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In this paper we extend two nowadays classical results to a nonlinearDirichlet problem to equations involving the fractional $p-$Laplacian. Thefirst result is a existence in a non-resonant range more specific between thefirst and second eigenvalue of the fractional $p-$Laplacian. The second resultis the anti-maximum principle for the fractional $p-$Laplacian.

Non-resonant Fredholm alternative and anti-maximum principle for the fractional $p-$Laplacian