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 α x ( t ) = f ( t , x ( t ) , D α − 1 x ( t ) ) , with the boundary value conditions to be either: x ( 0 ) = A , D α − 1 x ( 1 ) = B or D α − 1 x ( 0 ) = A , 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.