Solvability of a superlinear Neumann problem at resonance in the first eigenvalue

Zhang Yali

doi:10.11714/acta.snus.ZR20260036

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Solvability of a superlinear Neumann problem at resonance in the first eigenvalue

更新时间：2026-05-21

- Solvability of a superlinear Neumann problem at resonance in the first eigenvalue
- Acta Scientiarum Naturalium Universitatis Sunyatseni Pages: 1-9(2026)
- 作者机构：
  
  甘肃政法大学人工智能学院，甘肃兰州 730070
- 作者简介：
  
  Zhang Yali（zhangyali359@163.com）
- 基金信息：
- DOI：10.11714/acta.snus.ZR20260036
  CLC： O175.7
- Received：27 January 2026，
  
  Revised：2026-04-09，
  
  Accepted：09 April 2026，
  
  Online First：21 May 2026，
- 稿件说明：
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Zhang Yali. Solvability of a superlinear Neumann problem at resonance in the first eigenvalue[J/OL]. Acta Scientiarum Naturalium Universitatis Sunyatseni, 2026, 1-9.
DOI：

Zhang Yali. Solvability of a superlinear Neumann problem at resonance in the first eigenvalue[J/OL]. Acta Scientiarum Naturalium Universitatis Sunyatseni, 2026, 1-9. DOI： 10.11714/acta.snus.ZR20260036.

摘要

研究一类超线性二阶离散Neumann边值问题

</mo><mo> </mo><mo> </mo><mo> </mo><mi>t</mi><mo>∈</mo><msub><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>

</mo><mi>T</mi></mrow></mfenced></mrow><mrow><mi mathvariant="double-struck">Z</mi></mrow></msub><mo>

</mo></mtd></mtr><mtr><mtd><mi>Δ</mi><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>Δ</mi><mi>u</mi><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>

</mo></mtd></mtr></mtable></mrow></mfenced><mtext> </mtext></math>

http://notExist.jpg

70.44266510

11.76866722

（P）

解的存在性，其中

http://notExist.jpg

7.28133297

2.28600001

为整数，

http://notExist.jpg

7.19666624

2.96333337

，函数

http://notExist.jpg

2.20133328

2.96333337

满足

http://notExist.jpg

15.07066536

7.28133297

. 利用拓扑度理论，证明了问题（P）至少存在一个解. 此外，将相关方法推广至具有类似非线性结构的二阶离散方程组.

Abstract

The existence of solutions for a class of superlinear second-order discrete Neumann boundary value problems of the form

</mo><mo> </mo><mo> </mo><mo> </mo><mi>t</mi><mo>∈</mo><msub><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>

</mo><mi>T</mi></mrow></mfenced></mrow><mrow><mi mathvariant="double-struck">Z</mi></mrow></msub><mo>

</mo></mtd></mtr></mtable></mrow></mfenced><mtext> </mtext></math>

http://notExist.jpg

70.44266510

11.76866722

（P）

where

http://notExist.jpg

8.46666718

2.62466669

is an integer

http://notExist.jpg

8.38199997

3.47133350

and

http://notExist.jpg

2.28600001

3.47133350

satisfies the condition

http://notExist.jpg

17.44133186

8.29733276

is investigated. Based on the topological degree theory

we show that the problem (P) possesses at least one solution. Additionally

we extend our analysis to systems of second-order discrete equations with similar nonlinearities.

关键词

Keywords

references

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Arcoya D ， Villegas S ， 1995 . Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at <math id="M282"><mo>-</mo><mi mathvariant="normal">∞</mi></math> http://notExist.jpg http://notExist.jpg 4.06400013 2.28600001 and superlinear at <math id="M283"><mo>+</mo><mi mathvariant="normal">∞</mi></math> http://notExist.jpg http://notExist.jpg 4.06400013 2.28600001 ［J］. Math Z ， 219 （ 1 ）： 499 - 513 .

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Liu M Y ， Pei M H ， Wang L B ， 2024 . Solvability of a nonlinear second order <math id="M284"><mi>m</mi></math> http://notExist.jpg http://notExist.jpg 2.28600001 2.28600001 -point boundary value problem with p -Laplacian at resonance ［J］. Bound Value Probl ， 2024 （ 1 ）： 46 .

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Wang S L ， Liu J S ， Zhang J M ， et al ， 2013 . Existence of non-trivial solutions for resonant difference equations ［J］. J Differ Equ Appl ， 19 （ 2 ）： 209 - 222 .

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Zhu B S ， Yu J S ， 2009 . Multiple positive solutions for resonant difference equations ［J］. Math Comput Model ， 49 （ 9/10 ）： 1928 - 1936 .

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