Solutions of two-term time fractional order differential equations with nonlocal initial conditions

Abstract

We study the existence of mild solutions for the two-term fractional order abstract differential equation $${D}^{\alpha+1}_t u(t) + \mu {D}_t^{\beta} u(t) - Au(t) = D^{\alpha-1}_t f(t,u(t)), \quad t\in [0,1], \quad 0 < \alpha \leq \beta \leq 1, \mu \geq 0,$$ with nonlocal initial conditions and where $A$ is a linear operator of sectorial type. To achieve our goal, we use a new mixed method, which combines a generalization of the theory of $C_0$-semigroups, Hausdorff measure of noncompactness and a fixed point argument.

Keywords

• generalization of semigroups
• two-term time fractional derivative
• sectorial operators
• nonlocal initial conditions