On the fractional calculus in abstract spaces and their applications to the Dirichlet-type problem of fractional order

Abstract

In the following pages, based on the linear functional over a Banach space E and on the definition of fractional integrals of real-valued functions, we define the fractional Pettis-integrals of E -valued functions and the corresponding fractional derivatives. Also, we show that the well-known properties of fractional calculus over the domains of the Lebesgue integrable also hold in the Pettis space. To encompass the full scope of the paper, we apply this abstract result to investigate the existence of Pseudo-solutions to the following fractional-order boundary value problem { D α x ( t ) + λ a ( t ) f ( t , x ( t ) ) = 0 , t ∈ [ 0 , 1 ] , α ∈ ( n − 1 , n ] , n ≥ 2 , x ( 1 ) + ∫ 0 1 u ( τ ) x ( τ ) d τ = l , x ( k ) ( 0 ) = 0 , k = 0 , 1 , … , n − 2 , in the Banach space C [ I , E ] under Pettis integrability assumptions imposed on f . Our results extend all previous results of the same type in the Bochner integrability setting and in the Pettis integrability one. Here, λ ∈ R , u ∈ L p , a ∈ L q and l ∈ E .