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

Irina Astashova; Josef Diblík; Evgeniya Korobko

doi:10.3934/dcdss.2021133

Article Contents

2021, Volume 14, Issue 12: 4159-4178. Doi: 10.3934/dcdss.2021133 open access

This issue Previous Article Preface: Special issue on advances in partial differential equations Next Article Global existence and nonexistence for a class of finitely degenerate coupled parabolic systems with high initial energy level

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

1.
Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Department of Differential Equations, Leninskiye Gory 1, Main Building, 119991 Moscow, Russian Federation
2.
Plekhanov Russian University of Economics, Institute of Digital Economics and Information Technologies, Department of Higher Mathematics, Stremyanny lane 36, 117997 Moscow, Russian Federation
3.
Brno University of Technology, Faculty of Civil Engineering, Department of Mathematics and Descriptive Geometry, Veveří 331/95,602 00 Brno, Czech Republic
4.
Brno University of Technology, Faculty of Electrical Engineering and Communication, Department of Mathematics, Technická 2848/8,616 00 Brno, Czech Republic

^* Corresponding author: Josef Diblík
^* Corresponding author: Josef Diblík

Received: August 31, 2021

Revised: September 30, 2021

Early access: October 2021

Published: December 2021

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Abstract

The paper studies the asymptotic behaviour of solutions to a second-order non-linear discrete equation of Emden–Fowler type

$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0 $

where $ u\colon \{k_0, k_0+1, \dots\}\to \mathbb{R} $ is an unknown solution, $ \Delta^2 u(k) $ is its second-order forward difference, $ k_0 $ is a fixed integer and $ \alpha $, $ m $ are real numbers, $ m\not = 0, 1 $.

Keywords:

Mathematics Subject Classification: Primary: 39A33, 39A22; Secondary: 39A12.

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Figure 1. Solution of the system (69)

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Figure 2. Solution of the system (70), (71)

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Figure 3. Solution of the system (73)

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Figure 4. Solution of the system (74)

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Figure 5. Summary of admissible values

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