A mild solution approach to general fractional evolution equations with nonlocal constraints

Source abstract

We study the generalized conformable derivative NFα and the associated NFα--C0 semigroups on Banach spaces. Using the time change Gα(t)=integral from 0 to t of F(s,α)ds, we represent the generalized semigroup as TFα(t)=S(Gα(t)), where S is a classical C0--semigroup, and obtain the estimate ||TFα(t)|| ≤ M eωGα(t). An NFα--Laplace transform adapted to the function Gα is introduced together with a suitable convolution property. This allows us to derive a variation--of--constants formula for the nonlinear problem NFα x(t)=Ax(t)+f(t,x(t)) subject to a nonlocal initial condition x(0)=x0+g(x). Under appropriate compactness and growth assumptions on the semigroup and standard continuity hypotheses on the nonlinear terms, the existence of mild solutions is obtained via Schaefer’s fixed point theorem, while uniqueness may be obtained under an additional local Lipschitz condition. As an application, we establish the existence of a mild solution for a semilinear heat equation with a nonlocal initial condition driven by NF1/2. Several known conformable and classical results are recovered as special cases.