Differentiability in Fréchet spaces and delay differential equations

Walther, Hans‐Otto

doi:10.14232/ejqtde.2019.1.13

Cited by 4 publications

(10 citation statements)

References 20 publications

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“…Recently, Nishiguchi [17] showed the same for history spaces of general Sobolev type. Walther [21] discusses different kinds of C 1 -differentiability in Fréchet spaces. None of these articles touch upon the question of regularity beyond C 1 .…”

Section: Smooth Dependence On Initial History and Delaymentioning

confidence: 99%

Periodic delay orbits and the polyfold implicit function theorem

Albers

Seifert

2022

Comment. Math. Helv.

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We consider differential delay equations of the form \partial_tx(t) = X_{t}(x(t - \tau)) in \mathbb{R}^n , where (X_t)_{t\in S^1} is a time-dependent family of smooth vector fields on \mathbb{R}^n and \tau is a delay parameter. If there is a (suitably non-degenerate) periodic solution x_0 of this equation for \tau=0 , that is without delay, there are good reasons to expect existence of a family of periodic solutions for all sufficiently small delays, smoothly parametrized by \tau . However, it seems difficult to prove this using the classical implicit function theorem, since the equation above, considered as an operator, is not smooth in the delay parameter. In this paper, we show how to use the M-polyfold implicit function theorem by Hofer–Wysocki–Zehnder (2009, 2021) to overcome this problem in a natural setup.

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Section: Smooth Dependence On Initial History and Delaymentioning

confidence: 99%

Periodic delay orbits and the polyfold implicit function theorem

Albers

Seifert

2022

Comment. Math. Helv.

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“…In general C 1 F -smoothness is the stronger property. [16] and [15,Section 8] contain examples of maps which are C 1 MB -smooth but not C 1 F -smooth. The motivation to present results in both settings is that in work involving calculus in topological vector spaces C 1 MB -smoothness seems to be quite common whereas in our application to VIDEs we obtain an associated equation (1.3) with a map g which is in fact C 1 F -smooth.…”

Section: Introductionmentioning

confidence: 99%

“…given by continuously differentiable functions F and d : R → (0, ∞). For the latter continuously differentiable solution operators exist on submanifolds of the Fréchet space C 1 = C 1 ((−∞, 0], R) of continuously differentiable maps (−∞, 0] → R, with the topology of locally uniform convergence of maps and their derivatives [13,15].…”

Section: Introductionmentioning

confidence: 99%

“…Below in the appendix Section 8 we collect simple additional facts from calculus based on C 1 Fsmoothness. Proofs are given in [15].…”

Section: Introductionmentioning

confidence: 99%

“…[15, Proposition 3.2] Let F and G be Fréchet spaces, U ⊂ F open. A map g :U → G is C 1 F -smooth if and only if it is C 1 MB -smooth with U ∋ u → Dg(u) ∈ L c (F, G) β-continuous.Continuous linear maps L : F → G between Fréchet spaces are C 1 F -smooth since they are C 1 MB -smooth with constant derivative DL(u) = L for all u ∈ F , and differentiation g → Dg of C 1 F -maps U → G is linear.…”

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confidence: 99%

See 2 more Smart Citations

Delay differential equations with differentiable solution operators on open domains in $C((-\infty,0],\mathbb{R}^n)$, and processes for Volterra integro-differential equations

Walther¹

2018

Preprint

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For autonomous delay differential equations x ′ (t) = f (xt) we construct a continuous semiflow of continuously differentiable solution operators x 0 → xt, t ≥ 0, on open subsets of the Fréchet space C((−∞, 0], R n ). For nonautonomous equations this yields a continuous process of differentiable solution operators. As an application we obtain processes which incorporate all solutions of Volterra integro-differential equations x ′ (t) = t 0 k(t, s)h(x(s))ds.

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Maps That Are Continuously Differentiable in the Michal and Bastiani Sense But Not in the Fréchet Sense

Walther¹

2021

J Math Sci

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Differentiability in Fréchet spaces and delay differential equations

Cited by 4 publications

References 20 publications

Periodic delay orbits and the polyfold implicit function theorem

Periodic delay orbits and the polyfold implicit function theorem

Delay differential equations with differentiable solution operators on open domains in $C((-\infty,0],\mathbb{R}^n)$, and processes for Volterra integro-differential equations

Maps That Are Continuously Differentiable in the Michal and Bastiani Sense But Not in the Fréchet Sense

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