scholarly journals Riccati transformation and nonoscillation criterion for linear difference equations

2020 ◽  
Vol 148 (10) ◽  
pp. 4319-4332
Author(s):  
Kōdai Fujimoto ◽  
Petr Hasil ◽  
Michal Veselý
Keyword(s):  
Difference Equations ◽  
Riccati Transformation ◽  
Linear Difference Equations ◽  
Nonoscillation Criterion

scholarly journals New Oscillation Conditions for Second Order Half-Linear Advanced Difference Equations

2019 ◽  
Vol 4 (6) ◽  
pp. 1459-1470
Author(s):  
P. Dinakar ◽  
S. Selvarangam ◽  
E. Thandapani
Keyword(s):  
Difference Equations ◽  
Averaging Method ◽  
Second Order ◽  
Sufficient Condition ◽  
Riccati Transformation ◽  
Linear Difference Equations ◽  
All Solutions ◽  
Noncanonical Form

This paper aims to establish adequate conditions that are intended for the oscillation of every solution of the second order advanced type half-linear difference equations with noncanonical form. Initially, we derive a sufficient condition that ensures all solutions of the studied equation are either oscillatory or tending to zero. Secondly, we obtain a criteria for the oscillation of all solutions of the studied equation. These criteria are obtained by using Riccati transformation and summation averaging method. The results established in this paper in essence complement, extend and enhance the existing outcomes recorded in the literature. The improvement of our main results are illustrated through three examples.

scholarly journals Oscillation and Nonoscillation of Asymptotically Almost Periodic Half-Linear Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Michal Veselý ◽  
Petr Hasil
Keyword(s):  
Difference Equations ◽  
Riccati Transformation ◽  
Almost Periodic ◽  
Linear Difference Equations ◽  
Periodic Coefficients ◽  
Asymptotically Almost Periodic ◽  
Oscillation And Nonoscillation

We analyse half-linear difference equations with asymptotically almost periodic coefficients. Using the adapted Riccati transformation, we prove that these equations are conditionally oscillatory. We explicitly find a constant, determined by the coefficients of a given equation, which is the borderline between the oscillation and the nonoscillation of the equation. We also mention corollaries of our result with several examples.

scholarly journals Note on some representations of general solutions to homogeneous linear difference equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Stevo Stević ◽  
Bratislav Iričanin ◽  
Witold Kosmala ◽  
Zdeněk Šmarda
Keyword(s):  
General Solution ◽  
Difference Equation ◽  
Difference Equations ◽  
Fibonacci Sequence ◽  
Linear Difference Equation ◽  
Linear Difference Equations ◽  
Second Order Difference Equation ◽  
Constant Coefficients ◽  
Order Difference Equation ◽  
Homogeneous Linear

Abstract It is known that every solution to the second-order difference equation $x_{n}=x_{n-1}+x_{n-2}=0$ x n = x n − 1 + x n − 2 = 0 , $n\ge 2$ n ≥ 2 , can be written in the following form $x_{n}=x_{0}f_{n-1}+x_{1}f_{n}$ x n = x 0 f n − 1 + x 1 f n , where $f_{n}$ f n is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

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.

scholarly journals Oscillation of nonlinear third-order difference equations with mixed neutral terms

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jehad Alzabut ◽  
Martin Bohner ◽  
Said R. Grace
Keyword(s):  
Difference Equations ◽  
Riccati Transformation ◽  
Third Order ◽  
Order Difference ◽  
New Approach ◽  
Comparison Technique

AbstractIn this paper, new oscillation results for nonlinear third-order difference equations with mixed neutral terms are established. Unlike previously used techniques, which often were based on Riccati transformation and involve limsup or liminf conditions for the oscillation, the main results are obtained by means of a new approach, which is based on a comparison technique. Our new results extend, simplify, and improve existing results in the literature. Two examples with specific values of parameters are offered.

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