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Volume 2008, Article ID 586020,16pages doi:10.1155/2008/586020

Research Article

Existence and Multiple Solutions for

Nonlinear Second-Order Discrete Problems with

Minimum and Maximum

Ruyun Ma and Chenghua Gao

College of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, China

Correspondence should be addressed to Ruyun Ma,mary@nwnu.edu.cn

Received 15 March 2008; Revised 6 June 2008; Accepted 19 July 2008

Recommended by Svatoslav Stan ˇek

Consider the multiplicity of solutions to the nonlinear second-order discrete problems with minimum and maximum:Δ2uk−1 fk, uk,Δuk,kT, min{uk:kT}A, max{uk:

k∈T}B, wheref:T×R2R, a, bNare fixed numbers satisfyingba2, andA, BRare satisfyingB > A, T{a1, . . . , b−1},T{a, a1, . . . , b−1, b}.

Copyrightq2008 R. Ma and C. Gao. 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

Leta, b∈N, a2≤b, T{a1, . . . , b−1}, T{a, a1, . . . , b−1, b}. Let

E:u|u:T −→R, 1.1

and foru∈E, let

uE max

k∈T

uk. 1.2

Let

E:u|u:T−→R, 1.3

and foru∈E, let

uEmax k∈T

uk. 1.4

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In this paper, we discuss the nonlinear second-order discrete problems with minimum and maximum:

Δ2uk1 fk, uk,Δuk, kT,

1.5

minuk:k∈TA, maxuk:k∈TB, 1.6

wheref:T×R2Ris a continuous function,a, bNare fixed numbers satisfyingba2

andA, B∈RsatisfyingB > A.

Functional boundary value problem has been studied by several authors1–7 . But most of the papers studied the differential equations functional boundary value problem

1–6 . As we know, the study of difference equations represents a very important field in mathematical research8–12 , so it is necessary to investigate the corresponding difference equations with nonlinear boundary conditions.

Our ideas arise from 1, 3 . In 1993, Brykalov 1 discussed the existence of two different solutions to the nonlinear differential equation with nonlinear boundary conditions

xht, x, x, ta, b ,

minut:ta, b A, maxut:ta, b B,

1.7

wherehis a bounded function, that is, there exists a constantM >0, such that|ht, x, x| ≤ M. The proofs in1 are based on the technique of monotone boundary conditions developed in2 . From1,2 , it is clear that the results of1 are valid for functional differential equations in general form and for some cases of unbounded right-hand side of the equationsee1, Remark 3 and5 ,2, Remark 2 and8 .

In 1998, Stanˇek3 worked on the existence of two different solutions to the nonlinear differential equation with nonlinear boundary conditions

xt Fxt, a.e. t∈0,1 ,

minut:ta, b A, maxut:ta, b B,

1.8

whereFsatisfies the condition that there exists a nondecreasing functionf :0,∞→0,

satisfying∞0ds/fsba,0 s/fsds∞, such that

Futfut. 1.9

It is not difficult to see that when we takeFut ht, u, u,1.8is to be1.7, andFmay not be bounded.

But as far as we know, there have been no discussions about the discrete problems with minimum and maximum in literature. So, we use the Borsuk theorem13 to discuss the existence of two different solutions to the second-order difference equation boundary value problem1.5,1.6whenfsatisfies

H1f :T×R2Ris continuous, and there existp:TR, q:TR, r :TR, such

that

fk, u, vpk|u|qk|v|rk, k, u, vT×R2, 1.10

whereΓ:1−ba bia11|pi| − bia11|qi|>0.

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2. Preliminaries

then boundary condition1.6is equal to

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iiForη1< kη2, we get

In particular, it is not hard to obtain

max

k∈{c,...,d}ukdck∈{maxc,...,d−1}Δuk. 2.19

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Lemma 2.4. Supposec, d ∈N, c < d, u uc, uc1, . . . , ud. If there existsη1 ∈ {c, c

1, . . . , d−1, d}such thatuη1 0, then ukdc max

k∈{c,...,η1−1}Δuk, k

c, . . . , η1,

ukdc max k∈{η1,...,d−1}

Δuk, k

η11, . . . , d. 2.20

In particular, one has

max

k∈{c,...,d}ukdck∈{maxc,...,d−1}Δuk. 2.21

Lemma 2.5. Supposeγ∈ A0, c∈0,1 . Ifu∈Esatisfies

γuu 0, 2.22

then there existξ0, ξ1∈T, such thatuξ0≤0≤uξ1.

Proof. We only prove that there existsξ0∈T, such that uξ0≤0, and the other can be proved similarly.

Supposeuk > 0 fork ∈ T. Then γu > γ0 0, γu < γ0 0. Furthermore,

γuu>0, which contradicts withγuu 0.

Define functionalφ:va, va1, . . . , vb−1→Rby

φv max

d1

kc

vk:cd, c, d∈T\ {b}

. 2.23

Lemma 2.6. Supposeukis a solution of 1.5andωu 0. Then

minφΔu, φ−Δuba

b−1

ia1

ri. 2.24

Proof. Let

Ckuk>0, k∈T\ {b}, Ckuk<0, k∈T\ {b}, 2.25

andNCbe the number of elements inC, NC−the number of elements inC−.

IfC∅, thenφΔu 0; ifC−∅, thenφ−Δu 0. Equation2.24is obvious. Now, supposeC/∅andC/∅. It is easy to see that

minNC, NC

ba

2 . 2.26

At first, we prove the inequality

φΔuNC

Γ

b−1

ia1

ri. 2.27

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For anyαC, there exitsβsatisfying one of the following cases:

Case 1. βmin{k∈T\ {b} |Δuk≤0, k > α},

Case 2. βmax{k∈T\ {b} |Δuk≤0, k < α}.

We only prove that 2.27 holds when Case 1 occurs, if Case 2 occurs, it can be similarly proved.

If Case1holds, we divide the proof into two cases.

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Case 1.2uαuβ ≥0. Without loss of generality, we suppose≥ 0, uβ≥ 0. Thenξ1

will be discussed in different situations.

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Similarly, we can prove

φ−ΔuNΓC

b−1

ia1

ri. 2.56

From2.26,2.55, and2.56, the assertion is proved.

Remark 2.7. It is easy to see thatφis continuous, and

maxuk:k∈T−minuk:k∈TmaxφΔu, φ−Δu. 2.57

Lemma 2.8. LetCbe a positive constant as in2.3,ωas in2.3,φas in2.23. Set

Ω u, α, β|u, α, β∈E×R2,uE<C1ba,

|α|<C1ba,|β|< C1. 2.58

DefineΓi :Ω→E×R2 i1,2:

Γ1u, α, β

αβka, αωu, βφΔuC,

Γ2u, α, β

αβka, αωu, βφ−ΔuC. 2.59

Then

DI−Γi,Ω,0/0, i1,2, 2.60

whereDdenotes Brouwer degree, andIthe identity operator onE×R2.

Proof. Obviously,Ωis a bounded open and symmetric with respect toθ∈Ωsubset of Banach spaceE×R2.

DefineH, G:0,1 ×Ω→E×R2

Hλ, u, α, β αβka, αωu−1−λωu, βφΔu

φλ−1ΔuλC,

Gλ, u, α, β u, α, βHλ, u, α, β.

2.61

Foru, α, β∈Ω,

G1, u, α, β u, α, βαβka, αωu, βφΔuC

I−Γ1

u, α, β. 2.62

By Borsuk theorem, to prove DI −Γ1,Ω,0/0, we only need to prove that the following hypothesis holds.

aG0,·,·,·is an odd operator onΩ, that is,

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bHis a completely continuous operator;

cGλ, u, α, β/0 forλ, u, α, β∈0,1 ×Ω.

First, we takeu, α, β∈Ω, then

G0,u,α,β

u,α,β−−αβka,αωuωu,βφ−ΔuφΔu

u, α, βαβka, αωuωu, βφΔuφ−Δu

G0, u, α, β.

2.64

Thusais asserted.

Second, we proveb.

Letλn, un, αn, βn ⊂ 0,1 ×Ωbe a sequence. Then for eachn ∈ Z and the factk

T, |λn| ≤1, |αn| ≤C1ba, |βn| ≤C1, uEC1ba. The Bolzano-Weiestrass

theorem andE is finite dimensional show that, going if necessary to subsequences, we can assume limn→∞λnλ0, limn→∞αnα0, limn→∞βnβ0, limn→∞unu. Then

lim n→∞H

λn, un, αn, βn

lim n→∞

αnβnka, λnω

un

−1−λn

ωun

,

βnφ

Δun

φλn−1

Δun

λnC

α0β0ka, λ0ωu−1−λ0ωu,

β0φΔuφλ0−1Δuλ0C.

2.65

Sinceωandφare continuous,His a continuous operator. ThenHis a completely continuous operator.

At last, we provec.

Assume, on the contrary, that

Hλ0, u0, α0, β0u0, α0, β0, 2.66

for someλ0, u0, α0, β0∈0,1 ×Ω. Then

α0β0ka u0k, k∈T, 2.67

ωu0−1−λ0ωu00, 2.68

φΔu0φλ0−1Δu0λ0C. 2.69

By2.67andLemma 2.5takeuu0, c1−λ0, there existsξ∈T, such that u0ξ≤0. Also from2.67, we haveu0ξ α0β0ξa, then we get

u0k u0ξ β0kξ, 2.70

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Case 1. Ifβ00, thenu0k≤0. Now, we claimu0k≡0, k ∈T. In fact, u0k≤0 and2.68

show that there existsk0∈Tsatisfyingu0k0 0. This being combined withΔu0k β00,

u0k≡0, k∈T. 2.72

So,α0u0a 0, which contradicts withu0, α0, β0Ω.

Case 2. Ifβ0>0, then from2.67,Δu0k>0,and the definition ofφ, we have

φΔu0φλ0−1Δu0β0ba. 2.73

Together with2.69, we getφβ0 λ0C, and

β0 λ0C

ba < C1. 2.74

Furthermore, Δu0k > 0 shows that u0k is strictly increasing. From 2.68 and

Lemma 2.5, there existξ0, ξ1 ∈ T satisfyingu0ξ0 ≤ 0 ≤ u0ξ1. Thus,u0a ≤ 0 ≤ u0b. It is not difficult to see that

u0a u0ξ1

ξ1−1

ka

Δu0k≥ −

ξ1−1

ka

Δu0k, 2.75

that is,

u0a

ξ1−1

ka

Δu0k<C1ba. 2.76

Similarly,|u0b|<C1ba, then we getu0E <C1baand|α0||u0a|<

C1ba, which contradicts withu0, α0, β0Ω.

Case 3. Ifβ0<0, then from2.67, we getΔu0k β0<0 and

φΔu0φλ0−1Δu01−λ0β0ba. 2.77

By2.69, we have

1−λ0β0ba λ0C. 2.78

Ifλ00, thenβ0ba 0. Furthermore,β00, which contradicts withβ0<0. Ifλ01, thenλ0C0. Furthermore,C0, which contradicts withC >0. Ifλ∈0,1, then1−λ0β0ba<0, λ0C >0, a contradiction.

Thencis proved.

From the above discussion, the conditions of Borsuk theorem are satisfied. Then, we get

DI−Γ1,Ω,0/0. 2.79

Set

Hλ, u, α, β αβka, αωu−1−λωu,

βφ−Δuφ1−λΔuλC. 2.80

Similarly, we can prove

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3. The main results

Theorem 3.1. SupposeH1holds. Then1.5and1.6have at least two different solutions when

A0and

C > ba

b−1

ia1

ri. 3.1

Proof. LetA0, C >ba/iba11|ri|. Consider the boundary conditions

ωu 0, φΔu C, 3.2

ωu 0, φ−Δu C. 3.3

Supposeukis a solution of1.5. Then fromRemark 2.7,

maxuk:k∈T−minuk:k∈TmaxφΔu, φ−Δu}. 3.4

Now, if1.5and3.2have a solutionu1k, thenLemma 2.6and3.2show thatφ−Δu1<

Cand

maxu1k:k∈T−minu1k:k∈TC. 3.5

So,u1kis a solution of1.5and2.4, that is,u1kis a solution of1.5and1.6. Similarly, if1.5,3.3have a solutionu2k, thenφΔu2< Cand

maxu2k:k∈T−minu2k:k∈TC. 3.6

So,u2kis a solution of1.5and2.4.

Furthermore, sinceφΔu1 CandφΔu2< C, u1/u2.

Next, we need to prove BVPs 1.5, 3.2, and 1.5 and 3.3 have solutions, respectively.

Set

Ω u, α, β|u, α, β∈E×R2,uE <C1ba,

|α|<C1ba,|β|< C1. 3.7

Define operatorS1:0,1 ×Ω→E×R2,

S1λ, u, α, β

αβka λ

k−1

ia i

la1

fl, ul,Δul, αωu, βφΔuC

. 3.8

Obviously,

S10, u, α, β Γ1u, α, β, u, α, β∈Ω. 3.9

Consider the parameter equation

S1λ, u, α, β u, α, β, λ∈0,1 . 3.10

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ByLemma 2.8,DI−Γ1,Ω,0/0. Now we prove the following hypothesis.

aS1λ, u, α, βis a completely continuous operator;

b

S1λ, u, α, β/ u, α, β, λ, u, α, β∈0,1 ×Ω. 3.11

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Fork > k1, Combining Case1with Case2, we get

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Acknowledgments

This work was supported by the NSFCGrant no. 10671158, the NSF of Gansu Province

Grant no. 3ZS051-A25-016, NWNU-KJCXGC, the Spring-sun Programno. Z2004-1-62033, SRFDPGrant no. 20060736001, and the SRF for ROCS, SEM2006311 .

References

1 S. A. Brykalov, “Solutions with a prescribed minimum and maximum,”Differencial’nye Uravnenija, vol. 29, no. 6, pp. 938–942, 1993Russian, translation inDifferential Equations, vol. 29, no. 6, pp. 802– 805, 1993.

2 S. A. Brykalov, “Solvability of problems with monotone boundary conditions,” Differencial’nye Uravnenija, vol. 29, no. 5, pp. 744–750, 1993Russian, translation inDifferential Equations, vol. 29, no. 5, pp. 633–639, 1993.

3 S. Stanˇek, “Multiplicity results for second order nonlinear problems with maximum and minimum,”

Mathematische Nachrichten, vol. 192, no. 1, pp. 225–237, 1998.

4 S. Stanˇek, “Multiplicity results for functional boundary value problems,”Nonlinear Analysis: Theory, Methods & Applications, vol. 30, no. 5, pp. 2617–2628, 1997.

5 W. M. Whyburn, “Differential equations with general boundary conditions,”Bulletin of the American Mathematical Society, vol. 48, pp. 692–704, 1942.

6 R. Ma and N. Castaneda, “Existence of solutions of boundary value problems for second order functional differential equations,”Journal of Mathematical Analysis and Applications, vol. 292, no. 1, pp. 49–59, 2004.

7 A. Cabada, “Extremal solutions for the differenceϕ-Laplacian problem with nonlinear functional boundary conditions,”Computers & Mathematics with Applications, vol. 42, no. 3–5, pp. 593–601, 2001. 8 R. Ma, “Nonlinear discrete Sturm-Liouville problems at resonance,” Nonlinear Analysis: Theory,

Methods & Applications, vol. 67, no. 11, pp. 3050–3057, 2007.

9 R. Ma, “Bifurcation from infinity and multiple solutions for some discrete Sturm-Liouville problems,”

Computers & Mathematics with Applications, vol. 54, no. 4, pp. 535–543, 2007.

10 R. Ma, H. Luo, and C. Gao, “On nonresonance problems of second-order difference systems,”

Advances in Difference Equations, vol. 2008, Article ID 469815, 11 pages, 2008.

11 J.-P. Sun, “Positive solution for first-order discrete periodic boundary value problem,” Applied Mathematics Letters, vol. 19, no. 11, pp. 1244–1248, 2006.

12 C. Gao, “Existence of solutions top-Laplacian difference equations under barrier strips conditions,” Electronic Journal of Differential Equations, vol. 2007, no. 59, pp. 1–6, 2007.

References

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