Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 986575, 20 pages
doi:10.1155/2011/986575
Research Article
Existence of Positive Solutions for m-Point
Boundary Value Problem for Nonlinear Fractional
Differential Equation
Moustafa El-Shahed1 and Wafa M. Shammakh2
1
Department of Mathematics, Faculty of Arts and Sciences, College of Education, P.O. Box 3771,
Qassim-Unizah 51911, Saudi Arabia
2
Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, P.O. Box 30903,
Jeddah 21487, Saudi Arabia
Correspondence should be addressed to Wafa M. Shammakh, wfaa20053@hotmail.com
Received 23 January 2011; Revised 2 April 2011; Accepted 17 April 2011
Academic Editor: Paul Eloe
Copyright q 2011 M. El-Shahed and W. M. Shammakh. 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 investigate an m-point boundary value problem for nonlinear fractional differential equations.
The associated Green function for the boundary value problem is given at first, and some useful
properties of the Green function are obtained. By using the fixed point theorems of cone expansion
and compression of norm type and Leggett-Williams fixed point theorem, the existence of multiple
positive solutions is obtained.
1. Introduction
In recent years, the existence of positive solutions multipoint boundary value problems
of fractional order differential equations has been studied by many authors using various
methods see 1–7.
The study of multipoint boundary value problems for linear second-order ordinary
differential equations was initiated by II’in and Moiseev 8, 9.
Since then, nonlinear multipoint boundary value problems have been studied by
several authors see 10–14. Recently, in 15, the authors have studied the existence of at
least one positive solution for the following nth-order three-point boundary value problem:
un t htft, ut 0,
ua αu η ,
t ∈ a, b,
u a u a · · · un−2 a 0,
ub βu η ,
1.1
2
Abstract and Applied Analysis
where a < η < b, 0 ≤ α < 1, f ∈ Ca, b × 0, ∞, 0, ∞ and h ∈ Ca, b × 0, ∞ may be
singular at t a and t b.
Goodrich 16 considered the BVP for thehigher-dimensional fractional differential
equation as follows:
−D0ν yt f t, yt ,
0 < t < 1, n − 1 < ν ≤ n,
yi 0 0, 0 ≤ i ≤ n − 2,
α
1 ≤ α ≤ n − 2,
D0 yt t1 0,
1.2
and a Harnack-like inequality associated with the Green function related to the above
problem is obtained improving the results in 17.
Motivated by the aforementioned results and techniques in coping with those
boundary value problems of fractional differential equations, we then turn to investigate the
existence and multiplicity of positive solutions for the following BVP:
C
u a Daα ut ft, ut 0,
m−2
βi u ηi ,
1.3
a ≤ t ≤ b, n − 1 ≤ α < n, n > 2,
u a u a · · · un−1 a 0,
i1
ub m−2
γi u ηi ,
1.4
i1
m−2
C α
where a < η1 < η2 < · · · < ηm−2 < b, m−2
i1 βi < 1,
i1 γi < 1 and Da are the Caputo
fractional derivative.
In this paper, we study the existence of at least one positive solution, existence of two
positive solutions associated with the BVP 1.3-1.4 by applying the fixed point theorems
of cone expansion and compression of norm type, and the existence of at least three positive
solutions for BVP 1.3-1.4 by using Leggett-Williams fixed point theorem.
The rest of the paper is organized as follows. In Section 2, we introduce some basic
definitions and preliminaries used later. In Section 3, the existence of multipoint boundary
value problem 1.3-1.4 will be discussed.
2. Preliminaries
In this section, we introduce definitions and preliminary facts which are used throughout this
paper.
Definition 2.1 see 18. For a function y : a, ∞ → R, the Caputo derivative of fractional
order α > 0 is defined as
C
α
Da yt
1
Γn − α
t
a
t − sn−α−1 yn sds,
n − 1 < α ≤ n.
2.1
Abstract and Applied Analysis
3
Definition 2.2 see 18. The standard Riemann-Liouville fractional derivative of order α > 0
of a continuous function y : a, ∞ → R is given by
Daα yt
n t
d
1
t − sn−α−1 ysds,
Γn − α dt
a
2.2
where n α 1, provided that the integral on the right-hand side converges.
Definition 2.3 see 18. The Riemann-Liouville fractional integral of order α > 0 of a function
y : a, ∞ → R is given by
Iaα yt 1
Γα
t
t − sα−1 ysds
2.3
a
provided that the integral on the right-hand side converges.
Definition 2.4 see 19. Let E be a real Banach space. A nonempty closed convex set K ⊂ E
is called cone of E if it satisfies the following conditions:
1 x ∈ K, σ ≥ 0 implies σx ∈ K;
2 x ∈ K, −x ∈ K implies x 0.
Definition 2.5. An operator is called completely continuous if it is continuous and maps
bounded sets into precompact sets.
Theorem 2.6 see 20. Let E be a Banach space and K ⊂ E is a cone in E. Assume that Ω1 and
Ω2 are open subsets of E with 0 ∈ Ω1 and Ω1 ⊂ Ω2 . Let T : K ∩ Ω2 \ Ω1 → K be completely
continuous operator. In addition, suppose either
i T u
≤ u
, for all u ∈ K ∩ ∂Ω1 , and T u
≥ u
, for all u ∈ K ∩ ∂Ω2 or
ii T u
≤ u
, for all u ∈ K ∩ ∂Ω2 , and T u
≥ u
, for all u ∈ K ∩ ∂Ω1
holds. Then, T has a fixed point in K ∩ Ω2 \ Ω1 .
Lemma 2.7. For α > 0, the general solution of the fractional differential equation
given by
ut c0 c1 t − a c2 t − a2 · · · cn−1 t − an−1 ,
where ci ∈ R, i 0, 1, 2, . . . , n − 1.
C
α
Da ut 0 is
2.4
4
Abstract and Applied Analysis
Remark 2.8 see 18. In view of Lemma 2.7, it follows that
α
Iaα C Da ut ut c0 c1 t − a c2 t − a2 · · · cn−1 t − an−1 ,
2.5
for some ci ∈ R, i 0, 1, 2, . . . , n − 1.
Definition 2.9. The map θ is said to be a nonnegative continuous concave functional on a cone
P of a real Banach space E provided that θ : P → 0, ∞ is continuous and
θ λx 1 − λy ≥ λθx 1 − λθ y ,
2.6
for all x, y ∈ P, 0 ≤ λ ≤ 1.
Lemma 2.10 see 21. Let P be a cone in a real Banach space E, Pc {x ∈ P : x
< c}, θ
is a nonnegative continuous concave functional on P such that θx ≤ x
, for all x ∈ Pc , and
P θ, b, d {x ∈ P : b ≤ θx, x ≤ d}. Suppose that T : Pc → Pc is completely continuous and
there exist positive constants 0 < a < b < d ≤ c such that
C1 {x ∈ P θ, b, d : θx > b} /
φ and θx > b for x ∈ P θ, b, d,
C2 T x
< a for x ∈ Pa ,
C3 θT x > b for x ∈ P θ, b, d with T x
> d,
then T has at least three fixed points x1 , x2 , and x3 with
x1 < a,
b < θx2 ,
a < x3 2.7
with θx3 < b.
Lemma 2.11. For a given yt ∈ Ca, b and n − 1 ≤ α < n, the unique solution of the boundary
value problem
C
u a α
Da ut yt 0,
m−2
βi u ηi ,
2.8
a ≤ t ≤ b, n − 1 ≤ α < n, n > 2, n ∈ N,
u a u a · · · un−1 a 0,
ub i1
m−2
γi u ηi ,
2.9
i1
is given by
ut b
a
Gt, sysds b
a
H t, s; η1 , . . . , ηm−2 ysds,
2.10
Abstract and Applied Analysis
5
where
⎧
α−1
α−1
1 ⎨b − s − t − s ,
Gt, s Γα ⎩b − sα−1 ,
a ≤ s ≤ t ≤ b,
a ≤ t ≤ s ≤ b,
⎧
α−1
α−1
m−2
⎪
⎪
⎪ i1 γi b − s − ηi − s
⎪
⎪
⎪
⎪
δ2 Γα
⎪
⎪
⎪
⎪
⎨
α−2
μt m−2
i1 βi ηi − s
H t, s; η1 , . . . , ηm−2 , a ≤ s ≤ ηi , i 1, 2, . . . , m − 2,
⎪
⎪
δ1 δ2 Γα − 1
⎪
⎪
⎪
⎪
m−2
⎪
⎪
⎪
γi b − sα−1
⎪
⎪
⎩ i1
, ηi ≤ s ≤ b, i 1, 2, . . . , m − 2,
δ2 Γα
m−2
m−2
m−2
δ1 1 −
βi ,
δ2 1 −
γi ,
μt b −
γi ηi − δ2 t.
i1
i1
i1
2.11
Proof. Using Remark 2.8, for arbitrary constants ci ∈ R, i 0, 1, 2, . . . , n − 1, we have
−1
ut Γα
t
t − sα−1 ysds c0 c1 t − a c2 t − a2 · · · cn−1 t − an−1
a
2
n−1
−Iaα yt c0 c1 t − a c2 t − a · · · cn−1 t − a
β
2.12
.
αβ
In view of the relations C Daα Iaα ut ut and Iaα Ia ut Ia ut for α, β > 0, we obtain
n−2
,
u t −Iaα−1
yt c1 2c2 t − a · · · n − 1cn−1 t − a
n−3
u t −Iaα−2
,
yt 2c2 · · · n − 1n − 2cn−1 t − a
..
.
2.13
un−1 t −Iaα−n1
yt n − 1!cn−1 .
Applying the boundary conditions 2.9, we find that
c2 c3 · · · cn−1 0,
m−2
m−2
1−
βi c1 − βi Iaα−1
y ηi ,
i1
i1
2.14
6
Abstract and Applied Analysis
then c1 −
m−2
i1
βi Iaα−1
yηi /δ1 , and
m−2 α−1 m−2 α γi ηi − a
b − a − m−2
i1
i1 βi Ia y ηi
γ
I
y
η
Iaα yb
i
i
i1
a
−
c0 1 − m−2
1 − m−2
1 − m−2
1 − m−2
i1 γi
i1 γi
i1 βi
i1 γi
m−2
I α yb
a
−
δ2
i1
m−2 α−1 b − a − m−2
i1 γi ηi − a
i1 βi Ia y ηi
γi Iaα y ηi
.
δ2
δ1 δ2
2.15
Substituting the values of the constants ci , i 0, 1, 2, . . . , n − 1, in 2.12, we obtain
ut −Iaα yt m−2
i1
−
Iaα yb
δ2
m−2
−
I α yb
a
−
δ2
−Iaα yt Iaα yb −
a
−
−
i1
a
γi
δ2
t
δ2
m−2
i1
i1
γi
i1
δ2
η i
a
γi
b
a
m−2
i1
γi Iaα y ηi
μt m−2
βi I α−1
y ηi
δ2
δ1 δ2 i1 a
b − sα−1
ysds Γα
m−2
i1
δ2
γi
b
a
b − sα−1
ysds
Γα
α−1
ηi ηi − s α−2
ηi − s
μt m−2
ysds ysds
βi
Γα
δ1 δ2 i1
Γα − 1
a
a
δ2
m−2
δ1 δ2
ηi t − sα−1
ysds Γα
m−2
m−2 α−1 γ
−
a
η
b − a − m−2
i
i
i1
i1 βi Ia y ηi
γi Iaα y ηi
μt m−2
βi Iaα−1
y ηi
δ2
δ1 δ2 i1
1 − δ2 α
Ia yb −
δ2
t − sα−1
ysds Γα
m−2
ηi
βi Iaα−1
y ηi
t − a
δ1
−Iaα yt
t
α
i1 γi Ia y
b
ηi
b
a
b − sα−1
ysds
Γα
α−1 b − sα−1 − ηi − s
ysds
Γα
μt m−2
b − sα−1
ysds βi
Γα
δ1 δ2 i1
Lemma 2.12. μt b −
m−2
i1
α−2
ηi − s
ysds.
Γα − 1
ηi a
γi ηi − δ2 t ≥ 0, for t, ηi ∈ a, b, i 1, 2, . . . , m − 2.
2.16
Abstract and Applied Analysis
7
Proof. We have
μ
b−
m−2
− δ2 t
γi ηi
i1
≥
b−
m−2
γi ηi
−
1−
i1
≥
m−2
2.17
γi b
i1
m−2
γi b − ηi > 0.
i1
Lemma 2.13. The functions Gt, s, Ht, s; η1 , η2 , . . . , ηm−2 defined by 2.11 satisfy
i Gt, s ≥ 0, Ht, s; η1 , η2 , . . . , ηm−2 ≥ 0, for all t, s ∈ a, b,
ii minτ1 ≤t≤τ2 Gt, s ≥ τ0 maxa≤t≤b Gt, s τ0 Gs, s, for all t, s ∈ a, b, a < τ1 < τ2 <
b, τ0 minτ1 ≤t≤τ2 ϕt b − τ2 /b − a,
iii N2 qs ≤ Ht, s; η1 , η2 , . . . , ηm−2 ≤ N1 qs, where
b − sα−2
,
qs δ1 δ2 Γα
N1 α − 1 δ1 b − a
m−2
m−2
i1
i1
γi b
βi ,
m−2
δ1 m−2
i1 γi
N2 min
, δ1 γi b − ηi ,
α − 1
i1
2.18
iv minτ1 ≤t≤τ2 Ht, s; η1 , η2 , . . . , ηm−2 ≥ τ ∗ maxa≤t≤b Ht, s; η1 , η2 , . . . , ηm−2 , s ∈ a, b,
where
b − m−2
i1 γi ηi − δ2 τ2
τ ∗ < 1,
b − m−2
i1 γi ηi − δ2 a
a < τ1 < τ2 < b.
2.19
Proof. It is clear that i holds. So, we prove that ii is true.
ii For α > 1, in view of the expression for Gt, s, it follows that Gt, s ≤ Gs, s for
all s, t ∈ a, b, where Gs, s b − sα−1 /Γα.
8
Abstract and Applied Analysis
If a ≤ s ≤ t ≤ b, we have
b − sα−1 − t − sα−1
Gt, s
Gs, s
b − sα−1
b − sα−2 b − s − t − sα−2 t − s
b − sα−1
≥
2.20
b − sα−2 b − s − t − s
b − sα−1
b − t
: ϕt.
b − a
If a ≤ t ≤ s ≤ b, then we have
Gt, s
b − t
1≥
: ϕt.
Gs, s
b − a
2.21
Thus
max Gt, s Gs, s,
a≤t≤b
ϕtGs, s ≤ Gt, s ≤ Gs, s,
∀t, s ∈ a, b.
2.22
Therefore,
min Gt, s ≥ τ0 max Gt, s τ0 Gs, s,
τ1 ≤t≤τ2
a≤t≤b
∀t, s ∈ a, b, a < τ1 < τ2 < b.
2.23
iii If a ≤ s ≤ ηi , i 1, 2, . . . , m − 2, then
H t, s; η1 , η2 , . . . , ηm−2 ≤
≤
≤
α−1 m−2 α−1
γ
−
η
−
s
−
s
b
i
i
i1
α−2
βi ηi − s
δ2 Γα
δ1 δ2 Γα − 1
α−2
m−2
α−1
α − 1b m−2
i1 βi ηi − s
i1 γi b − s
δ2 Γα
δ1 δ2 Γα
m−2
m−2
α−2
1
α−1
δ1 γi b − s α − 1b βi ηi − s
δ1 δ2 Γα
i1
i1
m−2
m−2
α − 1b − sα−2
δ1 γi b − s b βi
δ1 δ2 Γα
i1
i1
m−2
m−2
α − 1b − sα−2
δ1 b − a γi b βi : N1 qs,
δ1 δ2 Γα
i1
i1
μt
m−2
i1
Abstract and Applied Analysis
H t, s; η1 , η2 , . . . , ηm−2 ≥
9
α−1 m−2 α−1
− ηi − s
i1 γi b − s
δ2 Γα
i1
α−2
βi ηi − s
δ1 δ2 Γα − 1
m−2
α−1 m−2 α−1
− ηi − s
i1 γi b − s
m−2
μt
i1
δ2 Γα
α−2 γi b − sα−2 b − s − ηi − s
ηi − s
δ2 Γα
b − sα−2 m−2
γi b − ηi ≥ N2 qs.
δ2 Γα i1
2.24
If ηi ≤ s ≤ b, i 1, 2, . . . , m − 2, then we have
H t, s; η1 , η2 , . . . , ηm−2 m−2
δ1
γi b − sα−1
δ2 Γα
i1
m−2
γi b − s
i1
≤ δ1
b − sα−2
δ1 δ2 Γα
m−2
γi b − aqs
i1
2.25
< N1 qs,
m−2
γi α − 1b − sα−1
H t, s; η1 , η2 , ..., ηm−2 i1
α − 1δ2 Γα
m−2
γi b − sα−2
≥ i1
α − 1δ2 Γα
δ1 m−2
i1 γi
≥
qs ≥ N2 qs.
α − 1
iv Since ∂Ht, s; η1 , η2 , . . . , ηm−2 /∂t −
Ht, s; η1 , η2 , . . . , ηm−2 is nonincreasing in t, so
m−2
i1
βi ηi − sα−2 /δ1 Γα − 1 ≤ 0, then
max H t, s; η1 , η2 , . . . , ηm−2 H a, s; η1 , η2 , . . . , ηm−2
a≤t≤b
α−1 m−2 α−1
− ηi − s
i1 γi b − s
δ2 Γα
α−2
m−2 b − m−2
−
δ
γ
η
a
i
i
2
i1
i1 βi ηi − s
δ1 δ2 Γα − 1
,
10
min H t, s; η1 , η2 , . . . , ηm−2 Abstract and Applied Analysis
α−1 m−2 α−1
− ηi − s
i1 γi b − s
δ2 Γα
α−2
m−2 i1 γi ηi − δ2 τ2
i1 βi ηi − s
τ1 ≤t≤τ2
b−
m−2
δ1 δ2 Γα − 1
α−1 m−2
α−1
γ
−
η
−
s
−
s
b
i
i
i1
δ2 Γα
α−2
m−2 b − m−2
−
δ
γ
η
a
i
i
2
i1
i1 βi ηi − s
τ∗
δ1 δ2 Γα − 1
> τ ∗ max H t; η1 , η2 , . . . , ηm−2 , s .
a≤t≤b
2.26
3. Main Results
Let us denote by E Ca, b the Banach space of all continuous real functions on a, b
endowed with the norm u
maxa≤t≤b |ut| and P the cone
P
u ∈ E : u ≥ 0, min ut ≥ τ
u
, t ∈ a, b ,
3.1
τ1 ≤t≤τ2
where τ min{τ0 , τ ∗ }, since τ0 , τ ∗ are constants do not depend on t.
Let the nonnegative continuous concave functional θ on the cone P be defined by
θu minτ1 ≤t≤τ2 ut.
Set T : P → E by
T ut b
Gt, s H t, s; η1 , η2 , . . . , ηm−2 fs, usds,
a ≤ t ≤ b,
3.2
a
where Gt, s, Ht, s; η1 , η2 , . . . , ηm−2 are defined as in Lemma 2.11.
From 3.2 and Lemma 2.13, we have
min T ut ≥
τ1 ≤t≤τ2
b
τ0 Gs, s τ ∗ max H t, s; η1 , . . . , ηm−2 fs, usds
a
a≤t≤b
3.3
≥ τ
T u
.
Hence, we have T P ⊂ P .
By standard argument, one can prove that T : P → P is a completely continuous
operator.
Abstract and Applied Analysis
11
The Existence of One Positive Solution
We introduce the following definitions:
fu : sup ft, u,
fu : inf ft, u,
t∈a,b
t∈a,b
f 0 lim supu → 0
fu
,
u
fu
,
f lim supu → ∞
u
b
−1
M
,
Gs, s N1 qs ds
∞
f0 lim infu → 0
fu
f∞ lim infu → ∞
N
τ
2
a
u
,
fu
u
3.4
,
−1
τ Gs, s N2 qs ds
.
τ1
Theorem 3.1. Let ft, u be continuous on a, b × 0, ∞ → 0, ∞. If there exist two positive
constants r2 > r1 > 0 such that
H1 ft, u ≤ Mr2 , for t, u ∈ a, b × 0, r2 ,
H2 ft, u ≥ Nr1 , for t, u ∈ a, b × 0, r1 ,
then the BVP 1.3-1.4 has at least a positive solution.
Proof. We know that the operator T : P → P defined by 3.2 is completely continuous.
a Let Ω2 {u ∈ E : u
< r2 }. For any u ∈ P ∩ ∂Ω2 , we have u
r2 which implies
that 0 ≤ ut ≤ r2 for every t ∈ a, b:
T ut b
Gt, s H t, s; η1 , . . . , ηm−2 fs, usds
a
≤ Mr2
b
Gs, s N1 qs ds
3.5
a
≤ r2 u
,
which implies that
T u
≤ u
,
u ∈ P ∩ ∂Ω2 .
3.6
12
Abstract and Applied Analysis
b Let Ω1 {u ∈ E : u
< r1 }. For any t ∈ τ1 , τ2 , u ∈ P ∩ ∂Ω1 . We have
T ut b
Gt, s H t, s; η1 , . . . , ηm−2 fs, usds
a
≥
b
ϕt Gs, s N2 qs fs, usds
a
≥ Nr1
τ2
3.7
τ Gs, s N2 qs ds
τ1
r1 u
,
which implies that
T u
≥ u
,
u ∈ P ∩ ∂Ω1 .
3.8
In view of Theorem 2.6, T has a fixed point u0 ∈ P ∩ Ω2 \ Ω1 which is a solution of the BVP
1.3-1.4.
The Existence of Two Positive Solutions
Theorem 3.2. Assume that all assumptions of Theorem 3.1, hold. Moreover, one assumes that ft, u
also satisfies
H3 f∞ ∞.
Then, the BVP 1.3-1.4 has at least two positive solutions.
Proof. At first, it follows from condition H1 that
T u
≤ u
,
u ∈ P ∩ ∂Ω2 .
3.9
u ∈ P ∩ ∂Ω1 .
3.10
Further, it follows from condition H2 that
T u
≥ u
,
Finally, since f∞ ∞, there exists ψ > τ 2
τ2
τ1
ft, u ≥ ψut,
−1
Gs, s N2 qsds
t ∈ a, b, u ≥ r3 .
and r3 > r2 such that
3.11
Abstract and Applied Analysis
13
Let r ∗ max{2r2 , τ −1 r3 } and set Ω3 {u ∈ E : u
< r ∗ }, then u ∈ P ∩ ∂Ω3 implies
minτ1 ≤t≤τ2 ut ≥ τ
u
≥ τr ∗ ≥ r3 ,
T ut b
Gt, s H t, s; η1 , η2 , . . . , ηm−2 fs, usds
a
≥
b
ϕt Gs, s N2 qs fs, usds
a
≥ψ
b
ϕt Gs, s N2 qs usds
3.12
a
≥ r∗ψ
τ2
τ 2 Gs, s N2 qs ds > r ∗ u
.
τ1
Therefore, we have
T u
≥ u
,
u ∈ P ∩ ∂Ω3 .
3.13
Thus, from 3.6, 3.8, 3.13, and Theorem 2.6, T has a fixed point u1 , in P ∩ Ω2 \ Ω1 and a
fixed point u2 , in P ∩ Ω3 \ Ω2 . Both are positive solutions of BVP 1.3-1.4 and satisfy
0 < u1 < r2 < u2 .
3.14
Theorem 3.3. Assume that ft, u be continuous on a, b × 0, ∞ → 0, ∞. If the following
assumptions hold:
H1 f0 > ψ,
H2 f∞ > ψ,
H3 there exists a constant ρ > 0 such that ft, u ≤ ρM, t, u ∈ a, b × 0, ρ,
then the BVP 1.3-1.4 has at least two positive solutions u1 and u2 such that
0 < u1 < ρ < u2 .
3.15
Proof. At first, it follows from condition H1 that we may choose ρ1 ∈ 0, ρ such that
ft, u > ψu,
0 < u ≤ ρ1 ,
3.16
14
Abstract and Applied Analysis
where ψ is defined as in Theorem 3.2. Set Ω1 {u ∈ E : u
< ρ1 }, and u ∈ P ∩ ∂Ω1 ; from 3.2
and Lemma 2.13, for a ≤ t ≤ b, we have
T ut b
Gt, s H t, s; η1 , η2 , . . . , ηm−2 fs, usds
a
≥
b
ϕt Gs, s N2 qs fs, usds
a
≥ψ
b
ϕt Gs, s N2 qs usds
3.17
a
≥ ρ1 ψ
τ2
τ 2 Gs, s N2 qs ds
τ1
> ρ1 u
.
Therefore, we have
T u
≥ u
,
u ∈ P ∩ ∂Ω1 .
3.18
Further, it follows from condition H2 that there exists ρ2 > ρ such that
ft, u > ψut,
u ≥ ρ2 .
3.19
Let ρ∗ max{2ρ, τ −1 ρ2 }, set Ω2 {u ∈ E : u
< ρ∗ }, then u ∈ P ∩ ∂Ω2 implies minτ1 ≤t≤τ2 ut ≥
τ
u
≥ τρ∗ ≥ ρ2 ,
T ut b
Gt, s H t, s; η1 , η2 , . . . , ηm−2 fs, usds
a
≥
b
ϕt Gs, s N2 qs fs, usds
a
≥ψ
b
ϕt Gs, s N2 qs usds
3.20
a
≥ ρ∗ ψ
τ2
τ 2 Gs, s N2 qs ds > ρ∗ u
.
τ1
Therefore, we have
T u
≥ u
,
u ∈ P ∩ ∂Ω2 .
3.21
Abstract and Applied Analysis
15
Finally, let Ω3 {u ∈ E : u
< ρ} and u ∈ P ∩ ∂Ω3 . By condition H3 , we have
T ut ≤
b
Gs, s N1 qs fs, usds
a
≤ Mρ
b
Gs, s N1 qs ds
3.22
a
ρ u
,
which implies
T u
≤ u
,
u ∈ P ∩ ∂Ω3 .
3.23
Thus, from 3.18, 3.21, 3.23, and Theorem 2.6, T has a fixed point u1 in P ∩ Ω3 \ Ω1 and
a fixed point u2 , in P ∩ Ω2 \ Ω3 . Both are positive solutions of BVP 1.3-1.4 and satisfy
0 < u1 < ρ < u2 .
3.24
Theorem 3.4. Assume that ft, u be continuous on a, b × 0, ∞ → 0, ∞. If the following
assumptions hold:
H1 f0 ∞,
H2 f∞ ∞,
H3 there exists a constant ρ > 0 such that ft, u ≤ ρ M, t, u ∈ a, b × 0, ρ ,
then the BVP 1.3-1.4 has at least two positive solutions u1 and u2 such that
0 < u1 < ρ < u2 .
3.25
The proof of Theorem 3.4 is very similar to that of Theorem 3.3 and therefore is omitted.
Theorem 3.5. Assume that ft, u be continuous on a, b × 0, ∞ → 0, ∞. If the following
assumptions hold:
H1 f 0 < M,
H2 f ∞ < M,
H3 there exists a constant l > 0 such that ft, u ≥ Nl, t, u ∈ a, b × τl, l,
then the BVP 1.3-1.4 has at least two positive solutions u1 and u2 such that
0 < u1 < l < u2 .
3.26
Proof. It follows from condition H1 that we may choose ρ3 ∈ 0, l such that
ft, u < Mu,
0 < u ≤ ρ3 .
3.27
16
Abstract and Applied Analysis
Set Ω4 {u ∈ E : u
< ρ3 }, and u ∈ P ∩ ∂Ω4 ; from 3.2 and Lemma 2.13, for a ≤ t ≤ b, we
have
T ut b
Gt, s H t, s; η1 , η2 , . . . , ηm−2 fs, usds
a
b
<M
Gs, s N1 qs ds · u
M · M−1 u
u
.
3.28
a
Therefore, we have
T u
≤ u
,
u ∈ P ∩ ∂Ω4 .
3.29
It follows from condition H2 that there exists ρ4 > l such that
ft, u < Mu,
u ≥ ρ4 ,
3.30
and we consider two cases.
Case 1. Suppose that f is unbounded, there exists l∗ > ρ4 such that ft, u ≤ ft, l∗ for 0 < u ≤ l∗ .
Then, for u ∈ P and u
l∗ , we have
T ut b
Gt, s H t, s; η1 , η2 , . . . , ηm−2 fs, usds
Gs, s N1 qs fs, l∗ ds
a
≤
b
a
< Ml
∗
b
3.31
Gs, s N1 qs ds l∗ u
.
a
Case 2. If f is bounded, that is, ft, u ≤ k for all u ∈ 0, ∞, taking l∗ ≥ max{2l, kM−1 }, for
u ∈ P and u
l∗ , then we have
T ut b
Gt, s H t, s; η1 , η2 , . . . , ηm−2 fs, usds
a
≤k
b
Gs, s N1 qs ds kM−1 ≤ l∗ u
.
3.32
a
Hence, in either case, we always may set Ω5 {u ∈ E : u
< l∗ } such that
T u
≤ u
,
u ∈ P ∩ ∂Ω5 .
3.33
Abstract and Applied Analysis
17
Finally, set Ω6 {u ∈ E : u
< l}, then u ∈ P ∩ ∂Ω6 and
min ut ≥ τ
u
τl,
τ1 ≤t≤τ2
3.34
and by condition H3 and 3.2, we have
T ut b
a
≥ Nl
≥ Nl
Gt, s H t, s; η1 , η2 , . . . , ηm−2 fs, usds
b
a
τ2
ϕt Gs, s N2 qs ds
3.35
τ Gs, s N2 qs ds l u
.
τ1
Hence, we have
T u
≥ u
,
u ∈ P ∩ ∂Ω6 .
3.36
Thus, from 3.29, 3.33, 3.36 and Theorem 2.6, T has a fixed point u1 in P ∩ Ω6 \ Ω4 and
a fixed point u2 in P ∩ Ω5 \ Ω6 . Both are positive solutions of BVP 1.3-1.4 and satisfy
0 < u1 < l < u2 .
3.37
Theorem 3.6. Assume that ft, u be continuous on a, b × 0, ∞ → 0, ∞. If the following
assumptions hold:
H1 f 0 0,
H2 f ∞ 0,
H3 there exists a constant ρ > 0 such that ft, u ≥ Nρ , t, u ∈ a, b × τρ , ρ ,
then the BVP 1.3-1.4 has at least two positive solutions u1 and u2 such that
0 < u1 < ρ < u2 .
3.38
The proof of Theorem 3.6 is very similar to that of Theorem 3.5 and therefore omitted.
The Existence of Three Positive Solutions
Theorem 3.7. Let ft, u be continuous on a, b × 0, ∞ → 0, ∞. If there exist constants 0 <
a1 < a2 ≤ a3 such that the following assumptions
i ft, u < Ma1 , t, u ∈ a, b × 0, a1 ,
ii ft, u ≤ Ma3 , t, u ∈ a, b × 0, a3 ,
iii ft, u ≥ Na2 , t, u ∈ τ1 , τ2 × a2 , a2 /τ,
18
Abstract and Applied Analysis
hold, then BVP 1.3-1.4 has at least three positive solutions u1 , u2 , and u3 with
u1 < a1 ,
a2 < θu2 < u2 ≤ a3 ,
a1 < u3 ,
θu3 < a2 .
3.39
Proof. We will show that all conditions of Lemma 2.10, are satisfied.
First, if u ∈ P a3 , then u
≤ a3 . So, 0 ≤ ut ≤ a3 , t ∈ a, b.
By condition ii, we have
T ut b
Gt, s H t, s; η1 , . . . , ηm−2 fs, usds
a
≤
b
Gs, s N1 qs Ma3 ds
a
Ma3
b
3.40
Gs, s N1 qs ds a3 ,
a
which implies that T u
≤ a3 , u ∈ P a3 . Hence T : P a3 → P a3 .
Next, by using the analogous argument, it follows from condition i that if u ∈ Pa1 ,
then T u
< a1 .
Choose ut a2 a2 /τ/2, t ∈ a, b, it is easy to see that ut a2 a2 /τ/2 ∈
P θ, a2 , a3 , θu a2 a2 /τ/2 > a2 .
φ. On the other hand, if u ∈ P θ, a2 , a2 /τ,
Therefore, {u ∈ P θ, a2 , a2 /τ | θu > a2 } /
then a2 ≤ ut ≤ a2 /τ, t ∈ τ1 , τ2 . By condition iii, we have ft, ut ≥ Na2 .
Hence,
θT ut min T ut
τ1 ≤t≤τ2
min
b
τ1 ≤t≤τ2
≥ Na2
Gt, s H t, s; η1 , . . . , ηm−2 fs, usds
a
b
τ Gs, s N2 qs ds
3.41
a
τ2
> Na2
τ Gs, s N2 qs ds a2 ,
τ1
which implies that θT u > a2 , for u ∈ P θ, a2 , a2 /τ.
Finally, if u ∈ P θ, a2 , a3 and T u
> a2 /τ, then
θu min T ut ≥ τ
T u
> a2 .
τ1 ≤t≤τ2
3.42
Abstract and Applied Analysis
19
Thus, all the conditions of the Leggett-Williams fixed point theorem are satisfied by taking
d a2 /τ. Hence, the BVPs have at least three solutions in P , that is, three positive solutions
ui i 1, 2, 3 such that
u1 < a1 ,
a2 < θu2 < u2 ≤ a3 ,
a1 < u3 ,
θu3 < a2 .
3.43
Example 3.8. Consider the problem
1
4.2
D0 ut 1 ueu 0, t ∈ 0, 1,
3
1
1 1
3
u 0 u
u1 u
,
u 0 u 0 u 0 0,
,
4
2
4
2
3.44
where α 4.2, a 0, b 1, β 0.25, γ 0.75, η 0.5, τ1 0.25, τ2 0.75, N1 2.6, N2 0.1758, N 168.9596, and M 1.6968, ft, u 1/31 ueu .
Since ft, u 1/31 ueu is a monotone increasing function on 0, ∞, we take
r1 0.001, r2 0.8. We can get
ft, u ≤ f0.8 0.9268 < Mr2 ,
ft, u ≥ f0 0.3333 > Nr1 .
3.45
So, conditions H1 and H2 hold. By Theorem 3.1, the BVP 3.44 has at least one positive
solution.
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