Nonoscillatory Solutions to Second-Order Neutral Difference Equations

Migda, Małgorzata; Migda, Janusz

doi:10.3390/sym10060207

Cited by 6 publications

(3 citation statements)

References 30 publications

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“…In [8], Migda has discussed the asymptotic properties of nonoscillatory solutions of neutral difference equations of the form (1.3)…”

Section: Introductionmentioning

confidence: 99%

“…∆ m (x n + p n x n−τ ) + f (n, x σ(n) ) = h n and has shown that any nonoscillatory solution x n has the property x n = cn m−1 + o(n m−1 ) for some c ∈ R. For m = 4, f (n, x σ(n) ) = q(n)G(x σ(n) ) and σ(n) = n − σ, (1.3) reduces to (1.2) for r(n) ≡ 1 and hence the papers [13] and [8] are comparable. But, more emphasis may be given to [13], which deals with the oscillatory, nonoscillatory and asymptotic characters.…”

Section: Introductionmentioning

confidence: 99%

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On oscillatory nonlinear fourth-order difference equations with delays

Tripathy¹

2017

Math.Boh.

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“…In [8], Migda has discussed the asymptotic properties of nonoscillatory solutions of neutral difference equations of the form (1.3)…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On oscillatory nonlinear fourth-order difference equations with delays

Tripathy¹

2017

Math.Boh.

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“…3) for k ∈ Z(a) and obtained a formulation of generalized zeros and (n, n)disconjugacy for (1.3). Migda [33] in 2004 studied an mth-order linear difference equation.…”

Section: Introductionmentioning

confidence: 99%

Existence of periodic solutions for 2$n$th-order nonlinear $p$-Laplacian difference equations

Shi

Liu

Zhang

2016

Rocky Mountain J. Math.

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By using the critical point theory, the existence of periodic solutions for 2nth-order nonlinear p-Laplacian difference equations is obtained. The main approaches used in our paper are variational techniques and the Saddle Point theorem. The problem is to solve the existence of periodic solutions for 2nth-order p-Laplacian difference equations. The results obtained successfully generalize and complement the existing ones. 2010 AMS Mathematics subject classification. Primary 39A11.

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Asymptotic Behavior of Solutions to Difference Equations of Neutral Type

Nockowska-Rosiak,

Migda

2023

Qual. Theory Dyn. Syst.

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We present sufficient conditions for the existence of a solution x to an equation $$\begin{aligned} \Delta ^m(x_n-u_nx_{n-k})=a_nf(x_{n-\tau })+b_n, \end{aligned}$$ Δ m ( x n - u n x n - k ) = a n f ( x n - τ ) + b n , which is “close” to a given solution y to the linear homogeneous equation of neutral type $$\Delta ^m(y_n-\lambda y_{n-k})=0$$ Δ m ( y n - λ y n - k ) = 0 , where $$\lambda $$ λ is the limit of the sequence u. Closeness of solutions to above equations is understood as $$x_n-y_n=\textrm{o}(\omega _n)$$ x n - y n = o ( ω n ) , where $$\omega $$ ω is a given nonincreasing sequence with positive values. Moreover, we establish under which conditions for a given solution x to $$\Delta ^m(x_n-u_nx_{n-k})=a_nf(x_{n-\tau })+b_n$$ Δ m ( x n - u n x n - k ) = a n f ( x n - τ ) + b n and a given nonincreasing sequence with positive values $$\omega $$ ω there exists a polynomial sequence $$\varphi $$ φ of degree less than m such that $$x_n=\varphi (n)+\textrm{o}(\omega _n)$$ x n = φ ( n ) + o ( ω n ) . Presented conditions strongly depend on $$\lambda $$ λ .

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Nonoscillatory Solutions to Second-Order Neutral Difference Equations

Abstract: We study asymptotic behavior of nonoscillatory solutions to second-order neutral difference equation of the form: ∆(r n ∆(x n + p n x n−τ)) = a n f (n, x n) + b n. The obtained results are based on the discrete Bihari type lemma and a Stolz type lemma.

Cited by 6 publications

References 30 publications

On oscillatory nonlinear fourth-order difference equations with delays

On oscillatory nonlinear fourth-order difference equations with delays

Existence of periodic solutions for 2$n$th-order nonlinear $p$-Laplacian difference equations

Asymptotic Behavior of Solutions to Difference Equations of Neutral Type

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