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

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) b 0 D β 0 u ( t ) + ∑ i = 1 m − 1 b i D β i u ( t ) + ∑ i = m n − 1 b i D α i u ( t ) + b n D α n u ( t ) = f ( t , u ( t ) ) , t ∈ ( 0 , T ) , x ∈ R , u ( 0 ) = f ( 0 ) , u t ( 0 ) = g ( 0 ) , 0 < β 1 < β 2 < ⋯ < β m − 1 < 1 < α m < α m + 1 < ⋯ < α n < 2 ( respectively 0 < β 1 < ⋯ < β m − 1 < α m < α m + 1 < ⋯ < α n < 1 , 1 < β 1 < ⋯ < β m − 1 < α m < α m + 1 < ⋯ < α n < 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.