In this paper, we are dedicated to researching the boundary value problems (BVPs) for equation Dαx(t)=f(t,x(t),Dα−1x(t))\documentclass[12pt]{minimal}
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\begin{document}$D^{\alpha }x(t)=f(t,x(t),D^{\alpha -1}x(t))$\end{document}, with the boundary value conditions to be either: x(0)=A\documentclass[12pt]{minimal}
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\begin{document}$x(0)=A$\end{document}, Dα−1x(1)=B\documentclass[12pt]{minimal}
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\begin{document}$D^{\alpha -1}x(1)=B$\end{document} or Dα−1x(0)=A\documentclass[12pt]{minimal}
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\begin{document}$D^{\alpha -1}x(0)=A$\end{document}, x(1)=B\documentclass[12pt]{minimal}
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\begin{document}$x(1)=B$\end{document}. 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.
