Existence-uniqueness result for a nonlinear n-term fractional equation

Abstract

We prove the existence-uniqueness of the solution to the nonlinear n-term time-fractional differential equation with constant coefficients in the Banach space C ([0, T]),(1)b0 Dβ0 u (t) + underover(∑, i = 1, m - 1) bi Dβi u (t) + underover(∑, i = m, n - 1) bi Dαi u (t) + bn Dαn u (t) = f (t, u (t)), t ∈ (0, T), x ∈ R, u (0) = f (0), ut (0) = g (0),0 < β1 < β2 < ⋯ < βm - 1 < 1 < αm < αm + 1 < ⋯ < αn < 2(respectively 0 < β1 < ⋯ < βm - 1 < αm < αm + 1 < ⋯ < αn < 1,1 < β1 < ⋯ < βm - 1 < αm < αm + 1 < ⋯ < αn < 2),f (t, u (t)) ∈ C ([0, T) × C ([0, T))) is a given function meant to be composed with a real valued function u and satisfies Assumption 1. © 2008 Elsevier Inc.