Skip to main content

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

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.

Introduction

In this paper, we are concerned with the following system involving one-dimensional \((p_{i},q_{i})\)-Laplacian operators (\(i=1,2\)):

$$\begin{aligned} \mbox{(S)}:\quad \textstyle\begin{cases} -( \vert u'(x) \vert ^{p_{1}-2}u'(x))'-( \vert u'(x) \vert ^{q_{1}-2}u'(x))'=f(x) \vert u(x) \vert ^{\alpha-2} \vert v(x) \vert ^{\beta}u(x),\\ -( \vert v'(x) \vert ^{p_{2}-2}v'(x))'-( \vert v'(x) \vert ^{q_{2}-2}v'(x))'=g(x) \vert u(x) \vert ^{\alpha} \vert v(x) \vert ^{\beta-2}v(x) \end{cases}\displaystyle \end{aligned}$$

on the interval \((a,b)\), under Dirichlet boundary conditions

$$\mbox{(DBC)}:\quad u(a)=u(b)=v(a)=v(b)=0. $$

System (S) is investigated under the assumptions

$$\alpha\geq 2,\qquad \beta\geq 2,\qquad p_{i}\geq 2,\qquad q_{i}\geq 2, \quad i=1,2, $$

and

$$ \frac{2\alpha}{p_{1}+q_{1}}+\frac{2\beta}{p_{2}+q_{2}}=1. $$
(1)

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

$$\begin{aligned} \textstyle\begin{cases} u''(t)+q(t)u(t)=0,\quad a< t< b,\\ u(a)=u(b)=0, \end{cases}\displaystyle \end{aligned}$$

has a nontrivial solution, where \(q: [a,b]\to \mathbb{R}\) is a continuous function, then

$$ \int_{a}^{b} \bigl\vert q(t) \bigr\vert \,dt> \frac{4}{b-a}. $$
(2)

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

$$\begin{aligned} \textstyle\begin{cases} ( \vert u' \vert ^{p-2}u')'+h(t) \vert u \vert ^{p-2}u=0,\quad a< t< b,\\ u(a)=u(b)=0, \end{cases}\displaystyle \end{aligned}$$

where \(1< p<\infty\) and \(h\in L^{1}(a,b)\), then

$$ \int_{a}^{b} \bigl\vert h(t) \bigr\vert \,dt> \frac{2^{p}}{(b-a)^{p-1}}. $$
(3)

Observe that for \(p=2\), (3) reduces to (2). Inequality (3) was extended in [5] to the following problem involving the φ-Laplacian operator:

$$\begin{aligned} \textstyle\begin{cases} (\varphi(u'))'+w(t)\varphi(u)=0,\quad a< t< b,\\ u(a)=u(b)=0, \end{cases}\displaystyle \end{aligned}$$

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

$$\begin{aligned} \textstyle\begin{cases} -( \vert u'(x) \vert ^{p-2}u'(x))'= f(x) \vert u(x) \vert ^{\alpha-2} \vert v(x) \vert ^{\beta}u(x),\\ -( \vert v'(x) \vert ^{q-2}v'(x))'= g(x) \vert u(x) \vert ^{\alpha} \vert v(x) \vert ^{\beta-2}v(x) \end{cases}\displaystyle \end{aligned}$$
(4)

on the interval \((a,b)\), with Dirichlet boundary conditions

$$ u(a)=u(b)=v(a)=v(b)=0. $$
(5)

Under the assumptions \(p,q>1\), \(f,g\in L^{1}(a,b)\), \(f,g\geq 0\), \(\alpha,\beta\geq 0\), and

$$\frac{\alpha}{p}+\frac{\beta}{q}=1, $$

it was proved (see [6], Theorem 1.5) that if (4)-(5) has a nontrivial solution, then

$$ 2^{\alpha+\beta} \leq (b-a)^{\frac{\alpha}{p'}+\frac{\beta}{q'}} \biggl( \int_{a}^{b} f(x)\,dx \biggr)^{\frac{\alpha}{p}} \biggl( \int_{a}^{b} g(x)\,dx \biggr)^{\frac{\beta}{q}}, $$
(6)

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 [716] and the references therein. For other results concerning Lyapunov-type inequalities, we refer the reader to [1729] 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.

Theorem 2.1

If (S)-(DBC) admits a nontrivial solution \((u,v)\in C^{2}[a,b]\times C^{2}[a,b]\), then

$$\begin{aligned} & \biggl[\min \biggl\{ \frac{2^{p_{1}}}{(b-a)^{p_{1}-1}},\frac{2^{q_{1}}}{(b-a)^{q_{1}-1}} \biggr\} \biggr]^{\frac{2\alpha}{p_{1}+q_{1}}} \biggl[\min \biggl\{ \frac{2^{p_{2}}}{(b-a)^{p_{2}-1}}, \frac{2^{q_{2}}}{(b-a)^{q_{2}-1}} \biggr\} \biggr]^{\frac{2\beta}{p_{2}+q_{2}}} \\ &\quad \leq \biggl(\frac{1}{2} \int_{a}^{b} f(x)\,dx \biggr)^{\frac{2\alpha}{p_{1}+q_{1}}} \biggl( \frac{1}{2} \int_{a}^{b} g(x)\,dx \biggr)^{\frac{2\beta}{p_{2}+q_{2}}}. \end{aligned}$$
(7)

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

$$\bigl\vert u(x_{0}) \bigr\vert =\max\bigl\{ \bigl\vert u(x) \bigr\vert :\, a\leq x\leq b\bigr\} $$

and

$$\bigl\vert v(y_{0}) \bigr\vert =\max\bigl\{ \bigl\vert v(x) \bigr\vert :\, a\leq x\leq b\bigr\} . $$

From the boundary conditions (DBC), we can write that

$$2u(x_{0})= \int_{a}^{x_{0}} u'(x)\,dx - \int_{x_{0}}^{b} u'(x)\,dx, $$

which yields

$$2 \bigl\vert u(x_{0}) \bigr\vert \leq \int_{a}^{b} \bigl\vert u'(x) \bigr\vert \,dx. $$

Using Hölder’s inequality with parameters \(p_{1}\) and \(p_{1}'=\frac{p_{1}}{p_{1}-1}\), we get

$$2 \bigl\vert u(x_{0}) \bigr\vert \leq (b-a)^{\frac{1}{p_{1}'}} \biggl( \int_{a}^{b} \bigl\vert u'(x) \bigr\vert ^{p_{1}}\,dx \biggr)^{\frac{1}{p_{1}}}, $$

that is,

$$ \frac{2^{p_{1}}}{(b-a)^{p_{1}-1}} \bigl\vert u(x_{0}) \bigr\vert ^{p_{1}} \leq \int_{a}^{b} \bigl\vert u'(x) \bigr\vert ^{p_{1}}\,dx. $$
(8)

Similarly, using Hölder’s inequality with parameters \(q_{1}\) and \(q_{1}'=\frac{q_{1}}{q_{1}-1}\), we get

$$ \frac{2^{q_{1}}}{(b-a)^{q_{1}-1}} \bigl\vert u(x_{0}) \bigr\vert ^{q_{1}} \leq \int_{a}^{b} \bigl\vert u'(x) \bigr\vert ^{q_{1}}\,dx. $$
(9)

By repeating the same argument for the function v, we obtain

$$ \frac{2^{p_{2}}}{(b-a)^{p_{2}-1}} \bigl\vert v(y_{0}) \bigr\vert ^{p_{2}} \leq \int_{a}^{b} \bigl\vert v'(x) \bigr\vert ^{p_{2}}\,dx $$
(10)

and

$$ \frac{2^{q_{2}}}{(b-a)^{q_{2}-1}} \bigl\vert v(y_{0}) \bigr\vert ^{q_{2}} \leq \int_{a}^{b} \bigl\vert v'(x) \bigr\vert ^{q_{2}}\,dx. $$
(11)

Now, multiplying the first equation of (S) by u and integrating over \((a,b)\), we obtain

$$ \int_{a}^{b} \bigl\vert u'(x) \bigr\vert ^{p_{1}}\,dx+ \int_{a}^{b} \bigl\vert u'(x) \bigr\vert ^{q_{1}}\,dx= \int_{a}^{b} f(x) \bigl\vert u(x) \bigr\vert ^{\alpha}\bigl\vert v(x) \bigr\vert ^{\beta}\,dx. $$
(12)

Multiplying the second equation of (S) by v and integrating over \((a,b)\), we obtain

$$ \int_{a}^{b} \bigl\vert v'(x) \bigr\vert ^{p_{2}}\,dx+ \int_{a}^{b} \bigl\vert v'(x) \bigr\vert ^{q_{2}}\,dx= \int_{a}^{b} g(x) \bigl\vert u(x) \bigr\vert ^{\alpha}\bigl\vert v(x) \bigr\vert ^{\beta}\,dx. $$
(13)

Using (8), (9) and (12), we obtain

$$\bigl\vert u(x_{0}) \bigr\vert ^{\alpha}\bigl\vert v(y_{0}) \bigr\vert ^{\beta}\int_{a}^{b} f(x) \,dx\geq \frac{2^{p_{1}}}{(b-a)^{p_{1}-1}} \bigl\vert u(x_{0}) \bigr\vert ^{p_{1}}+\frac{2^{q_{1}}}{(b-a)^{q_{1}-1}} \bigl\vert u(x_{0}) \bigr\vert ^{q_{1}}, $$

which yields

$$\bigl\vert u(x_{0}) \bigr\vert ^{\alpha}\bigl\vert v(y_{0}) \bigr\vert ^{\beta}\int_{a}^{b} f(x) \,dx\geq \min \biggl\{ \frac{2^{p_{1}}}{(b-a)^{p_{1}-1}},\frac{2^{q_{1}}}{(b-a)^{q_{1}-1}} \biggr\} \bigl( \bigl\vert u(x_{0}) \bigr\vert ^{p_{1}}+ \bigl\vert u(x_{0}) \bigr\vert ^{q_{1}} \bigr). $$

Using the inequality

$$A+B\geq 2\sqrt{A}\sqrt{B} $$

with \(A= \vert u(x_{0}) \vert ^{p_{1}}\) and \(B= \vert u(x_{0}) \vert ^{q_{1}}\), we get

$$ \min \biggl\{ \frac{2^{p_{1}+1}}{(b-a)^{p_{1}-1}},\frac{2^{q_{1}+1}}{(b-a)^{q_{1}-1}} \biggr\} \leq \bigl\vert u(x_{0}) \bigr\vert ^{\alpha-\frac{p_{1}+q_{1}}{2}} \bigl\vert v(y_{0}) \bigr\vert ^{\beta} \int_{a}^{b} f(x)\,dx. $$
(14)

Similarly, using (10), (11) and (13), we obtain

$$ \min \biggl\{ \frac{2^{p_{2}+1}}{(b-a)^{p_{2}-1}},\frac{2^{q_{2}+1}}{(b-a)^{q_{2}-1}} \biggr\} \leq \bigl\vert u(x_{0}) \bigr\vert ^{\alpha} \bigl\vert v(y_{0}) \bigr\vert ^{\beta-\frac{p_{2}+q_{2}}{2}} \int_{a}^{b} g(x)\,dx. $$
(15)

Raising inequality (14) to a power \(e_{1}>0\), inequality (15) to a power \(e_{2}>0\), and multiplying the resulting inequalities, we obtain

$$\begin{aligned} & \biggl[\min \biggl\{ \frac{2^{p_{1}+1}}{(b-a)^{p_{1}-1}},\frac{2^{q_{1}+1}}{(b-a)^{q_{1}-1}} \biggr\} \biggr]^{e_{1}} \biggl[\min \biggl\{ \frac{2^{p_{2}+1}}{(b-a)^{p_{2}-1}},\frac{2^{q_{2}+1}}{(b-a)^{q_{2}-1}} \biggr\} \biggr]^{e_{2}} \\ &\quad \leq \bigl\vert u(x_{0}) \bigr\vert ^{ (\alpha-\frac{p_{1}+q_{1}}{2} )e_{1}+\alpha e_{2}} \bigl\vert v(y_{0}) \bigr\vert ^{\beta e_{1}+ (\beta-\frac{p_{2}+q_{2}}{2} )e_{2}} \biggl( \int_{a}^{b} f(x)\,dx \biggr)^{e_{1}} \biggl( \int_{a}^{b} g(x)\,dx \biggr)^{e_{2}}. \end{aligned}$$

Next, we take \((e_{1},e_{2})\) any solution of the homogeneous linear system

$$\begin{aligned} \textstyle\begin{cases} (\alpha-\frac{p_{1}+q_{1}}{2} )e_{1}+\alpha e_{2}= 0,\\ \beta e_{1}+ (\beta-\frac{p_{2}+q_{2}}{2} )e_{2}=0. \end{cases}\displaystyle \end{aligned}$$

Using (1), we may take

$$\begin{aligned} \textstyle\begin{cases} e_{1}= \alpha,\\ e_{2}=\frac{\beta(p_{1}+q_{1})}{p_{2}+q_{2}}. \end{cases}\displaystyle \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} &2^{\alpha+\frac{\beta(p_{1}+q_{1})}{p_{2}+q_{2}}} \biggl[\min \biggl\{ \frac{2^{p_{1}}}{(b-a)^{p_{1}-1}},\frac{2^{q_{1}}}{(b-a)^{q_{1}-1}} \biggr\} \biggr]^{\alpha} \biggl[\min \biggl\{ \frac{2^{p_{2}}}{(b-a)^{p_{2}-1}}, \frac{2^{q_{2}}}{(b-a)^{q_{2}-1}} \biggr\} \biggr]^{\frac{\beta(p_{1}+q_{1})}{p_{2}+q_{2}}} \\ &\quad \leq \biggl( \int_{a}^{b} f(x)\,dx \biggr)^{\alpha} \biggl( \int_{a}^{b} g(x)\,dx \biggr)^{\frac{\beta(p_{1}+q_{1})}{p_{2}+q_{2}}}. \end{aligned}$$

Using again (1), we get

$$\begin{aligned} &2 \biggl[\min \biggl\{ \frac{2^{p_{1}}}{(b-a)^{p_{1}-1}},\frac{2^{q_{1}}}{(b-a)^{q_{1}-1}} \biggr\} \biggr]^{\frac{2\alpha}{p_{1}+q_{1}}} \biggl[\min \biggl\{ \frac{2^{p_{2}}}{(b-a)^{p_{2}-1}},\frac{2^{q_{2}}}{(b-a)^{q_{2}-1}} \biggr\} \biggr]^{\frac{2\beta}{p_{2}+q_{2}}} \\ &\quad \leq \biggl( \int_{a}^{b} f(x)\,dx \biggr)^{\frac{2\alpha}{p_{1}+q_{1}}} \biggl( \int_{a}^{b} g(x)\,dx \biggr)^{\frac{2\beta}{p_{2}+q_{2}}}, \end{aligned}$$

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

$$\begin{aligned} \textstyle\begin{cases} -( \vert u'(x) \vert ^{p-2}u'(x))'-( \vert u'(x) \vert ^{q-2}u'(x))'=f(x) \vert u(x) \vert ^{\frac{p+q}{2}-2}u(x),\quad x\in (a,b),\\ u(a)=u(b)=0, \end{cases}\displaystyle \end{aligned}$$

where \(p>1\), \(q>1\), \(f\geq 0\), and \(f\in L^{1}(a,b)\). Then

$$\min \biggl\{ \frac{2^{p}}{(b-a)^{p-1}},\frac{2^{q}}{(b-a)^{q-1}} \biggr\} \leq \frac{1}{2} \int_{a}^{b} f(x)\,dx. $$

Proof

An application of Theorem 2.1 with

$$p_{1}=p_{2}=p,\qquad q_{1}=q_{2}=q,\qquad \alpha=\frac{p+q}{2},\qquad \beta=0,\qquad v=u,\qquad g=f, $$

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).

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

$$\begin{aligned} \mbox{(S)}_{\lambda,\mu}:\quad \textstyle\begin{cases} -( \vert u'(x) \vert ^{p_{1}-2}u'(x))'-( \vert u'(x) \vert ^{q_{1}-2}u'(x))'=\lambda \alpha w(x) \vert u(x) \vert ^{\alpha-2} \vert v(x) \vert ^{\beta}u(x),\\ -( \vert v'(x) \vert ^{p_{2}-2}v'(x))'-( \vert v'(x) \vert ^{q_{2}-2}v'(x))'=\mu \beta w(x) \vert u(x) \vert ^{\alpha} \vert v(x) \vert ^{\beta-2}v(x), \end{cases}\displaystyle \end{aligned}$$

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

$$\alpha\geq 2,\qquad \beta\geq 2,\qquad p_{i}\geq 2,\qquad q_{i}\geq 2, \quad i=1,2, \qquad w\geq 0,\qquad w\in L^{1}(a,b), $$

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

$$ \mu\geq h(\lambda), $$
(16)

where \(h: (0,\infty)\to (0,\infty)\) is the function defined by

$$h(t)=\frac{1}{\beta} \biggl(\frac{C}{t^{\frac{2\alpha}{p_{1}+q_{1}}}\int_{a}^{b} w(x)\,dx} \biggr)^{\frac{p_{2}+q_{2}}{2\beta}},\quad t>0, $$

with

$$\begin{aligned} \alpha^{\frac{2\alpha}{p_{1}+q_{1}}} C =&2 \biggl[\min \biggl\{ \frac{2^{p_{1}}}{(b-a)^{p_{1}-1}}, \frac{2^{q_{1}}}{(b-a)^{q_{1}-1}} \biggr\} \biggr]^{\frac{2\alpha}{p_{1}+q_{1}}} \\ &{}\times\biggl[\min \biggl\{ \frac{2^{p_{2}}}{(b-a)^{p_{2}-1}},\frac{2^{q_{2}}}{(b-a)^{q_{2}-1}} \biggr\} \biggr]^{\frac{2\beta}{p_{2}+q_{2}}}. \end{aligned}$$

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

$$f(x)=\alpha \lambda w(x),\qquad g(x)=\beta \mu w(x), $$

and using condition (1), we obtain

$$2M\leq \alpha^{\frac{2\alpha}{p_{1}+q_{1}}} \lambda^{\frac{2\alpha}{p_{1}+q_{1}}}\beta^{\frac{2\beta}{p_{2}+q_{2}}} \mu^{\frac{2\beta}{p_{2}+q_{2}}} \int_{a}^{b} w(x)\,dx, $$

where

$$M= \biggl[\min \biggl\{ \frac{2^{p_{1}}}{(b-a)^{p_{1}-1}},\frac{2^{q_{1}}}{(b-a)^{q_{1}-1}} \biggr\} \biggr]^{\frac{2\alpha}{p_{1}+q_{1}}} \biggl[\min \biggl\{ \frac{2^{p_{2}}}{(b-a)^{p_{2}-1}},\frac{2^{q_{2}}}{(b-a)^{q_{2}-1}} \biggr\} \biggr]^{\frac{2\beta}{p_{2}+q_{2}}}. $$

Hence, we have

$$\mu^{\frac{2\beta}{p_{2}+q_{2}}}\geq \frac{C}{\lambda^{\frac{2\alpha}{p_{1}+q_{1}}}\beta^{\frac{2\beta}{p_{2}+q_{2}}}\int_{a}^{b} w(x)\,dx}, $$

which yields

$$\mu\geq \frac{1}{\beta} \biggl(\frac{C}{\lambda^{\frac{2\alpha}{p_{1}+q_{1}}}\int_{a}^{b} w(x)\,dx} \biggr)^{\frac{p_{2}+q_{2}}{2\beta}}, $$

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

$$B(0,c_{a,b})= \bigl\{ x=(x_{1},x_{2})\in \mathbb{R}^{2}:\, \Vert x \Vert _{\infty}< c_{a,b} \bigr\} , $$

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

$$ \lambda^{\frac{2\alpha}{p_{1}+q_{1}}} \mu^{\frac{2\beta}{p_{2}+q_{2}}}\geq \frac{C}{\beta^{\frac{2\beta}{p_{2}+q_{2}}}\int_{a}^{b} w(x)\,dx}. $$
(17)

On the other hand, using condition (1), we have

$$\lambda^{\frac{2\alpha}{p_{1}+q_{1}}} \mu^{\frac{2\beta}{p_{2}+q_{2}}}\leq \bigl\Vert (\lambda,\mu) \bigr\Vert _{\infty}^{\frac{2\alpha}{p_{1}+q_{1}}+\frac{2\beta}{p_{2}+q_{2}}}= \bigl\Vert (\lambda,\mu) \bigr\Vert _{\infty}. $$

Therefore, we obtain

$$\bigl\Vert (\lambda,\mu) \bigr\Vert _{\infty}\geq c_{a,b}, $$

where

$$c_{a,b}=\frac{C}{\beta^{\frac{2\beta}{p_{2}+q_{2}}}\int_{a}^{b} w(x)\,dx}. $$

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

$$b-a< \delta \quad \implies\quad \frac{C}{\int_{a}^{b} w(x)\,dx}>\lambda^{\frac{2\alpha}{p_{1}+q_{1}}} \mu^{\frac{2\beta}{p_{2}+q_{2}}} \beta^{\frac{2\beta}{p_{2}+q_{2}}}. $$

Let \(J=[a,a+\delta]\). Hence, if \(I\subset J\), we have

$$\frac{C}{\beta^{\frac{2\beta}{p_{2}+q_{2}}} \int_{a}^{b} w(x)\,dx}>\lambda^{\frac{2\alpha}{p_{1}+q_{1}}} \mu^{\frac{2\beta}{p_{2}+q_{2}}}, $$

which is a contradiction with (17). Therefore, if \(I\subset J\), there are no nontrivial solutions of \(\mbox{(S)}_{\lambda,\mu}\)-(DBC). □

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.

References

  1. Lyapunov, AM: Problème général de la stabilité du mouvement. Ann. Fac. Sci. Univ. Toulouse Sci. Math. Sci. Phys. 2, 203-407 (1907)

    Google Scholar 

  2. Borg, G: On a Liapounoff criterion of stability. Am. J. Math. 71, 67-70 (1949)

    MathSciNet  Article  MATH  Google Scholar 

  3. Elbert, A: A half-linear second order differential equation. Colloq. Math. Soc. János Bolyai 30, 158-180 (1979)

    Google Scholar 

  4. Pinasco, JP: Lower bounds for eigenvalues of the one-dimensional p-Laplacian. Abstr. Appl. Anal. 2004, 147-153 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  5. De Nápoli, PL, Pinasco, JP: A Lyapunov inequality for monotone quasilinear operators. Differ. Integral Equ. 18(10), 1193-1200 (2005)

    MathSciNet  MATH  Google Scholar 

  6. De Nápoli, PL, Pinasco, JP: Estimates for eigenvalues of quasilinear elliptic systems. J. Differ. Equ. 227, 102-115 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  7. Aktaş, MF, Çakmak, D, Tiryaki, A: Lyapunov-type inequality for quasilinear systems with anti-periodic boundary conditions. J. Math. Inequal. 8(2), 313-320 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  8. Çakmak, D: On Lyapunov-type inequality for a class of nonlinear systems. Math. Inequal. Appl. 16, 101-108 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Çakmak, D: Lyapunov-type inequalities for two classes of nonlinear systems with anti-periodic boundary conditions. Appl. Math. Comput. 223, 237-242 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Çakmak, D: On Lyapunov-type inequality for a class of quasilinear systems. Electron. J. Qual. Theory Differ. Equ. 2014, 9 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  11. Çakmak, D, Aktaş, MF, Tiryaki, A: Lyapunov-type inequalities for nonlinear systems involving the \((p_{1}, p_{2},\dots,p_{n})\)-Laplacian. Electron. J. Differ. Equ. 2013, 128 (2013)

    Article  MATH  Google Scholar 

  12. Çakmak, D, Tiryaki, A: Lyapunov-type inequality for a class of Dirichlet quasilinear systems involving the \((p_{1},p_{2},\ldots,p_{n})\)-Laplacian. J. Math. Anal. Appl. 369, 76-81 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  13. Çakmak, D, Tiryaki, A: On Lyapunov-type inequality for quasilinear systems. Appl. Math. Comput. 216, 3584-3591 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Tiryaki, A, Çakmak, D, Aktaş, MF: Lyapunov-type inequalities for a certain class of nonlinear systems. Comput. Math. Appl. 64(6), 1804-1811 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  15. Tiryaki, A, Çakmak, D, Aktaş, MF: Lyapunov-type inequalities for two classes of Dirichlet quasi-linear systems. Math. Inequal. Appl. 17(3), 843-863 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Yang, X, Kim, Y, Lo, K: Lyapunov-type inequality for a class of quasilinear systems. Math. Comput. Model. 53, 1162-1166 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  17. Canada, A, Montero, JA, Villegas, S: Lyapunov inequalities for partial differential equations. J. Funct. Anal. 237, 176-193 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  18. Cheng, SS: Lyapunov inequalities for differential and difference equations. Fasc. Math. 23, 25-41 (1991)

    MathSciNet  MATH  Google Scholar 

  19. Dhar, S, Kong, Q: Lyapunov-type inequalities for higher order half-linear differential equations. Appl. Math. Comput. 273, 114-124 (2016)

    MathSciNet  Google Scholar 

  20. Jleli, M, Kirane, M, Samet, B: Lyapunov-type inequalities for fractional partial differential equations. Appl. Math. Lett. 66, 30-39 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  21. Jleli, M, Samet, B: Lyapunov-type inequalities for fractional boundary-value problems. Electron. J. Differ. Equ. 2015, 88 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  22. Lee, C, Yeh, C, Hong, C, Agarwal, RP: Lyapunov and Wirtinger inequalities. Appl. Math. Lett. 17, 847-853 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  23. De Nápoli, PL, Pinasco, JP: Lyapunov-type inequalities for partial differential equations. J. Funct. Anal. 15(6), 1995-2018 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  24. Pachpatte, BG: On Lyapunov-type inequalities for certain higher order differential equations. J. Math. Anal. Appl. 195, 527-536 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  25. Parhi, N, Panigrahi, S: On Liapunov-type inequality for third-order differential equations. J. Math. Anal. Appl. 233(2), 445-460 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  26. Sim, I, Lee, YH: Lyapunov inequalities for one-dimensional p-Laplacian problems with a singular weight function. J. Inequal. Appl. 2010, Article ID 865096 (2010)

    MathSciNet  MATH  Google Scholar 

  27. Tang, X, Zhang, M: Lyapunov inequalities and stability for linear Hamiltonian systems. J. Differ. Equ. 252, 358-381 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  28. Wang, Y: Lyapunov-type inequalities for certain higher order differential equations with anti-periodic boundary conditions. Appl. Math. Lett. 25, 2375-2380 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  29. Yang, X: On inequalities of Lyapunov type. Appl. Math. Comput. 134, 293-300 (2003)

    MathSciNet  MATH  Google Scholar 

  30. Protter, M: The generalized spectrum of second order elliptic systems. Rocky Mt. J. Math. 9(3), 503-518 (1979)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The second author extends his appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bessem Samet.

Additional information

Competing interests

The authors declare to have no competing interests.

Funding

This project was funded by the Distinguished Scientist Fellowship Program (DSFP) at King Saud University, Riyadh, Kingdom of Saudi Arabia.

Authors’ contributions

All authors contributed equally in writing this paper.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13660-017-1377-0

MSC

  • 26D10
  • 15A42
  • 39B72

Keywords

  • Lyapunov-type inequalities
  • \((p,q)\)-Laplacian
  • system
  • generalized eigenvalues
  • generalized spectrum