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Zbl 0571.47051
Kato, Tosio
Locally coercive nonlinear equations, with applications to some periodic solutions. (English)
[J] Duke Math. J. 51, 923-936 (1984). ISSN 0012-7094

The equation $$ (NL)\quad Au=f, $$ where $u\in Y$ is the unknown and $f\in Y\sp*$ is a given element is considered. Namely, the setting of the problem is the following: \par i) $\{Y,Y\sp*\}$ is a pair of real Banach spaces in duality. This means that there is a nondegenerate continuous bilinear form $<\ ,\ >$ on $Y\times Y\sp*$. Moreover Y is reflexive and separable. \par ii) There is another pair $\{V,V\sp*\}$ of Banach spaces in duality, with V separable such that $V\subset Y$ and $V\sp*\supset Y\sp*$, with the injections continuous and dense. Moreover, the duality $<\ ,\ >$ on $V\times V\sp*$ is compatible with that of $\{Y,Y\sp*\}.$ \par iii) There is a bounded, closed and convex subset K of Y, containing the origin O as an internal point, and weakly sequentially continuous map A of K into $V\sp*$, such that $<v,Av>\ge \beta \ge 0$ for all $v\in V\cap \hat K$, where $\hat K$ denotes the set of bounding points of K. \par Then the main result is the following \par Theorem. Under the assumptions i), ii), iii), we have $AK\supset \beta K\sp 0$, where $K\sp 0\subset Y\sp*$ is the polar set of K. In other words, (NL) has a solution $u\in K$ for every $f\in \beta K\sp 0.$ \par Another theorem proves under some added assumptions uniqueness of the solution u. In the rest of the paper these theorems are used to prove existence, uniqueness and continuous dependence for periodic solutions to certain nonlinear PDE's.
[J.Siška]

MSC 2000:: *47J05 Equations involving nonlinear operators (general)
47H05 Monotone operators (with respect to duality)
35B10 Periodic solutions of PDE

Keywords: space in duality; higher order equations; polar; periodic solutions to certain nonlinear PDE's

Cited in: Zbl 0721.35019 Zbl 0654.47042

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