- Open Access
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.
- Lyapunov-type inequality
- fractional derivative
- turbulent flow
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.
Our main result is the following Lyapunov-type inequality.
Taking \(p=2\) in Corollary 3.2, we obtain the following result.
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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- Bognàr, G, Rontó, M: Numerical-analytic investigation of the radially symmetric solutions for some nonlinear PDEs. Comput. Math. Appl. 50, 983-991 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Glowinski, R, Rappaz, J: Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology. Math. Model. Numer. Anal. 37(1), 175-186 (2003) MathSciNetView ArticleMATHGoogle Scholar
- Kawohl, B: On a family of torsional creep problems. J. Reine Angew. Math. 410, 1-22 (1990) MathSciNetMATHGoogle Scholar
- Liu, C: Weak solutions for a viscous p-Laplacian equation. Electron. J. Differ. Equ. 2003, 63 (2003) MathSciNetView ArticleMATHGoogle Scholar
- Oruganti, S, Shi, J, Shivaji, R: Diffusive logistic equation with constant yield harvesting, I: steady-states. Trans. Am. Math. Soc. 354(9), 3601-3619 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Ramaswamy, M, Shivaji, R: Multiple positive solutions for classes of p-Laplacian equations. Differ. Integral Equ. 17(11-12), 1255-1261 (2004) MathSciNetMATHGoogle Scholar
- Diaz, JI, de Thelin, F: On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J. Math. Anal. 25(4), 1085-1111 (1994) MathSciNetView ArticleMATHGoogle Scholar
- Showalter, RE, Walkington, NJ: Diffusion of fluid in a fissured medium with microstructure. SIAM J. Math. Anal. 22, 1702-1722 (1991) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, X, Liu, L, Wu, Y: The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium. Appl. Math. Lett. 37, 26-33 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Liapunov, AM: Problème général de la stabilité du mouvement. Ann. Fac. Sci. Univ. Toulouse 2, 203-407 (1907) View ArticleGoogle Scholar
- Aktas, MF: Lyapunov-type inequalities for a certain class of n-dimensional quasilinear systems. Electron. J. Differ. Equ. 2013, 67 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Çakmak, D: On Lyapunov-type inequality for a class of nonlinear systems. Math. Inequal. Appl. 16, 101-108 (2013) MathSciNetMATHGoogle Scholar
- Hartman, P, Wintner, A: On an oscillation criterion of Liapounoff. Am. J. Math. 73, 885-890 (1951) MathSciNetView ArticleMATHGoogle Scholar
- He, X, Tang, XH: Lyapunov-type inequalities for even order differential equations. Commun. Pure Appl. Anal. 11, 465-473 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Pachpatte, BG: On Lyapunov-type inequalities for certain higher order differential equations. J. Math. Anal. Appl. 195, 527-536 (1995) MathSciNetView ArticleMATHGoogle Scholar
- Yang, X: On Lyapunov-type inequality for certain higher-order differential equations. Appl. Math. Comput. 134, 307-317 (2003) MathSciNetMATHGoogle Scholar
- Ferreira, RAC: A Lyapunov-type inequality for a fractional boundary value problem. Fract. Calc. Appl. Anal. 16(4), 978-984 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Ferreira, RAC: On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function. J. Math. Anal. Appl. 412, 1058-1063 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Jleli, M, Samet, B: Lyapunov-type inequalities for a fractional differential equation with mixed boundary conditions. Math. Inequal. Appl. 18(2), 443-451 (2015) MathSciNetMATHGoogle Scholar
- Rong, J, Bai, C: Lyapunov-type inequality for a fractional differential equation with fractional boundary conditions. Adv. Differ. Equ. 2015, 82 (2015) MathSciNetView ArticleGoogle Scholar
- Cabrera, I, Sadarangani, K, Samet, B: Hartman-Wintner-type inequalities for a class of nonlocal fractional boundary value problems. Math. Methods Appl. Sci. (2016). doi:10.1002/mma.3972 Google Scholar
- Jleli, M, Samet, B: A Lyapunov-type inequality for a fractional q-difference boundary value problem. J. Nonlinear Sci. Appl. 9, 1965-1976 (2016) MathSciNetMATHGoogle Scholar
- O’Regan, D, Samet, B: Lyapunov-type inequalities for a class of fractional differential equations. J. Inequal. Appl. 2015, 247 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) View ArticleMATHGoogle Scholar