Abstract
We establish Lyapunov-type inequalities for a system involving one-dimensional \((p_{i},q_{i})\)-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.
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1 Introduction
In this paper, we are concerned with the following system involving one-dimensional \((p_{i},q_{i})\)-Laplacian operators (\(i=1,2\)):
on the interval \((a,b)\), under Dirichlet boundary conditions
System (S) is investigated under the assumptions
and
We suppose also that f and g are two nonnegative real-valued functions such that \((f,g)\in L^{1}(a,b)\times L^{1}(a,b)\). 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 [1] (see also [2]) states that if the boundary value problem
has a nontrivial solution, where \(q: [a,b]\to \mathbb{R}\) is a continuous function, then
Inequality (2) 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 [3] (see also [4]), Elbert extended inequality (2) to the one-dimensional p-Laplacian equation. More precisely, he proved that, if u is a nontrivial solution of the problem
where \(1< p<\infty\) and \(h\in L^{1}(a,b)\), then
Observe that for \(p=2\), (3) reduces to (2). Inequality (3) was extended in [5] to the following problem involving the φ-Laplacian operator:
where \(\varphi: \mathbb{R}\to \mathbb{R}\) is a convex nondecreasing function satisfying a \(\Delta_{2}\) condition. In [6], Nápoli and Pinasco considered the quasilinear system of resonant type
on the interval \((a,b)\), with Dirichlet boundary conditions
Under the assumptions \(p,q>1\), \(f,g\in L^{1}(a,b)\), \(f,g\geq 0\), \(\alpha,\beta\geq 0\), and
it was proved (see [6], Theorem 1.5) that if (4)-(5) has a nontrivial solution, then
where \(p'=\frac{p}{p-1}\) and \(q'=\frac{q}{q-1}\). Some nice applications to generalized eigenvalues are also presented in [6]. Different generalizations and extensions of inequality (6) were obtained by many authors. In this direction, we refer the reader to [7–16] and the references therein. For other results concerning Lyapunov-type inequalities, we refer the reader to [17–29] and the references therein.
2 Lyapunov-type inequalities
A Lyapunov-type inequality for problem (S)-(DBC) is established in this section, and some particular cases are discussed.
Theorem 2.1
If (S)-(DBC) admits a nontrivial solution \((u,v)\in C^{2}[a,b]\times C^{2}[a,b]\), then
Proof
Let \((u,v)\in C^{2}[a,b]\times C^{2}[a,b]\) be a nontrivial solution to (S)-(DBC). Let \((x_{0},y_{0})\in (a,b)\times (a,b)\) be such that
and
From the boundary conditions (DBC), we can write that
which yields
Using Hölder’s inequality with parameters \(p_{1}\) and \(p_{1}'=\frac{p_{1}}{p_{1}-1}\), we get
that is,
Similarly, using Hölder’s inequality with parameters \(q_{1}\) and \(q_{1}'=\frac{q_{1}}{q_{1}-1}\), we get
By repeating the same argument for the function v, we obtain
and
Now, multiplying the first equation of (S) by u and integrating over \((a,b)\), we obtain
Multiplying the second equation of (S) by v and integrating over \((a,b)\), we obtain
Using (8), (9) and (12), we obtain
which yields
Using the inequality
with \(A= \vert u(x_{0}) \vert ^{p_{1}}\) and \(B= \vert u(x_{0}) \vert ^{q_{1}}\), we get
Similarly, using (10), (11) and (13), we obtain
Raising inequality (14) to a power \(e_{1}>0\), inequality (15) to a power \(e_{2}>0\), and multiplying the resulting inequalities, we obtain
Next, we take \((e_{1},e_{2})\) any solution of the homogeneous linear system
Using (1), we may take
Therefore, we obtain
Using again (1), we get
which proves Theorem 2.1. □
As a consequence of Theorem 2.1, we deduce the following result for the case of a single equation.
Corollary 1
Let us assume that there exists a nontrivial solution of
where \(p>1\), \(q>1\), \(f\geq 0\), and \(f\in L^{1}(a,b)\). Then
Proof
An application of Theorem 2.1 with
yields the desired result. □
Remark 1
Taking \(f=2h\) and \(q=p\) in Corollary 1, we obtain Lyapunov-type inequality (3) for the one-dimensional p-Laplacian equation.
Remark 2
Taking \(p_{1}=q_{1}=p\) and \(p_{2}=q_{2}=q\) in Theorem 2.1, we obtain Lyapunov-type inequality (6).
3 Generalized eigenvalues
The concept of generalized eigenvalues was introduced by Protter [30] for a system of linear elliptic operators. The first work dealing with generalized eigenvalues for p-Laplacian systems is due to Nápoli and Pinasco [6]. 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
on the interval \((a,b)\), with Dirichlet boundary conditions (DBC). If problem \(\mbox{(S)}_{\lambda,\mu}\)-(DBC) admits a nontrivial solution \((u,v)\in C^{2}[a,b]\times C^{2}[a,b]\), we say that \((\lambda,\mu)\) is a generalized eigenvalue of \(\mbox{(S)}_{\lambda,\mu}\)-(DBC). The set of generalized eigenvalues is called generalized spectrum, and it is denoted by σ.
We assume that
and (1) is satisfied.
The following result provides lower bounds of the generalized eigenvalues of \(\mbox{(S)}_{\lambda,\mu}\)-(DBC).
Theorem 3.1
Let \((\lambda,\mu)\) be a generalized eigenvalue of \(\mathrm{(S)}_{\lambda,\mu}\)-(DBC). Then
where \(h: (0,\infty)\to (0,\infty)\) is the function defined by
with
Proof
Let \((\lambda,\mu)\) be a generalized eigenpair, and let \(u,v\) be the corresponding nontrivial solutions. By replacing in Lyapunov-type inequality (7) the functions
and using condition (1), we obtain
where
Hence, we have
which yields
and the proof is finished. □
As consequences of the previous obtained result, we deduce the following Protter’s type results for the generalized spectrum.
Corollary 2
There exists a constant \(c_{a,b}>0\) that depends on a and b such that no point of the generalized spectrum σ is contained in the ball \(B(0,c_{a,b})\), where
and \(\Vert \cdot \Vert _{\infty}\) is the Chebyshev norm in \(\mathbb{R}^{2}\).
Proof
Let \((\lambda,\mu)\in \sigma\). From (16), we obtain easily that
On the other hand, using condition (1), we have
Therefore, we obtain
where
The proof is finished. □
Corollary 3
Let \((\lambda,\mu)\) be fixed. There exists an interval J of sufficiently small measure such that, if \(I=[a,b]\subset J\), then there are no nontrivial solutions of \(\mathrm{(S)}_{\lambda,\mu}\)-(DBC).
Proof
Suppose that \(\mbox{(S)}_{\lambda,\mu}\)-(DBC) admits a nontrivial solution. Since \(C\to +\infty\) as \(b-a\to 0^{+}\), where C is defined in Theorem 3.1, there exists \(\delta>0\) such that
Let \(J=[a,a+\delta]\). Hence, if \(I\subset J\), we have
which is a contradiction with (17). Therefore, if \(I\subset J\), there are no nontrivial solutions of \(\mbox{(S)}_{\lambda,\mu}\)-(DBC). □
4 Conclusion
Lyapunov-type inequalities for a system of differential equations involving one-dimensional \((p_{i},q_{i})\)-Laplacian operators (\(i=1,2\)) are derived. It was shown that such inequalities are very useful to obtain geometric characterizations of the generalized spectrum associated to the considered problem.
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The second author extends his appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).
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Jleli, M., Samet, B. On Lyapunov-type inequalities for \((p,q)\)-Laplacian systems. J Inequal Appl 2017, 100 (2017). https://doi.org/10.1186/s13660-017-1377-0
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DOI: https://doi.org/10.1186/s13660-017-1377-0