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Existence and stability of fixed point sets of multivalued mappings in b-metric spaces with applications
Thesis (PhD (Applied Mathematics))--University of Pretoria, 2017.
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| Format: | Thesis |
| Language: | English |
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2026
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| _version_ | 1867613582996275200 |
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| access_status_str | Open Access |
| author2 | Abbas, Mujahid |
| author_browse | Abbas, Mujahid |
| author_facet | Abbas, Mujahid |
| collection | Thesis |
| description | Thesis (PhD (Applied Mathematics))--University of Pretoria, 2017. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/110183 |
| institution | University of Pretoria (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:38:26.847Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace — University of Pretoria Institutional Repository |
| publishDate | 2026 |
| publishDateRange | 2026 |
| publishDateSort | 2026 |
| record_format | dspace |
| source_str | UPSpace — University of Pretoria Institutional Repository |
| spelling | oai:repository.up.ac.za:2263/110183 Existence and stability of fixed point sets of multivalued mappings in b-metric spaces with applications Abbas, Mujahid basit.aa@gmail.com Ali, Basit Metric spaces, b-metric spaces, multivalued mapping, fixed point, fixed point problems, Suzuki type, best approximations, data dependence, stability, Ulam-Hyers stability, well-posedness, multivalued fractals, delay differential equations. Thesis (PhD (Applied Mathematics))--University of Pretoria, 2017. The aim of this research is to investigate the existence and stability properties of fixed point sets of multivalued mappings in the setup of b-metric spaces. Czerwik (1993) introduced b-metric spaces as generalization of metric spaces. These spaces are semi-metrizable and hence are first countable. The distance function used in b-metric spaces is not continuous in general. Triangular inequality involves a constant greater than or equal to one (called b-metric constant). Unlike metric spaces, open balls are not open and closed balls are not closed sets in general in sequential topology on b-metric spaces (An et al., (2015)). The first main problem that is addressed in this thesis is the existence of fixed point sets of multivalued mappings that satisfy generalized Suzuki type contraction conditions in b-metric spaces. This class includes the class of mappings satisfying various contractions like Kannan, Edelstein strict contractions, Suzuki type contractions and strict contractions, Ciric quasi-contractions, admissible multivalued contractions, almost contractions and Banach contractions. Existence of fixed points of multivalued mappings further provided the existence of coincidence and common fixed points of hybrid pairs of mappings in b-metric spaces. After setting the problem of existence of fixed points of such mappings, the second part or problem is related to the stability properties of fixed point sets of multivalued mappings and fixed point inclusions. Regarding this we discuss the problem of data dependence of fixed point sets, that is to find an upper bound for the Hausdorff distance between the ii fixed point sets of two multivalued mappings. Moreover, we study the stability of fixed point sets of uniformly convergent sequence of multivalued contractions on b-metric spaces, that is if a sequence of multivalued operators converges uniformly to a limit operator, then the corresponding sequence of fixed point sets converges to the fixed point set of limit operator. Generalized Ulam-Hyers stability of fixed point problems related to admissible multivalued mappings of b-metric spaces has been discussed as well. Another important issue is the completeness problem, that is to know under what circumstances, the underlying space is complete. One approach is to study completeness characterization via fixed point theory. In this research, a fixed point theorem for multivalued mappings further yields a completeness characterization of underlying metric space. Note that Banach contraction principle does not characterize completeness of underlying metric space (Connel (1959), Elekes (2009)). Some results obtained in this research have found some interesting applications. The scope of the applications is not limited to the ones presented in this study. A result have been applied in best approximation theory to prove the existence of a point in the intersection of fixed point sets and the set of best approximations. Further, existence of multivalued fractals, solutions of functional equations that arise in connection with dynamic programming, delay differential equations, have been obtained as applications. A local fixed point theorem is obtained and applied in a homotopy result. To obtain the main results in this thesis, we use method of successive approximations, topological properties and tools from the theory of b-metric spaces and set valued analysis. Contraction conditions involve b-metric constant and some auxiliary functions such that the analogous metric version of these conditions still generalize the existing contraction conditions. Further, comparison based on examples and remarks shows that the obtained results generalize, extend, and unify the results in metric fixed point theory of multivalued valued mappings. Mathematics and Applied Mathematics PhD (Applied Mathematics) 2026-05-15T17:26:35Z 2026-05-15T17:26:35Z 17/02/06 2017 Thesis http://hdl.handle.net/2263/110183 en application/pdf |
| spellingShingle | Metric spaces, b-metric spaces, multivalued mapping, fixed point, fixed point problems, Suzuki type, best approximations, data dependence, stability, Ulam-Hyers stability, well-posedness, multivalued fractals, delay differential equations. Existence and stability of fixed point sets of multivalued mappings in b-metric spaces with applications |
| title | Existence and stability of fixed point sets of multivalued mappings in b-metric spaces with applications |
| title_full | Existence and stability of fixed point sets of multivalued mappings in b-metric spaces with applications |
| title_fullStr | Existence and stability of fixed point sets of multivalued mappings in b-metric spaces with applications |
| title_full_unstemmed | Existence and stability of fixed point sets of multivalued mappings in b-metric spaces with applications |
| title_short | Existence and stability of fixed point sets of multivalued mappings in b-metric spaces with applications |
| title_sort | existence and stability of fixed point sets of multivalued mappings in b metric spaces with applications |
| topic | Metric spaces, b-metric spaces, multivalued mapping, fixed point, fixed point problems, Suzuki type, best approximations, data dependence, stability, Ulam-Hyers stability, well-posedness, multivalued fractals, delay differential equations. |
| url | http://hdl.handle.net/2263/110183 |