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Periodic Solutions Of Volterra Integral Equations, Muhammad Islam
Periodic Solutions Of Volterra Integral Equations, Muhammad Islam
Mathematics Faculty Publications
Consider the system of equations
x(t)=f(t)+∫−∞tk(t,s)x(s)ds, (1)
and
x(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 obtained using the …