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Zbl 1094.34043
Teixeira, Eduardo V.
Strong solutions for differential equations in abstract spaces.
(English)
[J] J. Differ. Equations 214, No. 1, 65-91 (2005). ISSN 0022-0396

The author considers in a sequentially complete locally convex space $(E,{\Cal F})$ the initial value problem $$u_t(t)= f(t, u(t)),\quad u(t_0)= u_0.$$ The bounded elements of $E$ are denoted by $$E_b:= \Biggl\{x\in E:\Vert x\Vert_{{\Cal F}}= \sup_{\rho\in{\Cal F}}\,\rho(x)< \infty\Biggr\}.$$ $(E_b,{\Cal F})$ is supposed to be locally metrizable, and $B_{E_b}$ is supposed to be relatively compact with respect to the ${\Cal F}$-topology, $f: I\times E_b\to E_b$ is an ${\Cal F}$-Carathéodory map (i.e., an extension of the classical concept) satisfying a further inequality. By using the Schauder-Tychonoff fixed-point theorem, the local existence of a strong solution (i.e., differentiable with respect to the $\Vert\cdot\Vert_{{\Cal F}}$-norm) in $E_b$ with $x_0\in E_b$ is proved. The main tools used in this proof are developed in a preceding section. In the last section, the author studies by using the previously developed results a nonlinear differential equation involving the Hardy-Littlewood maximal operator and proves Lipschitz stability of the solution. I want to point out that the author makes an effort to explain his problem: In a long and interesting introduction, he gives a survey of the development of existence theorems for differential equations in Banach spaces. He compares assumptions and results and a long list of references is given. For his own problem, he gives motivations and discusses the assumptions.
[Marianne Reichert (Frankfurt / Main)]
MSC 2000:
*34G20 Nonlinear ODE in abstract spaces

Keywords: differential equations in locally convex spaces; strong solutions; regularity theory

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