doi:10.1155/2009/486895
Research Article
An Extension to Nonlinear Sum-Difference
Inequality and Applications
Wu-Sheng Wang
1and Xiaoliang Zhou
21Department of Mathematics, Hechi University, Yizhou, Guangxi 546300, China 2Department of Mathematics, Guangdong Ocean University, Zhanjiang 524088, China
Correspondence should be addressed to Xiaoliang Zhou,[email protected]
Received 31 March 2009; Revised 31 March 2009; Accepted 17 May 2009
Recommended by Martin J. Bohner
We establish a general form of sum-difference inequality in two variables, which includes both more than two distinct nonlinear sums without an assumption of monotonicity and a nonconstant term outside the sums. We employ a technique of monotonization and use a property of stronger monotonicity to give an estimate for the unknown function. Our result enables us to solve those discrete inequalities considered in the work of W.-S. Cheung2006. Furthermore, we apply our result to a boundary value problem of a partial difference equation for boundedness, uniqueness, and continuous dependence.
Copyrightq2009 W.-S. Wang and X. Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Gronwall-Bellman inequality 1, 2 is a fundamental tool in the study of existence, uniqueness, boundedness, stability, invariant manifolds and other qualitative properties of solutions of differential equations and integral equation. There are a lot of papers investigating them such as 3–15. Along with the development of the theory of integral inequalities and the theory of difference equations, more attentions are paid to some discrete versions of Bellman-Gronwall type inequalitiese.g.,16–18. Starting from the basic form
un≤an n−1 s0
fsus, 1.1
discussed in19, an interesting direction is to consider the inequality
u2n≤P2u20 2
n−1
s0
a discrete version of Dafermos’ inequality20, whereα, P, Qare nonnegative constants and
u, gare nonnegative functions defined on{1,2, . . . , T}and{1,2, . . . , T−1}, respectively. Pang and Agarwal21proved for1.2thatun≤1αnP u0 ns−01Qgsfor all 0≤n≤T. Another form of sum-difference inequality
u2n≤c22
u0 ds/ws. Recently, Pachpatte23,24discussed the inequalities of two variables
um, n≤c
wheregis nondecreasing. In25another form of inequality of two variables
u2m, n≤c2
was discussed. Later, this result was generalized in26to the inequality
upm, n≤c functions defined on a lattice inZ2
, andwis a continuous nondecreasing function satisfying wu>0 for allu >0.
In this paper we establish a more general form of sum-difference inequality with positive integersm, n,
ψum, n≤am, n k
ϕ1u and ϕ2u, respectively. Moreover, we consider more than two nonlinear terms and
monotonization to construct a sequence of functions which possesses stronger monotonicity than the previous one. Unlike the work in26for two sum terms, the maximal regions of validity for our estimate of the unknown functionuare decided by boundaries of more than two planar regions. Thus we have to consider the inclusion of those regions and find common regions. We demonstrate that inequalities1.6and other inequalities considered in26can also be solved with our result. Furthermore, we apply our result to boundary value problems of a partial difference equation for boundedness, uniqueness, and continuous dependence.
2. Main Result
Throughout this paper, letR −∞,∞,R 0,∞, andN0{0,1,2, . . .},m0, n0∈N0, X, Y ∈
N0∪ {∞}are given nonnegative integers. For any integerss < t, let diss, t {j : s ≤ j ≤
t, j ∈N0},I dism0, X, andJ disn0, Y. DefineΛ I×J ⊂N20, and letΛs,tdenote the sublattice dism0, s×disn0, tinΛfor anys, t∈Λ.
For functions gm, n, m, n ∈ N0, their first-order differences are defined by
Δ1gm, n gm1, n−gm, n and Δ2gm, n gm, n1−gm, n. Obviously, the
linear difference equationΔxm bmwith the initial conditionxm0 0 has the solution
m−1
sm0bs. In the sequel, for convenience, we complementarily define that
m0−1
sm0bs 0. We give the following basic assumptions for the inequality1.7.
H1ψ is a strictly increasing continuous function onRsatisfying thatψ∞ ∞and ψu>0 for allu >0.
H2Allϕii1,2, . . . , kare continuous and positive functions onR. H3am, n≥0 onΛ.
H4Allfii1,2, . . . , kare nonnegative functions onΛ×Λ.
With given functionsϕ1, ϕ2, andψ, we technically consider a sequence of functions
wis, which can be calculated recursively by
w1s: max
τ∈0,s
ϕ1τ
,
wi1s: max
τ∈0,s
ϕ
i1τ
wiτ
wis, i1, . . . , k−1.
2.1
For given constantsui>0 and variableu >0, we define
Wiu, ui:
u
ui dx wi ψ−1x
, i1,2, . . . , k. 2.2
Obviously,Wiis strictly increasing inu >0 and therefore the inversesWi−1are well defined,
continuous, and increasing. Let
fim, n, s, t: max τ,ξ∈m0,m×n0,n
fiτ, ξ, s, t, 2.3
which is nondecreasing in m and n for each fixed s and t and satisfies fix, y, t, s ≥
Theorem 2.1. Suppose thatH1–H4hold andum, nis a nonnegative function onΛsatisfying
whereΥkm, nis determined recursively by
Υ1m, n:am0, n0
do not affect our results, we simply letWiudenoteWiu, uiwhen there is no confusion. For positive constantsvi/ui, letWiu Moreover, by replacingWiwithWi, the condition in the definition ofUinTheorem 2.1reads
the left-hand side of which is equal to
and the right-hand side of which equals
ui
vi dx wi ψ−1x
∞
ui
dx wi ψ−1x
Wiui
∞
ui dx wi ψ−1x
. 2.10
The comparison between the both sides implies that2.8is equivalent to the condition given in the definition ofUinTheorem 2.1withm, n M1, N1.
Remark 2.3. If we choosek 2, ψu up,ϕ
1u uq,ϕ2u uqwu with p > q > 0,
f1m, n, s, t ds, tandf2m, n, s, t es, tand restrictam, nto be a constantcin1.7,
then we can applyTheorem 2.1to inequality1.6discussed in26.
3. Proof of Theorem
First of all, we monotonize some given functionsϕi,fi in the sums. Obviously, the sequence
wisdefined byϕii1, . . . , kin2.1consists of nondecreasing nonnegative functions and
satisfieswis≥ϕis, fori1, . . . , k. Moreover,
wi∝wi1, i1,2, . . . , k−1, 3.1
as defined in27for comparison of monotonicity of functionswis i 1, . . . , k, because every ratiowi1s/wisis nondecreasing. By the definitions of functionswi,fi, ψ, andΥ1,
from1.7we get
um, n≤ψ−1
Υ1m, n
k
i1
m−1
sm0 n−1
tn0
fim, n, s, twius, t
, ∀m, n∈Λ. 3.2
Then, we discuss the case thatam, n>0 for allm, n∈Λ. BecauseΥ1satisfies
Υ1m, n am0, n0
m−1
sm0
|as1, n0−as, n0|
n−1
tn0
|am, t1−am, t|
≥am, n,
3.3
it is positive and nondecreasing onΛ. We consider the auxiliary inequality to 3.2, for all
m, n∈ΛM,N,
um, n≤ψ−1
Υ1M, N
k
i1
m−1
sm0 n−1
tn0
fiM, N, s, twius, t
where M ∈ dism0, M1 and N ∈ disn0, N1 are chosen arbitrarily, and claim that, for
whereΥkM, N, m, n is determined recursively by
We note thatM2,N2can be chosen appropriately such that
M2M, N M1, N2M, N N1, ∀M, N∈ΛM1,N1. 3.8
In the following, we will use mathematical induction to prove3.5.
Fork 1, letzm, n ms−m10 nt−n10f1M, N, s, tw1us, t. Thenzis nonnegative
and nondecreasing in each variable onΛM,N. From3.4we observe that
Moreover, we note that w1 is nondecreasing and satisfies w1u > 0 for u > 0 and that
Υ1M, N zm, n>0. From3.10we have
Δ1Υ1M, N zm, n
w1 ψ−1Υ1M, N zm, n
n−1
tn0f1M, N, m, tw1um, t w1 ψ−1Υ1M, N zm, n
≤n−1 tn0
f1M, N, m, t.
3.11
On the other hand, by the Mean Value Theorem for integral and by the monotonicity ofw1
and ψ, for arbitrarily givenm, n, m1, n ∈ ΛM N there exists ξ in the open interval Υ1M, N zm, n,Υ1M, N zm1, nsuch that
W1Υ1M, N zm1, n−W1Υ1M, N zm, n
Υ1M,Nzm1,n
Υ1M,Nzm,n
du w1 ψ−1u
Δ1Υ1M, N zm, n
w1 ψ−1ξ
≤ Δ1Υ1M, N zm, n
w1 ψ−1Υ1M, N zm, n
.
3.12
It follows from3.11and3.12that
W1Υ1M, N zm1, n−W1Υ1M, N zm, n≤
n−1
tn0
f1M, N, m, t. 3.13
Substitutingmwithsand summing both sides of3.13fromsm0tom−1, we get, for all
m, n∈ΛM N,
W1Υ1M, N zm, n≤W1Υ1M, N
m−1
sm0 n−1
tn0
f1M, N, s, t. 3.14
We note from the definition ofzm, nin3.2and the definition ofsm0−m10 inSection 2that
zm0, n 0. By the monotonicity ofW−1and3.10we obtain
um, n≤ψ−1
W−1 1
W1Υ1M, N
m−1
sm0 n−1
tn0
f1M, N, s, t
, ∀m, n∈ΛM N, 3.15
Next, we make the inductive assumption that3.5is true forkl. Consider nonnegative and nondecreasing in each variable onΛM,N. Then3.16is equivalent to
um, n≤ψ−1 Υ1M, N ym, n
On the other hand, by the Mean Value Theorem for integrals and by the monotonicity of
Therefore, it follows from3.18and3.20that In order to demonstrate the basic condition of monotonicity, let hs ψ−1W−1
1 ψs,
obviously which is a continuous and nondecreasing function onR. Thus eachφihsis continuous and nondecreasing onRand satisfiesφihs>0 fors >0. Moreover,
φi1hs
whereΦiu :uuids/φihs, u > 0, u ψ−1W boundary of the lattice
U2 :
We further claim that the termW−1
1 ψθiM, N, m, nis the same asΥiM, N, m, n , defined
The remainder case is thatam, n 0 for somem, n∈Λ. Let
Υ1,εm, n Υ1m, n ε, 3.31
whereε >0 is an arbitrary small number. Obviously,Υ1,εm, n>0 for allm, n∈Λ. Using
the same arguments as above and replacingΥ1m, nwithΥ1,εm, n, we get
um, n≤ψ−1
W−1 2
W2
W−1 1
W1Υ1,m, n
m−1
sm0 n−1
tn0 f1s, t
m−1 sm0
n−1
tn0 f2s, t
3.32
for allm, n∈Λm1,n1.
Considering continuities ofWi andWi−1 fori 1,2 as well as ofΥi,εinεand letting ε → 0, we obtain2.4. This completes the proof.
We remark thatm1,n1 lie on the boundary of the latticeU. In particular,2.4is true
for allm, n∈Λwhen everywii1,2satisfies
∞
uidx/wiψ−1x ∞. Therefore, we may
takem1 M,n1N.
4. Applications to a Difference Equation
In this section we apply our result to the following boundary value problemsimply called BVPfor the partial difference equation:
Δ1Δ2ψzm, n Fm, n, zm, n, m, n∈Λ,
zm, n0 fm, zm0, n gn, m, n∈Λ,
4.1
whereΛ:I×Jis defined as in the beginning ofSection 2,ψ∈C0R,Ris strictly increasing
odd function satisfyingψu>0 foru >0,F :Λ×R → Rsatisfies
|Fm, n, u| ≤h1m, nϕ1|u| h2m, nϕ2|u|, 4.2
for given functionsh1, h2 : Λ → R andϕi ∈ C0R,R i 1,2satisfyingϕiu > 0 for u >0, and functions f :I → Randg : J → Rsatisfy thatfm0 gn0 0. Obviously,
Corollary 4.1. All solutions zm, n of BVP4.1have the following estimation for all m, n ∈
Proof. Clearly, the difference equation of BVP4.1is equivalent to
ψzm, n ψ fmψ gn Applying ourTheorem 2.1to inequality4.6, we obtain the estimate ofzm, nas given in this corollary.
Corollary 4.1gives a condition of boundedness for solutions. Concretely, if
Next, we discuss the uniqueness of solutions for BVP4.1.
Corollary 4.2. Suppose additionally that
|Fm, n, u1−Fm, n, u2| ≤h1m, nϕ1 ψu1−ψu2h2m, nϕ2 ψu1−ψu2
4.8
foru1, u2∈Randm, n∈Λ:I×J, whereI m0, M∩N0, J n0, N∩N0as assumed in the
beginning ofSection 2with natural numbersMandN, h1, h2are both nonnegative functions defined
on the latticeΛ,ϕ1, ϕ2∈C0R,Rare both nondecreasing with the nondecreasing ratioϕ2/ϕ1such
thatϕi0 0,ϕiu>0for allu >0and10ds/ϕis ∞fori1,2andψ∈C0R,Ris strictly
increasing odd function satisfyingψu>0foru >0. Then BVP4.1has at most one solution onΛ.
Proof. Assume that bothzm, nandzm, n are solutions of BVP4.1. From the equivalent form4.5of4.1we have
ψzm, n−ψzm, n ≤ m−1
sm0 n−1
tn0
h1s, tϕ1 ψzs, t−ψzs, t
m−1 sm0
n−1
tn0
h2s, tϕ2 ψzs, t−ψzs, t
4.9
for allm, n ∈ Λ, which is an inequality of the form 1.7, where am, n ≡ 0. Applying ourTheorem 2.1with the choice thatu1 u2 1, we obtain an estimate of the difference
|ψzm, n − ψzm, n| in the form 2.4, where Υ1m, n ≡ 0 because am, n ≡ 0.
Furthermore, by the definition ofWiwe see that
lim
u→0Wiu −∞, ulim→ −∞W −1
i u 0, i1,2. 4.10
It follows that
W1Υ1m, n
m−1
sm0 n−1
tn0
h1s, t −∞, 4.11
sincem < M, n < N. Thus, by4.10,
Υ2m, n W1−1
W1Υ1m, n
m−1
sm0 n−1
tn0 h1s, t
Similarly, we getW2Υ2m, n ms−m10
n−1
tn0 h2s, t −∞and therefore
W2−1
W2Υ2m, n
m−1
sm0 n−1
tn0 h2s, t
0. 4.13
Thus we conclude from 2.4 that |ψzm, n−ψzm, n| ≤ 0, implying thatzm, n
zm, nfor allm, n∈Λsinceψis strictly increasing. It proves the uniqueness.
Remark 4.3. Ifh1 ≡0 orh2≡0 in4.8, the conclusion of theCorollary 4.2also can be obtained.
Finally, we discuss the continuous dependence of solutions of BVP4.1on the given functionsF, f, andg. Consider a variation of BVP4.1
Δ1Δ2ψzm, n Fm, n, zm, n, m, n∈Λ,
zm, n0 fm, zm0, n gn, m, n∈Λ,
4.14
whereψ ∈ C0R,R is strictly increasing odd function satisfyingψu > 0 foru > 0, F ∈
C0Λ×R,R, andf:I → R,g:J → Rare functions satisfyingfm 0 gn 0 0.
Corollary 4.4. LetF be a function as assumed in the beginning ofSection 4and satisfy 4.2and
4.8on the same latticeΛas assumed inCorollary 4.2. Suppose that the three differences
max
m∈I
f−f, max
n∈J g−g, s,t,umax∈Λ×R
Fs, t, u−Fs, t, u 4.15
are all sufficiently small. Then solutionzm, n of BVP 4.14 is sufficiently close to the solution
zm, nof BVP4.1.
Proof. By Corollary 4.2, the solution zm, n is unique. By the continuity and the strict monotonicity ofψ, we suppose that
max
m∈I
ψfm −ψ fm< , max
n∈J
ψ gn
−ψ gn< ,
max s,t,u∈I×J×R
where >0 is a small number. By the equivalent difference equation4.5and the inequality
4.8we get
ψ zm, n −ψzm, n≤ψfm −ψ fmψ gn −ψ gn
m−1 sm0
n−1
tn0
Fs, t,zs, t −Fs, t, zs, t
≤2 m−1
sm0 n−1
tn0
Fs, t,zs, t −Fs, t,zs, t
m−1 sm0
n−1
tn0
|Fs, t,zs, t −Fs, t, zs, t|
≤ {2 m1−m0n1−n0}
m−1 sm0
n−1
tn0
h1s, tϕ1 ψzs, t −ψzs, t
m−1 sm0
n−1
tn0
h2s, tϕ2 ψzs, t −ψzs, t,
4.17
that is an inequality of the form 1.7. Applying Theorem 2.1 to 4.17, we obtain, for all
m, n∈Λm1,n1, that
ψ zm, n −ψzm, n≤W2−1
W2Υ2m, n
m−1
sm0 n−1
tn0 h2s, t
, 4.18
wherem1, n1are given as inTheorem 2.1,
Υ2m, n W1−1
W1Υ1m, n
m−1
sm0 n−1
tn0 h1t, s
,
Υ1m, n {2 m1−m0n1−n0}.
4.19
By 4.10 we see that Υim, n → 0i 1,2 as → 0. It follows from 4.18 that lim→0|ψzm, n −ψzm, n| 0 and hence zm, n depends continuously on F, f,and
g.
Acknowledgments
The authors thank Professor Weinian ZhangSichuan Universityfor his valuable discussion. The authors also thank the referees for their helpful comments and suggestions. This work is supported by the Natural Science Foundation of Guangxi Autonomous Region200991265, by Scientific Research Foundation of the Education Department of Guangxi Autonomous Region of China200707MS112and by Foundation of Natural Science of Hechi University
2006A-N001 and Key Discipline of Applied Mathematics of Hechi University of China
200725.
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