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 mm-point boundary value problem of fractional type Dαx(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=1m−2ζix(ηi), where 0<η1<⋯<ηm−2<1,ζi>00<η1<⋯<ηm−2<1,ζi>0 with ∑i=1m−2ζiηiα−1<1. It is assumed that qq is a real-valued continuous function and ff is a nonlinear Pettis integrable function.