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

Abstract

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

Keywords

• Boundary value problems
• Conformable fractional derivative
• Nonlinear alternative of Leray–Schauder