Lyapunov-type inequalities for a fractional p-Laplacian equation
© Al Arifi et al. 2016
Received: 21 January 2016
Accepted: 20 July 2016
Published: 2 August 2016
In this paper, we present new Lyapunov-type inequalities for a fractional boundary value problem that models a turbulent flow in a porous medium. The obtained inequalities are used to obtain a lower bound for the eigenvalues of corresponding equations.
The p-Laplacian operator arises in different mathematical models that describe physical and natural phenomena (see, for example, [1–6]). In particular, it is used in some models related to turbulent flows (see, for example, [7–9]).
Inequality (1.2) is useful in various applications, including oscillation theory, stability criteria for periodic differential equations, and estimates for intervals of disconjugacy.
Ferreira  established a fractional version of inequality (1.2) for a fractional boundary value problem involving the Caputo fractional derivative of order \(1<\alpha\leq2\). In both papers [17, 18], the author presented nice applications to obtain intervals where certain Mittag-Leffler functions have no real zeros.
The paper is organized as follows. In Section 2, we recall some basic concepts on fractional calculus and establish some preliminary results that will be used in Section 3, where we state and prove our main result. In Section 4, we present some applications of the obtained Lyapunov-type inequalities to eigenvalue problems.
Now, in order to obtain an integral formulation of (1.1), we need the following results.
The following estimates will be useful later.
Now, we are ready to state and prove our main result.
3 Main result
Our main result is the following Lyapunov-type inequality.
Taking \(p=2\) in Corollary 3.2, we obtain the following result.
4 Applications to eigenvalue problems
In this section, we present some applications of the obtained results to eigenvalue problems.
It follows from inequality (4.2) by taking \(p=2\). □
The authors extend their appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).
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