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On the Cesari fixed point method in a Banach space

Citation

Williams, Stephen Andrew (1967) On the Cesari fixed point method in a Banach space. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:03142016-112439237

Abstract

In a paper published in 1961, L. Cesari [1] introduces a method which extends certain earlier existence theorems of Cesari and Hale ([2] to [6]) for perturbation problems to strictly nonlinear problems. Various authors ([1], [7] to [15]) have now applied this method to nonlinear ordinary and partial differential equations. The basic idea of the method is to use the contraction principle to reduce an infinite-dimensional fixed point problem to a finite-dimensional problem which may be attacked using the methods of fixed point indexes.

The following is my formulation of the Cesari fixed point method:

Let B be a Banach space and let S be a finite-dimensional linear subspace of B. Let P be a projection of B onto S and suppose Г≤B such that pГ is compact and such that for every x in PГ, P-1x∩Г is closed. Let W be a continuous mapping from Г into B. The Cesari method gives sufficient conditions for the existence of a fixed point of W in Г.

Let I denote the identity mapping in B. Clearly y = Wy for some y in Г if and only if both of the following conditions hold:

(i) Py = PWy.

(ii) y = (P + (I - P)W)y.

Definition. The Cesari fixed paint method applies to (Г, W, P) if and only if the following three conditions are satisfied:

(1) For each x in PГ, P + (I - P)W is a contraction from P-1x∩Г into itself. Let y(x) be that element (uniqueness follows from the contraction principle) of P-1x∩Г which satisfies the equation y(x) = Py(x) + (I-P)Wy(x).

(2) The function y just defined is continuous from PГ into B.

(3) There are no fixed points of PWy on the boundary of PГ, so that the (finite- dimensional) fixed point index i(PWy, int PГ) is defined.

Definition. If the Cesari fixed point method applies to (Г, W, P) then define i(Г, W, P) to be the index i(PWy, int PГ).

The three theorems of this thesis can now be easily stated.

Theorem 1 (Cesari). If i(Г, W, P) is defined and i(Г, W, P) ≠0, then there is a fixed point of W in Г.

Theorem 2. Let the Cesari fixed point method apply to both (Г, W, P1) and (Г, W, P2). Assume that P2P1=P1P2=P1 and assume that either of the following two conditions holds:

(1) For every b in B and every z in the range of P2, we have that ‖b=P2b‖ ≤ ‖b-z‖

(2)P2Г is convex.

Then i(Г, W, P1) = i(Г, W, P2).

Theorem 3. If Ω is a bounded open set and W is a compact operator defined on Ω so that the (infinite-dimensional) Leray-Schauder index iLS(W, Ω) is defined, and if the Cesari fixed point method applies to (Ω, W, P), then i(Ω, W, P) = iLS(W, Ω).

Theorems 2 and 3 are proved using mainly a homotopy theorem and a reduction theorem for the finite-dimensional and the Leray-Schauder indexes. These and other properties of indexes will be listed before the theorem in which they are used.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Restricted to Caltech community only
Research Advisor(s):
  • Fuller, F. B.
Thesis Committee:
  • Unknown, Unknown
Defense Date:3 April 1967
Funders:
Funding AgencyGrant Number
NSFUNSPECIFIED
Record Number:CaltechTHESIS:03142016-112439237
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:03142016-112439237
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9613
Collection:CaltechTHESIS
Deposited By: Leslie Granillo
Deposited On:22 Mar 2016 15:41
Last Modified:22 Mar 2016 15:41

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