A Generalized Sum-Difference Inequality and Applications to Partial Difference Equations

Volume 2008, Article ID 695495,12pages doi:10.1155/2008/695495

Research Article

A Generalized Sum-Difference Inequality and Applications to Partial Difference Equations

Wu-Sheng Wang

Department of Mathematics, Hechi College, Guangxi, Yizhou 546300, China

Correspondence should be addressed to Wu-Sheng Wang,[email protected] Received 2 October 2007; Accepted 29 January 2008

Recommended by Rigoberto Medina

We establish a general form of sum-difference inequality in two variables, which includes both 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 inequal- ities considered by Cheung and Ren2006. Furthermore, we apply our result to a boundary value problem of a partial difference equation for boundedness, uniqueness, and continuous dependence.

Copyrightq2008 Wu-Sheng Wang. 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

Being an important tool in the study of differential equations and integral equation, various generalizations of Gronwall inequality1,2and their applications have attracted great inter- ests of many mathematicianssee 3–5. Some recent works can be found, for example, in 6–9and some references therein. Along with the development of the theory of integral in- equalities and the theory of difference equations, more attentions are paid to some discrete versions of Gronwall-type inequalitiessee, e.g.,10–12for some early works. Found in13, the unknown functionuin the fundamental form of sum-difference inequality

un≤an ⁿ⁻¹

s0

fsus 1.1

can be estimated byun≤an_n−1

s01fs. Pang and Agarwal14considered the inequal- ity

u²n≤P²u²0 2

n−1

s0

αu²s Qgsus

, 1.2

whereα,P, andQare nonnegative constants anduandgare nonnegative functions defined on {1,2, . . . , T}and{1,2, . . . , T−1},and they estimated thatun≤1αⁿP u0 _n−1

s0Qgs,

for all 0≤n≤T. Another form of sum-difference inequality, u²n≤c²2

n−1

s0

f₁susw us

f₂sus

, 1.3

wherecis a constant,f1 andf2 are both real-valued nonnegative functions defined onN0 {0,1,2, . . .}, andwis a continuous nondecreasing function defined onu0,∞such thatwu>

0 onu0,∞andwu0 0,for a real constantu₀, was estimated by Pachpatte15asun≤ Ω⁻¹Ωc_n−1

s0f₂s _n−1

s0f₁s, where Ωu : _u^u₀ds/ws. Recently, discretization see 16,17was also made for Ou-Yang’s inequality18. In16, the inequality of two variables,

u²m, n≤c^2m−1

sm0

n−1 tn0

as, tus, t ^m−1

sm0

n−1

tn0

bs, tus, tw us, t

, 1.4

was discussed. Later, this result was generalized in17to the inequality u^pm, n≤c^m−1

sm0

n−1

tn0

as, tu^qs, t ^m−1

sm0

n−1

tn0

bs, tu^qs, tw us, t

, 1.5

wherec≥0 andp > q >0 are all constant,aandbare both nonnegative real-valued functions defined on a lattice inZ², andwis a continuous nondecreasing function satisfyingwu>0, for allu >0.

In this paper, we establish a more general form of sum-difference inequality ψ

um, n

≤am, n ²

i1 m−1

sm0

n−1 tn0

f_is, tϕi

us, t

1.6 for nonnegative integersm,n. In1.6, we replace the constantcin1.5with a functionam, n and replace the functionsu^p,u^q,u^qwuin1.5with the more general form of functionsψu, ϕ₁u,ϕ₂u,respectively. Moreover, we do not require the monotonicity ofϕ₁andϕ₂. We em- ploy a technique of monotonization and use a property of stronger monotonicity to overcome the difficulty from nonmonotonicity so as to give an estimate for the unknown functionu. Our result enables us to solve the discrete inequality1.5and other inequalities considered in17.

Furthermore, we apply our result to a boundary value problem of a partial difference equation for boundedness, uniqueness, and continuous dependence.

2. Main result

Throughout this paper, let R denote the set of all real numbers, R 0,∞, and N0 {0,1,2, . . .}. Givenm0, n0 ∈N0, M, N ∈N0∪ {∞}, consider two latticesI m0, M∩N0and J n0, N∩N0of integer points inR. LetΛ I×J ⊂N²₀. For anys, t∈Λ,letΛs,tdenote the sublatticem0, s×n0, t∩ΛofΛ.

For functions wm, zm, n, m, n ∈ N0, their first-order differences are defined by Δwm wm1−wm,Δ1wm, n wm1, n−wm, n,andΔ2zm, n zm, n 1−zm, n. Obviously, the linear difference equationΔxm bmwith the initial condi- tionxm0 0 has the solution_m−1

sm0bs. For convenience, in the sequel we complementarily define that_m0−1

sm0bs 0.

Our basic assumptions for inequality1.6are given in the following.

H1ψ is a strictly increasing continuous function on R satisfying thatψu > 0,for all u >0.

H2Allϕ_i i1,2are continuous functions onRand positive on0,∞.

H3am, n≥0 onΛ.

H4Allf_ii1,2are nonnegative functions onΛ.

With given functionsϕ1,ϕ2, andψ, we define w₁u: max

τ∈0,u

ϕ₁τ

, 2.1

w₂u: max

τ∈0,u

ϕ2τ w1τ

w₁u, 2.2

W1

u, u1

: _u

u1

dx w1

ψ⁻¹x, 2.3

W2

u, u2

: _u

u2

dx w₂

ψ⁻¹x, 2.4

whereu_i > 0i 1,2are given constants. Sometimes we simply letW_iudenoteW_iu, ui when there is no confusion. Obviously,W₁ andW₂ are both strictly increasing inu > 0 and therefore the inversesW_i⁻¹i1,2are well defined, continuous, and increasing.

Theorem 2.1. Suppose that (H1)–(H4) hold andum, nis a nonnegative function onΛ satisfying 1.6. Then,

um, n≤ψ⁻¹

W₂⁻¹

W2

Υ2m, n ^m−1

sm0

n−1

tn0

f2s, t

2.5

form, n∈Λm1,n1, a sublattice inΛ, where

Υ2m, n:W₁⁻¹

W₁

Υ1m, n ^m−1

sm0

n−1

tn0

f₁s, t

,

Υ1m, n:a m₀, n₀

^m−1

sm0

a

s1, n₀

−a

s, n₀ⁿ⁻¹

tn0

am, t1−am, t 2.6

andm1, n₁∈Λis arbitrarily given on the boundary of the lattice

U:

m, n∈Λ:Wi

Υim, n ^m−1

sm0

n−1 tn0

fis, t≤ _∞

ui

dx wi

ψ⁻¹x, i1,2

. 2.7

Remark 2.2. Different choices of ui inWi i 1,2do not affect our results. For positive con- stants v_i / u_i, i 1,2, let W_iu _v^u

idx/w_iψ⁻¹x. Obviously, W_iu W_iu W_iui

and W_i⁻¹v W_i⁻¹v − Wiui. It follows that W_i⁻¹WiΥim, n _m−1

sm0

_n−1

tn0fis, t W_i⁻¹WiΥim, n _m−1

sm0

_n−1

tn0fis, t,that is, we obtain the same expression in2.5if we replaceWiwithWi. Moreover, by replacingWiwithWi, the condition in the definition ofUin our theorem reads

W_i Υi

m₁, n₁ ^m1−1

sm0

n1−1 tn0

f_is, t≤ _∞

vi

dx wi

ψ⁻¹x, 2.8

the left-hand side of which is equal to W_iui W_iΥim1, n_{1 m1−1}

sm0

n1−1

tn0f_is, tand the right-hand side of which equals

_ui

vi

dx wi

ψ⁻¹x _∞

ui

dx wi

ψ⁻¹xWiui

_∞

ui

dx wi

ψ⁻¹x. 2.9

Comparison between both sides implies that2.8is equivalent to the condition given in the definition ofUin our theorem withm, n m1, n₁.

Remark 2.3. If we choose ψu u^p, ϕ₁u u^q, ϕ₂u u^qwu, f1s, t as, t, and f₂s, t bs, twithp > q > 0 in1.6and restrict am, nto be a constantc, then we can applyTheorem 2.1to inequality1.5as discussed in17.

3. Proof of theorem

First of all, we monotonize some given functionsϕiin the sums. Obviously,w1sandw2s, defined byϕ1 and ϕ2 in2.1and 2.2, are nondecreasing and nonnegative functions and satisfyw_is≥ϕ_is, i1,2. Moreover, we can check that the ratiow₂s/w1sis also nonde- creasing. Therefore, from1.6we get

ψ

um, n

≤am, n ²

i1 m−1

sm0

n−1

tn0

f_is, twi

us, t

, ∀m, n∈Λ. 3.1 We first discuss in the case thatam, n>0,for allm, n∈Λ. It means thatΥ1m, n>0, for allm, n∈Λ. In such a circumstance,Υ1is positive and nondecreasing onΛand satisfies

Υ1m, n a m0, n0

^m−1

sm0

a

s1, n0

−a

s, n0ⁿ⁻¹

tn0

|am, t1−am, t| ≥am, n.

3.2 Becauseψis strictly increasing, from3.1we have

um, n≤ψ⁻¹

Υ1m, n ²

i1 m−1

sm0

n−1 tn0

fis, twi

us, t

ψ⁻¹

Υ1m, n zm, n

, ∀m, n∈Λ,

3.3

where

zm, n ²

i1 m−1

sm0

n−1

tn0

fis, twi

us, t

. 3.4

From the properties of fi and wi, we see that zis nonnegative and nondecreasing in each variable onΛ. SinceΥ1is nondecreasing, for arbitrarily fixed pair of integersK, L∈Λm1,n1, we observe from3.3that

um, n≤ψ⁻¹

Υ1K, L zm, n

, ∀m, n∈ΛK,L. 3.5 Moreover, we note thatw_iis nondecreasing and satisfiesw_iu>0,foru >0i1,2,and that Υ1K, L zm, n>0. It implies by3.5that

Δ1

Υ1K, L zm, n w1

ψ⁻¹

Υ1K, L zm, n _n−1

tn0f₁m, tw1

um, t w1

ψ⁻¹

Υ1K, L zm, n _n−1

tn0f₂m, tw2

um, t w1

ψ⁻¹

Υ1K, L zm, n

≤ⁿ⁻¹

tn0

f1m, t ⁿ⁻¹

tn0

f2m, tθ ψ⁻¹

Υ1K, L zm, t ,

3.6 where

θu: w₂u

w₁u. 3.7

On the other hand, by the mean-value theorem for integrals, for arbitrarily givenm, n,m 1, n∈ΛK,Lthere existsξin the open intervalΥ1K, L zm, n,Υ1K, L zm1, nsuch that

W₁

Υ1K, L zm1, n

−W₁

Υ1K, L zm, n

_zm1,nΥ1K,L

zm,nΥ1K,L

du w₁

ψ⁻¹u Δ1

Υ1K, L zm, n w₁

ψ⁻¹ξ ≤ Δ1

Υ1K, L zm, n w₁

ψ⁻¹

Υ1K, L zm, n 3.8

by the monotonicity ofw1andψ. It follows from3.6and3.8that W₁

Υ1K, L zm1, n

−W₁

Υ1K, L zm, n

≤ⁿ⁻¹

tn0

f₁m, t ⁿ⁻¹

tn0

f₂m, tθ ψ⁻¹

Υ1K, L zm, t

. 3.9

Keepnfixed and substitutemwithsin3.9. Then, taking the sum on both sides of3.9over sm₀, m₀1, m₀2, . . . , m−1, we get

W1

Υ1K, L zm, n

≤W1

Υ1K, L ^m−1

sm0

n−1

tn0

f1s, t ^m−1

sm0

n−1

tn0

f2s, tθ ψ⁻¹

Υ1K, L zs, t

, 3.10

for allm, n ∈ ΛK,L, where we note from the definition ofzm, nin3.3and the remark about_m0−1

sm0in the second paragraph ofSection 2thatzm0, n 0. For convenience, let Ξm, n:W1

Υ1K, L zm, n

, 3.11

σm, n:W1

Υ1K, L ^m−1

sm0

n−1

tn0

f1s, t. 3.12

Then,3.10can be rewritten as

Ξm, n≤σK, L ^m−1

sm0

n−1 tn0

f2s, tθ ψ⁻¹

W₁⁻¹

Ξs, t

, 3.13

for allm, n∈ΛK,L, where we note thatσK, L≥σm, n,for allm, n∈ΛK,L. Letgm, n denote the function on the right-hand side of3.13, which is obviously a positive function and nondecreasing in each variable. Since the compositionθψ⁻¹W₁⁻¹uis also nondecreasing inu, by3.13, that is the fact thatΞm, n≤gm, n, we have

Δ1

gm, n θ

ψ⁻¹ W₁⁻¹

gm, n

_n−1

tn0f2m, tθ ψ⁻¹

W₁⁻¹

Ξm, t θ

ψ⁻¹ W₁⁻¹

gm, n

≤ _n−1

tn0f₂m, tθ ψ⁻¹

W₁⁻¹

gm, t θ

ψ⁻¹ W₁⁻¹

gm, n

≤ⁿ⁻¹

tn0

f₂m, t.

3.14

In order to estimate the left-hand side of3.14further, we consider the following integral:

_gm1,n

gm,n

dx θ

ψ⁻¹

W₁⁻¹x

_gm1,n

gm,n

w1 ψ⁻¹

W₁⁻¹x dx w2

ψ⁻¹

W₁⁻¹x

_W⁻¹

1 gm1,n W₁⁻¹gm,n

dx w2

ψ⁻¹x W2

W₁⁻¹

gm1, n

−W2

W₁⁻¹

gm, n ,

3.15

where we note the definitions ofW1, W2,andθin2.3,2.4, and3.7. Applying the mean- value theorem to3.15, we see that for arbitrarily givenm, n,m1, n∈ΛK,L, there exists ηin the open intervalgm, n, gm1, nsuch that

_gm1,n

gm,n

dx θ

ψ⁻¹

W₁⁻¹x Δ1

gm, n θ

ψ⁻¹

W₁⁻¹η ≤ Δ1

gm, n θ

ψ⁻¹ W₁⁻¹

gm, n. 3.16

Thus, it follows from3.14,3.15, and3.16that W2

W₁⁻¹

gm1, n

−W2

W₁⁻¹

gm, n

≤ⁿ⁻¹

tn0

f2m, t, 3.17

for allm, n ∈ ΛK,L. Furthermore, using the same procedure as done for3.9, we keepn fixed and settingm sin3.17. Then, summing up both sides of3.17overs m0, m0 1, m₀2, . . . , m−1, we get

W2

W₁⁻¹

gm, n

≤W2

W₁⁻¹

σK, L ^m−1

sm0

n−1 tn0

f2s, t, 3.18

for allm, n∈ΛK,L, where we note the fact thatgm0, n σK, Land the definition ofσin 3.12. By the monotonicity ofW1andψ, the fact thatΞm, n≤ gm, n, given in3.13, and inequality3.18, we obtain from3.5that

um, n≤ψ⁻¹

Υ1K, L zm, n ψ⁻¹

W₁⁻¹

Ξm, n

≤ψ⁻¹ W₁⁻¹

gm, n

≤ψ⁻¹

W₂⁻¹

W₂ W₁⁻¹

σK, L ^m−1

sm0

n−1

tn0

f₂s, t

,

3.19

for allm, n∈ΛK,L, where we note the definitions ofΞin3.11andgjust after3.13. This result also implies the particular case that

uK, L≤ψ⁻¹

W₂⁻¹

W₂

W₁⁻¹

W₁

Υ1K, L ^K−1

sm0

L−1

tn0

f₁s, t

^K−1

sm0

L−1

tn0

f₂s, t

. 3.20 For the arbitrary choice of K, L ∈ Λm1,n1, it also implies that2.5 holds for allm, n ∈ Λm1,n1.

The remainder case is thatam, n 0,for somem, n∈Λ. Let

Υ1,εm, n Υ1m, n ε, 3.21

whereε >0 is an arbitrary small number. Obviously,Υ1,εm, n>0,for allm, n∈Λ. Using the same arguments as above, whereΥ1m, nis replaced withΥ1,εm, n, we get

um, n≤ψ⁻¹

W₂⁻¹

W2

W₁⁻¹

W1

Υ1,m, n ^m−1

sm0

n−1

tn0

f1s, t

^m−1

sm0

n−1 tn0

f2s, t

, 3.22 for allm, n ∈Λm1,n1. Lettingε →0, we obtain2.5because of continuity ofΥi,ε inεand continuity ofWiandW_i⁻¹, fori1,2. This completes the proof.

Remark thatm₁ andn₁ lie on the boundary of the latticeU. In particular,2.5is true for allm, n ∈ Λ when all w_isi 1,2 satisfy _u^∞

i dx/w_iψ⁻¹x ∞, so we may take m1M, 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:

Δ1Δ2ψ

zm, n F

m, n, zm, n

, m, n∈Λ, z

m, n0

fm, z m0, n

gn, m, n∈Λ, 4.1 whereΛ:I×Jis defined as in the beginning ofSection 2,ψ∈C⁰R,Ris a 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 1,2satisfying ϕiu > 0,for u >0, and functionsf :I →Randg :J →Rsatisfyfm0 gn0 0. Obviously,4.1is a generalization of the BVP problem considered in17, Section 3. So the results of17cannot be applied immediately. In what follows we first apply our main result to discuss boundedness of solutions of4.1.

Corollary 4.1. All solutionszm, nof BVP4.1have the estimate zm, n≤ψ⁻¹

W₂⁻¹

W₂

Υ2m, n ^m−1

sm0

n−1

tn0

h₂s, t

, 4.3

for allm, n∈Λm1,n1, wherem1, n1are given as inTheorem 2.1and W2u

_u

1

dx/

τ∈0,xmax

ϕ2

ψ⁻¹τ max_τ1∈0,τ

ϕ₁

ψ⁻¹τ1

τ∈0,xmax ϕ1

ψ⁻¹τ ,

W1u _u

1

dx/max

τ∈0,x

ϕ1

ψ⁻¹τ ,

Υ2m, n W₁⁻¹

W₁

Υ1m, n ^m−1

sm0

n−1

tn0

h₁t, s

,

Υ1m, n≤ ^m−1

sm0

ψ

fs1

−ψ

fsⁿ⁻¹

tn0

ψ

gt1

−ψ gt.

4.4

Proof. Clearly, the difference equation of BVP4.1is equivalent to ψ

zm, n ψ

fm

ψ

gn

^m−1

sm0

n−1

tn0

F

s, t, zs, t

. 4.5

It follows that ψ

zm, n≤ψ fm

ψ

gn^m−1

sm0

n−1 tn0

h1s, tϕ1zs, t^m−1

sm0

n−1 tn0

h2s, tϕ2zs, t 4.6 by4.2. Letam, n |ψfmψgn|. Since|ψzm, n|ψ|zm, n|,4.6is of the form 1.6. ApplyingTheorem 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 Υ1m, n<∞, ^m−1

sm0

n−1

tn0

h₁s, t<∞, ^m−1

sm0

n−1 tn0

h₂s, t<∞, 4.7 for allm, n∈Λm1,n1, then every solutionzm, nof BVP4.1is bounded onΛm1,n1.

Next, we discuss the uniqueness of solutions for BVP4.1.

Corollary 4.2. Suppose additionally that F

m, n, u1

−F

m, n, u2≤h1m, nϕ1ψ u1

−ψ

u2h2m, nϕ2ψ u1

−ψ u2,

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 thatϕ_i0 0,ϕ_iu > 0,for allu > 0 and ₀¹ds/ϕ_is ∞,fori 1,2,andψ ∈ C⁰R,Ris a strictly increasing odd function satisfyingψu > 0, foru > 0. Then, BVP4.1has at most one solution onΛ.

Proof. Assume that bothzm, nandzm, n are solutions of BVP4.1. From the equivalent form4.5of4.1, we have

ψ

zm, n

−ψ

zm, n≤ ^m−1

sm0

n−1

tn0

h1s, tϕ1ψ zs, t

−ψ zs, t

^m−1

sm0

n−1

tn0

h2s, tϕ2ψ zs, t

−ψ

zs, t,

4.9

for all m, n ∈ Λ, which is an inequality of the form 1.6, where am, n ≡ 0. Apply- ing Theorem 2.1 with the choice thatu1 u2 1, we obtain an estimate of the difference

|ψzm, n−ψzm, n|in the form2.5, whereΥ1m, n≡0,becauseam, n≡ 0. Further- more, by the definition ofWiwe see that

limu→0W_iu −∞, lim

u→−∞W_i⁻¹u 0, i1,2. 4.10

It follows that

W1

Υ1m, n ^m−1

sm0

n−1

tn0

h1s, t −∞ 4.11

sincem < M, n < N. Thus, by4.10 Υ2m, n W₁⁻¹

W1

Υ1m, n ^m−1

sm0

n−1 tn0

h1s, t

0. 4.12

Similarly, we getW₂Υ2m, n _m−1

sm0

_n−1

tn0h₂s, t −∞and therefore W₂⁻¹

W2

Υ2m, n ^m−1

sm0

n−1 tn0

h2s, t

0. 4.13

Thus, we conclude from2.5that|ψzm, n−ψzm, n| ≤0, implying thatzm, n zm, n, for allm, n∈Λsinceψis strictly increasing. It proves the uniqueness.

Remark 4.3. Ifh1≡0 orh2≡0 in4.8, the conclusion ofCorollary 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

Δ1Δ2ψ

zm, n F

m, n, zm, n

, m, n∈Λ, z

m, n0

fm, z m0, n

gn, m, n∈Λ, 4.14 where ψ ∈ C⁰R,Ris a 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. LetFbe a function as assumed in the beginning ofSection 4and satisfy4.2and4.8 on the same latticeΛas assumed inCorollary 4.2. Suppose that the three differences

maxm∈I f−f, max

n∈J g−g, max

s,t,u∈Λ×RFs, 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. ByCorollary 4.2, the solutionzm, nis unique. By the continuity and the strict mono- tonicity ofψ, we suppose that

maxm∈Iψ fm

−ψ

fm< , max

n∈J ψ

gn

−ψ

gn< ,

s,t,u∈I×J×Rmax Fs, t, u−Fs, t, u< , 4.16 where >0 is a small number. By the equivalent difference equation4.5and inequality4.8, we get

ψ zm, n

−ψ

zm, n

|

≤ψ fm

−ψ

fm

ψ

gn

−ψ

gn^m−1

sm0

n−1 tn0

F

s, t,zs, t

−F

s, t, zs, t

≤2 ^m−1

sm0

n−1 tn0

F

s, t,zs, t

−F

s, t,zs, t ^m−1

sm0

n−1

tn0

F

s, t,zs, t

−F

s, t, zs, t

≤ 2

m1−m0

n1−n0 ^m−1

sm0

n−1 tn0

h1s, tϕ1ψ zs, t

−ψ

zs, t ^m−1

sm0

n−1 tn0

h2s, tϕ2ψ zs, t

−ψ

zs, t,

4.17

that is, an inequality of the form1.6. ApplyingTheorem 2.1to4.17, we obtain ψ

zm, n

−ψ

zm, n≤W₂⁻¹

W2

Υ2m, n ^m−1

sm0

n−1 tn0

h2s, t

, 4.18

for allm, n∈Λm1,n1, wherem₁, n₁are given as inTheorem 2.1,

Υ2m, n W₁⁻¹

W₁

Υ1m, n ^m−1

sm0

n−1

tn0

h₁t, s

,

Υ1m, n 2

m₁−m₀

n₁−n₀ .

4.19

By 4.10 we see that Υim, n → 0 i 1,2 as → 0. It follows from 4.18 that lim→0|ψzm, n−ψzm, n| 0,and hence zm, ndepends continuously on F, f, andg sinceψis strictly increasing.

Our requirement on the small difference, F−F inCorollary 4.4, is stronger than the conditioniiiin17, Theorem 3.3, but ours may be easier to check because one has to verify the inequality in his conditioniiifor each solutionzm, n of BVP4.14.

Acknowledgments

The author thanks Professor Weinian ZhangSichuan Universityfor his valuable discussion.

The author also thanks the referees for their helpful comments and suggestions. This project is supported by Foundation of Guangxi Natural Science of China and by Foundation of Natural Science and Key Discipline of Applied Mathematics of Hechi College of China.

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