Existence of uncountably many periodic solutions for second-order superlinear difference equations with continuous time
Due to the essential difficulty of establishing an appropriate variational framework on a suitable working space, how to apply the critical point theory for showing the existence and multiplicity of periodic solutions of continuous-time difference equations remains a completely open problem. New ide...
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World Scientific Publishing
2025-04-01
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| Series: | Bulletin of Mathematical Sciences |
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| Online Access: | https://www.worldscientific.com/doi/10.1142/S1664360724500103 |
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| author | Genghong Lin Zhan Zhou Zupei Shen Jianshe Yu |
| author_facet | Genghong Lin Zhan Zhou Zupei Shen Jianshe Yu |
| author_sort | Genghong Lin |
| collection | DOAJ |
| description | Due to the essential difficulty of establishing an appropriate variational framework on a suitable working space, how to apply the critical point theory for showing the existence and multiplicity of periodic solutions of continuous-time difference equations remains a completely open problem. New ideas including gluing arguments are introduced in this work to overcome such a difficulty. This enables us to employ the critical point theory to construct uncountably many periodic solutions for a class of superlinear continuous-time difference equations without assuming symmetry properties on the nonlinear terms. The obtained solutions are piecewise differentiable in some cases, distinguishing continuous-time difference equations from ordinary differential equations qualitatively. To the best of our knowledge, this is the first time in the literature that the critical point theory has been used for such types of problems. Our work may open an avenue for studying discrete nonlinear systems with continuous time via the critical point theory. |
| format | Article |
| id | doaj-art-2cb716907cee4b749ad1c72d6e5d36dc |
| institution | OA Journals |
| issn | 1664-3607 1664-3615 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | World Scientific Publishing |
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| series | Bulletin of Mathematical Sciences |
| spelling | doaj-art-2cb716907cee4b749ad1c72d6e5d36dc2025-08-20T01:52:10ZengWorld Scientific PublishingBulletin of Mathematical Sciences1664-36071664-36152025-04-01150110.1142/S1664360724500103Existence of uncountably many periodic solutions for second-order superlinear difference equations with continuous timeGenghong Lin0Zhan Zhou1Zupei Shen2Jianshe Yu3Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, P. R. ChinaGuangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, P. R. ChinaSchool of Financial Mathematics and Statistics, Guangdong University of Finance, Guangzhou 510521, P. R. ChinaGuangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, P. R. ChinaDue to the essential difficulty of establishing an appropriate variational framework on a suitable working space, how to apply the critical point theory for showing the existence and multiplicity of periodic solutions of continuous-time difference equations remains a completely open problem. New ideas including gluing arguments are introduced in this work to overcome such a difficulty. This enables us to employ the critical point theory to construct uncountably many periodic solutions for a class of superlinear continuous-time difference equations without assuming symmetry properties on the nonlinear terms. The obtained solutions are piecewise differentiable in some cases, distinguishing continuous-time difference equations from ordinary differential equations qualitatively. To the best of our knowledge, this is the first time in the literature that the critical point theory has been used for such types of problems. Our work may open an avenue for studying discrete nonlinear systems with continuous time via the critical point theory.https://www.worldscientific.com/doi/10.1142/S1664360724500103Difference equationcontinuous timeperiodic solutionmultiplicitysuperlinear nonlinearitycritical point theory |
| spellingShingle | Genghong Lin Zhan Zhou Zupei Shen Jianshe Yu Existence of uncountably many periodic solutions for second-order superlinear difference equations with continuous time Bulletin of Mathematical Sciences Difference equation continuous time periodic solution multiplicity superlinear nonlinearity critical point theory |
| title | Existence of uncountably many periodic solutions for second-order superlinear difference equations with continuous time |
| title_full | Existence of uncountably many periodic solutions for second-order superlinear difference equations with continuous time |
| title_fullStr | Existence of uncountably many periodic solutions for second-order superlinear difference equations with continuous time |
| title_full_unstemmed | Existence of uncountably many periodic solutions for second-order superlinear difference equations with continuous time |
| title_short | Existence of uncountably many periodic solutions for second-order superlinear difference equations with continuous time |
| title_sort | existence of uncountably many periodic solutions for second order superlinear difference equations with continuous time |
| topic | Difference equation continuous time periodic solution multiplicity superlinear nonlinearity critical point theory |
| url | https://www.worldscientific.com/doi/10.1142/S1664360724500103 |
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