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 [1] first introduced the calculus on time scales. Motivated by the paper [1], 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 [2–7] and the references cited therein. At the same time, a few papers [8–13] 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, ℝ denotes the set of real numbers, ℝ+=[0,∞), ℤ denotes the set of integers, ℕ0={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, ℛ denotes the set of all regressive and rd-continuous functions, and ℛ+={p∈ℛ:1+μ(t)p(t)>0forallt∈𝕋}. 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 p≥q>0, p and q are real constants, and t≥t0,t0∈𝕋κ.

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

Lemma 2.1.

Let a≥0. Then
aq/p≤(qpKq-p/pa+p-qpKq/p)foranyK>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 t0∈𝕋κ and w:𝕋×𝕋κ→ℝ be continuous at (t,t), t∈𝕋κ 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|foralls∈U,
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).

Suppose u,b∈Crd, a∈ℛ+. Then
uΔ(t)≤a(t)u(t)+b(t)forallt∈𝕋κ
implies
u(t)≤u(t0)ea(t,t0)+∫t0tea(t,σ(τ))b(τ)Δτforallt∈𝕋κ.

Next, we establish our main results.

Theorem 2.4.

Assume that u,a,b,g,h∈Crd, 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)]Δτforallt∈𝕋κ
implies
u(t)≤{a(t)+b(t)∫t0teB(t,σ(τ))F(τ)Δτ}1/pforanyK>0,t∈𝕋κ,
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)/pforallt∈𝕋κ.

Proof.

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

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/p≤p-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,foranyK>0,t∈𝕋κ.

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,g∈Crd, and u(t) and g(t) are nonnegative. If α>0 is a constant, then
up(t)≤α+∫t0tg(τ)uq(τ)Δτforallt∈𝕋κ
implies
u(t)≤{α+∫t0teB̂(t,σ(τ))F̂(τ)Δτ}1/pforanyK>0,t∈𝕋κ,
where
F̂(t)=g(t)(p-qpKq/p+qαpK(p-q)/p),B̂(t)=qg(t)pK(p-q)/pforallt∈𝕋κ.

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=ℝ and 𝕋=ℤ, respectively, we have the following two results.

Corollary 2.8.

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

Corollary 2.9.

Let 𝕋=ℤ and assume that u(t),a(t),b(t),g(t), and h(t) are nonnegative functions defined for t∈ℕ0. Then the inequality
up(t)≤a(t)+b(t)∑s=0t-1[g(s)uq(s)+h(s)],t∈ℕ0
implies
u(t)≤[a(t)+b(t)∑θ=0t-1F(θ)∏s=θ+1t-1(1+B(s))]1/pforanyK>0,t∈ℕ0,
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,h∈Crd, 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 p≥qi>0,i=1,2,…,n, then
up(t)≤a(t)+b(t)∫t0t[∑i=1ngi(τ)uqi(τ)+h(τ)]Δτforallt∈𝕋κ
implies
u(t)≤{a(t)+b(t)∫t0teB*(t,σ(τ))F*(τ)Δτ}1/pforanyK>0,t∈𝕋κ,
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)/pforallt∈𝕋κ.

Theorem 2.11.

Assume that u,a,b,f,g,m∈Crd, 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,s∈𝕋 with s≤t. If, for any ε>0, there exists a neighborhood U of t, independent of τ∈[t0,σ(t)], such that for all s∈U,
|[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,t∈𝕋κ
implies
u(t)≤{a(t)+b(t)∫t0teA(t,σ(τ))G(τ)Δ(τ)}1/pforanyK>0,t∈𝕋κ,
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)Δτforallt∈𝕋κ,
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)]Δτ≤w(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)forallt∈𝕋κ,
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)Δτforallt∈𝕋κ.

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=ℝ and assume that u(t),a(t),b(t),f(t),g(t),m(t)∈C(ℝ+,ℝ+). If w(t,s) and its partial derivative (∂/∂t)w(t,s) are real-valued nonnegative continuous functions for t,s∈ℝ+ with s≤t, then the inequality
up(t)≤a(t)+b(t)∫0tw(t,s)[f(s)up(s)+g(s)uq(s)+m(s)]ds,t∈ℝ+
implies
u(t)≤{a(t)+b(t)∫0tG¯(s)exp(∫stA¯(τ)dτ)ds}1/pforanyK>0,t∈ℝ+,
where
A¯(t)=w(t,t)b(t)(f(t)+qg(t)pK(p-q)/p)+∫0t∂∂tw(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)]+∫0t∂∂tw(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=ℤ and assume that u(t),a(t),b(t),f(t),g(t) and m(t) are nonnegative functions defined for t∈ℕ0. If w(t,s) and Δ1w(t,s) are real-valued nonnegative functions for t,s∈ℕ0 with s≤t, then the inequality
up(t)≤a(t)+b(t)∑s=0t-1w(t,s)[f(s)up(s)+g(s)uq(s)+m(s)],t∈ℕ0,
implies
u(t)≤{a(t)+b(t)∑s=0t-1G̃(s)∏τ=s+1t-1(1+Ã(τ))}1/pforanyK>0,t∈ℕ0,
where Δ1w(t,s)=w(t+1,s)-w(t,s) for t,s∈ℕ0 with s≤t,
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 t∈𝕋κ. If, for any ε>0, there exists a neighborhood Uof t, independent of τ∈[t0,σ(t)], such that for all s∈U,
|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)Δτforallt∈𝕋κ
implies
u(t)≤{1q[(K(p-q)+qa(t))eA¯̃(t,t0)-K(p-q)]}1/pforanyK>0,t∈𝕋κ,
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)foranyK>0,t∈𝕋κ,
where the inequality holds because a(t) is nondecreasing for all t∈𝕋κ. 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)+1q∫t0teA¯̃(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/pforanyK>0,t∈𝕋κ.
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,m∈Crd, 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 p≥qi>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,s∈𝕋 with s≤t. If, for any ε>0, there exists a neighborhood U of t, independent of τ∈[t0,σ(t)], such that for all s∈U,
|[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,t∈𝕋κ
implies
u(t)≤{a(t)+b(t)∫t0teA*(t,σ(τ))G*(τ)Δ(τ)}1/pforanyK>0,t∈𝕋κ,
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,t∈𝕋κ,
where p and C are constants, p>0, and H:𝕋κ×ℝ→ℝ is a continuous function.

Assume that
|H(t,u(t))|≤g(t)|uq(t)|,
where g(t)∈Crd, g(t) is nonnegative, and 0<q≤p 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/pforanyK>0,t∈𝕋κ,
where B̂(t) is defined as in (2.21), and
J(t)=g(t)(p-qpKq/p+q|C|ppK(p-q)/p)forallt∈𝕋κ.

In fact, the solution u(t) of (3.1) satisfies the following equivalent equation:
up(t)=Cp+∫t0tH(τ,u(τ))Δτ,t∈𝕋κ.
Using the assumption (3.2), we have
|u(t)|p≤|C|p+∫t0tg(τ)|u(τ)|qΔτ,t∈𝕋κ.
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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