The Orbit-Summable Fixed Point Criterion and its Relation to Caristi's Theorem
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arXiv
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| Format: | Preprint |
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2025
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| author | Sette, Roblêdo Mak's Miranda |
| author_facet | Sette, Roblêdo Mak's Miranda |
| contents | The relationship between geometric and variational principles remains central to Nonlinear Analysis. This paper introduces the \textbf{Orbit-Summability Fixed Point Criterion}, a novel, purely dynamical condition, and establishes its profound connection to \textbf{Caristi`s Fixed Point Theorem} in complete metric spaces.
Our criterion, which requires only that the total displacement along a single orbit be finite (the orbit-summability property), provides a practical and concrete tool for checking the existence of fixed points without relying on the construction of an abstract potential function.
We demonstrate that, under a minimal regularity assumption involving the function $f$ and the metric $d$, the Orbit-Summability Criterion is \textbf{precisely equivalent} to Caristi`s Fixed Point Theorem. This equivalence is conceptually significant as it creates a direct bridge between the geometric principle of \textbf{dynamical gap summability} (akin to the core idea in Banach`s Contraction Principle) and the variational principle of Caristi.
As a direct consequence of this equivalence, the classical \textbf{Banach Contraction Principle is recovered as a straightforward corollary}. The methodology is designed to merge key elements from the Banach proof (orbit convergence via summability) with the structural requirements of Caristi (majorization by a well-behaved functional), effectively providing a unifying framework for these fundamental results. Thus, the Orbit-Summability Criterion offers an attractive and accessible perspective on these established cornerstones of Fixed Point Theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18153 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Orbit-Summable Fixed Point Criterion and its Relation to Caristi's Theorem Sette, Roblêdo Mak's Miranda Functional Analysis 47H10 The relationship between geometric and variational principles remains central to Nonlinear Analysis. This paper introduces the \textbf{Orbit-Summability Fixed Point Criterion}, a novel, purely dynamical condition, and establishes its profound connection to \textbf{Caristi`s Fixed Point Theorem} in complete metric spaces. Our criterion, which requires only that the total displacement along a single orbit be finite (the orbit-summability property), provides a practical and concrete tool for checking the existence of fixed points without relying on the construction of an abstract potential function. We demonstrate that, under a minimal regularity assumption involving the function $f$ and the metric $d$, the Orbit-Summability Criterion is \textbf{precisely equivalent} to Caristi`s Fixed Point Theorem. This equivalence is conceptually significant as it creates a direct bridge between the geometric principle of \textbf{dynamical gap summability} (akin to the core idea in Banach`s Contraction Principle) and the variational principle of Caristi. As a direct consequence of this equivalence, the classical \textbf{Banach Contraction Principle is recovered as a straightforward corollary}. The methodology is designed to merge key elements from the Banach proof (orbit convergence via summability) with the structural requirements of Caristi (majorization by a well-behaved functional), effectively providing a unifying framework for these fundamental results. Thus, the Orbit-Summability Criterion offers an attractive and accessible perspective on these established cornerstones of Fixed Point Theory. |
| title | The Orbit-Summable Fixed Point Criterion and its Relation to Caristi's Theorem |
| topic | Functional Analysis 47H10 |
| url | https://arxiv.org/abs/2512.18153 |