Abstract
This paper is devoted to establish Lyapunov-type inequalities for a class of fractional q-difference boundary value problem involving p-Laplace operator. Also, it is worth to mention that the Hartman-Wintner inequality for the qfractional p-Laplace boundary value problem is provided. The non existence of non trivial of 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 a construction of a Green function corresponding to the considered problem, and its properties as well as its maximum value. In order to illustrate this result, we provide an interval where an appropriate Mittag-Leffler function to the given eigenvalue fractional boundary problem has no real zeros.
Highlights
The field of fractional calculus and its applications to the class of partial differential equations, as well as ordinary equations, gained a rapid development
Some results focusing on the existence of positive solutions of boundary value problems for a class of fractional differential equations with the p-Laplacian operator have been raised in previous papers
Ren and Chen [15] and Su et al [17] established the existence of positive solutions to four-point boundary value problems for non-linear fractional differential equations with the p-Laplacian operator
Summary
The field of fractional calculus and its applications to the class of partial differential equations, as well as ordinary equations, gained a rapid development. Some results focusing on the existence of positive solutions of boundary value problems for a class of fractional differential equations with the p-Laplacian operator have been raised in previous papers (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22] and the references therein). Ren and Chen [15] and Su et al [17] established the existence of positive solutions to four-point boundary value problems for non-linear fractional differential equations with the p-Laplacian operator. We show that from this inequality derive several existing previous results in the literature as well as the standard Lyapunov inequality (1): those of Hartman and Wintner [52], Ferreira [39], and so on
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