Fractional Integro-Dierential Equations of Mixed Type with Solution Operator and Optimal Controls

Abstract

Local and global existence and uniqueness of mild solution for the fractional integro-differential equations of mixed type with delay are proved by using a family of solution operators and the contraction mapping principle on Banach space. The
Bolza optimal control problem of  a corresponding controlled  system is solved.  The Gronwall lemma with singular and time lag is derived to be tool for obtaining a priori estimate. In addition, the application to the fractional nonlinear heat equation is shown.