Some problems on ordinary differential equations in Banach spaces

Hájek, Petr; Vivi, Paola

doi:10.5052/racsam.2010.16

Cited by 3 publications

(2 citation statements)

References 27 publications

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“…A survey of existing results on solvability and uniqueness of solutions going beyond the classical Cauchy-Picard theorem from finite to infinite settings journey can be found in [20].…”

Section: Formulation Of the Problemmentioning

confidence: 99%

Dynamical System Related to Primal–Dual Splitting Projection Methods

2021

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We introduce a dynamical system to the problem of finding zeros of the sum of two maximally monotone operators. We investigate the existence, uniqueness and extendability of solutions to this dynamical system in a Hilbert space. We prove that the trajectories of the proposed dynamical system converge strongly to a primal–dual solution of the considered problem. Under explicit time discretization of the dynamical system we obtain the best approximation algorithm for solving coupled monotone inclusion problem.

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“…A survey of existing results on solvability and uniqueness of solutions going beyond the classical Cauchy-Picard theorem from finite to infinite settings journey can be found in [20].…”

Section: Formulation Of the Problemmentioning

confidence: 99%

Dynamical System Related to Primal–Dual Splitting Projection Methods

2021

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“…Such systematic investigations, however, are beyond the scope of this article; for that, the rich literature on this topic has to be consulted (see e.g. Tikhonov (1934); Valeev & Zhautykov (1974); Deimling (1977); Samoilenko & Teplinskii (2003); Hájek & Vivi (2010); Fabian et al (2011)). Instead, we will only make a small excursion into the uniqueness issue of these solutions when restricting the infinite system by a sufficient set of initial conditions, but only to show where still the problems lie and not on how to solve these problems.…”

Section: Introduction Motivation and A Brief Overview What To Expectmentioning

confidence: 99%

Application of Lie-group symmetry analysis to an infinite hierarchy of differential equations at the example of first order ODEs

Frewer

2015

Preprint

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This study will explicitly demonstrate by example that an unrestricted infinite and forward recursive hierarchy of differential equations must be identified as an unclosed system of equations, despite the fact that to each unknown function in the hierarchy there exists a corresponding determined equation to which it can be bijectively mapped to. As a direct consequence, its admitted set of symmetry transformations must be identified as a weaker set of indeterminate equivalence transformations. The reason is that no unique general solution can be constructed, not even in principle. Instead, infinitely many disjoint and thus independent general solution manifolds exist. This is in clear contrast to a closed system of differential equations that only allows for a single and thus unique general solution manifold, which, by definition, covers all possible particular solutions this system can admit. Herein, different first order Riccati-ODEs serve as an example, but this analysis is not restricted to them. All conclusions drawn in this study will translate to any first order or higher order ODEs as well as to any PDEs.

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On a Cauchy Problem in Banach Spaces

Cheng

Wen

2023

Results Math

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Some problems on ordinary differential equations in Banach spaces

Cited by 3 publications

References 27 publications

Dynamical System Related to Primal–Dual Splitting Projection Methods

Dynamical System Related to Primal–Dual Splitting Projection Methods

Application of Lie-group symmetry analysis to an infinite hierarchy of differential equations at the example of first order ODEs

On a Cauchy Problem in Banach Spaces

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