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

Research articles | 更新时间：2026-10-20

- Solvability of a superlinear Neumann problem at resonance in the first eigenvalue
- Acta Scientiarum Naturalium Universitatis Sunyatseni Vol. 65, Issue 5, Pages: 168-176(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：25 May 2026，
  
  Published：25 September 2026
- 稿件说明：
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张亚莉.在第一特征值处共振的超线性Neumann问题的可解性[J].中山大学学报(自然科学版)(中英文),2026,65(05):168-176.

Zhang Yali.Solvability of a superlinear Neumann problem at resonance in the first eigenvalue[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2026,65(05):168-176.
张亚莉.在第一特征值处共振的超线性Neumann问题的可解性[J].中山大学学报(自然科学版)(中英文),2026,65(05):168-176. DOI： 10.11714/acta.snus.ZR20260036.

Zhang Yali.Solvability of a superlinear Neumann problem at resonance in the first eigenvalue[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2026,65(05):168-176. 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></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640731&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640743&type=

69.68067169

12.10733318

解的存在性，其中

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640759&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640768&type=

8.46666718

2.62466669

为整数，

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640735&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640770&type=

8.38199997

3.47133350

，函数

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640764&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640763&type=

2.53999996

3.47133350

满足

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640767&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640766&type=

17.44133186

8.29733276

. 利用拓扑度理论，证明了问题至少存在一个解. 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></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640711&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640659&type=

69.68067169

12.10733318

where

is an integer

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640714&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640688&type=

8.38199997

3.47133350

and

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640726&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640691&type=

2.28600001

3.47133350

satisfies the condition

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640754&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120640718&type=

17.44133186

8.29733276

is investigated. Based on the topological degree theory

we show that the problem 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

Agarwal R P ， 2000 . Difference equations and inequalities： Theory， methods， and applications ［M］. Boca Raton ： CRC Press .

Arcoya D ， Villegas S ， 1995 . Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at <math id="M274"><mo>-</mo><mi mathvariant="normal">∞</mi></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120641943&type= https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120641922&type= 4.06400013 2.28600001 and superlinear at <math id="M275"><mo>+</mo><mi mathvariant="normal">∞</mi></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120641957&type= https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120641930&type= 4.06400013 2.28600001 ［J］. Math Z ， 219 （ 1 ）： 499 - 513 .

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de Paiva F O ， Rosa W ， 2017 . Neumann problems with resonance in the first eigenvalue ［J］. Math Nachr ， 290 （ 14/15 ）： 2198 - 2206 .

Freedman B ， Rodríguez J ， 2022 . On nonlinear discrete Sturm-Liouville boundary value problems with unbounded nonlinearities ［J］. J Differ Equ Appl ， 28 （ 1 ）： 1 - 9 .

Gao C H ， Yang E M ， Li H J ， 2024 . Solutions to a discrete resonance problem with eigenparameter-dependent boundary conditions ［J］. ERA ， 32 （ 3 ）： 1692 - 1707 .

Liu J S ， Wang S L ， Zhang J M ， 2011 . Multiple solutions for boundary value problems of second-order difference equations with resonance ［J］. J Math Anal Appl ， 374 （ 1 ）： 187 - 196 .

Liu M Y ， Pei M H ， Wang L B ， 2024 . Solvability of a nonlinear second order <math id="M276"><mi>m</mi></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120641946&type= https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=120641958&type= 2.28600001 2.28600001 -point boundary value problem with p -Laplacian at resonance ［J］. Bound Value Probl ， 2024 （ 1 ）： 46 .

Long Y H ， Chen J L ， 2018 . Existence of multiple solutions to second-order discrete Neumann boundary value problems ［J］. Appl Math Lett ， 83 ： 7 - 14 .

Ma R Y ， Gao C H ， Ma H L ， et al ， 2019 . Theory of difference equations and applications（in Chinese）［M］. Beijing ： Science Press .

Mawhin J ， 1979 . Topological degree methods in nonlinear boundary value problems ［M］. Providence ： AMS .

Papageorgiou N S ， Radulescu V D ， 2013 . Semilinear Neumann problems with indefinite and unbounded potential and crossing nonlinearity ［J］. Contemp Math ， 593 ： 293 - 315 .

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 .

Ward J R ， 1982 . Perturbations with some superlinear growth for a class of second order elliptic boundary value problems ［J］. Nonlinear Anal Theory Meth Appl ， 6 （ 4 ）： 367 - 374 .

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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