×

Principal matrix solutions and variation of parameters for a Volterra integro-differential equation and its adjoint. (English) Zbl 1115.45004

From the author’s abstract: We define the principal matrix solution \(Z(t,s)\) of the linear Volterra vector integro-differential equation \[ x'(t)=A(t)x(t)+\int_{s}^{t}B(t,u)x(u)\,du, \] in the same way that it is defined for \(x'=A(t)x\) and prove that it is the unique matrix solution of \[ \frac{\partial}{\partial t}Z(t,s)=A(t)Z(t,s)+\int_{s}^{t}B(t,u)Z(u,s)\,du, \quad Z(s,s)=I. \] Furthermore, we prove that the solution of \[ x'(t)=A(t)x(t)+\int_{\tau}^{t}B(t,u)x(u)\,du+f(t), \quad x(\tau)=x_{0}, \] is unique and given by the variation of parameters formula \[ x(t)=Z(t,\tau)x_{0}+\int_{\tau}^{t}Z(t,s)f(s)\,ds. \] We also define the principal matrix solution \(R(t,s)\) of the adjoint equation \[ r'(s)=-r(s)A(s)-\int_{s}^{t}r(u)B(u,s)\,du \] and prove that it is identical to Grossman and Miller’s resolvent, which is the unique matrix solution of \[ \frac{\partial}{\partial s}R(t,s)=-R(t,s)A(s)-\int_{s}^{t}R(t,u)B(u,s)\,du, \quad R(t,t)=I. \] Finally, we prove that despite the difference in their definitions \(R(t,s)\) and \(Z(t,s)\) are in fact identical.

MSC:

45J05 Integro-ordinary differential equations
47G20 Integro-differential operators
PDFBibTeX XMLCite
Full Text: DOI EuDML