Citation
Williams, Stephen Andrew (1967) On the Cesari Fixed Point Method in a Banach Space. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/m8ave522. https://resolver.caltech.edu/CaltechTHESIS:03142016112439237
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 infinitedimensional fixed point problem to a finitedimensional 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 finitedimensional 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^{1}x∩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^{1}x∩Г into itself. Let y(x) be that element (uniqueness follows from the contraction principle) of P^{1}x∩Г which satisfies the equation y(x) = Py(x) + (IP)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, P_{1}) and (Г, W, P_{2}). Assume that P_{2}P_{1}=P_{1}P_{2}=P_{1} and assume that either of the following two conditions holds:
(1) For every b in B and every z in the range of P_{2}, we have that ‖b=P_{2}b‖ ≤ ‖bz‖
(2)P_{2}Г is convex.
Then i(Г, W, P_{1}) = i(Г, W, P_{2}).
Theorem 3. If Ω is a bounded open set and W is a compact operator defined on Ω so that the (infinitedimensional) LeraySchauder index i_{LS}(W, Ω) is defined, and if the Cesari fixed point method applies to (Ω, W, P), then i(Ω, W, P) = i_{LS}(W, Ω).
Theorems 2 and 3 are proved using mainly a homotopy theorem and a reduction theorem for the finitedimensional and the LeraySchauder 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:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  3 April 1967  
Funders: 
 
Record Number:  CaltechTHESIS:03142016112439237  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:03142016112439237  
DOI:  10.7907/m8ave522  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  9613  
Collection:  CaltechTHESIS  
Deposited By:  INVALID USER  
Deposited On:  22 Mar 2016 15:41  
Last Modified:  20 Mar 2024 22:26 
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