We study the existence of multiple positive solutions for nth-order multipoint boundary value problem.. We obtained the existence of multiple positive solutions by applying the fixed poi
Trang 1Volume 2010, Article ID 708376, 13 pages
doi:10.1155/2010/708376
Research Article
Multipoint Boundary Value Problem
Yaohong Li1, 2 and Zhongli Wei2, 3
1 Department of Mathematics, Suzhou University, Suzhou, Anhui 234000, China
2 School of Mathematics, Shandong University, Jinan, Shandong 250100, China
3 School of Sciences, Shandong Jianzhu University, Jinan, Shandong 250101, China
Correspondence should be addressed to Yaohong Li,liz.zhanghy@163.com
Received 22 January 2010; Revised 9 April 2010; Accepted 3 June 2010
Academic Editor: Ivan T Kiguradze
Copyrightq 2010 Y Li and Z Wei 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
We study the existence of multiple positive solutions for nth-order multipoint boundary value problem u n t atfut 0, t ∈ 0, 1, u j−1 0 0j 1, 2, , n − 1, u1 m
i1 αiuηi,
where n ≥ 2, 0 < η1< η2< · · · < ηm < 1, αi > 0, i 1, 2, , m We obtained the existence of multiple
positive solutions by applying the fixed point theorems of cone expansion and compression
of norm type and Leggett-Williams fixed-point theorem The results obtained in this paper are different from those in the literature
1 Introduction
The existence of positive solutions for nth-order multipoint boundary problems has been
studied by some authorssee 1,2 In 1, Pang et al studied the expression and properties
of Green’s funtion and obtained the existence of at least one positive solution for nth-order
differential equations by applying means of fixed point index theory:
u n t atfut 0, t ∈ 0, 1,
, u1 m
i1
α i u
η i
where n ≥ 2, 0 < η1 < η2< · · · < η m < 1, α i > 0, i 1, 2, , m.
By using the fixed point theorems of cone expansion and compression of norm type, Yang and Wei in2 also obtained the existence of at least one positive solutions for the BVP
Trang 2of1 In addition, Eloe and Ahmad in 4 had solved successfully the existence of positive solutions to the BVP1.1 if m 1 Hao et al in 5 had discussed the existence of at least two
on cone expansion and compression if m 1 However, there are few papers dealing with the existence of multiple positive solutions for nth-order multipoint boundary value problem.
In this paper, we study the existence of at least two positive solutions associated with
norm type if m ≥ 2 and the existence of at least three positive solutions for BVP 1.1 by using Leggett-Williams fixed-point theorem The results obtained in this paper are different from
1 8
The rest of the paper is organized as follows InSection 2, we present several lemmas
InSection 3, we give some preliminaries and the fixed point theorems of cone expansion and compression of norm type The existence of at least two positive solutions for the BVP1.1
theorem and obtain the existence of at least three positive solutions for the BVP1.1
2 Several Lemmas
Definition 2.1 A function ut is said to be a position of the BVP 1.1 if ut satisfies the
following:
1 ut ∈ C0, 1 ∩ C n 0, 1;
2 ut > 0 for t ∈ 0, 1 and satisfies boundary value conditions 1.1;
3 u n t −atfut hold for t ∈ 0, 1.
Lemma 2.2 see 1 Suppose that
D
m
i1
then for any y ∈ C0, 1, the problem
u n t yt 0, t ∈ 0, 1,
, u1 m
i1
α i u
η i
has a unique solution:
n − 1!
t 0
t − s n−1
n − 1!1 − D
1 0
1 − s n−1
y sds
−n − 1!1 − D t n−1 m−2
i1
α i
η i 0
η i − sn−1
y sds
1 0
K t, sysds,
2.3
Trang 3K t, s K1t, s K2t, s,
⎧
⎨
⎩
t n−1 1 − s n−1 − t − s n−1
, 0 ≤ s < t ≤ 1,
t n−1 1 − s n−1
n − 1!1 − D t n−1 1 − s n−1−
1
n − 1!1 − D
s≤η i
α i t n−1
η i − sn−1
.
2.4
Lemma 2.3 see 1 Let D < 1; Green’s function Kt, s defined by 2.4 satisfies
0≤ Kt, s ≤ Ks, ∀t, s ∈ 0, 1,
min
where γ η n−1
t∈0,1 K1t, s max
t∈0,1 K2t, s s n−1 1 − s n−1
n − 1! 1− 1 − s n−1/n−2 2−n
K21, s 2.6
We omit the proofLemma 2.3here and you can see the detail of Theorem 2.2 in 1
Lemma 2.4 see 2 Let D < 1, y ∈ C0, 1, and y ≥ 0; the unique solution ut of the BVP 2.2
satisfies
min
where γ is defined by Lemma 2.3 , u max t∈0,1 |ut|.
3 Preliminaries
In this section, we give some preliminaries for discussing the existence of multiple positive solutions of the BVP1.1 in the next In real Banach space C0, 1 in which the norm is defined
by
u max
set
P
u ∈ C 0, 1 | u0 0, ut > 0 for 0 < t ≤ 1, min
t∈η1,1u t ≥ γu . 3.2
Obviously, P is a positive cone in C0, 1, where γ is fromLemma 2.3
Trang 4For convenience, we make the following assumptions:
A1 a : 0, 1 → 0, ∞ is continuous and at does not vanish identically, for t ∈
η1 , 1;
A2 f : 0, ∞ → 0, ∞ is continuous;
i1 α i η n−1
i < 1.
Let
Aut
1 0
where Kt, s is defined by 2.4
From Lemmas2.2–2.4, we have the following result
Lemma 3.1 see 2 Suppose that A1–A3 are satisfied, then A : C0, 1 → C0, 1 is a
completely continuous operator, AP ⊂ P , and the fixed points of the operator A in P are the positive solutions of the BVP1.1.
For convenience, one introduces the following notations Let
r n − 1!1 − D1
1 0
1 − s n−1
a sds,
R γ
i2 α i
n − 1!1 − D
η i
η1
η i − η i sn−1
−η i − sn−1
3.4
Problem 1 Inspired by the work of the paper2, whether we can obtain a similar conclusion
or not, if
lim
u → 0inff u
u > R
u → ∞inff u
u > R
or
lim
u → 0supf u
u < r
u → ∞supf u
u < r
The aim of the following section is to establish some simple criteria for the existence of multiple positive solutions of the BVP1.1, which gives a positive answer to the questions stated above The key tool in our approach is the following fixed point theorem, which is
a useful method to prove the existence of positive solutions for differential equations, for example2 5,9
Lemma 3.2 see 10,11 Suppose that E is a real Banach space and P is cone in E, and let Ω1 , Ω2
be two bounded open sets in E such that 0 ∈ Ω1, Ω1⊂ Ω2 Let operator A : P ∩ Ω2\ Ω1 → P be
completely continuous Suppose that one of two conditions holds:
i Au ≤ u, for all u ∈ P ∩ ∂Ω1;Au ≥ u, for all u ∈ P ∩ ∂Ω2;
ii Au ≥ u, for all u ∈ P ∩ ∂Ω1; Au ≤ u, for all u ∈ P ∩ ∂Ω2
then A has at least one fixed point in P ∩ Ω2\ Ω1
Trang 54 The Existence of Two Positive Solutions
Theorem 4.1 Suppose that the conditions A1–A3 are satisfied and the following assumptions
hold:
B1 lim u → 0inffu/u > R−1;
B2 lim u → ∞inffu/u > R−1;
B3 There exists a constant ρ > 0 such that fu ≤ r−1ρ, u ∈ 0, ρ.
Then the BVP1.1 has at least two positive solutions u1 and u2such that
Proof At first, it follows from the condition B1 that we may choose ρ1 ∈ 0, ρ such that
SetΩ1 {u ∈ C0, 1 : u < ρ1}, and u ∈ P ∩ ∂Ω1; from3.3 and 2.4 andLemma 2.4, for
0 < t ≤ 1, we have
Au1 n − 1!1 − D1
1 0
D 1 − s n−1 a sfusds − m−2
i1
α i
η i 0
η i − sn−1
a sfusds
≥
i1 α i
n − 1!1 − D
η i 0
η i − η i sn−1
−η i − sn−1
a sfusds
> R
i1 α i
n − 1!1 − D
η i 0
η i − η i sn−1
−η i − sn−1
a susds
> R
i2 α i
n − 1!1 − D
η i
η1
η i − η i sn−1
−η i − sn−1
a susds
> R
i2 α i
n − 1!1 − D
η i
η1
η i − η i sn−1
−η i − sn−1
a sds
R−1R u u.
4.3 Therefore, we have
Further, it follows from the conditionB2 that there exists ρ2 > ρ such that
Trang 6Let ρ∗ max{2ρ, γ−1ρ2}, set Ω2 {u ∈ C0, 1 : u < ρ∗}, then u ∈ P ∩ ∂Ω2 andLemma 2.4
imply
min
and by the conditionB2, 2.4, 3.3, andLemma 2.4, we have
Au1 n − 1!1 − D1
1 0
D 1 − s n−1
a sfusds −m
i1
α i
η i 0
η i − sn−1
a sfusds
≥
i1 α i
n − 1!1 − D
η i 0
η i − η i sn−1−η i − sn−1
a sfusds
> R
i1 α i
n − 1!1 − D
η i 0
η i − η i sn−1
−η i − sn−1
a susds
> R
i2 α i
n − 1!1 − D
η i
η1
η i − η i sn−1
−η i − sn−1
a susds
> R
i2 α i
n − 1!1 − D
η i
η1
η i − η i sn−1
−η i − sn−1
a sds
R−1R u u.
4.7 Therefore, we have
Finally, letΩ3 {u ∈ C0, 1 : u < ρ} and u ∈ P ∩ ∂Ω3 By2.3, 3.3, and the condition
B3, we have
n − 1!1 − D
1 0
1 − s n−1
a sfusds
≤ n − 1!1 − D r−1ρ
1 0
1 − s n−1 a sds r−1rρ u,
4.9
which implies
Thus from4.4–4.10 and Lemmas3.1and3.2, A has a fixed point u1 in P ∩ Ω3\ Ω1 and a
fixed u2 in P ∩ Ω2\ Ω3 Both are positive solutions of BVP 1.1 and satisfy
The proof is complete
Trang 7Corollary 4.2 Suppose that the conditions A1–A3 are satisfied and the following assumptions
hold:
B
1 limu → 0inffu/u ∞;
B
2 limu → ∞inffu/u ∞;
B
3 there exists a constant ρ > 0 such that fu ≤ r−1ρ , u ∈ 0, ρ .
Then the BVP1.1 has at least two positive solutions u1 and u2such that
Proof From the conditions B
i i 1, 2, there exist sufficiently big positive constants M i i
1, 2 such that
lim
u → 0supf u
u > M2, u → ∞lim supf u
u > M1 4.13
by the conditionB
3; so all the conditions ofTheorem 4.1are satisfied; by an application of
Theorem 4.1, the BVP1.1 has two positive solutions u1 and u2such that
Theorem 4.3 Suppose that the conditions A1–A3 are satisfied and the following assumptions
hold:
C1 lim u → 0supfu/u < r−1;
C2 lim u → ∞supfu/u < r−1;
C3 there exists a constant l > 0 such that fu ≥ R−1l, u ∈ γl, l.
Then the BVP1.1 has at least two positive solutions u1 and u2such that
Proof It follows from the condition C1 that we may choose ρ3 ∈ 0, l such that
SetΩ4 {u ∈ C0, 1 : u < ρ3}, and u ∈ P ∩ ∂Ω4; from3.2 and 2.4, for 0 < t ≤ 1, we have
Au t ≤ n − 1!1 − D t n−1
1 0
1 − s n−1
a sfusds
< r−1u
n − 1!1 − D
1 0
1 − s n−1 a sds r−1r u u.
4.17
Trang 8Therefore, we have
It follows from the conditionC2 that there exists ρ4 > l such that fu < r−1u for u ≥ ρ4, and
we consider two cases
Case i Suppose that f is unbounded; there exists l∗> ρ4such that fu ≤ fl∗ for 0 < u ≤ l∗
Then for u ∈ P and u l∗, we have
n − 1!1 − D
1 0
1 − s n−1 a sfusds
n − 1!1 − D
1 0
1 − s n−1
a sfl∗ds
< r
n − 1!1 − D
1 0
1 − s n−1
a sds r−1rl∗ l∗ u.
4.19
Case ii If f is bounded, that is, fu ≤ N for all u ∈ 0, ∞, taking l∗ ≥ max{2l, Nr}, for
u ∈ P and u l∗, we have
Au t ≤ n − 1!1 − D t n−1
1 0
1 − s n−1 a sfusds
≤ n − 1!1 − D N
1 0
1 − s n−1 a sds ≤ Nr ≤ l∗ u.
4.20
Hence, in either case, we always may setΩ5 {u ∈ C0, 1 : u < l∗} such that
Finally, setΩ6 {u ∈ C0, 1 : u < l}; then u ∈ P ∩ ∂Ω6andLemma 2.4imply
min
Trang 9and by the conditionC3, 2.4, and 3.3, we have
Au1 n − 1!1 − D1
1 0
D 1 − s n−1
a sfusds −m
i1
α i
η i 0
η i − sn−1
a sfusds
≥
i1 α i
n − 1!1 − D
η i 0
η i − η i sn−1
−η i − sn−1
a sfusds
≥ R−1l
i2 α i
n − 1!1 − D
η i
η1
η i − η i sn−1−η i − sn−1
a sds
≥ R−1lγ
i2 α i
n − 1!1 − D
η i
η1
η i − η i sn−1
−η i − sn−1
a sds
R−1lR u.
4.23 Hence, we have
From4.18–4.24 and Lemmas3.1and3.2, A has a fixed point u1 in P ∩ Ω6\Ω4 and a fixed
u2in P ∩ Ω5\ Ω6 Both are positive solutions of the BVP1.1 and satisfy
The proof is complete
Corollary 4.4 Suppose that the conditions A1–A3 are satisfied and the following assumptions
hold:
C
1 limu → 0supfu/u 0;
C
2 limu → ∞supfu/u 0;
C
3 there exists a constant ρ > 0 such that fu ≥ R−1ρ , u ∈ γρ , ρ .
Then BVP1.1 has at least two positive solutions u1 and u2such that
The proof ofCorollary 4.4is similar to that ofCorollary 4.2; so we omit it
Trang 105 The Existence of Three Positive Solutions
Let E be a real Banach space with cone P A map β : P → 0, ∞ is said to be a nonnegative continuous concave functional on P if β is continuous and
β
tx 1 − ty≥ tβx 1 − tβy
5.1
for all x, y ∈ P and t ∈ 0, 1 Let a, b be two numbers such that 0 < a < b and let β be a nonnegative continuous concave functional on P We define the following convex sets:
P
β, a, b
x ∈ P : a ≤ β x, x ≤ b. 5.2
Lemma 5.1 see 12 Let A : P c → P c be completely continuous and let β be a nonnegative continuous concave functional on P such that βx ≤ x for x ∈ P c Suppose that there exist
0 < d < a < b ≤ c such that
i {x ∈ Pβ, a, b : βx > a} / ∅ and βAx > a for x ∈ Pβ, a, b,
ii Ax < d for x ≤ d,
iii βAx > a for x ∈ Pβ, a, c with Ax > b.
Then A has at least three fixed points x1, x2, x3in P c such that
Now, we establish the existence conditions of three positive solutions for the BVP1.1
Theorem 5.2 Suppose that A1–A3 hold and there exist numbers a and d with 0 < d < a such
that the following conditions are satisfied:
D1 lim u → ∞ fu/u < 1/G,
D2 fu < d/G, u ∈ 0, d,
D3 fu > a/F, u ∈ a, a/γ,
where
F min
t∈η1,1
1
η1
t∈0,1
1 0
Then the boundary value problem1.1 has at least three positive solutions.
Proof Let P be defined by 3.2 and let A be defined by 3.3 For u ∈ P, let
Trang 11Then it is easy to check that β is a nonnegative continuous concave functional on P with
βu ≤ u for u ∈ P and A : P → P is completely continuous.
First, we prove that ifD1 holds, then there exists a number c > a/γ and A : P c → P c
To do this, byD1, there exist M > 0 and λ < 1/G such that
Set
δ max
it follows that fu < λu δ for all u ∈ 0, ∞ Take
c > max
δG
1− λG ,
a γ
If u ∈ P c, then
Aut ≤ max
1 0
K t, sasfusds < max
t∈0,1
1 0
K t, sasdsλu δ < λc δG < c,
5.9 that is,
Hence5.10 show that if D1 holds, then there exists a number c > a/γ such that A maps
P c into P c
P β, a, a/γ In fact, take xt ≡ a a/γ/2 > a, so x ∈ {u ∈ P β, a, a/γ : βu > a}.
Moreover, for u ∈ P β, a, a/γ, then βu > a, and we have
a
t∈η1,1
1 0
K t, sasfusds > a
F t∈ηmin1,1
1
η1
Next, we assert thatAu < d for u ≤ d In fact, if u ∈ P d, byD2 we have
Au < d
G
max
1 0
K t, sasds
Hence, A : P d → P d for u ∈ P d
Trang 12Finally, we assert that if u ∈ P β, a, c and Au > a/γ, then βAu > a To see this, if
u ∈ P β, a, c and Au > a/γ,then we have fromLemma 2.3that
t∈η1,1
1 0
K t, sasfusds
≥
1
0
min
t∈η1,1K t, sasfusds ≥ γ
1 0
≥ γ
1
0
max
t∈0,1
1 0
K t, sasfusds γAu.
5.14
So we have
β Au ≥ γAu > γ · a
To sum up5.10∼5.15, all the conditions of Lemma 5.1are satisfied by taking b a/γ.
Hence, A has at least three fixed points; that is, BVP1.1 has at least three positive solutions
The proof is complete
Acknowledgments
The authors are grateful to the referee’s valuable comments and suggestions The project is
Youth Foundation of Anhui Province Office of Education 2009SQRZ169, and the Natural
References
1 C Pang, W Dong, and Z Wei, “Green’s function and positive solutions of nth order m-point boundary value problem,” Applied Mathematics and Computation, vol 182, no 2, pp 1231–1239, 2006.
2 J Yang and Z Wei, “Positive solutions of nth order m-point boundary value problem,” Applied
Mathematics and Computation, vol 202, no 2, pp 715–720, 2008.
3 R Ma, “Positive solutions of a nonlinear three-point boundary-value problem,” Electronic Journal of
Di fferential Equations, vol 1999, no 34, pp 1–8, 1999.
4 P W Eloe and B Ahmad, “Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions,” Applied Mathematics Letters, vol 18, no 5, pp 521–527, 2005.
5 X Hao, L Liu, and Y Wu, “Positive solutions for nonlinear nth-order singular nonlocal boundary value problems,” Boundary Value Problems, vol 2007, Article ID 74517, 10 pages, 2007.
6 R Ma, “Existence of positive solutions for second order m-point boundary value problems,” Annales
Polonici Mathematici, vol 79, no 3, pp 265–276, 2002.
7 R Ma and N Castaneda, “Existence of solutions of nonlinear m-point boundary-value problems,”
Journal of Mathematical Analysis and Applications, vol 256, no 2, pp 556–567, 2001.
... nonlinear nth- order singular nonlocal boundary value problems,” Boundary Value Problems, vol 2007, Article ID 74517, 10 pages, 2007.6 R Ma, “Existence of positive solutions for second... ∩ Ω2\ Ω1
Trang 54 The Existence of Two Positive Solutions< /b>
Theorem... Ω3 Both are positive solutions of BVP 1.1 and satisfy
The proof is complete
Trang 7Corollary