Richard Avery, Douglas R. Anderson, and Johnny Henderson
Decomposing a Conjugate Fixed-Point Problem into Multiple Fixed-Point Problems
Dynamic Systems and Applications 32 (2023) 255-274

https://doi.org/10.46719/dsa2023.32.14

ABSTRACT.
Converting nonlinear boundary value problems to fixed point problems of an integral operator with a Green’s function kernal is a common technique to find or approximate solutions of boundary value problems. It is often difficult to apply Banach’s Theorem since it is challenging to find an initial estimate with a contractive constant less than one. We decompose the integral operator associated to a conjugate boundary value problem creating multiple fixed point problems which have contractive constants less than one. We then provide conditions for the original boundary value problem to have a solution that can be found by iteration using the decomposition through a fixed point of a real valued function which matches the fixed points of our decomposition.

AMS (MOS) Subject Classification. 47H10, 34B18

Key Words and Phrases. Fixed point theorems, alternative inversion, iteration.