Sobolev Type Fractional Dynamic Equations and Optimal Multi-Integral   Controls with Fractional Nonlocal Conditions
  We prove existence and uniqueness of mild solutions to Sobolev type fractional nonlocal dynamic equations in Banach spaces. The Sobolev nonlocal condition is considered in terms of a Riemann-Liouville fractional derivative. A Lagrange optimal control problem is considered, and existence of a multi-integral solution obtained. Main tools include fractional calculus, semigroup theory, fractional power of operators, a singular version of Gronwall's inequality, and Leray-Schauder fixed point theorem. An example illustrating the theory is given. 
Amar Debbouche, Delfim F. M. Torres
math.OC math.CA