Perimetric Contraction on Polygons and Related Fixed Point Theorems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909367633182720 |
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| author | Zhou, Mi Petrov, Evgeniy |
| author_facet | Zhou, Mi Petrov, Evgeniy |
| contents | In the present paper, a new type of mappings called perimetric contractions on $k$-polygons is introduced. These contractions can be viewed as a generalization of mappings that contracts perimeters of triangles. A fixed point theorem for this type of mappings in a complete metric space is established. Achieving a fixed point necessitates the avoidance of periodic points of prime period $2,3,\cdots, k-1$. The class of contraction mappings is encompassed by perimeter-based mappings, leading to the recovery of Banach's fixed point theorem as a direct outcome from our main result. A sufficient condition to guarantee the uniqueness of the fixed point is also provided. Moreover, we introduce the Kannan type perimetric contractions on $k$-polygons, establishing a fixed point theorem and a sufficient uniqueness condition. The relationship between these contractions, generalized Kannan type mappings, and mappings contracting the perimeters on $k$-polygons is investigated. Several examples are illustrated to support the validity of our main results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_20449 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Perimetric Contraction on Polygons and Related Fixed Point Theorems Zhou, Mi Petrov, Evgeniy General Topology Primary 47H09, Secondary 47H10 In the present paper, a new type of mappings called perimetric contractions on $k$-polygons is introduced. These contractions can be viewed as a generalization of mappings that contracts perimeters of triangles. A fixed point theorem for this type of mappings in a complete metric space is established. Achieving a fixed point necessitates the avoidance of periodic points of prime period $2,3,\cdots, k-1$. The class of contraction mappings is encompassed by perimeter-based mappings, leading to the recovery of Banach's fixed point theorem as a direct outcome from our main result. A sufficient condition to guarantee the uniqueness of the fixed point is also provided. Moreover, we introduce the Kannan type perimetric contractions on $k$-polygons, establishing a fixed point theorem and a sufficient uniqueness condition. The relationship between these contractions, generalized Kannan type mappings, and mappings contracting the perimeters on $k$-polygons is investigated. Several examples are illustrated to support the validity of our main results. |
| title | Perimetric Contraction on Polygons and Related Fixed Point Theorems |
| topic | General Topology Primary 47H09, Secondary 47H10 |
| url | https://arxiv.org/abs/2410.20449 |