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Erschienen in:
2007 | OriginalPaper | Buchkapitel
Topological and Monotonicity Methods
Erschienen in: Methods of Nonlinear Analysis
Verlag: Birkhäuser Basel
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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.