# Some Difference Inequalities for Iterated Sums with Applications

Authors: Wu-Sheng Wang; Shanhe Wu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The main objective of this paper is to establish two new nonlinear sum-difference
inequalities with multiple iterated sums. Under several practical assumptions, the inequalities
are solved through rigorous analysis, and explicit bounds for the unknown
functions are given clearly. These new inequalities can be used as handy tools in the
study of the estimation of solutions of difference equations.


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## Body

## 1. Introduction

One of the best known and widely used inequalities in the study of nonlinear differential equations is Gronwall-Bellman inequality [1, 2], which can be stated as follows: if u and f are nonnegative continuous functions on an interval [a,b] satisfying
(1)u(t)≤c+∫at‍f(s)u(s)ds,t∈[a,b],
for some constant c≥0, then
(2)u(t)≤cexp(∫atf(s)ds),t∈[a,b].It has become one of the very few classic and most influential results in the theory and applications of inequalities. Because of its fundamental importance, over the years, many generalizations and analogous results of (2) have been established, such as [3–14].Among these references, Baĭnov and Simeonov [4, P. 107] considered the following interesting Gronwall-type inequality:
(3)u(t)≤a(t)+∑i=1n‍∫αt‍f1(t,t1)×(∫αt1‍f2(t1,t2)⋯(∫αti-1‍fi(ti-1,ti)u(ti)dti)⋯)dt1.
Kim [8] considered analogous Gronwall-type integral inequalities involving iterated integrals,
(4)u(t)≤a+b(t)(∫αt‍f1(t1)u(t1)logu(t1)dt1+∑i=2n‍∫αt‍g1(t1)(∫αt1‍g2(t2)+∑i=2n‍∫αt‍g1(t1)(h×(∫αt1‍⋯(∫αti-2‍gi-1(ti-1)+∑i=2n‍∫αt‍g1(t1)(∫αt1‍g2(t2)×(∫αti-1‍fi(ti)u(ti)+∑i=2n‍∫αt‍g1(t1)(∫αt1‍g2(t2)×(×log(u(ti))dti∫αti-1‍)dti-1)+∑i=2n‍∫αt‍g1(t1)(∫αt1‍g2(t2)×(⋯∫αti-1‍)dt2∫αt‍)dt1).In 2011, Abdeldaim and Yakout [12] studied some new integral inequalities of Gronwall-Bellman-Pachpatte type such as
(5)u(t)≤u0+∫0t‍f(s)u(s)u0+∫0t‍×[u(s)+∫0s‍h(τ)[∫0τ‍u(τ)u0+∫0t‍×[u(s)+∫0s‍h(τ)[+∫0τ‍g(ξ)u(ξ)dξ]dτ]ds,u(t)≤u0+∫0t‍[f(s)u(s)+q(s)]ds+∫0t‍f(s)u(s)[u(s)+∫0sg(τ)u(τ)dτ]ds.Along with the development of the theory of integral inequalities and the theory of difference equations, more and more attentions are paid to discrete versions of Gronwall-type inequalities; for detailed information, please refer to the literatures [15–35]. For instance, Pachpatte [19] considered the following discrete inequality:
(6)u(n)≤u0+∑s=n0n-1‍f(s)u(s)+∑s=n0n-1‍g(s)×(∑t=n0s-1‍h(t)(∑τ=n0t-1‍k(τ)up(τ))).In 2006, Cheung and Ren [24] studied
(7)up(m,n)≤c+∑s=m0m-1‍∑t=n0n-1‍a(s,t)uq(s,t)+∑s=m0m-1‍∑t=n0n-1‍b(s,t)uq(s,t)w(u(s,t)).Later, Zheng et al. [31] discussed the following discrete inequality:
(8)u(n)≤a(n)+∑i=1k‍∑s=0n-1‍fi(n,s)wi(u(s)).In 2012, Zhou et al. [33] studied the following inequalities:
(9)u(n)≤a(n)+∑s=n0n-1‍f1(n,s)w(u(s))+∑s=n0n-1‍f1(n,s)w(u(s))∑τ=n0s-1‍f2(s,τ)w(u(τ))+∑s=n0n-1‍f1(n,s)w(u(s))×∑τ=n0s-1‍f2(s,τ)∑ξ=n0τ-1‍f3(τ,ξ)w(u(ξ)),u(n)≤a(n)+∑s=n0n-1‍f1(n,s)w1(u(s))+∑s=n0n-1‍f1(n,s)w1(u(s))∑τ=n0s-1‍f2(s,τ)w2(u(τ))+∑s=n0n-1‍f1(n,s)w1(u(s))×∑τ=n0s-1‍f2(s,τ)w2(u(τ))∑ξ=n0τ-1‍f3(τ,ξ)w3(u(ξ)).However, the above results are not applicable to some certain inequalities with multiple iterated sums. Hence, it is desirable to consider more general difference inequalities of these extended types. They can be used in the study of certain classes of difference equations or applied in many practical engineering problems.Motivated by the results given in [7, 8, 12, 19, 24, 25, 29, 33], in this paper we discuss the following two types of inequalities:
(10)u(n)≤a(n)+∑t1=n0n-1‍f1(n,t1)×(∑t2=n0t1-1‍f2(t1,t2)⋯(∑tk=n0tk-1‍fk(tk-1,tk)up(tk))⋯),(11)u(n)≤a(n)+c(n)[∑t1=n0n-1‍f1(n,t1)g(u(t1))+∑i=2k‍∑t1=n0n-1‍f1(n,t1)+c(n)[×(∑t2=n0t1-1‍f2(t1,t2)⋯×(∑ti=n0ti-1‍fi(ti-1,ti)+c(n)[(∑t2=n0t1-1‍f2(t1,t2)⋯×(×g(u(ti))∑ti=n0ti-1‍)⋯)].All the assumptions on (10) and (11) are given in the next sections. The inequalities (10) and (11) consist of multiple iterated sums. Under several practical assumptions, the inequalities are solved through rigorous analysis, and explicit bounds for the unknown functions are given clearly. Further, the derived results are applied to study the estimation of solutions of difference equations.


## 2. Main Results

Throughout this paper, letℕn0:={n0,n0+1,n0+2,…} and ℕn0b:={n0,n0+1,n0+2,…,n0+n=b}(n0∈ℕ0,n,b∈ℕ). For function u(n), its difference is defined by Δu(n)=u(n+1)-u(n). Obviously, the linear difference equation Δu(n)=f(n) with the initial condition u(n0)=0 has the solution u(n)=∑s=n0n-1‍f(s). For convenience, in the sequel we complementarily define that ∑s=n0n0-1‍f(s)=0.Lemma 1.
Letu(n), a(n) and c(n) be real-valued nonnegative functions defined on ℕ0 and satisfy the inequality
(12)Δu(n)≤a(n)u(n)+c(n),∀n∈ℕn0,
where un0 is a nonnegative constant. Then,
(13)u(n)≤(un0+∑s=n0n-1‍c(s)∏t=n0s‍(1+a(t))-1)hhh×∏s=n0n-1‍(1+a(s)),∀n∈ℕn0.Proof.
From (12), we have
(14)u(n+1)-(1+a(n))u(n)≤c(n),∀n∈ℕn0.
Multiplying by∏s=n0n-1‍(1+a(s))-1 on both sides of the above inequality (14) and summing up both sides from n0 to n-1, we obtain
(15)u(n)∏s=n0n-1‍(1+a(s))-1-un0≤∑s=n0n-1‍c(s)∏t=n0s‍(1+a(t))-1,∀n∈ℕn0.
From (15), we obtain the desired estimate (13).Theorem 2.
Letu(n) and a(n) be nonnegative functions defined on ℕn0 with a(n) nondecreasing on ℕn0. Moreover, let fi(n,s), i=1,2,…,k, be nonnegative functions for n0≤s≤n  (n0,n,s∈ℕn0) and nondecreasing in n for fixed s∈ℕn0. If p≥0 and p is not equal to 1, then the discrete inequality (10) gives
(16)u(n)≤V1(n,n),∀n∈ℕn0b1,
where V1(n,n) can be successively determined from the formulas
(17)Vk(M,n)=exp(W1-1(W1(ln(a(M))+∑s=n0M-1‍(∑i=1k-1‍fi(M,s)))=exp(W1-1(ii+∑s=n0n-1‍fk(M,s))),(18)Vj(M,n)≤(a(M)+∑s=n0n-1‍fj(M,s)Vj+1(M,s)≤(h×∏t=n0s‍(1+(∑i=1j-1‍fi(M,t)-fj(M,t)))-1)×∏s=n0n-1‍(1+(∑i=1j-1‍fi(M,s)-fj(M,s)))=Vj(M,n),
for j=k-1,…,2,1, n∈ℕn0M,
(19)W1(x)=∫x0xdsexp((p-1)s),x0>0,
where W1-1 is the inverse functions of W1, M∈ℕn0, M≤b1 is chosen arbitrarily, and b1 is the largest natural number such that
(20)W1(ln(a(b1))+∑s=n0b1-1‍(∑i=1k-1‍fi(b1,s)))+∑s=n0b1-1‍fk(b1,s)∈Dom(W1-1).Remark 3.
Firstly, from (17) and (18), we obtain V1(M,n); then let M=n, and we get V1(n,n) since M is chosen arbitrarily.Remark 4.
We can obtainb1 using MATLAB program: firstly let b1=n0, when W1(ln(a(b1))+∑s=n0b1-1‍(∑i=1k-1‍fi(b1,s)))+∑s=n0b1-1‍fk(b1,s)<W1(∞); let b1=b1+1, when W1(ln(a(b1))+∑s=n0b1-1‍(∑i=1k-1‍fi(b1,s)))+∑s=n0b1-1‍fk(b1,s)<W1(∞); let b1=b1+1, and so on until W1(ln(a(b1))+∑s=n0b1-1‍(∑i=1k-1‍fi(b1,s)))+∑s=n0b1-1‍fk(b1,s)≥W1(∞). If W1(ln(a(b1))+∑s=n0b1-1‍(∑i=1k-1‍fi(b1,s)))+∑s=n0b1-1‍fk(b1,s)<W1(∞), for all b1∈ℕn0, then b1=∞.Proof.
FixM∈ℕn0b1, where M is chosen arbitrarily and b1 is defined by (20). For n∈ℕn0M, from (10) we have
(21)u(n)≤a(M)+∑t1=n0n-1‍f1(M,t1)×(∑t2=n0t1-1‍f2(M,t2)⋯(∑tk=n0tk-1‍fk(M,tk)up(tk))⋯).
Now we introduce the functions(22)v1(n)=a(M)+∑t1=n0n-1‍f1(M,t1)×(∑t2=n0t1-1‍f2(M,t2)⋯(∑tk=n0tk-1‍fk(M,tk)up(tk))⋯),(23)vj(n)=vj-1(n)+∑tj=n0n-1‍fj(M,tj)×(∑tj+1=n0tj-1‍fj+1(M,tj+1)⋯×(h×(∑tk=n0tk-1‍fk(M,tk)up(tk))⋯).
Forn∈ℕn0M and j=2,3,…,k, then vj, j=1,2,…,k, are all positive and nondecreasing functions on ℕn0M with vj(n0)=a(M), j=1,2,…,k, and the inequalities (22) and (23) imply that
(24)u(n)≤v1(n)≤v2(n)≤⋯≤vk(n),∀n∈ℕn0M.
From (22), we observe that
(25)Δv1(n)=v1(n+1)-v1(n)=f1(M,n)×(∑t2=n0n-1‍f2(M,t2)⋯(∑tk=n0tk-1‍fk(M,tk)up(tk))⋯)≤f1(M,n)(v2(n)-v1(n))=-f1(M,n)v1(n)+f1(M,n)v2(n),∀n∈ℕn0M.
We claim that(26)Δvj(n)≤(∑i=1j-1‍fi(M,n)-fj(M,n))vj(n)+fj(M,n)vj+1(n),(27)Δvk(n)≤(∑i=1k-1‍fi(M,n))vk(n)+fk(M,n)vkp(n),
for n∈ℕn0M, j=2,3,…,k-1.
Now we prove (26) and (27) by induction. Obviously, (26) is true for j=1 by (26). We make the inductive assumption that (26) is true for j-1. By the inductive assumption and (24), from (23) we obtain
(28)Δvj(n)≤Δvj-1(n)+fj(M,n)×(∑tj+1=n0tj-1‍fj+1(M,tj+1)⋯×(h×(∑tk=n0tk-1‍fk(M,tk)up(tk))⋯)≤(∑i=1j-2‍fi(M,n)-fj-1(M,n))vj-1(n)+fj-1(M,n)vj(n)+fj(M,n)(vj+1(n)-vj(n))≤(∑i=1j-2‍fi(M,n))vj(n)+fj-1(M,n)vj(n)+fj(M,n)vj+1(n)-fj(M,n)vj(n)=(∑i=1j-1‍fi(M,n)-fj(M,n))vj(n)+fj(M,n)vj+1(n),∀n∈ℕn0M.
It actually proves (26) by induction. From (23) and (26), we have
(29)Δvk(n)=Δvk-1(n)+fk(M,n)up(n)≤(∑i=1k-2‍fi(M,n)-fk-1(M,n))vk-1(n)+fk-1(M,n)vk(n)+fk(M,n)vkp(n)≤(∑i=1k-1‍fi(M,n))vk(n)+fk(M,n)vkp(n),+fk-1(M,n)vk(n)+fk(M,n)vkp∀n∈ℕn0M.
It proves (27). From (27), we have
(30)Δvk(n)vk(n)≤(∑i=1k-1‍fi(M,n))+fk(M,n)vkp-1(n),hhhhhhhhhhhhhhhhhhhhhhhhhhh∀n∈ℕn0M.
On the other hand, by the mean-value theorem for integrals, for arbitrarily given integersn,n+1∈ℕn0M, there exists ξ in the open interval (vk(n),vk(n+1)) such that
(31)ln(vk(n+1))-ln(vk(n))=∫vk(n)vk(n+1)‍dss=Δvk(n)ξ≤Δvk(n)vk(n),=Δvk(n)ξh∀n∈ℕn0M.
By settingn=s in (31) and substituting s=n0,n0+1,n0+2,…,n-1, successively, we obtain
(32)ln(vk(n))≤ln(vk(n0))+∑s=n0n-1‍(∑i=1k-1‍fi(M,s))+∑s=n0n-1‍fk(M,s)vkp-1(s)≤ln(vk(n0))+∑s=n0M-1‍(∑i=1k-1‍fi(M,s))+∑s=n0n-1‍fk(M,s)vkp-1(s),∀n∈ℕn0M.
Letw1(n) denote the right-hand side of (32), which is a positive and nondecreasing function on ℕn0M with
(33)w1(n0)=ln(vk(n0))+∑s=n0M-1‍(∑i=1k-1‍fi(M,s)).
Then, (32) is equivalent to
(34)vk(n)≤exp(w1(n)),∀n∈ℕn0M.
By the definition ofw1, we obtain
(35)Δw1(n)=fk(M,n)vkp-1(n)≤fk(M,n)exp((p-1)w1(n)),≤fk(M,n)exp(p-)hh∀n∈ℕn0M.
From (34) and (35), we get
(36)Δw1(n)exp((p-1)w1(n))≤fk(M,n),∀n∈ℕn0M.
Once again, performing the similar procedure from (30) to (32), (36) gives
(37)W1(w1(n))≤W1(w1(n0))+∑s=n0n-1‍fk(M,s),≤W1(w1(n0))+∑s=n0n-1h‍∀n∈ℕn0M,
where W1 is defined by (19). By combining (33), (34), and (37), we can obtain that
(38)vk(n)≤exp(w1(n))≤exp(W1-1(W1(w1(n0))+∑s=n0n-1‍fk(M,s)))≤exp(W1-1(W1(∑s=n0M-1‍ln(a(M))≤exp(W1-1(W1(hl+∑s=n0M-1‍(∑i=1k-1‍fi(M,s)))≤exp(W1-1(hl+∑s=n0n-1‍fk(M,s)))=Vk(M,n),∀n∈ℕn0M,
where Vk(M,n) is defined by (17). Applying Lemma 1 to (26) for j=k-1,…,2,1, we have
(39)vj(n)≤(a(M)+∑s=n0n-1‍fj(M,s)Vj+1(M,s)≤(×∏t=n0s‍(1+(∑i=1j-1‍fi(M,t)-fj(M,t)))-1)×∏s=n0n-1‍(1+(∑i=1j-1‍fi(M,s)-fj(M,s)))=Vj(M,n),∀n∈ℕn0M,
where Vj(M,n) is defined by (18). From (24) and (39), we have
(40)u(n)≤v1(n)≤V1(M,n),∀n∈ℕn0M.
SinceM∈ℕn0 is arbitrary, from (40), we get the required estimate
(41)u(n)≤V1(n,n),∀n∈ℕn0b1,
where b1 is defined by (20). Theorem 2 is proved.Theorem 5.
Letu(n), a(n), and c(n) be nonnegative functions defined on ℕn0 with a(n) and c(n) nondecreasing on ℕn0, and let fi(n,s), i=1,2,…,k, be nonnegative functions for n,s∈ℕn0, n0≤s≤n, which are nondecreasing in n for fixed s∈ℕn0. Suppose that g(u) is a nondecreasing continuous function on [0,∞) with g(u)>0 for u>0. The inequality (11) implies that
(42)u(n)≤G-1(G(a(n))+c(n)∑s=n0n-1‍f1(n,s)E(n,s)),≤G-1(G(a(n))+c(n)∑s=n0n-1‍f1hhhh)∀n∈ℕn0b2,
where G-1 is the inverse function of G,
(43)G(u)=∫u0u‍dsg(s),u0>0,(44)E(n,s)≔{1+f2(n,s)[1+f3(n,s)≔{1+f2(n,s)h×(⋯(1+fk-1(n,s)⋯≔{1+f2(n,s)hhhhhh×(1+fk(n,s)))⋯)]},
and b2 is the largest natural number such that
(45)G(a(b2))+c(b2)∑s=n0b2-1f1(b2,s)E(b2,s)∈Dom(G-1).Remark 6.
We can obtainb2 using Matlab program similar to Remark 4.Proof.
Let the functiona(n) be positive. Fix M∈ℕn0b2, where M is chosen arbitrarily and b2 is defined by (45). For n∈ℕn0M, from (11) we have
(46)u(n)≤a(M)+c(M)[∑t1=n0n-1‍f1(M,t1)g(u(t1))≤a(M)+c(M)[+∑i=2k‍∑t1=n0n-1‍f1(M,t1)≤a(M)+c(M)[×(∑t2=n0t1-1‍f2(M,t2)⋯≤a(M)+c(M)[(h×(∑ti=n0ti-1‍fi(M,ti)g(u(ti)))⋯)].
We denote the right-hand side of (46) by y(n) for n∈ℕn0M. Then y(n0)=a(M), the function y(n) is positive and nondecreasing in n∈ℕn0M, u(n)≤y(n), and
(47)Δy(n)=c(M)[∑ti=n0ti-1‍f1(M,n)g(u(n))+f1(M,n)=c(M)[h×(∑t2=n0n-1‍f2(M,t2)g(u(t2)))+∑i=3k‍f1(M,n)=c(M)[h×(∑t2=n0n-1‍f2(M,t2)⋯=c(M)[(hh×(∑ti=n0ti-1‍fi(M,ti)g(u(ti)))⋯)]=c(M)f1(M,n)[g(u(n))+∑t2=n0n-1‍f2(M,t2)g(u(t2))=c(M)f1(M,n)[+∑i=3k‍(∑t2=n0n-1‍f2(M,t2)⋯=c(M)f1(M,n)[+∑i=3k‍(×(∑ti=n0ti-1‍fi(M,ti)g(u(ti)))⋯)],=c(M)f1(M,n)[g(u(n))+∑t2=n0n-1‍f2(M,t2)g(u)∀n∈ℕn0M.
Define a functiony1(n) by
(48)y1(n)=∑t2=n0n-1‍f2(M,t2)g(u(t2))+∑i=3k‍(∑t2=n0n-1‍f2(M,t2)⋯+∑i=3k‍(h×(∑ti=n0ti-1‍fi(M,ti)g(u(ti)))⋯∑ti=n0ti-1‍),
for all n∈ℕn0M. From (47) and (48), we have
(49)Δy(n)=c(M)f1(M,n)[g(u(n))+y1(n)],∀n∈ℕn0M.
From (48), we have
(50)Δy1(n)=f2(M,n)[g(u(n))+∑t3=n0n-1‍f3(M,t3)g(u(t3))=f2(M,n)[+∑i=4k‍(∑t3=n0n-1‍f3(M,t3)⋯=f2(M,n)[+∑i=4k‍(×(∑ti=n0ti-1‍fi(M,ti)g(u(ti)))⋯)],=f2(M,n)[g(u(n))+∑t3=n0n-1‍f3(M,t3)g(u)hh∀n∈ℕn0M.
From (50), we get
(51)Δy1(n)=f2(M,n)[g(u(n))+y2(n)],∀n∈ℕn0M,
where
(52)y2(n)=∑t3=n0n-1‍f3(M,t3)g(u(t3))+∑i=4k‍(∑t2=n0n-1‍f3(M,t3)⋯(∑ti=n0ti-1‍fi(M,ti)g(u(ti)))⋯),
for all n∈ℕn0M.
Continuing in this way, we obtain(53)Δyk-2(n)=fk-1(M,n)[g(u(n))+yk-1(n)],∀n∈ℕn0M,
where
(54)yk-1(n)=∑tn=n0n-1‍fk(M,tk)g(u(tk)),∀n∈ℕn0M.
From (54) and the inequality u(n)≤y(n), we have
(55)Δyk-1(n)g(y(n))≤fk(M,n),∀n∈ℕn0M.
We define the functionsy~(s),y~i(s)(i=1,2…,k-1), which are nondecreasing and continuously differentiable on [n0,∞) with y~(n)=y(n), y~i(n)=yi(n)(i=1,2…,k-1) on ℕn0M.
On the other hand, by the formula of partial integration, we have(56)∫nn+1‍y~k-1′(s)g(y~(s))ds=yk-1(n)g(y(n))+∫nn+1‍y~k-1(s)g′(y~(s))y~′(s)g2(y~(s))ds,∀n∈ℕn0M.
By the monotonicity ofg, y, from (56) we have
(57)∫nn+1‍y~k-1′(s)g(y~(s))ds≥yk-1(n)g(y(n)),∀n∈ℕn0M.
By the mean-value theorem for integrals, for arbitrarily given integersn,n+1∈ℕn0M, there exists ξ in the open interval (n,n+1) such that
(58)∫nn+1‍y~k-1′(s)g(y~(s))ds=∫nn+1‍d(y~k-1(s))g(y~(s))=1g(y(ξ))∫nn+1‍d(y~k-1(s))≤Δyk-1(n)g(y(n)),∀n∈ℕn0M.
From (55), (57), and (58), we have
(59)yk-1(n)g(y(n))≤Δyk-1(n)g(y(n))≤fk(M,n),∀n∈ℕn0M.
Next, from the inequalities (53) and (59), we have
(60)Δyk-2(n)g(y(n))≤fk-1(M,n)[1+fk(M,n)],∀n∈ℕn0M.
Once again, applying the same procedure from (56) to (59) to the inequality (60), we get
(61)yk-2(n)g(y(n))≤Δyk-2(n)g(y(n))≤fk-1(M,n)[1+fk(M,n)],hhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀n∈ℕn0M.
Proceeding in this way, we obtain(62)y1(n)g(y(n))≤Δy1(n)g(y(n))≤f2(M,n)≤×{1+f3(M,n)≤{hh×[1+f4(M,n)≤{hh×[×(⋯(1+fk-1(M,n)≤{hh×[×(⋯(×(1+fk(M,n)))⋯)]},≤{hhh×(1+))⋯)]}hhhhhhhlh,∀n∈ℕn0M.
Using the inequalities (49) and (62), we have
(63)Δy(n)g(y(n))≤c(M)f1(M,n)×{1+f2(M,n)[1+f3(M,n)×{1+f2(M,n)[×(⋯(1+fk-1(M,n)×{1+f2(M,n)[×(×1+fk(M,n))⋯)]}=c(M)f1(M,n)E(M,n),∀n∈ℕn0M,
where E(M,n) is defined by (44).
Once again, using the mean-value theorem for integrals, for arbitrarily given integersn,n+1∈ℕn0M, there exists ξ in the open interval (y(n),y(n+1)) such that
(64)G(y(n+1))-G(y(n))=∫y(n)y(n+1)‍dsg(s)=Δy(n)g(ξ)≤Δy(n)g(y(n)),=Δy(n)g(ξ)hh∀n∈ℕn0M,
where G is defined by (43). Using (63), (64), and y(n0)=a(M), we obtain
(65)u(n)≤y(n)≤G-1(∑s=n0n-1‍G(a(M))y(n)≤G-1(+c(M)∑s=n0n-1‍f1(M,s)E(M,s)),≤y(n)≤G-1(∑s=n0n-1‍G(a(M))+c(M)∑s=n0n-1‍∀n∈ℕn0M.
In (65), let n=M; we have
(66)u(M)≤G-1(G(a(M))+c(M)∑s=n0M-1‍f1(M,s)E(M,s)),(G(a(M))+c(M)∑s=n0M-1‍f1(M,s)E(M,s)h)∀n∈ℕn0M.
Due to the randomness ofT, (42) is achieved immediately from (66).


## 3. Application

In this section, we apply Theorem5 to the following difference equation:
(67)Δx(n)=F(n,x(n),∑s=n0n-1‍z(s,x(s))),∀n∈ℕn0.Corollary 7.
Assume thatF is defined on ℕn0×[0,∞)×[0,∞), and there exist nonnegative functions di(n), i=1,2, such that
(68)|F(n,x,y)|≤d1(n)g(|x|)+d1(n)y,|z(s,x)|≤d2(n)g(|x|),
where the function g is as in Theorem 5. If x(n) is any solution of the problem (67), then
(69)|x(n)|≤G-1(G(|x(n0)|)+∑s=n0n-1‍d1(s)E(s)),∀n∈ℕn0b3,
where the functions G, G-1 are as in Theorem 5,
(70)E(n)=1+d1(n)(1+d2(n)),
and b3 is the largest natural number such that
(71)G(|x(n0)|)+∑s=n0b3-1‍d1(s)E(s)∈Dom(G-1).Proof.
The difference equation (67) is equivalent to
(72)x(n)=x(n0)+∑s=n0n-1‍F(s,x(s),∑t=n0s-1‍z(t,x(t))),∑s=n0n-1‍F(s,x(s),∑t=n0s-1‍z(t,x(t)))lh∀n∈ℕn0.
Using (68), from (72), we have
(73)|x(n)|≤|x(n0)|+∑s=n0n-1‍d1(s)g(|x(s)|)+∑s=n0n-1‍d1(s)(∑t=n0s-1‍d2(t)g(|x(s)|)),(s)(∑t=n0s-1‍d2(t)g(|x(s)|))∀n∈ℕn0.
Notice that, by the assumption, all functions in (73) satisfy the conditions of Theorem 5. Applying Theorem 5 to the inequality (73), (69) is immediately derived. This completes the proof of Corollary 7.



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*Source: 804152-2013-12-19.xml*