Abstract
In the present paper using precise results on the solutions of linear elliptic differential operators with Holder continuous coefficient as well as a variant of the Lery - Schauder method and the gal of this paper to find an adequate degree theory for the infinite dimensional setting and to extend the theory of homotopy classes of maps form <img width="17" height="16" src="http://article.sciencepublishinggroup.com/journal/141/1411129/image001.png" /> to <img width="17" height="16" src="http://article.sciencepublishinggroup.com/journal/141/1411129/image001.png" /> to homotopy classes of maps on infinite dimensional spaces.
Highlights
The infinite dimensional space under consideration are normed linear vector spaces are their subsets
We can alter the notion of homotopy invariance in order to a degree theory, or limit the types of maps for which a degree is well defined the Leray Schauder degree does both by considering specific types of mappings
Homotopies are considered in the same class, denote the function class by efDg' ΩK( = = l − T|T ∈ H and by efDg' ΩK( for mapping defined on Properties of the Leray – Schauder degree: Theorem (2-5): For Leray –Schauder degree we have the following properties: (A1) if ∈ Ω, Qj°±'l, Ω, `( = 1 (A2) for Ωu, Ω6 ⊂ Ω, disjoint open subsets of Ω, and p ∉ Ωu ∪ Ω6, it holds that
Summary
The infinite dimensional space under consideration are normed linear vector spaces are their subsets. R are mapped to bounded set under This is not the case in general Banach spaces. Lemma (1-2): A uniformly continuous map is bounded. Differentiability: Definition (1-3): A mapping C' , *( is called Gateaux differentiable in the direction h ∈ , at a point D , if there exists a ∈ * such that %=⟵ D‖ ' D + h( − ' D( − ‖ = 0 , With D + h defined in a neighborhood F D. A mapping : → is compact if GG'GGΩG2GG( is compact for any bounded subset Ω2 ⊂. Definition (1-4): A continuous map K:2 Ω ⊂ → is called compact if GG'GGΩGKJGG( is compact. Lemma (1-5): Let k ∈ K'ΩK(, for any, there exists a finite rank map k.
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