# Explicit Bounds to Some New Gronwall-Bellman-Type Delay Integral Inequalities in Two Independent Variables on Time Scales

Authors: Fanwei Meng; Qinghua Feng; Bin Zheng
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

Some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales are established, which provide a handy tool in the research of qualitative and quantitative properties of solutions of delay dynamic equations on time scales. The established inequalities generalize some of the results in the work of Zhang and Meng 2008, Pachpatte 2002, and Ma 2010.


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## Body

## 1. Introduction

During the past decades, with the development of the theory of differential and integral equations, a lot of integral and difference inequalities have been discovered, which play an important role in the research of boundedness, global existence, stability of solutions of differential and integral equations as well as difference equations. In these established inequalities, Gronwall-Bellman-type inequalities are of particular importance as these inequalities provide explicit bounds for unknown functions, and much effort has been done for developing such inequalities (e.g., see [1–13] and the references therein). On the other hand, Hilger [14] initiated the theory of time scales as a theory capable containing both difference and differential calculus in a consistent way. Since then many authors have expounded on various aspects of the theory of dynamic equations on time scales (e.g., see [15–17] and the references therein). In these investigations, integral inequalities on time scales have been paid much attention by many authors, and a lot of integral inequalities on time scales have been established (e.g., see [18–26]), which have been designed to unify continuous and discrete analysis  and play an important role in the research of qualitative and quantitative properties of solutions of certain dynamic equations on time scales. But to our knowledge, Gronwall-Bellman-type delay integral inequalities on time scales have been paid little attention in the literature so far. Recent results in this direction include the works of Li [27] and Ma and Pečarić [28] to our best knowledge. Furthermore, nobody has studied Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales.Our aim in this paper is to establish some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales, which unify some known continuous and discrete analysis. New explicit bounds for unknown functions are obtained due to the presented inequalities. We will also present some applications for our results.First we will give some preliminaries on time scales and some universal symbols for further use.Throughout the paper,ℝ denotes the set of real numbers and ℝ+=[0,∞), while ℤ denotes the set of integers. For two given sets G, H, we denote the set of maps from G to H by (G,H).A time scale is an arbitrary nonempty closed subset of the real numbers. In this paper,𝕋 denotes an arbitrary time scale. On 𝕋 we define the forward and backward jump operators σ∈(𝕋,𝕋) and ρ∈(𝕋,𝕋) by σ(t)=inf{s∈𝕋,s>t}andρ(t)=sup{s∈𝕋,s<t}.Definition 1.1.
The graininessμ∈(𝕋,ℝ+) is defined by μ(t)=σ(t)-t.Remark 1.2.
Obviously,μ(t)=0 if 𝕋=ℝ while μ(t)=1 if 𝕋=ℤ.Definition 1.3.
A pointt∈𝕋 is said to be left-dense if ρ(t)=t and t≠inf𝕋, right-dense if σ(t)=t and t≠sup𝕋, left-scattered if ρ(t)<t, and right-scattered if σ(t)>t.Definition 1.4.
The set𝕋κ is defined to be 𝕋 if 𝕋 does not have a left-scattered maximum, otherwise it is 𝕋 without the left-scattered maximum.Definition 1.5.
A functionf∈(𝕋,ℝ) is called rd-continuous if it is continuous at right-dense points and if the left-sided limits exist at left-dense points, while f is called regressive if 1+μ(t)f(t)≠0. Crd denotes the set of rd-continuous functions, while ℜ denotes the set of all regressive and rd-continuous functions, and ℜ+={f∣f∈ℜ,1+μ(t)f(t)>0,∀t∈𝕋}.Definition 1.6.
For somet∈𝕋κ and a function f∈(𝕋,ℝ), the delta derivative of f is denoted by fΔ(t) and satisfies
(1.1)|f(σ(t))-f(s)-fΔ(t)(σ(t)-s)|≤ɛ|σ(t)-s|for∀ɛ>0,
where s∈𝔘, and 𝔘 is a neighborhood of t which can depend on ɛ.
Similarly, for somey∈𝕋κ and a function f∈(𝕋×𝕋,ℝ), the partial  delta derivative of f with respect to y is denoted by (f(x,y))yΔ or fyΔ(x,y) and satisfies
(1.2)|f(x,σ(y))-f(x,s)-fyΔ(x,y)(σ(y)-s)|≤ɛ|σ(y)-s|for∀ɛ>0,
where s∈𝔘, and 𝔘 is a neighborhood of y which can depend on ɛ.Remark 1.7.
If𝕋=ℝ, then fΔ(t) becomes the usual derivative f′(t), while fΔ(t)=f(t+1)-f(t) if 𝕋=ℤ, which represents the forward difference.Definition 1.8.
Fora,b∈𝕋 and a function f∈(𝕋,ℝ), the Cauchy  integral of f is defined by
(1.3)∫abf(t)Δt=F(b)-F(a),
where FΔ(t)=f(t),t∈𝕋κ.
Similarly, fora,b∈𝕋 and a function f∈(𝕋×𝕋,ℝ), the Cauchy  partial integral of f with respect to y is defined by
(1.4)∫abf(x,y)Δy=F(x,b)-F(x,a),
where FyΔ(x,y)=f(x,y),y∈𝕋κ.Definition 1.9.
The cylinder transformationξh is defined by
(1.5)ξh(z)={Log(1+hz)h,ifh≠0(forz≠-1h),z,ifh=0,
where Log is the principal logarithm function.Definition 1.10.
Forp(x,y)∈ℜ with respect to y, the exponential  function is defined by
(1.6)ep(y,s)=exp⁡(∫syξμ(τ)(p(x,τ))Δτ),fors,y∈T.Remark 1.11.
If𝕋=ℝ, then for y∈ℝ the following formula holds:
(1.7)ep(y,s)=exp⁡(∫syp(x,τ)dτ),fors∈T.
If 𝕋=ℤ, then, for y∈ℤ, ep(y,s)=∏τ=sy-1[1+p(x,τ)],fors∈ℤ and    s<y.The following two theorems include some known properties on theexponential function.Theorem 1.12.
Ifp(x,y)∈ℜ with respect to y, then the following conclusions hold:(i)
ep(y,y)≡1 and e0(s,y)≡1,(ii)
ep(s,σ(y))=(1+μ(y)p(x,y))ep(s,y),(iii)
ifp∈ℜ+ with respect to y, then ep(s,y)>0 for all s,y∈𝕋,(iv)
ifp∈ℜ+ with respect to y, then ⊖p∈ℜ+,(v)
ep(s,y)=1/ep(y,s)=e⊖p(y,s), where (⊖p)(x,y)=-(p(x,y)/1+μ(y)p(x,y)).Theorem 1.13.
Ifp(x,y)∈ℜ with respect to y,y0∈𝕋 is a fixed number, then the exponential function ep(y,y0) is the unique solution of the following initial value problem:
(1.8)zyΔ(x,y)=p(x,y)z(x,y),z(x,y0)=1.Theorems1.12-1.13 are similar to [24, Theorems 5.1-5.2]. For more details about the calculus of time scales, we advise to refer to [29].In the rest of this paper, for the convenience of notation, we always assume that𝕋0=[x0,∞)⋂𝕋,𝕋̃0=[y0,∞)⋂𝕋, where x0,y0∈𝕋κ, and furthermore assume that  𝕋0⊆𝕋κ,𝕋̃0⊆𝕋κ.


## 2. Main Results

We will give some lemmas for further use.Lemma 2.1.
Suppose thatX∈𝕋0 is a fixed number and u(X,y),b(X,y)∈Crd,m(X,y)∈ℜ+ with respect to y, m(X,y)≥0; then
(2.1)u(X,y)≤a(X,y)+b(X,y)∫y0ym(X,t)u(X,t)Δt,y∈T̃0,
implies
(2.2)u(X,y)≤a(X,y)+b(X,y)∫y0yem¯(y,σ(t))a(X,t)m(X,t)Δt,y∈T̃0,
where m¯(X,y)=m(X,y)b(X,y), and em¯(y,y0) is the unique solution of the following IVP
(2.3)zyΔ(X,y)=m(X,y)z(X,y),z(X,y0)=1.The proof of Lemma2.1 is similar to that of [24, Theorem 5.6], and we omit it here.Lemma 2.2.
Under the conditions of Lemma2.1 and furthermore assuming that a(x,y) is nondecreasing in y for every fixed x,b(x,y)≡1, then one has
(2.4)u(X,y)≤a(X,y)em(y,y0).Proof.
Sincea(x,y) is nondecreasing in y for every fixed x, then from Lemma 2.1 we have
(2.5)u(X,y)≤a(X,y)+∫y0yem(y,σ(t))a(X,t)m(X,t)Δt≤a(X,y)[1+∫y0yem(y,σ(t))m(X,t)Δt].
On the other hand, from [29, Theorems 2.39 and 2.36 (i)] we have 1+∫y0yem(y,σ(t))m(X,t)Δt=em(y,y0).
Then collecting the above information we can obtain the desired inequality.Lemma 2.3 (see [30]).
Assume thata≥0,p≥q≥0, and p≠0, then for any K>0(2.6)aq/p≤qpK(q-p)/pa+p-qpKq/p.Theorem 2.4.
Suppose thatu,f,g,h,a,b∈Crd(𝕋0×𝕋̃0,ℝ+) and a,b are nondecreasing. p,q,r,m are constants, and p≥q≥0,p≥r≥0,p≥m≥0,p≠0.τ1∈(𝕋0,𝕋),τ1(x)≤x,-∞<α=inf{τ1(x),x∈𝕋0}≤x0. τ2∈(𝕋̃0,𝕋),τ2(y)≤y,-∞<β=inf{τ2(y),y∈𝕋̃0}≤y0. ϕ∈Crd(([α,x0]×[β,y0])⋂𝕋2,ℝ+). If for (x,y)∈𝕋0×𝕋̃0, u(x,y) satisfies the following inequality:
(2.7)up(x,y)≤a(x,y)+b(x,y)∫y0y∫x0x[f(s,t)uq(τ1(s),τ2(t))+g(s,t)ur(s,t)]ΔsΔt+b(x,y)∫y0y∫x0x∫y0t∫x0sh(ξ,η)um(τ1(ξ),τ2(η))ΔξΔηΔsΔt,
with the initial condition
(2.8)u(x,y)=ϕ(x,y),ifx∈[α,x0]⋂T,ory∈[β,y0]⋂T,ϕ(τ1(x),τ2(y))≤α1/p(x,y),ifτ1(x)≤x0orτ2(y)≤y0,∀(x,y)∈T0×T̃0,
then
(2.9)u(x,y)≤[B1(x,y)+b(x,y)∫y0yeB¯2(y,σ(t))B2(x,t)B1(x,t)Δt]1/p,(x,y)∈T0×T̃0,
where
(2.10)B1(x,y)=a(x,y)+b(x,y)∫y0y∫x0x[f(s,t)p-qpKq/p+g(s,t)p-rpKr/p]ΔsΔt+b(x,y)∫y0y∫x0x∫y0t∫x0sh(ξ,η)p-mpKm/pΔξΔηΔsΔt,∀K>0,(2.11)B2(x,y)=∫x0x[f(s,y)qpK(q-p)/p+g(s,y)rpK(r-p)/p+∫y0y∫x0sh(ξ,η)mpK(m-p)/pΔξΔη]Δs,∀K>0,(2.12)B¯2(x,y)=b(x,y)B2(x,y).Proof.
Let the right side of (2.7) be v(x,y). Then
(2.13)u(x,y)≤v1/p(x,y),(x,y)∈T0×T̃0.
If τ1(x)≥x0 and τ2(y)≥y0, then τ1(x)∈𝕋0,τ2(y)∈𝕋̃0, and since a,b are nondecreasing we have
(2.14)u(τ1(x),τ2(y))≤v1/p(τ1(x),τ2(y))≤v1/p(x,y).
If τ1(x)≤x0 or τ2(y)≤y0, then from (2.8) we have
(2.15)u(τ1(x),τ2(y))=ϕ(τ1(x),τ2(y))≤a1/p(x,y)≤v1/p(x,y).
From (2.14) and (2.15) we have
(2.16)u(τ1(x),τ2(y))≤v1/p(x,y),(x,y)∈T0×T̃0.
Fix X∈𝕋0, and let x∈[x0,X]⋂𝕋,y∈𝕋̃0; then
(2.17)v(X,y)=a(X,y)+b(X,y)∫y0y∫x0X[f(s,t)uq(τ1(s),τ2(t))+g(s,t)ur(s,t)]ΔsΔt+b(X,y)∫y0y∫x0X∫y0t∫x0sh(ξ,η)um(τ1(ξ),τ2(η))ΔξΔηΔsΔt≤a(X,y)+b(X,y)∫y0y∫x0X[∫x0shf(s,t)vq/p(s,t)+g(s,t)vr/p(s,t)+∫y0t∫x0sh(ξ,η)vm/p(ξ,η)ΔξΔη]ΔsΔt.
From Lemma 2.3, we have
(2.18)vq/p(x,y)≤qpK(q-p)/pv(x,y)+p-qpKq/p,vr/p(x,y)≤rpK(r-p)/pv(x,y)+p-rpKr/p,vm/p(x,y)≤mpK(m-p)/pv(x,y)+p-mpKm/p,∀K>0.
So combining (2.17) and (2.18), it follows that
(2.19)v(X,y)≤a(X,y)+b(X,y)∫y0y∫x0Xf(s,t)(qpK(q-p)/pv(s,t)+p-qpKq/p)ΔsΔt+b(X,y)∫y0y∫x0Xg(s,t)(rpK(r-p)/pv(s,t)+p-rpKr/p)ΔsΔt+b(X,y)∫y0y∫x0X∫y0t∫x0sh(ξ,η)(mpK(m-p)/pv(ξ,η)+p-mpKm/p)ΔξΔηΔsΔt≤a(X,y)+b(X,y)∫y0y∫x0X[∫x0shf(s,t)p-qpKq/p+g(s,t)p-rpKr/p+∫y0t∫x0sh(ξ,η)p-mpKm/pΔξΔη]ΔsΔt+b(X,y)∫y0y{∫x0X[∫x0sf(s,t)qpK(q-p)/p+g(s,t)rpK(r-p)/p+∫y0t∫x0sh(ξ,η)mpK(m-p)/pΔξΔη]Δs}v(X,t)Δt=B1(X,y)+b(X,y)∫y0yB2(X,t)v(X,t)Δt,
where B1(x,y),B2(x,y) are defined in (2.10) and (2.11), respectively. Considering B¯2(X,y)=b(X,y)B2(X,y), by application of Lemma 2.1, we have
(2.20)v(X,y)≤B1(X,y)+b(X,y)∫y0yeB¯2(y,σ(t))B2(X,t)B1(X,t)Δt,y∈T̃0.
Since X∈𝕋0 is arbitrary, then in fact (2.20) holds for all x∈𝕋0, that is,
(2.21)v(x,y)≤B1(x,y)+b(x,y)∫y0yeB¯2(y,σ(t))B2(x,t)B1(x,t)Δt,(x,y)∈(T0×T̃0).
Combining (2.13) and (2.21), we obtain
(2.22)u(x,y)≤[B1(x,y)+b(x,y)∫y0yeB¯2(y,σ(t))B2(x,t)B1(x,t)Δt]1/p,(x,y)∈(T0×T̃0),
which is the desired inequality.If we apply Lemma2.2 instead of Lemma 2.1 at the end of the proof of Theorem 2.4, we obtain the following theorem.Theorem 2.5.
Suppose thatu,f,g,h,a,p,q,r,m,τ1,τ2,α,β,ϕ are defined as in Theorem 2.4. If that for (x,y)∈𝕋0×𝕋̃0,u(x,y) satisfies the following inequality:
(2.23)up(x,y)≤a(x,y)+∫y0y∫x0x[∫x0sf(s,t)uq(τ1(s),τ2(t))+g(s,t)ur(s,t)+∫y0t∫x0sh(ξ,η)um(τ1(ξ),τ2(η))ΔξΔη]ΔsΔt
with the initial condition (2.8), then
(2.24)u(x,y)≤[B1(x,y)eB2(y,y0)]1/p,(x,y)∈T0×T̃0,
where
(2.25)B1(x,y)=a(x,y)+∫y0y∫x0x[f(s,t)p-qpKq/p+g(s,t)p-rpKr/p]ΔsΔt+∫y0y∫x0x∫y0t∫x0sh(ξ,η)p-mpKm/pΔξΔηΔsΔt,∀K>0,B2(x,y)=∫x0x[f(s,y)qpK(q-p)/p+g(s,y)rpK(r-p)/p+∫y0y∫x0sh(ξ,η)mpK(m-p)/pΔξΔη]Δs,∀K>0.From Theorems2.4 and 2.5 we can obtain two direct corollaries.Corollary 2.6.
Under the conditions of Theorem2.4, if, for (x,y)∈𝕋0×𝕋̃0, u(x,y) satisfies the following inequality:
(2.26)u(x,y)≤a(x,y)+∫y0y∫x0x[f(s,t)u(τ1(s),τ2(t))+g(s,t)u(s,t)]ΔsΔt+∫y0y∫x0x∫y0t∫x0sh(ξ,η)u(τ1(ξ),τ2(η))ΔξΔηΔsΔt,
with the initial condition (2.8) (p=1), then
(2.27)u(x,y)≤a(x,y)+∫y0yeB2(y,σ(t))B2(x,t)a(x,t)Δt,(x,y)∈T0×T̃0,
where
(2.28)B2(x,y)=∫x0x[f(s,y)+g(s,y)+∫y0y∫x0sh(ξ,η)ΔξΔη]Δs.Corollary 2.7.
Under the conditions of Theorem2.5, if, for (x,y)∈𝕋0×𝕋̃0, u(x,y) satisfies the following inequality:
(2.29)u(x,y)≤a(x,y)+∫y0y∫x0x[f(s,t)u(τ1(s),τ2(t))+g(s,t)u(s,t)]ΔsΔt+∫y0y∫x0x∫y0t∫x0sh(ξ,η)u(τ1(ξ),τ2(η))ΔξΔηΔsΔt,
with the initial condition (2.8) (p=1), then
(2.30)u(x,y)≤a(x,y)eB2(y,y0),(x,y)∈T0×T̃0,
where
(2.31)B2(x,y)=∫x0x[f(s,y)+g(s,y)+∫y0y∫x0sh(ξ,η)ΔξΔη]Δs.Theorem 2.8.
Suppose thatu∈Crd(𝕋0×𝕋̃0,ℝ+), f,g,h,τ1,τ2 are defined as in Theorem 2.4, and τ1(x)≥x0, τ2(y)≥y0. If, for (x,y)∈𝕋0×𝕋̃0,u(x,y) satisfies the following inequality:
(2.32)u(x,y)≤∫y0y∫x0x[f(s,t)u(τ1(s),τ2(t))+g(s,t)u(s,t)+∫y0t∫x0sh(ξ,η)u(τ1(ξ),τ2(η))ΔξΔη]ΔsΔt,
then u(x,y)≡0.The proof of Theorem2.8 is similar to Theorem 2.4, and we omit it here.Based on Theorem2.4, we will establish a class of Volterra-Fredholm-type integral inequality on time scales.Theorem 2.9.
Suppose thatu,fi,gi,hi∈Crd(𝕋0×𝕋̃0,ℝ+),i=1,2.a,p,q,r,m,ϕ,τ1,τ2,α,β are the same as in Theorem 2.4, and M∈𝕋0,N∈𝕋̃0 are two fixed numbers. If, for (x,y)∈([x0,M]⋂𝕋)×([y0,N]⋂𝕋),u(x,y) satisfies the following inequality:
(2.33)up(x,y)≤a(x,y)+∫y0y∫x0x[f1(s,t)uq(τ1(s),τ2(t))+g1(s,t)ur(τ1(s),τ2(t))]ΔsΔt+∫y0y∫x0x∫y0t∫x0sh1(ξ,η)um(τ1(ξ),τ2(η))ΔξΔηΔsΔt+∫y0N∫x0M[f2(s,t)uq(τ1(s),τ2(t))+g2(s,t)ur(τ1(s),τ2(t))]ΔsΔt+∫y0N∫x0M∫y0t∫x0sh2(ξ,η)um(τ1(ξ),τ2(η))ΔξΔηΔsΔt,
with the initial condition (2.8), then one has
(2.34)u(x,y)≤{[λ+B̃61-B̃5]B̃3(x,y)+B̃4(x,y)}1/p,(x,y)∈([x0,M]⋂T)×([y0,N]⋂T),
provided that B̃5<1, where
(2.35)λ=∫y0N∫x0M[f2(s,t)p-qpKq/p+g2(s,t)p-rpKr/p+∫y0t∫x0sh2(ξ,η)p-mpKm/pΔξΔη]ΔsΔt,(2.36)B̃1(x,y)=a(x,y)+∫y0y∫x0x[f1(s,t)p-qpKq/p+g1(s,t)p-rpKr/p]ΔsΔt+∫y0y∫x0x∫y0t∫x0sh1(ξ,η)p-mpKm/pΔξΔηΔsΔt,∀K>0,(2.37)B̃2(x,y)=∫x0x[∫y0yf1(s,y)qpK(q-p)/p+g1(s,y)rpK(r-p)/p+∫y0y∫x0sh1(ξ,η)mpK(m-p)/pΔξΔη]Δs,∀K>0,(2.38)B̃3(x,y)=1+∫y0yeB̃2(y,σ(t))B̃2(x,t)Δt,(2.39)B̃4(x,y)=B̃1(x,y)+∫y0yeB̃2(y,σ(t))B̃2(x,t)B̃1(x,t)Δt,(2.40)B̃5=∫y0N∫x0M[f2(s,t)qpK(q-p)/pB̃3(s,t)+g2(s,t)rpK(r-p)/pB̃3(s,t)]ΔsΔt+∫y0N∫x0M∫y0t∫x0sh2(ξ,η)mpK(m-p)/pB̃3(ξ,η)ΔξΔηΔsΔt,(2.41)B̃6=∫y0N∫x0M[f2(s,t)qpK(q-p)/pB̃4(s,t)+g2(s,t)rpK(r-p)/pB̃4(s,t)]ΔsΔt+∫y0N∫x0M∫y0t∫x0sh2(ξ,η)mpK(m-p)/pB̃4(ξ,η)ΔξΔηΔsΔt.Proof.
Let the right side of (2.33) be v(x,y) and
(2.42)μ=∫y0N∫x0M[f2(s,t)uq(τ1(s),τ2(t))+g2(s,t)ur(τ1(s),τ2(t))]ΔsΔt+∫y0N∫x0M∫y0t∫x0sh2(ξ,η)um(τ1(ξ),τ2(η))ΔξΔηΔsΔt.
Then
(2.43)u(x,y)≤v1/p(x,y),(x,y)∈([x0,M]⋂T)×([y0,N]⋂T).
Similar to the process of (2.14)–(2.16) one has
(2.44)u(τ1(x),τ2(y))≤v1/p(x,y),(x,y)∈([x0,M]⋂T)×([y0,N]⋂T).
Fix X∈[x0,M]⋂𝕋, and let x∈[x0,X]⋂𝕋,y∈[y0,N]⋂𝕋. Then
(2.45)v(X,y)=a(X,y)+μ+∫y0y∫x0X[f1(s,t)uq(τ1(s),τ2(t))+g1(s,t)ur(τ1(s),τ2(t))]ΔsΔt+∫y0y∫x0X∫y0t∫x0sh1(ξ,η)um(τ1(ξ),τ2(η))ΔξΔηΔsΔt≤a(X,y)+μ+∫y0y∫x0X[f1(s,t)vq/p(s,t)+g1(s,t)vr/p(s,t)]ΔsΔt+∫y0y∫x0X∫y0t∫x0sh1(ξ,η)vm/p(ξ,η)ΔξΔηΔsΔt.
Considering the structure of (2.45) is similar to (2.17), then following in a same manner as the process of (2.17)–(2.20) we can deduce
(2.46)v(X,y)≤μ+B̃1(X,y)+∫y0yeB̃2(y,σ(t))B̃2(X,t)(μ+B̃1(X,t))Δt=μ[1+∫y0yeB̃2(y,σ(t))B̃2(X,t)Δt]+B̃1(X,y)+∫y0yeB̃2(y,σ(t))B̃2(X,t)B̃1(X,t)Δt,y∈[y0,N]⋂T,
where B̃1(x,y),B̃2(x,y) are defined in (2.36) and (2.37), respectively.
SinceX is selected from [x0,M]⋂𝕋 arbitrarily, then in fact (2.46) holds  for all x∈𝕋0, that is,
(2.47)v(x,y)≤μ[1+∫y0yeB̃2(y,σ(t))B̃2(x,t)Δt]+B̃1(x,y)+∫y0yeB̃2(y,σ(t))B̃2(x,t)B̃1(x,t)Δt=μB̃3(x,y)+B̃4(x,y),(x,y)∈([x0,M]⋂T)×([y0,N]⋂T),
where B̃3(x,y),B̃4(x,y) are defined in (2.38) and (2.39), respectively.
On the other hand, from (2.18), (2.42), and (2.44) we obtain
(2.48)μ≤∫y0N∫x0M[f2(s,t)vq/p(s,t)+g2(s,t)vr/p(s,t)+∫y0t∫x0sh2(ξ,η)vm/p(ξ,η)ΔξΔη]ΔsΔt≤∫y0N∫x0M[f2(s,t)(qpK(q-p)/pv(s,t)+p-qpKq/p)+g2(s,t)(rpK(r-p)/pv(s,t)+p-rpKr/p)]ΔsΔt+∫y0N∫x0M∫y0t∫x0sh2(ξ,η)(mpK(m-p)/pv(ξ,η)+p-mpKm/p)ΔξΔηΔsΔt=λ+∫y0N∫x0M[f2(s,t)qpK(q-p)/pv(s,t)+g2(s,t)rpK(r-p)/pv(s,t)]ΔsΔt+∫y0N∫x0M∫y0t∫x0sh2(ξ,η)mpK(m-p)/pv(ξ,η)ΔξΔηΔsΔt,
where λ is defined in (2.35). Then using (2.47) in (2.48) yields
(2.49)μ≤λ+∫y0N∫x0M{f2(s,t)qpK(q-p)/p[μB̃3(s,t)+B̃4(s,t)]+g2(s,t)rpK(r-p)/p[μB̃3(s,t)+B̃4(s,t)]}ΔsΔt+∫y0N∫x0M∫y0t∫x0sh2(ξ,η)mpK(m-p)/p[μB̃3(ξ,η)+B̃4(ξ,η)]ΔξΔηΔsΔt=λ+μ{∫y0N∫x0M[f2(s,t)qpK(q-p)/pB̃3(s,t)+g2(s,t)rpK(r-p)/pB̃3(s,t)]ΔsΔt+∫y0N∫x0M∫y0t∫x0sh2(ξ,η)mpK(m-p)/pB̃3(ξ,η)ΔξΔηΔsΔt}+∫y0N∫x0M[f2(s,t)qpK(q-p)/pB̃4(s,t)+g2(s,t)rpK(r-p)/pB̃4(s,t)]ΔsΔt+∫y0N∫x0M∫y0t∫x0sh2(ξ,η)mpK(m-p)/pB̃4(ξ,η)ΔξΔηΔsΔt=λ+μB̃5+B̃6,
which implies(2.50)μ≤λ+B̃61-B̃5.
Combining (2.43), (2.47), and (2.50) we can obtain the desired inequality (2.34).In the proof of Theorem2.9, if we let the right side of (2.33) be a(x,y)+v(x,y) in the first statement, then following in a same process as in Theorem 2.9 we obtain another bound on the function u(x,y), which is shown in the following theorem.Theorem 2.10.
Under the conditions of Theorem2.9, if, for (x,y)∈([x0,M]⋂𝕋)×([y0,N]⋂𝕋), u(x,y) satisfies (2.33) with the initial condition (2.8), then the following inequality holds:
(2.51)u(x,y)≤{a(x,y)+μ̃+J1(x,y)1-λ̃eJ2(y,y0)}1/p,(x,y)∈([x0,M]⋂T)×([y0,N]⋂T),
provided that λ̃<1, where
(2.52)λ̃=∫y0N∫x0M[f2(s,t)qpK(q-p)/peJ2(t,y0)+g2(s,t)rpK(r-p)/peJ2(t,y0)]ΔsΔt+∫y0N∫x0M∫y0t∫x0sh2(ξ,η)mpK(m-p)/peJ2(η,y0)ΔξΔηΔsΔt,μ̃=∫y0N∫x0M[f2(s,t)(qpK(q-p)/pa(s,t)+p-qpKq/p)+g2(s,t)(rpK(r-p)/pa(s,t)+p-rpKr/p)+∫y0t∫x0sh2(ξ,η)(mpK(m-p)/p(a(ξ,η)+v(ξ,η))+p-mpKm/p)ΔξΔη]ΔsΔt,J1(x,y)=∫y0y∫x0x[f1(s,t)(qpK(q-p)/pa(s,t)+p-qpKq/p)+g1(s,t)(rpK(r-p)/pa(s,t)+p-rpKr/p)]ΔsΔt+∫y0y∫x0X∫y0t∫x0sh1(ξ,η)(mpK(m-p)/pa(ξ,η)+p-mpKm/p)ΔξΔηΔsΔt,J2(x,y)=∫x0x[f1(s,y)qpK(q-p)/p+g1(s,y)rpK(r-p)/p+∫y0y∫x0sh1(ξ,η)mpK(m-p)/pΔξΔη]Δs.Finally, we will establish a more general inequality than that in Theorems2.9-2.10. Consider the following inequality:(2.53)up(x,y)≤a(x,y)+∫y0y∫x0x[L(s,t,u(τ1(s),τ2(t)))+∫y0t∫x0sh1(ξ,η)uq(τ1(ξ),τ2(η))ΔξΔη]ΔsΔt+∫y0N∫x0M[L(s,t,u(τ1(s),τ2(t)))+∫y0t∫x0sh2(ξ,η)uq(τ1(ξ),τ2(η))ΔξΔη]ΔsΔt,
with the initial condition (2.8), where u,a,p,q,ϕ,α,β,τi,hi,i=1,2 are the same as in Theorem 2.4, M∈𝕋0,N∈𝕋̃0 are two fixed numbers. L∈(𝕋0×𝕋̃0×ℝ+,ℝ+), and 0≤L(s,t,x)-L(s,t,y)≤A(s,t,y)(x-y) for x≥y≥0, where A∈(𝕋0×𝕋̃0×ℝ+,ℝ+).Theorem 2.11.
If, for(x,y)∈([x0,M]⋂𝕋)×([y0,N]⋂𝕋),u(x,y) satisfies (2.53), then the following inequality holds:
(2.54)u(x,y)≤{[λ̂+B̂61-B̂5]B̂3(x,y)+B̂4(x,y)}1/p,(x,y)∈([x0,M]⋂T)×([y0,N]⋂T),
provided that B̃5<1, where
(2.55)λ̂=∫y0N∫x0M[L(s,t,p-1pK1/p)+∫y0t∫x0sh2(ξ,η)p-qpKq/pΔξΔη]ΔsΔt,(2.56)B̂1(x,y)=a(x,y)+∫y0y∫x0x[L(s,t,p-1pK1/p)+∫y0t∫x0sh1(ξ,η)p-qpKq/pΔξΔη]ΔsΔt,∀K>0,(2.57)B̂2(x,y)=∫x0x[A(s,y,p-1pK1/p)1pK(1-p)/p+∫y0y∫x0sh1(ξ,η)qpK(q-p)/pΔξΔη]Δs,∀K>0,(2.58)B̂3(x,y)=1+∫y0yeB̂2(y,σ(t))B̂2(x,t)Δt,(2.59)B̂4(x,y)=B̂1(x,y)+∫y0yeB̂2(y,σ(t))B̂2(x,t)B̂1(x,t)Δt,(2.60)B̂5=∫y0N∫x0M[∫x0sh2A(s,t,p-1pK1/p)1pK(1-p)/pB̂3(s,t)+∫y0t∫x0sh2(ξ,η)qpK(q-p)/pB̂3(ξ,η)ΔξΔη]ΔsΔt,(2.61)B̂6=∫y0N∫x0M[A(s,t,p-1pK1/p)1pK(1-p)/pB̂4(s,t)+∫y0t∫x0sh2(ξ,η)qpK(q-p)/pB̂4(ξ,η)ΔξΔη]ΔsΔt.Proof.
Let the right side of (2.53) be v(x,y) and
(2.62)μ̂=∫y0N∫x0M[L(s,t,u(τ1(s),τ2(t)))+∫y0t∫x0sh2(ξ,η)uq(τ1(ξ),τ2(η))ΔξΔη]ΔsΔt.
Then
(2.63)u(x,y)≤v1/p(x,y),(x,y)∈([x0,M]⋂T)×([y0,N]⋂T).
Similar to the process of (2.14)–(2.16) we have
(2.64)u(τ1(x),τ2(y))≤v1/p(x,y),(x,y)∈([x0,M]⋂T)×([y0,N]⋂T).
Fix X∈[x0,M]⋂𝕋, and let x∈[x0,X]⋂𝕋,y∈[y0,N]⋂𝕋. Then
(2.65)v(X,y)=a(X,y)+μ̂+∫y0y∫x0X[L(s,t,u(τ1(s),τ2(t)))+∫y0t∫x0sh1(ξ,η)uq(τ1(ξ),τ2(η))ΔξΔη]ΔsΔt≤a(X,y)+μ̂+∫y0y∫x0X[L(s,t,v1/p(s,t))+∫y0t∫x0sh1(ξ,η)vq/p(ξ,η)ΔξΔη]ΔsΔt.
From Lemma 2.3, we have
(2.66)vq/p(x,y)≤qpK(q-p)/pv(x,y)+p-qpKq/p,v1/p(x,y)≤1pK(1-p)/pv(x,y)+p-1pK1/p,∀K>0.
Combining (2.65) and (2.66), it follows that(2.67)v(X,y)≤a(X,y)+μ̂+∫y0y∫x0XL(s,t,1pK(1-p)/pv(s,t)+p-1pK1/p)ΔsΔt+∫y0y∫x0X∫y0t∫x0sh1(ξ,η)(qpK(q-p)/pv(ξ,η)+p-qpKq/p)ΔξΔηΔsΔt=a(X,y)+μ̂+∫y0y∫x0X[L(s,t,1pK(1-p)/pv(s,t)+p-1pK1/p)-L(s,t,p-1pK1/p)+L(s,t,p-1pK1/p)]ΔsΔt+∫y0y∫x0X∫y0t∫x0sh1(ξ,η)(qpK(q-p)/pv(ξ,η)+p-qpKq/p)ΔξΔηΔsΔt≤a(X,y)+μ̂+∫y0y∫x0X[A(s,t,p-1pK1/p)1pK(1-p)/pv(s,t)+L(s,t,p-1pK1/p)]ΔsΔt+∫y0y∫x0X[∫y0t∫x0sh1(ξ,η)qpK(q-p)/pΔξΔη]v(X,t)ΔsΔt+∫y0y∫x0X∫y0t∫x0sh1(ξ,η)p-qpKq/pΔξΔηΔsΔt≤a(X,y)+μ̂+∫y0y[∫x0XA(s,t,p-1pK1/p)1pK(1-p)/pΔs]v(X,t)Δt+∫y0y∫x0XL(s,t,p-1pK1/p)ΔsΔt+∫y0y[∫x0X∫y0t∫x0sh1(ξ,η)qpK(q-p)/pΔξΔηΔs]v(X,t)Δt+∫y0y∫x0X∫y0t∫x0sh1(ξ,η)p-qpKq/pΔξΔηΔsΔt=μ̂+B̂1(X,y)+∫y0yB̂2(X,t)v(X,t)Δt,
where B̂1(x,y),B̂2(x,y) are defined in (2.56) and (2.57), respectively.
We notice the structure of (2.67) is similar to (2.19), so following in a same manner as in (2.19)–(2.21) we obtain(2.68)v(x,y)≤μ̂[1+∫y0yeB̂2(y,σ(t))B̂2(x,t)Δt]+B̂1(x,y)+∫y0yeB̂2(y,σ(t))B̂2(x,t)B̂1(x,t)Δt=μ̂B̂3(x,y)+B̂4(x,y),(x,y)∈([x0,M]⋂T)×([y0,N]⋂T),
where B̂3(x,y),B̂4(x,y) are defined in (2.58) and (2.59), respectively.
On the other hand, from (2.62), (2.64), and (2.66) we have
(2.69)μ̂≤∫y0N∫x0M[L(s,t,v1/p)+∫y0t∫x0sh2(ξ,η)vq/p(ξ,η)ΔξΔη]ΔsΔt≤∫y0N∫x0M[∫y0t∫x0sL(s,t,1pK(1-p)/pv(s,t)+p-1pK1/p)+∫y0t∫x0sh2(ξ,η)(qpK(q-p)/pv(ξ,η)+p-qpKq/p)ΔξΔη]ΔsΔt=∫y0N∫x0M[L(s,t,1pK(1-p)/pv(s,t)+p-1pK1/p)-L(s,t,p-1pK1/p)+L(s,t,p-1pK1/p)]ΔsΔt+∫y0N∫x0M∫y0t∫x0sh2(ξ,η)(qpK(q-p)/pv(ξ,η)+p-qpKq/p)ΔξΔηΔsΔt≤∫y0N∫x0M[A(s,t,p-1pK1/p)1pK(1-p)/pv(s,t)+L(s,t,p-1pK1/p)]ΔsΔt+∫y0N∫x0M∫y0t∫x0sh2(ξ,η)(qpK(q-p)/pv(ξ,η)+p-qpKq/p)ΔξΔηΔsΔt=λ̂+∫y0N∫x0M[∫x0sh2A(s,t,p-1pK1/p)1pK(1-p)/pv(s,t)+∫y0t∫x0sh2(ξ,η)qpK(q-p)/pv(ξ,η)ΔξΔη]ΔsΔt,
where λ̂ is defined in (2.55). Then using (2.68) in (2.69) yields(2.70)μ̂≤λ̂+∫y0N∫x0MA(s,t,p-1pK1/p)1pK(1-p)/p[μ̂B̂3(s,t)+B̂4(s,t)]ΔsΔt+∫y0N∫x0M∫y0t∫x0sh2(ξ,η)qpK(q-p)/p[μ̂B̂3(ξ,η)+B̂4(ξ,η)]ΔξΔηΔsΔt=λ̂+μ̂{∫y0N∫x0MA(s,t,p-1pK1/p)1pK(1-p)/pB̂3(s,t)+∫y0t∫x0sh2(ξ,η)qpK(q-p)/pB̂3(ξ,η)ΔξΔηΔsΔt}+∫y0N∫x0M[∫y0tA(s,t,p-1pK1/p)1pK(1-p)/pB̂4(s,t)+∫y0t∫x0sh2(ξ,η)qpK(q-p)/pB̂4(ξ,η)ΔξΔη]ΔsΔt=λ̂+μ̂B̂5+B̂6,
which implies
(2.71)μ̂≤λ̂+B̂61-B̂5.
Combining (2.63), (2.68), and (2.71), we obtain the desired inequality (2.54).Remark 2.12.
The established above inequalities generalize many known results including both integral inequalities for continuous functions and discrete inequalities. For example, if we take𝕋=ℝ,p=q=1,g(x,y)=h(x,y)≡0, then Theorem 2.4 reduces to [1, Theorem 2.2], which is one case of integral inequality for continuous function. If we take 𝕋=ℝ,h(x,y)≡0,a(x,y)≡C, then Corollary 2.7 reduces to [2, Theorem 3 (c1)], which is another case of inequality for continuous function. If we take 𝕋=ℝ,p=q=1,g1(x,y)=g2(x,y)=f2(x,y)=h1(x,y)=h2(x,y)≡0, then Theorem 2.10 reduces to [1, Theorem 2.2] with slight difference. If we take 𝕋=ℤ,g1(x,y)=h1(x,y)=f2(x,y)=h2(x,y)≡0,τ1(x)=x,τ2(y)=y, then Theorem 2.10 reduces to [3, Theorem 2.1], which is a discrete inequality.Remark 2.13.
Since𝕋 is an arbitrary time scale, then if we take 𝕋 for some peculiar cases, such as 𝕋=ℝ or 𝕋=ℤ,   we can deduce a series of corollaries according to Theorem 2.4–2.11. Due to the limited space, we omit them here.


## 3. Some Applications

In this section, we will present some applications for the results we have established previously. New explicit bounds for solutions of certain dynamic equations are derived in the first two examples, while the quantitative property of solutions is  focused on in the final example.Example 3.1.
Consider the following delay dynamic differential equation:(3.1)(up(x,y))yxΔΔ=F(s,t,u(τ1(s),τ2(t)),∫y0t∫x0sW(ξ,η,u(τ1(ξ),τ2(η)))ΔξΔη),(x,y)∈T0×T̃0,
with the initial condition
(3.2)(up(x0,y))yΔ=bΔ(y),up(x,y0)=a(x),u(x,y)=ϕ(x,y),ifx∈[α,x0]⋂T,ory∈[β,y0]⋂T,|ϕ(τ1(x),τ2(y))|≤|k(x,y)|1/p,ifτ1(x)≤x0,orτ2(y)≤y0,∀(x,y)∈T0×T̃0,
where u∈Crd(𝕋0×𝕋̃0,ℝ), a∈Crd(𝕋0,ℝ), b∈Crd(𝕋̃0,ℝ), b is delta differential, and b(y0)=0,k∈Crd(𝕋0×𝕋̃0,ℝ+), p>0 is a constant, ϕ∈Crd(([α,x0]×[β,y0])⋂𝕋2,ℝ),F∈(𝕋0×𝕋̃0×ℝ2,ℝ),W∈(𝕋0×𝕋̃0×ℝ,ℝ). α,β,τ1,τ2 are the same as in Theorem 2.4.Theorem 3.2.
Suppose thatu(x,y) is a solution of (3.1)-(3.2), |a(x)+b(y)|≤k(x,y), and |F(s,t,x,y)|≤f(s,t)|x|q+|y|, |W(ξ,η,x)|≤h(ξ,η)|x|m, where f,h,q,m are defined as in Theorem 2.4; then
(3.3)|u(x,y)|≤[B1(x,y)+∫y0yeB2(y,σ(t))B2(x,t)B1(x,t)Δt]1/p,(x,y)∈T0×T̃0,
where
(3.4)B1(x,y)=k(x,y)+∫y0y∫x0x[f(s,t)p-qpKq/p+∫y0t∫x0sh(ξ,η)p-mpKm/pΔξΔη]ΔsΔt,∀K>0,
and B2(x,y) is defined as in Theorem 2.4 (with g(x,y)≡0).Proof.
The equivalent integral equation of (3.1) can be denoted by(3.5)up(x,y)=a(x)+b(y)+∫y0y∫x0xF(s,t,u(τ1(s),τ2(t)),∫y0t∫x0sW(ξ,η,u(τ1(ξ),τ2(η)))ΔξΔη)ΔsΔt.
Then
(3.6)|up(x,y)|≤k(x,y)+∫y0y∫x0x|F(s,t,u(τ1(s),τ2(t)),∫y0t∫x0sW(ξ,η,u(τ1(ξ),τ2(η)))ΔξΔη)|ΔsΔt≤k(x,y)+∫y0y∫x0x[f(s,t)|u(τ1(s),τ2(t))|q+|∫y0t∫x0sW(ξ,η,u(τ1(ξ),τ2(η)))ΔξΔη|]ΔsΔt≤k(x,y)+∫y0y∫x0x[∫x0sh(ξ,η)f(s,t)|u(τ1(s),τ2(t))|q+∫y0t∫x0sh(ξ,η)|u(τ1(ξ),τ2(η))|mΔξΔη]ΔsΔt,
and a suitable application of Theorem 2.4 to (3.6) yields the desired inequality (3.3).Theorem 3.3.
Under the conditions of Theorem3.2, one has
(3.7)|u(x,y)|≤[B1(x,y)eB2(y,y0)]1/p,(x,y)∈T0×T̃0,
where B1,B2 are defined as in Theorem 3.2.Proof.
The desired inequality can be obtained by an application of Theorem2.5 to (3.6).Example 3.4.
Consider the following delay dynamic integral equation:(3.8)up(x,y)=C+∫y0y∫x0xF1(s,t,u(τ1(s),τ2(t)),∫y0t∫x0sW1(ξ,η,u(τ1(ξ),τ2(η)))ΔξΔη)ΔsΔt+∫y0N∫x0MF2(s,t,u(τ1(s),τ2(t)),∫y0t∫x0sW2(ξ,η,u(τ1(ξ),τ2(η)))ΔξΔη)ΔsΔt,(x,y)∈([x0,M]⋂T)×([y0,N]⋂T),
with the initial condition
(3.9)u(x,y)=ϕ(x,y),ifx∈[α,x0]⋂T,ory∈[β,y0]⋂T,|ϕ(τ1(x),τ2(y))|≤|C|1/p,ifτ1(x)≤x0,orτ2(y)≤y0,∀(x,y)∈T0×T̃0,
where u∈Crd(𝕋0×𝕋̃0,ℝ), p>0 is a constant, C=up(x0,y0), M∈𝕋0,N∈𝕋̃0 are two fixed numbers. ϕ∈Crd(([α,x0]×[β,y0])⋂𝕋2,ℝ),Fi∈(𝕋0×𝕋̃0×ℝ2,ℝ),Wi∈(𝕋0×𝕋̃0×ℝ,ℝ),i=1,2. α,β,τ1,τ2 are the same as in Theorem 2.4.Theorem 3.5.
Supposeu(x,y) is a solution of (3.8)-(3.9) and |Fi(s,t,x,y)|≤L(s,t,|x|)+|y|,|Wi(ξ,η,x)|≤hi(ξ,η)|x|q,i=1,2, where L,hi,i=1,2,q are defined the same as in Theorem 2.11; then the following inequality holds:
(3.10)|u(x,y)|≤{[λ̂+B̂61-B̂5]B̂3(x,y)+B̂4(x,y)}1/p,(x,y)∈([x0,M]⋂T)×([y0,N]⋂T),
provided that B̂5<1, where λ̂,B̂2(x,y),B̂3(x,y),B̂4(x,y),B̂5,B̂6 are defined the same as in Theorem 2.11, and
(3.11)B̂1(x,y)=|C|+∫y0y∫x0x[L(s,t,p-1pK1/p)+∫y0t∫x0sh1(ξ,η)p-qpKq/pΔξΔη]ΔsΔt,∀K>0.Proof.
From (3.8) we have
(3.12)|up(x,y)|≤|C|+∫y0y∫x0x|F1(s,t,u(τ1(s),τ2(t)),∫y0t∫x0sW1(ξ,η,u(τ1(ξ),τ2(η)))ΔξΔη)|ΔsΔt+∫y0N∫x0M|F2(s,t,u(τ1(s),τ2(t)),∫y0t∫x0sW2(ξ,η,u(τ1(ξ),τ2(η)))ΔξΔη)|ΔsΔt≤|C|+∫y0y∫x0x[L(s,t,|u(τ1(s),τ2(t))|)+|∫y0t∫x0sW1(ξ,η,u(τ1(ξ),τ2(η)))ΔξΔη|]ΔsΔt+∫y0N∫x0M[L(s,t,|u(τ1(s),τ2(t))|)+|∫y0t∫x0sW2(ξ,η,u(τ1(ξ),τ2(η)))ΔξΔη|]ΔsΔt≤|C|+∫y0y∫x0x[L(s,t,|u(τ1(s),τ2(t))|)+∫y0t∫x0sh1(ξ,η)|u(τ1(ξ),τ2(η))|qΔξΔη]ΔsΔt+∫y0N∫x0M[L(s,t,|u(τ1(s),τ2(t))|)+∫y0t∫x0sh2(ξ,η)|u(τ1(ξ),τ2(η))|qΔξΔη]ΔsΔt.
So by use of Theorem 2.11 we obtain the desired inequality (3.10).Example 3.6.
Consider the following delay dynamic integral equation:(3.13)u(x,y)=C+∫y0y∫x0xF(s,t,u(τ1(s),τ2(t)),∫y0t∫x0sW(ξ,η,u(τ1(ξ),τ2(η)))ΔξΔη)ΔsΔt,
where u∈Crd(𝕋0×𝕋̃0,ℝ), C=up(x0,y0),F∈(𝕋0×𝕋̃0×ℝ2,ℝ),W∈(𝕋0×𝕋̃0×ℝ,ℝ). τ1,τ2 are the same as in Theorem 2.4.Theorem 3.7.
Assume that|F(s,t,u1,v1)-F(s,t,u2,v2)|≤f(s,t)|u1-u2|+|v1-v2|,|W(s,t,u1)-W(s,t,u2)|≤h(s,t)|u1-u2|, where f,h are defined as in Theorem 2.4, and; furthermore, assume that τ1(x)≥x0,τ2(y)≥y0, then (3.13) has at most one solution.Proof.
Suppose thatu1(x,y),u2(x,y) are two solutions of (3.13); then we have
(3.14)|u1(x,y)-u2(x,y)|≤|∫y0y∫x0x[F(s,t,u1(τ1(s),τ2(t))∫y0t∫x0sW(ξ,η,u1(τ1(ξ),τ2(η)))ΔξΔη)-F(s,t,u2(τ1(s),τ2(t))∫y0t∫x0sW(ξ,η,u2(τ1(ξ),τ2(η)))ΔξΔη)]ΔsΔt|≤∫y0y∫x0x|F(s,t,u1(τ1(s),τ2(t))∫y0t∫x0sW(ξ,η,u1(τ1(ξ),τ2(η)))ΔξΔη)-F(s,t,u2(τ1(s),τ2(t))∫y0t∫x0sW(ξ,η,u2(τ1(ξ),τ2(η)))ΔξΔη)|ΔsΔt≤∫y0y∫x0xf(s,t)|u1(τ1(s),τ2(t))-u2(τ1(s),τ2(t))|ΔsΔt+∫y0y∫x0x∫y0t∫x0s|W(ξ,η,u1(τ1(ξ),τ2(η)))-W(ξ,η,u2(τ1(ξ),τ2(η)))|ΔξΔηΔsΔt≤∫y0y∫x0xf(s,t)|u1(τ1(s),τ2(t))-u2(τ1(s),τ2(t))|ΔsΔt+∫y0y∫x0x∫y0t∫x0sh(ξ,η)|u1(τ1(ξ),τ2(η))-u2(τ1(ξ),τ2(η))|ΔξΔηΔsΔt,.
Then a suitable application of Theorem 2.8 yields |u1(x,y)-u2(x,y)|≤0, that is, u1(x,y)≡u2(x,y), and the proof is complete.


## 4. Conclusions

In this paper, we established some new Gronwall-Bellman type integral inequalities on time scales. As one can see, the presented results provide a handy tool in the quantitative as well as qualitative analysis of solutions of certain delay dynamic equations on time scales. The established inequalities unify some known continuous and discrete inequalities.



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*Source: 754350-2011-10-13.xml*