# Bounds for Certain Nonlinear Dynamic Inequalities on Time Scales

Authors: Wei Nian Li; Maoan Han
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

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.


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## Body

## 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.
Leta≥0.  Then
(2.1)aq/p≤(qpKq-p/pa+p-qpKq/p)foranyK>0.Proof.
Ifa=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(2.2)f(K)=qpKq-p/pa+p-qpKq/p,K>0,
we have
(2.3)f′(K)=q(p-q)p2Kq-2p/p(K-a).
It is easy to see that
(2.4)f′(K)≥0,K>a,f′(K)=0,K=a,f′(K)≤0,0<K<a.
Therefore,
(2.5)f(K)≥f(a)=aq/p.
The proof of Lemma 2.1 is complete.Lemma 2.2 (see [6]).
Lett0∈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
(2.6)|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
(2.7)υ(t):=∫t0tw(t,τ)Δτ
implies
(2.8)υΔ(t)=∫t0tw1Δ(t,τ)Δτ+w(σ(t),t).Lemma 2.3 (Comparison theorem [6]).
Supposeu,b∈Crd, a∈ℛ+. Then
(2.9)uΔ(t)≤a(t)u(t)+b(t)forallt∈𝕋κ
implies
(2.10)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 thatu,a,b,g,h∈Crd, and u(t),a(t),b(t),g(t) and h(t) are nonnegative. Then
(E1)up(t)≤a(t)+b(t)∫t0t[g(τ)uq(τ)+h(τ)]Δτforallt∈𝕋κ
implies
(2.11)u(t)≤{a(t)+b(t)∫t0teB(t,σ(τ))F(τ)Δτ}1/pforanyK>0,t∈𝕋κ,
where
(2.12)F(t)=g(t)(p-qpKq/p+qa(t)pK(p-q)/p)+h(t),
and also
(2.13)B(t)=qb(t)g(t)pK(p-q)/pforallt∈𝕋κ.Proof.
Obviously, ift=t0, then the inequality (2.11) holds. Therefore, in the next proof, we always assume that t>t0,t∈𝕋κ.
Define a functionz(t) by
(2.14)z(t)=∫t0t[g(τ)uq(τ)+h(τ)]Δτ.
Then (E1) can be restated as
(2.15)up(t)≤a(t)+b(t)z(t).
Using Lemma 2.1, from (2.15), for any K>0, we easily obtain
(2.16)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
(2.17)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
(2.18)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 lettingp=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 Theorem2.4, we immediately obtain the following result.Corollary 2.6.
Assume thatu,g∈Crd, and u(t) and g(t) are nonnegative. If α>0 is a constant, then
(E′1)up(t)≤α+∫t0tg(τ)uq(τ)Δτforallt∈𝕋κ
implies
(2.19)u(t)≤{α+∫t0teB̂(t,σ(τ))F̂(τ)Δτ}1/pforanyK>0,t∈𝕋κ,
where
(2.20)F̂(t)=g(t)(p-qpKq/p+qαpK(p-q)/p),(2.21)B̂(t)=qg(t)pK(p-q)/pforallt∈𝕋κ.Remark 2.7.
The result of Theorem2.4 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.
Let𝕋=ℝ and assume that u(t),a(t),b(t),g(t),h(t)∈C(ℝ+,ℝ+). Then the inequality
(2.22)up(t)≤a(t)+b(t)∫0t[g(s)uq(s)+h(s)]ds,t∈ℝ+
implies
(2.23)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.
Let𝕋=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
(2.24)up(t)≤a(t)+b(t)∑s=0t-1[g(s)uq(s)+h(s)],t∈ℕ0
implies
(2.25)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 Theorem2.4 carefully, we can obtain the following result.Theorem 2.10.
Assume thatu,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
(E′′1)up(t)≤a(t)+b(t)∫t0t[∑i=1ngi(τ)uqi(τ)+h(τ)]Δτforallt∈𝕋κ
implies
(2.26)u(t)≤{a(t)+b(t)∫t0teB*(t,σ(τ))F*(τ)Δτ}1/pforanyK>0,t∈𝕋κ,
where
(2.27)F*(t)=∑i=1ngi(t)(p-qipKqi/p+qia(t)pK(p-qi)/p)+h(t),(2.28)B*(t)=∑i=1nqib(t)gi(t)pK(p-qi)/pforallt∈𝕋κ.Theorem 2.11.
Assume thatu,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,
(2.29)|[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
(E2)up(t)≤a(t)+b(t)∫t0tw(t,τ)[f(τ)up(τ)+g(τ)uq(τ)+m(τ)]Δτ,t∈𝕋κ
implies
(2.30)u(t)≤{a(t)+b(t)∫t0teA(t,σ(τ))G(τ)Δ(τ)}1/pforanyK>0,t∈𝕋κ,
where
(2.31)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
(2.32)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 functionz(t) by
(2.33)z(t)=∫t0tk(t,τ)Δτforallt∈𝕋κ,
where
(2.34)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
(2.35)k(σ(t),t)=w(σ(t),t)[f(τ)up(τ)+g(τ)uq(τ)+m(τ)],
and also
(2.36)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
(2.37)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
(2.38)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.
Lettingp=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 Theorem2.11.Corollary 2.13.
Let𝕋=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
(2.39)up(t)≤a(t)+b(t)∫0tw(t,s)[f(s)up(s)+g(s)uq(s)+m(s)]ds,t∈ℝ+
implies
(2.40)u(t)≤{a(t)+b(t)∫0tG¯(s)exp(∫stA¯(τ)dτ)ds}1/pforanyK>0,t∈ℝ+,
where
(2.41)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
(2.42)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.
Lettingp=q=1, f(t)=0 in Corollary 2.13, we easily obtain [14, Theorem 1.4.3].Corollary 2.15.
Let𝕋=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
(2.43)up(t)≤a(t)+b(t)∑s=0t-1w(t,s)[f(s)up(s)+g(s)uq(s)+m(s)],t∈ℕ0,
implies
(2.44)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,
(2.45)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),
(2.46)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 lettingp=q=1, f(t)=0 in Corollary 2.15, it is very easy to obtain [15, Theorem 1.3.4].Corollary 2.17.
Suppose thatu(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,
(2.47)|uq(τ)[w(σ(t),τ)-w(s,τ)-w1Δ(t,τ)(σ(t)-s)]|≤ε|σ(t)-s|,
then
(E′2)up(t)≤a(t)+∫t0tw(t,τ)uq(τ)Δτforallt∈𝕋κ
implies
(2.48)u(t)≤{1q[(K(p-q)+qa(t))eA¯̃(t,t0)-K(p-q)]}1/pforanyK>0,t∈𝕋κ,
where
(2.49)A¯̃(t)=qpK(p-q)/p(w(σ(t),t)+∫t0tw1Δ(t,τ)Δτ).Proof.
Lettingb(t)=1,f(t)=0,g(t)=1, and m(t)=0 in Theorem 2.11, we obtain
(2.50)A(t)=qpK(p-q)/p(w(σ(t),t)+∫t0tw1Δ(t,τ)Δτ):=A¯̃(t),
and also
(2.51)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
(2.52)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 Corollary2.17, letting w(t,s)=w(s),p=q=1, we immediately obtain [12, Theorem 3.1].From the proof procedure of Theorem2.11, we can obtain the following result.Theorem 2.19.
Assume thatu,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,
(2.53)|[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
(E′′2)up(t)≤a(t)+b(t)∫t0tw(t,τ)[f(τ)up(τ)+∑i=1ngi(τ)uqi(τ)+m(τ)]Δτ,t∈𝕋κ
implies
(2.54)u(t)≤{a(t)+b(t)∫t0teA*(t,σ(τ))G*(τ)Δ(τ)}1/pforanyK>0,t∈𝕋κ,
where
(2.55)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,(2.56)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 Corollary2.6 to obtain the explicit estimates on the solutions of a dynamic equation on time scales.Example 3.1.
Consider the dynamic equation(3.1)(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(3.2)|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
(3.3)|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
(3.4)J(t)=g(t)(p-qpKq/p+q|C|ppK(p-q)/p)forallt∈𝕋κ.
In fact, the solutionu(t) of (3.1) satisfies the following equivalent equation:
(3.5)up(t)=Cp+∫t0tH(τ,u(τ))Δτ,t∈𝕋κ.
Using the assumption (3.2), we have
(3.6)|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).



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