equations that can be useful for the analysis of solutions of the N-th-order difference equations. In particular for the third-order linear difference equation the nonlinear second-order difference equation that plays the same role as the Riccati equation for second-order linear difference equation is obtained. The new form of the Ndimensional system of first order equations can also be used to find the WKB solutions of the linear difference equation with coefficients that vary slowly with index.

Oscillation theorems for certain second-order nonlinear retarded difference equations

2021 ◽  
Vol 71 (4) ◽  
pp. 871-880
Author(s):  
George E. Chatzarakis ◽  
Said R. Grace ◽  
Irena Jadlovská
Keyword(s):  
Difference Equations ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Neutral Equations ◽  
Oscillation Criteria ◽  
First Order ◽  
Oscillation Theorems

Abstract This paper deals with the oscillation of second-order nonlinear retarded difference equations. We present some new oscillation criteria via comparison with first-order equations whose oscillatory behavior are known. The results are generalized to be applicable to different kinds of neutral equations. An example is also given to demonstrate the applicability of the obtained conditions.

scholarly journals Explicit bounds for third-order difference equations

2006 ◽  
Vol 47 (3) ◽  
pp. 359-366
Author(s):  
Kenneth S. Berenhaut ◽  
Eva G. Goedhart ◽  
Stevo Stević
Keyword(s):  
Recent Work ◽  
Difference Equations ◽  
Wide Class ◽  
Higher Order ◽  
Second Order ◽  
Third Order ◽  
Order Difference ◽  
Explicit Bounds ◽  
Second Order Equations

AbstractThis paper gives explicit, applicable bounds for solutions of a wide class of third-order difference equations with nonconstant coefficients. The techniques used are readily adaptable for higher-order equations. The results extend recent work of the authors for second-order equations.

scholarly journals Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

AbstractIn this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian $$\begin{aligned} \Delta (\left| \Delta u(k-1)\right| ^{p-2}\Delta u(k-1))+a(k)\left| u(k)\right| ^{p-2}u(k)=0 \end{aligned}$$ Δ ( Δ u ( k - 1 ) p - 2 Δ u ( k - 1 ) ) + a ( k ) u ( k ) p - 2 u ( k ) = 0 with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.

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