Premium
Periodic Solutions of Linear Integro‐Differential Equations
Author(s) -
Burton T. A.,
Eloe P. W.,
Islam M. N.
Publication year - 1990
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19901470120
Subject(s) - mathematics , bounded function , a priori and a posteriori , mathematical analysis , differential equation , homotopy analysis method , degree (music) , upper and lower bounds , uniform boundedness , homotopy , pure mathematics , physics , epistemology , acoustics , philosophy
Using a degree‐theoretic result of Granas, a homotopy is constructed enabling us to show that if there is an a priori bound on all possible T ‐periodic solutions of a Volterra equation, then there is a T ‐periodic solution. The a priori bound is established by means of a Liapunov functional. The latter result is unusual in that no bounds on the Liapunov functional are required. Thus, in addition to the periodic solution, the equation may have both bounded and unbounded Solutions.