Asymptotic properties of solutions of the discrete analogue of the Emden-Fowler equation

Diblı́k, Josef; Hlavičková, Irena

doi:10.2969/aspm/05310023

Cited by 4 publications

(4 citation statements)

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“…Although the literature devoted to continuous Emden-Fowler equations and their modifications and generalizations is very rich, there are not many papers related to discrete Emden-Fowler type equation ( 4) or to similar discrete equations. We mention just the results in [2,19,24,26,37,38,40], which are the closest to our investigation. Let us remark as well that the asymptotic behaviour of solutions to linear discrete equations is studied in [10].…”

supporting

confidence: 67%

“…But this is not the case if discrete equations of this type are considered. The literature related to the investigation of properties of solutions to discrete equations of Emden-Fowler type being much scarcer, we have only found a paper by L. Erbe, J. Baoguo, and J. Peterson [24] on non-oscillatory solutions of Emden-Fowler type discrete equations providing the asymptotic properties of a similar equation on time scales in addition to paper [19] co-authored by the second author dealing with the "−" case in (4) for α = −2, m > 1 and describing the boundary between the orange and violet hathed open domains in Figure 5 not covered by our results. This means that the investigation of the existence and properties of non-oscillating solutions is a promising field.…”

Section: System Of Difference Equations Equivalent To Equation (4)mentioning

confidence: 99%

“…In other words, if u(k) in equation ( 4) is replaced by u app (k), then, in the left-hand side of (23), the coefficients of the terms k −s , k −s−1 , k −s−2 and k −s−3 will be eliminated and it is possible to set g(k) = k s+3 in (19).…”

mentioning

confidence: 99%

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Existence of a solution of discrete Emden-Fowler equation caused by continuous equation

Асташова¹,

Diblı́k²,

Korobko³

2021

DCDS-S

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The paper studies the asymptotic behaviour of solutions to a second-order non-linear discrete equation of Emden–Fowler type<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0 $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ u\colon \{k_0, k_0+1, \dots\}\to \mathbb{R} $\end{document}</tex-math></inline-formula> is an unknown solution, <inline-formula><tex-math id="M2">\begin{document}$ \Delta^2 u(k) $\end{document}</tex-math></inline-formula> is its second-order forward difference, <inline-formula><tex-math id="M3">\begin{document}$ k_0 $\end{document}</tex-math></inline-formula> is a fixed integer and <inline-formula><tex-math id="M4">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ m $\end{document}</tex-math></inline-formula> are real numbers, <inline-formula><tex-math id="M6">\begin{document}$ m\not = 0, 1 $\end{document}</tex-math></inline-formula>.

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supporting

confidence: 67%

Section: System Of Difference Equations Equivalent To Equation (4)mentioning

confidence: 99%

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Existence of a solution of discrete Emden-Fowler equation caused by continuous equation

Асташова¹,

Diblı́k²,

Korobko³

2021

DCDS-S

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“…The work done in this area focuses on finding conditions that guarantee the existence of a solution of such discrete equations. In [8], the authors consider the special case of (1.6) where α = −2, write it as a system of two difference equations, and prove a general theorem for this that gives sufficient conditions that guarantee the existence of at least one solution.…”

Section: 5)mentioning

confidence: 99%

Asymptotic analysis of Emden-Fowler type equation with an application to power flow models

Christianen¹,

Janssen²,

Vlasiou³

et al. 2022

Preprint

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Emden-Fowler type equations are nonlinear differential equations that appear in many fields such as mathematical physics, astrophysics and chemistry. In this paper, we perform an asymptotic analysis of a specific Emden-Fowler type equation that emerges in a queuing theory context as an approximation of voltages under a well-known power flow model. Thus, we place Emden-Fowler type equations in the context of electrical engineering. We derive properties of the continuous solution of this specific Emden-Fowler type equation and study the asymptotic behavior of its discrete analog. We conclude that the discrete analog has the same asymptotic behavior as the classical continuous Emden-Fowler type equation that we consider.

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Asymptotic Behavior of Solutions of Delayed Difference Equations

Diblı́k

Hlavičková²

2011

Abstract and Applied Analysis

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This contribution is devoted to the investigation of the asymptotic behavior of delayed difference equations with an integer delay. We prove that under appropriate conditions there exists at least one solution with its graph staying in a prescribed domain. This is achieved by the application of a more general theorem which deals with systems of first-order difference equations. In the proof of this theorem we show that a good way is to connect two techniques—the so-called retract-type technique and Liapunov-type approach. In the end, we study a special class of delayed discrete equations and we show that there exists a positive and vanishing solution of such equations.

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Asymptotic properties of solutions of the discrete analogue of the Emden-Fowler equation

Cited by 4 publications

References 2 publications

Existence of a solution of discrete Emden-Fowler equation caused by continuous equation

Existence of a solution of discrete Emden-Fowler equation caused by continuous equation

Asymptotic analysis of Emden-Fowler type equation with an application to power flow models

Asymptotic Behavior of Solutions of Delayed Difference Equations

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