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,k∈T, min{uk:k∈T}A, max{uk:
k∈T}B, wheref:T×R2→R, a, b∈Nare fixed numbers satisfyingb≥a2, andA, B∈Rare 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
In this paper, we discuss the nonlinear second-order discrete problems with minimum and maximum:
Δ2uk−1 fk, uk,Δuk, k∈T,
1.5
minuk:k∈TA, maxuk:k∈TB, 1.6
wheref:T×R2→Ris a continuous function,a, b∈Nare fixed numbers satisfyingb≥a2
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, t∈a, b ,
minut:t∈a, b A, maxut:t∈a, 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:t∈a, b A, maxut:t∈a, b B,
1.8
whereFsatisfies the condition that there exists a nondecreasing functionf :0,∞→0,∞
satisfying∞0ds/fs≥b−a, ∞0 s/fsds∞, such that
Fut≤fut. 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×R2→Ris continuous, and there existp:T→R, q:T→R, r :T→R, such
that
fk, u, v≤pk|u|qk|v|rk, k, u, v∈T×R2, 1.10
whereΓ:1−b−a bi−a11|pi| − bi−a11|qi|>0.
2. Preliminaries
then boundary condition1.6is equal to
iiForη1< k≤η2, we get
In particular, it is not hard to obtain
max
k∈{c,...,d}uk≤d−ck∈{maxc,...,d−1}Δuk. 2.19
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 uk≤d−c max
k∈{c,...,η1−1}Δuk, k∈
c, . . . , η1,
uk≤d−c max k∈{η1,...,d−1}
Δuk, k∈
η11, . . . , d. 2.20
In particular, one has
max
k∈{c,...,d}uk≤d−ck∈{maxc,...,d−1}Δuk. 2.21
Lemma 2.5. Supposeγ∈ A0, c∈0,1 . Ifu∈Esatisfies
γu−cγ−u 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, γ−u < γ0 0. Furthermore,
γu−cγ−u>0, which contradicts withγu−cγ−u 0.
Define functionalφ:va, va1, . . . , vb−1→Rby
φv max
d−1
kc
vk:c≤d, c, d∈T\ {b}
. 2.23
Lemma 2.6. Supposeukis a solution of 1.5andωu 0. Then
minφΔu, φ−Δu≤ b−a
2Γ b−1
ia1
ri. 2.24
Proof. Let
Ck|Δuk>0, k∈T\ {b}, C−k|Δuk<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−
≤ b−a
2 . 2.26
At first, we prove the inequality
φΔu≤ NC
Γ
b−1
ia1
ri. 2.27
For anyα∈C, 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.
Case 1.2uαuβ ≥0. Without loss of generality, we supposeuα≥ 0, uβ≥ 0. Thenξ1
will be discussed in different situations.
Similarly, we can prove
φ−Δu≤ NΓC−
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<C1b−a,
|α|<C1b−a,|β|< C1. 2.58
DefineΓi :Ω→E×R2 i1,2:
Γ1u, α, β
αβk−a, αωu, βφΔu−C,
Γ2u, α, β
αβk−a, αωu, βφ−Δu−C. 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 αβk−a, αωu−1−λω−u, βφΔu
−φλ−1Δu−λC,
Gλ, u, α, β u, α, β−Hλ, u, α, β.
2.61
Foru, α, β∈Ω,
G1, u, α, β u, α, β−αβk−a, αωu, βφΔu−C
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,
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−−α−βk−a,−αω−u−ωu,−βφ−Δu−φΔu
−u, α, β−αβk−a, αωu−ω−u, βφΔu−φ−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| ≤C1b−a, |βn| ≤C1, uE ≤C1b−a. 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→∞
αnβnk−a, λnω
un
−1−λn
ω−un
,
βnφ
Δun
−φλn−1
Δun
−λnC
α0β0k−a, λ0ωu−1−λ0ω−u,
β0φΔu−φλ0−1Δu−λ0C.
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
α0β0k−a u0k, k∈T, 2.67
ωu0−1−λ0ω−u00, 2.68
φΔu0−φλ0−1Δu0λ0C. 2.69
By2.67andLemma 2.5takeuu0, c1−λ0, there existsξ∈T, such that u0ξ≤0. Also from2.67, we haveu0ξ α0β0ξ−a, then we get
u0k u0ξ β0k−ξ, 2.70
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, β0∈∂Ω.
Case 2. Ifβ0>0, then from2.67,Δu0k>0,and the definition ofφ, we have
φΔu0−φλ0−1Δu0β0b−a. 2.73
Together with2.69, we getφβ0 λ0C, and
β0 λ0C
b−a < 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<C1b−a. 2.76
Similarly,|u0b|<C1b−a, then we getu0E <C1b−aand|α0||u0a|<
C1b−a, which contradicts withu0, α0, β0∈∂Ω.
Case 3. Ifβ0<0, then from2.67, we getΔu0k β0<0 and
φΔu0−φλ0−1Δu01−λ0β0b−a. 2.77
By2.69, we have
1−λ0β0b−a λ0C. 2.78
Ifλ00, thenβ0b−a 0. Furthermore,β00, which contradicts withβ0<0. Ifλ01, thenλ0C0. Furthermore,C0, which contradicts withC >0. Ifλ∈0,1, then1−λ0β0b−a<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 αβk−a, αωu−1−λω−u,
βφ−Δu−φ1−λΔu−λC. 2.80
Similarly, we can prove
3. The main results
Theorem 3.1. SupposeH1holds. Then1.5and1.6have at least two different solutions when
A0and
C > b−a
2Γ b−1
ia1
ri. 3.1
Proof. LetA0, C >b−a/2Γ ib−a11|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 <C1b−a,
|α|<C1b−a,|β|< C1. 3.7
Define operatorS1:0,1 ×Ω→E×R2,
S1λ, u, α, β
αβk−a λ
k−1
ia i
la1
fl, ul,Δul, αωu, βφΔu−C
. 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
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
Fork > k1, Combining Case1with Case2, we get
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.