A Class of Functionals on the Sequence Space $s$ Satisfying the Palais-Smale Condition
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914087686897664 |
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| author | Eftekharinasab, Kaveh |
| author_facet | Eftekharinasab, Kaveh |
| contents | We introduce a class of functionals on the space of rapidly decreasing sequences $s$, called $\mathcal{F}_s$-functionals, defined as decomposable sums of quadratic and convex terms with quadratic growth. We prove that such functionals satisfy the Palais-Smale condition and admit a unique global minimum. Furthermore, we show that the Palais-Smale condition is preserved under linear homeomorphisms. This allows us to construct corresponding functionals satisfying the Palais-Smale condition on Fréchet spaces isomorphic to $s$. We then show how this framework provides a tool for the proof of existence and uniqueness of solutions for specific operator problems, where coupled infinite-dimensional systems are transformed into diagonalized problems in the space $s$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10146 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Class of Functionals on the Sequence Space $s$ Satisfying the Palais-Smale Condition Eftekharinasab, Kaveh Functional Analysis 46A45, 47J30, 47J05 We introduce a class of functionals on the space of rapidly decreasing sequences $s$, called $\mathcal{F}_s$-functionals, defined as decomposable sums of quadratic and convex terms with quadratic growth. We prove that such functionals satisfy the Palais-Smale condition and admit a unique global minimum. Furthermore, we show that the Palais-Smale condition is preserved under linear homeomorphisms. This allows us to construct corresponding functionals satisfying the Palais-Smale condition on Fréchet spaces isomorphic to $s$. We then show how this framework provides a tool for the proof of existence and uniqueness of solutions for specific operator problems, where coupled infinite-dimensional systems are transformed into diagonalized problems in the space $s$. |
| title | A Class of Functionals on the Sequence Space $s$ Satisfying the Palais-Smale Condition |
| topic | Functional Analysis 46A45, 47J30, 47J05 |
| url | https://arxiv.org/abs/2510.10146 |