Solutions of 2th Order Boundary Value Problem for Difference Equation via Variational Method

doi:10.1155/2009/730484

Research Article

Solutions of 2

n

th-Order Boundary Value Problem

for Difference Equation via Variational Method

Qingrong Zou and Peixuan Weng

School of Mathematics, South China Normal University, Guangzhou 510631, China

Correspondence should be addressed to Peixuan Weng,wengpx@scnu.edu.cn

Received 7 July 2009; Accepted 15 October 2009

Recommended by Kanishka Perera

The variational method and critical point theory are employed to investigate the existence of solutions for 2nth-order difference equationΔn_p

k−nΔnyk−n −1n1fk, yk 0 fork∈1, N with boundary value conditiony1−ny2−n · · ·y00, yN1· · ·yNn 0 by constructing a functional, which transforms the existence of solutions of the boundary value problemBVP to the existence of critical points for the functional. Some criteria for the existence of at least one solution and two solutions are established which is the generalization for BVP of the even-order difference equations.

Copyrightq2009 Q. Zou and P. Weng. 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

Difference equations have been applied as models in vast areas such as finance insurance, biological populations, disease control, genetic study, physical field, and computer applica-tion technology. Because of their importance, many literature deals with its existence and uniqueness problems. For example, see1–10 .

We notice that the existing results are usually obtained by various analytical techniques, for example, the conical shell fixed point theorem 1, 6 , Banach contraction map method 7 , Leray-Schauder fixed point theorem 2, 10 , and the upper and lower solution method 3 . It seems that the variational technique combining with the critical point theory11 developed in the recent decades is one of the effective ways to study the boundary value problems of difference equations. However because the variational method requires a “symmetrical” functional, it is hard for the odd-order difference equations to create a functional satisfying the “symmetrical” property. Therefore, the even-order difference equations have been investigated in most references.

with some traditional analytical skills:

Δn_p

k−nΔnyk−n

−1n1fk, yk

0 k∈1, N , 1.1

y1−n y2−n· · ·y00, yN1· · ·yNn0, 1.2

whereΔn_y

k Δn−1yk1−Δn−1yk n1is the forward difference operator;pk ∈Rfork ∈

1−n, N andf∈C1, N ×R,R.A variational functional for BVP1.1-1.2is constructed which transforms the existence of solutions of the boundary value problem BVP to the existence of critical points of this functional. In order to prove the existence criteria of critical points of the functional, some lemmas are given inSection 2. Two criteria for the existence of at least one solution and two solutions for BVP1.1-1.2are established inSection 3which is the generalization for BVP of the even-order difference equations. The existence results obtained in this paper are not found in the references, to the best of our knowledge.

For convenience, we will use the following notations in the following sections:

Fk, u

_u

0

fk, sds, p max

k∈1−n,N pk, pk∈min1−n,N

pk. 1.3

2. Variational Structure and Preliminaries

We need two lemmas from12 or11 .

Lemma 2.1. LetH be a real reflexive Banach space with a norm · , and letφbe a weakly lower (upper) semicontinuous functional, such that

lim

x → ∞φx ∞

or lim

x → ∞φx −∞

, 2.1

then there existsx0∈Hsuch that

φx0 inf x∈Hφx0

or φx0 sup x∈H

φx0 . 2.2

Furthermore, ifφhas bounded linear Gˆateaux derivative, thenφx0 0.

Lemma 2.2 mountain-pass lemma. Let H be a real Banach space, and let φ : H →

R be continuously differential, satisfying the P-S condition. Assume that x0, x1 ∈ H and Ω

is an open neighborhood of x0, but x1/∈Ω. If max{φx0, φx1} < infx∈∂Ωφx, then c

infh∈Γmaxt∈0,1 φhtis the critical value ofφ,where

Γ {h|h:0,1 −→H, his continuous, h0 x0, h1 x1}. 2.3

This means that there existsx2∈H, s.t.φx2 0,φx2 c.

Lemma 2.3. IfAm×mis a symmetric and positive-defined real matrix,Bm×nis a real matrix,BTis the

transposed matrix ofB.ThenBT_{ABis positive defined if and only ifrankBn.}

Proof. SinceAis positive defined, then

BTABis positive-defined⇐⇒ ∀x /0, xTBTABx >0

⇐⇒ ∀x /0, BxTABx>0⇐⇒ ∀x /0, Bx /0⇐⇒rankBn.

2.4

LetHbe a Hilbert space defined by

Hy:1−n, Nn −→R|y1−ny2−n· · ·y00, yN1 · · ·yNn0

Lemma 2.4. There is

Repeating the above process, we obtain

all eigenvalues of BT_{Bare real and positive. Let λ be the minimal eigenvalue of these N}

eigenvalues, thenλ >0.ThereforexT_BT_BxλxT_{x,that is,}N

1−nΔnxk2λx2.

However, how to find the λ in Lemma 2.4 is a skillful and challenging task. The following lemma from13 offers some help for the estimation ofλ.

Lemma 2.5Brualdi13 . IfA aij ∈MNis weak irreducible, then each eigenvalue is contained

InLemma 2.5,Cis the complex number set, and the denotations CA,γ,Pi,R_i can

be found in 13 . SinceBT_{Bis positive defined, all eigenvalues are positive real numbers.}

Therefore, byLemma 2.5, let

B calculate the eigenvalues directly.

Define the functionalφonHby

φy

we have

Repeating the above process, we obtain

N

Sincex ∈ H is arbitrary, we know that the solution of BVP1.1-1.2corresponds to the critical point ofφ.

3. Main Results

Now we present our main results of this paper.

Theorem 3.1. If there existM1>0,a1>0,a2∈R, andσ >2s.t.

Fk, ua1|u|σa2, ∀ |u|> M1, 3.1

then BVP1.1-1.2has at least one solution.

Proof. In fact, we can choose a suitablea2<0 such that

Since there existsσ1>0 withσ1yyσ,we have

N

1−n Fk, yk

a1

N

1

ykσa2Nna1σ1σy σ_a

2Nn. 3.3

Then byLemma 2.4, we obtain

φy p

24

n_y2₋

a1σσ₁yσ−a2Nn. 3.4

Noticing thatσ >2,we have limy →∞φy −∞.FromLemma 2.1, the conclusion of this

lemma follows.

Corollary 3.2. If there existsM2 >0s.t.ufk, u>0for all|u|> M2, and

inf

k∈1−n,N ulim→ ∞

fk, u

|u|r r1, 3.5

where r,r1 satisfy eitherr 1,r1 > 4np orr > 1, r1 > 0, then BVP1.1-1.2has at least one

solution.

Proof. Assume thatr 1,r1 > 4np.Then for1 r1−4np/2 > 0,there exists M3 > M2,

such that|fk, y|r1−1|y|as|y|> M3. We have from the continuity offk, uthat there

is aK > 0 such that−K ≤ fk, u ≤ K for allk ∈ 1, N ,|u| ≤ M3. When y > 0, one has fk, yr1−1y >0 fory∈M3,∞,then

y

0

fk, sds

M3

0

fk, sds

y

M3

fk, sds−KM3 r1−1

2 y

2₋r1−1

2 M

2

3; 3.6

wheny <0, one hasfk, yr1−1y <0 fory∈−∞,−M3,then

y

0

fk, sds

₋M3

0

fk, sds

y

−M3

fk, sds−KM3r1−1

2 y

2₋r1−1

2 M

2

3. 3.7

Letc :−KM3−r1−1/2M2₃, then we have

*_y

0fk, sds r1−1/2y2cfory∈R.

Therefore, we have

φyp

24

n_y2₋r1−1

2

N

1

yk2−c

4n_p−r 11

2 y

2₋ c−1

2y

2₋

c, 3.8

which implies limy →∞φy −∞,and byLemma 2.1, the conclusion of this lemma follows.

Assume thatr > 1, r1 > 0.Then for 2 r1/2 > 0, there exists M4 > M2, such

a K > 0 such that−K ≤ fk, u ≤ K for allk ∈ 1, N ,|u| ≤ M4. Wheny > 0, one has fk, yr1/2yr >0,y∈M4,∞,then we have

_y

0

fk, sds

_M4

0

fk, sds

_y

M4

fk, sds

−KM4

r1

2r1y

r1₋ r1

2r1M r1 4 ,

3.9

wheny <0, one hasfk, y−r1/2|y|r −r1/2−yr <0,y∈−∞,−M4,then we have

_y

0

fk, sds

_−M4

0

fk, sds

_y

−M4

fk, sds

−KM4r1

2

y

−M4

−srd−s

−KM4₂ r1 r1y

r1₋ r1

2r1M4 r1_.

3.10

Letd : −KM4−r1/2r1M₄r1, then we have

*_y

0fk, sds r1/2r1|y|r1dfor

|y|> M4.Therefore, byTheorem 3.1, the conclusion of this lemma follows.

Theorem 3.3. Assume thatpk>0,k1−n, . . . , N,and

isup_k∈1−n,N limu→0fk, u/ur2< pλ,λ >0is defined inLemma 2.4;

iiFsatisfies3.1inTheorem 3.1orfsatisfies the assumptions inCorollary 3.2. Then BVP1.1-1.2has at least two solutions.

Proof. We first show thatφsatisfies the P-S condition. Let{ym}∞_m1⊂Hsatisfy that{φym}

is bounded and limm→ ∞φym 0. If {ym} is unbounded, it possesses a divergent

subseries, say ymk → ∞ _{as k} → ∞_{. However from}ii, we get _{3.4 or 3.8}, hence

φymk → −∞_ask → ∞_{, which is contradictory to the the fact that}{_φym}_{is bounded.} Next we use the mountain-pass lemma to finish the proof. Byi, for3 pλ−r2/2>

0, there existsR1 > 0 such thatfk, y/y r23 for|y| R1. Then

*y

0fk, sds r2 3/2y2for|y|R1. Now together withLemma 2.3,we have

φyp

2λy

2₋r23

2

N

1−n

|yk|2y2

_pλ−r

2−3

2

3

2y

2_{>0 for yR} 1,

3.11

which implies that

φy 3

2R

2

1>0φθ, y∈∂Ω, 3.12

whereθis the zero element inH, andΩ {y ∈ H | y < R1}.Since we have from3.4

φy1 < φθ 0.Using Lemma 2.2, we have shown thatξ infh∈Γmaxt∈0,1 φhtis the

critical value ofφ,withΓdefined as

Γ h|h:0,1 −→H, his continuous, h0 θ, h1 y1

. 3.13

We denoteyas its corresponding critical point.

On the other hand, byTheorem 3.1orCorollary 3.2, we know that there existsy∗∈H,

s.t.φy∗ sup_y∈Hφy.If y∗/y, the theorem is proved. If on the contrary, y∗ y, that is, sup_y∈Hφy infh∈Γmaxt∈0,1 φht, that implies for any h ∈ Γ, maxt∈0,1 φht

sup_y∈Hφy.Takingh1/h2 inΓ with maxt∈0,1 φh1t maxt∈0,1 φh2t sup_y∈Hφy,

by the continuity of φht, there exist t1, t2 ∈ 0,1 s.t. φh1t1 maxt∈0,1 φh1t, φh2t2 maxt∈0,1 φh2t.Henceh1t1,h2t2are two different critical points ofφ,that

is, BVP1.1-1.2has at least two different solutions.

4. An Example

Consider the 6th-order boundary value problem for difference equation

Δ6_y

k−3y3_key

2

k−90, k∈1,300 ,

y−2y−1y00, y301y302y3030.

4.1

Let fk, u u3_eu2₋₉

, we have limu→0fk, u/u 0, limu→ ∞fk, u/u ∞. Hence

fk, usatisfies the conditions inTheorem 3.3, the boundary value problem4.1has at least two solutions.

Acknowledgments

This research is partially supported by the NSF of China and NSF of Guangdong Province.

References

1 R. P. Agarwal and J. Henderson, “Positive solutions and nonlinear eigenvalue problems for third-order difference equations,”Computers & Mathematics with Applications, vol. 36, no. 10–12, pp. 347–355, 1998.

2 R. P. Agarwal and D. O’Regan, “Singular discrete n, p boundary value problems,” Applied Mathematics Letters, vol. 12, no. 8, pp. 113–119, 1999.

3 R. P. Agarwal and F.-H. Wong, “Upper and lower solutions method for higher-order discrete boundary value problems,”Mathematical Inequalities & Applications, vol. 1, no. 4, pp. 551–557, 1998.

4 R. P. Agarwal, K. Perera, and D. O’Regan, “Multiple positive solutions of singular and nonsingular discrete problems via variational methods,”Nonlinear Analysis: Theory, Methods & Applications, vol. 58, no. 1-2, pp. 69–73, 2004.

5 Z. M. Guo and J. S. Yu, “Existence of periodic and subharmonic solutions for two-order superlinear difference equations,”Science in China Series A, vol. 33, pp. 226–235, 2003.

6 D. Jiang, J. Chu, D. O’Regan, and R. P. Agarwal, “Positive solutions for continuous and discrete boundary value problems to the one-dimensionp-Laplacian,”Mathematical Inequalities & Applications, vol. 7, no. 4, pp. 523–534, 2004.

8 H. H. Liang and P. X. Weng, “Existence and multiple solutions for a second-order difference boundary value problem via critical point theory,”Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 511–520, 2007.

9 H. H. Liang and P. X. Weng, “Existence of solutions for a fourth-order difference boundary value problem and a critical point method,”Applied Mathematics A Journal of Chinese Universities, Series A, vol. 23, no. 1, pp. 67–72, 2008.

10 P. J. Y. Wong and R. P. Agarwal, “Existence theorems for a system of difference equations withn, p -type conditions,”Applied Mathematics and Computation, vol. 123, no. 3, pp. 389–407, 2001.

11 P. H. Rabinowitz,Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65, American Mathematical Society, Providence, RI, USA, 1986.

12 A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,”Journal of Functional Analysis, vol. 14, pp. 349–381, 1973.

13 R. A. Horn and C. R. Johnson,Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.