Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 359251, 15 pages
doi:10.1155/2012/359251
Research Article
On Six Solutions for m-Point Differential
Equations System with Two Coupled Parallel
Sub-Super Solutions
Jian Liu,1, 2 Hua Su,2 and Shuli Wang3
1
School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics,
Jinan, Shandong 250000, China
3
College of Science, Yanshan University at Qinhuangdao, Qinhuangdao 066004, China
2
Correspondence should be addressed to Jian Liu, kkword@126.com
Received 15 August 2011; Accepted 5 October 2011
Academic Editor: Rudong Chen
Copyright q 2012 Jian Liu et al. 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.
Under the assumption of two coupled parallel subsuper solutions, the existence of at least six
solutions for a kind of second-order m-point differential equations system is obtained using the
fixed point index theory. As an application, an example to demonstrate our result is given.
1. Introduction
In this paper, we consider the following second-order m-point boundary value problems of
nonlinear equations system
−ϕ t f1 ϕt f2 ψt ,
−ψ t g1 ψt g2 ϕt ,
ϕ 0 0,
ϕ1 t ∈ 0, 1,
t ∈ 0, 1,
m−2
αi ϕξi ,
i1
ψ 0 0,
ψ1 m−2
αi ψξi ,
i1
where fi , gi : R1 → R1 i 1, 2 are continuous and αi , ξi satisfying
1.1
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H0 m−2
i1 αi ∈ 0, 1 with αi ∈ 0, ∞ for i 1, 2, . . . , m − 2 and 0 < ξ1 < ξ2 < · · · <
ξm−2 < 1.
Multipoint boundary value problems arise in many applied sciences for example, the
vibrations of a guy wire composed of N parts with a uniform cross-section throughout, but
different densities in different parts can be set up as a multipoint boundary value problems
see 1. Many problems in the theory of elastic stability can be modelled by multipoint
boundary value problems see 2. The study of multipoint boundary value problems for
linear second-order ordinary differential equations was initiated by Il’in and Moiseev 3.
Subsequently, Gupta 4 studied certain three-point boundary value problems for nonlinear
second-order ordinary differential equations. Since then, the solvability of more general
nonlinear multipoint boundary value problems has been discussed by several authors using
various methods. We refer the readers to 5–12 and the references therein.
In the recent years, many authors have studied existence and multiplicity results for
solutions of multipoint boundary value problems via the well-ordered upper and lower
solutions method, see 8, 13, 14 and the references therein. However, only in very recent
years, some authors considered the multiplicity of solutions under conditions of non-wellordered upper and lower solutions. For some abstract results concerning conditions of nonwell-ordered upper and lower solutions, the readers are referred to recent works 15–18.
In 19, Xu et al. considered the following second-order three-point boundary value
problem
y t f t, y 0,
y0 0,
t ∈ 0, 1,
y1 − αy η 0,
1.2
where 0 < η < 1, 0 < α < 1, f ∈ C0, 1 × R1 , R1 . He obtained the following result. First, let
us give the following condition H0 to be used later.
H0 There exists M > 0 such that
ft, x2 − ft, x1 ≥ −Mx2 − x1 ,
t ∈ 0, 1, x2 ≥ x1 .
1.3
Let the function e be e et t for t ∈ 0, 1.
Theorem 1.1. Suppose that H0 holds, u1 and u2 are two strict lower solutions of 1.2, v1 and v2
are two strict upper solutions of 1.2, and u1 < v1 , u2 < v2 , u2 v1 , u1 v2 . Moreover, assume
−ξ0 e ≤ u2 − u1 ≤ ξ0 e,
−ξ0 e ≤ v2 − v1 ≤ ξ0 e,
for some ξ0 > 0. Then, the three-point boundary value problem 1.2 has at least six solutions.
1.4
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3
We would also like to mention the result of Yang 20, in 20. Yang studied the following integral boundary value problem
− au bu gtft, u,
1
uτdατ,
cos γ0 u0 − sin γ0 u 0 1.5
0
cos γ1 u1 sin γ1 u 1 1
uτdβτ,
0
1
1
where γ0 ∈ 0, π/2 and γ1 ∈ 0, π/2, 0 uτdατ and 0 uτdβτ denote the RiemannStieltjes integrals of u with respect to α and β, respectively. Some sufficient conditions for
the existence of either none, or one, or more positive solutions of the problem 1.5 were established. The main tool used in the proofs of existence results is a fixed point theorem in a cone,
due to Krasnoselskii and Zabreiko.
At the same time, we note that Webb and Lan 21 have considered the first eigenvalue
of the following linear problem
u t λut 0,
u0 0,
u1 0 < t < 1,
1.6
m−2
αi u ηi ,
i1
they also investigated the existence and multiplicity of positive solutions of several related
nonlinear multipoint boundary value problems. Furthermore, Ma and O’Regan 22 studied
the spectrum structure of the problem 1.6, and the authors obtained the concrete computational method and the corresponding properties of real eigenvalue of 1.6 by constructing an
auxiliary function. Their work is very fundamental to further study for multipoint boundary
value problems. By extending and improving the work in 22, Rynne 23 showed that the
associated Sturm-Liouville problem consisting of 1.6 has a strictly increasing sequence of
with
eigenfunctions
φ
t
sin
λn t.
simple eigenvalues {λn }∞
n
n0
Very recently, Kong et al. 24 were concerned with the general boundary value problem with a variable w
u t wtfu 0,
cos αua − sin αu a 0,
t ∈ a, b,
α ∈ 0, π,
ub m−2
k i u ηi
1.7
i1
By relating 1.7 to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, the existence and nonexistence of nodal solutions of 1.7 were obtained. We also point out that Webb 25 made the excellent remark on some existence results
of symmetric positive solutions obtained in some recent papers and the author also corrected
the values of the principle eigenvalue previously given in some examples.
In this paper, by means of two coupled parallel subsuper solutions, we obtain some
sufficient conditions for the existence of six solutions for 1.1 and our main tool is based on
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the fixed point index theory. At the end of this paper, we will give an example which illustrates that our work is true. Our method stems from the paper 18.
2. Preliminaries and a Lemma
In the section, we shall give some preliminaries and a lemma which are fundamental to prove
our main result.
Let E be an ordered Banach space in which the partial ordering ≤ is induced by a cone
P ⊂ E. A cone P is said to be normal if there exists a constant N > 0, such that θ ≤ x ≤ y implies x ≤ Ny, the smallest N is called the normal constant of P . P is called solid, if
intP / ∅, that is, P has nonempty interior. Every cone P in E defines a partial ordering in E
given by x ≤ y if and only if y − x ∈ P . If x ≤ y and x /
y, we write x < y; if cone P is solid and
y − x ∈ intP , we write x y. P is called total if E P − P . Let B : E → E be a bounded linear
operator. B is said to be positive if BP ⊂ P . An operator A is strongly increasing, that is,
x < y implies Ax Ay. If A is a linear operator, A is strongly increasing implying A is
strongly positive.
Let E be an ordered Banach space, P a total cone in E, the partial ordering ≤ induced
by P . B : E → E is a positive completely continuous linear operator. Let rB > 0 the spectral
radius of B, B ∗ the conjugated operator of B, and P ∗ the conjugated cone of P . Since P ⊂ E is
a total cone i.e., E P − P , according to the famous Krein-Rutman theorem see 26, we
infer that if rB /
0, then there exist ϕ ∈ P \ {θ} and g ∗ ∈ P ∗ \ {θ}, such that
Bϕ rBϕ,
2.1
B∗ g ∗ rBg ∗ .
Fixed ϕ ∈ P \ {θ}, g ∗ ∈ P ∗ \ {θ} such that 2.1 holds. For δ > 0, let
P g ∗ , δ x ∈ P, g ∗ x ≥ δx ,
2.2
then P g ∗ , δ is also a cone in E. One can refer 26–28 for definition and properties about the
cones.
Definition 2.1 see 29. Let B be a positive linear operator. The operator B is said to satisfy
condition H, if there exist ϕ ∈ P \ {θ}, g ∗ ∈ P ∗ \ {θ}, and δ > 0 such that 2.1 holds, and B
maps P into P g ∗ , δ.
Lemma 2.2 see 10. Suppose that d 1 −
m−2
i1
−u t 0,
u 0 0,
αi /
0. Then, the BVP
t ∈ 0, 1,
u1 m−2
αi uξi ,
i1
2.3
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5
has Green’s function
s Gt, s Gt,
m−2
i , s
αi Gξ
,
m−2
1 − i1 αi
i1
2.4
where
s Gt,
⎧
⎨1 − t,
0 ≤ s ≤ t ≤ 1,
⎩1 − s,
0 ≤ t ≤ s ≤ 1.
2.5
For convenience, we list the following hypotheses which will be used in our main
result.
H1 fi , gi i 1, 2 are strictly increasing;
H2 there exist constants k > 0, l > 0 and D > 0 such that
i |f1 ±k ± l| < N −1 k,
ii |g1 ±k ± D| < N −1 k,
iii |f2 ±k| ≤ l, and
iv |g2 ±k| ≤ D, where N maxt∈0,1
1
0
Gt, sds;
H3 there exist constants 0 < c1 < k, −k < c2 < 0, 0 < c3 < k, −k < c4 < 0, such that, for
all t ∈ 0, 1, we have
i c1 <
ii c2 >
iii c3 <
iv c4 >
1
0
Gt, sf1 c1 ds − Nl,
0
Gt, sg1 c2 ds ND,
0
Gt, sg1 c3 ds − ND, and
0
Gt, sf1 c4 ds Nl;
1
1
1
H4 i lim|ϕ| → ∞ f1 ϕ f2 ψ/ϕ ≥ 2λ1 uniformly for ψ ∈ R,
ii lim|Ψ| → ∞ g1 ψ g2 ϕ/ψ ≥ 2λ1 uniformly for ϕ ∈ R, where λ1 is the first
eigenvalue of the following boundary value problem:
−u t λu,
u 0 0,
t ∈ 0, 1,
u1 m−2
αi uξi .
2.6
i1
It is well known that λ1 r −1 H, where linear operator H : C0, 1 → C0, 1 is defin1
ed as Hut 0 Gt, susds.
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3. Main Results
Theorem 3.1. Assume H0 , H1 –H4 hold, then BVP 1.1 has at least six distinct continuous
solutions.
Proof. It is easy to check that BVP 1.1 is equivalent to the following integral equation systems:
ϕt 1
Gt, s f1 ϕs f2 ψs ds,
0
ψt 1
3.1
Gt, s g1 ψs g2 ϕs ds,
0
where Gt, s is defined as in Lemma 2.2. By H0 , we know that Gt, s ≥ 0, for all t, s ∈
0, 1.
Let E C0, 1 × C0, 1, define the norm in E as ϕ, ψ ϕ ψ. Then, E is a
Banach space with this norm. Let P {ϕ, ψ ∈ E | ϕt ≥ 0, ψt ≥ 0, for all t ∈ 0, 1}, Q {ϕ ∈ C0, 1 | ϕt ≥ 0, for all t ∈ 0, 1}. Then, P Q × Q is a normal and solid cone. Set
T : E → E, such that
T ϕ, ψ 1
Gt, s f1 ϕs f2 ψs ds,
1
0
Gt, s g1 ψs g2 ϕs ds ,
3.2
0
it is clear that the solutions of 1.1 are equivalent to the fixed points of T .
Set 0 < ξ1 ≤ λ1 , let
f1 ϕ f2 ψ , g1 ψ g2 ϕ
f ϕ, ψ ,
λ1 ξ1
K ϕ, ψ λ1 ξ1 Hϕ, λ1 ξ1 Hψ H1 ϕ, H1 ψ ,
3.3
where H1 λ1 ξ1 H, then T Kf. It is easy to see that H is a strongly positive completely
continuous operator, and it follows from Gt, s ≥ 0 and the continuity of Gt, s that K is
a strongly positive completely continuous operator. Since fi , gi : R1 → R1 i 1, 2 are
strictly increasing continuous functions, we know that f is a strictly increasing continuous
bounded operator. By T Kf, we can prove that T is completely continuous. We infer from
the increasing properties of K and f that T is increasing.
Let ϕ1 ≡ c1 , ψ1 ≡ −k, ϕ2 ≡ k, ψ2 ≡ c2 , ϕ3 ≡ −k, ψ3 ≡ c3 , ϕ4 ≡ c4 , ψ4 ≡ k, then
ϕi , ψi i 1, 2, 3, 4 satisfy
ϕ1 , ψ1 < ϕ2 , ψ2 ,
ϕ3 , ψ3 < ϕ4 , ψ4 ,
ϕ1 , ψ1 ϕ4 , ψ4 ,
ϕ3 , ψ3 ϕ2 , ψ2 .
3.4
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By H2 iii and H3 i, we have
1
Gt, s f1 ϕ1 f2 ψ1 ds ≥
1
0
Gt, sf1 c1 ds − Nl > c1 ϕ1 .
3.5
0
It follows from H2 ii,iv, and the increasing property of g2 that
1
Gt, s g1 ψ1 g2 ϕ1 ds > −kN −1
0
1
Gt, sds ≥ −kN −1 N −k ψ1 .
3.6
0
Equations 3.2, 3.5, and 3.6 imply that
ϕ1 , ψ1 < T ϕ1 , ψ1 .
3.7
Similarly, by H1 –H3 , we obtain
T ϕ2 , ψ2 < ϕ2 , ψ2 ,
ϕ3 , ψ3 < T ϕ3 , ψ3 ,
T ϕ4 , ψ4 < ϕ4 , ψ4 .
3.8
By 20, Lemma 3, we get that H1 satisfies condition H. Therefore, there exist j0∗ ∈
Q∗ \ {θ}, δ > 0, such that
H1∗ j0∗ rH1 j0∗ ,
j0∗ H1 ϕ ≥ δH1 ϕ, ∀ϕ ∈ Q.
3.9
3.10
By the definition of spectral radius of completely continuous operator, we have rK rH1 ,
and combining 3.9, we infer that
H1∗ j0∗ rKj0∗ .
3.11
Let j ∗ ϕ, ψ j0∗ ϕ j0∗ ψ, for all ϕ, ψ ∈ E, then j ∗ ∈ P ∗ \ {θ}. According to the proof in
18, we can get that K satisfies condition H.
By condition H4 , we obtain that there exists C > 0, such that
f1 ϕ f2 ψ ≥ λ1 ξ1 ϕ,
ϕ ≥ C, ψ ∈ R,
f1 ϕ f2 ψ ≤ λ1 ξ1 ϕ, ϕ ≤ −C, ψ ∈ R,
g1 ψ g2 ϕ ≥ λ1 ξ1 ψ, ψ ≥ C, ϕ ∈ R,
g1 ψ g2 ϕ ≤ λ1 ξ1 ψ, ψ ≤ −C, ϕ ∈ R.
3.12
3.13
3.14
3.15
Equations 3.12–3.15 imply
f1 ϕ f2 ψ , g1 ψ g2 ϕ ≥ λ1 ξ1 ϕ, ψ ,
f1 ϕ f2 ψ , g1 ψ g2 ϕ ≤ λ1 ξ1 ϕ, ψ ,
∀ ϕ, ψ ≥ C,
3.16
∀ϕ, ψ ≤ −C.
3.17
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Since f1 ϕ, g1 ψ are continuous in −c1 k, C, so they are bounded, then there exists h > 0
such that
f1 ϕ ≥ −h,
g1 ψ ≥ −h,
− ϕ1 , ψ1 −c1 k ≤ ϕ,
ψ ≤ C.
3.18
By virtue of 3.18 and the increasing properties of f2 and g2 , one shows
f1 ϕ f2 ψ , g1 ψ g2 ϕ
≥ −h − f2 − ϕ1 , ψ1 , −h − g2 − ϕ1 , ψ1 ,
− ϕ1 , ψ1 ≤ ϕ,
ψ ≤ C.
3.19
In addition, if ϕ, ψ satisfy ϕ ≥ C, −ϕ1 , ψ1 ≤ ψ ≤ C, then it follows from 3.12,
3.18, and the increasing property of g2 that
f1 ϕ f2 ψ , g1 ψ g2 ϕ
≥ λ1 ξ1 ϕ, −h g2 − ϕ1 , ψ1 ≥ λ1 ξ1 ϕ, ψ − λ1 ξ1 C h g2 − ϕ1 , ψ1 , λ1 ξ1 C h g2 − ϕ1 , ψ1 λ1 ξ1 ϕ, ψ − d1 , d1 ,
3.20
where
d1 λ1 ξ1 C h g2 − ϕ1 , ψ1 .
3.21
Similarly, if ϕ, ψ satisfy ψ ≥ C, −ϕ1 , ψ1 ≤ ϕ ≤ C, then combining the increasing property
of f2 with 3.14 and 3.18, we know that
f1 ϕ f2 ψ , g1 ψ g2 ϕ ≥ λ1 ξ1 ϕ, ψ − d2 , d2 ,
3.22
where d2 λ1 ξ1 C h |f2 −ϕ1 , ψ1 |. Let d max{d1 , d2 }. By 3.21 and 3.22, we get
that if ϕ ≥ C, −ϕ1 , ψ1 ≤ ψ ≤ C or ψ ≥ C, −ϕ1 , ψ1 ≤ ϕ ≤ C, it is obvious that
f1 ϕ f2 ψ , g1 ψ g2 ϕ ≥ λ1 ξ1 ϕ, ψ − d, d.
3.23
It follows from 3.16, 3.19, and 3.23 that
f1 ϕ f2 ψ , g1 ψ g2 ϕ ≥ λ1 ξ1 ϕ, ψ − d, d,
ϕ, ψ ≥ − ϕ1 , ψ1 .
3.24
In a similar way, from 3.12 and 3.14, we can show that there exists e > 0 such that
f1 ϕ f2 ψ , g1 ψ g2 ϕ ≥ λ1 ξ1 ϕ, ψ − e, e,
ϕ, ψ ≥ − ϕ3 , ψ3 .
3.25
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Let a max{d, e}. It follows from 3.24 and 3.25 that if ϕ, ψ ≥ −ϕ1 , ψ1 or ϕ, ψ ≥
−ϕ3 , ψ3 , then
f1 ϕ f2 ψ , g1 ψ g2 ϕ ≥ λ1 ξ1 ϕ, ψ − a, a.
3.26
In a similar way, from 3.13 and 3.15, we can prove that there exists constant a such that if
ϕ, ψ ≤ ϕ2 , ψ2 or ϕ, ψ ≤ ϕ4 , ψ4 , then
a, a.
f1 ϕ f2 ψ , g1 ψ g2 ϕ ≤ λ1 ξ1 ϕ, ψ − 3.27
Let P ϕ1 , ψ1 {u, v ∈ E | u, v ≥ ϕ1 , ψ1 }, P ϕ3 , ψ3 {u, v ∈ E | u, v ≥ ϕ3 , ψ3 },
P ϕ2 , ψ2 {u, v ∈ E | u, v ≤ ϕ2 , ψ2 }, P ϕ4 , ψ4 {u, v ∈ E | u, v ≤ ϕ4 , ψ4 },
for all ϕ, ψ ∈ P ϕ1 , ψ1 ∪ P ϕ3 , ψ3 , then ϕt, ψt ≥ −ϕ1 , ψ1 or ϕt, ψt ≥
−ϕ3 , ψ3 , for all t ∈ 0, 1; therefore, in virtue of expression of B, f, and 3.26, we have
Kf ϕt, ψt
1
1
λ1 ξ1 a
≥ K ϕt, ψt −
Gt, sds, Gt, sds
λ1 ξ1 0
0
r −1 K K ϕt, ψt − u1 t, v1 t,
where 1 − r −1 K, u1 t, v1 t a
show
0< 1−
1
0
Gt, sds,
1
0
3.28
Gt, sds. Since 0 < ξ1 ≤ λ1 , one can
ξ1
λ1
λ1
≤
r −1 K.
λ1 ξ1 λ1 ξ1 λ1 ξ1
3.29
This implies that there exist u1 , v1 ∈ E and 0 < ξ2 ≤ r −1 K such that
Kf ϕ, ψ ≥ r −1 K ξ2 K ϕ, ψ − u1 , v1 ,
∀ ϕ, ψ ∈ P ϕ1 , ψ1 ∪ P ϕ3 , ψ3 . 3.30
Similarly, we get by 3.27 that there exist u2 , v2 ∈ E and 0 < ξ3 ≤ r −1 K such that
Kf ϕ, ψ ≤ r −1 K ξ3 K ϕ, ψ − u2 , v2 ,
∀ ϕ, ψ ∈ P ϕ2 , ψ2 ∪ P ϕ4 , ψ4 . 3.31
We get by 3.7 that
T :P
ϕ1 , ψ1
−→ P
ϕ1 , ψ1 .
3.32
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Let E1 {u, v ∈ E | ϕ1 , ψ1 ≤ u, v ≤ ϕ2 , ψ2 }. Since P is normal, then E1 is bounded see
28. Choose M1 > 0 such that
M1 > max
1 ,
ξ2 δ ϕ1 , ψ1 ξ2 δT ϕ1 , ψ1 r −1 Kg ∗ u1 , v1 − ξ2 g ∗ T ϕ1 , ψ1
ξ2 δ
sup u, v ϕ1 , ψ1 .
u,v∈E1
3.33
Let Ω1 {u, v ∈ P ϕ1 , ψ1 | u, v − ϕ1 , ψ1 < M1 , u, v/
≥ϕ3 , ψ3 }, then E1 ⊂ Ω1 and Ω1
is a bounded open set. By the proof of Theorem 2.1 in 18, we can show that
λKu, v,
u, v − T u, v /
∀λ ≥ 0, u, v ∈ ∂Ω1 ,
3.34
where u, v satisfies Ku, v rKu, v.
Equation 3.34 implies that T has no fixed point on ∂Ω1 . It is easy to prove that
P ϕ1 , ψ1 is a retract of E, which together with 3.32 implies that the fixed point index
iT, Ω1 , P ϕ1 , ψ1 over Ω1 with respect to P ϕ1 , ψ1 is well defined, and a standard proof
yields
i T sKu, v, Ω1 , P ϕ1 , ψ1
0.
3.35
Set Ht, u, v 1 − tT u, v tT u, v sKu, v, t, u, v ∈ 0, 1 × Ω1 , then for any
u, v ∈ Ω1 , t ∈ 0, 1, we have Ht, u, v ∈ P ϕ1 , ψ1 . It follows from 3.34 that
Ht, u, v /
u, v, for all t, u, v ∈ 0, 1 × ∂Ω1 , and by 3.35 and the homotopy invariance of the fixed point index, we get
i T sKu, v, Ω1 , P ϕ1 , ψ1
0.
i T, Ω1 , P ϕ1 , ψ1
3.36
Let W1 {u, v ∈ P ϕ1 , ψ1 | u, v ϕ2 , ψ2 }. By means of usual method see 30, we
get that
i T, W1 , P ϕ1 , ψ1
1.
3.37
It is evident that A has no fixed point on ∂W1 , by 3.36, 3.37, and the additivity of the fixed
point index, we have
i T, Ω1 , P ϕ1 , ψ1
i T, Ω1 \ W1 , P ϕ1 , ψ1
− i T, W1 , P ϕ1 , ψ1
0 − 1 −1.
3.38
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11
Set E2 {u, v ∈ E | ϕ3 , ψ3 ≤ u, v ≤ ϕ4 , ψ4 }, and choose M2 > 0 such that
M2 > max
1 ,
ξ2 δ ϕ3 , ψ3 ξ2 δT ϕ3 , ψ3 r −1 Kg ∗ u1 , v1 − ξ2 g ∗ T ϕ3 , ψ3
ξ2 δ
sup u, v ϕ3 , ψ3
.
u,v∈E2
3.39
Let
≥ ϕ1 , ψ1 ,
Ω2 u, v ∈ P ϕ3 , ψ3 | u, v − ϕ3 , ψ3 < M2 , u, v/
W2 u, v ∈ P ϕ3 , ψ3 | u, v ϕ4 , ψ4 .
3.40
Similarly to the proof of 3.37 and 3.38, we get that
i T, W2 , P ϕ3 , ψ3
1,
−1.
i T, Ω2 \ W2 , P ϕ3 , ψ3
3.41
Choose M3 , M4 > 0 such that
M3 > max
1 ξ3 δ ϕ2 , ψ2 ξ3 δT ϕ2 , ψ2 r −1 Kg ∗ u2 , v2 − ξ3 g ∗ T ϕ2 , ψ2
ξ3 δ
,
sup u, v ϕ2 , ψ2
u,v∈E1
M4 > max
1 ,
ξ3 δ ϕ4 , ψ4 ξ3 δT ϕ4 , ψ4 r −1 Kg ∗ u2 , v2 − ξ3 g ∗ T ϕ4 , ψ4
ξ3 δ
sup u, v ϕ4 , ψ4
.
u,v∈E2
3.42
Set
Ω3 u, v ∈ P ϕ2 , ψ2 | u, v − ϕ2 , ψ2 < M3 , u, v ϕ4 , ψ4 ,
Ω4 u, v ∈ P ϕ4 , ψ4 | u, v − ϕ4 , ψ4 < M4 , u, v ϕ2 , ψ2 ,
W3 u, v ∈ P ϕ2 , ψ2 | ϕ1 , ψ1 u, v ,
W4 u, v ∈ P ϕ4 , ψ4 | ϕ2 , ψ2 u, v .
3.43
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By virtue of 3.31 and the same method as that for 3.34, we have
− μKu, v,
u, v − T u, v /
∀μ ≥ 0, u, v ∈ ∂Ω3 ∪ ∂Ω4 .
3.44
By 3.44, similarly to the proof of 3.38, we can prove that
i T, Ω4 \ W4 , P ϕ4 , ψ4
−1.
i T, Ω3 \ W3 , P ϕ2 , ψ2
3.45
Equations 3.37–3.41, 3.45 imply that T has at least six distinct fixed points, that is, the
system of differential equations 1.1 has at least six solution in C0, 1 × C0, 1.
4. An Example
In this section, we present a simple example to explain our results.
Consider the following second-order three-point BVP for nonlinear equations system:
−ϕ t f1 ϕt f2 ψt ,
−ψ t g1 ψt g2 ϕt ,
ϕ 0 0,
ψ 0 0,
t ∈ 0, 1,
t ∈ 0, 1,
1
1
,
ϕ1 ϕ
2
4
1
1
,
ψ1 ψ
2
4
4.1
√
1/3
where f1 ϕ ϕ3 /30000 5ϕ
, g1 ψ ψ 3 /30000 4ψ 1/5 , f2 ψ 1/6 · 5 100 √
1/12 arctan 6ψ, g2 ϕ 1/16 · 3 100 1/30 arctan 7ϕ, α1 1/2, ξ1 1/4, N maxt∈0,1
1
Gt, sds 31/32, fi , gi : R1 → R1 are strictly increasing continuous functions, and condi0
tion H1 is satisfied. Choose k 100, l 5/12 π/24, D 5/16 π/60. Some direct calculations show
6
√
f1 ±100 ± l ≤ 10 5 · 3 100 5 π
12 24
3 · 104
32
· 100 N −1 k,
31
√
g1 ±100 ± D ≤ 100 4 · 5 100 5 π
3
16 60
<
< 34 4 ·
5 32
<
· 100 N −1 k,
2 31
√
f2 ±100 ≤ 1 · 5 100 1 arctan 6 · 100
6
12
Journal of Applied Mathematics
13
1 π
5
π
1 5
· · l,
6 2 12 2
12 24
√
g2 ±100 ≤ 1 · 3 100 1 arctan 7 · 100
16
30
<
<
1
1 π
5
π
·5
· D,
16
30 2
16 60
4.2
Therefore, condition H2 is satisfied.
Choosing c1 1/8, c2 −1, c3 1/4, c4 −1, it is easy to check that
1
1
31 5
π
5
15
−
Gt, sf1 c1 ds − Nl ≥
32 83 · 3 · 104 2
32 12 24
0
5
1
75
7
1
15 5
· −
−
> c1 ,
≥
32 2
12 6
64 12 8
1
−1
π
31 5
15
Gt, s g1 c2 ds ND ≤
−
4
32 3 · 104
32 16 60
0
15
6
3
· −4 − < −1 c2 ,
32
16
2
1
1/5 31 5
1
π
1
15
−
Gt, sg1 c3 ds − ND ≥
4
32 43 · 3 · 104
4
32 16 60
0
15
3 15
3 3
5 1
≥
·4
− ≥
·4· −
32
4 8 32
4 8
≤
23 1
> c3 ,
32 4
1
π
31 5
15
−1
Gt, sf1 c4 ds Nl ≤
−
5
32 3 · 104
32 12 24
0
≤
−75
7
15
7
< −1 c4 .
−5 32
12
32
12
4.3
Therefore, condition H3 is satisfied. At last, we will check condition H4 , by the method of
9, 31, and we consider the linear eigenvalue problem
u λu 0,
u 0 0,
0 < t < 1,
1
1
.
u1 u
2
4
4.4
Let Γs cos s−1/2 coss/4. By the paper 31, we know that the sequence of positive eigenvalue of 4.4 is exactly given by λn s2n , n 1, 2, . . ., where sn is the sequence of positive
14
Journal of Applied Mathematics
solutions of Γs 0. In 31, Han obtained that s1 1.0675, moreover λ1 s21 1.1396. It is
easy to know that
√
3
f1 x f2 y
x /30000 5x1/3 1/6 · 5 100 1/12 arctan 6y
lim
lim
|x| → ∞
|x| → ∞
x
x
4.5
∞ ≥ 2 · 1.1396 2λ1 ,
uniformly for y ∈ R,
√
3
g1 y g2 x
y /30000 4y1/5 1/16 · 3 100 1/30 arctan 7x
lim
lim
|y| → ∞
|y| → ∞
y
y
4.6
∞ ≥ 2 · 1.1396 2λ1 ,
uniformly for x ∈ R. Therefore, condition H4 is also satisfied. Consequently, all conditions
of Theorem 3.1 are satisfied, and we get the system of differential equations 4.1 has at least
six solutions in C0, 1 × C0, 1.
Acknowledgments
The authors are very grateful to the anonymous referees for their valuable suggestions. This
project is supported by the National Natural Science Foundation of China 10971046, the
University Science and Technology Foundation of Shandong Provincial Education Department J10LA62, the Natural Science Foundation of Shandong Province ZR2009AM004,
ZR2010AL014, and the Doctor of Scientific Startup Foundation for Shandong University of
Finance 08BSJJ32.
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