D ifferential E quations & A pplications Volume 9, Number 3 (2017), 353–368 doi:10.7153/dea-2017-09-25 FRACTIONAL LYAPUNOV INEQUALITIES ON SPHERICAL SHELLS YOUSEF GHOLAMI AND KAZEM GHANBARI (Communicated by Qingkai Kong) Abstract. This paper, deals with Lyapunov inequalities of conformable fractional boundary value problems on an N-dimensional spherical shell. Applicability of these Lyapunov inequalities will be examined by establishing the disconjugacy as a nonexistence criterion for nontrivial solutions, lower bound estimation for eigenvalues of the corresponding fractional eigenvalue problem, upper bound estimation for maximum number of zeros of the nontrivial solutions and distance between consecutive zeros of an oscillatory solution. 1. Introduction The theory of fractional calculus that acts on the arbitrary order differentiation and integration, tries to generalize the ordinary calculus. But, unfortunately expected gener- alization has not occurred by now. As we know, fractional calculus essentially was con- structed based on the Riemann-Liouville fractional operators, see [16],[18],[19],[22]. On the other hand, V E. Tarasov in references [25],[26] proves that all of Riemann- Liouville based differentiation operators do not satisfy in the Leibniz and chain rules as in the ordinary calculus. Overcomming this inconvenience, more recently R. Khalil et al, in [15], introduced a new definition for fractional order differentiation operators that generalizes the limit approach of the classic differentiation. They called these operators conformable fractional differentiation operators that will be presented in the next sec- tion. But about the Lyapunov inequalities, this is well known that the concept of the Lya- punov inequality turns to the deep studies of the Russian mathematician A. M. Lya- punov on stability of motion, in the late XIX century. Maybe the best interpretation of the Lyapunov inequalities can be stated as follows: THEOREM 1.1. (cf. [5]-[9],[13]) If the boundary value problem  y ′′ (t )+ q(t )y(t )= 0, a < t < b, y(a)= 0 = y(b), (1.1) has a nontrivial solution, where q is a real and continuous function, then  b a |q(s)|ds > 4 b - a . (1.2) Mathematics subject classification (2010): 34A08, 26A33, 26D15. Keywords and phrases: Conformable fractional derivative, Lyapunov inequality, Spherical shell, Qual- itative dynamic. c  , Zagreb Paper DEA-09-25 353