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Title Principal matrix solutions and variation of parameters for Volterra integro-dynamic equations on time scales

Name Murat Adivar
Country Turkey
Email murat.adivar@ieu.edu.tr
Submit Time 2010-02-22 09:12:04
Special Session 48: Differential, Difference, and Dynamic Equations
We introduce the principal matrix solution $Z(t,s)$ of the linear Volterra vector integro-dynamic equation \[ x^{\Delta}(t)=A(t)x(t)+\int_{s}^{t}B(t,u)x(u)\Delta u \] and prove that it is the unique matrix solution of \[ Z^{\Delta_{t}}(t,s)=A(t)Z(t,s)+\int_{s}^{t}B(t,u)Z(u,s)\Delta u,\ \ Z(s,s)=I. \] We also show that the solution of \[ x^{\Delta}(t)=A(t)x(t)+\int_{s}^{t}B(t,u)x(u)\Delta u+f(t),\ \ \ x(\tau)=x_{0} \] is unique and given by the variation of parameters formula \[ x(t)=Z(t,\tau)x_{0}+\int_{\tau}^{t}Z(t,\sigma(s))f(s)\Delta s. \] MSC. 34N05, 45D05, 39A13, 45J05 Keywords: Fundamental matrix of solutions, principal matrix solution, time scales, variation of parameters, Volterra type integro dynamic equation