Existence of solutions for some two-point fractional boundary value problems under barrier strip conditions

In this paper, we are dedicated to researching the boundary value problems (BVPs) for equation $D^{\alpha }x(t)=f(t,x(t),D^{\alpha -1}x(t))$Dαx(t)=f(t,x(t),Dα−1x(t)), with the boundary value conditions to be either: $x(0)=A$x(0)=A, $D^{\alpha -1}x(1)=B$Dα−1x(1)=B or $D^{\alpha -1}x(0)=A$Dα−1x(0)=A, $x(1)=B$x(1)=B. Let the nonlinear term f satisfy some sign conditions, then by making use of the Leray–Schauder nonlinear alternative, some existence results are obtained. In the end, an example is given to verify the main results.