On the Weierstrass Theorem on extrema and its application for finding fixed points and zeros of mappings
In this paper, we examine the applicability of the Lagrange Multiplier Rule, specifically the Karush–Kuhn–Tucker Theorem, to investigate the existence of fixed points and zeros of certain potential mappings between finite and infinite spaces. Such zeros are investigated as constrained extrema, whose...
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World Scientific Publishing
2025-08-01
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| Series: | Bulletin of Mathematical Sciences |
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| Online Access: | https://www.worldscientific.com/doi/10.1142/S1664360725500031 |
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| author | Marek Galewski |
| author_facet | Marek Galewski |
| author_sort | Marek Galewski |
| collection | DOAJ |
| description | In this paper, we examine the applicability of the Lagrange Multiplier Rule, specifically the Karush–Kuhn–Tucker Theorem, to investigate the existence of fixed points and zeros of certain potential mappings between finite and infinite spaces. Such zeros are investigated as constrained extrema, whose existence is established via the Weierstrass Theorem. By assuming the potentiality of the operator, we are able to relax the required invariance conditions. The paper is divided into two parts. In the first part, we provide simple proofs of the Lagrange Multiplier Rule and the Karush–Kuhn–Tucker Theorem in some special cases. In the second part, these results are used to investigate potential counterparts of several known theorems, such as the Brouwer Fixed Point Theorem and the Schauder Fixed Point Theorem. |
| format | Article |
| id | doaj-art-36f09b68fc184fb0b7223a6799e672ae |
| institution | Kabale University |
| issn | 1664-3607 1664-3615 |
| language | English |
| publishDate | 2025-08-01 |
| publisher | World Scientific Publishing |
| record_format | Article |
| series | Bulletin of Mathematical Sciences |
| spelling | doaj-art-36f09b68fc184fb0b7223a6799e672ae2025-08-22T07:46:40ZengWorld Scientific PublishingBulletin of Mathematical Sciences1664-36071664-36152025-08-01150210.1142/S1664360725500031On the Weierstrass Theorem on extrema and its application for finding fixed points and zeros of mappingsMarek Galewski0Institute of Mathematics, Lodz University of Technology, Al. Politechniki 8, 93-590 Lodz, PolandIn this paper, we examine the applicability of the Lagrange Multiplier Rule, specifically the Karush–Kuhn–Tucker Theorem, to investigate the existence of fixed points and zeros of certain potential mappings between finite and infinite spaces. Such zeros are investigated as constrained extrema, whose existence is established via the Weierstrass Theorem. By assuming the potentiality of the operator, we are able to relax the required invariance conditions. The paper is divided into two parts. In the first part, we provide simple proofs of the Lagrange Multiplier Rule and the Karush–Kuhn–Tucker Theorem in some special cases. In the second part, these results are used to investigate potential counterparts of several known theorems, such as the Brouwer Fixed Point Theorem and the Schauder Fixed Point Theorem.https://www.worldscientific.com/doi/10.1142/S1664360725500031Lagrange multiplier ruleKuhn–Tucker theoremSchauder theoremBrouwer lemmaKrasnoselskii theorem |
| spellingShingle | Marek Galewski On the Weierstrass Theorem on extrema and its application for finding fixed points and zeros of mappings Bulletin of Mathematical Sciences Lagrange multiplier rule Kuhn–Tucker theorem Schauder theorem Brouwer lemma Krasnoselskii theorem |
| title | On the Weierstrass Theorem on extrema and its application for finding fixed points and zeros of mappings |
| title_full | On the Weierstrass Theorem on extrema and its application for finding fixed points and zeros of mappings |
| title_fullStr | On the Weierstrass Theorem on extrema and its application for finding fixed points and zeros of mappings |
| title_full_unstemmed | On the Weierstrass Theorem on extrema and its application for finding fixed points and zeros of mappings |
| title_short | On the Weierstrass Theorem on extrema and its application for finding fixed points and zeros of mappings |
| title_sort | on the weierstrass theorem on extrema and its application for finding fixed points and zeros of mappings |
| topic | Lagrange multiplier rule Kuhn–Tucker theorem Schauder theorem Brouwer lemma Krasnoselskii theorem |
| url | https://www.worldscientific.com/doi/10.1142/S1664360725500031 |
| work_keys_str_mv | AT marekgalewski ontheweierstrasstheoremonextremaanditsapplicationforfindingfixedpointsandzerosofmappings |