We establish several new Lyapunov-type inequalities
for some quasilinear dynamic system involving the (p1,p2,…,pm)-Laplacian
on an arbitrary time scale 𝕋, which generalize and improve some related existing results including the continuous and discrete cases.

1. Introduction

In recent years, the theory of time scales (or measure chains) has been developed by several authors with one goal being the unified treatment of differential equations (the continuous case) and difference equations (the discrete case). A time scale is an arbitrary nonempty closed subset of the real numbers ℝ. We assume that 𝕋 is a time scale and 𝕋 has the topology that it inherits from the standard topology on the real numbers 𝕋. The two most popular examples are 𝕋=ℝ and 𝕋=ℤ. In Section 2, we will briefly introduce the time scale calculus and some related basic concepts of Hilger [1–3]. For further details, we refer the reader to the books independently by Kaymakcalan et al. [4] and by Bohner and Peterson [5, 6].

Consider the following quasilinear dynamic system involving the (p1,p2,…,pm)-Laplaci-an on an arbitrary time scale 𝕋:-(r1(t)|u1Δ(t)|p1-2u1Δ(t))Δ=f1(t)|u1(σ(t))|α1-2|u2(σ(t))|α2⋯|um(σ(t))|αmu1(σ(t)),-(r2(t)|u2Δ(t)|p2-2u2Δ(t))Δ=f2(t)|u1(σ(t))|α1|u2(σ(t))|α2-2⋯|um(σ(t))|αmu2(σ(t)),⋮-(rm(t)|umΔ(t)|pm-2umΔ(t))Δ=fm(t)|u1(σ(t))|α1|u2(σ(t))|α2⋯|um(σ(t))|αm-2um(σ(t)).

It is obvious that system (1.1) covers the continuous quasilinear system and the corresponding discrete case, respectively, when 𝕋=ℝ and 𝕋=ℤ; that is,-(r1(t)|u1′(t)|p1-2u1′(t))′=f1(t)|u1(t)|α1-2|u2(t)|α2⋯|um(t)|αmu1(t),-(r2(t)|u2′(t)|p2-2u2′(t))′=f2(t)|u1(t)|α1|u2(t)|α2-2⋯|um(t)|αmu2(t),⋮-(rm(t)|um′(t)|pm-2um′(t))′=fm(t)|u1(t)|α1|u2(t)|α2⋯|un(t)|αm-2um(t),-Δ(r1(n)|Δu1(n)|p1-2Δu1(n))=f1(n)|u1(n+1)|α1-2|u2(n+1)|α2⋯|um(n+1)|αmu1(n+1),-Δ(r2(n)|Δu2(n)|p2-2Δu2(n))=f2(n)|u1(n+1)|α1|u2(n+1)|α2-2⋯|um(n+1)|αmu2(n+1),⋮-Δ(rm(n)|Δum(n)|pm-2Δum(n))=fm(n)|u1(n+1)|α1|u2(n+1)|α2⋯|um(n+1)|αm-2um(n+1).

In 1907, Lyapunov [7] established the first so-called Lyapunov inequality(b-a)∫ab|q(t)|dt>4,
if the Hill equationx′′(t)+q(t)x(t)=0
has a real solution x(t) such thatx(a)=x(b)=0,x(t)≢0,t∈[a,b].
Moreover the constant 4 in (1.3) cannot be replaced by a larger number, where q(t) is a piece-wise continuous and nonnegative function defined on ℝ.

It is a classical topic for us to study Lyapunov-type inequalities which have proved to be very useful in oscillation theory, disconjugacy, eigenvalue problems, and numerous other applications in the theory of differential and difference equations. So far, there are many literatures which improved and extended the classical Lyapunov including continuous and discrete cases. For example, inequality (1.3) has been generalized to discrete linear Hamiltonian system by Zhang and Tang [8], to second-order nonlinear differential equations by Eliason [9] and by Pachpatte [10], to second-order nonlinear difference system by He and Zhang [11], to the second-order delay differential equations by Eliason [12] and by Dahiya and Singh [13], to higher-order differential equations by Pachpatte [14], Yang [15, 16], Yang and Lo [17] and Cakmak and Tiryaki [18, 19]. Lyapunov-type inequalities for the Emden-Fowler-type equations can be found in Pachpatte [10], and for the half-linear equations can be found in Lee et al. [20] and Pinasco [21]. Recently, there has been much attention paid to Lyapunov-type inequalities for dynamic systems on time scales and some authors including Agarwal et al. [22], Jiang and Zhou [23], He [24], He et al. [25], Saker [26], Bohner et al. [27], and Ünal and Cakmak [28] have contributed the above results.

In this paper, we use the methods in [29] to establish some Lyapunov-type inequalities for system (1.1) on an arbitrary time scale 𝕋.

2. Preliminaries about the Time Scales Calculus

We introduce some basic notions connected with time scales.

Definition 2.1 (see [<xref ref-type="bibr" rid="B3">6</xref>]).

Let t∈𝕋. We define the forward jump operator σ:𝕋→𝕋 by
σ(t):=inf{s∈T:s>t},∀t∈T,
while the backward jump operator ρ:𝕋→𝕋 by
ρ(t):=sup{s∈T:s<t},∀t∈T.
In this definition, we put inf∅=sup𝕋 (i.e., σ(M)=M if 𝕋 has a maximum M) and sup∅=inf𝕋 (i.e., ρ(m)=m if 𝕋 has a minimum m), where ∅ denotes the empty set. If σ(t)>t, we say that t is right-scattered, while if ρ(t)<t, we say that t is left-scattered. Also, if t<sup𝕋 and σ(t)=t, then t is called right-dense, and if t>inf𝕋 and ρ(t)=t, then t is called left-dense. Points that are right-scattered and left-scattered at the same time are called isolated. Points that are right-dense and left-dense at the same time are called dense. If 𝕋 has a left-scattered maximum M, then we define 𝕋k=𝕋-{M} otherwise; 𝕋k=𝕋. The graininess function u:𝕋→[0,∞) is defined by
μ(t):=σ(t)-t,∀t∈T.
We consider a function f:𝕋→ℝ and define so-called delta (or Hilger) derivative of f at a point t∈𝕋k.

Definition 2.2 (see [<xref ref-type="bibr" rid="B3">6</xref>]).

Assume that f:𝕋→ℝ is a function, and let t∈𝕋k. Then, we define fΔ(t) to be the number (provided it exists) with the property that given any ɛ>0, there is a neighborhood U of t (i.e., U=(t-δ,t+δ)∩𝕋 for some δ>0) such that
|f(σ(t))-f(s)-fΔ(t)(σ(t)-s)|≤ɛ|σ(t)-s|,∀s∈U.
We call fΔ(t) the delta (or Hilger) derivative of f at t.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B3">6</xref>]).

Assume that f,g:𝕋→ℝ are differential at t∈𝕋k, then,

for any constant a and b, the sum af+bg:𝕋→ℝ is differential at t with

if g(t)g(σ(t))≠0, then f/g is differential at t and

(fg)Δ(t)=fΔ(t)g(t)-f(t)gΔ(t)g(t)g(σ(t)).Definition 2.4 (see [<xref ref-type="bibr" rid="B3">6</xref>]).

A function f:𝕋→ℝ is called rd-continuous, provided it is continuous at right-dense points in 𝕋 and left-sided limits exist (finite) at left-dense points in 𝕋 and denotes by Crd=Crd(𝕋)=Crd(𝕋,ℝ).

Definition 2.5 (see [<xref ref-type="bibr" rid="B3">6</xref>]).

A function F:𝕋→ℝ is called an antiderivative of f:𝕋→ℝ, provided FΔ(t)=f(t) holds for all t∈𝕋k. We define the Cauchy integral by
∫τsf(t)Δt=F(s)-F(τ),∀s,τ∈T.

The following lemma gives several elementary properties of the delta integral.

Lemma 2.6 (see [<xref ref-type="bibr" rid="B3">6</xref>]).

If a,b,c∈𝕋,k∈ℝ and f,g∈Crd, then

∫ab[f(t)+g(t)]Δt=∫abf(t)Δt+∫abg(t)Δt,

∫ab(kf)(t)Δt=k∫abf(t)Δt,

∫abf(t)Δt=∫acf(t)Δt+∫cbf(t)Δt,

∫abf(σ(t))gΔ(t)Δt=(fg)(b)-(fg)(a)-∫abfΔ(t)g(t)Δt,

∫tσ(t)f(s)Δs=μ(t)f(t)fort∈𝕋k,

if |f(t)|≤g(t)on[a,b), then

|∫abf(t)Δt|≤∫abg(t)Δt.

The notation [a,b],[a,b) and [a,+∞) will denote time scales intervals. For example, [a,b)={t∈𝕋∣a≤t<b}. To prove our results, we present the following lemma.

Lemma 2.7 (see [<xref ref-type="bibr" rid="B3">6</xref>]).

Let a,b∈𝕋 and 1<p,q<∞ with 1/p+1/q=1. For f,g∈Crd, one has
∫ab|f(t)g(t)|Δt≤{∫ab|f(t)|pΔt}1/p{∫ab|g(t)|qΔt}1/q.

Lemma 2.8 (see [<xref ref-type="bibr" rid="B3">6</xref>]).

Let a,b∈𝕋 and 1<rk<∞ with ∑k=1m(1/rk)=1 for k=1,2,…,m. For fk∈Crd,k=1,2,…,m, one has
∫ab∏k=1m|fk(t)|Δt≤∏k=1m{∫ab|fk(t)|rkΔt}1/rk.

ri(t) and fi(t) are rd-continuous real functions and ri(t)>0 for i=1,2,…,m and t∈𝕋. Furthermore, 1<pi<∞ and αi>0 satisfy ∑i=1m(αi/pi)=1 for i=1,2,…,m.

Theorem 3.1.

Let a,b∈𝕋k with σ(a)≤b. Suppose that hypothesis (H1) is satisfied. If (1.1) has a real solution (u1(t),u2(t),…,um(t)) satisfying the boundary value conditions
ui(a)=ui(b)=0,ui(t)≢0,∀t∈[a,b],i=1,2,…,m,
then one has
∏i=1m∏j=1m(∫abζi(t)ηi(t)ζi(t)+ηi(t)fj+(t)Δt)αiαj/pipj≥1,
where and in what follows fi+(t)=max{fi(t),0} for i=1,2,…,m.

Proof.

By (1.1) and Lemma 2.3(iv), we obtain
-(ri(t)|uiΔ(t)|pi-2uiΔ(t)ui(t))Δ+ri(t)|uiΔ(t)|pi=fi(t)∏k=1m|uk(σ(t))|αk,
where i=1,2,…,m. From Definition 2.5, integrating (3.5) from a to b, together with (3.3), we get
∫abri(t)|uiΔ(t)|piΔt=∫abfi(t)∏k=1m|uk(σ(t))|αkΔt,i=1,2,…,m.
It follows from (3.1), (3.3), and Lemma 2.7 that
|ui(σ(t))|pi=|∫aσ(t)uiΔ(τ)Δτ|pi≤(∫aσ(t)[ri(τ)]1/(1-pi)Δτ)pi-1∫aσ(t)ri(τ)|uiΔ(τ)|piΔτ=ζi(t)∫aσ(t)ri(τ)|uiΔ(τ)|piΔτ,a≤t≤b,i=1,2,…,m.
Similarly, it follows from (3.2), (3.3), and Lemma 2.7 that
|ui(σ(t))|pi=|∫σ(t)buiΔ(τ)Δτ|pi≤(∫σ(t)b[ri(τ)]1/(1-pi)Δτ)pi-1∫σ(t)bri(τ)|uiΔ(τ)|piΔτ=ηi(t)∫σ(t)bri(τ)|uiΔ(τ)|piΔτ,a≤t≤b,i=1,2,…,m.
From (3.7) and (3.8), we have
|ui(σ(t))|pi≤ζi(t)ηi(t)ζi(t)+ηi(t)∫abri(τ)|uiΔ(τ)|piΔτ,a≤t≤b,i=1,2,…,m.
So, from (3.3), (3.6), (3.9), (H1), and Lemma 2.8, we have
∫abfi+(t)|ui(σ(t))|piΔt≤∫abζi(t)ηi(t)ζi(t)+ηi(t)fi+(t)Δt∫abri(t)|uiΔ(t)|piΔt=Mij∫abfi(t)∏k=1m|uk(σ(t))|αkΔt≤Mij∫abfi+(t)∏k=1m|uk(σ(t))|αkΔt≤Mij∏k=1m(∫abfi+(t)|uk(σ(t))|pkΔt)αk/pk,
where
Mij=∫abζi(t)ηi(t)ζi(t)+ηi(t)fj+(t)Δt,i,j=1,2,…,m.

Next, we prove that∫abfi+(t)|uk(σ(t))|pkΔt>0.
If (3.12) is not true, there exist i0,k0∈{1,2,…,m} such that
∫abfi0+(t)|uk0(σ(t))|pk0Δt=0.
From (3.6), (3.13), and Lemma 2.8, we have
0≤∫abri0(t)|ui0Δ(t)|pi0Δt=∫abfi0(t)∏k=1m|uk(σ(t))|αkΔt≤∏k=1m(∫abfi0+(t)|uk(σ(t))|pkΔt)αk/pk=0.
It follows from the fact that ri0(t)>0 that
ui0Δ(t)≡0,a≤t≤b.
Combining (3.7) with (3.15), we obtain that ui0(t)≡0 for a≤t≤b, which contradicts (3.3). Therefore, (3.12) holds. From (3.10), (3.12), and (H1), we have
∏i=1m∏j=1mMijαiαj/pipj≥1.
It follows from (3.11) and (3.16) that (3.4) holds.

Corollary 3.2.

Let a,b∈𝕋k with σ(a)≤b. Suppose that hypothesis (H1) is satisfied. If (1.1) has a real solution (u1(t),u2(t),…,um(t)) satisfying the boundary value conditions (3.3), then one has
∏i=1m∏j=1m(∫abfj+(t)[ζi(t)ηi(t)]1/2Δt)αiαj/pipj≥2.

Proof.

Since
ζi(t)+ηi(t)≥2[ζi(t)ηi(t)]1/2,i=1,2,…,m,
it follows from (3.4) and (H1) that (3.17) holds.

Corollary 3.3.

Let a,b∈𝕋k with σ(a)≤b. Suppose that hypothesis (H1) is satisfied. If (1.1) has a real solution (u1(t),u2(t),…,um(t)) satisfying the boundary value conditions (3.3), then one has
∏i=1m(∫ab[ri(t)]1/(1-pi)Δt)αi(pi-1)/pi∏j=1m(∫abfi+(t)Δt)αj/pj≥2A,
where 𝒜=∑i=1mαi.

Proof.

Since
[ζi(t)ηi(t)]1/2=(∫aσ(t)[ri(τ)]1/(1-pi)Δτ∫σ(t)b[ri(τ)]1/(1-pi)Δτ)(pi-1)/2≤12pi-1(∫ab[ri(τ)]1/(1-pi)Δτ)pi-1,i=1,2,…,m,
it follows from (3.20) and (H1) that (3.19) holds.

When m=1,p1=α1=γ>1,r1(t)=r(t)>0,u1(σ(t))=u(σ(t)), u1(t)=u(t), and f1(t)=ϱ(t), system (1.1) reduces to a second-order half-linear dynamic equation, and denote by(r(t)|uΔ(t)|γ-2uΔ(t))Δ+ϱ(t)|u(σ(t))|γ-2u(σ(t))=0.

We can easily derive the following corollary for (3.21).

Corollary 3.4.

Let a,b∈𝕋k with σ(a)≤b. If (3.21) has a solution u(t) satisfying
u(a)=u(b)=0,u(t)≢0,∀t∈[a,b],
then
∫ab(∫aσ(t)[r(τ)]1/(1-γ)Δτ)γ-1(∫σ(t)b[r(τ)]1/(1-γ)Δτ)γ-1(∫aσ(t)[r(τ)]1/(1-γ)Δτ)γ-1+(∫σ(t)b[r(τ)]1/(1-γ)Δτ)γ-1ϱ+(t)Δt≥1.

Especially, while m=1,p1=α1=2,r1(t)=1,u1(σ(t))=u(σ(t)), u1(t)=u(t), and f1(t)=ϱ(t), system (1.1) reduces to a second-order linear dynamic equation and denote by(uΔ(t))Δ+ϱ(t)u(σ(t))=0.

Obviously, (3.24) is a special case of (3.21). One can also obtain a corollary immediately.

Corollary 3.5.

Let a,b∈𝕋k with σ(a)≤b. If (3.24) has a solution u(t) satisfying
u(a)=u(b)=0,u(t)≢0,∀t∈[a,b],
then
(b-a)∫abϱ+(t)Δt≥4.

Acknowledgments

This work is partially supported by the NNSF (no. 11171351) of China and by Scientific Research Fund of Hunan Provincial Education Department (no. 10C0655 and no. 11A095).

HilgerS.HilgerS.Analysis on measure chains-a unified approach to continuous and discrete calculusHilgerS.Differential and difference calculus-unified!KaymakcalanB.LakshmikanthamV.SivasundaramS.BohnerM.PetersonA.AhlbrandtC. D.PetersonA. C.LyapunovA. M.Problème général de la stabilité du mouvementZhangQ.TangX. H.Lyapunov inequalities and stability for discrete linear Hamiltonian systemEliasonS. B.A Lyapunov inequality for a certain second order nonlinear differential equationPachpatteB. G.Inequalities related to the zeros of solutions of certain second order differential equationsHeX.ZhangQ.A discrete analogue of Lyapunov-type inequalities for nonlinear difference systemsEliasonS. B.Lyapunov type inequalities for certain second order functional differential equationsDahiyaR. S.SinghB.A Lyapunov inequality and nonoscillation theorem for a second order non-linear differential-difference equation197371631700350151PachpatteB. G.On Lyapunov-type inequalities for certain higher order differential equationsYangX.On Liapunov-type inequality for certain higher-order differential equationsYangX.On inequalities of Lyapunov typeYangX.LoK.Lyapunov-type inequality for a class of even-order differential equationsCakmakD.TiryakiA.On Lyapunov-type inequality for quasilinear systemsCakmakD.TiryakiA.Lyapunov-type inequality for a class of Dirichlet quasilinear systems involving the p1,p2,…,pn-LaplacianLeeC.YehC.HongC.AgarwalR. P.Lyapunov and Wirtinger inequalitiesPinascoJ. P.Lower bounds for eigenvalues of the one-dimensional p-LaplacianAgarwalR. P.BohnerM.RehakP.Half-linear dynamic equationsJiangL. Q.ZhouZ.Lyapunov inequality for linear Hamiltonian systems on time scalesHeZ.Existence of two solutions of m-point boundary value problem for second order dynamic equations on time scalesHeX.ZhangQ.TangX. H.On inequalities of Lyapunov for linear Hamiltonian systems on time scalesSakerS. H.Oscillation of nonlinear dynamic equations on time scalesBohnerM.ClarkS.RidenhourJ.Lyapunov inequalities for time scalesÜnalM.CakmakD.Lyapunov-type inequalities for certain nonlinear systems on time scalesTangX. H.HeX.Lower bounds for generalized eigenvalues of the quasilinear systems