Skip to main content

Lyapunov-type inequalities for generalized one-dimensional Minkowski-curvature problems

Abstract

In this paper, we consider some types of scalar equations and systems of generalized one-dimensional Minkowski-curvature problems. Using an inequality technique, we establish several new Lyapunov-type inequalities for the problems considered. Our results extend the existing work in the literature.

1 Introduction

In [1], Russian mathematician Lyapunov proved the following result: If \(y(t)\) is a solution of

$$\begin{aligned} y''+q(t)y=0 \end{aligned}$$
(1)

satisfying \(y(a)=y(b)=0\) (\(a< b\)) and \(y(t)\neq 0\) for \(t\in (a,b)\), then

$$\begin{aligned} \int _{a}^{b} \bigl\vert q(t) \bigr\vert \,\mathrm{d}t>\frac{4}{b-a}. \end{aligned}$$
(2)

The above result is known as the Lyapunov inequality.

This result plays an important role in the study of various properties of solutions of Eq. (1) such as oscillation theory, disconjugacy and eigenvalue problems. After this seminal paper, the Lyapunov inequality and many of its generalizations have been studied by many researchers; see [2–42] and the references therein.

For example, Yang [43] obtained a Lyapunov-type inequality for the second order half-linear equation

$$\begin{aligned}& \bigl(r(t) \bigl\vert y'(t) \bigr\vert ^{p-1}y'(t) \bigr)'+q(t) \bigl\vert y(t) \bigr\vert ^{p-1}y(t)=0, \end{aligned}$$
(3)
$$\begin{aligned}& \ y(a)=(b)=0,\qquad y(t)\neq 0, \quad t\in (a,b), \end{aligned}$$
(4)

where \(q,r\in C([a,b],\mathbb{R})\) such that \(r(t)>0\) for \(t\in [a,b]\), and \(p>0\) is a constant.

Theorem 1.1

([43])

Assume Eq. (3) has a solution\(y(t)\), then the inequality

$$\begin{aligned} \int _{a}^{b}q_{+}(t)\,\mathrm{d}t\geq \frac{2^{p+1}}{ (\int _{a}^{b}r^{-\frac{1}{p}}(t)\,\mathrm{d}t )^{p}} \end{aligned}$$
(5)

holds, where\(q_{+}(t):=\max \{q(t),0\}\).

Recently, Yang et al. [44] investigated a Lyapunov-type inequality for one-dimensional Minkowski-curvature problem with singular weight

$$\begin{aligned}& - \biggl(\frac{y'(t)}{\sqrt{1- \vert y'(t) \vert ^{2}}} \biggr)'=k(t)y(t), \end{aligned}$$
(6)
$$\begin{aligned}& \ y(a)=y(b)=0,\qquad y(t)\neq 0, \quad t\in (a,b). \end{aligned}$$
(7)

They presented the following result.

Theorem 1.2

([44])

If the problem (6) has a positive solution, then one has

$$\begin{aligned} \int _{a}^{b}(t-a) (b-t)k(t)\,\mathrm{d}t>b-a, \end{aligned}$$
(8)

where\(k(t)\geq 0\)for all\(t\in (a,b)\), \(k\not \equiv 0\)in any compact subinterval of\([a,b]\)and\(k\in U=\{k\in L^{1}_{\mathrm{loc}}((a,b),[0,\infty )):\int _{a}^{b}(t-a)(b-t)k(t) \,\mathrm{d}t<\infty \}\).

Motivated by this work, in this paper, we will establish Lyapunov-type inequalities for the generalized one-dimensional Minkowski-curvature problems with singular weight function

$$\begin{aligned}& - \biggl(\frac{r(t) \vert y'(t) \vert ^{p-2}y'(t)}{\sqrt{1- \vert y'(t) \vert ^{p}}} \biggr)'=q(t) \bigl\vert y(t) \bigr\vert ^{p-2}y(t), \end{aligned}$$
(9)
$$\begin{aligned}& \ y(a)=y(b)=0,\qquad y(t)\neq 0, \quad t\in (a,b), \end{aligned}$$
(10)

and

$$\begin{aligned}& - \biggl(\frac{r(t) \vert y'(t) \vert ^{p-2}y'(t)}{\sqrt{1- \vert y'(t) \vert ^{p}}} \biggr)'=l(t) \bigl\vert y(t) \bigr\vert ^{ \alpha -2}y(t)-h(t) \bigl\vert y(t) \bigr\vert ^{\beta -2}y(t), \end{aligned}$$
(11)
$$\begin{aligned}& \ y(a)=y(b)=0,\qquad y(t)\neq 0, \quad t\in (a,b), \end{aligned}$$
(12)

and the cycled systems of a generalized one-dimensional Minkowski-curvature problem with singular weight functions

$$\begin{aligned}& \textstyle\begin{cases} ( \frac{r_{1}(t) \vert y_{1}'(t) \vert ^{p-2}y_{1}'(t)}{\sqrt{1- \vert y_{1}'(t) \vert ^{p}}} )'+q_{1}(t) \vert y_{2}(t) \vert ^{p-2}y_{2}(t)=0,\quad t\in (a,b), \\ ( \frac{r_{2}(t) \vert y_{2}'(t) \vert ^{p-2}y_{2}'(t)}{\sqrt{1- \vert y_{2}'(t) \vert ^{p}}} )'+q_{2}(t) \vert y_{3}(t) \vert ^{p-2}y_{3}(t)=0,\quad t\in (a,b), \\ ( \frac{r_{3}(t) \vert y_{3}'(t) \vert ^{p-2}y_{3}'(t)}{\sqrt{1- \vert y_{3}'(t) \vert ^{p}}} )'+q_{3}(t) \vert y_{4}(t) \vert ^{p-2}y_{4}(t)=0,\quad t\in (a,b), \\ \quad \cdots , \\ ( \frac{r_{n}(t) \vert y_{n}'(t) \vert ^{p-2}y_{n}'(t)}{\sqrt{1- \vert y_{n}'(t) \vert ^{p}}} )'+q_{n}(t) \vert y_{1}(t) \vert ^{p-2}y_{1}(t)=0,\quad t\in (a,b), \end{cases}\displaystyle \end{aligned}$$
(13)
$$\begin{aligned}& y_{1}(a)=\cdots =y_{n}(a)=0=y_{1}(b)=\cdots =y_{n}(b),\quad y_{i}(t)\neq 0, t\in (a,b), \\& \quad \text{for }i=1,2,\ldots ,n, \end{aligned}$$
(14)

and

$$\begin{aligned}& \textstyle\begin{cases} ( \frac{r_{1}(t) \vert y_{1}'(t) \vert ^{p-2}y_{1}'(t)}{\sqrt{1- \vert y_{1}'(t) \vert ^{p}}} )'+l_{1}(t) \vert y_{2}(t) \vert ^{\alpha -2}y_{2}(t)-h_{1}(t) \vert y_{2}(t) \vert ^{ \beta -2}y_{2}(t)=0,\quad t\in (a,b), \\ ( \frac{r_{2}(t) \vert y_{2}'(t) \vert ^{p-2}y_{2}'(t)}{\sqrt{1- \vert y_{2}'(t) \vert ^{p}}} )'+l_{2}(t) \vert y_{3}(t) \vert ^{\alpha -2}y_{3}(t)-h_{2}(t) \vert y_{3}(t) \vert ^{ \beta -2}y_{3}(t)=0,\quad t\in (a,b), \\ ( \frac{r_{3}(t) \vert y_{3}'(t) \vert ^{p-2}y_{3}'(t)}{\sqrt{1- \vert y_{3}'(t) \vert ^{p}}} )'+l_{3}(t) \vert y_{4}(t) \vert ^{\alpha -2}y_{4}(t)-h_{3}(t) \vert y_{4}(t) \vert ^{ \beta -2}y_{4}(t)=0,\quad t\in (a,b), \\ \quad \cdots , \\ ( \frac{r_{n}(t) \vert y_{n}'(t) \vert ^{p-2}y_{n}'(t)}{\sqrt{1- \vert y_{n}'(t) \vert ^{p}}} )'+l_{n}(t) \vert y_{1}(t) \vert ^{\alpha -2}y_{1}(t)-h_{n}(t) \vert y_{1}(t) \vert ^{ \beta -2}y_{1}(t)=0,\quad t\in (a,b), \end{cases}\displaystyle \end{aligned}$$
(15)
$$\begin{aligned}& y_{1}(a)=\cdots =y_{n}(a)=0=y_{1}(b)=\cdots =y_{n}(b),\quad y_{i}(t)\neq 0, t\in (a,b), \\& \quad \text{for } i=1,2,\ldots ,n, \end{aligned}$$
(16)

where \(p>1\), \(1< p<\alpha <\beta \) or \(1<\beta <\alpha <p\), \(r, r_{i}\in C([a,b],(0,+\infty ))\), \(q(t), q_{i}(t)\geq 0\) for all \(t\in (a,b)\), \(q, q_{i}\not \equiv 0\) in any compact subinterval of \([a,b]\) and

$$ q, q_{i}\in \mathfrak{D}=\biggl\{ f\in L^{1}_{\mathrm{loc}} \bigl((a,b),[0,\infty \bigr)): \int _{a}^{b}(t-a)^{p-1}(b-t)^{p-1}f(t) \,\mathrm{d}t< \infty \biggr\} , $$

\(i=1,2,\ldots ,n\). \(l(t), h(t), l_{i}(t), h_{i}(t)> 0\) for all \(t\in (a,b)\) such that

$$\begin{aligned}& A(t)=\frac{l(t)(\beta -\alpha )}{\beta -p} \biggl( \frac{(\beta -p)h(t)}{(\alpha -p)l(t)} \biggr)^{(\alpha -p)/(\alpha - \beta )}, \\& A_{i}(t)=\frac{l_{i}(t)(\beta -\alpha )}{\beta -p} \biggl( \frac{(\beta -p)h_{i}(t)}{(\alpha -p)l_{i}(t)} \biggr)^{(\alpha -p)/( \alpha -\beta )}, \end{aligned}$$

satisfy \(A, A_{i}\in \mathfrak{D}\), \(i=1,2,\ldots , n\). Class \(\mathfrak{D}\) admits rather stronger singular functions at the boundary. For example, \(q(t)=t^{-(2p-1)/p}\in \mathfrak{D}\) with \(a=0\), \(b=1\) but not in \(L^{1}(0,1)\).

Our results not only extend the existing work in the literature, but also give necessary conditions for the existence of positive solutions for scalar equations and systems of generalized one-dimensional Minkowski-curvature problems with singular weight functions.

2 Preliminaries

In this section, we give some definitions and lemmas which are needed in the sequel.

Definition 2.1

We say y is a solution of problem (9)–(10) (or (11)–(12)) if \(y\in C^{1}[a,b]\), \(\|y'\|_{\infty }<1\), and \(\frac{r(\cdot )|y'(\cdot )|^{p-2}y'(\cdot )}{\sqrt{1-|y'(\cdot )|^{p}}}\) is absolutely continuous in any compact subinterval of \((a,b)\), and y satisfies the equation and the boundary conditions in problem (9)–(10) (or (11)–(12)).

Definition 2.2

We say \((y_{1},y_{2}, \ldots , y_{n})\) is a solution of problem (13)–(14) (or (15)–(16)) if \(y_{i}\in C^{1}[a,b]\), \(\|y'_{i}\|_{\infty }<1\), and \(\frac{r(\cdot )|y'_{i}(\cdot )|^{p-2}y'_{i}(\cdot )}{\sqrt{1-|y'_{i}(\cdot )|^{p}}}\) is absolutely continuous in any compact subinterval of (a,b), and \(y_{i}\) satisfies the equations and the boundary conditions in problem (13)–(14) (or (15)–(16)).

Lemma 2.1

([12])

Suppose that\(a, b\in \mathbb{R}\), \(\gamma >0\). Then

$$\begin{aligned} \frac{1}{\widetilde{K}(\gamma )}\bigl( \vert a \vert + \vert b \vert \bigr)^{\gamma }\leq \vert a \vert ^{\gamma }+ \vert b \vert ^{\gamma }, \end{aligned}$$

where

$$\begin{aligned} \widetilde{K}(\gamma )= \textstyle\begin{cases} 1, & 0< \gamma \leq 1, \\ 2^{\gamma -1},&\gamma >1. \end{cases}\displaystyle \end{aligned}$$

Lemma 2.2

If\(y\in C^{1}[a,b]\), \(y(a)=y(b)=0\)and\(p>1\), then we have

$$\begin{aligned} \bigl\vert y(t) \bigr\vert ^{p}\leq K(p) \biggl(\frac{(t-a)(b-t)}{b-a} \biggr)^{p-1} \biggl( \int _{a}^{b} \bigl\vert y'(s) \bigr\vert ^{p} \,\mathrm{d}s \biggr), \end{aligned}$$
(17)

where

$$\begin{aligned} K(p)= \textstyle\begin{cases} 1, &1< p\leq 2, \\ 2^{p-2},&p>2. \end{cases}\displaystyle \end{aligned}$$
(18)

Proof

From Hölder’s inequality, we get \(\forall t\in [a,b]\),

$$\begin{aligned} \bigl\vert y(t) \bigr\vert \leq \int _{a}^{t} \bigl\vert y'(s) \bigr\vert \,\mathrm{d}s\leq (t-a )^{1/{p_{*}}} \biggl( \int _{a}^{t} \bigl\vert y'(s) \bigr\vert ^{p}\,\mathrm{d}s \biggr)^{{1}/{p}}, \end{aligned}$$

where \(p^{*}=\frac{p}{p-1}\). In view of \((b-t)/(b-a)\geq 0\), we obtain

$$\begin{aligned} \biggl(\frac{b-t}{b-a} \biggr)^{1/{p_{*}}} \bigl\vert y(t) \bigr\vert \leq \biggl(\frac{b-t}{b-a} \biggr)^{1/{p_{*}}} (t-a )^{1/{p_{*}}} \biggl( \int _{a}^{t} \bigl\vert y'(s) \bigr\vert ^{p} \,\mathrm{d}s \biggr)^{{1}/{p}}. \end{aligned}$$

Thus

$$\begin{aligned} \biggl(\frac{b-t}{b-a} \biggr)^{p/{p_{*}}} \bigl\vert y(t) \bigr\vert ^{p}\leq \biggl( \frac{b-t}{b-a} \biggr)^{p/{p_{*}}} (t-a )^{p/{p_{*}}} \biggl( \int _{a}^{t} \bigl\vert y'(s) \bigr\vert ^{p} \,\mathrm{d}s \biggr). \end{aligned}$$
(19)

Similarly, from \((t-a)/(b-a)\geq 0\), and

$$\begin{aligned} \bigl\vert y(t) \bigr\vert \leq \int _{t}^{b} \bigl\vert y'(s) \bigr\vert \,\mathrm{d}s\leq (b-t )^{1/{p_{*}}} \biggl( \int _{t}^{b} \bigl\vert y'(s) \bigr\vert ^{p}\,\mathrm{d}s \biggr)^{{1}/{p}}, \end{aligned}$$

we obtain

$$\begin{aligned} \biggl(\frac{t-a}{b-a} \biggr)^{1/{p_{*}}} \bigl\vert y(t) \bigr\vert \leq \biggl(\frac{t-a}{b-a} \biggr)^{1/{p_{*}}} (b-t )^{1/{p_{*}}} \biggl( \int _{t}^{b} \bigl\vert y'(s) \bigr\vert ^{p} \,\mathrm{d}s \biggr)^{{1}/{p}}. \end{aligned}$$

Thus

$$\begin{aligned} \biggl(\frac{t-a}{b-a} \biggr)^{p/{p_{*}}} \bigl\vert y(t) \bigr\vert ^{p}\leq \biggl( \frac{t-a}{b-a} \biggr)^{p/{p_{*}}} (b-t )^{p/{p_{*}}} \biggl( \int _{t}^{b} \bigl\vert y'(s) \bigr\vert ^{p} \,\mathrm{d}s \biggr). \end{aligned}$$
(20)

Adding (19) and (20), we have

$$\begin{aligned} \biggl( \biggl(\frac{b-t}{b-a} \biggr)^{p/{p_{*}}}+ \biggl(\frac{t-a}{b-a} \biggr)^{p/{p_{*}}} \biggr) \bigl\vert y(t) \bigr\vert ^{p}\leq \biggl(\frac{(t-a)(b-t)}{b-a} \biggr)^{p/{p_{*}}} \biggl( \int _{a}^{b} \bigl\vert y'(s) \bigr\vert ^{p}\,\mathrm{d}s \biggr). \end{aligned}$$

According to \(\frac{p}{p^{*}}=p-1\), we get

$$\begin{aligned} \biggl( \biggl(\frac{b-t}{b-a} \biggr)^{p-1}+ \biggl( \frac{t-a}{b-a} \biggr)^{p-1} \biggr) \bigl\vert y(t) \bigr\vert ^{p}\leq \biggl(\frac{(t-a)(b-t)}{b-a} \biggr)^{p-1} \biggl( \int _{a}^{b} \bigl\vert y'(s) \bigr\vert ^{p} \,\mathrm{d}s \biggr). \end{aligned}$$
(21)

On the other hand, from Lemma 2.1 we obtain

$$\begin{aligned} \biggl(\frac{b-t}{b-a} \biggr)^{p-1}+ \biggl( \frac{t-a}{b-a} \biggr)^{p-1}\geq \frac{1}{K(p)} \biggl( \frac{b-t}{b-a}+\frac{t-a}{b-a} \biggr)^{p-1} = \frac{1}{K(p)}. \end{aligned}$$
(22)

Therefore, by (21) and (22), we get

$$\begin{aligned} \bigl\vert y(t) \bigr\vert ^{p}\leq K(p) \biggl(\frac{(t-a)(b-t)}{b-a} \biggr)^{p-1} \biggl( \int _{a}^{b} \bigl\vert y'(s) \bigr\vert ^{p} \,\mathrm{d}s \biggr). \end{aligned}$$
(23)

The proof is complete. □

Lemma 2.3

([45])

Letm, n, p, αandβbe positive constants, then, for each\(x\geq 0\),

$$\begin{aligned} mx^{\alpha }-nx^{\beta }\leq \frac{m(\beta -\alpha )}{\beta -p} \biggl( \frac{(\beta -p)n}{(\alpha -p)m} \biggr)^{(\alpha -p)/(\alpha -\beta )}x^{p} \end{aligned}$$
(24)

holds for the cases when\(0< p<\alpha <\beta \)or\(0<\beta <\alpha <p\).

3 Main results

Theorem 3.1

If\(y(t)\)is a positive solution of problem (9)–(10), then

$$\begin{aligned} \int _{a}^{b}q(t) (t-a)^{p-1}(b-t)^{p-1} \,\mathrm{d}t> \frac{(b-a)^{p-1} }{K(p) } \min_{a\leq b}\bigl\{ r(t)\bigr\} , \end{aligned}$$
(25)

where\(K(p)\)is defined as in (18).

Proof

Multiplying (9) by \(y(t)\) and integrating from a to b by parts yield

$$\begin{aligned} \int _{a}^{b} \frac{r(t) \vert y'(t) \vert ^{p}}{\sqrt{1- \vert y'(t) \vert ^{p}}}\,\mathrm{d}t= \int _{a}^{b}q(t) \bigl\vert y(t) \bigr\vert ^{p}\,\mathrm{d}t. \end{aligned}$$
(26)

By Lemma 2.2, we get

$$\begin{aligned} \int _{a}^{b}q(t) \bigl\vert y(t) \bigr\vert ^{p}\,\mathrm{d}t \leq& \int _{a}^{b}q(t)K(p) \biggl(\frac{(t-a)(b-t)}{b-a} \biggr)^{p-1} \biggl( \int _{a}^{b} \bigl\vert y'(t) \bigr\vert ^{p} \,\mathrm{d}t \biggr)\,\mathrm{d}t \\ =&K(p) \int _{a}^{b} \bigl\vert y'(t) \bigr\vert ^{p}\,\mathrm{d}t \int _{a}^{b}q(t) \biggl( \frac{(t-a)(b-t)}{b-a} \biggr)^{p-1}\,\mathrm{d}t. \end{aligned}$$
(27)

On the other hand,

$$\begin{aligned} \int _{a}^{b} \frac{r(t) \vert y'(t) \vert ^{p}}{\sqrt{1- \vert y'(t) \vert ^{p}}}\,\mathrm{d}t \geq \min_{a\leq b}\bigl\{ r(t)\bigr\} \int _{a}^{b} \frac{ \vert y'(t) \vert ^{p}}{\sqrt{1- \vert y'(t) \vert ^{p}}}\,\mathrm{d}t. \end{aligned}$$
(28)

It follows from (26)–(28) and \(\|y'\|_{\infty }<1\) that

$$\begin{aligned}& \min_{a\leq b}\bigl\{ r(t)\bigr\} \int _{a}^{b} \frac{ \vert y'(t) \vert ^{p}}{\sqrt{1- \vert y'(t) \vert ^{p}}}\,\mathrm{d}t \\& \quad \leq K(p) \int _{a}^{b} \bigl\vert y'(t) \bigr\vert ^{p}\,\mathrm{d}t \int _{a}^{b}q(t) \biggl( \frac{(t-a)(b-t)}{b-a} \biggr)^{p-1}\,\mathrm{d}t \\& \quad < K(p) \biggl( \int _{a}^{b}\frac{ \vert y'(t) \vert ^{p}}{\sqrt{1- \vert y'(t) \vert ^{p}}} \,\mathrm{d}t \biggr) \int _{a}^{b}q(t) \biggl(\frac{(t-a)(b-t)}{b-a} \biggr)^{p-1} \,\mathrm{d}t . \end{aligned}$$
(29)

Now, we claim that

$$ \int _{a}^{b} \frac{ \vert y'(t) \vert ^{p}}{\sqrt{1- \vert y'(t) \vert ^{p}}}\,\mathrm{d}t>0. $$

In fact, if the above inequality is not true, then we have

$$ \int _{a}^{b} \frac{ \vert y'(t) \vert ^{p}}{\sqrt{1- \vert y'(t) \vert ^{p}}}\,\mathrm{d}t=0. $$

Then \(y'(t)=0\) for \(t\in [a, b]\). By condition (10), we obtain \(y(t)=0\) for \(t\in [a, b]\), which contradicts to \(y(t)\not \equiv 0\), \(t\in [a, b]\). Thus dividing both sides of (29) by

$$ \int _{a}^{b} \frac{ \vert y'(t) \vert ^{p}}{\sqrt{1- \vert y'(t) \vert ^{p}}}\,\mathrm{d}t, $$

we obtain

$$\begin{aligned} K(p) \int _{a}^{b}q(t) \biggl(\frac{(t-a)(b-t)}{b-a} \biggr)^{p-1}\,\mathrm{d}t> \min_{a\leq b}\bigl\{ r(t)\bigr\} , \end{aligned}$$
(30)

from which (25) is obtained. The proof is complete. □

Remark 3.1

If we take \(p=2\) and \(r(t)\equiv 1\), then Theorem 3.1 reduces to [44, Theorem 2.1].

Theorem 3.2

If\(y(t)\)is a positive solution of problem (11)–(12), then

$$\begin{aligned} \int _{a}^{b}A(t) \biggl(\frac{(t-a)(b-t)}{b-a} \biggr)^{p-1}\,\mathrm{d}t> \frac{1}{K(p) }\min_{a\leq b}\bigl\{ r(t)\bigr\} , \end{aligned}$$
(31)

where

$$ A(t)=\frac{l(t)(\beta -\alpha )}{\beta -p} \biggl( \frac{(\beta -p)h(t)}{(\alpha -p)l(t)} \biggr)^{(\alpha -p)/(\alpha - \beta )}, $$

and\(K(p)\)is defined as in (18).

Proof

Multiplying (11) by \(y(t)\) and integrating from a to b by parts yield

$$\begin{aligned} \int _{a}^{b} \frac{r(t) \vert y'(t) \vert ^{p}}{\sqrt{1- \vert y'(t) \vert ^{p}}}\,\mathrm{d}t= \int _{a}^{b} \bigl(l(t) \bigl\vert y(t) \bigr\vert ^{\alpha }-h(t) \bigl\vert y(t) \bigr\vert ^{\beta } \bigr) \,\mathrm{d}t. \end{aligned}$$
(32)

By Lemma 2.3, the right side of (32) satisfies

$$\begin{aligned} \int _{a}^{b} \bigl(l(t) \bigl\vert y(t) \bigr\vert ^{\alpha }-h(t) \bigl\vert y(t) \bigr\vert ^{\beta } \bigr) \,\mathrm{d}t\leq \int _{a}^{b}A(t) \bigl\vert y(t) \bigr\vert ^{p}\,\mathrm{d}t. \end{aligned}$$
(33)

From Lemma 2.2, we have

$$\begin{aligned} \int _{a}^{b}A(t) \bigl\vert y(t) \bigr\vert ^{p}\,\mathrm{d}t \leq& \int _{a}^{b}A(t)K(p) \biggl(\frac{(t-a)(b-t)}{b-a} \biggr)^{p-1} \biggl( \int _{a}^{b} \bigl\vert y'(s) \bigr\vert ^{p} \,\mathrm{d}s \biggr)\,\mathrm{d}t \\ =&K(p) \int _{a}^{b} \bigl\vert y'(t) \bigr\vert ^{p}\,\mathrm{d}t \int _{a}^{b}A(t) \biggl( \frac{(t-a)(b-t)}{b-a} \biggr)^{p-1}\,\mathrm{d}t. \end{aligned}$$
(34)

On the other hand,

$$\begin{aligned} \int _{a}^{b} \frac{r(t) \vert y'(t) \vert ^{p}}{\sqrt{1- \vert y'(t) \vert ^{p}}}\,\mathrm{d}t \geq \min_{a\leq b}\bigl\{ r(t)\bigr\} \int _{a}^{b} \frac{ \vert y'(t) \vert ^{p}}{\sqrt{1- \vert y'(t) \vert ^{p}}}\,\mathrm{d}t. \end{aligned}$$
(35)

It follows from (32)–(35) and \(\|y'\|_{\infty }<1\) that

$$\begin{aligned}& \min_{a\leq b}\bigl\{ r(t)\bigr\} \int _{a}^{b} \frac{ \vert y'(t) \vert ^{p}}{\sqrt{1- \vert y'(t) \vert ^{p}}}\,\mathrm{d}t \\& \quad \leq K(p) \int _{a}^{b} \bigl\vert y'(t) \bigr\vert ^{p}\,\mathrm{d}t \int _{a}^{b}A(t) \biggl( \frac{(t-a)(b-t)}{b-a} \biggr)^{p-1}\,\mathrm{d}t \\& \quad < K(p) \biggl( \int _{a}^{b}\frac{ \vert y'(t) \vert ^{p}}{\sqrt{1- \vert y'(t) \vert ^{p}}} \,\mathrm{d}t \biggr) \int _{a}^{b}A(t) \biggl(\frac{(t-a)(b-t)}{b-a} \biggr)^{p-1} \,\mathrm{d}t . \end{aligned}$$
(36)

The rest of the proof is similar to that of Theorem 3.1, and therefore is omitted. The proof is complete. □

Theorem 3.3

If\((y_{1}(t),y_{2}(t), \ldots , y_{n}(t))\)is a positive solution of problem (13)–(14), then

$$\begin{aligned} \prod_{i=1}^{n} \biggl( \int _{a}^{b}q_{i}(t) (t-a)^{p-1}(b-t)^{p-1} \,\mathrm{d}t \biggr)> \frac{(b-a)^{n(p-1)}}{[K(p)]^{n} }\prod_{i=1}^{n} \min_{a\leq b}\bigl\{ r_{i}(t)\bigr\} , \end{aligned}$$
(37)

where\(K(p)\)is defined as in (18).

Proof

From Lemma 2.2, we get

$$\begin{aligned} \bigl\vert y_{i}(t) \bigr\vert ^{p}\leq K(p) \biggl( \frac{(t-a)(b-t)}{b-a} \biggr)^{p-1} \biggl( \int _{a}^{b} \bigl\vert y'_{i}(t) \bigr\vert ^{p}\,\mathrm{d}t \biggr),\quad i=1,2, \ldots ,n, \end{aligned}$$
(38)

i.e.,

$$\begin{aligned} \bigl\vert y_{i}(t) \bigr\vert \leq \bigl[K(p) \bigr]^{{1}/{p}} \biggl(\frac{(t-a)(b-t)}{b-a} \biggr)^{(p-1)/{p}} \biggl( \int _{a}^{b} \bigl\vert y'_{i}(t) \bigr\vert ^{p}\,\mathrm{d}t \biggr)^{{1}/{p}},\quad i=1,2, \ldots ,n. \end{aligned}$$
(39)

Multiplying (13) by \(y_{1}(t)\) and integrating from a to b by parts, we have

$$\begin{aligned} \int _{a}^{b} \frac{r_{1}(t) \vert y_{1}'(t) \vert ^{p}}{\sqrt{1- \vert y_{1}'(t) \vert ^{p}}}\,\mathrm{d}t= \int _{a}^{b}q_{1}(t) \bigl\vert y_{1}(t) \bigr\vert \bigl\vert y_{2}(t) \bigr\vert ^{p-1}\,\mathrm{d}t. \end{aligned}$$
(40)

Together with (39), we obtain

$$\begin{aligned} \min_{a\leq t\leq b}\bigl\{ r_{1}(t)\bigr\} \int _{a}^{b} \bigl\vert y_{1}'(t) \bigr\vert ^{p} \,\mathrm{d}t < & \int _{a}^{b} \frac{r_{1}(t) \vert y_{1}'(t) \vert ^{p}}{\sqrt{1- \vert y_{1}'(t) \vert ^{p}}}\,\mathrm{d}t \\ =& \int _{a}^{b}q_{1}(t) \bigl\vert y_{1}(t) \bigr\vert \bigl\vert y_{2}(t) \bigr\vert ^{p-1}\,\mathrm{d}t \\ \leq& K(p) \biggl( \int _{a}^{b} \bigl\vert y'_{1}(t) \bigr\vert ^{p}\,\mathrm{d}t \biggr)^{{1}/{p}} \biggl( \int _{a}^{b} \bigl\vert y'_{2}(t) \bigr\vert ^{p}\,\mathrm{d}t \biggr)^{(p-1)/{p}} \\ &{}\times \int _{a}^{b}q_{1}(t) \biggl( \frac{(t-a)(b-t)}{b-a} \biggr)^{p-1} \,\mathrm{d}t . \end{aligned}$$
(41)

Repeating this procedure to each equation in problem (13)–(14), for \(i=2,3,\ldots ,n\), we have

$$\begin{aligned}& \min_{a\leq t\leq b}\bigl\{ r_{i}(t)\bigr\} \int _{a}^{b} \bigl\vert y_{i}'(t) \bigr\vert ^{p}\,\mathrm{d}t \\& \quad < K(p) \biggl( \int _{a}^{b} \bigl\vert y'_{i}(t) \bigr\vert ^{p}\,\mathrm{d}t \biggr)^{1/p} \biggl( \int _{a}^{b} \bigl\vert y'_{i+1}(t) \bigr\vert ^{p} \,\mathrm{d}t \biggr)^{(p-1)/{p}} \\& \qquad {}\times \int _{a}^{b}q_{i}(t) \biggl( \frac{(t-a)(b-t)}{b-a} \biggr)^{p-1}\,\mathrm{d}t, \end{aligned}$$
(42)

where \(y_{n+1}(t)=y_{1}(t)\). Multiplying all inequalities, and from the fact \(\int _{a}^{b}|y'_{i}(t)|^{p}\,\mathrm{d}t>0\), \(i=1,2,\ldots ,n\), we obtain (37). The proof is complete. □

Remark 3.2

If we take \(p=2\) and \(r_{i}(t)\equiv 1\), \(i=1,2,\ldots ,n\), then Theorem 3.3 reduces to [44, Theorem 4.1].

Theorem 3.4

If\((y_{1}(t),y_{2}(t), \ldots , y_{n}(t))\)is a positive solution of problem (15)–(16), then

$$\begin{aligned} \prod_{i=1}^{n} \biggl( \int _{a}^{b}A_{i}(t) (t-a)^{p-1}(b-t)^{p-1} \,\mathrm{d}t \biggr)> \frac{(b-a)^{n(p-1)}}{[K(p)]^{n} }\prod_{i=1}^{n} \min_{a\leq b}\bigl\{ r_{i}(t)\bigr\} , \end{aligned}$$
(43)

where

$$ A_{i}(t)=\frac{l_{i}(t)(\beta -\alpha )}{\beta -p} \biggl( \frac{(\beta -p)h_{i}(t)}{(\alpha -p)l_{i}(t)} \biggr)^{(\alpha -p)/( \alpha -\beta )},\quad i=1,2,\ldots ,n, $$

and\(K(p)\)is defined as in (18).

Proof

Multiplying (15) by \(y_{1}(t)\) and integrating from a to b by parts, we have

$$\begin{aligned} \int _{a}^{b} \frac{r_{1}(t) \vert y_{1}'(t) \vert ^{p}}{\sqrt{1- \vert y_{1}'(t) \vert ^{p}}}\,\mathrm{d}t =& \int _{a}^{b} \bigl(l_{1}(t) \bigl\vert y_{2}(t) \bigr\vert ^{\alpha -1}-h_{1}(t) \bigl\vert y_{2}(t) \bigr\vert ^{ \beta -1} \bigr)y_{1}(t) \,\mathrm{d}t \\ \leq& \int _{a}^{b}A_{1}(t) \bigl\vert y_{2}(t) \bigr\vert ^{p-1} \bigl\vert y_{1}(t) \bigr\vert \,\mathrm{d}t. \end{aligned}$$
(44)

The rest of the proof is similar to that of Theorem 3.3, and therefore is omitted. The proof is complete. □

References

  1. Lyapunov, A.M.: Probleme général de la stabilité du mouvement (French translation of a Russian paper dated 1893). Ann. Fac. Sci. Univ. Toulouse Sci. Math. Sci. Phys. 2, 27–247 (1907) Reprinted as Ann. Math. Studies, No. 17, Princeton (1947)

    Google Scholar 

  2. Pinasco, J.: Lower bounds for eigenvalues of the one-dimensional p-Laplacian. Abstr. Appl. Anal. 2, 147–153 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Lee, C.F., Yeh, C.C., Hong, C.H., Agarwal, R.P.: Lyapunov and Wirtinger inequalities. Appl. Math. Lett. 17, 847–853 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Agarwal, R.P., Özbekler, A.: Disconjugacy via Lyapunov and Vallée–Poussin type inequalities for forced differential equations. Appl. Math. Comput. 265, 456–468 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Çakmak, D.: Lyapunov-type integral inequalities for certain higher order differential equations. Appl. Math. Comput. 216, 368–373 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Liu, H.D.: Lyapunov-type inequalities for certain higher-order half-linear differential equations. J. Math. Inequal. 13(4), 1159–1170 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Dhar, S., Kong, Q.K.: Lyapunov-type inequalities for α-th order fractional differential equations with \(2<\alpha \leq 3\) and fractional boundary conditions. Electron. J. Differ. Equ. 2017, 203 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Eliason, S.B.: Lyapunov inequalities and bounds on solutions of certain second order equations. Can. Math. Bull. 17(4), 499–504 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu, X.G., Tang, M.L.: Lyapunov-type inequality for higher order difference equations. Appl. Math. Comput. 232, 666–669 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Liu, H.D.: Half-linear Volterra–Fredholm type integral inequalities on time scales and their applications. J. Appl. Anal. Comput. 10(1), 234–248 (2020)

    Google Scholar 

  11. Zhang, Q.M., Tang, X.H.: Lyapunov-type inequalities for even order difference equations. Appl. Math. Lett. 25, 1830–1834 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Shao, J., Meng, F.W.: Nonlinear impulsive differential and integral inequalities with integral jump conditions. Adv. Differ. Equ. 2016, 112 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Yang, X.J., Lo, K.M.: Lyapunov-type inequalities for a class of higher-order linear differential equations with anti-periodic boundary conditions. Appl. Math. Lett. 34, 33–36 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, H.D., Meng, F.W.: Some new generalized Volterra–Fredholm type discrete fractional sum inequalities and their applications. J. High Energy Phys. 2016, 213 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhao, D.L., Yuan, S.L., Liu, H.D.: Stochastic dynamics of the delayed chemostat with Lévy noises. Int. J. Biomath. 12(5), 1950056 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jleli, M., Samet, B.: On Lyapunov-type inequalities for (p,q)-Laplacian systems. J. Inequal. Appl. 2017, 100 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, H.D.: Some new integral inequalities with mixed nonlinearities for discontinuous functions. Adv. Differ. Equ. 2018, 22 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zou, C., Xia, Y., Pinto, M., Shi, J., Bai, Y.: Boundness and linearisation of a class of differential equations with piecewise constant argument. Qual. Theory Dyn. Syst. 18(2), 495–531 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, H.D.: Lyapunov-type inequalities for second-order boundary value problems with a parameter. Discrete Dyn. Nat. Soc. 2020, Article ID 1209260 (2020)

    MathSciNet  Google Scholar 

  20. Zhang, B., Zhuang, J.S., Liu, H.D., Cao, J.D., Xia, Y.H.: Master-slave synchronization of a class of fractional-order Takagi–Sugeno fuzzy neural networks. Adv. Differ. Equ. 2018, 473 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, H.D., Yin, C.C.: Some generalized Volterra–Fredholm type dynamical integral inequalities in two independent variables on time scale pairs. Adv. Differ. Equ. 2020, Article ID 31 (2020)

    Article  MathSciNet  Google Scholar 

  22. Guseinov, G.S., Kaymakcalan, B.: Lyapunov inequalities for discrete linear Hamiltonian systems. Comput. Math. Appl. 45, 1399–1416 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu, H.D., Meng, F.W., Liu, P.C.: Oscillation and asymptotic analysis on a new generalized Emden–Fowler equation. Appl. Math. Comput. 219(5), 2739–2748 (2012)

    MathSciNet  MATH  Google Scholar 

  24. Zhao, D.L.: Study on the threshold of a stochastic SIR epidemic model and its extensions. Commun. Nonlinear Sci. Numer. Simul. 38, 172–177 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu, H.D.: Some new half-linear integral inequalities on time scales and applications. Discrete Dyn. Nat. Soc. 2019, Article ID 9860302 (2019)

    MathSciNet  Google Scholar 

  27. Unal, M., Çakmak, D., Tiryaki, A.: A discrete analogue of Lyapunov-type inequalities for nonlinear systems. Comput. Math. Appl. 55, 2631–2642 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, H.D., Meng, F.W.: Existence of positive periodic solutions for a predator-prey system of Holling type IV function response with mutual interference and impulsive effects. Discrete Dyn. Nat. Soc. 2015, 138984 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Zhao, D.L., Liu, H.D.: Coexistence in a two species chemostat model with Markov switchings. Appl. Math. Lett. 94, 266–271 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, H.D.: An improvement of the Lyapunov inequality for certain higher order differential equations. J. Inequal. Appl. 2018, 215 (2018)

    Article  MathSciNet  Google Scholar 

  31. Cheng, S.S.: Lyapunov inequalities for differential and difference equations. Fasc. Math. 23, 25–41 (1991)

    MathSciNet  MATH  Google Scholar 

  32. Zhao, D.L., Yuan, S.L., Liu, H.D.: Random periodic solution for a stochastic SIS epidemic model with constant population size. Adv. Differ. Equ. 2018, 64 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. Liu, H.D., Li, C.Y., Shen, F.C.: A class of new nonlinear dynamic integral inequalities containing integration on infinite interval on time scales. Adv. Differ. Equ. 2019, 311 (2019)

    Article  MathSciNet  Google Scholar 

  34. Tunç, E., Liu, H.D.: Oscillatory behavior for second-order damped differential equation with nonlinearities including Riemann–Stieltjes integrals. Electron. J. Differ. Equ. 2018, 54 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  35. Liu, H.D.: A class of retarded Volterra–Fredholm type integral inequalities on time scales and their applications. J. Inequal. Appl. 2017, 293 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Feng, Q.H., Meng, F.W., Zheng, B.: Gronwall–Bellman type nonlinear delay integral inequalities on time scale. J. Math. Anal. Appl. 382, 772–784 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Liu, H.D., Meng, F.W.: Interval oscillation criteria for second-order nonlinear forced differential equations involving variable exponent. Adv. Differ. Equ. 2016, 291 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Xia, Y.H., Chen, L., Kou, K.I.: Holder regularity of Grobman–Hartman theorem for dynamic equations on measure chains. Bull. Malays. Math. Sci. Soc. 41(3), 1153–1180 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  39. Liu, H.D., Meng, F.W.: Nonlinear retarded integral inequalities on time scales and their applications. J. Math. Inequal. 12(1), 219–234 (2018)

    MathSciNet  MATH  Google Scholar 

  40. Tiryaki, A., Çakmak, D., Aktas, M.F.: Lyapunov-type inequalities for a certain class of nonlinear systems. Comput. Math. Appl. 64, 1804–1811 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  41. Liu, H.D.: Lyapunov-type inequalities for certain higher-order difference equations with mixed non-linearities. Adv. Differ. Equ. 2018, 229 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang, Q.M., Tang, X.H.: Lyapunov inequalities and stability for discrete linear Hamiltonian system. Appl. Math. Comput. 218, 574–582 (2011)

    MathSciNet  MATH  Google Scholar 

  43. Yang, X.J.: On inequalities of Lyapunov type. Appl. Math. Comput. 134(2–3), 293–300 (2003)

    MathSciNet  MATH  Google Scholar 

  44. Yang, R., Sim, I., Lee, Y.-H.: Lyapunov-type inequalities for one-dimensional Minkowski-curvature problems. Appl. Math. Lett. 91, 188–193 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  45. Liu, H.D.: Lyapunov-type inequalities for higher-order half-linear difference equations. J. Inequal. Appl. 2020, Article ID 80 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is indebted to the anonymous referees for their valuable suggestions and helpful comments which helped improve the paper significantly.

Availability of data and materials

Not applicable.

Funding

This research was supported by the Natural Science Foundation of Shandong Province (China) (No.: ZR2018MA018), and the National Natural Science Foundation of China (No.: 61873144).

Author information

Authors and Affiliations

Authors

Contributions

The author carried out the results, and read and approved the current version of the manuscript.

Corresponding author

Correspondence to Haidong Liu.

Ethics declarations

Competing interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, H. Lyapunov-type inequalities for generalized one-dimensional Minkowski-curvature problems. J Inequal Appl 2020, 169 (2020). https://doi.org/10.1186/s13660-020-02431-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13660-020-02431-8

Keywords