On Lyapunov-Type Inequalities for Nonlinear Dynamic Systems on Time Scales✩

Computers and Mathematics with Applications 62 (2011) 4028–4038

Contents lists available at SciVerse ScienceDirect

Computers and Mathematics with Applications

journal homepage: www.elsevier.com/locate/camwa

On Lyapunov-type inequalities for nonlinear dynamic systems on time scales✩

Qi-Ming Zhang a,∗, Xiaofei He b, Jianchu Jiang c a College of Science, Hunan University of Technology, Zhuzhou 412007, Hunan, PR China b College of Mathematics and Computer Science, Jishou University, Jishou, 416000, Hunan, PR China c Department of Mathematics, Humanities Science and Technology, University of Hunan, Loudi, 417000, Hunan, PR China article info a b s t r a c t

Article history: In this paper, by using elementary analysis, we establish several new Lyapunov type Received 25 April 2011 inequalities for the following nonlinear dynamic system on an arbitrary time scale T Received in revised form 16 September − 2011 x1(t) = α(t)x(σ (t)) + β(t)|y(t)|p 2y(t), − Accepted 19 September 2011 y1(t) = −γ (t)|x(σ (t))|q 2x(σ (t)) − α(t)y(t),

Keywords: when the end-points are not necessarily usual zeros, but rather, generalized zeros, which Nonlinear dynamic system generalize and improve all related existing ones including the continuous and discrete Lyapunov inequality cases. Generalized zero © 2011 Elsevier Ltd. All rights reserved. Time scales

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 Bohner and Peterson [5]. Consider the following nonlinear dynamic system on an arbitrary time scale T − x1(t) = α(t)x(σ (t)) + β(t)|y(t)|p 2y(t), − (1.1) y1(t) = −γ (t)|x(σ (t))|q 2x(σ (t)) − α(t)y(t), 1 + 1 = where p, q > 1 and p q 1, the coefficients α(t), β(t) and γ (t) are real-valued rd-continuous functions on T. Furthermore, α(t) satisfies the condition 1 − µ(t)α(t) > 0, ∀ t ∈ T, (1.2) the definition of µ(t) (see Section 2) and f 1(t) is called the delta derivative of the function f at some point t. Throughout this paper, we always assume that β(t) ≥ 0, ∀ t ∈ T. (1.3)

✩ 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) and by Hunan Provincial Natural Science Foundation of China (No. 11JJ2005). ∗ Corresponding author. E-mail address: [email protected] (Q. Zhang).

When p = q = 2, system (1.1) reduces to the Hamiltonian system (see [5]) which contains two scalar linear dynamic equations

x1(t) = α(t)x(σ (t)) + β(t)y(t), (1.4) y1(t) = −γ (t)x(σ (t)) − α(t)y(t).

We also remark that both the Emden–Fowler type dynamic equation

− − (p(t)|x1(t)|p 2x1(t))1 + q(t)|x(σ (t))|q 2x(σ (t)) = 0, (1.5) and the half-linear dynamic equation

− − (p(t)|x1(t)|r 2x1(t))1 + q(t)|x(σ (t))|r 2x(σ (t)) = 0, (1.6) can be written as special cases of system (1.1), where r > 1, p and q are the same as (1.1), p(t) and q(t) are real-valued rd-continuous functions and p(t) > 0 for all t ∈ T. For example, let p = r/(r − 1), q = r and 1 α(t) = 0, β(t) = , γ (t) = q(t), (1.7) [p(t)]1/(r−1) then Eq. (1.6) can be written as an equivalent form of (1.1). Especially, while r = 2, Eq. (1.6) reduces to a second-order linear dynamic equation

(p(t)x1(t))1 + q(t)x(σ (t)) = 0. (1.8)

In 1907, Lyapunov [6] established the first so called Lyapunov inequality

∫ b (b − a) q(t)dt > 4, (1.9) a if Hill’s equation ′′ x (t) + q(t)x(t) = 0 has a real solution x(t) such that x(a) = x(b) = 0, x(t) ̸≡ 0, t ∈ [a, b], and the constant 4 cannot be replaced by a larger number, where q(t) is a piece-wise continuous and nonnegative function defined on R. Since then, it has become 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 is a great deal of literature which improves and extends the classical Lyapunov inequality including continuous and discrete cases. For example, inequality (1.9) has been generalized to second- order nonlinear differential equations by Eliason [7] and Pachpatte [8], to the second-order delay differential equations of by Eliason [9] and Dahiya and Singh [10], and to higher order differential equations by Pachpatte [11]. Lyapunov-type inequalities for the Emden–Fowler type equations can be found in [8], and for the half-linear equations can be found in [12,13]. At the same time, we refer the reader to [14–29] for more comprehensive understanding. Recently, there has been much attention paid to Lyapunov-type inequality for linear Hamiltonian systems on time scales and some authors including Agarwal [30], Jiang [31], He [32] and Saker [33] have contributed the above results. In paper [34], Ünal has obtained some interesting Lyapunov-type inequalities on time scales and these results have almost covered the corresponding continuous and discrete versions that may be found in [25,26], respectively.

Theorem 1.1 ([34]). Suppose

1 − µ(t)α(t) > 0, β(t) > 0, γ (t) > 0, ∀ t ∈ [a, σ (b)]. (1.10)

Let a, b ∈ T with σ (a) < b. Assume (1.1) has a real solution (x(t), y(t)) such that x(a)x(σ (a)) < 0, and x(b)x(σ (b)) < 0. Then the inequality

∫ b ∫ σ (b) 1/p ∫ b 1/q |α(t)| 1t + β(t)1t γ (t)1t > 1 (1.11) a a a holds.

Theorem 1.2 ([34]). Suppose

1 − µ(t)α(t) > 0, β(t) > 0, ∀ t ∈ [a, σ (b)]. (1.12) 4030 Q. Zhang et al. / Computers and Mathematics with Applications 62 (2011) 4028–4038

Let a, b ∈ T with σ (a) < b. Assume (1.1) has a real solution (x(t), y(t)) such that x(a)x(σ (a)) < 0, and x(σ (b)) = 0. Then the inequality

∫ b ∫ σ (b) 1/p ∫ b 1/q + |α(t)| 1t + β(t)1t γ (t)1t > 1 (1.13) σ (a) σ (a) a holds, where and in the sequel

+ γ (t) = max{γ (t), 0}. (1.14)

In this paper, by using some simpler methods different from [34], we obtain the following Lyapunov-type inequalities

∫ b ∫ σ (b) 1/p ∫ b 1/q + |α(t)| 1t + β(t)1t γ (t)1t ≥ 2, (1.15) a a a and

∫ ρ(b) ∫ b 1/p ∫ b 1/q + |α(t)|1t + β(t)1t γ (t)1t ≥ 2, (1.16) a a a only under assumptions (1.2) and (1.3), which are better than (1.11) and (1.14), respectively. In addition, when the end-point b satisfies some general conditions, (Theorem 3.5 in Section 3), that is to say, b is not necessarily a generalized zero, we also establish a better Lyapunov-type inequality than (1.15)

∫ b ∫ b 1/p ∫ b 1/q + |α(t)| 1t + β(t)1t γ (t)1t ≥ 2. (1.17) a a a At first, we adopt the following concept of generalized zero on time scales.

Definition 1.3. A function f : T → R is said to have a generalized zero at t0 ∈ T provided either f (t0) = 0 or f (t0)f (σ (t0)) < 0.

2. Preliminaries about the time scales calculus

We introduce some basic notions connected to time scales.

Definition 2.1 ([5]). Let t ∈ T. We define the forward jump operator σ : T → T by

σ (t) := inf{s ∈ T : s > t} for all t ∈ T, while the backward jump operator ρ : T → T by

ρ(t) := sup{s ∈ T : s < t} for all t ∈ T.

In this definition, we put inf ∅ = sup T (i.e., σ (M) = M if T has a maximum M) and sup ∅ = inf T (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 < sup T and σ (t) = t, then t is called right-dense, and if t > inf T 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 k left-dense at the same time are called dense. If T has a left-scattered maximum M, then we define T = T − {M}, otherwise k T = T. The graininess function u : T → [0, ∞) is defined by

µ(t) := σ (t) − t, ∀ t ∈ T.

k We consider a function f : T → R and define so-called delta (or Hilger) derivative of f at a point t ∈ T .

k 1 Definition 2.2 ([5]). Assume f : T → R is a function and let t ∈ T . Then we define f (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 1(t)(σ (t) − s)| ≤ ε|σ (t) − s|, ∀ s ∈ U.

We call f 1(t) the delta (or Hilger) derivative of f at t. Q. Zhang et al. / Computers and Mathematics with Applications 62 (2011) 4028–4038 4031

k Lemma 2.3 ([5]). Assume f , g : T → R are differential at t ∈ T . Then (i) For any constant a and b, the sum af + bg : T → R is differential at t with (af + bg)1(t) = af 1(t) + bg1(t). (ii) If f 1(t) exists, then f is continuous at t. (iii) If f 1(t) exists, then f (σ (t)) = f (t) + µ(t)f 1(t). (iv) The product fg : T → R is differential at t with (fg)1(t) = f 1(t)g(t) + f (σ (t))g1(t) = f (t)g1(t) + f 1(t)g(σ (t)). (v) If g(t)g(σ (t)) ̸= 0, then f /g is differential at t and  f 1 f 1(t)g(t) − f (t)g1(t) (t) = . g g(t)g(σ (t))

Definition 2.4 ([5]). 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).

1 Definition 2.5 ([5]). A function F : T → R is called an antiderivative of f : T → R provided F (t) = f (t) holds for all k t ∈ T . We define the Cauchy integral by ∫ s f (t)1t = F(s) − F(τ), ∀ s, τ ∈ T. τ

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

Lemma 2.6 ([5]). If a, b, c ∈ T, k ∈ R and f , g ∈ Crd, then  b[ + ] =  b +  b (i) a f (t) g(t) 1t a f (t)1t a g(t)1t;  b =  b (ii) a (kf )(t)1t k a f (t)1t;  b =  c +  b (iii) a f (t)1t a f (t)1t c f (t)1t;  b 1 = − −  b 1 (iv) a f (σ (t))g (t)1t (fg)(b) (fg)(a) a f (t)g(t)1t;  σ (t) = ∈ k (v) t f (s)1s µ(t)f (t) for t T ; (vi) if |f (t)| ≤ g(t) on [a, b), then ∫ b  ∫ b    f (t)1t ≤ g(t)1t.  a  a

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 ([5], Cauchy–Schwarz Inequality). Let a, b ∈ T. For f , g ∈ Crd, we have

1 ∫ b ∫ b ∫ b  2 |f (t)g(t)| 1t ≤ f 2(t)1t · g2(t)1t . a a a

Lemma 2.8 ([5]). Let

A = {t ∈ T | t is left-dense and right-scattered}, B = {t ∈ T | t is right-dense and left-scattered}. Then

σ (ρ(t)) = t, ∀ t ∈ T \ A; ρ(σ (t)) = t, ∀ t ∈ T \ B.

3. Lyapunov type inequalities

In this section, we establish some new Lyapunov type inequalities on time scales T. 4032 Q. Zhang et al. / Computers and Mathematics with Applications 62 (2011) 4028–4038

k Theorem 3.1. Suppose (1.2) holds and let a, b ∈ T with σ (a) ≤ b. Assume (1.1) has a real solution (x(t), y(t)) such that x(t) has generalized zeros at end-points a and b and x(t) is not identically zero on [a, b], i.e., x(a) = 0 or x(a)x(σ (a)) < 0; x(b) = 0 or x(b)x(σ (b)) < 0; max |x(t)| > 0. (3.1) a≤t≤b Then one has the following inequality

∫ b ∫ σ (b) 1/p ∫ b 1/q + |α(t)| 1t + β(t)1t γ (t)1t ≥ 2. (3.2) a a a

Proof. It follows from (3.1) that there exist ξ, η ∈ [0, 1) such that (1 − ξ)x(a) + ξx(σ (a)) = 0, (3.3) and (1 − η)x(b) + ηx(σ (b)) = 0. (3.4) Multiplying the first equation of (1.1) by y(t) and the second one by x(σ (t)), and then adding, we get

[x(t)y(t)]1 = β(t)|y(t)|p − γ (t)|x(σ (t))|q. (3.5) Integrating Eq. (3.5) from a to b, we can obtain

∫ b ∫ b x(b)y(b) − x(a)y(a) = β(t)|y(t)|p1t − γ (t)|x(σ (t))|q1t. (3.6) a a From the first equation of (1.1) and using Lemma 2.3(iii), we have

− [1 − µ(t)α(t)]x(σ (t)) = x(t) + µ(t)β(t)|y(t)|p 2y(t). (3.7) Combining (3.7) with (3.3), we have

ξµ(a)β(a) − x(a) = − |y(a)|p 2y(a). (3.8) 1 − (1 − ξ)µ(a)α(a) Similarly, it follows from (3.7) and (3.4) that

ηµ(b)β(b) − x(b) = − |y(b)|p 2y(b). (3.9) 1 − (1 − η)µ(b)α(b) Substituting (3.8) and (3.9) into (3.6), we have ∫ b ∫ b ηµ(b)β(b) ξµ(a)β(a) β(t)|y(t)|p1t − γ (t)|x(σ (t))|q1t = − |y(b)|p + |y(a)|p; a a 1 − (1 − η)µ(b)α(b) 1 − (1 − ξ)µ(a)α(a) by using Lemma 2.6(v), we get (1 − ξ)[1 − µ(a)α(a)] ∫ b ηµ(b)β(b) µ(a)β(a)|y(a)|p + β(t)|y(t)|p1t + |y(b)|p 1 − (1 − ξ)µ(a)α(a) σ (a) 1 − (1 − η)µ(b)α(b) ∫ b = γ (t)|x(σ (t))|q1t. (3.10) a Denote that (1 − ξ)[1 − µ(a)α(a)] β(˜ a) = β(a), (3.11) 1 − (1 − ξ)µ(a)α(a) η β(˜ b) = β(b), (3.12) 1 − (1 − η)µ(b)α(b) and

β(˜ t) = β(t), σ (a) ≤ t ≤ ρ(b). (3.13) Then we can rewrite (3.10) as

∫ σ (b) ∫ b β(˜ t)|y(t)|p1t = γ (t)|x(σ (t))|q1t. (3.14) a a Q. Zhang et al. / Computers and Mathematics with Applications 62 (2011) 4028–4038 4033

On the other hand, integrating the first equation of (1.1) from a to τ and using (3.8), (3.11), (3.13) and Lemma 2.6(v), we obtain ∫ τ ∫ τ − x(τ) = x(a) + α(t)x(σ (t))1t + β(t)|y(t)|p 2y(t)1t a a ∫ τ ∫ τ ξµ(a)β(a) − − = − |y(a)|p 2y(a) + α(t)x(σ (t))1t + β(t)|y(t)|p 2y(t)1t 1 − (1 − ξ)µ(a)α(a) a a ∫ τ ∫ τ (1 − ξ)[1 − µ(a)α(a)] − − = α(t)x(σ (t))1t + µ(a)β(a)|y(a)|p 2y(a) + β(t)|y(t)|p 2y(t)1t a 1 − (1 − ξ)µ(a)α(a) σ (a) ∫ τ ∫ τ − = α(t)x(σ (t))1t + β(˜ t)|y(t)|p 2y(t)1t, σ (a) ≤ τ ≤ b. (3.15) a a Similarly, integrating the first equation of (1.1) from τ to b and using (3.9), (3.12), (3.13) and Lemma 2.6(v), we have

∫ b ∫ b − x(τ) = x(b) − α(t)x(σ (t))1t − β(t)|y(t)|p 2y(t)1t τ τ ∫ b ∫ b ηµ(b)β(b) − − = − |y(b)|p 2y(b) − α(t)x(σ (t))1t − β(t)|y(t)|p 2y(t)1t 1 − (1 − η)µ(b)α(b) τ τ ∫ b ∫ σ (b) − = − α(t)x(σ (t))1t − β(˜ t)|y(t)|p 2y(t)1t, σ (a) ≤ τ ≤ b. (3.16) τ τ It follows from (3.15), (3.16) and Lemma 2.6 that ∫ τ ∫ τ − |x(τ)| ≤ |α(t)| |x(σ (t))|1t + β(˜ t)|y(t)|p 11t, σ (a) ≤ τ ≤ b, a a and ∫ b ∫ σ (b) − |x(τ)| ≤ |α(t)| |x(σ (t))|1t + β(˜ t)|y(t)|p 11t, σ (a) ≤ τ ≤ b. τ τ Adding the above two inequalities, we have

∫ b ∫ σ (b) − 2|x(τ)| ≤ |α(t)| |x(σ (t))|1t + β(˜ t)|y(t)|p 11t, σ (a) ≤ τ ≤ b. (3.17) a a ∗ Let |x(τ )| = maxσ (a)≤t≤b |x(t)|. There are two possible cases: (1) end-point b is left-scattered; (2) end-point b is left-dense. Case (1). In this case, it follows from Lemma 2.8 that σ (ρ(b)) = b. Hence,

∫ b ∫ ρ(b) ∫ b |α(t)| |x(σ (t))|1t = |α(t)| |x(σ (t))|1t + |α(t)| |x(σ (t))|1t a a ρ(b) ∫ ρ(b) ∫ σ (ρ(b)) = |α(t)| |x(σ (t))|1t + |α(t)| |x(σ (t))|1t a ρ(b) ∫ ρ(b) = |α(t)| |x(σ (t))|1t + u(ρ(b))|α(ρ(b))| |x(σ (ρ(b)))| a ∫ ρ(b) = |α(t)| |x(σ (t))|1t + u(ρ(b))|α(ρ(b))| |x(b)| a [∫ ρ(b) ] ∗ ≤ |x(τ )| |α(t)|1t + u(ρ(b))|α(ρ(b))| a ∫ b ∗ ≤ |x(τ )| |α(t)|1t. (3.18) a That is

∫ b ∫ b ∗ |α(t)| |x(σ (t))|1t ≤ |x(τ )| |α(t)|1t. (3.19) a a 4034 Q. Zhang et al. / Computers and Mathematics with Applications 62 (2011) 4028–4038

Similarly, we have

∫ b ∫ b + ∗ + γ (t)|x(σ (t))|q1t ≤ |x(τ )|q γ (t)1t. (3.20) a a

Case (2). In this case, there exists a sequence {bn} of T such that

a < b1 < b2 < b3 < ··· < bn < ··· < b, lim bn = b. n→∞ Hence

∫ b ∫ bn ∫ b |α(t)| |x(σ (t))|1t = |α(t)| |x(σ (t))|1t + |α(t)| |x(σ (t))|1t a a bn ∫ bn ∫ b ∗ ≤ |x(τ )| |α(t)|1t + |α(t)| |x(σ (t))|1t a bn ∫ b ∫ b ∗ ≤ |x(τ )| |α(t)|1t + |α(t)| |x(σ (t))|1t a bn ∫ b ∗ ≤ |x(τ )| |α(t)|1t, n → ∞, a which implies that (3.19) holds. Similarly, we can prove that (3.20) holds as well. Applying Lemma 2.7 and using (3.14), (3.19) and (3.20), we have ∫ b ∫ σ (b) ∗ − 2|x(τ )| ≤ |α(t)| |x(σ (t))|1t + β(˜ t)|y(t)|p 11t a a ∫ b ∫ σ (b) 1/p ∫ σ (b) 1/q ∗ ≤ |x(τ )| |α(t)|1t + β(˜ t)1t β(˜ t)|y(t)|p1t a a a ∫ b ∫ σ (b) 1/p ∫ b 1/q ∗ = |x(τ )| |α(t)|1t + β(˜ t)1t γ (t)|x(σ (t))|q1t a a a ∫ b ∫ σ (b) 1/p ∫ b 1/q ∗ + ≤ |x(τ )| |α(t)|1t + β(˜ t)1t γ (t)1t . (3.21) a a a

Dividing the latter inequality of (3.21) by |x(τ ∗)|, we obtain

∫ b ∫ σ (b) 1/p ∫ b 1/q + |α(t)|1t + β(˜ t)1t γ (t)1t ≥ 2. (3.22) a a a Since µ(t) ≥ 0, 1 − µ(t)α(t) > 0 and ξ, η ∈ [0, 1), we have

β(˜ t) ≤ β(t), a ≤ t < σ (b),

(3.2) follows immediately from (3.22).

Remark 3.2. It is obvious that the Lyapunov type inequality (3.2) of Theorem 3.1 is better than (1.10) of Theorem 1.1 for the bound 2 in the right side of (3.2) is better than that of (1.10). Furthermore, the assumptions of the former is weaker than the ones of the latter.

In the case x(b) = 0, i.e. η = 0, we have the following equation

∫ b ∫ b β(˜ t)|y(t)|p1t = γ (t)|x(σ (t))|q1t (3.23) a a and inequality

∫ ρ(b) ∫ b − 2|x(τ)| ≤ |α(t)| |x(σ (t))|1t + β(˜ t)|y(t)|p 11t, σ (a) ≤ τ ≤ b (3.24) a a instead of (3.14) and (3.17), respectively. It is easy to see that (3.23) holds because β(˜ b) = 0. Next, we prove (3.24) is true. If b is left-dense, then ρ(b) = b, and so (3.24) holds. If b is left-scattered, then it follows from Lemma 2.8 that σ (ρ(b)) = b. Q. Zhang et al. / Computers and Mathematics with Applications 62 (2011) 4028–4038 4035

Thus, it follows from Lemma 2.6(iii) and (v) and the assumption x(b) = 0 that ∫ b ∫ ρ(b) ∫ b |α(t)| |x(σ (t))|1t = |α(t)| |x(σ (t))|1t + |α(t)| |x(σ (t))|1t a a ρ(b) ∫ ρ(b) = |α(t)| |x(σ (t))|1t + u(ρ(b))|α(ρ(b))| |x(b)| a ∫ ρ(b) = |α(t)| |x(σ (t))|1t, a which, together with (3.17) and the fact that β(˜ b) = 0, implies that (3.24) holds. Similar to the proof of (3.22), we have

∫ ρ(b) ∫ b 1/p ∫ b 1/q + |α(t)|1t + β(˜ t)1t γ (t)1t ≥ 2. (3.25) a a a

Since β(˜ t) ≤ β(t) for a ≤ t ≤ b, it follows that

∫ ρ(b) ∫ b 1/p ∫ b 1/q + |α(t)|1t + β(t)1t γ (t)1t ≥ 2. (3.26) a a a Therefore, we can obtain the following theorem.

k Theorem 3.3. Suppose that (1.2) holds and let a, b ∈ T with σ (a) ≤ ρ(b). Assume (1.1) has a real solution (x(t), y(t)) such that x(t) has a generalized zero at end-point a but a usual zero at end-point b and x(t) is not identically zero on [a, b], i.e.,

x(a) = 0 or x(a)x(σ (a)) < 0; x(b) = 0; max |x(t)| > 0. a≤t≤b

Then inequality (3.26) holds.

Remark 3.4. In view of the proof of Theorem 3.3, in the case when both end-points a and b are usual zeros, i.e., x(a) = x(b) = 0, assumption (1.2) can be dropped in Theorem 3.3.

While the end-point b is not necessarily a generalized zero of x(t), we can establish the following more general theorem.

k Theorem 3.5. Suppose that (1.2) holds and let a, b ∈ T with σ (a) ≤ b. Assume (1.1) has a real solution (x(t), y(t)) such that p/(p−1) x(t) has a generalized zero at end-point a and (x(b), y(b)) = (λ1x(a), λ2y(a)) with 0 < |λ1| ≤ λ1λ2 ≤ 1 and x(t) is not identically zero on [a, b]. Then one has the following inequality

∫ b ∫ b 1/p ∫ b 1/q + |α(t)| 1t + β(t)1t γ (t)1t ≥ 2. (3.27) a a a

Proof. It follows from the assumption x(t) = 0 or x(t)x(σ (t)) < 0 that there exists ξ ∈ [0, 1) such that (3.3) holds. Further, by the proof of Theorem 3.1, (3.5)–(3.8) hold. Since (x(b), y(b)) = (λ1x(a), λ2y(a)), then by (3.6), we have

∫ b ∫ b p q (λ1λ2 − 1)x(a)y(a) = β(t)|y(t)| 1t − γ (t)|x(σ (t))| 1t. (3.28) a a Substituting (3.8) into (3.28), we have

∫ b ∫ b (1 − λ1λ2)ξµ(a)β(a) β(t)|y(t)|p1t − γ (t)|x(σ (t))|q1t = |y(a)|p, a a 1 − (1 − ξ)µ(a)α(a) which implies that

∫ b ∫ b p p q κ1µ(a)β(a)|y(a)| + β(t)|y(t)| 1t = γ (t)|x(σ (t))| 1t, (3.29) σ (a) a where

1 − (1 − λ1λ2)ξ − (1 − ξ)µ(a)α(a) κ1 = . (3.30) 1 − (1 − ξ)µ(a)α(a) 4036 Q. Zhang et al. / Computers and Mathematics with Applications 62 (2011) 4028–4038

On the other hand, integrating the first equation of (1.1) from a to τ and using (3.8) and Lemma 2.6(v), we can obtain the following inequality which is similar to (3.15). ∫ τ (1 − ξ)[1 − µ(a)α(a)] − x(τ) = µ(a)β(a)|y(a)|p 2y(a) + α(t)x(σ (t))1t 1 − (1 − ξ)µ(a)α(a) a ∫ τ − + β(t)|y(t)|p 2y(t)1t, σ (a) ≤ τ ≤ b. (3.31) σ (a)

Similarly, integrating the first equation of (1.1) from τ to b and using (3.8), Lemma 2.6(v) and the fact that x(b) = λ1x(a), we have ∫ b ∫ b − x(τ) = x(b) − α(t)x(σ (t))1t − β(t)|y(t)|p 2y(t)1t τ τ ∫ b ∫ b p−2 = λ1x(a) − α(t)x(σ (t))1t − β(t)|y(t)| y(t)1t τ τ ∫ b λ1ξµ(a) − = − β(a)|y(a)|p 2y(a) − α(t)x(σ (t))1t 1 − (1 − ξ)µ(a)α(a) τ ∫ b − − β(t)|y(t)|p 2y(t)1t, σ (a) ≤ τ ≤ b. (3.32) τ From (3.31) and (3.32), we obtain ∫ τ (1 − ξ)[1 − µ(a)α(a)] − |x(τ)| ≤ µ(a)β(a)|y(a)|p 1 + |α(t)| |x(σ (t))|1t 1 − (1 − ξ)µ(a)α(a) a ∫ τ − + β(t)|y(t)|p 11t, σ (a) ≤ τ ≤ b, σ (a) and ∫ b ∫ b |λ1|ξ − − |x(τ)| ≤ µ(a)β(a)|y(a)|p 1 + |α(t)| |x(σ (t))|1t + β(t)|y(t)|p 11t, σ (a) ≤ τ ≤ b. 1 − (1 − ξ)µ(a)α(a) τ τ Adding the above two inequalities, we have

∫ b ∫ b p−1 p−1 2|x(τ)| ≤ κ2µ(a)β(a)|y(a)| + |α(t)| |x(σ (t))|1t + β(t)|y(t)| 1t, σ (a) ≤ τ ≤ b, (3.33) a σ (a) where

1 − (1 − |λ1|)ξ − (1 − ξ)µ(a)α(a) κ2 = . (3.34) 1 − (1 − ξ)µ(a)α(a) ∗ Let |x(τ )| = maxσ (a)≤τ≤b |x(τ)|. Applying Lemmas 2.6 and 2.7 and using (3.19), (3.20), (3.29) and (3.33), we have

∫ b ∫ b ∗ p−1 p−1 2|x(τ )| ≤ κ2µ(a)β(a)|y(a)| + |α(t)| |x(σ (t))|1t + β(t)|y(t)| 1t a σ (a) ∫ b ∗ ≤ |x(τ )| |α(t)|1t a  1/p 1/q κ p ∫ b  ∫ b  + 2 a a + t t p a a |y a |p + t |y t |p t p−1 µ( )β( ) β( )1 κ1 ( )β( ) ( ) β( ) ( ) 1 κ1 σ (a) σ (a) 1/p ∫ b  p ∫ b  ∫ b 1/q ∗ κ = |x | | t | t + 2 a a + t t t |x t |q t (τ ) α( ) 1 p−1 µ( )β( ) β( )1 γ ( ) (σ ( )) 1 a κ1 σ (a) a

 1/p  ∫ b  p ∫ b  ∫ b 1/q ∗ κ + ≤ |x | | t | t + 2 a a + t t t t (3.35) (τ )  α( ) 1 u−1 µ( )β( ) β( )1 γ ( )1  . a κ1 σ (a) a Q. Zhang et al. / Computers and Mathematics with Applications 62 (2011) 4028–4038 4037

Dividing the latter inequality of (3.35) by |x(τ ∗)|, we obtain

1/p ∫ b  p ∫ b  ∫ b 1/q κ + | t | t + 2 a a + t t t t ≥ 2 (3.36) α( ) 1 p−1 µ( )β( ) β( )1 γ ( )1 . a κ1 σ (a) a Set d = 1 − (1 − ξ)µ(a)α(a). Since (1 − ξ)[1 − µ(a)α(a)] > 0, it follows that d > ξ ≥ 0. Let

p/(p−1) f (t) ≤ max{f (0), f (1)} = max{0, |λ1| − λ1λ2} = 0, ∀ t ∈ [0, 1]. That is

p/(p−1) [1 − (1 − |λ1|)t] ≤ 1 − (1 − λ1λ2)t, ∀ t ∈ [0, 1], which implies that [ ξ ]p/(p−1) ξ 1 − (1 − |λ1|) ≤ 1 − (1 − λ1λ2) . d d This, together with (3.30) and (3.34), implies that p  1− 1−| | − 1− a  p ( λ1 )ξ ( ξ)α( ) κ 1−(1−ξ)α(a) 2 = p−1 p−1  1−(1−λ λ )ξ−(1−ξ)α(a)  κ1 1 2 1−(1−ξ)α(a) [d − (1 − |λ |)ξ]p = 1 p−1 d[d − (1 − λ1λ2)ξ]  − − | | ξ p 1 (1 λ1 ) d = −  − − ξ p 1 1 (1 λ1λ2) d ≤ 1.

Substituting this into (3.36), we obtain (3.27). Applying Theorems 3.1, 3.3 and 3.5 to the second-order half-linear difference equation (1.6), we have the following corollaries.

k Corollary 3.6. Let r > 1 and a, b ∈ T with σ (a) ≤ b. Suppose that p(t) > 0, ∀ t ∈ T. (3.37) Assume (1.6) has a real solution x(t) such that x(a) = 0 or x(a)x(a + 1) < 0 and x(b) = 0 or x(b)x(b + 1) < 0 and x(t) is not identically zero on [a, b]. Then one has the following inequality

1− 1 1 ∫ σ (b)  r ∫ b  r 1 + 1t q (t)1t ≥ 2. (3.38) 1/(r−1) a [p(t)] a

k Corollary 3.7. Suppose that (3.37) holds and let a, b ∈ T with σ (a) ≤ ρ(b). Assume (1.6) has a real solution x(t) such that x(a) = 0 or x(a)x(a + 1) < 0 and x(b) = 0 and x(t) is not identically zero on [a, b]. Then one has the following inequality

1− 1 1 ∫ b  r ∫ b  r 1 + 1t q (t)1t ≥ 2. (3.39) 1/(r−1) a [p(t)] a 4038 Q. Zhang et al. / Computers and Mathematics with Applications 62 (2011) 4028–4038

k Corollary 3.8. Suppose that (3.37) holds and let a, b ∈ T with σ (a) ≤ b. Assume (1.6) has a real solution x(t) such that

1 r−2 1 1 r−2 1 x(a)x(a + 1) < 0, x(b) = λ1x(a), p(b)|x (b)| x (b) = λ2p(a)|x (a)| x (a), (3.40) r where 0 ≤ |λ1| ≤ λ1λ2 ≤ 1, and x(t) is not identically zero on [a, b]. Then (3.39) holds.

References

[1] S. Hilger, Einßmakettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. Thesis, Universität Würzburg, 1988 (in German). [2] S. Hilger, Analysis on measure chain-a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18–56. [3] S. Hilger, Differential and difference calculus-unified!, Nonlinear Anal. 30 (1997) 2683–2694. [4] B. Kaymakcalan, V. Lakshimikantham, S. Sivasundaram, Dynamic System on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996. [5] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston, Inc., Boston, MA, 2003. [6] A.M. Lyapunov, Probléme général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse Math. 2 (1907) 203–407. [7] S.B. Eliason, A Lyapunov inequality for a certain nonlinear differential equation, J. Lond. Math. Soc. 2 (1970) 461–466. [8] B.G. Pachpatte, Inequalities related to the zeros of solutions of certain second order differential equations, Facta Univ. Ser. Math. Inform. 16 (2001) 35–44. [9] S.B. Eliason, Lyapunov type inequalities for certain second order functional differential equations, SIAM J. Appl. Math. 27 (1) (1974) 180–199. [10] R.S. Dahiya, B. Singh, A Lyapunov inequality and nonoscillation theorem for a second order nonlinear differential–difference equations, J. Math. Phys. Sci. 7 (1973) 163–170. [11] B.G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations, J. Math. Anal. Appl. 195 (1995) 527–536. [12] C. Lee, C. Yeh, C. Hong, R.P. Agarwal, Lyapunov and Wirtinger inequalities, Appl. Math. Lett. 17 (2004) 847–853. [13] J.P. Pinasco, Lower bounds for eigenvalues of the one-dimensional p-Laplacian, Abstr. Appl. Anal. 2004 (2004) 147–153. [14] C.D. Ahlbrandt, A.C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Academic, Boston, MA, 1996. [15] R. Agarwal, C.D. Ahlbrandt, M. Bohner, A.C. Peterson, Discrete linear Hamiltonian systems: a survey, Dynam. Systems Appl. 8 (1999) 307–333. [16] M. Bohner, S. Clark, J. Ridenhour, Lyapunov inequalities for time scales, J. Inequal. Appl. 7 (2002) 61–77. [17] S.S. Cheng, A discrete analogue of the inequality of Lyapunov, Hokkaido Math. J. 12 (1983) 105–112. [18] S.S. Cheng, Lyapunov inequalities for differential and difference equations, Fasc. Math. 23 (1991) 25–41. [19] S. Clark, D.B. Hinton, Discrete Lyapunov inequalities for linear Hamiltonian systems, Math. Inequal. Appl. 1 (1998) 201–209. [20] S. Clark, D.B. Hinton, Discrete Lyapunov inequalities, Dynam. Systems Appl. 8 (1999) 369–380. [21] G.Sh. Guseinov, A. Zafer, Stability criteria for linear periodic impulsive Hamiltonian systems, J. Math. Anal. Appl. 335 (2007) 1195–1206. [22] G.Sh. Guseinov, B. Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian systems, Comput. Math. Appl. 45 (2003) 1399–1416. [23] Z. He, Existence of two solutions of m-point boundary value problem for second dynamic equations on time scales, J. Math. Anal. Appl. 296 (2004) 97–109. [24] M.G. Krein, Foundations of the theory of λ-zones of stability of canonical system of linear differential equations with periodic coefficients, Amer. Math. Soc. Transl. Ser. 2 (120) (1983) 1–70. In memory of A.A. Andronov, Izdat. Acad. Nauk SSSR, Moscow, 1955, 413–498. [25] A. Tiryaki, M. Ünal, D. Cakmak, Lyapunov-type inequalities for nonlinear systems, J. Math. Anal. Appl. 332 (2007) 497–511. [26] M. Ünal, D. Cakmak, A. Tiryaki, A discrete analogue of Lyapunov-type inequalities for nonlinear systems, Comput. Math. Appl. 55 (2008) 2631–2642. [27] X. Wang, Stability criteria for linear periodic Hamiltonian systems, J. Math. Anal. Appl. 367 (2010) 329–336. [28] Q. Zhang, X.H. Tang, Lyapunov inequalities and stability for discrete linear Hamiltonian system. doi:10.1016/j.amc.2011.05.101. [29] Q. Zhang, X.H. Tang, Lyapunov inequalities and stability for discrete linear Hamiltonian systems, J. Difference Equ. Appl., in press (doi:10.1080/10236198.2011.572071). [30] R. Agarwal, M. Bohner, P. Rehak, Half-linear dynamic equations, Nonlinear Anal. Appl. 1 (2003) 1–56. [31] L.Q. Jiang, Z. Zhou, Lyapunov inequality for linear Hamiltonian systems on time scales, J. Math. Anal. Appl. 310 (2005) 579–593. [32] X. He, Q. Zhang, X.H. Tang, On inequalities of Lyapunov for linear Hamiltonian systems on time scales, J. Math. Anal. Appl. (2011) doi:10.1016/j.jmaa.2011.03.036. [33] S. Saker, Oscillation of nonlinear dynamic equations on time scales, Appl. Math. Comput. 148 (2004) 81–91. [34] M. Ünal, D. Cakmak, Lyapunov-type inequalities for certain nonlinear systems on time scales, Turkish J. Math. 32 (2008) 255–275.