By A. N. Kolmogorov, S. V. Fomin
ISBN-10: 9998063787
ISBN-13: 9789998063785
According to the authors' classes and lectures, this two-part advanced-level textual content is now to be had in one quantity. subject matters comprise metric and normed areas, non-stop curves in metric areas, degree idea, Lebesque durations, Hilbert area, and extra. each one part includes routines. Lists of symbols, definitions, and theorems. 1957 variation.
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Extra info for Elements of the Theory of Functions and Functional Analysis, Volume 2
Example text
This open problem was studied in T. Bartsch-M. Willem [48]. 7 is an improvement and generalization of the results in [48]. Another example is f{x,u) =i^|i^|^~^ln(2 + |i^|), a G (1,2); g{x,u) =/ii^ln(l + |i^|). Then (Di), (D2), (D3) and (D4)-(2) hold if /i < 0; (Di), (D2), (D3) and (D4)-(3) hold with o^ = 2 if /i > 0. If we choose g{x,u) = u'^ for |i^| < 1; g{x,u) = c|^|-i/2ln(l + 1^1) for |^| > 1, then (Di), (D2), (D3) and (D4)-(l) hold. 7. Assume that (DiJ-fD^) hold. Then equation (D) has infinitely many solutions {uk] satisfying ^{uk) '•=-\\uk\\^ — I F{x,Uk)dx — I G{x,Uk)dx ^ 0~ 2 JQ JQ as k ^ 00, where \\u\\ = (J^ | y i^p(ix)^/^.
1) {I\u^v)-I\u^w),v-w) > \\v - w\\h{\\v - w\\) for all u G Ei^v^w G ^2, then we have the following results. (1) There exists a continuous function (j) : Ei ^ E2 such that I{u -\- (j){u)) = min I{u + v). veE2 Moreover, (j){u) is the unique member of E2 such that {I'{u^(j){u)),v) =0, \JveE2. veEi. (3) An element u e Ei is a critical point of J if and only if u -\- (j){u) is a critical point of L Proof. (1) For each u G ^ 1 , define H^ : E2 ^ K by Hu{v) = I{u + v). 1), Hu is of C^ and has at most one critical point.
Li-Z. Q. Wang [217], sign-changing small energy solutions were obtained. 7 of the present chapter were obtained by W. Zou in [385]. Chapter 4 Linking and Homoclinic Type Solutions In this chapter, we first prove a weak finking tfieorem wfiicfi, to some extent, unifies the classical linking theorems. Moreover, it produces a bounded Palais-Smale sequence for a non-even functional. Applications will be given on the existence of homoclinic orbits for Hamiltonian systems and solutions to Schrodinger equations.
Elements of the Theory of Functions and Functional Analysis, Volume 2 by A. N. Kolmogorov, S. V. Fomin
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