On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies

This paper is devoted to investigate the existence of Pseudo solutions for the nonlinear m -point boundary value problem of fractional type D α x ( t ) + q ( t ) f ( t , x ( t ) ) = 0 , a.e. on  [ 0 , 1 ] , α ∈ ( n − 1 , n ] , n ≥ 2 , x ( 0 ) = x ′ ( 0 ) = x ″ ( 0 ) = ⋯ = x ( n − 2 ) ( 0 ) = 0 , x ( 1 ) = ∑ i = 1 m − 2 ζ i x ( η i ) , where 0 < η 1 < ⋯ < η m − 2 < 1 , ζ i > 0 with ∑ i = 1 m − 2 ζ i η i α − 1 < 1 . It is assumed that q is a real-valued continuous function and f is a nonlinear Pettis integrable function.