On Lyapunov-type inequalities for (p,q)$(p,q)$-Laplacian systems

*Correspondence: bsamet@ksu.edu.sa Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia Abstract We establish Lyapunov-type inequalities for a system involving one-dimensional (pi ,qi)-Laplacian operators (i = 1, 2). Next, the obtained inequalities are used to derive some geometric properties of the generalized spectrum associated to the considered problem.


Introduction
In this paper, we are concerned with the following system involving one-dimensional (p i , q i )-Laplacian operators (i = , ): We suppose also that f and g are two nonnegative real-valued functions such that (f , g) ∈ L  (a, b) × L  (a, b). We establish a Lyapunov-type inequality for problem (S)-(DBC). Next, we use the obtained inequality to derive some geometric properties of the generalized spectrum associated to the considered problem.
The standard Lyapunov inequality [] (see also []) states that if the boundary value problem Inequality () was successfully applied to oscillation theory, stability criteria for periodic differential equations, estimates for intervals of disconjugacy, and eigenvalue bounds for ordinary differential equations. In [] (see also []), Elbert extended inequality () to the one-dimensional p-Laplacian equation. More precisely, he proved that, if u is a nontrivial solution of the problem Observe that for p = , () reduces to (). Inequality () was extended in [] to the following problem involving the ϕ-Laplacian operator: where ϕ : R → R is a convex nondecreasing function satisfying a  condition. In [], Nápoli and Pinasco considered the quasilinear system of resonant type on the interval (a, b), with Dirichlet boundary conditions Under the assumptions p, q > , f , g ∈ L  (a, b), f , g ≥ , α, β ≥ , and it was proved (see [], Theorem .) that if ()-() has a nontrivial solution, then where p = p p- and q = q q- . Some nice applications to generalized eigenvalues are also presented in []. Different generalizations and extensions of inequality () were obtained by many authors. In this direction, we refer the reader to [-] and the references therein. For other results concerning Lyapunov-type inequalities, we refer the reader to [-] and the references therein.

Lyapunov-type inequalities
A Lyapunov-type inequality for problem (S)-(DBC) is established in this section, and some particular cases are discussed.
From the boundary conditions (DBC), we can write that Using Hölder's inequality with parameters p  and p  = p  p  - , we get Similarly, using Hölder's inequality with parameters q  and q  = q  q  - , we get By repeating the same argument for the function v, we obtain Now, multiplying the first equation of (S) by u and integrating over (a, b), Multiplying the second equation of (S) by v and integrating over (a, b), Using (), () and (), we obtain Using the inequality Similarly, using (), () and (), we obtain Raising inequality () to a power e  > , inequality () to a power e  > , and multiplying the resulting inequalities, we obtain Next, we take (e  , e  ) any solution of the homogeneous linear system Using (), we may take Therefore, we obtain Using again (), we get which proves Theorem ..
As a consequence of Theorem ., we deduce the following result for the case of a single equation.

Corollary  Let us assume that there exists a nontrivial solution of
where p > , q > , f ≥ , and f ∈ L  (a, b). Then Proof An application of Theorem . with yields the desired result.
Remark  Taking f = h and q = p in Corollary , we obtain Lyapunov-type inequality () for the one-dimensional p-Laplacian equation.
Remark  Taking p  = q  = p and p  = q  = q in Theorem ., we obtain Lyapunov-type inequality ().

Generalized eigenvalues
The concept of generalized eigenvalues was introduced by Protter [] for a system of linear elliptic operators. The first work dealing with generalized eigenvalues for p-Laplacian systems is due to Nápoli and Pinasco []. Inspired by that work, we present in this section some applications to generalized eigenvalues related to problem (S)-(DBC). Let us consider the generalized eigenvalue problem (S) λ,μ : , we say that (λ, μ) is a generalized eigenvalue of (S) λ,μ -(DBC). The set of generalized eigenvalues is called generalized spectrum, and it is denoted by σ . We assume that and () is satisfied.
The following result provides lower bounds of the generalized eigenvalues of (S) λ,μ -(DBC).
Corollary  Let (λ, μ) be fixed. There exists an interval J of sufficiently small measure such that, if I = [a, b] ⊂ J, then there are no nontrivial solutions of (S) λ,μ -(DBC).
Proof Suppose that (S) λ,μ -(DBC) admits a nontrivial solution. Since C → +∞ as ba →  + , where C is defined in Theorem ., there exists δ >  such that which is a contradiction with (). Therefore, if I ⊂ J, there are no nontrivial solutions of (S) λ,μ -(DBC).

Conclusion
Lyapunov-type inequalities for a system of differential equations involving one-dimensional (p i , q i )-Laplacian operators (i = , ) are derived. It was shown that such inequalities are very useful to obtain geometric characterizations of the generalized spectrum associated to the considered problem.