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volume 6, issue 1, article 6, 2005.

Received 13 December, 2004;

accepted 04 January, 2005.

Communicated by:D. Hinton

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Journal of Inequalities in Pure and Applied Mathematics

PACHPATTE INEQUALITIES ON TIME SCALES

ELVAN AKIN-BOHNER, MARTIN BOHNER AND FAYSAL AKIN

University of Missouri–Rolla Department of Mathematics Rolla, MO 65409-0020, USA EMail:akine@umr.edu EMail:bohner@umr.edu Dicle University

Department of Mathematics

Diyarbakir, Turkey EMail:akinff@dicle.edu.tr

2000c Victoria University ISSN (electronic): 1443-5756 239-04

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Pachpatte Inequalities on Time Scales

Elvan Akin-Bohner, Martin Bohner and Faysal Akin

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J. Ineq. Pure and Appl. Math. 6(1) Art. 6, 2005

Abstract

In the study of dynamic equations on time scales we deal with certain dynamic inequalities which provide explicit bounds on the unknown functions and their derivatives. Most of the inequalities presented are of comparison or Gronwall type and, more specifically, of Pachpatte type.

2000 Mathematics Subject Classification:Primary: 34A40; Secondary: 39A10.

Key words: Time scales, Pachpatte inequalities, Dynamic equations and inequali- ties.

1. Introduction

In this paper we present a number of dynamic inequalities that are essentially based on Gronwall’s inequality. Most of these inequalities are also known as being of Pachpatte type. For a summary of related continuous inequalities, the monograph [4] by Pachpatte is an authoritative source. For the corresponding discrete inequalities, we refer the interested reader to the excellent monograph [5] by Pachpatte.

Our dynamic inequalities unify and extend the (linear) inequalities presented in the first chapters of [4, 5]. The setup of this paper is as follows: In Section 2we give some preliminary results with respect to the calculus on time scales, which can also be found in the books by Bohner and Peterson [2,3]. Some basic dynamic inequalities are given as established in the paper by Agarwal, Bohner, and Peterson [1]. The remaining sections deal with our dynamic inequalities.

Note that they contain differential and difference inequalities as special cases, and they also contain all other dynamic inequalities, such as, for example, q- difference inequalities, as special cases.

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2. Calculus On Time Scales

A time scale Tis an arbitrary nonempty closed subset of the real numbers R. We define the forward jump operatorσ onTby

σ(t) := inf{s ∈T: s > t} ∈T for all t∈T.

In this definition we putσ(∅) = supT, where∅is the empty set. Ifσ(t) > t, then we say that t is right-scattered. Ifσ(t) = t andt < supT, then we say that t is right-dense. The backward jump operator and left-scattered and left- dense points are defined in a similar way. The graininess µ : T → [0,∞)is defined byµ(t) :=σ(t)−t. The setT^κ is derived fromTas follows: If Thas a left-scattered maximum m, then T^κ = T− {m}; otherwise, T^κ = T. For f : T → Randt ∈ T^κ, we definef^∆(t)to be the number (provided it exists) such that given anyε >0, there is a neighorhoodU oftwith

[f(σ(t))−f(s)]−f^∆(t)[σ(t)−s]

≤ε|σ(t)−s| for all s∈U.

We callf^∆(t)the delta derivative off att, andf^∆is the usual derivativef⁰ if T = Rand the usual forward difference ∆f (defined by∆f(t) = f(t+ 1)− f(t)) ifT=Z.

Theorem 2.1. Assumef, g :T → Rand lett ∈ T^κ. Then we have the follow- ing:

(i) Iff is differentiable att, thenf is continuous att.

(ii) If f is continuous attandtis right-scattered, thenf is differentiable att with

f^∆(t) = f(σ(t))−f(t)

µ(t) .

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(iii) Iff is differentiable attandtis right-dense, then f^∆(t) = lim

s→t

f(t)−f(s) t−s . (iv) Iff is differentiable att, then

f^σ(t) = f(t) +µ(t)f^∆(t), where f^σ :=f◦σ.

(v) Iff andgare differentiable att, then so isf gwith (f g)^∆(t) =f^∆(t)g(t) +f^σ(t)g^∆(t).

We say that f : T → R is rd-continuous providedf is continuous at each right-dense point ofTand has a finite left-sided limit at each left-dense point of T. The set of rd-continuous functions will be denoted in this paper byC_rd, and the set of functions that are differentiable and whose derivative is rd-continuous is denoted by C¹_rd. A function F : T → R is called an antiderivative of f : T →RprovidedF^∆(t) =f(t)holds for allt ∈T^κ. In this case we define the integral off by

Z t

s

f(τ)∆τ =F(t)−F(s) for s, t∈T.

We say thatp : T → Ris regressive provided 1 +µ(t)p(t) 6= 0for allt ∈ T. We denote byRthe set of all regressive and rd-continuous functions. We define

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the set of all positively regressive functions byR⁺ ={p∈ R : 1 +µ(t)p(t)>

0for allt ∈T}. Ifp, q ∈ R, then we define p⊕q =p+q+µpq, q =− q

1 +µq, and p q=p⊕( q).

If p : T → R is rd-continuous and regressive, then the exponential function ep(·, t0)is for each fixedt0 ∈Tthe unique solution of the initial value problem

x^∆=p(t)x, x(t₀) = 1 on T.

We use the following four theorems which are proved in Bohner and Peterson [2].

Theorem 2.2. Ifp, q ∈ R, then (i) e_p(t, t)≡1ande₀(t, s)≡1;

(ii) e_p(σ(t), s) = (1 +µ(t)p(t))e_p(t, s);

(iii) _e ¹

p(t,s) =e_p(t, s) =e_p(s, t);

(iv) ^e_e^p^(t,s)

q(t,s) =ep q(t, s);

(v) e_p(t, s)e_q(t, s) = ep⊕q(t, s);

(vi) ifp∈ R⁺, thene_p(t, t₀)>0for allt ∈T.

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Example 2.1. In order to allow for a comparison with the continuous Pachpatte inequalities given in [4], we note that, if T = R, the exponential function is given by

e_p(t, s) =e^R^s^t^p(τ)dτ, e_α(t, s) = e^α(t−s), e_α(t,0) =e^αt

fors, t ∈R, whereα∈Ris a constant andp:R→Ris a continuous function.

To compare with the discrete Pachpatte inequalities given in [5], we also give the exponential function forT=Zas

e_p(t, s) =

t−1

Y

τ=s

[1 +p(τ)], e_α(t, s) = (1 +α)^t−s, e_α(t,0) = (1 +α)^t for s, t ∈ Z with s < t, where α 6= −1 is a constant and p : Z → R is a sequence satisfyingp(t) 6= −1for allt ∈ Z. Further examples of exponential functions can be found in [2, Section 2.3].

Theorem 2.3. Ifp∈ Randa, b, c∈T, then Z b

a

p(t)e_p(c, σ(t))∆t=e_p(c, a)−e_p(c, b).

Theorem 2.4. Ifa, b, c ∈Tandf ∈Crd such thatf(t)≥ 0for alla ≤t < b, then

Z b

a

f(t)∆t≥0.

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Theorem 2.5. Lett₀ ∈T^κ and assumek : T×T →Ris continuous at(t, t), wheret∈T^κwitht > t0. Also assume thatk(t,·)is rd-continuous on[t0, σ(t)].

Suppose that for eachε >0there exists a neighborhoodU oft, independent of τ ∈[t₀, σ(t)], such that

k(σ(t), τ)−k(s, τ)−k^∆(t, τ)(σ(t)−s)

≤ε|σ(t)−s| for all s∈U, wherek^∆denotes the derivative ofkwith respect to the first variable. Then

g(t) :=

Z t

t0

k(t, τ)∆τ implies g^∆(t) = Z t

t0

k^∆(t, τ)∆τ +k(σ(t), t).

The next four results are proved by Agarwal, Bohner and Peterson [1]. For convenience of notation we let throughout

t₀ ∈T, T⁰ = [t₀,∞)∩T, and T⁻0 = (−∞, t₀]∩T. Also, for a functionb ∈C_rd we write

b ≥0 if b(t)≥0for allt∈T.

Theorem 2.6 (Comparison Theorem). Supposeu, b∈C_rdanda∈ R⁺. Then u^∆(t)≤a(t)u(t) +b(t) for all t∈T⁰

implies

u(t)≤u(t₀)e_a(t, t₀) + Z t

t

e_a(t, σ(τ))b(τ)∆τ for all t∈T0.

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Theorem 2.7 (Gronwall’s Inequality). Supposeu, a, b ∈C_rdandb ≥0. Then u(t)≤a(t) +

Z t

t0

b(τ)u(τ)∆τ for all t ∈T0

implies

u(t)≤a(t) + Z t

t0

a(τ)b(τ)e_b(t, σ(τ))∆τ for all t∈T0.

Remark 1. In the next section we show that Gronwall’s inequality can be stated in different forms (see Theorem 3.1, Theorem3.5, Theorem3.8, and Theorem 3.10).

The next two results follow from Theorem 2.7 with a = 0 and a = u0, respectively.

Corollary 2.8. Supposeu, b∈C_rdandb ≥0. Then u(t)≤

Z t

t0

u(τ)b(τ)∆τ for all t∈T0

implies

u(t)≤0 for all t∈T0.

Corollary 2.9. Supposeu, b∈C_rd,u₀ ∈R, andb≥0. Then u(t)≤u₀+

Z t

t0

implies

u(t)≤u₀e_b(t, t₀) for all t∈T0.

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The continuous version [4, Th. 1.2.2] of Corollary 2.9 was first proved by Bellman, while the corresponding discrete version [5, Th. 1.2.2] is due to Sugiyama.

The remaining results in this section will be needed later on in this paper.

Corollary 2.10. Ifp∈ R⁺andp(t)≤q(t)for allt∈T, then e_p(t, t₀)≤e_q(t, t₀) for all t∈T0. Proof. Letu(t) =ep(t, t0). Then

u^∆(t) =p(t)u(t)≤q(t)u(t).

Now note thatq∈ R⁺, so using Theorem2.6witha =qandb= 0, we obtain e_p(t, t₀) =u(t)≤u(t₀)e_q(t, t₀) =e_q(t, t₀)

for allt ∈T0.

Remark 2. The following statements hold:

(i) Ifp≥0, thene_p(t, t₀)≥e₀(t, t₀) = 1by Corollary2.10and Theorem2.2.

Thereforee_p(t, t₀)≤1.

(ii) Ifp≥0, thene_p(·, t₀)is nondecreasing sincee^∆_p(t, t₀) = p(t)e_p(t, t₀)≥0.

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3. Dynamic Inequalities

Note that whenp= 1andq= 0in Theorem3.1below, then we obtain Theorem 2.7. ForT = R, see [4, Th. 1.3.4]. For T = Z, we refer to [5, Th. 1.3.1 and Th. 1.2.3]. The proof of Theorem 3.1below is similar to the proof of Theorem 2.7and hence is omitted.

Theorem 3.1. Supposeu, a, b, p, q∈C_rdandb, p≥0. Then u(t)≤a(t) +p(t)

Z t

t0

[b(τ)u(τ) +q(τ)]∆τ for all t ∈T0

implies

u(t)≤a(t) +p(t) Z t

t0

[a(τ)b(τ) +q(τ)]e_bp(t, σ(τ))∆τ for all t∈T0. The next result follows from Theorem3.1witha=q = 0.

Corollary 3.2. Supposeu, b, p∈C_rd andb, p≥0. Then u(t)≤p(t)

Z t

t0

u(τ)b(τ)∆τ for all t∈T0

implies

u(t)≤0 for all t∈T0. Remark 3. The following statements hold:

(i) Ifp= 1in Corollary3.2, then we get Corollary2.8.

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(ii) Ifq = 0in Theorem3.1andais nondecreasing onT, then u(t)≤a(t) +p(t)

Z t

t0

b(τ)u(τ)∆τ for all t∈T0

implies

u(t)≤a(t)

1 +p(t) Z t

t0

b(τ)ebp(t, σ(τ))∆τ

for all t∈T⁰. For the casesT = R and T = Z, see [4, Th. 1.3.3] and [5, Th. 1.2.4], respectively.

The next result follows from Theorem 3.1. While the continuous version [4, Th. 1.5.1] of Theorem 3.3below is due to Gamidov, its discrete version [5, Th. 1.3.2] has been established by Pachpatte.

Theorem 3.3. Suppose u, a, b_i, p_i ∈ C_rd and u, b_i, p := max1≤j≤np_j ≥ 0for 1≤i≤n. Then

u(t)≤a(t) +

n

X

i=1

p_i(t) Z t

t0

b_i(τ)u(τ)∆τ for all t∈T0

implies withb :=Pn i=1b_i u(t)≤a(t) +p(t)

Z t

t0

a(τ)b(τ)e_bp(t, σ(τ))∆τ for all t∈T0.

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The comparison theorem motivates us to consider the following result whose proof is similar to that of Theorem2.6.

Theorem 3.4 (Comparison Theorem). Letu, b∈C_rdanda∈ R⁺. Then u^∆(t)≤ −a(t)u^σ(t) +b(t) for all t∈T0

implies

u(t)≤u(t₀)e a(t, t₀) + Z t

t0

b(τ)e a(t, τ)∆τ for all t ∈T0, and

u^∆(t)≤ −a(t)u^σ(t) +b(t) for all t∈T⁻0

implies

u(t)≥u(t₀)e a(t, t₀) + Z t

t0

b(τ)e a(t, τ)∆τ for all t ∈T⁻0. Proof. We calculate

[ue_a(·, t₀)]^∆(t) =u^∆(t)e_a(t, t₀) +u^σ(t)a(t)e_a(t, t₀)

=

u^∆(t) +a(t)u^σ(t)

e_a(t, t₀)

≤b(t)e_a(t, t₀) for allt ∈T0 so that

u(t)e_a(t, t₀)−u(t₀)e_a(t₀, t₀)≤ Z t

t0

e_a(τ, t₀)b(τ)∆τ

for allt∈T0, and hence the first claim follows. For the second claim, note that the latter inequality is reversed ift∈T⁻0.

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For the continuous and discrete versions of the following three theorems, we refer the reader to [4, Th. 1.3.4, Th. 1.3.3, and Th. 1.3.5] and [5, Th. 1.2.5, Th. 1.2.6, and Th. 1.2.8], respectively.

Theorem 3.5. Supposeu, b, p, q∈C_rdandb, p ≥0. Then u(t)≤a(t) +p(t)

Z t0

t

[b(τ)u^σ(τ) +q(τ)] ∆τ for all t∈T⁻0

implies

u(t)≤a(t) +p(t) Z t0

t

[b(τ)a^σ(τ) +q(τ)]e (bp^σ)(t, τ)∆τ for all t∈T⁻0. Proof. Definez(t) :=−Rt0

t [b(τ)u^σ(τ) +q(τ)] ∆τ. Then for allt ∈T⁻0

z^∆(t) =b(t)u^σ(t) +q(t)

≤b(t) [a^σ(t)−p^σ(t)z^σ(t)] +q(t)

=−b(t)p^σ(t)z^σ(t) +b(t)a^σ(t) +q(t).

Sinceb, p≥0, we havebp^σ ∈ R⁺, and we may apply Theorem3.4to obtain z(t)≥z(t₀)e (bp^σ)(t, t₀) +

Z t

t0

e (bp^σ)(t, τ) [b(τ)a^σ(τ) +q(τ)] ∆τ

=− Z t0

t

e (bp^σ)(t, τ) [b(τ)a^σ(τ) +q(τ)] ∆τ

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for allt ∈T⁻0, and therefore u(t)≤a(t)−p(t)z(t)

≤a(t) +p(t) Z t0

t

e (bp^σ)(t, τ) [b(τ)a^σ(τ) +q(τ)] ∆τ for allt ∈T⁻0.

Theorem 3.6. Supposeu, b∈C_rd,b≥0, anda∈C¹_rd. Then u(t)≤a(t) +

Z t

t0

implies

u(t)≤a(t₀)e_b(t, t₀) + Z t

t0

a^∆(τ)e_b(t, σ(τ))∆τ for all t ∈T0. Proof. Definez(t) := a(t) +Rt

t0b(τ)u(τ)∆τ. Then we obtainz^∆(t)≤a^∆(t) + b(t)z(t). Applying Theorem2.6completes the proof.

Theorem 3.7. Supposeφ, u, b, p∈C_rdandb, p ≥0. Then u(t)≥φ(s)−p(t)

Z t

s

b(τ)φ^σ(τ)∆τ for all s, t ∈T, s≤t implies

u(t)≥φ(s)e (p(t)b)(t, s) for all s, t∈T, s≤t.

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Proof. Fixt₀ ∈T. Then φ(t)≤u(t₀) +p(t₀)

Z t0

t

b(τ)φ^σ(τ)∆τ for all t∈T⁻0. By Theorem3.5, we find

φ(t)≤u(t₀) +p(t₀) Z t0

t

b(τ)u(t₀)e (bp(t₀))(t, τ)∆τ

=u(t₀) +u(t₀) Z t0

t

b(τ)p(t₀)e_bp(t₀₎(τ, t)∆τ

=u(t₀) +u(t₀)

e_bp(t₀₎(t₀, t)−1

=u(t₀)e_bp(t₀₎(t₀, t) for allt ∈T⁻0 and thus

u(t₀)≥φ(t)e (bp(t₀))(t₀, t) for all t∈T⁻0. Sincet₀ ∈Twas arbitrary, the claim follows.

Remark 4. The following statements hold:

(i) The continuous version of Theorem3.7is due to Gollwitzer.

(ii) WhenT=R,

e (p(t)b)(t, s) = e^−p(t)

Rt sb(τ)dτ

,

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and whenT=Z,

e (p(t)b)(t, s) =

t−1

Y

τ=s

[1 +p(t)b(τ)]⁻¹.

(iii) Ifp= 1in Theorem3.7, then we obtainu(t)≥φ(s)e b(t, s).

The following Volterra type inequality reduces to Theorem2.7ifk =p= 1 andq = 0. ForT= R, it is due to Norbury and Stuart and can be found in [4, Th. 1.4.3]. ForT=Z, see [5, Th. 1.3.4 and Th. 1.3.3].

Theorem 3.8. Suppose u, a, b, p, q ∈ C_rd and u, b, p, q ≥ 0. Let k(t, s) be defined as in Theorem2.5such thatk(σ(t), t)≥0andk^∆(t, s)≥0fors, t ∈T withs≤t. Then

t0

k(t, τ) [b(τ)u(τ) +q(τ)] ∆τ for all t∈T0

implies

t0

¯b(τ)e_¯_a(t, σ(τ))∆τ for all t∈T0, where

¯

a(t) =k(σ(t), t)b(t)p(t) + Z t

t0

k^∆(t, τ)b(τ)p(τ)∆τ

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and

¯b(t) =k(σ(t), t) [a(t)b(t) +q(t)] + Z t

t0

k^∆(t, τ) [a(τ)b(τ) +q(τ)] ∆τ.

Proof. Definez(t) :=Rt

t0k(t, τ) [b(τ)u(τ) +q(τ)] ∆τ. Then for allt∈T0

z^∆(t) = k(σ(t), t) [b(t)u(t) +q(t)] + Z t

t0

k^∆(t, τ) [b(τ)u(τ) +q(τ)] ∆τ

≤

k(σ(t), t)b(t)p(t) + Z t

t0

k^∆(t, τ)b(τ)p(τ)∆τ

z(t)

+k(σ(t), t) [a(t)b(t) +q(t)] + Z t

t0

k^∆(t, τ) [a(τ)b(τ) +q(τ)] ∆τ

= ¯a(t)z(t) + ¯b(t).

In view of¯a∈ R⁺, we may apply Theorem2.6to obtain z(t)≤z(t0)e¯a(t, t0) +

Z t

t0

e¯a(t, σ(τ))¯b(τ)∆τ = Z t

t0

e¯a(t, σ(τ))¯b(τ)∆τ for all t ∈ T⁰. Since u(t) ≤ a(t) +p(t)z(t)holds for all t ∈ T⁰, the claim follows.

Corollary 3.9. In addition to the assumptions of Theorem 3.8withp = b = 1 andq= 0, suppose thatais nondecreasing. Then

u(t)≤a(t) + Z t

k(t, τ)u(τ)∆τ for all t∈T0

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implies

u(t)≤a(t)ea¯(t, t0) for all t ∈T⁰, where

¯

a(t) = k(σ(t), t) + Z t

t0

k^∆(t, τ)∆τ.

Proof. By Theorem3.8with

¯b(t) =k(σ(t), t)a(t) + Z t

t0

k^∆(t, τ)a(τ)∆τ

≤

k(σ(t), t) + Z t

t0

k^∆(t, τ)∆τ

a(t)

= ¯a(t)a(t), we obtain for allt∈T⁰

u(t)≤a(t) + Z t

t0

¯b(τ)e¯a(t, σ(τ))∆τ

≤a(t)

1 + Z t

t0

¯

a(τ)e_¯_a(t, σ(τ))∆τ

=a(t){1 +e_a_¯(t, t₀)−e_¯_a(t, t)}

=a(t)e¯a(t, t0),

where we have also used Theorem2.2and Theorem2.3.

The following theorem withk= 1reduces to Theorem3.5.

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Theorem 3.10. Suppose u, a, b, p, q ∈ C_rd and u, b, p, q ≥ 0. Let k(t, s) be defined as in Theorem2.5such thatk(σ(t), t)≥0for allt∈T⁻0 andk^∆(t, s)≤ 0fors, t ∈T⁻0 withs≥t. Then

t

k(t, τ) [b(τ)u^σ(τ) +q(τ)] ∆τ for all t∈T⁻0

implies

t

¯b(τ)e ¯a(t, τ)∆τ for all t∈T⁻0, where

¯

a(t) = k(σ(t), t)b(t)p(σ(t))− Z t0

t

k^∆(t, τ)b(τ)p^σ(τ)∆τ and

¯b(t) = k(σ(τ), t) [b(t)a^σ(t) +q(t)]− Z t0

t

k^∆(t, τ) [b(τ)a^σ(τ) +q(τ)] ∆τ.

Proof. Define z(t) := −Rt0

t k(t, τ) [b(τ)u^σ(τ) +q(τ)] ∆τ. Then for all t ∈ T⁻0 \ {t₀}

z^∆(t) = k(σ(t), t) [b(t)u^σ(t) +q(t)]− Z t0

t

k^∆(t, τ) [b(τ)u^σ(τ) +q(τ)] ∆τ

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≤ −

k(σ(t), t)b(t)p^σ(t)− Z t0

t

k^∆(t, τ)b(τ)p^σ(τ)∆τ

z^σ(t) +k(σ(t), t) [b(t)a^σ(t) +q(t)]−

Z t0

t

k^∆(t, τ) [b(τ)a^σ(τ) +q(τ)] ∆τ

=−¯a(t)z^σ(t) + ¯b(t).

In view of¯a∈ R⁺, we may apply Theorem3.4to obtain for allt∈T⁻0

z(t)≥z(t0)e ¯a(t, t0)− Z t0

t

e ¯a(t, τ)¯b(τ)∆τ =− Z t0

t

e ¯a(t, τ)¯b(τ)∆τ.

Sinceu(t)≤a(t)−p(t)z(t)for allt∈T⁻0, the claim follows.

Corollary 3.11. In addition to the assumptions of Theorem3.10withp=b= 1 andq= 0, suppose thatais nondecreasing. Then

u(t)≤a(t) + Z t0

t

k(t, τ)u^σ(τ)∆τ for all t ∈T⁻0

implies

u(t)≤a(t)e¯a(t0, t) for all t∈T⁻0, where

¯

a(t) = k(σ(t), t)− Z t0

t

k^∆(t, τ)∆τ.

Proof. The proof is similar to the proof of Corollary 3.9, this time using The- orem 3.10 instead of Theorem 3.8. Note also that this time we have ¯b(t) ≤

¯

a(t)a^σ(t).

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The continuous versions of our next two results are essentially due to Greene and can be found in [4, Th. 1.6.2 and Th. 1.6.1]. Their discrete versions [5, Th. 1.3.8 and Th. 1.3.7] are proved by Pachpatte. Note that for the discrete versions, “normal” exponential functions are used, while we employ time scales exponential functions below.

Theorem 3.12. Suppose u, v, f, g, p, q, b_i ∈ C_rd and u, v, f, p, q, b_i ≥ 0, i ∈ {1,2,3,4}. Then

u(t)≤f(t) +p(t) Z t

t0

b1(τ)u(τ)∆τ+ Z t

t0

eq(τ, t0)b2(τ)v(τ)∆τ

for all t∈T⁰ and

v(t)≤g(t) +p(t) Z t

t0

e_q(τ, t₀)b₃(τ)u(τ)∆τ+ Z t

t0

b₄(τ)v(τ)∆τ

for all t∈T⁰ imply

u(t)≤e_q(t, t₀)Q(t) and v(t)≤Q(t) for t∈T0, where

Q(t) =f(t) +g(t) +p(t) Z t

t0

[f(τ) +g(τ)]b(τ)e_bp(t, σ(τ))∆τ with

b(t) = max{b₁(t) +b₃(t), b₂(t) +b₄(t)}.

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Proof. We definew(t) = e q(t, t₀)u(t) +v(t). By Remark2we obtain for all t ∈T⁰

w(t)≤e_q(t, t₀)f(t) +g(t) +p(t)

Z t

t0

n

[e q(t, t₀)b₁(τ) +e q(τ, t₀)b₃(τ)]u(τ) + [e_q(t, τ)b₂(τ) +b₄(τ)]v(τ)o

∆τ

≤e q(t, t₀)f(t) +g(t) +p(t)

Z t

t0

n

e q(τ, t₀) [b₁(τ) +b₃(τ)]u(τ) + [b₂(τ) +b₄(τ)]v(τ)o

∆τ

≤e q(t, t0)f(t) +g(t) +p(t) Z t

t0

b(τ)w(τ)∆τ

≤f(t) +g(t) +p(t) Z t

t0

b(τ)w(τ)∆τ.

Nowb, p≥0so that Theorem3.1yields for allt∈T⁰ w(t)≤f(t) +g(t) +p(t)

Z t

t0

[f(τ) +g(τ)]b(τ)e_bp(t, σ(τ))∆τ =Q(t).

Hence

u(t) =e_q(t, t₀)w(t)−e_q(t, t₀)v(t)≤e_q(t, t₀)Q(t) and

v(t) = w(t)−e q(t, t0)u(t)≤Q(t) for allt ∈T⁰.

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Corollary 3.13. In addition to the assumptions of Theorem3.12withf(t)≡c₁, g(t)≡c2, andp(t)≡1, supposec1, c2 ∈R. Then

u(t)≤c₁+ Z t

t0

[b₁(τ)u(τ) +e_q(τ, t₀)b₂(τ)v(τ)] ∆τ for all t ∈T⁰ and

v(t)≤c2+ Z t

t0

[e q(τ, t0)b3(τ)u(τ) +b4(τ)v(τ)] ∆τ for all t∈T⁰ imply withc=c₁+c₂

u(t)≤ceb⊕q(t, t0) and v(t)≤ceb(t, t0) for all t∈T⁰. Proof. In this case we find, using Theorem2.2and Theorem2.3, that

Q(t) = c+ Z t

t0

cb(τ)eb(t, σ(τ))∆τ =ceb(t, t0).

Hence u(t) ≤ e_q(t, t₀)ce_b(t, t₀) = ceb⊕q(t, t₀) and v(t) ≤ ce_b(t, t₀) for all t ∈T0by Theorem3.12.

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4. Further Dynamic Inequalities

Our first few results are, even for the cases T = R andT = Z, more general than any result given in [4,5].

Theorem 4.1. Suppose u, a, b, c, d, p, w ∈ C_rd such that u, a, b, c, p, w ≥ 0.

Then

u(t)≤w(t) +p(t) Z t

t0

[a(τ) +b(τ)]u(τ) +b(τ)p(τ)

Z τ

t0

[c(s)u(s) +d(s)]∆s

∆τ for allt ∈T⁰ implies

t0

[a(τ) +b(τ)]

×

w(τ) +p(τ) Z τ

t0

e_p(a+b+c)(τ, σ(s))[(a+b+c)w+d](s)∆s

∆τ for allt ∈T0.

Proof. Define z(t) :=

Z t

t0

[a(τ) +b(τ)]u(τ) +b(τ)p(τ) Z τ

t0

[c(s)u(s) +d(s)]∆s

∆τ and

r(t) :=z(t) + Z t

t0

n

c(τ) [w(τ) +p(τ)z(τ)] +d(τ)o

∆τ.

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Then we haveu(t)≤w(t) +p(t)z(t),z(t)≤r(t), and z^∆(t) = [a(t) +b(t)]u(t) +b(t)p(t)

Z t

t0

[c(τ)u(τ) +d(τ)]∆τ

≤[a(t) +b(t)]w(t) +a(t)p(t)z(t) +b(t)p(t)r(t)

≤[a(t) +b(t)][w(t) +p(t)r(t)]

and therefore

r^∆(t) = z^∆(t) +c(t)[w(t) +p(t)z(t)] +d(t)

≤[a(t) +b(t)][w(t) +p(t)r(t)] +c(t)[w(t) +p(t)r(t)] +d(t)

= [(a+b+c)p](t)r(t) + [(a+b+c)w+d](t).

By Theorem2.6we find r(t)≤

Z t

t0

e_(a+b+c)p(t, σ(τ))[(a+b+c)w+d](τ)∆τ

since r(t₀) = 0. Using this in z^∆(t) ≤ [a(t) + b(t)][w(t) + p(t)r(t)] and integrating the resulting inequality completes the proof.

In certain cases it will be possible to further evaluate the integral occurring in Theorem 4.1. To this end we present the following useful auxiliary result, which is an extension of Theorem2.3.

Theorem 4.2. Supposef : T → Ris differentiable. Ifp ∈ Randa, b, c ∈ T, then

Z b

f(t)p(t)e_p(c, σ(t))∆t =e_p(c, a)f(a)−e_p(c, b)f(b)+

Z b

e_p(c, σ(t))f^∆(t)∆t.

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Proof. We use Theorem2.2and integration by parts:

Z b

a

e_p(c, σ(t))p(t)f(t)∆t

= Z b

a

e p(σ(t), c)p(t)f(t)∆t

= Z b

a

1

1 +µ(t)p(t)e_p(t, c)p(t)f(t)∆t

=− Z b

a

( p)(t)e p(t, c)f(t)∆t

=− Z b

a

e^∆_p(t, c)f(t)∆t

=−

e p(b, c)f(b)−e p(a, c)f(a)− Z b

a

e p(σ(t), c)f^∆(t)∆t

=e_p(c, a)f(a)−e_p(c, b)f(b) + Z b

a

e_p(c, σ(t))f^∆(t)∆t, which completes the proof.

Using Theorem4.2, we now present the following result.

Theorem 4.3. Suppose u, a, b, c, d, p, w ∈ C_rd such that u, a, b, c, p, w ≥ 0.

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Furthermore assume thatwis differentiable and thatpis nonincreasing. Then u(t)≤w(t) +p(t)

Z t

t0

[a(τ) +b(τ)]u(τ) +b(τ)p(τ)

Z τ

t0

[c(s)u(s) +d(s)]∆s

∆τ for allt ∈T0 implies

t0

[a(τ) +b(τ)]

e_(a+b+c)p(τ, t₀)w(t₀) +

Z τ

t0

e_p(a+b+c)(τ, σ(s))[w^∆(s) +p(τ)d(s)]∆s

Proof. Using Theorem4.2and the fact thatpis nonincreasing, we employ The- orem4.1to find

t0

[a(τ) +b(τ)]z(τ)∆τ, where

z(t) :=w(t) +p(t) Z t

t0

e_(a+b+c)p(t, σ(τ))[(a+b+c)w+d](τ)∆τ

≤w(t) +p(t) Z t

t0

e_(a+b+c)p(t, σ(τ))d(τ)∆τ +

Z t

e_(a+b+c)p(t, σ(τ))(a+b+c)(τ)p(τ)w(τ)∆τ

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=p(t) Z t

t0

e_(a+b+c)p(t, σ(τ))d(τ)∆τ+e_(a+b+c)p(t, t₀)w(t₀) +

Z t

t0

e_(a+b+c)p(t, σ(τ))w^∆(τ)∆τ, and this completes the proof.

Corollary 4.4. Under the same assumptions of Theorem4.3we can conclude u(t)≤e_(a+b+c)p(t, t₀)w(t₀) +

Z t

t0

w^∆(τ) + [a(τ) +b(τ)]

× Z τ

t0

e_(a+b+c)p(τ, σ(s))[w^∆(s) +p(τ)d(s)]∆s

Proof. The estimate p(t)

Z t

t0

[a(τ) +b(τ)]e_(a+b+c)p(τ, t₀)∆τ

≤ Z t

t0

[a(τ) +b(τ) +c(τ)]p(τ)e_(a+b+c)p(τ, t0)∆τ completes the proof as the latter integral may be evaluated directly.

The following two results (for T = R and T = Z, see [4, Th. 1.7.2 (iv) and Th. 1.7.4] and [5, Th. 1.4.4 and Th. 1.4.2], respectively) are immediate consequences of Theorem4.1.

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Corollary 4.5. Supposeu, b, c, p, w∈C_rdsuch thatu, b, c, p, w≥0. Then u(t)≤w(t) +p(t)

Z t

t0

b(τ)

u(τ) +p(τ) Z τ

t0

c(s)u(s)∆s

∆τ for allt ∈T0 implies

t0

b(τ)

w(τ) +p(τ)

Z τ

t0

e_p(b+c)(τ, σ(s))(b+c)(s)w(s)∆s

Proof. Puta=d= 0in Theorem4.1.

Corollary 4.6. If we suppose in addition to the assumptions of Corollary 4.5 thatpis nonincreasing andwis nondecreasing, then

u(t)≤w(t)

1 +p(t) Z t

t0

b(τ)e_(b+c)p(τ, t₀)∆τ

for all t∈T0. Proof. We have

p(t) Z t

t0

e_(b+c)p(t, σ(τ))w(τ)(b+c)(τ)∆τ

≤w(t) Z t

t

e_(b+c)p(t, σ(τ))p(τ)(b+c)(τ)∆τ,

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and the latter integral can be directly evaluated using Theorem2.3, hence yield- ing the result.

Remark 5. The right-hand side of the inequality in Corollary4.6can be further estimated and then evaluated by Theorem2.3so that the statement of Corollary 4.6can be replaced by

u(t)≤w(t)e_(b+c)p(t, t₀) for all t ∈T0.

In the following theorem we state some easy consequences of Theorem4.3.

See [4, Th. 1.7.2] forT=Rand [5, Th. 1.4.6] forT=Z.

Theorem 4.7. Suppose u, a, b, c, d, q ∈ C_rd andu, a, b, c, q ≥ 0. Let u₀ be a nonnegative constant. Then

(i) u(t)≤u₀+ Z t

t0

b(τ)

u(τ) +q(τ) + Z τ

t0

c(s)u(s)∆s

∆τ, t∈T0

implies

u(t)≤u₀+ Z t

t0

Q(τ)∆τ for all t∈T0, where

Q(t) = b(t)

u₀e_b+c(t, t₀) + Z t

t0

b(τ)q(τ)e_b+c(t, σ(τ))∆τ +p(t)

;

(ii) u(t)≤u₀+ Z t

t0

b(τ)

u(τ) + Z τ

t0

[c(s)u(s) +d(s)] ∆s

∆τ, t∈T0

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implies

u(t)≤u₀+ Z t

t0

b(τ)

u₀e_b+c(τ, t₀) + Z τ

t0

e_b+c(τ, σ(s))d(s)∆s

∆τ

for all t ∈T⁰;

(iii) u(t)≤u₀+ Z t

t0

a(s)u(s)∆s

+ Z t

t0

b(s)

u(s) + Z s

t0

c(τ)u(τ)∆τ

∆s, t∈T0

implies

u(t)≤u₀e_a+b+c(t, t₀) for all t∈T0. Proof. In each case we use Theorem4.3, for (i) with

a=d= 0, p= 1, and w(t) =u₀+ Z t

t0

b(τ)q(τ)∆τ, for (ii) with

a= 0, p= 1, and w=u₀, and for (iii) with

d= 0, p= 1, and w=u₀.

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In (i) and (ii), the claim follows directly, while the calculation

u(t)≤u₀

1 + Z t

t0

[a(τ) +b(τ)]e_a+b+c(τ, t₀)∆τ

≤u₀

1 + Z t

t0

[a(τ) +b(τ) +c(τ)]e_a+b+c(τ, t₀)∆τ

=u₀e_a+b+c(t, t₀) completes the proof of statement (iii).

For further reference we state the following corollary, whose continuous and discrete versions can be found in [4, Th. 1.7.1] and [5, Th. 1.4.1], respectively.

Corollary 4.8. Supposeu, b, c ∈ C_rd andu, b, c ≥ 0. Letu₀ be a nonnegative constant. Then

u(t)≤u₀+ Z t

t0

b(τ)

u(τ) + Z τ

t0

c(s)u(s)∆s

∆τ for all t ∈T0

implies

u(t)≤u0

1 +

Z t

t0

b(τ)eb+c(τ, t0)∆τ

for all t∈T⁰.

Proof. This follows from Theorem4.7(i) withq= 0or from Theorem4.7(iii) witha= 0.

Remark 6. As in Remark5, we can replace the conclusion in Corollary4.8by u(t)≤u₀e_b+c(t, t₀) for all t ∈T0.

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ForT=Zin the following result, we refer to [5, Th. 1.4.3].

Theorem 4.9. Suppose u, a, b, c ∈ C_rd, a > 0, and u, b, c ≥ 0. Letu₀ be a nonnegative constant. Then

u(t)≤a(t)

u₀+ Z t

t0

b(τ)

u(τ) + Z τ

t0

c(s)u(s)∆s

∆τ

for all t ∈T0

implies

(i) u(t)≤u₀a(t)

1 + Z t

t0

b(τ)e_b+c(τ, t₀)∆τ

for all t∈T0

if0< a(t)≤1holds for allt∈T, and (ii) u(t)≤a(t)u0

1 +

Z t

t0

a(τ)b(τ)ea(b+c)(τ, t0)∆τ

for all t ∈T⁰ ifa(t)≥1holds for allt ∈T.

Proof. Sincea(t)>0, we have u(t)

a(t) ≤u₀+ Z t

t0

b(τ)

u(τ) + Z τ

t0

c(s)u(s)∆s

∆τ.

First we assume that0< a(t)≤1holds for allt∈T. Then u(t)

a(t) ≤u₀+ Z t

b(τ) u(τ)

a(τ) + Z τ

c(s)u(s) a(s)∆s

∆τ.

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We apply Corollary4.8to obtain u(t)

a(t) ≤u₀

1 + Z t

t0

b(τ)e_b+c(τ, t₀)∆τ

for all t∈T⁰ so that (i) follows. Next we assume thata(t)≥1holds for allt∈T. Then

u(t)

a(t) ≤u0+ Z t

t0

b(τ)

u(τ) + Z τ

t0

c(s)u(s)∆s

∆τ

=u₀+ Z t

t0

b(τ) u(τ)

a(τ)a(τ) + Z τ

t0

c(s)u(s)

a(s)a(s)∆s

∆τ

≤u₀+ Z t

t0

b(τ) u(τ)

a(τ)a(τ) +a(τ) Z τ

t0

c(s)u(s)

a(s)a(s)∆s

∆τ

=u₀+ Z t

t0

b(τ)a(τ) u(τ)

a(τ) + Z τ

t0

c(s)u(s)

a(s)a(s)∆s

∆τ.

We again apply Corollary4.8to obtain u(t)

a(t) ≤u₀

1 + Z t

t0

a(τ)b(τ)e_a(b+c)(τ, t₀)∆τ

for all t∈T⁰ so that (ii) follows. Hence the proof is complete.

Remark 7. Ifc= 0in the above theorem witha≥0andu0 ∈R, then we can use Theorem3.1, Theorem2.3, and Theorem2.2to conclude

u(t)≤u₀a(t)e_ab(t, t₀) for all t∈T0.

This improves [5, Th. 1.2.7] (Ma’s inequality) for the caseT=Z, where under the assumptionsa >0andu0 ≥0a similar result as in Theorem4.9is shown.