Periodic solutions of Volterra integral equations

• Main Author: M. N. Islam
• Format: Article
• Language: English
• Published: Hindawi Limited 1988-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Subjects: Volterra integral equation; periodic solution; resolvent; integrability of resolvent; limit equation.
• Online Access: http://dx.doi.org/10.1155/S016117128800095X

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.