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Volume 2010, Article ID 973731, 22 pagesdoi:10.1155/2010/973731 Research Article Solutions and Green’s Functions for Boundary Value Problems of Second-Order Four-Point Functional Differe

Volume 2010, Article ID 973731, 22 pages

doi:10.1155/2010/973731

Research Article

Solutions and Green’s Functions for

Boundary Value Problems of Second-Order

Four-Point Functional Difference Equations

Yang Shujie and Shi Bao

Institute of Systems Science and Mathematics, Naval Aeronautical and Astronautical University, Yantai, Shandong 264001, China

Correspondence should be addressed to Yang Shujie,yangshujie@163.com

Received 23 April 2010; Accepted 11 July 2010

Academic Editor: Irena Rach ˚unkov´a

Copyrightq 2010 Y Shujie and S Bao This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

We consider the Green’s functions and the existence of positive solutions for a second-orderfunctional difference equation with four-point boundary conditions

1 Introduction

In recent years, boundary value problems BVPs of differential and difference equationshave been studied widely and there are many excellent resultssee Gai et al 1, Guo andTian2, Henderson and Peterson 3, and Yang et al 4 By using the critical point theory,Deng and Shi5 studied the existence and multiplicity of the boundary value problems to aclass of second-order functional difference equations

Lu n  fn, u n1, u n , u n−1 1.1with boundary value conditions

where the operator L is the Jacobi operator

Lu n  a n u n1 a n−1u n−1 b n u n 1.3

Ntouyas et al.6 and Wong 7 investigated the existence of solutions of a BVP forfunctional differential equations

Yang et al 9 considered two-point BVP of the following functional difference

equation with p-Laplacian operator:

For any real function x defined on the interval −τ, T and any t ∈ 0, T with T ∈ N,

we denote by x t an element of C τ defined by x t k  xt  k, k ∈ −τ, 0.

In this paper, we consider the following second-order four-point BVP of a nonlinearfunctional difference equation:

At this point, it is necessary to make some remarks on the first boundary condition in

1.9 This condition is a generalization of the classical condition

u 0  αuη

from ordinary difference equations Here this condition connects the history u0with the single

u η This is suggested by the well-posedness of BVP 1.9, since the function f depends on the term u t i.e., past values of u.

As usual, a sequence{u−τ, , uT  1} is said to be a positive solution of BVP 1.9

if it satisfies BVP1.9 and uk ≥ 0 for k ∈ −τ, T with uk > 0 for k ∈ 1, T.

2 The Green’s Function of  1.9 

First we consider the nonexistence of positive solutions of1.9 We have the following result

Lemma 2.1 Assume that

or

α

T  1 − η> T  1. 2.2

Then1.9 has no positive solution.

Proof FromΔ2u t − 1  −rtft, u t  ≤ 0, we know that ut is convex for t ∈ 0, T  1 Assume that xt is a positive solution of 1.9 and 2.1 holds

1 Consider that γ  0.

If xT  1 > 0, then xξ > 0 It follows that

which is a contradiction to the convexity of xt.

If xT  1  0, then xξ  0 If x0 > 0, then we have

which is a contradiction to the convexity of xt.

If t0∈ ξ, T, similar to the above proof, we can also get a contradiction.

2 Consider that γ > 0.

which is a contradiction to the convexity of xt.

Assume that xt is a positive solution of 1.9 and 2.2 holds

which is a contradiction to the convexity of xt.

If xη > 0, similar to the above proof, we can also get a contradiction.

If xT  1  xη  0, and so x0  0, then there exists a t0 ∈ 1, η ∪ η, T such that

x t0 > 0 Otherwise, xt ≡ 0 is a trivial solution Assume that t0 ∈ 1, η, then

A contradiction to the convexity of xt follows.

If t0∈ η, T, we can also get a contradiction.

2 Consider that h0 > 0.

which is a contradiction to the convexity of xt.

Next, we consider the existence of the Green’s function of equation

Motivated by Zhao10, we have the following conclusions

Theorem 2.2 The Green’s function for second-order four-point linear BVP 2.13 is given by

It is easy to find that the solution of BVP2.16 is given by

where c and d are constants that will be determined.

From2.18 and 2.20, we have

The four-point BVP2.13 can be obtained from replacing uT  1  0 by uT  1  βuξ in

2.19 Thus we suppose that the solution of 2.13 can be expressed by

w t  vt  a  btvξ, 2.25

where a and b are constants that will be determined.

From2.24 and 2.25, we get

Remark 2.3 ByH1, we can see that G1t, s > 0 for t, s ∈ 0, T  12 Let

m min

t,s∈1,T2G1t, s, M max

t,s∈1,T2G1t, s. 2.30

Then M ≥ m > 0.

solution which is given in2.29.

Proof We need only to show the uniqueness.

Obviously, wt in 2.29 is a solution of BVP 2.13 Assume that vt is another

αηc1− 1 − αc2 0,



T  1 − βξc11− βc2  0. 2.36

ConditionH1 implies that 2.36 has a unique solution c1 c2 0 Therefore vt ≡ wt for

t ∈ −τ, T  1 This completes the proof of the uniqueness of the solution.

3 Existence of Positive Solutions

In this section, we discuss the BVP1.9

K p  {u ∈ K | u  p} Furthermore, assume that Φ : K → K is a completely continuous operator

and Φu / u for u ∈ ∂K p  {u ∈ K | u  p} Thus, one has the following conclusions:

(1) if u ≤ Φu for u ∈ ∂K p , then i Φ, K p , K   0;

(2) if u ≥ Φu for u ∈ ∂K p , then i Φ, K p , K   1.

Assume that f≡ 0 Then 3.1 may be rewritten as

Let ut be a solution of BVP 3.1 and yt  ut − ut Then for t ∈ 1, T we have

Then E is a Banach space endowed with norm · and K is a cone in E.

For y ∈ K, we have by H1 and the definition of K,

Proof For s ∈ τ  1, T, k ∈ −τ, 0, and s  k ∈ 1, T, by the definitions of · τand · , wehave

Lemma 3.3 Consider that ΦK ⊂ K.

Proof If t ∈ −τ, 0 and t  T 1, Φyt  αΦη and ΦyT 1  βΦξ, respectively Thus,

Lemma 3.4 Suppose that (H1) holds Then Φ : K → K is completely continuous.

We have the following main results

Theorem 3.5 Assume that (H1)–(H3) hold Then BVP3.1 has at least one positive solution if the

following conditions are satisfied:

H4 there exists a p1> h such that, for s ∈ 1, T, if φ τ ≤ p1 h, then fs, φ ≤ R1p1;

H5 there exists a p2> p1such that, for s ∈ 1, T, if φ τ ≥ m/Mp2, then f s, φ ≥ R2p2

or

H61 > α > 0;

H7 there exists a 0 < r1 < p1such that, for s ∈ 1, T, if φ τ ≤ r1, then f s, φ ≥ R2r1;

H8 there exists an r2 ≥ max{p2 h, Mh/mα}, such that, for s ∈ 1, T, if φ τ ≥

For every y ∈ ∂K p2, by3.8–3.10 andLemma 3.2, we have, for s ∈ τ  1, T, y s τ ≥

m/My  m/Mp2 Then by3.13 and H5, we have

Consequently, u1 y1 u or u2 y2 u is a positive solution of BVP 3.1

Theorem 3.6 Assume that (H1)–(H3) hold Then BVP3.1 has at least one positive solution if (H4) and (H7) or (H5) and (H8) hold.

Theorem 3.7 Assume that (H1)–( H3) hold Then BVP3.1 has at least two positive solutions if

(H4), (H5), and (H7) or (H4), (H5), and (H8) hold.

Theorem 3.8 Assume that (H1)–(H3) hold Then BVP3.1 has at least three positive solutions if

Then by3.27, H9, H10, and the definition of Ht, we have Ft, w t  > 0 for t ∈

1, T Thus, the BVP 1.9 can be changed into the following BVP:

Theorem 4.1 Assume that (H1), (H2), (H6),

Proof Assume that conditionH12 holds If λ > 1/mrf0and f0 < ∞, there exists an  > 0

sufficiently small, such that

r > max {r1, h/μα }, such that, for t ∈ 1, T and φ ≥ r,

f

t, φ

<

f∞ 1φ. 4.16

We now show that there is r2 ≥ r, such that, for y ∈ ∂K r2,Φy ≤ y In fact, for

s ∈ 1, T r2 ≥ Mr/mα and every y ∈ ∂K r , δy ≥ y s  u s τ ≥ r; hence in a similar way,

It follows that we can take λ0  r − h/MLT

s1r s > 0 such that, for all 0 < λ ≤ λ0and all

For every y ∈ ∂K R, by the definition of·, ·τand the definition ofLemma 3.2, there

exists a t0∈ τ  1, T such that y  y t0τ  R and u t0  0, thus y t0 u t0τ ≥ R Hence

which by combining with4.21 completes the proof

Example 4.3 Consider the BVP3.33 inExample 3.9with

where s  φ τ , A is some positive constant, p2 40, m  21/24, and M  163/40.

By calculation, f0 A, f∞ πA/2000, and r  1/120; let δ  1 Then by Theorem4.1,

for λ ∈ 2608/49A, 640000/163πA, the above equation has at least one positive solution.

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