International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 4, Pages 781-792
doi:10.1155/S016117128800095X
Periodic solutions of Volterra integral equations
Department of Mathematics, University of Dayton, Dayton 45469, OH, USA
Received 2 December 1986; Revised 4 April 1987
Copyright © 1988 M. N. Islam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Consider the system of equationsx(t)=f(t)+∫−∞tk(t,s)x(s)ds, (1)andx(t)=f(t)+∫−∞tk(t,s)g(s,x(s))ds. (2)Existence of continuous periodic solutions of (1) is shown using the resolvent function of the kernel k. Some important properties of the resolvent function including its uniqueness are obtained in the process. In obtaining periodic solutions of (1) it is necessary that the resolvent of k is integrable in some sense. For a scalar convolution kernel k some explicit conditions are derived to determine whether or not the resolvent of k is integrable. Finally, the existence and uniqueness of continuous periodic solutions of (1) and (2) are btained using the contraction mapping principle as the basic tool.