(?, c)-periodic and asymptotically (?, c)-periodic mild solutions to fractional Cauchy problems

In this paper, we establish some new properties of (?,c)-periodic and asymptotically functions, then apply them to study the existence uniqueness mild solutions these types following semilinear fractional differential equations: (1) {cDt?u(t)=Au(t)+cDt??1f(t,u(t)),1<?<2,t?R,u(0)=0(1) (2) {cDt?u(t)=Au(t)+cDt??1f(t,u(t?h)),1<?<2,t,h?R+,u(0)=0(2) where cDt?(?)(1<?<2) stands for Caputo derivative A is a linear densely defined operator sectorial type on complex Banach space X function f(t,x) or with respect first variable. Our results are obtained using Leray–Schauder alternative theorem, fixed point principle Schauder theorem. Then illustrate our main an application diffusion-wave equations.