Solvability of a Class of Generalized Neumann Boundary Value Problems for

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Volume 2011, Article ID 308362,11pages doi:10.1155/2011/308362

Research Article

Solvability of a Class of Generalized Neumann Boundary Value Problems for

Second-Order Nonlinear Difference Equations

Jianye Xia

¹

and Yuji Liu

²

1Department of Applied Mathematics, Guangdong University of Finance, Guangzhou 510000, China

2Department of Mathematics, Guangdong University of Business Studies, Guangzhou 510000, China

Correspondence should be addressed to Jianye Xia,jianye [email protected] Received 18 April 2011; Accepted 17 June 2011

Academic Editor: Hassan A. El-Morshedy

Copyrightq2011 J. Xia and Y. Liu. 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.

This paper is motivated by Rachnkovab and Tisdell2006and Anderson et al.2007. New suf- ficient conditions for the existence of at least one solution of the generalized Neumann boundary value problems for second order nonlinear diﬀerence equations∇Δxk fk, xk, xk1, k∈1, n−1,x0 ax1,xn bxn−1, are established.

1. Introduction

Recently, there have been many papers discussed the solvability of two-point or multipoint boundary value problems for second-order or higher-order diﬀerence equations, we refer the readers to the text books1,2 and papers3–8 and the references therein.

In a recent paper3 , Anderson et al. studied the following problem:

∇Δyk f

k, yk,Δyk

, k1, . . . , n−1,

Δy0 Δyn 0, 1.1

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where

Δyk

⎧⎨

⎩

yk1−yk, fork0, . . . , n−1,

0, forkn,

∇Δyk

⎧⎨

⎩

yk1−2yk yk−1, fork1, . . . , n−1,

0, fork0 orn.

1.2

The following result was proved.

Theorem ART

Suppose thatfis continuous and there exist constantsα≤0,K≥0 such that f

t, p, q

−p≤α 2pf

t, p, q

q² K, t, p, q

∈ {1, . . . , n−1} ×R². ∗

Then BVP1.1has at least one solution.

The methods in3 involved new inequalities on the right-hand side of the diﬀerence equation and Schaefer’s Theorem in the finite-dimensional space setting.

In7 , the following discrete boundary value problemBVPinvolving second order diﬀerence equations and two-point boundary conditions

∇Δyk

h² f

t_k, y_k,Δyk

h

, k1, . . . , n−1, y₀0, y_n0,

1.3

was studied, wheren ≥ 2 an integer,f is continuous, scalar-valued function, the step size ish N/nwith N a positive constant, the grid points aretk kh for k 0, . . . , n. The diﬀerences are given by

Δyk

⎧⎨

⎩

y_k1−y_k, k0, . . . , n−1,

0, kn,

∇Δyk

⎧⎨

⎩

y_k1−2yky_k−1, k1, . . . , n−1,

0, k0 orkn.

1.4

The following two results were proved in7 .

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Theorem RT

Letf be continuous on0, N ×R²andα, β, and,Kbe nonnegative constants. If there exist c, d∈0,1such that

ft, u, v≤α|u|^cβ|v|^dK, t, u, v∈0, N ×R², 1.5

then the discrete BVP1.3has at least one solution.

Theorem RT

Letfbe continuous on0, N ×R²andα, β, andKnonnegative constants. If

ft, u, v≤α|u|β|v|K, t, u, v∈0, N ×R², 1.6 αN²

8 βN

2 <1, 1.7

then the discrete BVP1.3has at least one solution.

In paper8 , Cabada and Otero-Espinar studied the existence of solutions of a class of nonlinear second-order diﬀerence problem with Neumann boundary conditions by using upper and lower solution methods. Assuming the existence of a pair of ordered lower and upper solutionsγandβ, they obtained optimal existence results for the caseγ ≤βand even forγ≥β.

In this paper, we study the following boundary value problem for second-order nonlinear diﬀerence equation

∇Δxk fk, xk, xk1, k∈1, n−1 ,

x0 ax1, xn bxn−1, 1.8

wherea, b∈R,n≥2 is an integer, andfis continuous, scalar-valued function. We note that whenab1, BVP1.8becomes the following BVP:

∇Δxk fk, xk, xk1, t∈1, T −1 ,

Δx0 0 Δxn−1, 1.9

which is called Neumann boundary value problem of diﬀerence equation and is a special case of BVP1.1. Whenab0, BVP1.8is changed to

∇Δxk fk, xk, xk1, t∈1, T −1 ,

x0 0xn, 1.10

which is the so-called Dirichlet problem for discrete diﬀerence equations and is a special case of BVP1.3.

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The purpose of this paper is to improve the assumptions∗,1.5, and 1.6 in the results in paper3,5,7–9 , by using Mawhin’s continuation theorem of coincidence degree, to establish suﬃcient conditions for the existence of at least one solution of BVP1.8. It is interesting that we allowfto be sublinear, at most linear or superlinear.

This paper is organized as follows. In Section 2, we make the main results, and in Section 3, we give some examples, which cannot be solved by theorems in5,7,9 , to illustrate the main results presented inSection 3.

2. Main Results

To get the existence results for solutions of BVP1.8, we need the following fixed point theorems.

LetX and Y be Banach spaces, L :DL ⊂ X → Y a Fredholm operator of index zero, andP :X → X,Q :Y → Y projectors such that ImP KerL, KerQ ImL, X KerL⊕ KerP, Y ImL⊕ ImQ. It follows thatL|DL∩KerP :DL∩KerP → ImLis invertible;

we denote the inverse of that map byK_p.

IfΩis an open bounded subset ofX,DL∩Ω/∅, the mapN:X → Y will be called L-compact onΩifQNΩis bounded andKpI−QN:Ω → Xis compact.

Lemma 2.1see9 . LetL be a Fredholm operator of index zero, and letN beL-compact on Ω.

Assume that the following conditions are satisfied:

iLx /λNxfor everyx, λ∈DL\KerL∩∂Ω ×0,1;

iiNx /∈ImLfor everyx∈KerL∩∂Ω;

iiideg∧QN|KerL,Ω∩KerL,0/0, where∧: KerL → Y/ImLis the isomorphism.

Then the equationLxNxhas at least one solution inDL∩Ω.

Lemma 2.2see9 . LetX andY be Banach spaces. SupposeL:DL ⊂X → Y is a Fredholm operator of index zero with KerL {0},N:X → Y isL-compact on any open bounded subset of X. If 0 ∈Ω ⊂ Xis an open bounded subset andLx /λNxfor allx ∈DL∩∂Ωand λ∈ 0,1 , then there is at least onex∈Ωso thatLxNx.

LetXRⁿ¹, Y Rⁿ⁻¹be endowed with the norms

x max

n∈0,n |xn|, y max

k∈1,n−1 yk 2.1

forx∈Xandy∈Y, respectively. It is easy to see thatXandY are Banach spaces. ChooseDL {x∈ X :x0 ax1, xn bxn−1}. LetL :X → Y, Lxk ∇Δxk, x∈DL, and N:X → Y byNxk fk, xk, xk1.

Consider the following problem:

∇Δxk 0, x0 ax1, xn bxn−1. 2.2

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It is easy to see that problem2.2has a unique solutionxk 0 if and only if

1−an−1b−n /a1−b. 2.3 If2.3holds, we call BVP1.8at nonresonance case. If1−an−1b−n a1−b, then problem2.2has infinite nontrivial solutions. At this case, we call BVP1.8at resonance case.

In this paper, we establish suﬃcient conditions for the existence of solutions of BVP1.8at nonresonance case, that is,1−an−1b−n /a1−b, and at resonance case, a b 1. It is similar to get existence results for the existence of solutions at resonance case when 1−an−1b−n a1−banda /1, b /1.

Lemma 2.3. Supposeab1. Then the following results are valid.

iKerL{x c, . . . , c∈X:c∈R}.

iiImL{y∈Y :_n−1

i1 yi 0}.

iiiLis a Fredholm operator of index zero.

ivThere are projectorsP :X → XandQ : Y → Y such that KerL ImP, KerQ ImL. Furthermore, letΩ⊂Xbe an open bounded subset withΩ∩DL/∅; thenNisL-compact on Ω.

vx∈DLis a solution ofLx Nxwhich implies thatxis a solution of BVP1.8.

The projectorsP :X → X andQ :Y → Y, the isomorphism∧ : KerL → Y/ImL, and the generalized inverseK_p: ImL → DL∩ImPare as follows:

P xn x1,

Qyn 1

n−1

n−1

i1

yi,

∧c c, K_pyn ^k

s1

s i1

yi.

2.4

Lemma 2.4. Suppose1−an−1b−n /a1−b. Then the following results are valid.

ix∈DLis a solution ofLx Nxwhich implies thatxis a solution of BVP1.8.

ii KerL{0}.

iiiLis a Fredholm operator of index zero,NisL-compact on each open bounded subset ofX.

Suppose

Athere exist numbersβ >0,θ≥1, nonnegative sequencespn, qn, rn, functions gn, x, y,hn, x, ysuch thatfn, x, y gn, x, y hn, x, yand

g n, x, y

x≥β|x|^θ1, h

n, x, y≤pn|x|^θqny^θrn, 2.5

for alln∈ {1, . . . , n−1}, x, y∈R²;

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Bthere exists a constantM >0 so that

c _n−1

i1

fn, c, c

>0 2.6

for all|c|> Mor

c _n−1

i1

fn, c, c

<0 2.7

for all|c|> M.

Theorem L

Supposea²≤1, b²≤1, and thatAandBhold. Then BVP1.8has at least one solution if pqmax

|b|^θ1,1

< β. 2.8

Proof. To applyLemma 2.1, we considerLxλNxforλ∈0,1 . Step 1. LetΩ1{x∈X:LxλNx, λ∈0,1 }. Forx∈Ω1, we have

xk1−2xk xk−1 λfk, xk, xk1, k∈1, n−1 , x0 ax1,

xn bxn−1.

2.9

So

xk1−2xk xk−1 xk λfk, xk, xk1xk, k∈1, n−1 . 2.10

It is easy to see that

2

n−1

n1

xk1−2xk xk−1 xk

ⁿ⁻¹

n1

−xk1 ²2xkxk1−xk ²−xk−1 ²2xk−1xk−xk ²

xk1 ²−2xk ² xk−1 ² ⁿ⁻¹

n1

−xk1−xk ²−xk−1−xk ² xk1 ²−2xk ² xk−1 ²

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ⁿ⁻¹

n1

−xk1−xk ²−xk−1−xk ²

xn ²−xn−1 ²−x1 ² x0 ²

ⁿ⁻¹

n1

−xk1−xk ²−xk−1−xk ²

b²−1

xn−1 ² a²−1

x1 ² .

2.11

Sincea²≤1, b²≤1, we get

n−1

n1

xk1−2xk xk−1 xk≤0. 2.12

So, we get

λ n−1 n1

fk, xk, xk1xk≤0. 2.13

Then

n−1

n1

gk, xk, xk1 hk, xk, xk1

xk≤0. 2.14

It follows that

β

n−1

k1

|xk|^θ1≤ −ⁿ⁻¹

n1

hk, xk, xk1xk

≤ⁿ⁻¹

n1

pk|xk|^θ1qk|xk1|^θ|xk|rk|xk|

≤pⁿ⁻¹

n1

|xk|^θ1qⁿ⁻¹

k1

|xk1|^θ|xk|ⁿ⁻¹

k1

rk|xk|.

2.15

Forxi≥0, yi≥0, Holder’s inequality implies s

i1

xiyi≤ _s

i1

x^p_i

_1/p _s

i1

y_i^q _1/q

, 1 p 1

q 1, q >0, p >0. 2.16

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It follows that

β

n−1

k1

|xk|^θ1

≤pⁿ⁻¹

n1

|xk|^θ1q_n−1

k1

|xk1|^θ1

_θ/θ1_n−1

k1

|xk|^θ1

_1/θ1

rⁿ⁻¹

k1

|xk|

rn−1^θ/θ1 _n−1

k1

|xk|^θ1

_1/θ1

pⁿ⁻¹

n1

|xk|^θ1 q

|b|^θ1|x1|^θ1ⁿ⁻²

k1

|xk1|^θ1

_θ/θ1_n−1

k1

|xk|^θ1

_1/θ1

≤ rn−1^θ/θ1 _n−1

k1

|xk|^θ1

_1/θ1

pⁿ⁻¹

n1

|xk|^θ1 qmax

|b|^θ1,1n−1

k1

|xk|^θ1

_θ/θ1_n−1

k1

|xk|^θ1

_1/θ1

rn−1^θ/θ1 _n−1

k1

|xk|^θ1

_1/θ1

pⁿ⁻¹

n1

|xk|^θ1 qmax

|b|^θ1,1ⁿ⁻¹

k1

|xk|^θ1.

2.17

It follows from2.8that there exists a constantM₁>0 such that

n−1

k1

|xk|^θ1≤M₁. 2.18

Hence|xk| ≤M1/n−1^1/θ1for allk∈ {1, . . . , n−1}. Hence||x|| ≤M1/n−1^1/θ1. SoΩ1is bounded.

Step 2. Prove that the setΩ2{x∈ KerL:Nx∈ ImL}is bounded.

For x ∈ KerL, we havexk c fork ∈ 0, n . Thus we have Nxk fk, c, c.

Nx, y∈ ImLimplies that

n−1

k1

fn, c, c 0. 2.19

It follows from conditionBthat|c| ≤M. ThusΩ2is bounded.

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Step 3. Prove the setΩ3{x∈ KerL:±λ∧x 1−λQNx 0, ∃λ∈0,1 }is bounded.

If the first inequality ofBholds, let

Ω3{x∈KerL:λ∧x 1−λQNx 0, ∃λ∈0,1 }. 2.20

We will prove thatΩ3 is bounded. Forxk cfork ∈ 0, n such thatx ∈ Ω3, and λ∈0,1 , we have

−1−λⁿ⁻¹

k1

fn, c, c λcn−1. 2.21

Ifλ1, thenc0. Ifλ /1, then

0>−1−λcⁿ⁻¹

k1

fn, c, c λc²T ≥0, 2.22

a contradiction.

If the second inequality ofBholds, let

Ω3{x∈KerL:−λ∧x 1−λQNx 0, ∃λ∈0,1 }. 2.23

Similarly, we can get a contradiction. SoΩ3is bounded.

Step 4. Obtain open bounded setΩsuchi,ii, andiiiofLemma 2.1.

In the following, we will show that all conditions ofLemma 2.1are satisfied. SetΩan open bounded subset ofXsuch thatΩ⊃₃

i1Ωi. We know thatLis a Fredholm operator of index zero andNisL-compact onΩ. By the definition ofΩ, we haveΩ⊃Ω1andΩ⊃Ω2, thus Lx/λNxforx∈DL\KerL ∩ ∂Ωandλ∈0,1;Nx∈/ImLforx∈ KerL ∩ ∂Ω.

In fact, letHx, λ ±λ∧x 1−λQNx. According the definition ofΩ, we know Ω⊃Ω3, thusHx, λ/0 forx∈∂Ω∩KerL, thus by homotopy property of degree,

degQN|_Ker_L,Ω∩KerL,0 degH·,0,Ω∩KerL,0 degH·,1,Ω∩KerL,0 deg±∧,Ω∩KerL,0/0.

2.24

Thus byLemma 2.1,Lx Nxhas at least one solution inDL∩Ω, which is a solution of BVP1.8. The proof is completed.

Theorem L

Supposea²≤1, b²≤1,1−an−1b−n /a1−b, and thatAholds. Then BVP1.8has at least one solution if2.8holds.

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Proof. To applyLemma 2.2, we consider Lx λNxforλ ∈ 0,1 . LetΩ1 {x ∈ X : Lx λNx, λ ∈ 0,1 }. Forx ∈ Ω1, we get 2.9and 2.10. using the methods in the proof of Theorem LX1, we get thatΩ1 is bounded. LetΩbe a nonempty open bounded subset ofX such thatΩ⊃Ω1centered at zero. It is easy to see thatLis a Fredholm operator of index zero andNis L-compact onΩ. One can see thatLx /λNxfor allx ∈DL∩∂Ωand λ∈ 0,1 . Thus, fromLemma 2.2,LxNxhas at least one solutionx∈DL∩Ω, soxis a solution of BVP1.8. The proof is complete.

3. An Example

In this section, we present an example to illustrate the main results inSection 2.

Example 3.1. Consider the following problem:

xk1−2xk xk−1 βxk ^2m1pkxk ^2m1qkxk1 ^2m1rk, k∈1, n−1 , x0 ax1,

xn bxn−1,

3.1

wheren≥2, m≥1 are integers andβ >0,pn, qn, rnare sequences. Corresponding to the assumptions of Theorem L1, we set

f k, x, y

βx^2m1pkx^2m1qky^2m1rk, g

k, x, y

βx^2m1, h

k, x, y

pkx^2m1qky^2m1rk,

3.2

andθ2m1. It is easy to see thatAholds, and

fn, c, c c^2m1βpkc^2m1qkc^2m1rk 3.3

implies that there isM >0 such thatc_n−1

i1c^2m1βpkc^2m1qkc^2m1rk >0 for all n∈1, n−1 and|c|> M.

It follows from Theorem L2 that3.1has at least one solution ifa² ≤1, b² ≤ 1, 1− an−1b−n /a1−band||p||||q||max{|b|^θ1,1}< β. BVP3.1has at least one solution ifab1 and||p||||q||max{|b|^θ1,1}< β.

Remark 3.2. It is easy to see that BVP3.1whenab0 cannot be solved by using theorems obtained in paper7 . BVP3.1whenab1 cannot be solved by the results obtained in paper3 .

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Acknowledgments

This paper is supported by Natural Science Foundation of Hunan province, China no.

06JJ5008and Natural Science Foundation of Guangdong provinceno. 7004569.

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