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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Sup 1~(t)-w(t) I~. 3). Y is a closed Note that we did not require v e E to be continuous in the Y-norm. We shall now show that @ : E ~ E is chosen sufficiently small. is a strict contraction map if To this end we use the perturbation theorem (Theorem 3), which gives d(@w, *v) ~< e~T Since T' '( JlfW-fvlj1,X + i l(AW-Av) ~vl 11,x). l~w(t)-fv(t) i~ ~< ~2 [~v(t)-v(t) I~ by (fl), i ~wfv tl1'x ~< ~T'd(w,v). < ~(llyoil+i~)T~i(w,v). Thus we obtain d(~w,@v) which shows that @ is a contraction map if T' is sufficiently small.

Amer. Math. Soc. No. 134 (1973). 2. E . A . Coddington~ Selfadjoint subspace extensions of nondensely defined symmetric operators, Bull. Amer. Math. Soc. 79 (1973)~ 712-715; complete version with proofs to appear in Advances in Math. 3. E . A . Coddington, Eigenfunction expansions for nondensely defined operators generated by symmetric ordinary differential expressions~ Bull. Amer. Math. Soc. 79 (1973), 964-968; complete version will appear in Advances in Math. under the title: Selfadjoint problems for nondensely defined ordinary differential operators and their eigenfunction expansions.

8. H . J . Zimmerberg, Linear integro-differential-boundary-parameter problems, to appear in Ann. di Mat. Pura ed Appl. ~uasi-linear equations of evolution, with applications to partial differential equations Tosio Kato Introduction The purpose of this paper is to present a unified treatment of the Cauchy problem (local in time) for various quasi-linear partial differential equations that appear in mathematical physics, based on the theory of abstract equations of evolution. We shall prove two rather general theorems on abstract equations, one on the existence and uniqueness, and the other on the continuous dependence on the data, of the solution.

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