Existence of solutions for multi point boundary value problems for fractional differential equations

In this paper, by employing the Leggett–Williams fixed point theorem, we study the existence of three solutions in the multi point fractional boundary value problemD0+αu(t)=f(t,u(t),u′(t)),t∈[0,1],u(0)=u′(0)=0,u(1)-∑i=1maiu(ξi)=λwhere 2 < α ⩽ 3 and m ⩾ 1 are integers, 0 < ξ1 < ξ2 < ⋯ < ξn < 1 are constants, λ ∈ (0,∞) is a parameter, ai > 0 for 1 ⩽ i ⩽ m   and ∑i=1maiξiα-1<1,f∈C([0,1]×[0,∞)×[0,∞);[0,∞)).