학술논문
An example of application of the Nielsen theory to integro-differential equations.
Document Type
Journal
Author
Andres, Jan (CZ-PLCKS-MA) AMS Author Profile; Fürst, Tomáš (CZ-PLCKS-MA) AMS Author Profile
Source
Subject
34 Ordinary differential equations -- 34K Functional-differential and differential-difference equations
34K13Periodic solutions
47Operator theory -- 47N Miscellaneous applications of operator theory
47N20Applications to differential and integral equations
54General topology -- 54H Connections with other structures, applications
54H25Fixed-point and coincidence theorems
55Algebraic topology -- 55M Classical topics
55M20Fixed points and coincidences
34K13
47
47N20
54
54H25
55
55M20
Language
English
Abstract
Nielsen's fixed point theory aims at ascertaining a lower bound for the number of fixed points of a compact (continuous) self-map of a metric ANR-space. However, the nontrivial application of this theory to differential equations is considered a difficult task. The main purpose of this paper is to present a nontrivial example of an application of Nielsen's fixed point theory to integro-differential equations. In continuation of the first author's work [Proc. Amer. Math. Soc. {\bf 128} (2000), no.~10, 2921--2931; MR1664285 (2000m:47074)] and the observations made by M. J. Capiński\ and K. Wójcik\ [Proc. Amer. Math. Soc. {\bf 131} (2003), no.~8, 2443--2451 (electronic); MR1974642 (2003m:37019)], the authors make a breakthrough and establish the existence of at least three periodic solutions to an integro-differential system in the plane. Technical results are too complicated to mention. This work should prove highly useful to researchers looking for applications of the Nielsen theory.