On the structure of the set of solutions of equations involving 𝐴-proper mappings

Fitzpatrick, P. M.

doi:10.1090/S0002-9947-1974-0336475-5

On the structure of the set of solutions of equations involving $A$-proper mappings
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Trans. Amer. Math. Soc. 189 (1974), 107-131 Request permission

Abstract:

Let X and Y be Banach spaces having complete projection schemes (say, for example, they have Schauder bases). We consider various properties of mappings $T:D \subset X \to Y$ which are either Approximation-proper (A-proper) or the uniform limit of such mappings. In §1 general properties, including those of the generalized topological degree, of such mappings are discussed. In §2 we give sufficient conditions in order that the solutions of an equation involving a nonlinear mapping be a continuum. The conditions amount to requiring that the generalized topological degree not vanish, and that the mapping involved be the uniform limit of well structured mappings. We devote §3 to proving a result connecting the topological degree of an A-proper Fréchet differentiable mapping to the degree of its derivative. Finally, in §4, various Lipschitz-like conditions are discussed in an A-proper framework, and constructive fixed point and surjectivity results are obtained.

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