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

Abstract

We study the generalized conformable derivative N_F^α and the associated N_F^α--C₀ semigroups on Banach spaces. Using the time change G_α(t)=integral from 0 to t of F(s,α)ds, we represent the generalized semigroup as T_F^α(t)=S(G_α(t)), where S is a classical C₀--semigroup, and obtain the estimate ||T_F^α(t)|| ≤ M e^ωG_α(t). An N_F^α--Laplace transform adapted to the function G_α is introduced together with a suitable convolution property. This allows us to derive a variation--of--constants formula for the nonlinear problem N_F^α 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 N_F^1/2. Several known conformable and classical results are recovered as special cases.