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

Abstract

AbstractIn 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)=Bor$D^{\alpha -1}x(0)=A$Dα−1x(0)=A,$x(1)=B$x(1)=B. Let the nonlinear termfsatisfy 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.