DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation89708710.1155/2009/897087897087Research ArticleBounds for Certain Nonlinear Dynamic Inequalities on Time ScalesLiWei Nian1, 2HanMaoan2ZhangGuang1Department of MathematicsBinzhou UniversityShandong 256603Chinabzu.edu.cn2Department of Applied MathematicsShanghai Normal UniversityShanghai 200234Chinashtu.edu.cn200924112009200929072009191120092009Copyright © 2009This 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 investigate some new nonlinear dynamic inequalities on time scales. Our results unify and extend some integral inequalities and their corresponding discrete analogues. The inequalities given here can be used to investigate the properties of certain dynamic equations on time scales.

1. Introduction

To unify the theory of continuous and discrete dynamic systems, in 1988, Hilger  first introduced the calculus on time scales. Motivated by the paper , many authors have expounded on various aspects of the theory of dynamic equations on time scales. For example, we refer the reader to the literatures  and the references cited therein. At the same time, a few papers  have studied the theory of dynamic inequalities on time scales.

The main purpose of this paper is to investigate some nonlinear dynamic inequalities on time scales, which unify and extend some integral inequalities and their corresponding discrete analogues. Our work extends some known results of dynamic inequalities on time scales.

Throughout this paper, a knowledge and understanding of time scales and time-scale notation is assumed. For an excellent introduction to the calculus on time scales, we refer the reader to monographes [6, 7].

2. Main Results

In what follows, denotes the set of real numbers, +=[0,), denotes the set of integers, 0={0,1,2,} denotes the set of nonnegative integers, C(M,S) denotes the class of all continuous functions defined on set M with range in the set S, 𝕋 is an arbitrary time scale, Crd denotes the set of rd-continuous functions, denotes the set of all regressive and rd-continuous functions, and +={p:1+μ(t)p(t)>0for  all  t𝕋}. We use the usual conventions that empty sums and products are taken to be 0 and 1, respectively. Throughout this paper, we always assume that pq>0, p and q are real constants, and tt0,  t0𝕋κ.

Firstly, we introduce the following lemmas, which are useful in our main results.

Lemma 2.1.

Let a0. Then aq/p(qpKq-p/pa+p-qpKq/p)forany    K>0.

Proof.

If a=0, then we easily see that the inequality (2.1) holds. Thus we only prove that the inequality (2.1) holds in the case of a>0.

Letting f(K)=qpKq-p/pa+p-qpKq/p,K>0, we have f(K)=q(p-q)p2Kq-2p/p(K-a). It is easy to see that f(K)0,K>a,f(K)=0,K=a,f(K)0,0<K<a. Therefore, f(K)f(a)=aq/p. The proof of Lemma 2.1 is complete.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

Let t0𝕋κ and w:𝕋×𝕋κ be continuous at (t,t), t𝕋κ with t>t0. Assume that w1Δ(t,·) is rd-continuous on [t0,σ(t)]. If, for any ε>0, there exists a neighborhood U of t, independent of τ[t0,σ(t)], such that |w(σ(t),τ)-w(s,τ)-w1Δ(t,τ)(σ(t)-s)|ε|σ(t)-s|        for  allsU, where w1Δ denotes the derivative of w with respect to the first variable, then υ(t):=t0tw(t,τ)Δτ implies υΔ(t)=t0tw1Δ(t,τ)Δτ+w(σ(t),t).

Lemma 2.3 (Comparison theorem [<xref ref-type="bibr" rid="B6">6</xref>]).

Suppose u,bCrd, a+. Then uΔ(t)a(t)u(t)+b(t)        for  allt𝕋κ implies u(t)u(t0)ea(t,t0)+t0tea(t,σ(τ))b(τ)Δτfor  allt𝕋κ.

Next, we establish our main results.

Theorem 2.4.

Assume that u,a,b,g,hCrd, and u(t),a(t),b(t),g(t) and h(t) are nonnegative. Then up(t)a(t)+b(t)t0t[g(τ)uq(τ)+h(τ)]Δτfor  allt𝕋κ implies u(t){a(t)+b(t)t0teB(t,σ(τ))F(τ)Δτ}1/p        for  anyK>0,t𝕋κ, where F(t)=g(t)(p-qpKq/p+qa(t)pK(p-q)/p)+h(t), and also B(t)=qb(t)g(t)pK(p-q)/p        for  allt𝕋κ.

Proof.

Obviously, if t=t0, then the inequality (2.11) holds. Therefore, in the next proof, we always assume that t>t0,t𝕋κ.

Define a function z(t) by z(t)=t0t[g(τ)uq(τ)+h(τ)]Δτ. Then (E1) can be restated as up(t)a(t)+b(t)z(t). Using Lemma 2.1, from (2.15), for any K>0, we easily obtain uq(t)(a(t)+b(t)z(t))q/pp-qpKq/p+qa(t)pK(p-q)/p+qb(t)z(t)pK(p-q)/p. It follows from (2.14) and (2.16) that zΔ(t)g(t)(p-qpKq/p+qa(t)pK(p-q)/p+qb(t)z(t)pK(p-q)/p)+h(t)=F(t)+B(t)z(t), where F(t) and B(t) are defined as in (2.12) and (2.13), respectively. Using Lemma 2.3 and noting z(t0)=0, from (2.17) we have z(t)t0teB(t,σ(τ))F(τ)Δτ,      for  anyK>0,t𝕋κ.

Therefore, the desired inequality (2.11) follows from (2.15) and (2.18). This completes the proof of Theorem 2.4.

Remark 2.5.

By letting p=q=1 in Theorem 2.4, it is easy to observe that the bound obtained in (2.11) reduces to the bound obtained in [9, Theorem 3.1].

As a particular case of Theorem 2.4, we immediately obtain the following result.

Corollary 2.6.

Assume that u,gCrd, and u(t) and g(t) are nonnegative. If α>0 is a constant, then up(t)α+t0tg(τ)uq(τ)Δτ        for  allt𝕋κ implies u(t){α+t0teB̂(t,σ(τ))F̂(τ)Δτ}1/p        for  anyK>0,t𝕋κ, where F̂(t)=g(t)(p-qpKq/p+qαpK(p-q)/p),B̂(t)=qg(t)pK(p-q)/p        for  allt𝕋κ.

Remark 2.7.

The result of Theorem 2.4 holds for an arbitrary time scale. Therefore, using Theorem 2.4, we immediately obtain many results for some peculiar time scales. For example, letting 𝕋= and 𝕋=, respectively, we have the following two results.

Corollary 2.8.

Let 𝕋= and assume that u(t),a(t),b(t),g(t),h(t)C(+,+). Then the inequality up(t)a(t)+b(t)0t[g(s)uq(s)+h(s)]ds,      t+ implies u(t)[a(t)+b(t)0tF(θ)exp(θtB(s)ds)dθ]1/p        for  anyK>0,t+, where F(t) and B(t) are defined as in Theorem 2.4.

Corollary 2.9.

Let 𝕋= and assume that u(t),a(t),b(t),g(t), and h(t) are nonnegative functions defined for t0. Then the inequality up(t)a(t)+b(t)s=0t-1[g(s)uq(s)+h(s)],t0 implies u(t)[a(t)+b(t)θ=0t-1F(θ)s=θ+1t-1(1+B(s))]1/p        for  anyK>0,t0, where F(t) and B(t) are defined as in Theorem 2.4.

Investigating the proof procedure of Theorem 2.4 carefully, we can obtain the following result.

Theorem 2.10.

Assume that u,a,b,gi,hCrd, and u(t),a(t),b(t),gi(t), and h(t) are nonnegative, i=1,2,,n. If there exists a series of positive real numbers q1,q2,,qn such that pqi>0,i=1,2,,n, then up(t)a(t)+b(t)t0t[i=1ngi(τ)uqi(τ)+h(τ)]Δτ        for  allt𝕋κ implies u(t){a(t)+b(t)t0teB*(t,σ(τ))F*(τ)Δτ}1/p        for  anyK>0,t𝕋κ, where F*(t)=i=1ngi(t)(p-qipKqi/p+qia(t)pK(p-qi)/p)+h(t),B*(t)=i=1nqib(t)gi(t)pK(p-qi)/p        for  allt𝕋κ.

Theorem 2.11.

Assume that u,a,b,f,g,mCrd, u(t),a(t),b(t),f(t),g(t), and m(t) are nonnegative, and w(t,s) is defined as in Lemma 2.2 such that w(t,s)0 and w1Δ(t,s)0 for t,s𝕋 with st. If, for any ε>0, there exists a neighborhood U of t, independent of τ[t0,σ(t)], such that for all sU, |[w(σ(t),τ)-w(s,τ)-w1Δ(t,τ)(σ(t)-s)][f(τ)up(τ)+g(τ)uq(τ)+m(τ)]|ε|σ(t)-s|, then up(t)a(t)+b(t)t0tw(t,τ)[f(τ)up(τ)+g(τ)uq(τ)+m(τ)]Δτ,t𝕋κ implies u(t){a(t)+b(t)t0teA(t,σ(τ))G(τ)Δ(τ)}1/p        for  anyK>0,t𝕋κ, where A(t)=w(σ(t),t)b(t)(f(t)+qg(t)pK(p-q)/p)+t0tw1Δ(t,τ)b(τ)(f(τ)+qg(τ)pK(p-q)/p)Δτ, and also G(t)=w(σ(t),t)[a(t)f(t)+g(t)((p-q)Kq/pp+qa(t)pK(p-q)/p)+m(t)]+t0tw1Δ(t,τ)[a(τ)f(τ)+g(τ)((p-q)Kq/pp+qa(τ)pK(p-q)/p)+m(τ)]Δτ.

Proof.

Define a function z(t) by z(t)=t0tk(t,τ)Δτ        for  allt𝕋κ, where k(t,τ)=w(t,τ)[f(τ)up(τ)+g(τ)uq(τ)+m(τ)]. Then z(t0)=0. As in the proof of Theorem 2.4, we easily obtain (2.15) and (2.16).

It follows from (2.34) that k(σ(t),t)=w(σ(t),t)[f(τ)up(τ)+g(τ)uq(τ)+m(τ)], and also k1Δ(t,τ)=w1Δ(t,τ)[f(τ)up(τ)+g(τ)uq(τ)+m(τ)]. Therefore, noting the condition (2.29), using Lemma 2.2 and combining (2.33)–(2.36), (2.15), and (2.16), we have zΔ(t)=k(σ(t),t)+t0tk1Δ(t,τ)Δτ=w(σ(t),t)[f(t)up(t)+g(t)uq(t)+m(t)]+t0tw1Δ(t,τ)[f(τ)up(τ)+g(τ)uq(τ)+m(τ)]Δτw(σ(t),t)[a(t)f(t)+g(t)((p-q)Kq/pp+qa(t)pK(p-q)/p)+m(t)+b(t)(f(t)+qg(t)pK(p-q)/p)z(t)]+t0tw1Δ(t,τ)[a(τ)f(τ)+g(τ)((p-q)Kq/pp+qa(τ)pK(p-q)/p)+m(τ)+b(τ)(f(τ)+qg(τ)pK(p-q)/p)z(τ)]Δτ[w(σ(t),t)b(t)(f(t)+qg(t)pK(p-q)/p)+t0tw1Δ(t,τ)b(τ)(f(τ)+qg(τ)pK(p-q)/p)Δτ]z(t)+w(σ(t),t)[a(t)f(t)+g(t)((p-q)Kq/pp+qa(t)pK(p-q)/p)+m(t)]+t0tw1Δ(t,τ)[a(τ)f(τ)+g(τ)((p-q)Kq/pp+qa(τ)pK(p-q)/p)+m(τ)]Δτ=A(t)z(t)+G(t)        for  allt𝕋κ, where A(t) and G(t) are defined as in (2.31) and (2.32), respectively. Therefore, using Lemma 2.3 and noting z(t0)=0, we get z(t)t0teA(t,σ(τ))G(τ)Δτ        for  allt𝕋κ.

It is easy to see that the desired inequality (2.30) follows from (2.15) and (2.38). This completes the proof of Theorem 2.11.

Remark 2.12.

Letting p=q=1, f(t)=0 in Theorem 2.11, we easily obtain [9, Theorem 3.10].

The following two corollaries are easily established by using Theorem 2.11.

Corollary 2.13.

Let 𝕋= and assume that u(t),a(t),b(t),f(t),g(t),m(t)C(+,+). If w(t,s) and its partial derivative (/t)w(t,s) are real-valued nonnegative continuous functions for t,s+ with st, then the inequality up(t)a(t)+b(t)0tw(t,s)[f(s)up(s)+g(s)uq(s)+m(s)]ds,t+ implies u(t){a(t)+b(t)0tG¯(s)exp(stA¯(τ)dτ)ds}1/p        for  anyK>0,t+, where A¯(t)=w(t,t)b(t)(f(t)+qg(t)pK(p-q)/p)+0ttw(t,s)b(s)(f(s)+qg(s)pK(p-q)/p)ds, and also G¯(t)=w(t,t)[a(t)f(t)+g(t)((p-q)Kq/pp+qa(t)pK(p-q)/p)+m(t)]+0ttw(t,s)[a(s)f(s)+g(s)((p-q)Kq/pp+qa(s)pK(p-q)/p)+m(s)]ds.

Remark 2.14.

Letting p=q=1, f(t)=0 in Corollary 2.13, we easily obtain [14, Theorem 1.4.3].

Corollary 2.15.

Let 𝕋= and assume that u(t),a(t),b(t),f(t),g(t) and m(t) are nonnegative functions defined for t0. If w(t,s) and Δ1w(t,s) are real-valued nonnegative functions for t,s0 with st, then the inequality up(t)a(t)+b(t)s=0t-1w(t,s)[f(s)up(s)+g(s)uq(s)+m(s)],t0, implies u(t){a(t)+b(t)s=0t-1G̃(s)τ=s+1t-1(1+Ã(τ))}1/p        for  anyK>0,t0, where Δ1w(t,s)=w(t+1,s)-w(t,s) for t,s0 with st, Ã(t)=w(t+1,t)b(t)(f(t)+qg(t)pK(p-q)/p)  +s=0t-1Δ1w(t,s)b(s)(f(s)+qg(s)pK(p-q)/p),        G̃(t)=w(t+1,t)[a(t)f(t)+g(t)((p-q)Kq/pp+qa(t)pK(p-q)/p)+m(t)]+s=0t-1Δ1w(t,s)[a(s)f(s)+g(s)((p-q)Kq/pp+qa(s)pK(p-q)/p)+m(s)].

Remark 2.16.

By letting p=q=1, f(t)=0 in Corollary 2.15, it is very easy to obtain [15, Theorem 1.3.4].

Corollary 2.17.

Suppose that u(t),a(t), and w(t,s) are defined as in Theorem 2.11, and let a(t) be nondecreasing for all t𝕋κ. If, for any ε>0, there exists a neighborhood Uof t, independent of τ[t0,σ(t)], such that for all sU, |uq(τ)[w(σ(t),τ)-w(s,τ)-w1Δ(t,τ)(σ(t)-s)]|ε|σ(t)-s|, then up(t)a(t)+t0tw(t,τ)uq(τ)Δτ        forallt𝕋κ implies u(t){1q[(K(p-q)+qa(t))eA¯̃(t,t0)-K(p-q)]}1/p        for  anyK>0,t𝕋κ, where A¯̃(t)=qpK(p-q)/p(w(σ(t),t)+t0tw1Δ(t,τ)Δτ).

Proof.

Letting b(t)=1,f(t)=0,g(t)=1, and m(t)=0 in Theorem 2.11, we obtain A(t)=qpK(p-q)/p(w(σ(t),t)+t0tw1Δ(t,τ)Δτ):=A¯̃(t), and also G(t)=1pK(p-q)/p{w(σ(t),t)[K(p-q)+qa(t)]+t0tw1Δ(t,τ)[K(p-q)+qa(τ)]Δτ}K(p-q)+qa(t)pK(p-q)/p{w(σ(t),t)+t0tw1Δ(t,τ)Δτ}=1q[K(p-q)+qa(t)]A¯̃(t)for  anyK>0,t𝕋κ, where the inequality holds because a(t) is nondecreasing for all t𝕋κ. Therefore, using Theorem 2.11 and noting (2.50) and (2.51), we easily have u(t){a(t)+t0teA(t,σ(τ))G(τ)Δτ}1/p{a(t)+1qt0teA¯̃(t,σ(τ))[K(p-q)+qa(τ)]A¯̃(τ)Δτ}1/p{a(t)+1q[K(p-q)+qa(t)]t0teA¯̃(t,σ(τ))A¯̃(τ)Δτ}1/p={a(t)+1q[K(p-q)+qa(t)],[eA¯̃(t,t0)-eA¯̃(t,t)]}1/p={1q[(K(p-q)+qa(t))eA¯̃(t,t0)-K(p-q)]}1/pfor  anyK>0,t𝕋κ. The proof of Corollary 2.17 is complete.

Remark 2.18.

In Corollary 2.17, letting w(t,s)=w(s),p=q=1, we immediately obtain [12, Theorem 3.1].

From the proof procedure of Theorem 2.11, we can obtain the following result.

Theorem 2.19.

Assume that u,a,b,f,gi,mCrd, u(t),a(t),b(t),f(t),gi(t), and m(t) are nonnegative, i=1,2,,n, and there exists a series of positive real numbers q1,q2,,qn such that pqi>0,i=1,2,,n. Let w(t,s) be defined as in Lemma 2.2 such that w(t,s)0 and w1Δ(t,s)0 for t,s𝕋 with st. If, for any ε>0, there exists a neighborhood U of t, independent of τ[t0,σ(t)], such that for all sU, |[w(σ(t),τ)-w(s,τ)-w1Δ(t,τ)(σ(t)-s)][f(τ)up(τ)+i=1ngi(τ)uqi(τ)+m(τ)]|ε|σ(t)-s|, then up(t)a(t)+b(t)t0tw(t,τ)[f(τ)up(τ)+i=1ngi(τ)uqi(τ)+m(τ)]Δτ,t𝕋κ implies u(t){a(t)+b(t)t0teA*(t,σ(τ))G*(τ)Δ(τ)}1/p        for  anyK>0,t𝕋κ, where A*(t)=w(σ(t),t)b(t)(f(t)+i=1nqigi(t)pK(p-qi)/p)+t0tw1Δ(t,τ)b(τ)(f(τ)+i=1nqigi(τ)pK(p-qi)/p)Δτ,G*(t)=w(σ(t),t)[a(t)f(t)+i=1ngi(t)((p-qi)Kqi/pp+qia(t)pK(p-qi)/p)+m(t)]+t0tw1Δ(t,τ)[a(τ)f(τ)+i=1ngi(τ)((p-qi)Kqi/pp+qia(τ)pK(p-qi)/p)+m(τ)]Δτ.

Remark 2.20.

Using our main results, we can obtain many dynamic inequalities for some peculiar time scales. Due to limited space, their statements are omitted here.

3. An Application

In this section, we present an application of Corollary 2.6 to obtain the explicit estimates on the solutions of a dynamic equation on time scales.

Example 3.1.

Consider the dynamic equation (up(t))Δ=H(t,u(t)),u(t0)=C,t𝕋κ, where p and C are constants, p>0, and H:𝕋κ× is a continuous function.

Assume that |H(t,u(t))|g(t)|uq(t)|, where g(t)Crd, g(t) is nonnegative, and 0<qp is a constant. If u(t) is a solution of (3.1), then |u(t)|{|C|p+t0teB̂(t,σ(τ))J(τ)Δτ}1/p        for  anyK>0,t𝕋κ, where B̂(t) is defined as in (2.21), and J(t)=g(t)(p-qpKq/p+q|C|ppK(p-q)/p)        for  allt𝕋κ.

In fact, the solution u(t) of (3.1) satisfies the following equivalent equation: up(t)=Cp+t0tH(τ,u(τ))Δτ,      t𝕋κ. Using the assumption (3.2), we have |u(t)|p|C|p+t0tg(τ)|u(τ)|qΔτ,t𝕋κ. Now a suitable application of Corollary 2.6 to (3.6) yields (3.3).

Acknowledgments

This work is supported by the National Natural Science Foundation of China (10971018, 10971139), the Natural Science Foundation of Shandong Province (Y2009A05), China Postdoctoral Science Foundation Funded Project (20080440633), Shanghai Postdoctoral Scientific Program (09R21415200), the Project of Science and Technology of the Education Department of Shandong Province (J08LI52), and the Doctoral Foundation of Binzhou University (2006Y01).

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