# Some New Volterra-Fredholm-Type Nonlinear Discrete Inequalities with Two Variables Involving Iterated Sums and Their Applications

Authors: Run Xu
Journal: Journal of Applied Mathematics
 (2017)

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## Abstract

Some generalized discrete Volterra-Fredholm-type inequalities were developed, which can be used as effective tools in the qualitative analysis of the solution to difference equations.


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## Body

## 1. Introduction

In recent years, various forms of inequalities played increasingly important roles in the study of quantitative properties of solutions of differential and integral equations [1–15]. Discrete inequalities, especially the discrete Volterra-Fredholm-type inequalities, have been applied to study the discrete equations widely. For example, see [1–3, 9–11] and the references therein. In this paper, some new Volterra-Fredholm-type discrete inequalities involving four iterated infinite sums were established. Furthermore, to illustrate the usefulness of the established results, some examples were provided for the studying of their solutions on the boundedness, uniqueness, and continuous dependence.We design the needed symbols as follows:(a)
N0 denotes the set of nonnegative integers and Z denotes the set of integers, while R denotes the set of real numbers R+=0,∞.(b)
LetΩ≔m0,M×n0,N∩Z2, where m0,n0∈Z, and M,N∈Z∪∞ are two constants.(c)
Ki>0i=1,2,3,4 are all constants, and l1,l2∈Z are two constants.(d)
IfU 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.(e)
For a functiong∈℘+U, we have ∑s=m0m1gs=0 provided m0>m1.We need the following lemmas in the discussions of our main results.Lemma 1 (see [4]).
Letum,n∈℘+Ω,bs,t,m,n∈℘+Ω2 be nondecreasing in the third variable; k≥0 is a constant. For m,n∈Ω, if (1)um,n≤k+∑s=m0m-1∑t=n0n-1bs,t,m,nus,t,then (2)um,n≤kexp⁡∑s=m0m-1∑t=n0n-1bs,t,m,n.Lemma 2 (see [4]).
Letum,n,am,n,cm,n∈℘+Ω. If am,n is nondecreasing in the first variable, then, for m,n∈Ω, (3)um,n≤am,n+∑s=m0m-1cs,nus,n,then(4)um,n≤am,n∏s=m0m-11+cs,n.Lemma 3 (see [5]).
Leta≥0,p≥q≥0, and p≠0; then, for any K>0, (5)aq/p≤qpKq-p/pa+p-qpKq/p.


## 2. Main Results

Theorem 4.
Suppose thatum,n,am,n,b1m,n, b2m,n∈℘+Ω, cis,t,m,n, dis,t,m,n,eis,t,m,n,fjs,t,m,n,gjs,t,m,n, wjs,t,m,n∈℘+Ω2, and p,qi,ri,hj,vj are nonnegative constants with p≥qi>0,p≥ri>0i=1,2,…,l1, p≥hj>0, p≥vj>0j=1,2,…,l2, and ci,di,ei,fj,gj,wj being nondecreasing in the last two variables, b1m,n and b2m,n are also nondecreasing. If(6)upm,n≤am,n+b1m,n∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nuqis,t+dis,t,m,nuris,t+eis,t,m,n+b2m,n∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nuhjs,t+gjs,t,m,nuvjs,t+wjs,t,m,n,then, for m,n∈Ω, we have(7)um,n≤am,n+bm,nJM,N1-λM,NCm,n1/p,provided that λM,N<1, where(8)bm,n=max⁡b1m,n,b2m,n,(9)Cm,n=exp⁡∑s=m0m-1∑t=n0n-1Bs,t,m,n,(10)Bs,t,m,n=∑i=1l1cis,t,m,nqipK1qi-p/p+dis,t,m,nripK2ri-p/pbs,t,(11)Jm,n=∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nqipK1qi-p/pas,t+p-qipK1qi/p+dis,t,m,nripK2ri-p/pas,t+p-ripK2ri/p+eis,t,m,n+∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nhjpK3hj-p/pas,t+p-hjpK3hj/p+gjs,t,m,nvjpK4vj-p/pas,t+p-vjpK4vj/p+wjs,t,m,n,(12)λm,n=∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nhjpK3hj-p/p+gjs,t,m,nvjpK4vj-p/pbs,tCs,t.Proof.
Givenbm,n=max⁡b1m,n,b2m,n, for m,n∈Ω, we have(13)upm,n≤am,n+bm,n∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nuqis,t+dis,t,m,nuris,t+eis,t,m,n+bm,n∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nuhjs,t+gjs,t,m,nuvjs,t+wjs,t,m,n.Define a function zm,n by(14)zm,n=∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nuqis,t+dis,t,m,nuris,t+eis,t,m,n+∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nuhjs,t+gjs,t,m,nuvjs,t+wjs,t,m,n.Then(15)upm,n≤am,n+bm,nzm,n,or(16)um,n≤am,n+bm,nzm,n1/p.By using Lemma 3, for any Ki>0i=1,2,3,4, we have(17)zm,n≤∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nqipK1qi-p/pas,t+bs,tzs,t+p-qipK1qi/p+dis,t,m,nripK2ri-p/pas,t+bs,tzs,t+p-ripK2ri/p+eis,t,m,n+∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nhjpK3hj-p/pas,t+bs,tzs,t+p-hjpK3hj/p+gjs,t,m,nvjpK4vj-p/pas,t+bs,tzs,t+p-vjpK4vj/p+wjs,t,m,n=Rm,n+∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nqipK1qi-p/p+dis,t,m,nripK2ri-p/pbs,tzs,t,where(18)Rm,n=Jm,n+∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nhjpK3hj-p/p+gjs,t,m,nvjpK4vj-p/pbs,tzs,t,and Jm,n is defined in (11). Then, using that Rm,n is nondecreasing in every variable, we get(19)zm,n≤RM,N+∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nqipK1qi-p/p+dis,t,m,nripK2ri-p/pbs,tzs,t=RM,N+∑s=m0m-1∑t=n0n-1Bs,t,m,nzs,t,where Bs,t,m,n is defined in (10).
Sincebm,n is nondecreasing and cis,t,m,n,dis,t,m,n are nondecreasing in the last two variables, then Bs,t,m,n is also nondecreasing in the last two variables, and, by Lemma 1 and (19), we get(20)zm,n≤RM,Nexp⁡∑s=m0m-1∑t=n0n-1Bs,t,m,n=RM,NCm,n,where Cm,n is defined in (9). Considering the definition of Rm,n and (20), we have(21)RM,N=JM,N+∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,M,NhjpK3hj-p/p+gjs,t,M,NvjpK4vj-p/pbs,tzs,t≤JM,N+RM,N∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,M,NhjpK3hj-p/p+gjs,t,M,NvjpK4vj-p/pbs,tCs,t=JM,N+RM,NλM,N,where λm,n is defined in (12). Then,(22)RM,N≤JM,N1-λM,N.Combining (20) and (22), we deduce(23)zm,n≤JM,N1-λM,NCm,n,where Cm,n,λm,n are defined in (9) and (12).
Then, combining (16) and (23), we obtain the desired result.Corollary 5.
Letr1im,n,d1im,n,c1im,n,e1im,n∈℘+Ω,i=1,2,…,l1,f1jm,n,g1jm,n, w1jm,n,r2jm,n∈℘+Ω,j=1,2,…,l2,r1im,n,r2jm,n,b1m,n and b2m,n be nondecreasing in every variable. um,n,am,n,b1m,n,b2m,n,p,qi,ri,hj,vj are defined as in Theorem 4. If(24)upm,n≤am,n+b1m,n∑i=1l1r1im,n∑s=m0m-1∑t=n0n-1c1is,tuqis,t+d1is,turis,t+e1is,t+b2m,n∑j=1l2r2jm,n∑s=m0M-1∑t=n0N-1f1js,tuhjs,t+g1js,tuvjs,t+w1js,t,then, for m,n∈Ω, we have(25)um,n≤am,n+bm,nJM,N1-λM,NCm,n1/p,provided that λM,N<1, where (26)bm,n=max⁡b1m,n,b2m,n,Cm,n=exp⁡∑s=m0m-1∑t=n0n-1Bs,t,m,n,Bs,t,m,n=∑i=1l1r1im,nc1is,tqipK1qi-p/p+d1is,tripK2ri-p/pbs,t,Jm,n=∑i=1l1r1im,n∑s=m0m-1∑t=n0n-1c1is,tqipK1qi-p/pas,t+p-qipK1qi/p+d1is,tripK2ri-p/pas,t+p-ripK2ri/p+e1is,t+∑j=1l2r2jm,n∑s=m0M-1∑t=n0N-1f1js,thjpK3hj-p/pas,t+p-hjpK3hj/p+g1js,tvjpK4vj-p/pas,t+p-vjpK4vj/p+w1js,t,λm,n=∑j=1l2r2jm,n∑s=m0M-1∑t=n0N-1f1js,thjpK3hj-p/p+g1js,tvjpK4vj-p/pbs,tCs,t.The proof of Corollary5 can be completed by setting cis,t,m,n=r1im,nc1is,t, dis,t,m,n=r1im,nd1is,t,eis,t,m,n=r1im,ne1is,t,fjs,t,m,n=r2jm,nf1js,t,gjs,t,m,n=r2jm,ng1js,t,wjs,t,m,n=r2jm,nw1js,t in Theorem 4.Lettingp=1, we get the following corollary.Corollary 6.
Letum,n,am,n,b1m,n,b2m,n,cis,t,m,n,dis,t,m,n,eis,t,m,n, fjs,t,m,n, gjs,t,m,n,wjs,t,m,n be defined as in Theorem 4. If(27)um,n≤am,n+b1m,n∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nus,t+dis,t,m,nus,t+eis,t,m,n+b2m,n∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nus,t+gjs,t,m,nus,t+wjs,t,m,n,then, for m,n∈Ω, we have(28)um,n≤am,n+bm,nJM,N1-λM,NCm,n,provided that λM,N<1, where (29)bm,n=max⁡b1m,n,b2m,n,Cm,n=exp⁡∑s=m0m-1∑t=n0n-1Bs,t,m,n,Bs,t,m,n=∑i=1l1cis,t,m,n+dis,t,m,nbs,t,Jm,n=∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,n+dis,t,m,nas,t+eis,t,m,n+∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,n+gjs,t,m,nas,t+wjs,t,m,n,λm,n=∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,n+gjs,t,m,nbs,tCs,t.Theorem 7.
Letφm,n∈℘+Ω, um,n,am,n, b1m,n,b2m,n, cis,t,m,n, dis,t,m,n, eis,t,m,n, fjs,t,m,n,gjs,t,m,n,wjs,t,m,n, p,qi,ri,hj,vj be defined as in Theorem 4. Assume that am,n is nondecreasing in the first variable. If(30)upm,n≤am,n+∑s=m0m-1φs,nups,n+b1m,n∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nuqis,t+dis,t,m,nuris,t+eis,t,m,n+b2m,n∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nuhjs,t+gjs,t,m,nuvjs,t+wjs,t,m,n,then, for m,n∈Ω, we have(31)um,n≤am,n+bm,nJ~M,N1-λ~M,NC~m,nφ~m,n1/p,provided that λ~M,N<1, where(32)φ~m,n=∏s=m0m-11+φs,n,(33)bm,n=max⁡b1m,n,b2m,n,(34)c~is,t,m,n=cis,t,m,nφ~s,tqi/p,d~is,t,m,n=dis,t,m,nφ~s,tri/p,i=1,2,…,l1,f~js,t,m,n=fjs,t,m,nφ~s,thj/p,g~js,t,m,n=gjs,t,m,nφ~s,tvj/p,j=1,2,…,l2,(35)C~m,n=exp⁡∑s=m0m-1∑t=n0n-1B~s,t,m,n,(36)B~s,t,m,n=∑i=1l1c~is,t,m,nqipK1qi-p/p+d~is,t,m,nripK2ri-p/pbs,t,(37)J~m,n=∑i=1l1∑s=m0m-1∑t=n0n-1c~is,t,m,nqipK1qi-p/pas,t+p-qipK1qi/p+d~is,t,m,nripK2ri-p/pas,t+p-ripK2ri/p+eis,t,m,n+∑j=1l2∑s=m0M-1∑t=n0N-1f~js,t,m,nhjpK3hj-p/pas,t+p-hjpK3hj/p+g~js,t,m,nvjpK4vj-p/pas,t+p-vjpK4vj/p+wjs,t,m,n,(38)λ~m,n=∑j=1l2∑s=m0M-1∑t=n0N-1f~js,t,m,nhjpK3hj-p/p+g~js,t,m,nvjpK4vj-p/pbs,tC~s,t.Proof.
Givenbm,n=max⁡b1m,n,b2m,n, for m,n∈Ω, we have(39)upm,n≤am,n+∑s=m0m-1φs,nups,n+bm,n∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nuqis,t+dis,t,m,nuris,t+eis,t,m,n+bm,n∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nuhjs,t+gjs,t,m,nuvjs,t+wjs,t,m,n.Define function z~m,n by(40)z~m,n=am,n+bm,n∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nuqis,t+dis,t,m,nuris,t+eis,t,m,n+bm,n∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nuhjs,t+gjs,t,m,nuvjs,t+wjs,t,m,n.Then,(41)upm,n≤z~m,n+∑s=m0m-1φs,nups,n.Clearly zm,n is nondecreasing in the first variable. Then, by Lemma 2, we get(42)upm,n≤z~m,n∏s=m0m-11+φs,n=z~m,nφ~m,n,where φ~m,n is defined in (32). Define function(43)vm,n=∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nuqis,t+dis,t,m,nuris,t+eis,t,m,n+∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nuhjs,t+gjs,t,m,nuvjs,t+wjs,t,m,n.From (40), we get(44)z~m,n=am,n+bm,nvm,n.Then (42) becomes(45)um,n≤am,n+bm,nvm,nφ~m,n1/p.By (45) and Lemma 3, from (43), we have(46)vm,n≤∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nφ~s,tqi/pqipK1qi-p/pas,t+bs,tvs,t+p-qipK1qi/p+dis,t,m,nφ~s,tri/pripK2ri-p/pas,t+bs,tvs,t+p-ripK2ri/p+eis,t,m,n+∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nφ~s,thj/phjpK3hj-p/pas,t+bs,tvs,t+p-hjpK3hj/p+gjs,t,m,nφ~s,tvj/pvjpK4vj-p/pas,t+bs,tvs,t+p-vjpK4vj/p+wjs,t,m,n=R~m,n+∑i=1l1∑s=m0m-1∑t=n0n-1c~is,t,m,nqipK1qi-p/p+d~is,t,m,nripK2ri-p/pbs,tvs,t,where (47)R~m,n=J~m,n+∑j=1l2∑s=m0M-1∑t=n0N-1f~js,t,m,nhjpK3hj-p/p+g~js,t,m,nvjpK4vj-p/pbs,tvs,t,c~i,d~i,f~j,g~j and J~m,n are defined in (34) and (37), respectively.
Similar to the process of (17)–(23), we deduce that(48)vm,n≤J~M,N1-λ~M,NC~m,n,where C~m,n,λ~m,n are defined in (35) and (38).
Combining (45) and (48), we get the desired result.Theorem 8.
Letum,n,am,n, b1m,n,b2m,n, cis,t,m,n, dis,t,m,n, eis,t,m,n, fjs,t,m,n,gjs,t,m,n,wjs,t,m,n, p,qi,ri,hj,vj be defined as in Theorem 4. Hj,Lj:Ω×R+→R+j=1,2,…,l2 satisfies 0≤Hjm,n,u-Hjm,n,v≤Ljm,n,vu-v for u≥v≥0. If(49)upm,n≤am,n+b1m,n∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nuqis,t+dis,t,m,nuris,t+eis,t,m,n+b2m,n∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nHjs,t,uhjs,t+gjs,t,m,nHjs,t,uvjs,t+wjs,t,m,n,then, for m,n∈Ω, we have(50)um,n≤am,n+bm,nJ¯M,N1-λ¯M,NC¯m,n1/p,provided that λ¯M,N<1, where (51)bm,n=max⁡b1m,n,b2m,n,(52)C¯m,n=exp⁡∑s=m0m-1∑t=n0n-1B¯s,t,m,n,(53)B¯s,t,m,n=∑i=1l1cis,t,m,nqipK1qi-p/p+dis,t,m,nripK2ri-p/pbs,t,(54)J¯m,n=∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nqipK1qi-p/pas,t+p-qipK1qi/p+dis,t,m,nripK2ri-p/pas,t+p-ripK2ri/p+eis,t,m,n+∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nHjs,t,hjpK3hj-p/pas,t+p-hjpK3hj/p+gjs,t,m,nHjs,t,vjpK4vj-p/pas,t+p-vjpK4vj/p+wjs,t,m,n,(55)f¯js,t,m,n=fjs,t,m,nLjs,t,hjpK3hj-p/pas,t+p-hjpK3hj/p,j=1,2,…,l2,(56)g¯js,t,m,n=gjs,t,m,nLjs,t,vjpK4vj-p/pas,t+p-vjpK4vj/p,j=1,2,…,l2,(57)λ¯m,n=∑j=1l2∑s=m0M-1∑t=n0N-1f¯js,t,m,nhjpK3hj-p/p+g¯js,t,m,nvjpK4vj-p/pbs,tC¯s,t.Proof.
Givenbm,n=max⁡b1m,n,b2m,n, for m,n∈Ω, we have(58)upm,n≤am,n+bm,n∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nuqis,t+dis,t,m,nuris,t+eis,t,m,n+bm,n∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nHjs,t,uhjs,t+gjs,t,m,nHjs,t,uvjs,t+wjs,t,m,n.Define function v¯m,n by(59)v¯m,n=∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nuqis,t+dis,t,m,nuris,t+eis,t,m,n+∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nHjs,t,uhjs,t+gjs,t,m,nHjs,t,uvjs,t+wjs,t,m,n.Then(60)upm,n≤am,n+bm,nv¯m,n,or(61)um,n≤am,n+bm,nv¯m,n1/p.By Lemma 3, we have(62)v¯m,n≤∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nas,t+bs,tv¯s,tqi/p+dis,t,m,nas,t+bs,tv¯s,tri/p+eis,t,m,n+∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nHjs,t,as,t+bs,tv¯s,thj/p+gjs,t,m,nHjs,t,as,t+bs,tv¯s,tvj/p+wjs,t,m,n≤∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nqipK1qi-p/pas,t+bs,tv¯s,t+p-qipK1qi/p+dis,t,m,nripK2ri-p/pas,t+bs,tv¯s,t+p-ripK2ri/p+eis,t,m,n+∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nHjs,t,hjpK3hj-p/pas,t+bs,tv¯s,t+p-hjpK3hj/p-Hjs,t,hjpK3hj-p/pas,t+p-hjpK3hj/p+Hjs,t,hjpK3hj-p/pas,t+p-hjpK3hj/p+gjs,t,m,nHjs,t,vjpK4vj-p/pas,t+bs,tv¯s,t+p-vjpK4vj/p-Hjs,t,vjpK4vj-p/pas,t+p-vjpK4vj/p+Hjs,t,vjpK4vj-p/pas,t+p-vjpK4vj/p+wjs,t,m,n≤∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nqipK1qi-p/pas,t+bs,tv¯s,t+p-qipK1qi/p+dis,t,m,nripK2ri-p/pas,t+bs,tv¯s,t+p-ripK2ri/p+eis,t,m,n+∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nLjs,t,hjpK3hj-p/pas,t+p-hjpK3hj/phjpK3hj-p/pbs,tv¯s,t+Hjs,t,hjpK3hj-p/pas,t+p-hjpK3hj/p+gjs,t,m,nLjs,t,vjpK4vj-p/pas,t+p-vjpK4vj/pvjpK4vj-p/pbs,tv¯s,t+Hjs,t,vjpK4vj-p/pas,t+p-vjpK4vj/p+wjs,t,m,n=R¯m,n+∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nqipK1qi-p/p+dis,t,m,nripK2ri-p/pbs,tv¯s,t,where(63)R¯m,n=J¯m,n+∑j=1l2∑s=m0M-1∑t=n0N-1f¯js,t,m,nhjpK3hj-p/p+g¯js,t,m,nvjpK4vj-p/pbs,tv¯s,t,and J¯m,n,f¯js,t,m,n,g¯js,t,m,n are defined in (54)–(56).
Similar to the process of (17)–(23), we get(64)v¯m,n≤J¯M,N1-λ¯M,NC¯m,n,where C¯m,n,λ¯m,n are defined in (52) and (57).
Combining (61) and (64), we get the desired result.Theorem 9.
Letφm,n∈℘+Ω,um,n,am,n, b1m,n,b2m,n, cis,t,m,n, dis,t,m,n, eis,t,m,n, fjs,t,m,n,gjs,t,m,n,wjs,t,m,n, p,qi,ri,hj,vj be defined as in Theorem 4. Assume that am,n is nondecreasing in the first variable. Hj,Ljj=1,2,…,l2 are defined as in Theorem 7. If(65)upm,n≤am,n+∑s=m0m-1φs,nups,n+b1m,n∑i=1l1∑s=m0m-1∑t=n0n-1cis,t,m,nuqis,t+dis,t,m,nuris,t+eis,t,m,n+b2m,n∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nHjs,t,uhjs,t+gjs,t,m,nHjs,t,uvjs,t+wjs,t,m,n,then, for m,n∈Ω, we have(66)um,n≤am,n+bm,nJ^M,N1-λ^M,NC^m,n1/p,provided that λ^M,N<1, where (67)φ^m,n=∏s=m0m-11+φs,n,bm,n=max⁡b1m,n,b2m,n,C^m,n=exp⁡∑s=m0m-1∑t=n0n-1B^s,t,m,n,B^s,t,m,n=∑i=1l1c^is,t,m,nqipK1qi-p/p+d^is,t,m,nripK2ri-p/pbs,t,J^m,n=∑i=1l1∑s=m0m-1∑t=n0n-1c^is,t,m,nqipK1qi-p/pas,t+p-qipK1qi/p+d^is,t,m,nripK2ri-p/pas,t+p-ripK2ri/p+eis,t,m,n+∑j=1l2∑s=m0M-1∑t=n0N-1fjs,t,m,nHjs,t,φ^s,thj/phjpK3hj-p/pas,t+p-hjpK3hj/p+gjs,t,m,nHjs,t,φ^s,tvj/pvjpK4vj-p/pas,t+p-vjpK4vj/p+wjs,t,m,n,λ^m,n=∑j=1l2∑s=m0M-1∑t=n0N-1f^js,t,m,nhjpK3hj-p/p+g^js,t,m,nvjpK4vj-p/pbs,tC^s,t,c~is,t,m,n=cis,t,m,nφ^s,tqi/p,d~is,t,m,n=dis,t,m,nφ^s,tri/p,i=1,2,…,l1,f^js,t,m,n=fjs,t,m,nφ^s,thj/pLjs,t,φ^s,thj/phjpK3hj-p/pas,t+p-hjpK3hj/p,g^js,t,m,n=gjs,t,m,nφ^s,tvj/pLjs,t,φ^s,tvj/pvjpK4vj-p/pas,t+p-vjpK4vj/p,j=1,2,…,l2.The proof for Theorem9 is similar to the combination of Theorems 7 and 8, and we omit the details here.


## 3. Applications

In this section, we will present some applications for the established results to study boundedness, uniqueness, and continuous dependence of solutions of certain difference equations.Consider the following Volterra-Fredholm sum-difference equations:(68)upm,n=am,n+∑s=m0m-1∑t=n0n-1Cs,t,m,n,us,t+Ds,t,m,n,us,t+Es,t,m,n+∑s=m0M-1∑t=n0N-1Fs,t,m,n,us,t+Gs,t,m,n,us,t+Ws,t,m,n,where um,n,am,n∈℘Ω,p≥1 is an odd number, C,D,F,G:Ω2×R→R,E,W∈℘Ω2.Theorem 10.
Assume that functionsC,D,E,F,G,W in equation (68) satisfy the following conditions:(69)Cs,t,m,n,u1≤c1s,t,m,nu1q,Ds,t,m,n,u1≤d1s,t,m,nu1r,Es,t,m,n≤e1s,t,m,n,Fs,t,m,n,u1≤f1s,t,m,nu1h,Gs,t,m,n,u1≤g1s,t,m,nu1v,Ws,t,m,n≤w1s,t,m,nfor m,n∈Ω,u1∈R, where q,r,h,v are nonnegative constants satisfying p≥q>0,p≥r>0,p≥h>0,p≥v>0,c1,d1,e1,f1,g1,w1∈℘+Ω2 which are nondecreasing in the last two variables; then one has(70)um,n≤am,n+J1M,N1-λ1M,NC1m,n1/p,provided that λ1M,N<1, where (71)C1m,n=exp⁡∑s=m0m-1∑t=n0n-1B1s,t,m,n,B1s,t,m,n=c1s,t,m,nqpK1q-p/p+d1s,t,m,nrpK2r-p/p,J1m,n=∑s=m0m-1∑t=n0n-1c1s,t,m,nqpK1q-p/pas,t+p-qpK1q/p+d1s,t,m,nrpK2r-p/pas,t+p-rpK2r/p+e1s,t,m,n+∑s=m0M-1∑t=n0N-1f1s,t,m,nhpK3h-p/pas,t+p-hpK3h/p+g1s,t,m,nvpK4v-p/pas,t+p-vpK4v/p+w1s,t,m,n,λ1m,n=∑s=m0M-1∑t=n0N-1f1s,t,m,nhpK3h-p/p+g1s,t,m,nvpK4v-p/pC1s,t.Proof.
Using conditions (69) to (68), we have(72)upm,n≤am,n+∑s=m0m-1∑t=n0n-1Cs,t,m,n,us,t+Ds,t,m,n,us,t+Es,t,m,n+∑s=m0M-1∑t=n0N-1Fs,t,m,n,us,t+Gs,t,m,n,us,t+Ws,t,m,n≤am,n+∑s=m0m-1∑t=n0n-1c1s,t,m,nuqs,t+d1s,t,m,nurs,t+e1s,t,m,n+∑s=m0M-1∑t=n0N-1f1s,t,m,nuhs,t+g1s,t,m,nuvs,t+w1s,t,m,n.Then a suitable application of Theorem 4 (with l1=l2=1) to (72) yields the desired result.
The following theorem deals with the uniqueness of the solutions of (68).Theorem 11.
Supposing that(73)Cs,t,m,n,u1-Cs,t,m,n,u2≤c1s,t,m,nu1p-u2p,Fs,t,m,n,u1-Fs,t,m,n,u2≤f1s,t,m,nu1p-u2p,Ds,t,m,n,u1-Ds,t,m,n,u2≤d1s,t,m,nu1p-u2p,Gs,t,m,n,u1-Gs,t,m,n,u2≤g1s,t,m,nu1p-u2phold for u1,u2∈R, where c1,d1,f1,g1∈℘+Ω2 are nondecreasing in the last two variables, (74)λM,N=∑s=m0M-1∑t=n0N-1f1s,t,M,N+g1s,t,M,NCs,t<1,Bs,t,m,n=c1s,t,m,n+d1s,t,m,n,Cs,t=exp⁡∑s=m0m-1∑t=n0n-1Bs,t,m,n,then (68) has at most one solution.Proof.
Assume thatum,n,u¯m,n are two solutions of (68). Then (75)upm,n-u¯pm,n≤∑s=m0m-1∑t=n0n-1Cs,t,m,n,us,t-Cs,t,m,n,u¯s,t+Ds,t,m,n,us,t-Ds,t,m,n,u¯s,t+∑s=m0M-1∑t=n0N-1Fs,t,m,n,us,t-Fs,t,m,n,u¯s,t+Gs,t,m,n,us,t-Gs,t,m,n,u¯s,t≤∑s=m0m-1∑t=n0n-1c1s,t,m,n+d1s,t,m,nups,t-u¯ps,t+∑s=m0M-1∑t=n0N-1f1s,t,m,n+g1s,t,m,nups,t-u¯ps,t.
Treatupm,n-u¯pm,n as one variable, and a suitable application of Corollary 6 yields upm,n-u¯pm,n≤0, which implies that upm,n≡u¯pm,n. Since p is an odd number, then we have upm,n=u¯pm,n, and the proof is complete.Finally we study the continuous dependence of the solutions of (68) on functions a,C,D,E,F,G,W. For this, we consider the following variation of (68):(76)u~pm,n=a~m,n+∑s=m0m-1∑t=n0n-1C~s,t,m,n,u~s,t+D~s,t,m,n,u~s,t+E~s,t,m,n+∑s=m0M-1∑t=n0N-1F~s,t,m,n,u~s,t+G~s,t,m,n,u~s,t+W~s,t,m,n,where C~,D~,F~,G~:Ω2×R→R,E~,W~∈℘Ω2 and p≥1 is an odd number.Theorem 12.
Consider (68) and (76). If (77)Cs,t,m,n,u1s,t-Cs,t,m,n,u2s,t≤c1s,t,m,nu1p-u2p,Ds,t,m,n,u1s,t-Ds,t,m,n,u2s,t≤d1s,t,m,nu1p-u2p,Fs,t,m,n,u1s,t-Fs,t,m,n,u2s,t≤f1s,t,m,nu1p-u2p,Gs,t,m,n,u1s,t-Gs,t,m,n,u2s,t≤g1s,t,m,nu1p-u2p,hold for u1,u2∈R, where c1,d1,f1,g1∈℘+Ω2, and are nondecreasing in the last two variables, furthermore, for all solution u~ of (76), the following conditions hold for m,n∈Ω: (78)am,n-a~m,n≤ε4,∑s=m0m-1∑t=n0n-1Es,t,m,n-E~s,t,m,n≤ε8,∑s=m0M-1∑t=n0N-1Ws,t,m,n-W~s,t,m,n≤ε8,∑s=m0m-1∑t=n0n-1Fs,t,m,n,u~-F~s,t,m,n,u~≤ε8,∑s=m0M-1∑t=n0N-1Cs,t,m,n,u~-C~s,t,m,n,u~≤ε8,∑s=m0m-1∑t=n0n-1Ds,t,m,n,u~-D~s,t,m,n,u~≤ε8,∑s=m0M-1∑t=n0N-1Gs,t,m,n,u~-G~s,t,m,n,u~≤ε8,where ε>0 is an arbitrary constant. Then(79)upm,n-u~pm,n≤ε1+J2M,N1-λ2M,NC2m,n,where λ2M,N<1, and (80)C2m,n=exp⁡∑s=m0m-1∑t=n0n-1B2s,t,m,n,B2s,t,m,n=c1s,t,m,n+d1s,t,m,n,λ2m,n=∑s=m0M-1∑t=n0N-1f1s,t,m,n+g1s,t,m,nC2s,t,J2m,n=∑s=m0m-1∑t=n0n-1c1s,t,m,n+d1s,t,m,n+∑s=m0M-1∑t=n0N-1f1s,t,m,n+g1s,t,m,nfor m,n∈Ω. That is, up depends continuously on the functions a,C,D,E,F,G,W.Proof.
Letum,n and u~m,n be solutions of (68) and (76), respectively. Then um,n satisfies (68) and u~m,n satisfies (76). Hence (81)upm,n-u~pm,n≤am,n-a~m,n+∑s=m0m-1∑t=n0n-1Cs,t,m,n,us,t-C~s,t,m,n,u~s,t+Ds,t,m,n,us,t-D~s,t,m,n,u~s,t+Es,t,m,n-E~s,t,m,n+∑s=m0M-1∑t=n0N-1Fs,t,m,n,us,t-F~s,t,m,n,u~s,t+Gs,t,m,n,us,t-G~s,t,m,n,u~s,t+Ws,t,m,n-W~s,t,m,n≤am,n-a~m,n+∑s=m0m-1∑t=n0n-1Cs,t,m,n,us,t-Cs,t,m,n,u~s,t+Cs,t,m,n,u~s,t-C~s,t,m,n,u~s,t+Ds,t,m,n,us,t-Ds,t,m,n,u~s,t+Ds,t,m,n,u~s,t-D~s,t,m,n,u~s,t+Es,t,m,n-E~s,t,m,n+∑s=m0M-1∑t=n0N-1Fs,t,m,n,us,t-Fs,t,m,n,u~s,t+Fs,t,m,n,u~s,t-F~s,t,m,n,u~s,t+Gs,t,m,n,us,t-Gs,t,m,n,u~s,t+Gs,t,m,n,u~s,t-G~s,t,m,n,u~s,t+Ws,t,m,n-W~s,t,m,n≤ε+∑s=m0m-1∑t=n0n-1c1s,t,m,n+d1s,t,m,nup-u~p+∑s=m0M-1∑t=n0N-1f1s,t,m,n+g1s,t,m,nup-u~p.Treat upm,n-u~pm,n as one variable, and a suitable application of Corollary 6 (with l1=l2=1) yields the desired result (79). Hence up depends continuously on a,C,D,E,F,G,W.


## 4. Conclusions

The author carried out some new Volterra-Fredholm-type discrete inequalities involving four iterated infinite sums and their corresponding applications. The results are more effective to qualitative analysis of solutions for sum-difference equations, such as the boundedness, uniqueness, and continuous dependence on solutions.



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*Source: 1474052-2017-09-27.xml*