# Some New Volterra-Fredholm-Type Discrete Inequalities and Their Applications in the Theory of Difference Equations

Authors: Bin Zheng; Qinghua Feng
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Some new Volterra-Fredholm-type discrete inequalities in two independent variables are
established, which provide a handy tool in the study of qualitative and quantitative properties of solutions
of certain difference equations. The established results extend some known results in the literature.


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## Body

## 1. Introduction

In the research of solutions of certain differential and difference equations, if the solutions are unknown, then it is necessary to study their qualitative and quantitative properties such as boundedness, uniqueness, and continuous dependence on initial data. The Gronwall-Bellman inequality [1, 2] and its various generalizations which provide explicit bounds play a fundamental role in the research of this domain. Many such generalized inequalities (e.g., see [3–30] and the references therein) have been established in the literature including the known Ou-Liang's inequality [3]. In [8], Ma generalized the discrete version of Ou-Liang's inequality in two variables to Volterra-Fredholm form for the first time, which has proved to be very useful in the study of qualitative as well as quantitative properties of solutions of certain Volterra-Fredholm-type difference equations. But since then, few results on Volterra-Fredholm-type discrete inequalities have been established. Recent results in this direction include the work of Ma [9] to our knowledge. We notice, in the analysis of some certain Volterra-Fredholm-type difference equations with more complicated forms, that the bounds provided by the earlier inequalities are inadequate and it is necessary to seek some new Volterra-Fredholm-type discrete inequalities in order to obtain a diversity of desired results.Our aim in this paper is to establish some new generalized Volterra-Fredholm-type discrete inequalities, which extend Ma's work in [9], and provide new bounds for unknown functions lying in these inequalities. We will illustrate the usefulness of the established results by applying them to study the boundedness, uniqueness, and continuous dependence on initial data of solutions of certain more complicated Volterra-Fredholm-type difference equations.Throughout this paper,ℝ denotes the set of real numbers  ℝ+=[0,∞), and ℤ denotes the set of integers, while ℕ0 denotes the set of nonnegative integers. Let Ω:=([m0,M]×[n0,N])⋂ℤ2, where m0,n0∈ℤ and M,N∈ℤ⋃{∞} are two constants. l1,l2∈ℤ are two constants, and Ki>0,i=1,2,3,4, are all constants. If U is a lattice, then we denote the set of all ℝ-valued functions on U by ℘(U) and denote the set of all ℝ+-valued functions on U by ℘+(U). Finally, for a function f∈℘+(U), we have ∑s=m0m1f=0 provided m0>m1.


## 2. Main Results

Lemma 2.1 (see [15]).
Assume thata≥0,p≥q≥0, and p≠0 then for any K>0(2.1)aq/p≤qpK(q-p)/pa+p-qpKq/p.Lemma 2.2.
Suppose thatu(m,n)∈℘+(Ω),b(s,t,m,n)∈℘+(Ω2), α≥0 is a constant. If b is nondecreasing in the third variable, then, for (m,n)∈Ω,
(2.2)u(m,n)≤α+∑s=m0m-1∑t=n0n-1b(s,t,m,n)u(s,t)
implies that
(2.3)u(m,n)≤αexp⁡{∑s=m0m-1∑t=n0n-1b(s,t,m,n)}.Lemma 2.3.
Suppose thatu(m,n),a(m,n),b(m,n)∈℘+(Ω). If a(m,n) is nondecreasing in the first variable, then, for (m,n)∈Ω,
(2.4)u(m,n)≤a(m,n)+∑s=m0m-1b(s,n)u(s,n)
implies that
(2.5)u(m,n)≤a(m,n)∏s=m0m-1[1+b(s,n)].Remark 2.4.
Lemma2.3 is a direct variation of [19, Lemma 2.5(β1)], and we note a(m,n)≥0 here.Theorem 2.5.
Suppose thatu(m,n),a(m,n)∈℘+(Ω), bi(s,t,m,n),ci(s,t,m,n)∈℘+(Ω2), i=1,2,…,l1, di(s,t,m,n),ei(s,t,m,n)∈℘+(Ω2), i=1,2,…,l2with bi,ci,di,ei nondecreasing in the last two variables. p,qi,ri are nonnegative constants with p≥qi,p≥ri,i=1,2,…,l1,p≠0, while hi,ji are nonnegative constants with p≥hi,p≥ji,i=1,2,…,l2. If, for (m,n)∈Ω, u(m,n) satisfies
(2.6)up(m,n)≤a(m,n)+∑i=1l1∑s=m0m-1∑t=n0n-1[bi(s,t,m,n)uqi(s,t)+∑ξ=m0s∑η=n0tci(ξ,η,m,n)uri(ξ,η)]+∑i=1l2∑s=m0M-1∑t=n0N-1[di(s,t,m,n)uhi(s,t)+∑ξ=m0s∑η=n0tei(ξ,η,m,n)uji(ξ,η)],
then
(2.7)u(m,n)≤{a(m,n)+J(M,N)1-μ(M,N)C(m,n)}1/p,
provided that μ(M,N)<1, where
(2.8)J(m,n)=∑i=1l1∑s=m0m-1∑t=n0n-1{∑ξ=m0s∑η=n0tci(ξ,η,m,n)[ripK2(ri-p)/pa(ξ,η)+p-ripK2ri/p]bi(s,t,m,n)[qipK1(qi-p)/pa(s,t)+p-qipK1qi/p]+∑ξ=m0s∑η=n0tci(ξ,η,m,n)[ripK2(ri-p)/pa(ξ,η)+p-ripK2ri/p]}+∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0tei(ξ,η,m,n)[jipK4(ji-p)/pa(ξ,η)+p-jipK4ji/p]di(s,t,m,n)[hipK3(hi-p)/pa(s,t)+p-hipK3hi/p]+∑ξ=m0s∑η=n0tei(ξ,η,m,n)[jipK4(ji-p)/pa(ξ,η)+p-jipK4ji/p]},(2.9)μ(m,n)=∑i=1l2∑s=m0M-1∑t=n0N-1{di(s,t,m,n)hipK3(hi-p)/pC(s,t)+∑ξ=m0s∑η=n0tei(ξ,η,m,n)jipK4(ji-p)/pC(ξ,η)},(2.10)C(m,n)=exp⁡{∑s=m0m-1∑t=n0n-1B(s,t,m,n)},(2.11)B(s,t,m,n)=∑i=1l1[bi(s,t,m,n)qipK1(qi-p)/p+∑ξ=m0s∑η=n0tci(ξ,η,m,n)ripK2(ri-p)/p].Proof.
Denote(2.12)v(m,n)=∑i=1l1∑s=m0m-1∑t=n0n-1[bi(s,t,m,n)uqi(s,t)+∑ξ=m0s∑η=n0tci(ξ,η,m,n)uri(ξ,η)]+∑i=1l2∑s=m0M-1∑t=n0N-1[di(s,t,m,n)uhi(s,t)+∑ξ=m0s∑η=n0tei(ξ,η,m,n)uji(ξ,η)].
Then, we have
(2.13)u(m,n)≤[a(m,n)+v(m,n)]1/p,
and, furthermore, from Lemma 2.1 we have
(2.14)uqi(m,n)≤[a(m,n)+v(m,n)]qi/p≤qipK1(qi-p)/p[a(m,n)+v(m,n)]+p-qipK1qi/p,i=1,2,…,l1,uri(m,n)≤[a(m,n)+v(m,n)]ri/p≤ripK2(ri-p)/p[a(m,n)+v(m,n)]+p-ripK2ri/p,i=1,2,…,l1,uhi(m,n)≤[a(m,n)+v(m,n)]hi/p≤hipK3(hi-p)/p[a(m,n)+v(m,n)]+p-hipK3hi/p,i=1,2,…,l2,uji(m,n)≤[a(m,n)+v(m,n)]ji/p≤jipK4(ji-p)/p[a(m,n)+v(m,n)]+p-jipK4ji/p,i=1,2,…,l2.
So
(2.15)v(m,n)≤∑i=1l1∑s=m0m-1∑t=n0n-1{∑ξ=m0s∑η=n0tci(ξ,η,m,n)[ripK2(ri-p)/p(a(ξ,η)+v(ξ,η))+p-ripK2ri/p]bi(s,t,m,n)[qipK1(qi-p)/p(a(s,t)+v(s,t))+p-qipK1qi/p]+∑ξ=m0s∑η=n0tci(ξ,η,m,n)[ripK2(ri-p)/p(a(ξ,η)+v(ξ,η))+p-ripK2ri/p]}+∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0tei(ξ,η,m,n)[jipK4(ji-p)/p(a(ξ,η)+v(ξ,η))+p-jipK4ji/p]di(s,t,m,n)[hipK3(hi-p)/p(a(s,t)+v(s,t))+p-hipK3hi/p]+∑ξ=m0s∑η=n0tei(ξ,η,m,n)[jipK4(ji-p)/p(a(ξ,η)+v(ξ,η))+p-jipK4ji/p]}=H(m,n)+∑i=1l1∑s=m0m-1∑t=n0n-1[∑ξ=m0s∑η=n0tci(ξ,η,m,n)ripK2(ri-p)/pv(ξ,η)bi(s,t,m,n)qipK1(qi-p)/pv(s,t)+∑ξ=m0s∑η=n0tci(ξ,η,m,n)ripK2(ri-p)/pv(ξ,η)],
where H(m,n)=J(m,n)+∑i=1l2∑s=m0M-1∑t=n0N-1{di(s,t,m,n)(hi/p)K3(hi-p)/pv(s,t)+∑ξ=m0s∑η=n0tei(ξ,η,m,n)(ji/p)K4(ji-p)/pv(ξ,η)} and J(m,n) is defined in (2.8). Then, using that H(m,n) is nondecreasing in every variable, we obtain
(2.16)v(m,n)≤H(M,N)+∑i=1l1∑s=m0m-1∑t=n0n-1[∑ξ=m0s∑η=n0tci(ξ,η,m,n)ripK2(ri-p)/pv(ξ,η)bi(s,t,m,n)qipK1(qi-p)/pv(s,t)+∑ξ=m0s∑η=n0tci(ξ,η,m,n)ripK2(ri-p)/pv(ξ,η)]≤H(M,N)+∑i=1l1∑s=m0m-1∑t=n0n-1[∑ξ=m0s∑η=n0tci(ξ,η,m,n)ripK2(ri-p)/pbi(s,t,m,n)qipK1(qi-p)/p+∑ξ=m0s∑η=n0tci(ξ,η,m,n)ripK2(ri-p)/p]v(s,t)=H(M,N)+∑s=m0m-1∑t=n0n-1B(s,t,m,n)v(s,t),
where B(s,t,m,n) is defined in (2.11).
Sincebi(s,t,m,n),ci(s,t,m,n) are nondecreasing in the last two variables, then B(s,t,m,n) is also nondecreasing in the last two variables, and by a suitable application of Lemma 2.2 we obtain
(2.17)v(m,n)≤H(M,N)exp⁡{∑s=m0m-1∑t=n0n-1B(s,t,m,n)}=H(M,N)C(m,n),
where C(m,n) is defined in (2.10). Furthermore, considering the definition of H(m,n) and (2.17) we have
(2.18)H(M,N)=J(M,N)+∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0tei(ξ,η,M,N)jipK4(ji-p)/pv(ξ,η)di(s,t,M,N)hipK3(hi-p)/pv(s,t)+∑ξ=m0s∑η=n0tei(ξ,η,M,N)jipK4(ji-p)/pv(ξ,η)}≤J(M,N)+∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0tei(ξ,η,m1,n1)jipK4(ji-p)/pH(M,N)C(ξ,η)di(s,t,M,N)hipK3(hi-p)/pH(M,N)C(s,t)+∑ξ=m0s∑η=n0tei(ξ,η,m1,n1)jipK4(ji-p)/pH(M,N)C(ξ,η)}=J(M,N)+H(M,N)∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0tei(ξ,η,M,N)jipK4(ji-p)/pC(ξ,η)di(s,t,M,N)hipK3(hi-p)/pC(s,t)+∑ξ=m0s∑η=n0tei(ξ,η,M,N)jipK4(ji-p)/pC(ξ,η)}=J(M,N)+H(M,N)μ(M,N),
where μ(m,n) is defined in (2.9). Then,
(2.19)H(M,N)≤J(M,N)1-μ(M,N).
Combining (2.17) and (2.19) we deduce
(2.20)v(m,n)≤J(M,N)1-μ(M,N)C(m,n).
Then, combining (2.13) and (2.20), we obtain the desired result.Corollary 2.6.
Suppose thatg1i(m,n),g2i(m,n),b1i(m,n),c1i(m,n)∈℘+(Ω),i=1,2,…,l1 with g1i,g2i nondecreasing in every variable. d1i(m,n),e1i(m,n)∈℘+(Ω),i=1,2,…,l2. u(m,n),a(m,n),p,qi,ri,hi,ji are defined as in Theorem 2.5. If, for (m,n)∈Ω, u(m,n) satisfies
(2.21)up(m,n)≤a(m,n)+∑i=1l1g1i(m,n)∑s=m0m-1∑t=n0n-1[b1i(s,t)uqi(s,t)+∑ξ=m0s∑η=n0tc1i(ξ,η)uri(ξ,η)]+∑i=1l2g2i(m,n)∑s=m0M-1∑t=n0N-1[d1i(s,t)uhi(s,t)+∑ξ=m0s∑η=n0te1i(ξ,η)uji(ξ,η)],
then
(2.22)u(m,n)≤{a(m,n)+J(M,N)1-μ(M,N)C(M,N)}1/p,
provided that μ(M,N)<1, where
(2.23)J(m,n)=∑i=1l1g1i(m,n)∑s=m0m-1∑t=n0n-1{∑ξ=m0s∑η=n0tc1i(ξ,η)[ripK2(ri-p)/pa(ξ,η)+p-ripK2ri/p]b1i(s,t)[qipK1(qi-p)/pa(s,t)+p-qipK1qi/p]+∑ξ=m0s∑η=n0tc1i(ξ,η)[ripK2(ri-p)/pa(ξ,η)+p-ripK2ri/p]}+∑i=1l2g2i(m,n)∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0te1i(ξ,η)[jipK4(ji-p)/pa(ξ,η)+p-jipK4ji/p]d1i(s,t)[hipK3(hi-p)/pa(s,t)+p-hipK3hi/p]+∑ξ=m0s∑η=n0te1i(ξ,η)[jipK4(ji-p)/pa(ξ,η)+p-jipK4ji/p]},μ(m,n)=∑i=1l2{g2i(m,n)∑s=m0M-1∑t=n0N-1[∑ξ=m0s∑η=n0te1i(ξ,η)jipK4(ji-p)/pC(ξ,η)d1i(s,t)hipK3(hi-p)/pC(s,t)+∑ξ=m0s∑η=n0te1i(ξ,η)jipK4(ji-p)/pC(ξ,η)]},C(m,n)=exp⁡{∑s=m0m-1∑t=n0n-1B(s,t,m,n)},B(s,t,m,n)=∑i=1l1g1i(m,n)[b1i(s,t)qipK1(qi-p)/p+∑ξ=m0s∑η=n0tc1i(ξ,η)ripK2(ri-p)/p].The proof of Corollary2.6 can be completed by setting bi(s,t,m,n)=g1i(m,n)b1i(s,t), ci(s,t,m,n)=g1i(m,n)c1i(s,t),di(s,t,m,n)=g2i(m,n)d1i(s,t),ei(s,t,m,n)=g2i(m,n)e1i(s,t) in Theorem 2.5.Corollary 2.7.
Suppose thatu(m,n),a(m,n),bi(s,t,m,n),ci(s,t,m,n),di(s,t,m,n),ei(s,t,m,n) are defined as in Theorem 2.5. If, for (m,n)∈Ω, u(m,n) satisfies
(2.24)u(m,n)≤a(m,n)+∑i=1l1∑s=m0m-1∑t=n0n-1[bi(s,t,m,n)u(s,t)+∑ξ=m0s∑η=n0tci(ξ,η,m,n)u(ξ,η)]+∑i=1l2∑s=m0M-1∑t=n0N-1[di(s,t,m,n)u(s,t)+∑ξ=m0s∑η=n0tei(ξ,η,m,n)u(ξ,η)],
then
(2.25)u(m,n)≤a(m,n)+J(M,N)1-μ(M,N)C(m,n),
provided that μ(M,N)<1, where
(2.26)J(m,n)=∑i=1l1∑s=m0m-1∑t=n0n-1{bi(s,t,m,n)a(s,t)+∑ξ=m0s∑η=n0tci(ξ,η,m,n)a(ξ,η)}+∑i=1l2∑s=m0M-1∑t=n0N-1{di(s,t,m,n)a(s,t)+∑ξ=m0s∑η=n0tei(ξ,η,m,n)a(ξ,η)},μ(m,n)=∑i=1l2∑s=m0M-1∑t=n0N-1{di(s,t,m,n)C(s,t)+∑ξ=m0s∑η=n0tei(ξ,η,m,n)C(ξ,η)},C(m,n)=exp⁡{∑s=m0m-1∑t=n0n-1B(s,t,m,n)},B(s,t,m,n)=∑i=1l1[bi(s,t,m,n)+∑ξ=m0s∑η=n0tci(ξ,η,m,n)].Theorem 2.8.
Suppose thatw(m,n)∈℘+(Ω), u,a,bi,ci,di,ei,p,qi,ri,hi,ji are defined as in Theorem 2.5. Furthermore, assume that a(m,n) is nondecreasing in the first variable. If, for (m,n)∈Ω, u(m,n) satisfies
(2.27)up(m,n)≤a(m,n)+∑s=m0m-1w(s,n)up(m,n)+∑i=1l1∑s=m0m-1∑t=n0n-1[bi(s,t,m,n)uqi(s,t)+∑ξ=m0s∑η=n0tci(ξ,η,m,n)uri(ξ,η)]+∑i=1l2∑s=m0M-1∑t=n0N-1[di(s,t,m,n)uhi(s,t)+∑ξ=m0s∑η=n0tei(ξ,η,m,n)uji(ξ,η)],
then
(2.28)u(m,n)≤{[a(m,n)+J¯(M,N)1-μ¯(M,N)C¯(m,n)]w¯(m,n)}1/p,
provided that μ¯(M,N)<1, where
(2.29)J¯(m,n)=∑i=1l1∑s=m0m-1∑t=n0n-1{∑ξ=m0s∑η=n0tc¯i(ξ,η,m,n)[ripK2(ri-p)/pa(ξ,η)+p-ripK2ri/p]b¯i(s,t,m,n)[qipK1(qi-p)/pa(s,t)+p-qipK1qi/p]+∑ξ=m0s∑η=n0tc¯i(ξ,η,m,n)[ripK2(ri-p)/pa(ξ,η)+p-ripK2ri/p]}+∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0te¯i(ξ,η,m,n)[jipK4ji-ppa(ξ,η)+p-jipK4jip]d¯i(s,t,m,n)[hipK3(hi-p)/pa(s,t)+p-hipK3hi/p]+∑ξ=m0s∑η=n0te¯i(ξ,η,m,n)[jipK4(ji-p)/pa(ξ,η)+p-jipK4ji/p]},(2.30)b¯i(s,t,m,n)=bi(s,t,m,n)(w¯(s,t))qi/p,c¯i(s,t,m,n)=ci(s,t,m,n)(w¯(s,t))ri/p,i=1,2,…,l1,(2.31)d¯i(s,t,m,n)=di(s,t,m,n)(w¯(s,t))hi/p,e¯i(s,t,m,n)=ei(s,t,m,n)(w¯(s,t))ji/p,i=1,2,…,l2,(2.32)w¯(m,n)=∏s=m0m-1[1+w(s,n)],μ¯(m,n)=∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0te¯i(ξ,η,m,n)jipK4(ji-p)/pC¯(ξ,η)d¯i(s,t,m,n)hipK3(hi-p)/pC¯(s,t)+∑ξ=m0s∑η=n0te¯i(ξ,η,m,n)jipK4(ji-p)/pC¯(ξ,η)}(2.33)C(m,n)=exp⁡{∑s=m0m-1∑t=n0n-1B¯(s,t,m,n)},(2.34)B¯(s,t,m,n)=∑i=1l1[b¯i(s,t,m,n)qipK1(qi-p)/p+∑ξ=m0s∑η=n0tc¯i(ξ,η,m,n)ripK2(ri-p)/p].Proof.
Denote(2.35)z(m,n)=a(m,n)+∑i=1l1∑s=m0m-1∑t=n0n-1[bi(s,t,m,n)uqi(s,t)+∑ξ=m0s∑η=n0tci(ξ,η,m,n)uri(ξ,η)]+∑i=1l2∑s=m0M-1∑t=n0N-1[di(s,t,m,n)uhi(s,t)+∑ξ=m0s∑η=n0tei(ξ,η,m,n)uji(ξ,η)].
Then, we have
(2.36)up(m,n)≤z(m,n)+∑s=m0m-1w(s,n)up(m,n).
Obviously z(m,n) is nondecreasing in the first variable. So by Lemma 2.3 we obtain
(2.37)up(m,n)≤z(m,n)∏s=m0m-1[1+w(s,n)]=z(m,n)w¯(m,n),
where w¯(m,n)=∏s=m0m-1[1+w(s,n)]. Define
(2.38)v(m,n)=∑i=1l1∑s=m0m-1∑t=n0n-1[bi(s,t,m,n)uqi(s,t)+∑ξ=m0s∑η=n0tci(ξ,η,m,n)uri(ξ,η)]+∑i=1l2∑s=m0M-1∑t=n0N-1[di(s,t,m,n)uhi(s,t)+∑ξ=m0s∑η=n0tei(ξ,η,m,n)uji(ξ,η)].
Then,
(2.39)u(m,n)≤[(a(m,n)+v(m,n))w¯(m,n)]1/p,
and, furthermore, by (2.39) and Lemma 2.1 we have
(2.40)v(m,n)≤∑i=1l1∑s=m0m-1∑t=n0n-1{∑ξ=m0s∑η=n0tci(ξ,η,m,n)[(a(ξ,η)+v(ξ,η))w¯(ξ,η)]ri/pbi(s,t,m,n)[(a(s,t)+v(s,t))w¯(s,t)]qi/p+∑ξ=m0s∑η=n0tci(ξ,η,m,n)[(a(ξ,η)+v(ξ,η))w¯(ξ,η)]ri/p}+∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0tei(ξ,η,m,n)[(a(ξ,η)+v(ξ,η))w¯(ξ,η)]ji/pdi(s,t,m,n)[(a(s,t)+v(s,t))w¯(s,t)]hi/p+∑ξ=m0s∑η=n0tei(ξ,η,m,n)[(a(ξ,η)+v(ξ,η))w¯(ξ,η)]ji/p}≤∑i=1l1∑s=m0m-1∑t=n0n-1{bi(s,t,m,n)(w¯(s,t))qi/p[qipK1(qi-p)/p(a(s,t)+v(s,t))+p-qipK1qi/p]+∑ξ=m0s∑η=n0tci(ξ,η,m,n)(w¯(ξ,η))ri/p×[ripK2(ri-p)/p(a(ξ,η)+v(ξ,η))+p-ripK2ri/p]}+∑i=1l2∑s=m0M-1∑t=n0N-1{di(s,t,m,n)(w¯(s,t))hi/p[hipK3(qi-p)/p(a(s,t)+v(s,t))+p-hipK3hi/p]+∑ξ=m0s∑η=n0tei(ξ,η,m,n)(w¯(ξ,η))ji/p×[jipK4ji-p/p(a(ξ,η)+v(ξ,η))+p-jipK4ji/p]}=∑i=1l1∑s=m0m-1∑t=n0n-1{∑ξ=m0s∑η=n0tc¯i(ξ,η,m,n)[ripK2ri-p/p(a(ξ,η)+v(ξ,η))+p-ripK2ri/p]b¯i(s,t,m,n)[qipK1(qi-p)/p(a(s,t)+v(s,t))+p-qipK1qi/p]+∑ξ=m0s∑η=n0tc¯i(ξ,η,m,n)[ripK2(ri-p)/p(a(ξ,η)+v(ξ,η))+p-ripK2ri/p]}+∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0te¯i(ξ,η,m,n)[jipK4(ji-p)/p(a(ξ,η)+v(ξ,η))+p-jipK4ji/p]d¯i(s,t,m,n)[hipK3(qi-p)/p(a(s,t)+v(s,t))+p-hipK3hi/p]+∑ξ=m0s∑η=n0te¯i(ξ,η,m,n)[jipK4(ji-p)/p(a(ξ,η)+v(ξ,η))+p-jipK4ji/p]}=H¯(m,n)+∑i=1l1∑s=m0m-1∑t=n0n-1[∑ξ=m0s∑η=n0tc¯i(ξ,η,m,n)ripK2(ri-p)/pv(ξ,η)b¯i(s,t,m,n)qipK1(qi-p)/pv(s,t)+∑ξ=m0s∑η=n0tc¯i(ξ,η,m,n)ripK2(ri-p)/pv(ξ,η)],
where  H¯(m,n)=J¯(m,n)+∑i=1l2∑s=m0M-1∑t=n0N-1{d¯i(s,t,m,n)(hi/p)K3(hi-p)/pv(s,t)+∑ξ=m0s∑η=n0te¯i(ξ,η,m,n)ji/pK4(ji-p)/pv(ξ,η)}, and J¯(m,n),b¯i,c¯i,d¯i,e¯i are defined in (2.29)–(2.31) respectively.
Similar to the process of (2.15)–(2.20) we deduce
(2.41)v(m,n)≤J¯(M,N)1-μ¯(M,N)C¯(m,n),
where μ¯(m,n),C¯(m,n) are defined in (2.32) and (2.33).
Combining (2.39) and (2.41), we get the desired result.Remark 2.9.
If we setbi(s,t,m,n)=g1i(m,n)b1i(s,t),ci(s,t,m,n)=g1i(m,n)c1i(s,t),di(s,t,m,n)= g2i(m,n)d1i(s,t),ei(s,t,m,n)=g2i(m,n)e1i(s,t) or set p=qi=ri=hi=ji=1 in Theorem 2.8, then immediately we get two corollaries which are similar to Corollaries 2.6 and 2.7, and we omit the details for them.Theorem 2.10.
Suppose thatu,a,bi,ci,di,ei,p,qi,ri,hi,ji are defined as in Theorem 2.5. Li,Ti:Ω×ℝ+→ℝ+,i=1,2,…,l2, satisfies 0≤Li(m,n,u)-Li(m,n,v)≤Ti(m,n,v)(u-v) for u≥v≥0. If, for (m,n)∈Ω, u(m,n) satisfies
(2.42)up(m,n)≤a(m,n)+∑i=1l1∑s=m0m-1∑t=n0n-1[bi(s,t,m,n)uqi(s,t)+∑ξ=m0s∑η=n0tci(ξ,η,m,n)uri(ξ,η)]+∑i=1l2∑s=m0M-1∑t=n0N-1[di(s,t,m,n)Li(s,t,uhi(s,t))+∑ξ=m0s∑η=n0tei(ξ,η,m,n)uji(ξ,η)],
then
(2.43)u(m,n)≤{a(m,n)+Ĵ(M,N)1-μ̂(M,N)Ĉ(m,n)}1/p,
provided that μ̂(M,N)<1, where
(2.44)Ĵ(m,n)=∑i=1l1∑s=m0m-1∑t=n0n-1{∑ξ=m0s∑η=n0tci(ξ,η,m,n)[ripK2(ri-p)/pa(ξ,η)+p-ripK2ri/p]bi(s,t,m,n)[qipK1(qi-p)/pa(s,t)+p-qipK1qi/p]+∑ξ=m0s∑η=n0tci(ξ,η,m,n)[ripK2(ri-p)/pa(ξ,η)+p-ripK2ri/p]}+∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0tei(ξ,η,m,n)[jipK4(ji-p)/pa(ξ,η)+p-jipK4ji/p]di(s,t,m,n)Li[s,t,hipK3(hi-p)/pa(s,t)+p-hipK3hi/p]+∑ξ=m0s∑η=n0tei(ξ,η,m,n)[jipK4(ji-p)/pa(ξ,η)+p-jipK4ji/p]},(2.45)μ̂(m,n)=∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0tei(ξ,η,m,n)jipK4(ji-p)/pC(ξ,η)d̂i(s,t,m,n)hipK3(hi-p)/pC(s,t)+∑ξ=m0s∑η=n0tei(ξ,η,m,n)jipK4(ji-p)/pC(ξ,η)},(2.46)d̂i(s,t,m,n)=di(s,t,m,n)Ti[s,t,hipK3(hi-p)/pa(s,t)+p-hipK3hi/p],i=1,2,…,l2,(2.47)Ĉ(m,n)=exp⁡{∑s=m0m-1∑t=n0n-1B̂(s,t,m,n)},(2.48)B̂(s,t,m,n)=∑i=1l1[bi(s,t,m,n)qipK1(qi-p)/p+∑ξ=m0s∑η=n0tci(ξ,η,m,n)ripK2(ri-p)/p].Proof.
Denote(2.49)v(m,n)=∑i=1l1∑s=m0m-1∑t=n0n-1[bi(s,t,m,n)uqi(s,t)+∑ξ=m0s∑η=n0tci(ξ,η,m,n)uri(ξ,η)]+∑i=1l2∑s=m0M-1∑t=n0N-1[di(s,t,m,n)Li(s,t,uhi(s,t))+∑ξ=m0s∑η=n0tei(ξ,η,m,n)uji(ξ,η)].
Then
(2.50)u(m,n)≤[a(m,n)+v(m,n)]1/p,
and, furthermore, from Lemma 2.1 we have
(2.51)v(m,n)≤∑i=1l1∑s=m0m-1∑t=n0n-1{∑ξ=m0s∑η=n0tci(ξ,η,m,n)[ripK2(ri-p)/p(a(ξ,η)+v(ξ,η))+p-ripK2ri/p]bi(s,t,m,n)[qipK1(qi-p)/p(a(s,t)+v(s,t))+p-qipK1qi/p]+∑ξ=m0s∑η=n0tci(ξ,η,m,n)[ripK2(ri-p)/p(a(ξ,η)+v(ξ,η))+p-ripK2ri/p]}+∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0tei(ξ,η,m,n)[jipK4(ji-p)/p(a(ξ,η)+v(ξ,η))+p-jipK4ji/p]di(s,t,m,n)Li[s,t,hipK3(hi-p)/p(a(s,t)+v(s,t))+p-hipK3hi/p]+∑ξ=m0s∑η=n0tei(ξ,η,m,n)[jipK4(ji-p)/p(a(ξ,η)+v(ξ,η))+p-jipK4ji/p]}≤∑i=1l1∑s=m0m-1∑t=n0n-1{∑ξ=m0s∑η=n0tci(ξ,η,m,n)[ripK2(ri-p)/p(a(ξ,η)+v(ξ,η))+p-ripK2ri/p]bi(s,t,m,n)[qipK1(qi-p)/p(a(s,t)+v(s,t))+p-qipK1qi/p]+∑ξ=m0s∑η=n0tci(ξ,η,m,n)[ripK2(ri-p)/p(a(ξ,η)+v(ξ,η))+p-ripK2ri/p]}+∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0tei(ξ,η,m,n)[jipK4(ji-p)/p(a(ξ,η)+v(ξ,η))+p-jipK4ji/p]di(s,t,m,n){Li[s,t,hipK3(hi-p)/p(a(s,t)+v(s,t))+p-hipK3hi/p]-Li[s,t,hipK3(hi-p)/pa(s,t)+p-hipK3hi/p]+Li[s,t,hipK3(hi-p)/pa(s,t)+p-hipK3hi/p]}+∑ξ=m0s∑η=n0tei(ξ,η,m,n)[jipK4(ji-p)/p(a(ξ,η)+v(ξ,η))+p-jipK4ji/p]}≤∑i=1l1∑s=m0m-1∑t=n0n-1{∑ξ=m0s∑η=n0tci(ξ,η,m,n)[ripK2(ri-p)/p(a(ξ,η)+v(ξ,η))+p-ripK2ri/p]bi(s,t,m,n)[qipK1(qi-p)/p(a(s,t)+v(s,t))+p-qipK1qi/p]+∑ξ=m0s∑η=n0tci(ξ,η,m,n)[ripK2(ri-p)/p(a(ξ,η)+v(ξ,η))+p-ripK2ri/p]}+∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0tei(ξ,η,m,n)[jipK4(ji-p)/p(a(ξ,η)+v(ξ,η))+p-jipK4ji/p]di(s,t,m,n){Ti[s,t,hipK3(hi-p)/pa(s,t)+p-hipK3hi/p]hipK3(hi-p)/p×v(s,t)+Li[s,t,hipK3(hi-p)/pa(s,t)+p-hipK3hi/p]}+∑ξ=m0s∑η=n0tei(ξ,η,m,n)[jipK4(ji-p)/p(a(ξ,η)+v(ξ,η))+p-jipK4ji/p]}=Ĥ(m,n)+∑i=1l1∑s=m0m-1∑t=n0n-1[∑ξ=m0s∑η=n0tci(ξ,η,m,n)ripK2(ri-p)/pv(ξ,η)bi(s,t,m,n)qipK1(qi-p)/pv(s,t)+∑ξ=m0s∑η=n0tci(ξ,η,m,n)ripK2(ri-p)/pv(ξ,η)],
where Ĥ(m,n)= Ĵ(m,n)+∑i=1l2∑s=m0M-1∑t=n0N-1{d̂i(s,t,m,n)(hi/p)K3(hi-p)/pv(s,t)+∑ξ=m0s∑η=n0tei(ξ,η,m,n)(ji/p)K4(ji-p)/pv(ξ,η)} and Ĵ(m,n),d̂i(s,t,m,n) are defined in (2.44) and (2.46) respectively.
Similar to the process of (2.15)–(2.20) we deduce
(2.52)v(m,n)≤Ĵ(M,N)1-μ̂(M,N)Ĉ(M,N),
where μ̂(m,n),Ĉ(m,n) are defined in (2.45) and (2.47) respectively.
Combining (2.50) and (2.52), we get the desired result.Theorem 2.11.
Suppose thatw(m,n)∈℘+(Ω), u,a,bi,ci,di,ei,p,qi,ri,hi,ji are defined as in Theorem 2.5. Furthermore, assume a(m,n) is nondecreasing in the first variable. Li,Ti,i=1,2,…,l2, are defined as in Theorem 2.10. If, for (m,n)∈Ω, u(m,n) satisfies
(2.53)up(m,n)≤a(m,n)+∑s=m0m-1w(s,n)up(m,n)+∑i=1l1∑s=m0m-1∑t=n0n-1[bi(s,t,m,n)uqi(s,t)+∑ξ=m0s∑η=n0tci(ξ,η,m,n)uri(ξ,η)]+∑i=1l2∑s=m0M-1∑t=n0N-1[di(s,t,m,n)Li(s,t,uhi(s,t))+∑ξ=m0s∑η=n0tei(ξ,η,m,n)uji(ξ,η)],
then
(2.54)u(m,n)≤{[a(m,n)+J̃(M,N)1-μ̃(M,N)C̃(m,n)]w̃(m,n)}1/p,
provided that μ̃(M,N)<1, where
(2.55)J̃(m,n)=∑i=1l1∑s=m0m-1∑t=n0n-1{∑ξ=m0s∑η=n0tc̃i(ξ,η,m,n)[ripK2(ri-p)/pa(ξ,η)+p-ripK2ri/p]b̃i(s,t,m,n)[qipK1(qi-p)/pa(s,t)+p-qipK1qi/p]+∑ξ=m0s∑η=n0tc̃i(ξ,η,m,n)[ripK2(ri-p)/pa(ξ,η)+p-ripK2ri/p]}+∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0tẽi(ξ,η,m,n)[jipK4(ji-p)/pa(ξ,η)+p-jipK4ji/p]di(s,t,m,n)Li[s,t,(w̃(s,t))hi/p(hipK3(hi-p)/pa(s,t)+p-hipK3hi/p)]+∑ξ=m0s∑η=n0tẽi(ξ,η,m,n)[jipK4(ji-p)/pa(ξ,η)+p-jipK4ji/p]},b̃i(s,t,m,n)=bi(s,t,m,n)(w̃(s,t))qi/p,c̃i(s,t,m,n)=ci(s,t,m,n)(w̃(s,t))ri/p,i=1,2,…,l1,d̃i(s,t,m,n)=di(s,t,m,n)(w̃(s,t))hi/pTi[s,t,(w̃(s,t))hi/p(hipK3(hi-p)/pa(s,t)+p-hipK3hi/p)],ẽi(s,t,m,n)=ei(s,t,m,n)(w̃(s,t))ji/p,i=1,2,…,l2,w̃(m,n)=∏s=m0m-1[1+w(s,n)],μ̃(m,n)=∑i=1l2∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0tẽi(ξ,η,m,n)jipK4(ji-p)/pC̃(ξ,η)d̃i(s,t,m,n)hipK3(hi-p)/pC̃(s,t)+∑ξ=m0s∑η=n0tẽi(ξ,η,m,n)jipK4(ji-p)/pC̃(ξ,η)},C̃(m,n)=exp⁡{∑s=m0m-1∑t=n0n-1B̃(s,t,m,n)},B̃(s,t,m,n)=∑i=1l1[b¯i(s,t,m,n)qipK1(qi-p)/p+∑ξ=m0s∑η=n0tc¯i(ξ,η,m,n)ripK2(ri-p)/p].The proof for Theorem2.11 is similar to the combination of Theorems 2.8 and 2.10, and we omit the details here.Remark 2.12.
If we takeg1i(m,n)≡1,c1i(m,n)≡0,i=1,2,…,l1, and g2i(m,n)≡1,e1i(m,n)≡0,i=1,2,…,l2 in Corollary 2.6, then Corollary 2.6 reduces to [9, Theorem 2.5]. If furthermore l1=l2=1, then Corollary 2.6 reduces to [9, Theorem 2.1]. If we take bi(s,t,m,n)=b1i(s,t),ci(s,t,m,n)≡0,i=1,2,…,l1 and di(s,t,m,n)=d1i(s,t), ei(s,t,m,n)≡0,hi=1,i=1,2,…,l2 in Theorem 2.10, then Theorem 2.10 reduces to [9, Theorem 2.7]. If furthermore l1=l2=1, then Theorem 2.10 reduces to [9, Theorem 2.6].


## 3. Applications

In this section, we will present some applications for the established results above and show that they are useful in the study of boundedness, uniqueness, and continuous dependence of solutions of certain difference equations.Example 3.1.
Consider the following Volterra-Fredholm sum-difference equation:(3.1)up(m,n)=a(m,n)+∑s=m0m-1∑t=n0n-1[F1(s,t,m,n,u(s,t))+∑ξ=m0s∑η=n0tF2(ξ,η,m,n,u(ξ,η))]+∑s=m0M-1∑t=n0M-1[G1(s,t,m,n,u(s,t))+∑ξ=m0s∑η=n0tG2(ξ,η,m,n,u(ξ,η))],
where u(m,n),a(m,n)∈℘(Ω), p≥1 is an odd number, Fi,Gi:Ω2×ℝ→ℝ,i=1,2,M,N are two integers defined the same as in Theorem 2.5.Theorem 3.2.
Suppose thatu(m,n) is a solution of (3.1), and |F1(s,t,m,n,u)|≤f1(s,t,m,n)|u|q, |F2(s,t,m,n,u)|≤f2(s,t,m,n)|u|r,|G1(s,t,m,n,u)|≤g1(s,t,m,n)|u|h,|G2(s,t,m,n,u)|≤g2(s,t,m,n)|u|j, where q,r,h,j are nonnegative constants satisfying p≥q,p≥r,p≥h,p≥j, fi,gi∈℘+(Ω2),i=1,2 and fi,gi are nondecreasing in the last two variables; then one has
(3.2)|u(m,n)|≤{|a(m,n)|+J(M,N)1-μ(M,N)C(m,n)}1/p,
provided that μ(M,N)<1, where
(3.3)J(m,n)=∑s=m0m-1∑t=n0n-1{∑ξ=m0s∑η=n0tf2(ξ,η,m,n)[rpK2(r-p)/p|a(ξ,η)|+p-rpK2r/p]f1(s,t,m,n)[qpK1(q-p)/p|a(s,t)|+p-qpK1q/p]+∑ξ=m0s∑η=n0tf2(ξ,η,m,n)[rpK2(r-p)/p|a(ξ,η)|+p-rpK2r/p]}+∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0tg2(ξ,η,m,n)[jpK4(j-p)/p|a(ξ,η)|+p-jpK4j/p]g1(s,t,m,n)[hpK3(h-p)/p|a(s,t)|+p-hpK3h/p]+∑ξ=m0s∑η=n0tg2(ξ,η,m,n)[jpK4(j-p)/p|a(ξ,η)|+p-jpK4j/p]},μ(m,n)=∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0tg2(ξ,η,m,n)jpK4(j-p)/pC(ξ,η)g1(s,t,m,n)hpK3(h-p)/pC(s,t)+∑ξ=m0s∑η=n0tg2(ξ,η,m,n)jpK4(j-p)/pC(ξ,η)},C(m,n)=exp⁡{∑s=m0m-1∑t=n0n-1B(s,t,m,n)},B(s,t,m,n)=f1(s,t,m,n)qpK1(q-p)/p+∑ξ=m0s∑η=n0tf2(ξ,η,m,n)rpK2(r-p)/p.Proof.
From (3.1) we have
(3.4)|u(m,n)|p≤|a(m,n)|+∑s=m0m-1∑t=n0n-1[|F2(ξ,η,m,n,u(ξ,η))F1(s,t,m,n,u(s,t))|+∑ξ=m0s∑η=n0t|F2(ξ,η,m,n,u(ξ,η))|]+∑s=m0M-1∑t=n0M-1[|G2(ξ,η,m,n,u(ξ,η))G1(s,t,m,n,u(s,t))|+∑ξ=m0s∑η=n0t|G2(ξ,η,m,n,u(ξ,η))|]≤|a(m,n)|+∑s=m0m-1∑t=n0n-1[f1(ξ,η,m,ns,t,m,n)|u(ξ,η)u(s,t)|q+∑ξ=m0s∑η=n0tf2(ξ,η,m,n)|u(ξ,η)|r]+∑s=m0M-1∑t=n0M-1[g1(ξ,η,m,ns,t,m,n)|u(ξ,η)u(s,t)|h+∑ξ=m0s∑η=n0tg2(ξ,η,m,n)|u(ξ,η)|j].
Then a suitable application of Theorem 2.5 (with l1=l2=1) to (3.4) yields the desired result.The following theorem deals with the uniqueness of the solutions of (3.1).Theorem 3.3.
Suppose that|Fi(s,t,m,n,u)-Fi(s,t,m,n,v)|≤fi(s,t,m,n)|up-vp|,|Gi(s,t,m,n,u)-Gi(s,t,m,n,v)|≤gi(s,t,m,n)|up-vp|,i=1,2 hold for ∀u,v∈ℝ, where fi,gi∈℘+(Ω2),i=1,2 with fi,gi nondecreasing in the last two variables, and μ(M,N)=∑s=m0M-1∑t=n0N-1{g1(s,t,M,N)C(s,t)+∑ξ=m0s∑η=n0tg2(ξ,η,M,N)C(ξ,η)}<1, where C(m,n)=exp{∑s=m0m-1∑t=n0n-1B(s,t,m,n)}, and B(s,t,m,n)=f1(s,t,m,n)+∑ξ=m0s∑η=n0tf2(ξ,η,m,n), then (3.1) has at most one solution.Proof.
Suppose thatu1(m,n),u2(m,n) are two solutions of (3.1). Then,
(3.5)|u1p(m,n)-u2p(m,n)|≤∑s=m0m-1∑t=n0n-1[∑ξ=m0s∑η=n0t|F2(ξ,η,m,n,u1(ξ,η))-F2(ξ,η,m,n,u2(ξ,η))||F2(ξ,η,m,n,u1(ξ,η))-F2(ξ,η,m,n,u2(ξ,η))F1(s,t,m,n,u1(s,t))-F1(s,t,m,n,u2(s,t))|+∑ξ=m0s∑η=n0t|F2(ξ,η,m,n,u1(ξ,η))-F2(ξ,η,m,n,u2(ξ,η))|]+∑s=m0M-1∑t=n0M-1[∑ξ=m0s∑η=n0t|G2(ξ,η,m,n,u1(ξ,η))-G2(ξ,η,m,n,u2(ξ,η))||G2(ξ,η,m,n,u1(ξ,η))-G2(ξ,η,m,n,u2(ξ,η))G1(s,t,m,n,u1(s,t))-G1(s,t,m,n,u2(s,t))|+∑ξ=m0s∑η=n0t|G2(ξ,η,m,n,u1(ξ,η))-G2(ξ,η,m,n,u2(ξ,η))|]≤∑s=m0m-1∑t=n0n-1[∑ξ=m0s∑η=n0tf2(ξ,η,m,n)|u1p(ξ,η)-u2p(ξ,η)|f1(s,t,m,n)|u1p(s,t)-u2p(s,t)|+∑ξ=m0s∑η=n0tf2(ξ,η,m,n)|u1p(ξ,η)-u2p(ξ,η)|]+∑s=m0M-1∑t=n0M-1[∑ξ=m0s∑η=n0tg2(ξ,η,m,n)|u1p(ξ,η)-u2p(ξ,η)|g1(s,t,m,n)|u1p(s,t)-u2p(s,t)|+∑ξ=m0s∑η=n0tg2(ξ,η,m,n)|u1p(ξ,η)-u2p(ξ,η)|].
Treat|u1p(m,n)-u2p(m,n)| as one variable, and a suitable application of Corollary 2.7 yields |u1p(m,n)-u2p(m,n)|≤0, which implies that u1p(m,n)≡u2p(m,n). Since p is an odd number, then we have u1(m,n)≡u2(m,n), and the proof is complete.Finally we study the continuous dependence of the solutions of (3.1) on the functions a,F1,F2,G1,G2.Theorem 3.4.
Suppose thatu(m,n) is a solution of (3.1), |Fi(s,t,m,n,u)-Fi(s,t,m,n,v)|≤fi(s,t,m,n)|up-vp|, |Gi(s,t,m,n,u)-Gi(s,t,m,n,v)|≤gi(s,t,m,n)|up-vp|,i=1,2 hold for ∀u,v∈ℝ, where fi,gi∈℘+(Ω2),i=1,2 with fi,gi nondecreasing in the last two variables, and, furthermore,
(3.6)|a(m,n)-a¯(m,n)|+∑s=m0m-1∑t=n0n-1{∑η=n0t|F2(ξ,η,u¯(ξ,η))-F¯2(ξ,η,u¯(ξ,η))||F1(s,t,u¯(s,t))-F¯1(s,t,u¯(s,t))|+∑ξ=m0s∑η=n0t|F2(ξ,η,u¯(ξ,η))-F¯2(ξ,η,u¯(ξ,η))|}+∑s=m0M-1∑t=n0N-1{+∑ξ=m0s∑η=n0t|G2(ξ,η,u¯(ξ,η))-G¯2(ξ,η,u¯(ξ,η))||G1(s,t,u¯(s,t))-G¯1(s,t,u¯(s,t))|+∑ξ=m0s∑η=n0t|G2(ξ,η,u¯(ξ,η))-G¯2(ξ,η,u¯(ξ,η))|}≤ε,
where ε>0 is a constant, and u¯(m,n)∈℘(Ω) is the solution of the following difference equation:
(3.7)u¯p(m,n)=a¯(m,n)+∑s=m0m-1∑t=n0n-1[F¯1(s,t,m,n,u¯(s,t))+∑ξ=m0s∑η=n0tF¯2(ξ,η,m,n,u¯(ξ,η))]+∑s=m0M-1∑t=n0M-1[G¯1(s,t,m,n,u¯(s,t))+∑ξ=m0s∑η=n0tG¯2(ξ,η,m,n,u¯(ξ,η))],
where F¯i,G¯i:Ω2×ℝ→ℝ,i=1,2; then one has
(3.8)|up(m,n)-u¯p(m,n)|≤ε[1+J(M,N)1-μ(M,N)C(m,n)],
provided that μ(M,N)<1, where
(3.9)J(m,n)=∑s=m0m-1∑t=n0n-1{f1(s,t,m,n)+∑ξ=m0s∑η=n0tf2(ξ,η,m,n)}+∑s=m0M-1∑t=n0N-1{g1(s,t,m,n)+∑ξ=m0s∑η=n0tg2(ξ,η,m,n)},μ(m,n)=∑s=m0M-1∑t=n0N-1{g1(s,t,m,n)C(s,t)+∑ξ=m0s∑η=n0tg2(ξ,η,m,n)C(ξ,η)},C(m,n)=exp⁡{∑s=m0m-1∑t=n0n-1B(s,t,m,n)},B(s,t,m,n)=f1(s,t,m,n)+∑ξ=m0s∑η=n0tf2(ξ,η,m,n).Proof.
From (3.1) and (3.7) we deduce
(3.10)|up(m,n)-u¯p(m,n)|≤|a(m,n)-a¯(m,n)|+∑s=m0m-1∑t=n0n-1{∑ξ=m0s∑η=n0t|F2(ξ,η,m,n,u(ξ,η))-F¯2(ξ,η,m,n,u¯(ξ,η))||F1(s,t,m,n,u(s,t))-F¯1(s,t,m,n,u¯(s,t))|+∑ξ=m0s∑η=n0t|F2(ξ,η,m,n,u(ξ,η))-F¯2(ξ,η,m,n,u¯(ξ,η))|}+∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0t|G2(ξ,η,m,n,u(ξ,η))-G¯2(ξ,η,m,n,u¯(ξ,η))||G1(s,t,m,n,u(s,t))-G¯1(s,t,m,n,u¯(s,t))|+∑ξ=m0s∑η=n0t|G2(ξ,η,m,n,u(ξ,η))-G¯2(ξ,η,m,n,u¯(ξ,η))|}≤|a(m,n)-a¯(m,n)|+∑s=m0m-1∑t=n0n-1{|F2(ξ,η,m,n,u¯(ξ,η))-F¯2(ξ,η,m,n,u¯(ξ,η))||F2(ξ,η,m,n,u¯(ξ,η))-F¯2(ξ,η,m,n,u¯(ξ,η))|||F2(ξ,η,m,n,u¯(ξ,η))-F¯2(ξ,η,m,n,u¯(ξ,η))|F1(s,t,m,n,u(s,t))-F1(s,t,m,n,u¯(s,t))|+|F1(s,t,m,n,u¯(s,t))-F¯1(s,t,m,n,u¯(s,t))|+∑ξ=m0s∑η=n0t|F2(ξ,η,m,n,u(ξ,η))-F2(ξ,η,m,n,u¯(ξ,η))|+|F2(ξ,η,m,n,u¯(ξ,η))-F¯2(ξ,η,m,n,u¯(ξ,η))|}+∑s=m0M-1∑t=n0N-1{|G2(ξ,η,m,n,u¯(ξ,η))-G¯2(ξ,η,m,n,u¯(ξ,η))G1(s,t,m,n,u(s,t))-G1(s,t,m,n,u¯(s,t))|+|G1(s,t,m,n,u¯(s,t))-G¯1(s,t,m,n,u¯(s,t))|+∑ξ=m0s∑η=n0t|G2(ξ,η,m,n,u(ξ,η))-G2(ξ,η,m,n,u¯(ξ,η))|+|G2(ξ,η,m,n,u¯(ξ,η))-G¯2(ξ,η,m,n,u¯(ξ,η))|}≤ε+∑s=m0m-1∑t=n0n-1{|F2(ξ,η,m,n,u(ξ,η))-F2(ξ,η,m,n,u¯(ξ,η))F1(s,t,m,n,u(s,t))-F1(s,t,m,n,u¯(s,t))|+|F2(ξ,η,m,n,u(ξ,η))-F2(ξ,η,m,n,u¯(ξ,η))|}+∑s=m0M-1∑t=n0N-1{∑ξ=m0s∑η=n0t|G2(ξ,η,m,n,u(ξ,η))-G2(ξ,η,m,n,u¯(ξ,η))||G2(ξ,η,m,n,u(ξ,η))-G2(ξ,η,m,n,u¯(ξ,η))G1(s,t,m,n,u(s,t))-G1(s,t,m,n,u¯(s,t))|+∑ξ=m0s∑η=n0t|G2(ξ,η,m,n,u(ξ,η))-G2(ξ,η,m,n,u¯(ξ,η))|}≤ε+∑s=m0m-1∑t=n0n-1[∑ξ=m0s∑η=n0tf2(ξ,η,m,n)|up(ξ,η)-u¯p(ξ,η)|f1(s,t,m,n)|up(s,t)-u¯p(s,t)|+∑ξ=m0s∑η=n0tf2(ξ,η,m,n)|up(ξ,η)-u¯p(ξ,η)|]+∑s=m0M-1∑t=n0M-1[∑ξ=m0s∑η=n0tg2(ξ,η,m,n)|up(ξ,η)-u¯p(ξ,η)|g1(s,t,m,n)|up(s,t)-u¯p(s,t)|+∑ξ=m0s∑η=n0tg2(ξ,η,m,n)|up(ξ,η)-u¯p(ξ,η)|].
Then a suitable application of Corollary 2.7 yields the desired result.Remark 3.5.
We note that the results in [1–30] are not available here to establish the analysis above.



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*Source: 584951-2011-08-29.xml*