The Convergence and Boundedness of Solutions to SFDEs with the G-Framework
Generally, stochastic functional differential equations (SFDEs) pose a challenge as they often lack explicit exact solutions. Consequently, it becomes necessary to seek certain favorable conditions under which numerical solutions can converge towards the exact solutions. This article aims to delve i...
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| Format: | Article |
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MDPI AG
2024-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/2/279 |
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| author | Rahman Ullah Faiz Faizullah Quanxin Zhu |
| author_facet | Rahman Ullah Faiz Faizullah Quanxin Zhu |
| author_sort | Rahman Ullah |
| collection | DOAJ |
| description | Generally, stochastic functional differential equations (SFDEs) pose a challenge as they often lack explicit exact solutions. Consequently, it becomes necessary to seek certain favorable conditions under which numerical solutions can converge towards the exact solutions. This article aims to delve into the convergence analysis of solutions for stochastic functional differential equations by employing the framework of G-Brownian motion. To establish the goal, we find a set of useful monotone type conditions and work within the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">C</mi><mi>r</mi></msub><mrow><mo>(</mo><mrow><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mn>0</mn><mo>]</mo></mrow><mo>;</mo><msup><mi>R</mi><mi>n</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The investigation conducted in this article confirms the mean square boundedness of solutions. Furthermore, this study enables us to compute both <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">L</mi><mi>G</mi><mn>2</mn></msubsup></semantics></math></inline-formula> and exponential estimates. |
| first_indexed | 2024-03-08T10:41:54Z |
| format | Article |
| id | doaj.art-036f2f5c2d9c45b2ba4c0e9c402e5282 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T10:41:54Z |
| publishDate | 2024-01-01 |
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| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-036f2f5c2d9c45b2ba4c0e9c402e52822024-01-26T17:32:49ZengMDPI AGMathematics2227-73902024-01-0112227910.3390/math12020279The Convergence and Boundedness of Solutions to SFDEs with the G-FrameworkRahman Ullah0Faiz Faizullah1Quanxin Zhu2School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, ChinaCollege of Electrical and Mechanical Engineering, National University of Sciences and Technology (NUST), Islamabad 44000, PakistanMOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, ChinaGenerally, stochastic functional differential equations (SFDEs) pose a challenge as they often lack explicit exact solutions. Consequently, it becomes necessary to seek certain favorable conditions under which numerical solutions can converge towards the exact solutions. This article aims to delve into the convergence analysis of solutions for stochastic functional differential equations by employing the framework of G-Brownian motion. To establish the goal, we find a set of useful monotone type conditions and work within the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">C</mi><mi>r</mi></msub><mrow><mo>(</mo><mrow><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mn>0</mn><mo>]</mo></mrow><mo>;</mo><msup><mi>R</mi><mi>n</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The investigation conducted in this article confirms the mean square boundedness of solutions. Furthermore, this study enables us to compute both <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">L</mi><mi>G</mi><mn>2</mn></msubsup></semantics></math></inline-formula> and exponential estimates.https://www.mdpi.com/2227-7390/12/2/279G-Brownian motionexponential and <named-content content-type="inline-formula"><inline-formula> <mml:math display="block" id="mm300"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi mathvariant="double-struck">L</mml:mi> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content> estimatesboundednessconvergence |
| spellingShingle | Rahman Ullah Faiz Faizullah Quanxin Zhu The Convergence and Boundedness of Solutions to SFDEs with the G-Framework Mathematics G-Brownian motion exponential and <named-content content-type="inline-formula"><inline-formula> <mml:math display="block" id="mm300"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi mathvariant="double-struck">L</mml:mi> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content> estimates boundedness convergence |
| title | The Convergence and Boundedness of Solutions to SFDEs with the G-Framework |
| title_full | The Convergence and Boundedness of Solutions to SFDEs with the G-Framework |
| title_fullStr | The Convergence and Boundedness of Solutions to SFDEs with the G-Framework |
| title_full_unstemmed | The Convergence and Boundedness of Solutions to SFDEs with the G-Framework |
| title_short | The Convergence and Boundedness of Solutions to SFDEs with the G-Framework |
| title_sort | convergence and boundedness of solutions to sfdes with the g framework |
| topic | G-Brownian motion exponential and <named-content content-type="inline-formula"><inline-formula> <mml:math display="block" id="mm300"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi mathvariant="double-struck">L</mml:mi> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content> estimates boundedness convergence |
| url | https://www.mdpi.com/2227-7390/12/2/279 |
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