Dynamic Systems and Applications 32 (2023) 255-274 DECOMPOSING A CONJUGATE FIXED-POINT PROBLEM INTO MULTIPLE FIXED-POINT PROBLEMS RICHARD AVERY 1 , DOUGLAS R. ANDERSON 2 , JOHNNY HENDERSON 3 1 College of Arts and Sciences, Dakota State University, Madison SD 57042, USA e-mail: Rich.Avery@dsu.edu 2 Department of Mathematics, Concordia College, Moorhead, MN 56562, USA email: andersod@cord.edu 3 Department of Mathematics, Baylor University, Waco TX 76798, USA email: Johnny Henderson@Baylor.edu 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. 1. Introduction The Banach Fixed Point Theorem [2] is a powerful tool that can be used to find solutions of nonlinear initial and boundary value problems that have been converted to fixed point problems. The Picard-Lindel¨of Theorem (see the fixed point books by Zeidler [5] or Dugundji-Granas [3]) is used to find unique solutions for a first order nonlinear initial value problem where the key is to restrict the interval so the operator whose fixed points are solutions on the interval is k-contractive. This is the first manuscript that we are aware of that follows an approach similar to the initial value problem approach by Picard-Lindel¨of for boundary value problems, that is, restricting the interval so an associated operator is k-contractive. Received August 2, 2023 ISSN 1056-2176(Print); ISSN 2693-5295 (online) www.dynamicpublishers.org https://doi.org/10.46719/dsa2023.32.14 $15.00 c ⃝ Dynamic Publishers, Inc.