Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 506716, 15 pages
doi:10.1155/2012/506716
Research Article
Existence of Positive Solutions for
Third-Order m-Point Boundary Value Problems
with Sign-Changing Nonlinearity on Time Scales
Fatma Tokmak and Ilkay Yaslan Karaca
Department of Mathematics, Ege University, Izmir, 35100 Bornova, Turkey
Correspondence should be addressed to Ilkay Yaslan Karaca, ilkay.karaca@ege.edu.tr
Received 2 August 2012; Accepted 8 November 2012
Academic Editor: Yongfu Su
Copyright q 2012 F. Tokmak and I. Y. Karaca. 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.
A four-functional fixed point theorem and a generalization of Leggett-Williams fixed point
theorem are used, respectively, to investigate the existence of at least one positive solution and at
least three positive solutions for third-order m-point boundary value problem on time scales with
an increasing homeomorphism and homomorphism, which generalizes the usual p-Laplacian
operator. In particular, the nonlinear term ft, u is allowed to change sign. As an application,
we also give some examples to demonstrate our results.
1. Introduction
The theory of time scales was introduced by Hilger in his Ph.D. thesis in 1990 1. Theoretically, this new theory not only unifies continuous and discrete equations, but has also
exhibited much more complicated dynamics on time scales. Moreover, the study of dynamic
equations on time scales has led to several important applications, for example, insect
population models, biology, neural networks, heat transfer, and epidemic models see 2, 3
and references therein. Some preliminary definitions and theorems on time scales also can
be found in books 2, 3, which are excellent references for calculus on time scales.
Multipoint boundary value problem BVP arises in a variety of different areas of
applied mathematics and physics 4. The study of multipoint BVPs for linear second-order
ordinary differential equations was initiated by Ilin and Moiseev 5, since then many authors
studied more general nonlinear multipoint boundary value problems 6, 7 and references
therein. Recently, when ϕp is p-Laplacian operator, that is, ϕp u |u|p−2 u p > 1 and
the nonlinear term does not depend on the first-order derivative, the existence problems of
positive solutions of boundary value problems have attracted much attention. One can notice
2
Abstract and Applied Analysis
that the oddness of the p-Laplacian operator is key to the proof. However, in the operator φ is
not necessary odd, so it improves and generalizes the p-Laplacian operator.
There are few works that investigate the existence of positive solutions by using
four functionals fixed point theorem 8 or a generalization of Leggett-Williams fixed point
theorem 9. Avery et al. 8 present a four-functional fixed point theorem, which is a major
generalization of the original Leggett-Williams fixed point theorem 10. To our knowledge,
this fixed point theorem so far has only been used by Sun et al. 11. By using the fourfunctional fixed point theorem and five-functional fixed point theorem, they obtain the
existence criteria of at least one positive solution and three positive solutions for the nonlocal
BVP with p-Laplacian. Liu and Sun 12 discussed the existence of at least three and arbitrary
odd number positive solutions for m-point boundary value problems on time scale. They
used a generalization of Leggett-Williams fixed point theorem 9.
There have been extensive studies on BVP with sign-changing nonlinearity 13–17.
Most of these results are obtained by using the fixed point theorem on cone 18, 19. To
the best of our knowledge, positive solutions of third-order m-point BVP for an increasing
homeomorphism and homomorphism with sign-changing nonlinearity on time scales by
using four functionals fixed point theorem 8 or a generalization of Leggett-Williams fixed
point theorem 9 have not been considered till now. Our aim in this paper is to fill the gap.
In this paper, motivated by above results, we consider the existence of at least one or
three positive solutions of the following third-order m-point boundary value problem BVP
on time scales
∇
atft, ut 0,
φ uΔ∇ t
u0 m−2
bi uξi ,
uΔ T 0,
t ∈ 0, T T ,
m−2
ci φ uΔ∇ ξi ,
φ uΔ∇ 0 i1
1.1
i1
where φ : R → R is an increasing homeomorphism and homomorphism with φ0 0. A
projection φ : R → R is called an increasing homeomorphism and homomorphism if the
following conditions are satisfied.
i If x ≤ y, then φx ≤ φy, for all x, y ∈ R;
ii φ is continuous bijection and its inverse mapping is also continuous;
iii φxy φxφy, for all x, y ∈ R.
We will assume that the following conditions are satisfied throughout this paper:
m−2
H1 bi , ci ∈ 0, ∞, 0 < ξ1 < ξ2 < · · · < ξm−2 < ρT , 0 < m−2
i1 bi < 1, 0 <
i1 ci < 1;
H2 f : 0, T T × 0, ∞ → −∞, ∞ is continuous, a ∈ Cld 0, T T , 0, ∞, and there
exist t0 ∈ 0, T T such that at0 > 0;
H3 ζ min{t ∈ T : t > T/2} exist.
By using a four-functional fixed point theorem 8, we establish the existence of at least
one positive solution, and by using a generalization of Leggett-Williams fixed point theorem
9, we establish the existence of at least three positive solutions for the above boundary value
problem. In particular, the nonlinear term ft, ut is allowed to change sign. The remainder
of this paper is organized as follows. Section 2 is devoted to some preliminary lemmas. We
give and prove our main results in Section 3. Finally, in Section 4, we give two examples to
demonstrate our results.
Abstract and Applied Analysis
3
2. Preliminary Lemmas
In this section, we present some definitions and lemmas, which will be needed in the proof of
the main results.
Lemma 2.1 see 13. If condition (H1) holds, then for h ∈ Cld 0, T T , the boundary value problem
(BVP),
uΔ∇ ht 0,
t ∈ 0, T T ,
m−2
u0 2.1
uΔ T 0,
bi uξi ,
i1
has a unique solution
ut t
m−2
T − shs∇s i1
0
ξi
T − shs∇s
.
m−2
1 − i1 bi
bi
2.2
0
Lemma 2.2. If condition (H1) holds, then for h ∈ Cld 0, T T , the boundary value problem (BVP),
∇
φ uΔ∇ t
ht 0,
u0 m−2
Δ
u T 0,
bi uξi ,
φ u
t ∈ 0, T T ,
Δ∇
0 i1
m−2
Δ∇
ci φ u
ξi ,
2.3
i1
has a unique solution
s
m−2 ξi
s
−1
− 0 hr∇r D ∇s
i1 bi 0 s − T φ
ut s − T φ
,
− hr∇r D ∇s 1 − m−2
0
0
i1 bi
2.4
t
where D −
−1
m−2
i1
ci
ξi
0
hr∇r/1 −
m−2
i1
ci .
Proof. Integrating both sides of equation in 2.3 on 0, t, we have
t
φ uΔ∇ t φ uΔ∇ 0 − hr∇r.
2.5
0
So,
Δ∇
φ u
ξi φ u
Δ∇
0 −
ξi
hr∇r.
0
2.6
4
Abstract and Applied Analysis
m−2
By boundary value condition φuΔ∇ 0 Δ∇
φ u
i1
ci φuΔ∇ ξi , we have
ξ
ci 0i hr∇r
.
1 − m−2
i1 ci
m−2
0 −
i1
2.7
By 2.5 and 2.7, we know
⎞
⎛ ξi
t
m−2
c
hr∇r
i
i1
0
− hr∇r ⎠.
uΔ∇ t φ−1 ⎝−
1 − m−2
c
0
i
i1
2.8
This together with Lemma 2.1 implies that
s
m−2 ξi
s
−1
− 0 hr∇r D ∇s
i1 bi 0 s − T φ
ut s − T φ
,
− hr∇r D ∇s 1 − m−2
0
0
i1 bi
2.9
t
−1
where D −
m−2
i1
ci
ξi
0
hr∇r/1 −
m−2
i1
ci . The proof is complete.
Lemma 2.3. Let condition H1 hold. If h ∈ Cld 0, T T and ht ≥ 0, then the unique solution ut
of 2.3 satisfies
i ut ≥ 0, t ∈ 0, T T ,
ii inft∈0,T T ut ≥ σu,
where σ m−2
i1
bi ξi /T −
m−2
i1
Proof. i Let ψ0 s : φ−1 −
−
s
s
0
bi T − ξi , u supt∈0,T T |ut|.
hr∇r D. Consider the following equation:
hr∇r D −
0
s
0
ξ
ci 0i hr∇r
≤ 0.
1 − m−2
i1 ci
m−2
hr∇r −
i1
2.10
So, we have ψ0 s ≤ 0. Hence ut ≥ 0, t ∈ 0, T T .
t
ii By uΔ∇ t φ−1 − 0 hr∇r D ≤ 0, we can know that the graph of ut is concave
down on 0, T T and uΔ t is nonincreasing on 0, T T . This together with the assumption that
the boundary condition uΔ T 0 implies that uΔ t ≥ 0 for t ∈ 0, T T . This implies that
u uT ,
mint∈0,T T ut u0.
2.11
For all i ∈ {1, 2, . . . , m − 2}, we have from the concavity of u that
uξi − u0 uT − u0
,
≥
ξi
T
2.12
Abstract and Applied Analysis
5
that is,
uξi − u0 ξi
ξi
u0 ≥ uT .
T
T
This together with the boundary condition u0 m−2
i1
m−2
i1
mint∈0,T T ut ≥
T−
m−2
i1
2.13
bi uξi implies that
bi ξi
bi T − ξi uT .
2.14
This completes the proof.
Let E Cld 0, T T be equipped with the norm u maxt∈0,T T |ut|. Clearly, it follows
that E, u is a Banach space. For the convenience, let
⎞
ξ
s
ci 0i ar∇r
− ar∇r ⎠.
1 − m−2
0
i1 ci
⎛ m−2
ψs φ
−1 ⎝
−
i1
2.15
We define two cones by
P {u : u ∈ E, ut ≥ 0, t ∈ 0, T T },
K
u : u ∈ E, ut is concave, nondecreasing, nonnegative on 0, T T ,
2.16
min ut ≥ σu ,
t∈0,T T
where σ is as in Lemma 2.3. Define the operators F : P → E and S : K → E by setting
Fut t
s
s − T φ−1 − arfr, ur∇r A ∇s
0
0
m−2
where A −
m−2
i1
ci
ξi
0
i1
bi
ξi
0
s
s − T φ−1 − 0 arfr, ur∇r A ∇s
,
1 − m−2
i1 bi
arfr, ur∇r/1 −
Sut t
0
m−2
i1
ci ,
m−2
s − T ϕs∇s i1
2.17
ξi
s − T ϕs∇s
,
m−2
1 − i1 bi
bi
0
2.18
s
A
−m−2 ci ξi arf r, ur∇r/1 −
where ϕs φ−1 − 0 arf r, ur∇r A,
i1
0
m−2
i1 ci , and f t, ut max{ft, ut, 0}. Obviously, u is a solution of the BVP 1.1 if
and only if u is a fixed point of operator F.
6
Abstract and Applied Analysis
Lemma 2.4. S : K → K is completely continuous.
Proof. It is easy to see that SK ⊂ K by f ≥ 0 and Lemma 2.3. By Arzela-Ascoli theorem and
Lebesgue dominated convergence theorem, we can easily prove that operator S is completely
continuous.
3. Main Results
In this section, we prove the existence of at least one positive solution to the BVP 1.1
by applying a four-functional fixed point theorem 8. Also, by using the generalization of
Leggett-Williams fixed point theorem 9, we prove the existence of at least three positive
solutions for the BVP 1.1.
Let α and Ψ be nonnegative continuous concave functionals on P , and let β and θ be
nonnegative continuous convex functionals on P , then for positive numbers r, j, v, and R,
we define the following sets:
Q α, β, r, R x ∈ P : r ≤ αx, βx ≤ R ,
U Ψ, j x ∈ Q α, β, r, R : j ≤ Ψx ,
V θ, v x ∈ Q α, β, r, R : θx ≤ v .
3.1
Theorem 3.1 see 8. If P is a cone in a real Banach space E, α and Ψ are nonnegative continuous
concave functionals on P , β, and θ are nonnegative continuous convex functionals on P and there
exist positive numbers r, j, v, and R, such that
A : Q α, β, r, R −→ P
3.2
is a completely continuous operator, and Qα, β, r, R is a bounded set. If
i {x ∈ UΨ, j : βx < R} ∩ {x ∈ V θ, v : r < αx} /
∅,
ii αAx ≥ r, for all x ∈ Qα, β, r, R, with αx r and v < θAx,
iii αAx ≥ r, for all x ∈ V θ, v, with αx r,
iv βAx ≤ R, for all x ∈ Qα, β, r, R, with βx R and ΨAx < j,
v βAx ≤ R, for all x ∈ UΨ, j, with βx R,
then A has a fixed point x in Qα, β, r, R.
Suppose τ, η ∈ T with 0 < τ < η < T . For convenience, we take the notations
w
m−2
1−
i1 bi
m−2 ,
i1 bi
Nτ Nη η
Nm−2 s − T ψs∇s,
0
τ
N0 τ
0
s − T ψs∇s,
0
NT s − T ψs∇s,
0
ξm−2
T
0
s − T ψs∇s,
s
− ar∇r ∇s.
s − T φ
−1
0
3.3
Abstract and Applied Analysis
7
Theorem 3.2. Assume that (H1), (H2) hold if there exist constants r, j, v, R with R > max{NT Nm−2 w/Nτ , T/τ}j, v ≥ max{Nη Nm−2 w/Nτ j, η/τr}, r < j, and suppose that ft, u
satisfies the following conditions:
B1 ft, u ≤ φR/NT Nm−2 w for all t, u ∈ 0, T T × 0, R;
B2 ft, u ≥ φr/N0 for all t, u ∈ 0, τT × r, v;
B3 ft, u ≥ 0 for all t, u ∈ 0, T T × σr, R.
Then the BVP 1.1 has a fixed point u ∈ P such that
min ut ≥ r,
t∈τ,ηT
max ut ≤ R.
3.4
t∈0,T T
Define maps
αu Ψu min ut,
t∈τ,ηT
θu max ut,
t∈τ,ηT
βu max ut.
t∈0,T T
3.5
Let Qα, β, r, R, UΨ, j, and V θ, v be defined by 3.1.
Proof. We first show that Qα, β, r, R is bounded and S : Qα, β, r, R → K is completely
continuous. For all u ∈ Qα, β, r, R, we have u maxt∈0,T T ut βu ≤ R, which means
that Qα, β, r, R is a bounded set. According to Lemma 2.4, it is clear that S : Qα, β, r, R →
K is completely continuous.
Let
⎛
⎞
m−2 ξi
j ⎝ t
i1 bi 0 s − T ψs∇s ⎠
.
u0 s − T ψs∇s Nτ
1 − m−2
0
i1 bi
Clearly, u0 ∈ K. By direct calculation,
⎛
⎞
m−2 ξi
j ⎝ τ
i1 bi 0 s − T ψs∇s ⎠
Ψu0 u0 τ s − T ψs∇s Nτ
1 − m−2
0
i1 bi
≥
j
Nτ
τ
s − T ψs∇s j,
0
⎛
⎞
m−2 ξi
j ⎝ T
i1 bi 0 s − T ψs∇s ⎠
βu0 s − T ψs∇s Nτ
1 − m−2
0
i1 bi
⎛
⎞
m−2 ξm−2
s − T ψs∇s
j ⎝ T
i1 bi 0
⎠
≤
s − T ψs∇s Nτ
1 − m−2
0
i1 bi
j
NT ωNm−2 < R,
Nτ
3.6
8
Abstract and Applied Analysis
⎛
⎞
m−2 ξi
j ⎝ η
i1 bi 0 s − T ψs∇s ⎠
θu0 s − T ψs∇s Nτ
1 − m−2
0
i1 bi
⎛
⎞
m−2 ξm−2
b
−
T
s
ψs∇s
j ⎝ η
i 0
i1
⎠
≤
s − T ψs∇s Nτ
1 − m−2
0
i1 bi
j Nη ωNm−2 ≤ v,
Nτ
αu0 Ψu0 ≥ j > r.
3.7
So, u0 ∈ {u ∈ UΨ, j : βu < R} ∩ {u ∈ V θ, v : r < αu}, which means that i in
Theorem 3.1 is satisfied.
For all u ∈ Qα, β, r, R, with αu r and v < θSu, we have
αSu Suτ ≥
τ
τ
τ
Su η θSu > v ≥ r,
η
η
η
3.8
and for all u ∈ Qα, β, r, R, with βu R and ΨSu < j, we have
βSu SuT ≤
T
T
T
Suτ ΨSu < j < R.
τ
τ
τ
3.9
Hence ii and iv in Theorem 3.1 are fulfilled.
Lastly, we consider Theorem 3.1 iii and v. For all V θ, v, with αu r,
αSu τ
m−2
s − T ϕs∇s 0
≥
τ
0
≥
τ
⎛
s − T φ−1 ⎝−
s
0
i1
ξi
s − T ϕs∇s
m−2
1 − i1 bi
bi
0
arf r, ur∇r −
m−2
i1
ci
⎞
arf
ur∇r
r,
0
⎠∇s
1 − m−2
c
i
i1
ξi
s
s
r τ
s − T φ−1 − arf r, ur∇r ∇s ≥
s − T φ−1 − ar∇r ∇s r,
N0 0
0
0
0
3.10
Abstract and Applied Analysis
9
and for all u ∈ UΨ, j, with βu R,
T
m−2
ξi
s − T ϕs∇s
m−2
1 − i1 bi
0
m−2 ξm−2
T
s − T ϕs∇s
i1 bi 0
≤
s − T ϕs∇s m−2
1 − i1 bi
0
T
ξm−2
R
≤
s − T ψs∇s w
s − T ψs∇s R.
NT Nm−2 w
0
0
βSu s − T ϕs∇s i1
bi
0
3.11
Thus, all conditions of Theorem 3.1 are satisfied. S has a fixed point u in Qα, β, r, R. Clearly,
σr ≤ ut ≤ R, t ∈ 0, T T . By condition B3, we have ft, ut ≥ 0, t ∈ 0, T T , that is,
f t, ut ft, ut. Hence, Fu Su. This means that u is a fixed point of operator F.
Therefore, the BVP 1.1 has at least one positive solution.
Suppose α and β are two nonnegative continuous convex functionals satisfying
x ≤ L max αx, βx ,
x ∈ P,
3.12
where L is a positive constant, and
Ω x ∈ P : αx < r, βx < l /
∅,
r > 0, l > 0.
3.13
Let r > a > 0, l > 0 be given, α, β nonnegative continuous convex functionals on P
satisfying the relation 3.12 and 3.13, and γ a nonnegative continuous concave functional
on P . We define the following convex sets:
P α, r; β, l u ∈ P : αu < r, βu < l ,
P α, r; β, l u ∈ P : αu ≤ r, βu ≤ l ,
P α, r; β, l; γ, a u ∈ P : αu < r, βu < l, γu > a ,
P α, r; β, l; γ, a u ∈ P : αu ≤ r, βu ≤ l, γu ≥ a .
3.14
Theorem 3.3 see 9. Let B be a Banach space, P ⊂ B a cone, and r2 ≥ d > b > r1 > 0, l2 ≥ l1 > 0.
Assume that α and β are nonnegative continuous convex functionals satisfying 3.12 and 3.13, γ
is a nonnegative continuous concave functional on P such that γu ≤ αu for all u ∈ P α, r2 ; β, l2 ,
and A : P α, r2 ; β, l2 → P α, r2 ; β, l2 is a completely continuous operator. Suppose
i {u ∈ P α, d; β, l2 ; γ, b : γu > b} / ∅, γAu > b for u ∈ P α, d; β, l2 ; γ, b;
ii αAu < r1 , βAu < l1 for u ∈ P α, r1 ; β, l1 ;
iii γAu > b for u ∈ P α, r2 ; β, l2 ; γ, b with αAu > d.
10
Abstract and Applied Analysis
Then A has at least three fixed points u1 , u2 , u3 ∈ P α, r2 ; β, l2 with
u1 ∈ P α, r1 ; β, l1 ,
u2 ∈ P α, r2 ; β, l2 ; γ, b : γu > b ,
u3 ∈ P α, r2 ; β, l2 \ P α, r2 ; β, l2 ; γ, b ∪ P α, r1 ; β, l1 .
3.15
It is easy to see that for u ∈ K, ut ≥ t/T u, t ∈ 0, T T . So we get
ut ≥
ζ
1
u > u,
T
2
3.16
t ∈ ζ, T T ,
where ζ is as in H3. Let α, β, and γ be defined on the cone K by
αu max ut,
t∈0,T T
βu min ut,
t∈0,T T
γu min ut,
t∈ζ,T T
3.17
respectively. Then it is easy to see that u max{αu, βu} and 3.12, 3.13 hold. Now,
ζ
for convenience, let 3.3 and M 0 s − T ψζ∇s.
Theorem 3.4. Assume (H1)–(H3) hold, ft, 0 is not equivalent to 0 for t ∈ 0, T T . If there
are positive numbers r2 ≥ 2b > b > r1 > 0, l2 ≥ l1 > 0 with b/M ≤ min{r2 /NT Nm−2 w, l2 /Nm−2 w}, such that the following conditions are satisfied.
C1 0 ≤ ft, u ≤ min{φr2 /NT Nm−2 w, φl2 /Nm−2 w} for t, u ∈ 0, T T × 0, r2 ,
C2 ft, u > φb/M for t, u ∈ ζ, T T × b, 2b,
C3 ft, u < min{φr1 /NT Nm−2 w, φl1 /Nm−2 w} for t, u ∈ 0, T T × 0, r1 .
Then the BVP 1.1 has at least three positive solutions u1 , u2 , u3 satisfying
max u1 t < r1 ,
t∈0,T T
min u1 t < l1 ,
t∈0,T T
b < min u2 t ≤ max u2 t ≤ r2 ,
t∈ζ,T T
t∈0,T T
max u3 t < 2b,
t∈0,T T
min u2 t ≤ l2 ,
t∈0,T T
3.18
with min u3 t < b, min u3 t ≤ l2 .
t∈ζ,T T
t∈0,T T
Proof. By the definition of S, it suffices to show that the conditions of Theorem 3.3 hold with
respect to the operator S. Firstly, we show that if the condition C1 is satisfied, then
S : K α, r2 ; β, l2 −→ K α, r2 ; β, l2 .
3.19
Abstract and Applied Analysis
11
If u ∈ Kα, r2 ; β, l2 , then assumption C1 implies
ft, u ≤ min φ
r2
l2
,φ
, t ∈ 0, T T ,
NT Nm−2 w
Nm−2 w
m−2 ξi
T
i1 bi 0 s − T ϕs∇s
αSu s − T ϕs∇s 1 − m−2
0
i1 bi
m−2
T
ξm−2
s − T ϕs∇s
i1 bi 0
≤
s − T ϕs∇s m−2
1 − i1 bi
0
ξm−2
T
r2
≤
s − T ψs∇s w
s − T ψs∇s r2 ,
NT Nm−2 w 0
0
m−2 ξi
i1 bi 0 s − T ϕs∇s
βSu 1 − m−2
i1 bi
ξm−2
l2
≤
w
s − T ψs∇s l2 .
Nm−2 w
0
3.20
Therefore, 3.19 holds. In the same way, if u ∈ Kα, r1 ; β, l1 , then condition C3 implies
ft, u < min φ
l1
r1
,φ
,
NT Nm−2 w
Nm−2 w
t ∈ 0, T T .
3.21
As the argument above, we have S : Kα, r1 ; β, l1 → Kα, r1 ; β, l1 . Thus, condition ii of
Theorem 3.3 holds. To check the condition i of Theorem 3.3, we choose
u0 t 2bt
,
T
3.22
t ∈ 0, T T .
It is easy to see that u0 t ∈ Kα, 2b; β, l2 ; γ, b, γu0 > b. Therefore, for u ∈ Kα, 2b; β, l2 ; γ, b,
there are b ≤ ut ≤ 2b, t ∈ ζ, T T . Hence, condition C2 implies ft, u > φb/M, t ∈
ζ, T T . So, by the definition of γ,
γSu ζ
m−2
s − T ϕs∇s 0
ζ
0
⎛
s − T φ−1 ⎝−
s
0
i1
ξi
s − T ϕs∇s
≥
m−2
1 − i1 bi
bi
0
arf r, ur∇r −
m−2
i1
ζ
s − T ϕs∇s
0
ci
⎞
arf
ur∇r
r,
0
⎠∇s
1 − m−2
i1 ci
ξi
12
Abstract and Applied Analysis
⎛
⎞
m−2 ξi
ζ
ζ
i1 ci 0 arf r, ur∇r ⎠
−1 ⎝
− arf r, ur∇r −
∇s
≥
s − T φ
1 − m−2
0
0
i1 ci
b
>
M
ζ
⎛
s − T φ−1 ⎝−
ζ
0
⎞
ξ
ci 0i ar∇r
⎠∇s b.
1 − m−2
c
i
i1
m−2
i1
ar∇r −
0
3.23
Hence, condition i of Theorem 3.3 holds. We finally prove that iii in Theorem 3.3 holds. In
fact, for u ∈ Kα, 2b; β, l2 ; γ, b with αSu > 2b, we have γSu > b. Thus from Theorem 3.3,
the BVP 1.1 has at least three fixed points u1 , u2 , u3 ∈ Kα, r2 ; β, l2 . Clearly, 0 ≤ u1 t ≤ r1 ,
σb < u2 t ≤ r2 , 0 ≤ u3 t ≤ 2b, t ∈ 0, T T . By condition C1, we have ft, ut ≥ 0, t ∈ 0, T T ,
that is, f t, ut ft, ut. Hence, Fu Su. This means that u1 , u2 , u3 are fixed points of
operator F. Therefore, the BVP 1.1 has at least three positive solutions.
4. Example
Example 4.1. Let T 0, 1/2 ∪ {0.6} ∪ 0.7, 1. If we choose T 1, at 1,b1 1/3, c1 1/2,
ξ1 1/4 in the boundary value problem 1.1, then we have the following BVP on time scale
T:
∇
ft, ut 0,
φ uΔ∇ t
1
1
,
u0 u
3
4
t ∈ 0, 1T ,
1
1
,
φ uΔ∇ 0 φ uΔ∇
2
4
uΔ 1 0,
4.1
where
φu ft, u ⎧
⎪
⎨u,
u ≤ 0,
⎪
⎩u2 ,
u > 0,
⎧
⎪
⎨95u − 0.0001 0.0007,
0, 1T × 0, 0.0001,
⎪
⎩0.0007,
0, 1T × 0.0001, ∞.
4.2
Set τ 1/3, η 0.6, by calculation,
w
1
,
2
NT 1721
,
6000
Nm−2 2533
Nη ,
12000
31
,
384
Nτ 1
σ ,
9
73
,
648
N0 7
,
162
4.3
Abstract and Applied Analysis
13
and let r 0.001, j r 10−4 , R 0.012, and v 0.0035 with 0.012 > 0.0033, 0.0035 ≥
0.002455. Clearly, we can verify that the conditions in Theorem 3.2 are fulfilled. Thus, by
Theorem 3.2, the BVP 4.1 has a fixed point u such that
min
t∈1/3,0.6T
ut ≥ 0.001,
max ut ≤ 0.012.
4.4
t∈0,1T
Example 4.2. Let T 0, 1 ∪ {1.2} ∪ {1.5} ∪ 1.8, 2. If we choose T 2, at 1, b1 1/4,
c1 1/3, ξ1 2/3 in the boundary value problem 1.1, then we have the following BVP on
time scale T :
∇
φ uΔ∇ t
ft, ut 0,
2
1
,
u0 u
4
3
Δ
u 2 0,
Δ∇
φ u
t ∈ 0, 2T ,
1
Δ∇ 2
,
0 φ u
3
3
4.5
where
u, u ≤ 0,
φu u2 , u > 0,
⎧
1
1
⎪
⎪
t u3 ,
⎪
⎪
80
10
⎪
⎪
⎪
⎨
1
8
ft, u t u,
⎪
80
5
⎪
⎪
⎪
⎪
⎪
⎪
⎩ 1 t 520 − u,
80
t ∈ 0, 2T , u ∈ 0, 4,
4.6
t ∈ 0, 2T , u ∈ 4, 200,
t ∈ 0, 2T , u ∈ 200, ∞.
Obviously the hypotheses H1, H2 hold and ft, 0 is not equivalent to 0 on 0, 2T . By
simple calculations, we have
ζ 1.2,
w
1
,
3
NT 1.859,
Nm−2 0.716,
M 2.5398.
4.7
If we choose r2 200, b 2, r1 1/2, and l2 80, l1 20, then ft, u satisfies
1 0 ≤ ft, u ≤ 320.025 ≤ min{φr2 /NT Nm−2 w, φl2 /Nm−2 w} 9090 for t, u ∈
0, 2T × 0, 200;
2 ft, u ≥ 0.815 > φb/M 0.62 for t, u ∈ 1.2, 2T × 2, 4,
3 ft, u ≤ 0.0375 < min{φr1 /NT Nm−2 w, φl1 /Nm−2 w} 0.0568.
14
Abstract and Applied Analysis
So, all conditions of Theorem 3.4 hold. Thus by Theorem 3.4, the BVP 4.5 has at least three
positive solutions u1 , u2 , u3 such that
max u1 t <
t∈0,2T
1
,
2
min u1 t < 20,
t∈0,2T
2 < min u2 t ≤ max u2 t ≤ 200,
t∈1.2,2T
max u3 t < 4,
t∈0,2T
t∈0,2T
min u2 t ≤ 80,
t∈0,2T
4.8
with min u3 t < 2, min u3 t ≤ 80.
t∈1.2,2T
t∈0,2T
References
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