Solvability of Coupled Forward-Backward Volterra Integral Equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917858419671040 |
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| author | Li, Wenyang Wang, Hanxiao Yong, Jiongmin |
| author_facet | Li, Wenyang Wang, Hanxiao Yong, Jiongmin |
| contents | Motivated by the optimality system associated with controlled (forward) Volterra integral equations (FVIEs, for short), the well-posedness of coupled forward-backward Voterra integral equations (FBVIEs, for short) is studied. The main feature of FBVIEs is that the unknown $\{(X(t,s),Y(t,s))\}$ has two arguments. By taking $t$ as a parameter and $s$ as a (time) variable, one can regard FBVIE as a system of ordinary differential equations (ODEs, for short), with infinite-dimensional space values $\{(X(\cdot,s),Y(\cdot,s));\,s\in[0,T]\}$. To establish the well-posedness of such an FBVIE, a new non-local monotonicity condition is introduced, by which a bridge in infinite-dimensional spaces is constructed. Then by generalizing the method of continuation developed by \cite{Hu-Peng1995,Yong1997,Peng-Wu1999} for differential equations, we have established the well-posedness of FBVIEs.The key is to apply the chain rule to the mapping $t\mapsto\big[\int_\cdot^T\langle Y(s,s),X(s,\cdot)\rangle ds +\langle G(X(T,T)),X(T,\cdot)\rangle\big](t)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_04268 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solvability of Coupled Forward-Backward Volterra Integral Equations Li, Wenyang Wang, Hanxiao Yong, Jiongmin Optimization and Control Classical Analysis and ODEs Motivated by the optimality system associated with controlled (forward) Volterra integral equations (FVIEs, for short), the well-posedness of coupled forward-backward Voterra integral equations (FBVIEs, for short) is studied. The main feature of FBVIEs is that the unknown $\{(X(t,s),Y(t,s))\}$ has two arguments. By taking $t$ as a parameter and $s$ as a (time) variable, one can regard FBVIE as a system of ordinary differential equations (ODEs, for short), with infinite-dimensional space values $\{(X(\cdot,s),Y(\cdot,s));\,s\in[0,T]\}$. To establish the well-posedness of such an FBVIE, a new non-local monotonicity condition is introduced, by which a bridge in infinite-dimensional spaces is constructed. Then by generalizing the method of continuation developed by \cite{Hu-Peng1995,Yong1997,Peng-Wu1999} for differential equations, we have established the well-posedness of FBVIEs.The key is to apply the chain rule to the mapping $t\mapsto\big[\int_\cdot^T\langle Y(s,s),X(s,\cdot)\rangle ds +\langle G(X(T,T)),X(T,\cdot)\rangle\big](t)$. |
| title | Solvability of Coupled Forward-Backward Volterra Integral Equations |
| topic | Optimization and Control Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2412.04268 |