By Earl A. Coddington (auth.), Prof. William N. Everitt (eds.)

**Read or Download Spectral Theory and Differential Equations: Proceedings of the Symposium held at Dundee, Scotland, 1–19 July, 1974 PDF**

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**Extra resources for Spectral Theory and Differential Equations: Proceedings of the Symposium held at Dundee, Scotland, 1–19 July, 1974**

**Example text**

Furthermore, the new norm may In this way we may generalize condition 49 Consider the space in an obvious way. II II and N = {11 i{vl of all equivalent norms in X, metrized For example, we may define the distance between two norms II II by log max{ sup Ilxli v / X We shall denote by X the space llxll X # ~ sup X hBxl[#/ Hxll~}. renormed by II IIv. We may now assume that there are maps v : [O,T] × W ~ : [O,T] × W -~ (-co, co) assume that v and ~ N and such that A(t,y) E G ( X (t,y),1, ~(t,y)). We should also are smooth in a certain, possibly complicated sense.

Here we list two simple lemmas that follow immediately from (X). LEMMA 7°3 If a subset of Y is convex, closed and bounded, then it is also closed in X. 4 If a function continuous in X-norm, then g g on [O,T] to Y is bounded in Y-norm and is weakly continuous (hence strongly measurable) as a Y-valued function. 8. An example. 1) -Go ~ co, u t = 8u/St, u x = 8u/Sx. ] simplicity, The real-valued functions to be as smooth as one likes. 2) k = 0,1,2, .... ~) I~-II~ ~< S. 3) Y with center 0 and radius R.

12) It should be noted that in the place of H s-l, but the equation du/dt E H s-1. The method sketched in Lax [14] is basically similar to ours. 53 2. 13) J in which ao is also s~m~metric and positive-definite. indicated in §11, 1, will be applicable. 13) was studied in detail in [12] as a separate problem, using the same idea. 13) was also considered by Fischer and Marsden [15] under more restrictive assumptions. 13. Korteweg-de Vries equation. 1) is assumed to be a smooth f~nction. 2) need not be an integer.