Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like Inequalities
The main aim of this paper is twofold. Our first objective is to study a new system of generalized multivalued variational-like inequalities in Banach spaces and to establish its equivalence with a system of fixed point problems utilizing the concept of <i>P</i>-<inline-formula><...
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MDPI AG
2022-06-01
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| Online Access: | https://www.mdpi.com/2227-7390/10/12/2016 |
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| author | Javad Balooee Shih-Sen Chang Lin Wang Zhaoli Ma |
| author_facet | Javad Balooee Shih-Sen Chang Lin Wang Zhaoli Ma |
| author_sort | Javad Balooee |
| collection | DOAJ |
| description | The main aim of this paper is twofold. Our first objective is to study a new system of generalized multivalued variational-like inequalities in Banach spaces and to establish its equivalence with a system of fixed point problems utilizing the concept of <i>P</i>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-proximal mapping. The obtained alternative equivalent formulation is used and a new iterative algorithm for finding its approximate solution is suggested. Under some appropriate assumptions imposed on the mappings and parameters involved in the system of generalized multivalued variational-like inequalities, the existence of solution for the system mentioned above is proved and the convergence analysis of the sequences generated by our proposed iterative algorithm is discussed. The second objective of this work is to investigate and analyze the notion <i>M</i>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-proximal mapping defined in the literature. Taking into account of the assumptions considered for such a mapping, we prove that every <i>M</i>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-proximal mapping is actually <i>P</i>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-proximal and is not a new one. At the same time, some comments relating to some existing results are pointed out. |
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| language | English |
| last_indexed | 2024-03-09T23:09:12Z |
| publishDate | 2022-06-01 |
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| series | Mathematics |
| spelling | doaj.art-cef529eeb98c42b8b223a4f2b974e0e22023-11-23T17:48:20ZengMDPI AGMathematics2227-73902022-06-011012201610.3390/math10122016Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like InequalitiesJavad Balooee0Shih-Sen Chang1Lin Wang2Zhaoli Ma3School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran 1417935840, IranCenter for General Education, China Medical University, Taichung 40402, TaiwanCollege of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, ChinaCollege of Public Foundation, Yunnan Open University, Kunming 650223, ChinaThe main aim of this paper is twofold. Our first objective is to study a new system of generalized multivalued variational-like inequalities in Banach spaces and to establish its equivalence with a system of fixed point problems utilizing the concept of <i>P</i>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-proximal mapping. The obtained alternative equivalent formulation is used and a new iterative algorithm for finding its approximate solution is suggested. Under some appropriate assumptions imposed on the mappings and parameters involved in the system of generalized multivalued variational-like inequalities, the existence of solution for the system mentioned above is proved and the convergence analysis of the sequences generated by our proposed iterative algorithm is discussed. The second objective of this work is to investigate and analyze the notion <i>M</i>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-proximal mapping defined in the literature. Taking into account of the assumptions considered for such a mapping, we prove that every <i>M</i>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-proximal mapping is actually <i>P</i>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-proximal and is not a new one. At the same time, some comments relating to some existing results are pointed out.https://www.mdpi.com/2227-7390/10/12/2016system of generalized multivalued variational-like inequalitiesgraph convergence<i>P</i>-<i>η</i>-proximal mappingiterative algorithmconvergence analysis |
| spellingShingle | Javad Balooee Shih-Sen Chang Lin Wang Zhaoli Ma Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like Inequalities Mathematics system of generalized multivalued variational-like inequalities graph convergence <i>P</i>-<i>η</i>-proximal mapping iterative algorithm convergence analysis |
| title | Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like Inequalities |
| title_full | Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like Inequalities |
| title_fullStr | Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like Inequalities |
| title_full_unstemmed | Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like Inequalities |
| title_short | Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like Inequalities |
| title_sort | algorithmic aspect and convergence analysis for system of generalized multivalued variational like inequalities |
| topic | system of generalized multivalued variational-like inequalities graph convergence <i>P</i>-<i>η</i>-proximal mapping iterative algorithm convergence analysis |
| url | https://www.mdpi.com/2227-7390/10/12/2016 |
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