Lyapunov-Type Inequalities for Some Quasilinear Dynamic System Involving the (𝑝₁,𝑝₂,…,𝑝_𝑚)-Laplacian on Time Scales

Abstract

We establish several new Lyapunov-type inequalities for some quasilinear dynamic system involving the (𝑝₁,𝑝₂,…,𝑝_𝑚)-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,…,𝑝_𝑚)-Laplaci-an on an arbitrary time scale 𝕋:−𝑟₁(𝑡)𝑢^Δ₁(𝑡)^𝑝1−2𝑢^Δ₁(𝑡)^Δ=𝑓₁(𝑡)𝑢₁(𝜎(𝑡))^𝛼1−2𝑢₂(𝜎(𝑡))^𝛼2⋯𝑢_𝑚(𝜎(𝑡))^𝛼𝑚𝑢₁(𝜎(𝑡,−𝑟))₂(𝑡)𝑢^Δ₂(𝑡)^𝑝2−2𝑢^Δ₂(𝑡)^Δ=𝑓₂(𝑡)𝑢₁(𝜎(𝑡))^𝛼1𝑢₂(𝜎(𝑡))^𝛼2−2⋯𝑢_𝑚(𝜎(𝑡))^𝛼𝑚𝑢₂(𝜎(𝑡,⋮−𝑟))_𝑚(𝑡)𝑢^Δ_𝑚(𝑡)^𝑝𝑚−2𝑢^Δ_𝑚(𝑡)^Δ=𝑓_𝑚(𝑡)𝑢₁(𝜎(𝑡))^𝛼1𝑢₂(𝜎(𝑡))^𝛼2⋯𝑢_𝑚(𝜎(𝑡))^𝛼𝑚−2𝑢_𝑚(𝜎(𝑡)). (1.1)

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)𝑢^′₁(𝑡)^𝑝1−2𝑢^′₁(𝑡)^′=𝑓₁(𝑡)𝑢₁(𝑡)^𝛼1−2𝑢₂(𝑡)^𝛼2⋯𝑢_𝑚(𝑡)^𝛼𝑚𝑢₁(𝑡),−𝑟₂(𝑡)𝑢^′₂(𝑡)^𝑝2−2𝑢^′₂(𝑡)^′=𝑓₂(𝑡)𝑢₁(𝑡)^𝛼1𝑢₂(𝑡)^𝛼2−2⋯𝑢_𝑚(𝑡)^𝛼𝑚𝑢₂(𝑡),⋮−𝑟_𝑚(𝑡)𝑢^′_𝑚(𝑡)^𝑝𝑚−2𝑢^′_𝑚(𝑡)^′=𝑓_𝑚(𝑡)𝑢₁(𝑡)^𝛼1𝑢₂(𝑡)^𝛼2⋯𝑢_𝑛(𝑡)^𝛼𝑚−2𝑢_𝑚(𝑡),−Δ𝑟₁(𝑛)Δ𝑢₁(𝑛)^𝑝1−2Δ𝑢₁(𝑛)=𝑓₁(𝑛)𝑢₁(𝑛+1)^𝛼1−2𝑢₂(𝑛+1)^𝛼2⋯𝑢_𝑚(𝑛+1)^𝛼𝑚𝑢₁(),𝑟𝑛+1−Δ₂(𝑛)Δ𝑢₂(𝑛)^𝑝2−2Δ𝑢₂(𝑛)=𝑓₂(𝑛)𝑢₁(𝑛+1)^𝛼1𝑢₂(𝑛+1)^𝛼2−2⋯𝑢_𝑚(𝑛+1)^𝛼𝑚𝑢₂(),⋮𝑟𝑛+1−Δ_𝑚(𝑛)Δ𝑢_𝑚(𝑛)^𝑝𝑚−2Δ𝑢_𝑚(𝑛)=𝑓_𝑚(𝑛)𝑢₁(𝑛+1)^𝛼1𝑢₂(𝑛+1)^𝛼2⋯𝑢_𝑚(𝑛+1)^𝛼𝑚−2𝑢_𝑚(𝑛+1). (1.2)

In 1907, Lyapunov [7] established the first so-called Lyapunov inequality(𝑏−𝑎)^𝑏_𝑎𝑞(𝑡)𝑑𝑡>4, (1.3) if the Hill equation𝑥^′′(𝑡)+𝑞(𝑡)𝑥(𝑡)=0 (1.4) has a real solution 𝑥(𝑡) such that𝑥(𝑎)=𝑥(𝑏)=0,𝑥(𝑡)≢0,𝑡∈[𝑎,𝑏]. (1.5) 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 [6]). Let 𝑡∈𝕋. We define the forward jump operator 𝜎∶𝕋→𝕋 by 𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡},∀𝑡∈𝕋, (2.1) while the backward jump operator 𝜌∶𝕋→𝕋 by 𝜌(𝑡)∶=sup{𝑠∈𝕋∶𝑠<𝑡},∀𝑡∈𝕋. (2.2) 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 𝕋^𝑘=𝕋−{𝑀} otherwise; 𝕋^𝑘=𝕋. The graininess function 𝑢∶𝕋→[0,∞) is defined by 𝜇(𝑡)∶=𝜎(𝑡)−𝑡,∀𝑡∈𝕋. (2.3) We consider a function 𝑓∶𝕋→ℝ and define so-called delta (or Hilger) derivative of 𝑓 at a point 𝑡∈𝕋^𝑘.

Definition 2.2 (see [6]). Assume that 𝑓∶𝕋→ℝ is a function, and let 𝑡∈𝕋^𝑘. Then, we define 𝑓^Δ(𝑡) to be the number (provided it exists) with the property that given any 𝜀>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)−𝑓^Δ(𝑡)(𝜎(𝑡)−𝑠)≤𝜀|𝜎(𝑡)−𝑠|,∀𝑠∈𝑈. (2.4) We call 𝑓^Δ(𝑡) the delta (or Hilger) derivative of 𝑓 at 𝑡.

Lemma 2.3 (see [6]). Assume that 𝑓,𝑔∶𝕋→ℝ are differential at 𝑡∈𝕋^𝑘, then, (i)for any constant 𝑎 and 𝑏, the sum 𝑎𝑓+𝑏𝑔∶𝕋→ℝ is differential at 𝑡 with 𝑎𝑓+𝑏𝑔^Δ(𝑡)=𝑎𝑓^Δ(𝑡)+𝑏𝑔^Δ(𝑡), (2.5)(ii)if 𝑓^Δ(𝑡) exists, then 𝑓 is continuous at 𝑡,(iii)if 𝑓^Δ(𝑡) exists, then 𝑓(𝜎(𝑡))=𝑓(𝑡)+𝜇(𝑡)𝑓^Δ(𝑡),(iv)the product 𝑓𝑔∶𝕋→ℝ is differential at 𝑡 with 𝑓𝑔^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)+𝑓(𝜎(𝑡))𝑔^Δ(𝑡)=𝑓(𝑡)𝑔^Δ(𝑡)+𝑓^Δ(𝑡)𝑔(𝜎(𝑡)), (2.6)(v)if 𝑔(𝑡)𝑔(𝜎(𝑡))≠0, then 𝑓/𝑔 is differential at 𝑡 and 𝑓𝑔^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)−𝑓(𝑡)𝑔^Δ(𝑡)𝑔(𝑡)𝑔(𝜎(𝑡.)) (2.7)

Definition 2.4 (see [6]). 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 𝐶_rd=𝐶_rd(𝕋)=𝐶_rd(𝕋,ℝ).

Definition 2.5 (see [6]). A function 𝐹∶𝕋→ℝ is called an antiderivative of 𝑓∶𝕋→ℝ, provided 𝐹^Δ(𝑡)=𝑓(𝑡) holds for all 𝑡∈𝕋^𝑘. We define the Cauchy integral by ^𝑠_𝜏𝑓(𝑡)Δ𝑡=𝐹(𝑠)−𝐹(𝜏),∀𝑠,𝜏∈𝕋. (2.8)

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

Lemma 2.6 (see [6]). If 𝑎,𝑏,𝑐∈𝕋,𝑘∈ℝ and 𝑓,𝑔∈𝐶_rd, then (i)∫^𝑏_𝑎[𝑓(𝑡)+𝑔(𝑡)]Δ𝑡=∫^𝑏_𝑎𝑓(𝑡)Δ𝑡+∫^𝑏_𝑎𝑔(𝑡)Δ𝑡, (ii)∫^𝑏_𝑎(𝑘𝑓)(𝑡)Δ𝑡=𝑘∫^𝑏_𝑎𝑓(𝑡)Δ𝑡, (iii)∫^𝑏_𝑎𝑓(𝑡)Δ𝑡=∫^𝑐_𝑎𝑓(𝑡)Δ𝑡+∫^𝑏_𝑐𝑓(𝑡)Δ𝑡, (iv)∫^𝑏_𝑎𝑓(𝜎(𝑡))𝑔^Δ(𝑡)Δ𝑡=(𝑓𝑔)(𝑏)−(𝑓𝑔)(𝑎)−∫^𝑏_𝑎𝑓^Δ(𝑡)𝑔(𝑡)Δ𝑡, (v)∫_𝑡^𝜎(𝑡)𝑓(𝑠)Δ𝑠=𝜇(𝑡)𝑓(𝑡)for𝑡∈𝕋^𝑘, (vi)if |𝑓(𝑡)|≤𝑔(𝑡) on[𝑎,𝑏), then ^𝑏_𝑎𝑓(𝑡)Δ𝑡≤^𝑏_𝑎𝑔(𝑡)Δ𝑡. (2.9)

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 [6]). Let 𝑎,𝑏∈𝕋 and 1<𝑝,𝑞<∞ with 1/𝑝+1/𝑞=1. For 𝑓,𝑔∈𝐶_rd, one has ^𝑏_𝑎𝑓(𝑡)𝑔(𝑡)Δ𝑡≤^𝑏_𝑎𝑓(𝑡)^𝑝Δ𝑡^1/𝑝𝑏_𝑎𝑔(𝑡)^𝑞Δ𝑡^1/𝑞. (2.10)

Lemma 2.8 (see [6]). Let 𝑎,𝑏∈𝕋 and 1<𝑟_𝑘<∞ with ∑^𝑚_𝑘=1(1/𝑟_𝑘)=1 for 𝑘=1,2,…,𝑚. For 𝑓_𝑘∈𝐶_rd,𝑘=1,2,…,𝑚, one has ^𝑏_𝑎^𝑚_𝑘=1𝑓_𝑘(𝑡)Δ𝑡≤^𝑚_𝑘=1^𝑏_𝑎𝑓_𝑘(𝑡)^𝑟𝑘Δ𝑡^1/𝑟𝑘. (2.11)

3. Lyapunov-Type Inequalities

Denote𝜁_𝑖(𝑡)∶=_𝑎^𝜎(𝑡)𝑟_𝑖(𝜏)^{1/(1−𝑝𝑖)}Δ𝜏^𝑝𝑖−1,𝑖=1,2,…,𝑚,𝜂_𝑖(𝑡)∶=^𝑏_𝜎(𝑡)𝑟_𝑖(𝜏)^{1/(1−𝑝𝑖)}Δ𝜏^𝑝𝑖−1,𝑖=1,2,…,𝑚. (3.1)(3.2)

First, we give the following hypothesis.(H1)𝑟_𝑖(𝑡) and 𝑓_𝑖(𝑡) are rd-continuous real functions and 𝑟_𝑖(𝑡)>0 for 𝑖=1,2,…,𝑚 and 𝑡∈𝕋. Furthermore, 1<𝑝_𝑖<∞ and 𝛼_𝑖>0 satisfy ∑^𝑚_𝑖=1(𝛼_𝑖/𝑝_𝑖)=1 for 𝑖=1,2,…,𝑚.

Theorem 3.1. Let 𝑎,𝑏∈𝕋^𝑘 with 𝜎(𝑎)≤𝑏. Suppose that hypothesis (H1) is satisfied. If (1.1) has a real solution (𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑚(𝑡)) satisfying the boundary value conditions 𝑢_𝑖(𝑎)=𝑢_𝑖(𝑏)=0,𝑢_𝑖(𝑡)≢0,∀𝑡∈[𝑎,𝑏],𝑖=1,2,…,𝑚, (3.3) then one has ^𝑚𝑚_𝑖=1𝑗=1^𝑏_𝑎𝜁_𝑖(𝑡)𝜂_𝑖(𝑡)𝜁_𝑖(𝑡)+𝜂_𝑖(𝑡)𝑓⁺_𝑗(𝑡)Δ𝑡^{𝛼𝑖𝛼𝑗/𝑝𝑖𝑝𝑗}≥1, (3.4) where and in what follows 𝑓⁺_𝑖(𝑡)=max{𝑓_𝑖(𝑡),0} for 𝑖=1,2,…,𝑚.

Proof. By (1.1) and Lemma 2.3(iv), we obtain −𝑟_𝑖(𝑡)𝑢^Δ_𝑖(𝑡)^𝑝𝑖−2𝑢^Δ_𝑖(𝑡)𝑢_𝑖(𝑡)^Δ+𝑟_𝑖(𝑡)𝑢^Δ_𝑖(𝑡)^𝑝𝑖=𝑓_𝑖(𝑡)^𝑚_𝑘=1𝑢_𝑘(𝜎(𝑡))^𝛼𝑘, (3.5) where 𝑖=1,2,…,𝑚. From Definition 2.5, integrating (3.5) from 𝑎 to 𝑏, together with (3.3), we get ^𝑏_𝑎𝑟_𝑖(𝑡)𝑢^Δ_𝑖(𝑡)^𝑝𝑖Δ𝑡=^𝑏_𝑎𝑓_𝑖(𝑡)^𝑚_𝑘=1𝑢_𝑘(𝜎(𝑡))^𝛼𝑘Δ𝑡,𝑖=1,2,…,𝑚. (3.6) It follows from (3.1), (3.3), and Lemma 2.7 that 𝑢_𝑖(𝜎(𝑡))^𝑝𝑖=_𝑎^𝜎(𝑡)𝑢^Δ_𝑖(𝜏)Δ𝜏^𝑝𝑖≤_𝑎^𝜎(𝑡)𝑟_𝑖(𝜏)^{1/(1−𝑝𝑖)}Δ𝜏^𝑝𝑖−1_𝑎^𝜎(𝑡)𝑟_𝑖(𝜏)𝑢^Δ_𝑖(𝜏)^𝑝𝑖Δ𝜏=𝜁_𝑖(𝑡)_𝑎^𝜎(𝑡)𝑟_𝑖(𝜏)𝑢^Δ_𝑖(𝜏)^𝑝𝑖Δ𝜏,𝑎≤𝑡≤𝑏,𝑖=1,2,…,𝑚. (3.7) Similarly, it follows from (3.2), (3.3), and Lemma 2.7 that 𝑢_𝑖(𝜎(𝑡))^𝑝𝑖=^𝑏_𝜎(𝑡)𝑢^Δ_𝑖(𝜏)Δ𝜏^𝑝𝑖≤^𝑏_𝜎(𝑡)𝑟_𝑖(𝜏)^{1/(1−𝑝𝑖)}Δ𝜏^{𝑝𝑖−1𝑏}_𝜎(𝑡)𝑟_𝑖(𝜏)𝑢^Δ_𝑖(𝜏)^𝑝𝑖Δ𝜏=𝜂_𝑖(𝑡)^𝑏_𝜎(𝑡)𝑟_𝑖(𝜏)𝑢^Δ_𝑖(𝜏)^𝑝𝑖Δ𝜏,𝑎≤𝑡≤𝑏,𝑖=1,2,…,𝑚. (3.8) From (3.7) and (3.8), we have 𝑢_𝑖(𝜎(𝑡))^𝑝𝑖≤𝜁_𝑖(𝑡)𝜂_𝑖(𝑡)𝜁_𝑖(𝑡)+𝜂_𝑖(𝑡)^𝑏_𝑎𝑟_𝑖(𝜏)𝑢^Δ_𝑖(𝜏)^𝑝𝑖Δ𝜏,𝑎≤𝑡≤𝑏,𝑖=1,2,…,𝑚. (3.9) So, from (3.3), (3.6), (3.9), (H1), and Lemma 2.8, we have ^𝑏_𝑎𝑓⁺_𝑖(𝑡)𝑢_𝑖(𝜎(𝑡))^𝑝𝑖Δ𝑡≤^𝑏_𝑎𝜁_𝑖(𝑡)𝜂_𝑖(𝑡)𝜁_𝑖(𝑡)+𝜂_𝑖(𝑡)𝑓⁺_𝑖(𝑡)Δ𝑡^𝑏_𝑎𝑟_𝑖(𝑡)𝑢^Δ_𝑖(𝑡)^𝑝𝑖Δ𝑡=𝑀_𝑖𝑗^𝑏_𝑎𝑓_𝑖(𝑡)^𝑚_𝑘=1𝑢_𝑘(𝜎(𝑡))^𝛼𝑘Δ𝑡≤𝑀_𝑖𝑗^𝑏_𝑎𝑓⁺_𝑖(𝑡)^𝑚_𝑘=1𝑢_𝑘(𝜎(𝑡))^𝛼𝑘Δ𝑡≤𝑀^𝑚_{𝑖𝑗𝑘=1}^𝑏_𝑎𝑓⁺_𝑖(𝑡)𝑢_𝑘(𝜎(𝑡))^𝑝𝑘Δ𝑡^{𝛼𝑘/𝑝𝑘}, (3.10) where 𝑀_𝑖𝑗=^𝑏_𝑎𝜁_𝑖(𝑡)𝜂_𝑖(𝑡)𝜁_𝑖(𝑡)+𝜂_𝑖(𝑡)𝑓⁺_𝑗(𝑡)Δ𝑡,𝑖,𝑗=1,2,…,𝑚. (3.11)
Next, we prove that^𝑏_𝑎𝑓⁺_𝑖(𝑡)𝑢_𝑘(𝜎(𝑡))^𝑝𝑘Δ𝑡>0. (3.12) If (3.12) is not true, there exist 𝑖₀,𝑘₀∈{1,2,…,𝑚} such that ^𝑏_𝑎𝑓⁺_𝑖0(𝑡)𝑢_𝑘0(𝜎(𝑡))^𝑝𝑘0Δ𝑡=0. (3.13) From (3.6), (3.13), and Lemma 2.8, we have 0≤^𝑏_𝑎𝑟_𝑖0(𝑡)𝑢^Δ_𝑖0(𝑡)^𝑝𝑖0Δ𝑡=^𝑏_𝑎𝑓_𝑖0(𝑡)^𝑚_𝑘=1𝑢_𝑘(𝜎(𝑡))^𝛼𝑘≤Δ𝑡_{𝑚𝑘=1𝑏𝑎}𝑓⁺_𝑖0(𝑡)𝑢_𝑘(𝜎(𝑡))^𝑝𝑘Δ𝑡^{𝛼𝑘/𝑝𝑘}=0. (3.14) It follows from the fact that 𝑟_𝑖0(𝑡)>0 that 𝑢^Δ_𝑖0(𝑡)≡0,𝑎≤𝑡≤𝑏. (3.15) Combining (3.7) with (3.15), we obtain that 𝑢_𝑖0(𝑡)≡0 for 𝑎≤𝑡≤𝑏, which contradicts (3.3). Therefore, (3.12) holds. From (3.10), (3.12), and (H1), we have ^𝑚𝑚_𝑖=1𝑗=1𝑀^{𝛼𝑖𝛼𝑗/𝑝𝑖𝑝𝑗}_𝑖𝑗≥1. (3.16) It follows from (3.11) and (3.16) that (3.4) holds.

Corollary 3.2. Let 𝑎,𝑏∈𝕋^𝑘 with 𝜎(𝑎)≤𝑏. Suppose that hypothesis (H1) is satisfied. If (1.1) has a real solution (𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑚(𝑡)) satisfying the boundary value conditions (3.3), then one has ^𝑚𝑚_𝑖=1𝑗=1^𝑏_𝑎𝑓⁺_𝑗(𝑡)𝜁_𝑖(𝑡)𝜂_𝑖(𝑡)^1/2Δ𝑡^{𝛼𝑖𝛼𝑗/𝑝𝑖𝑝𝑗}≥2. (3.17)

Proof. Since 𝜁_𝑖(𝑡)+𝜂_𝑖(𝑡)≥2𝜁_𝑖(𝑡)𝜂_𝑖(𝑡)^1/2,𝑖=1,2,…,𝑚, (3.18) it follows from (3.4) and (H1) that (3.17) holds.

Corollary 3.3. Let 𝑎,𝑏∈𝕋^𝑘 with 𝜎(𝑎)≤𝑏. Suppose that hypothesis (H1) is satisfied. If (1.1) has a real solution (𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑚(𝑡)) satisfying the boundary value conditions (3.3), then one has ^𝑚_𝑖=1^𝑏_𝑎𝑟_𝑖(𝑡)^{1/(1−𝑝𝑖)}Δ𝑡^{𝛼𝑖(𝑝𝑖−1)/𝑝𝑖𝑚}_𝑗=1^𝑏_𝑎𝑓⁺_𝑖(𝑡)Δ𝑡^{𝛼𝑗/𝑝𝑗}≥2^𝒜, (3.19) where 𝒜=∑^𝑚_𝑖=1𝛼_𝑖.

Proof. Since 𝜁_𝑖(𝑡)𝜂_𝑖(𝑡)^1/2=_𝑎^𝜎(𝑡)𝑟_𝑖(𝜏)^{1/(1−𝑝𝑖)}Δ𝜏^𝑏_𝜎(𝑡)𝑟_𝑖(𝜏)^{1/(1−𝑝𝑖)}Δ𝜏^{(𝑝𝑖−1)/2}≤12^{𝑝𝑖−1𝑏}_𝑎𝑟_𝑖(𝜏)^{1/(1−𝑝𝑖)}Δ𝜏^𝑝𝑖−1,𝑖=1,2,…,𝑚, (3.20) 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 by𝑟(𝑡)𝑢^Δ(𝑡)^𝛾−2𝑢^Δ(𝑡)^Δ+𝜚(𝑡)|𝑢(𝜎(𝑡))|^𝛾−2𝑢(𝜎(𝑡))=0. (3.21)

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

Corollary 3.4. Let 𝑎,𝑏∈𝕋^𝑘 with 𝜎(𝑎)≤𝑏. If (3.21) has a solution 𝑢(𝑡) satisfying 𝑢(𝑎)=𝑢(𝑏)=0,𝑢(𝑡)≢0,∀𝑡∈[𝑎,𝑏], (3.22) then ^𝑏_𝑎∫_𝑎^𝜎(𝑡)[𝑟(𝜏)]^1/(1−𝛾)Δ𝜏^𝛾−1∫^𝑏_𝜎(𝑡)[𝑟(𝜏)]^1/(1−𝛾)Δ𝜏^𝛾−1∫_𝑎^𝜎(𝑡)[𝑟(𝜏)]^1/(1−𝛾)Δ𝜏^𝛾−1+∫^𝑏_𝜎(𝑡)[𝑟(𝜏)]^1/(1−𝛾)Δ𝜏^𝛾−1𝜚⁺(𝑡)Δ𝑡≥1. (3.23)

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 by𝑢^Δ(𝑡)^Δ+𝜚(𝑡)𝑢(𝜎(𝑡))=0. (3.24)

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

Corollary 3.5. Let 𝑎,𝑏∈𝕋^𝑘 with 𝜎(𝑎)≤𝑏. If (3.24) has a solution 𝑢(𝑡) satisfying 𝑢(𝑎)=𝑢(𝑏)=0,𝑢(𝑡)≢0,∀𝑡∈[𝑎,𝑏], (3.25) then (𝑏−𝑎)^𝑏_𝑎𝜚⁺(𝑡)Δ𝑡≥4. (3.26)

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).