JAMJournal of Applied Mathematics1687-00421110-757XHindawi Publishing Corporation41813610.1155/2011/418136418136Research ArticleLyapunov-Type Inequalities for Some Quasilinear Dynamic System Involving the (p1,p2,,pm)-Laplacian on Time ScalesHeXiaofei1ZhangQi-Ming2LiuYansheng1College of Mathematics and Computer ScienceJishou UniversityJishou 416000HunanChinajsu.edu.cn2College of ScienceHunan University of TechnologyZhuzhou 412007HunanChinazhuzit.edu.cn20111112011201130062011250820112011Copyright © 2011 Xiaofei He and Qi-Ming Zhang.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 . 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= and 𝕋=. In Section 2, we will briefly introduce the time scale calculus and some related basic concepts of Hilger . For further details, we refer the reader to the books independently by Kaymakcalan et al.  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= and 𝕋=; 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  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 .

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 , to second-order nonlinear differential equations by Eliason  and by Pachpatte , to second-order nonlinear difference system by He and Zhang , to the second-order delay differential equations by Eliason  and by Dahiya and Singh , to higher-order differential equations by Pachpatte , Yang [15, 16], Yang and Lo  and Cakmak and Tiryaki [18, 19]. Lyapunov-type inequalities for the Emden-Fowler-type equations can be found in Pachpatte , and for the half-linear equations can be found in Lee et al.  and Pinasco . Recently, there has been much attention paid to Lyapunov-type inequalities for dynamic systems on time scales and some authors including Agarwal et al. , Jiang and Zhou , He , He et al. , Saker , Bohner et al. , and Ünal and Cakmak  have contributed the above results.

In this paper, we use the methods in  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 t𝕋. We define the forward jump operator σ:𝕋𝕋 by σ(t):=inf{sT:s>t},tT, while the backward jump operator ρ:𝕋𝕋 by ρ(t):=sup{sT:s<t},tT. 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,tT. We consider a function f:𝕋 and define so-called delta (or Hilger) derivative of f at a point t𝕋k.

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

Assume that f:𝕋 is a function, and let t𝕋k. 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|,sU. 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 are differential at t𝕋k, then,

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

(af+bg)Δ(t)=afΔ(t)+bgΔ(t),

if fΔ(t) exists, then f is continuous at t,

if fΔ(t) exists, then f(σ(t))=f(t)+μ(t)fΔ(t),

the product fg:𝕋 is differential at t with

(fg)Δ(t)=fΔ(t)g(t)+f(σ(t))gΔ(t)=f(t)gΔ(t)+fΔ(t)g(σ(t)),

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 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,).

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

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

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,c𝕋,k and f,gCrd, then

ab[f(t)+g(t)]Δt=abf(t)Δt+abg(t)Δt,

ab(kf)(t)Δt=kabf(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)  for  t𝕋k,

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)={t𝕋at<b}. To prove our results, we present the following lemma.

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

Let a,b𝕋 and 1<p,q< with 1/p+1/q=1. For f,gCrd, 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,b𝕋 and 1<rk< with k=1m(1/rk)  =1 for k=1,2,,m. For fkCrd,k=1,2,,m, one has abk=1m|fk(t)|Δtk=1m{ab|fk(t)|rkΔt}1/rk  .

3. Lyapunov-Type Inequalities

Denoteζi(t):=(aσ(t)[ri(τ)]1/(1-pi)Δτ)pi-1,i=1,2,,m,ηi(t):=(σ(t)b[ri(τ)]1/(1-pi)Δτ)pi-1,i=1,2,,m.

First, we give the following hypothesis.

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

Theorem 3.1.

Let a,b𝕋k 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=1mj=1m(abζi(t)ηi(t)ζi(t)+ηi(t)fj+(t)Δt)αiαj/pipj1, 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-1aσ(t)ri(τ)|uiΔ(τ)|piΔτ=ζi(t)aσ(t)ri(τ)|uiΔ(τ)|piΔτ,atb,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-1σ(t)bri(τ)|uiΔ(τ)|piΔτ=ηi(t)σ(t)bri(τ)|uiΔ(τ)|piΔτ,atb,i=1,2,,m. From (3.7) and (3.8), we have |ui(σ(t))|piζi(t)ηi(t)ζi(t)+ηi(t)abri(τ)|uiΔ(τ)|piΔτ,atb,i=1,2,,m. So, from (3.3), (3.6), (3.9), (H1), and Lemma 2.8, we have abfi+(t)|ui(σ(t))|piΔtabζi(t)ηi(t)ζi(t)+ηi(t)fi+(t)Δtabri(t)|uiΔ(t)|piΔt=Mijabfi(t)k=1m|uk(σ(t))|αkΔt    Mijabfi+(t)k=1m|uk(σ(t))|αkΔtMijk=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 thatabfi+(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 0abri0(t)|ui0Δ(t)|pi0Δt=abfi0(t)k=1m|uk(σ(t))|αkΔtk=1m(abfi0+(t)|uk(σ(t))|pkΔt)αk/pk=0. It follows from the fact that ri0(t)>0 that ui0Δ(t)0,atb. Combining (3.7) with (3.15), we obtain that ui0(t)0 for atb, which contradicts (3.3). Therefore, (3.12) holds. From (3.10), (3.12), and (H1), we have i=1mj=1mMijαiαj/pipj1. It follows from (3.11) and (3.16) that (3.4) holds.

Corollary 3.2.

Let a,b𝕋k 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=1mj=1m(abfj+(t)[ζi(t)ηi(t)]1/2Δt)αiαj/pipj2.

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,b𝕋k 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)/pij=1m(abfi+(t)Δt)αj/pj2A, where 𝒜=i=1mαi.

Proof.

Since [ζi(t)ηi(t)]1/2=(aσ(t)[ri(τ)]1/(1-pi)Δτσ(t)b[ri(τ)]1/(1-pi)Δτ)(pi-1)/212pi-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,b𝕋k 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-1  ϱ+(t)Δt1.

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,b𝕋k 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)Δt4.

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).

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