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) 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.

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