Schauder's fixed-point theorem

Schauder's fixed-point theorem

[¦shau̇d·ərz ¦fikst ‚pȯint ′thir·əm]
(mathematics)
A continuous mapping from a closed, compact, convex set in a Banach space into itself has at least one fixed point.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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On the other hand, the operator [[PSI].sub.k] : [U.sub.k] [right arrow] [U.sub.k] is completely continuous by Lemma 10; then by Schauder's fixed-point Theorem 11, each problem of (1)-(3) has a solution.
The proofs in [24] are based on Schauder's fixed-point theorem.
We next introduce Schauder's fixed-point theorem used to prove the existence of a solution to (4) and (5).
We next show the existence of a solution to (4) by the following Schauder's fixed-point theorem.
By Schauder's fixed-point theorem, we can conclude that problem (4) has at least one solution.