Topological and Monotonicity Methods
One of the most frequent problems in analysis, especially in its applications, consists in solving the equation F(x) = y where F is a mapping from a Banach space X into a Banach space Y.1 Such an equation can be reduced to the equation F(x) = o, or, provided X ⊂ Y, to the equation F(x) = x. (5.1.1) In this section we present two basic results on the solvability of (5.1.1) in a special case, namely, for a continuous mapping F and a finite dimensional X, and a compact mapping F in a general Banach space of infinite dimension — the Brouwer and the Schauder Fixed Point Theorems.
KeywordsBanach Space Compact Operator Monotone Operator Real Hilbert Space Real Banach Space
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