Spectral Theory and Differential Equations: Proceedings of by Earl A. Coddington (auth.), Prof. William N. Everitt (eds.)

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

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

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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.

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