Fixed point theory in metric and normed Spaces
- Authors: Naude, Luan
- Date: 2022-12
- Subjects: Metric spaces – South Africa , Banach spaces – South Africa , Mappings (Mathematics)
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10948/59898 , vital:62686
- Description: In this dissertation, we present major results in the theory of fixed points in metric and normed spaces. We start with a review of the Banach fixed point theorem and some of its applications (in systems of linear equations, differential equations, integral equations, and dynamical systems), and then discuss many of its extensions and generalizations. We look at the theorem of Edelstein ([8, Remark 3.1]) in compact metric spaces, and a generalizaton of it proved by Suzuki ([19, Theorem 3]) in 2009. We then give a detailed account of the work of Meyers: In [11], Meyers proved generalizations of the Banach fixed point theorem to uniform local contractions, and, in [10], a converse to the Banach fixed point theorem. Finally, we look at some of Browder’s work in fixed point theory. In [6], he showed the existence of fixed points for nonexpansive mappings on bounded, closed, and convex sets in uniformly convex Banach spaces, and, in [5], he proved similar results in Hilbert spaces using a connection between nonexpansive mappings and monotone operators. Keywords: Fixed point, Functional analysis, Metric spaces, Banach spaces, Hilbert spaces, contractions, Banach fixed point theorem, nonexpansive mappings. , Thesis (MSc) -- Faculty of Science, School of Computer Science, Mathematics, Physics and Statistics, 2022
- Full Text:
- Date Issued: 2022-12
- Authors: Naude, Luan
- Date: 2022-12
- Subjects: Metric spaces – South Africa , Banach spaces – South Africa , Mappings (Mathematics)
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10948/59898 , vital:62686
- Description: In this dissertation, we present major results in the theory of fixed points in metric and normed spaces. We start with a review of the Banach fixed point theorem and some of its applications (in systems of linear equations, differential equations, integral equations, and dynamical systems), and then discuss many of its extensions and generalizations. We look at the theorem of Edelstein ([8, Remark 3.1]) in compact metric spaces, and a generalizaton of it proved by Suzuki ([19, Theorem 3]) in 2009. We then give a detailed account of the work of Meyers: In [11], Meyers proved generalizations of the Banach fixed point theorem to uniform local contractions, and, in [10], a converse to the Banach fixed point theorem. Finally, we look at some of Browder’s work in fixed point theory. In [6], he showed the existence of fixed points for nonexpansive mappings on bounded, closed, and convex sets in uniformly convex Banach spaces, and, in [5], he proved similar results in Hilbert spaces using a connection between nonexpansive mappings and monotone operators. Keywords: Fixed point, Functional analysis, Metric spaces, Banach spaces, Hilbert spaces, contractions, Banach fixed point theorem, nonexpansive mappings. , Thesis (MSc) -- Faculty of Science, School of Computer Science, Mathematics, Physics and Statistics, 2022
- Full Text:
- Date Issued: 2022-12
Generalizations of some fixed point theorems in banach and metric spaces
- Niyitegeka, Jean Marie Vianney
- Authors: Niyitegeka, Jean Marie Vianney
- Date: 2015
- Subjects: Fixed point theory , Banach spaces , Mappings (Mathematics)
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10948/5265 , vital:20828
- Description: A fixed point of a mapping is an element in the domain of the mapping that is mapped into itself by the mapping. The study of fixed points has been a field of interests to mathematicians since the discovery of the Banach contraction theorem, i.e. if is a complete metric space and is a contraction mapping (i.e. there exists such that for all ), then has a unique fixed point. The Banach contraction theorem has found many applications in pure and applied mathematics. Due to fixed point theory being a mixture of analysis, geometry, algebra and topology, its applications to other fields such as physics, economics, game theory, chemistry, engineering and many others has become vital. The theory is nowadays a very active field of research in which many new theorems are published, some of them applied and many others generalized. Motivated by all of this, we give an exposition of some generalizations of fixed point theorems in metric fixed point theory, which is a branch of fixed point theory about results of fixed points of mappings between metric spaces, where certain properties of the mappings involved need not be preserved under equivalent metrics. For instance, the contractive property of mappings between metric spaces need not be preserved under equivalent metrics. Since metric fixed point theory is wide, we limit ourselves to fixed point theorems for self and non-self-mappings on Banach and metric spaces. We also take a look at some open problems on this topic of study. At the end of the dissertation, we suggest our own problems for future research.
- Full Text:
- Date Issued: 2015
- Authors: Niyitegeka, Jean Marie Vianney
- Date: 2015
- Subjects: Fixed point theory , Banach spaces , Mappings (Mathematics)
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10948/5265 , vital:20828
- Description: A fixed point of a mapping is an element in the domain of the mapping that is mapped into itself by the mapping. The study of fixed points has been a field of interests to mathematicians since the discovery of the Banach contraction theorem, i.e. if is a complete metric space and is a contraction mapping (i.e. there exists such that for all ), then has a unique fixed point. The Banach contraction theorem has found many applications in pure and applied mathematics. Due to fixed point theory being a mixture of analysis, geometry, algebra and topology, its applications to other fields such as physics, economics, game theory, chemistry, engineering and many others has become vital. The theory is nowadays a very active field of research in which many new theorems are published, some of them applied and many others generalized. Motivated by all of this, we give an exposition of some generalizations of fixed point theorems in metric fixed point theory, which is a branch of fixed point theory about results of fixed points of mappings between metric spaces, where certain properties of the mappings involved need not be preserved under equivalent metrics. For instance, the contractive property of mappings between metric spaces need not be preserved under equivalent metrics. Since metric fixed point theory is wide, we limit ourselves to fixed point theorems for self and non-self-mappings on Banach and metric spaces. We also take a look at some open problems on this topic of study. At the end of the dissertation, we suggest our own problems for future research.
- Full Text:
- Date Issued: 2015
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