This paper is devoted to establishing some criteria for the existence of non-trivial solutions for a class of fractional q-difference equations involving the p-Laplace operator, which is nowadays known as Lyapunov's inequality. The method employed for it is based on a construction of a Green's function and its maximum value. Parallel to this result, it is worth mentioning that the Hartman-Wintner inequality for the q-fractional p-Laplace boundary value problem is also provided. It covers all previous results known in the literature on the fractional case as well as that on the classical ordinary case. The non-existence of non-trivial solutions to the q-difference fractional p-Laplace equation subject to the Riemann-Liouville mixed boundary conditions will obey such integral inequalities. The tools mainly rely on an integral form of the solution construction of a Green function corresponding to the considered problem and its properties as well as its maximum value in consideration where the kernel is the Green's function. The example that we consider here for applying this result is an eigenvalue fractional problem. To be more specific, we provide an interval where an appropriate Mittag-Leffler function to the given eigenvalue fractional boundary problem has no real zeros.2010 Mathematics Subject Classification: 34A08, 34A40, 26D10, 33E12

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