On solutions of a class of three-point fractional boundary value problems

Existence results for the three-point fractional boundary value problem $$\begin{aligned}& D^{\alpha}x(t)= f \bigl(t, x(t), D^{\alpha-1} x(t) \bigr),\quad 0< t< 1, \\& x(0)=A, \qquad x(\eta)-x(1)=(\eta-1)B, \end{aligned}$$ Dαx(t)=f(t,x(t),Dα−1x(t)),0<t<1,x(0)=A,x(η)−x(1)=(η−1)B, are presented, where $A, B\in\mathbb{R}$A,B∈R, $0<\eta<1$0<η<1, $1<\alpha\leq2$1<α≤2. $D^{\alpha}x(t)$Dαx(t) is the conformable fractional derivative, and $f: [0, 1]\times\mathbb{R}^{2}\to\mathbb{R}$f:[0,1]×R2→R is continuous. The analysis is based on the nonlinear alternative of Leray–Schauder.