Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 647235, 14 pages
doi:10.1155/2012/647235
Research Article
Upper and Lower Solution Method for
Fourth-Order Four-Point Boundary Value Problem
on Time Scales
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 17 August 2011; Revised 20 October 2011; Accepted 3 November 2011
Academic Editor: Khalida Inayat Noor
Copyright q 2012 Ilkay Yaslan 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.
We consider a fourth-order four-point boundary value problem for dynamic equations on time
scales. By the upper and lower solution method, some results on the existence of solutions of the
fourth-order four-point boundary value problem on time scales are obtained. An example is also
included to illustrate our results.
1. Introduction
Let T be a closed nonempty subset of R, and let T have the subspace topology inherited from
the Euclidean topology on R. In some of the current literature, T is called a time scale or
measure chain. For notation, we shall use the convention that, for each interval J of R, J will
denote the time scales interval, that is, J : J ∩ T. Some preliminary definitions and theorems
on time scales can be found in the books 1, 2, which are excellent references for calculus of
time scales.
In this paper, let T be a time scale and σt the forward jump function in T. We are
concerned with the following fourth-order four-point boundary value problem on time scales
T:
4
2
2
yΔ t − qtyΔ σt f t, yσt, yΔ σt ,
y σ 2 b λyΔ σ 2 b 0,
ya 0,
ζyΔ ξ1 − ηyΔ ξ1 0,
2
3
γyΔ ξ2 δyΔ ξ2 0,
2
3
1.1
2
Abstract and Applied Analysis
for t ∈ a, b ⊂ T, a ≤ ξ1 < ξ2 ≤ σb. We will assume that the following conditions are
satisfied.
H1 ζ, γ, η, δ ≥ 0, λ ≥ σ 3 b − σ 2 b,
H2 qt ≥ 0. If qt ≡ 0, then ζ γ > 0,
H3 k ζδ ηγ ζγξ2 − ξ1 > 0, η − ζξ1 ≥ 0, δ − γσb − ξ2 ≥ 0.
The upper and lower solution method has been used to deal with the boundary value
problems for dynamic equations in recent years. In most of these studies, two-point boundary
value problem for second-order dynamic equations is considered 3–7.
Pang and Bai 8 studied the following fourth-order four-point BVP on time scales:
uΔΔΔΔ t f t, uσt, uΔΔ t ,
t ∈ 0, 1,
1.2
u0 u σ 4 1 0,
αyΔ ξ1 − βyΔ ξ1 0,
2
3
γyΔ ξ2 ηyΔ ξ2 0
2
3
for 0 ≤ ξ1 < ξ2 ≤ σb, and f ∈ C0, 1 × R × R, R and α, β, γ, η are nonnegative constants
satisfying αη βγ αγξ2 − ξ1 > 0. They establish criteria for the existence of a solution
by developing the upper and lower solution method and the monotone iterative technique.
Our problem is more general than the problems in 8, and our results are even new for the
differential equations as well as for dynamic equations on general time scales.
2. Preliminaries
To prove the main results in this paper, we will employ several lemmas. We consider the
linear boundary value problem
yΔ t − qtyσt ht,
2
ζyξ1 − ηyΔ ξ1 0,
t ∈ ξ1 , ρξ2 ,
γyξ2 δyΔ ξ2 0.
2.1
Denote by ϕ and ψ, the solutions of the corresponding homogeneous equation
yΔ t − qtyσt 0,
2
t ∈ ξ1 , ρξ2 2.2
under the initial conditions
ϕξ1 η,
ψξ2 δ,
ϕΔ ξ1 ζ,
ψ Δ ξ2 −γ,
2.3
so that ϕ and ψ satisfy the first and second boundary conditions of 2.1, respectively. Let us
set
D : ζψξ1 − ηψ Δ ξ1 δϕΔ ξ2 γϕξ2 .
2.4
Abstract and Applied Analysis
3
Using the initial conditions 2.3, we can deduce from 2.2 for ϕ and ψ the following
equations:
ϕt η ζt − ξ1 ψt δ γξ2 − t t τ
ξ1
ξ1
ξ2 ξ2
t
τ
qsϕσsΔsΔτ,
2.5
qsψσsΔsΔτ.
2.6
Lemma 2.1. Under the conditions (H1) and (H2), the following inequalities
ϕt ≥ 0,
t ∈ ξ1 , σξ2 ;
ϕΔ t ≥ 0,
t ∈ ξ1 , ξ2 ;
ψt ≥ 0,
t ∈ ξ1 , ξ2 ;
ψ Δ t ≤ 0,
t ∈ ξ1 , ξ2 ;
2.7
yield.
Proof. We apply the induction principle for time scales to the statement
ϕΔ t ≥ 0,
At : ϕt ≥ 0,
2.8
where t ∈ ξ1 , ξ2 .
I The statement Aξ1 is true, since ϕξ1 η and ϕΔ ξ1 ζ.
II Let t be right-scattered and let At be true, that is, ϕt ≥ 0 and ϕΔ t ≥ 0. We need
to show that ϕσt ≥ 0 and ϕΔ σt ≥ 0. By the definition of Δ-derivative, we have
ϕσt ϕt σt − tϕΔ t.
2.9
Further, by the definition of Δ-derivative and 2.2 for ϕt, we have
ϕΔ σt ϕΔ t σt − tϕΔ t ϕΔ t σt − tqtϕσt.
2
2.10
From 2.9, we get ϕσt ≥ 0, and then from 2.10, we get ϕΔ σt ≥ 0.
III Let t0 be right-dense, At0 be true and t1 ∈ ξ1 , ξ2 such that t1 > t0 and is
sufficiently close to t0 . We need to prove that At is true for t ∈ t0 , t1 .
From 2.2 with yt ϕt, the equations
Δ
Δ
ϕ t ϕ t0 t
t0
ϕt ϕt0 ϕΔ t0 t − t0 qsϕσsΔs,
t τ
t0
t0
qsϕσsΔsΔτ
2.11
2.12
4
Abstract and Applied Analysis
follow. To investigate the function ϕt appearing in 2.12, we consider the equation
yt ϕt0 ϕΔ t0 t − t0 t τ
t0
t0
qsyσsΔsΔτ,
2.13
where yt is the desired solution. Our aim is to show that with t1 sufficiently close to t0 ,
2.13 has a unique continuous solution yt satisfying inequality
yt ≥ ϕt0 ϕΔ t0 t − t0 ,
t ∈ t0 , t1 .
2.14
We solve 2.13 by the method of successive approximations, setting
y0 t ϕt0 ϕΔ t0 t − t0 ,
t τ
qsyj−1 σsΔsΔτ, j 1, 2, 3, . . . .
yj t t0
2.15
t0
If the series ∞
j0 yj t converges uniformly with respect to t ∈ t0 , t1 , then its sum will be,
obviously, a continuous solution of 2.13. To prove the uniform convergence of this series,
we let
Δ
M0 ϕt0 ϕ t0 t1 − t0 ,
M1 t1 τ
t0
t0
qsΔsΔτ.
2.16
Then the estimate
j
0 ≤ yj t ≤ M0 M1 ,
t ∈ t0 , t1 , j 0, 1, 2, . . .
2.17
can easily be obtained. Indeed, 2.17 evidently holds for j 0. Let it also hold for j n. Then
from 2.15, we get for t ∈ t0 , t1 ,
0 ≤ yn1 t ≤
≤
≤
t1 τ
t0
t1
t0
qsyn σsΔsΔτ
max yn σs
τ
t0 t0 ≤s≤ρτ
max yn σs
t0 ≤s≤ρ2 t1 t0
qsΔsΔτ
t1 τ
t0
t0
2.18
qsΔsΔτ
≤ M0 M1n M1 M0 M1n1 .
Therefore, by the usual mathematical induction principle, 2.17 holds for all j 0, 1, 2, . . ..
Abstract and Applied Analysis
5
Now choosing t1 appropriately, we obtain M1 < 1. Then 2.13 will have a continuous
solution
yt ∞
yj t for t ∈ t0 , t1 .
2.19
j0
Since yj t ≥ 0, it follows that yt ≥ y0 t thereby proving the validity of inequality 2.14.
To prove uniqueness of solution of 2.13 for t ∈ t0 , t1 , suppose it has two solutions y1 and
y2 , and passing on to the modulus, we get
y1 t − y2 t ≤
≤
t τ
t0
t0
qsy1 σs − y2 σsΔsΔτ
max y1 σs − y2 σs
t0 ≤s≤ρ2 t1 t1 τ
t0
t0
2.20
qsΔsΔτ.
Thus,
y1 t − y2 t ≤ M1 max y1 σs − y2 σs,
t0 ≤s≤ρ2 t1 t ∈ t0 , t1 .
2.21
Since M1 < 1, hence it follows that y1 t y2 t for t ∈ t0 , t1 .
From 2.12 and 2.13 in view of the uniqueness of solution, we get that ϕt yt, t ∈ t0 , t1 . Therefore,
ϕt ≥ ϕt0 ϕΔ t0 t − t0 ,
t ∈ t0 , t1 .
2.22
Hence by making use of the induction hypothesis At0 being true, we obtain ϕt ≥ 0 for
t ∈ t0 , t1 . Taking this into account, from 2.11, we also get ϕΔ t ≥ 0 for t ∈ t0 , t1 . Thus,
At is true for all t ∈ t0 , t1 .
IV Let t ∈ ξ1 , ξ2 and assume t is left-dense and such that As is true for all s < t,
that is,
ϕs ≥ 0,
ϕΔ s ≥ 0,
∀s ∈ ξ1 , t.
2.23
Passing on here to the limit as s → t, we get by the continuity of ϕs and ϕΔ s that ϕt ≥ 0
and ϕΔ t ≥ 0, thereby verifying the validity of At.
Consequently, by the induction principle on time scales, 2.8 holds for all t ∈ ξ1 , ξ2 .
From 2.9 and 2.8 for t ξ2 , we also get ϕσξ2 ≥ 0. So the statements 2.7 for ϕ
are proved.
We can prove the statements of the lemma for ψ similarly applying the backward
induction principle on time scales. The lemma is proved.
Lemma 2.2. Under the conditions (H1) and (H2), the inequality D > 0 holds.
6
Abstract and Applied Analysis
Proof. By 2.4 and 2.5, we have
D ζδ ηγ ζγξ2 − ξ1 δ
ξ2
ξ1
qsϕσsΔs γ
ξ2 τ
ξ1
ξ1
qsϕσsΔsΔτ.
2.24
Since ϕt ≥ 0 for t ∈ ξ1 , σξ2 , from 2.8, we have
D ≥ ζδ ηγ ζγξ2 − ξ1 .
2.25
If qt ≡ 0, then in 2.25 the equality holds. From the condition H2, we get D > 0. This proof
is completed.
Lemma 2.3. Assume that the conditions (H1) and (H2) are satisfied. If h ∈ Cξ1 , ρξ2 , then the
boundary value problem
yΔ t − qtyσt ht,
2
t ∈ ξ1 , ρξ2 ,
ζyξ1 − ηyΔ ξ1 0,
2.26
γyξ2 δyΔ ξ2 0
has a unique solution
yt −
ξ2
ξ1
Gt, shsΔs,
2.27
where
⎧
1 ⎨ψσsϕt,
Gt, s D ⎩ψtϕσs,
t ≤ s,
t ≥ σs.
2.28
Here D, ϕ, ψ are as in 2.4, 2.5, and 2.6, respectively.
Proof. Taking
1
zt −
D
t
ξ1
ϕσsψt − ϕtψσs hsΔs,
2.29
we have
1
ϕσtψ Δ σt − ϕΔ σtψσt ht
D
1 t
2
2
−
ϕσsψ Δ t − ϕΔ tψσs hsΔs.
D ξ1
zΔ t −
2
2.30
Abstract and Applied Analysis
7
By Corollary 3.14 in 1, since the Wronskian of any two solutions of 2.2 is independent of
t, we get
D W ψ, ϕ ψσtϕΔ σt − ϕσtψ Δ σt.
2.31
Hence we get
t
1
z t −
−Dht qt
ϕσsψσt − ϕσtψσs hsΔs
D
ξ1
1 t
ϕσsψσt − ϕσtψσs hsΔs
ht qt −
D ξ1
1 σt ϕσsψσt − ϕσtψσs hsΔs
ht qt −
D ξ1
Δ2
2.32
ht qtzσt.
So the general solution of equation
t ∈ ξ1 , ρξ2 ,
2.33
ϕσsψt − ϕtψσs hsΔs,
2.34
yΔ t − qtyσt ht,
2
has the form
yt c1 ϕt c2 ψt −
1
D
t
ξ1
where c1 and c2 are arbitrary constants. Substituting this expression for yt in the boundary
conditions of BVP 2.26, we can evaluate c1 and c2 . After some easy calculations, we can get
2.27 and 2.28.
Lemma 2.4. Under the conditions (H1) and (H2), the Green’s function of BVP 2.26 possesses the
following property:
Gt, s ≥ 0,
t, s ∈ ξ1 , ξ2 × ξ1 , ρξ2 .
2.35
Proof. The lemma follows from 2.28, Lemmas 2.1 and 2.2 immediately.
Lemma 2.5. Assume that the conditions (H1) and (H2) are satisfied. If h ∈ Ca, b, then the boundary
value problem
yΔ t − qtyΔ σt ht, t ∈ a, b,
ya 0,
y σ 2 b λyΔ σ 2 b 0,
4
2
ζyΔ ξ1 − ηyΔ ξ1 0,
2
3
γyΔ ξ2 δyΔ ξ2 0
2
3
2.36
8
Abstract and Applied Analysis
has a unique solution
yt σ 2 b
a
G1 t, ξ
ξ2
ξ1
G2 ξ, shsΔsΔξ,
2.37
where
⎧
⎨ σ 2 b − σs λ t − a,
1
G1 t, s 2
σ b − a λ ⎩σ 2 b − t λσs − a,
G2 t, s ⎧
1 ⎨ψσsϕt,
t ≤ s,
D ⎩ψtϕσs,
t ≥ σs.
t ≤ s,
t ≥ σs,
2.38
2.39
Here D, ϕ, ψ are as in 2.4, 2.5, and 2.6, respectively.
Proof. Let us consider the following BVP:
Δ2
y t −
ξ2
ξ1
ya 0,
G2 t, shsΔs,
t ∈ a, σ 2 b ,
y σ b λyΔ σ 2 b 0.
2.40
2
The Green’s function associated with the BVP 2.40 is G1 t, s. This completes the proof.
Lemma 2.6. Assume that the conditions (H1)–(H3) are satisfied. If y satisfies
yΔ − qtyΔ σt ≥ 0, t ∈ a, b,
ya ≥ 0,
y σ 2 b λyΔ σ 2 b ≥ 0,
4
2
ζyΔ ξ1 − ηyΔ ξ1 ≤ 0,
2
3
2.41
γyΔ ξ2 δyΔ ξ2 ≤ 0,
2
3
then yt ≥ 0, t ∈ a, σ 2 b and yΔ t ≤ 0, t ∈ a, σb.
2
Proof. Let
yΔ t − qtyΔ σt ht, t ∈ a, b,
y σ 2 b λyΔ σ 2 b t1 ,
ya t0 ,
4
2
ζyΔ ξ1 − ηyΔ ξ1 t2 ,
2
3
where t0 ≥ 0, t1 ≥ 0, t2 ≤ 0, t3 ≤ 0, h ≥ 0.
γyΔ ξ2 δyΔ ξ2 t3 ,
2
3
2.42
Abstract and Applied Analysis
9
It is easy to check that y and yΔ can be given by the expression
2
yt St −
σ 2 b
a
G1 t, ξRξΔξ yΔ t Rt −
2
σ 2 b
ξ2
ξ1
a
G1 t, ξ
ξ2
ξ1
G2 ξ, shsΔsΔξ,
2.43
G2 t, shsΔs,
where
St 1
2
,
σ
−
at
λ
−
t
t
t
b
0
1
σ 2 b − a λ
1 Rt ζt − ξ1 η t3 γξ2 − t δ t2 ,
k
2.44
and G1 t, s, G2 t, sare as in 2.38 and 2.39, respectively. The hypothesis of the lemma
implies that St ≥ 0 for t ∈ a, σ 2 b, Rt ≤ 0 for t ∈ a, σb, G1 t, s ≥ 0 for t, s ∈
a, σ 2 b × a, σb, and G2 t, s > 0 for t, s ∈ a, σb × a, b. Therefore, we get yt ≥ 0
2
for t ∈ a, σ 2 b and yΔ t ≤ 0 for t ∈ a, σb. The proof is completed.
3. Upper and Lower Solution Method
In this section, we present existence results for the BVP 1.1 by using the method of upper
and lower solutions. We define the set
D :
y:y
Δn
kn
∈ C a, σ b , k 0, 1, 2, 3, 4 .
4
3.1
Definition 3.1. Letting αt ∈ D on a, σ 4 b, we say α is a lower solution for the problem
1.1 if α satisfies
4
2
2
αΔ t − qtαΔ σt ≤ f t, ασt, αΔ σt ,
α σ 2 b λαΔ σ 2 b ≤ 0,
αa ≤ 0,
ζαΔ ξ1 − ηαΔ ξ1 ≥ 0,
2
3
3.2
γαΔ ξ2 δαΔ ξ2 ≥ 0.
2
3
Definition 3.2. Letting βt ∈ D, on a, σ 4 b, we say β is an upper solution for the problem
1.1 if β satisfies
4
2
2
βΔ t − qtβΔ σt ≥ f t, βσt, βΔ σt ,
β σ 2 b λβΔ σ 2 b ≥ 0,
βa ≥ 0,
ζβΔ ξ1 − ηβΔ ξ1 ≤ 0,
2
3
γβΔ ξ2 δβΔ ξ2 ≤ 0.
2
3
3.3
10
Abstract and Applied Analysis
We assume that the function ft, y, yΔ satisfies the following condition.
2
H4 f : a, b × R × R → R is continuous and satisfies
f t, y2 , z − f t, y1 , z ≥ 0,
f t, y, z2 − f t, y, z1 ≤ 0,
for ασt ≤ y1 ≤ y2 ≤ βσt, z ∈ R, t ∈ a, b,
for βΔ σt ≤ z1 ≤ z2 ≤ αΔ σt, y ∈ R, t ∈ a, b,
2
2
3.4
where α, β are lower and upper solutions, respectively, for the BVP 1.1, and satisfy α ≤
2
2
β, αΔ ≥ βΔ .
Theorem 3.3. Assume that the conditions (H1)–(H4) are satisfied. Then the problem 1.1 has a solution yt with
αt ≤ yt ≤ βt,
αΔ t ≥ yΔ t ≥ βΔ t
2
2
2
3.5
for t ∈ a, σ 3 b and t ∈ a, σb, respectively.
Proof. Consider the BVP,
4
2
2
yΔ t − qtyΔ σt F t, yσt, yΔ σt ,
y σ 2 b λyΔ σ 2 b 0,
ya 0,
ζyΔ ξ1 − ηyΔ ξ1 0,
2
t ∈ a, b,
3
3.6
γyΔ ξ2 δyΔ ξ2 0,
2
3
where
⎧ ⎪
f ∗ t, ασt, y , x < ασt,
⎪
⎪
⎨ F t, x, y f ∗ t, x, y ,
ασt ≤ x ≤ βσt,
⎪
⎪
⎪
⎩ ∗
f t, βσt, y , x > βσt,
⎧ 2
Δ2
⎪
, y < βΔ σt,
f
t,
x,
β
σt
⎪
⎪
⎪
⎨ 2
2
βΔ σt ≤ y ≤ αΔ σt,
f ∗ t, x, y f t, x, y ,
⎪
⎪
⎪
⎪
⎩f t, x, αΔ2 σt , y > αΔ2 σt.
3.7
By Lemma 2.5, it is clear that the solutions of the BVP 3.6 are the fixed points of the operator
Ayt σ 2 b
a
G1 t, ξ
ξ2
ξ1
2
G2 ξ, sF s, yσs, yΔ σs ΔsΔξ,
t ∈ a, σ 4 b ,
3.8
Abstract and Applied Analysis
11
where G1 t, s and G2 t, s are as in 2.38 and 2.39, respectively. It is clear that A is
2
continuous. Since the function ft, y, yΔ satisfies the conditions 3.4,
2
2
2
f t, ασt, αΔ σt ≤ F t, y, yΔ ≤ f t, βσt, βΔ σt
for t ∈ a, b.
3.9
Thus, there exists a positive constant M such that |Ft, y, yΔ | ≤ M, which implies that the
operator A is uniformly bounded. Moreover, the operator A is equicontinuous. Therefore,
from the Arzela-Ascoli theorem, the operator A is a compact operator. Thus, by Schauder’s
fixed point theorem, there exists a solution y of the BVP 3.6.
2
Suppose y∗ is a solution of the BVP 3.6. Since ft, y, yΔ satisfies the conditions 3.4,
we know that
2
2
Δ2
2
≤ f t, βσt, βΔ σt
f t, ασt, αΔ σt ≤ F t, y∗ , y∗
for t ∈ a, b.
3.10
Thus,
∗
Δ 2
t − qt β − y∗ σt
≥ f t, βσt, β2 σt
2
−F t, y∗ σt, yΔ σt ≥ 0,
β − y∗
Δ 4
β − y a ≥ 0,
β−y
∗
σ b λ β − y
2
Δ 2
Δ 3
ζ β − y∗ ξ1 − η β − y∗ ξ1 ≤ 0,
t ∈ a, b,
∗ Δ
σ b ≥ 0,
3.11
2
Δ 2
Δ 3
γ β − y∗ ξ2 δ β − y∗ ξ2 ≤ 0.
Δ2
By virtue of Lemma 2.6, y∗ t ≤ βt for t ∈ a, σ 2 b and y∗ t ≥ βΔ t for t ∈ a, σb.
If σ 2 b is right-scattered, by using the inequality β − y∗ σ 2 b λβ − y∗ Δ σ 2 b ≥ 0, we
get the inequality λ/σ 3 b − σ 2 bβ − y∗ σ 3 b ≥ λ/σ 3 b − σb − 1β − y∗ σ 2 b.
So we get y∗ t ≤ βt on a, σ 3 b. If σ 2 b is right-dense, it is trivial that the inequality
y∗ t ≤ βt holds on a, σ 3 b. Similarly, one can show that αt ≤ y∗ t for t ∈ a, σ 3 b and
2
Δ2
αΔ t ≥ y∗ t for t ∈ a, σb. This completes the proof.
2
Theorem 3.4. Assume that the conditions (H1)–(H4) are satisfied. Then there exist two monotone
sequences {αn } and {βn }, nonincreasing and nondecreasing, respectively, with α0 α and β0 β,
which converge to the extremal solutions in β, α of the problem 1.1.
Proof. For any function ϑ which satisfies αt ≤ ϑt ≤ βt for t ∈ a, σ 2 b, consider the
following problem:
4
2
2
yΔ t − qtyΔ σt f t, ϑσt, ϑΔ σt ,
y σ 2 b λyΔ σ 2 b 0,
ya 0,
ζyΔ ξ1 − ηyΔ ξ1 0,
2
t ∈ a, b,
3
γyΔ ξ2 δyΔ ξ2 0.
2
3
3.12
12
Abstract and Applied Analysis
Clearly, this problem is type 2.12. Obviously, it has a unique solution given by the
expression
yt σ 2 b
a
G2 t, ξ
ξ2
ξ1
2
G1 ξ, sf s, ϑσs, ϑΔ σs ΔsΔξ ≡ Aϑt,
t ∈ a, σ 4 b .
3.13
Step 1. α ≤ Aα, Aβ ≤ β on a, σ 3 b and αΔ ≥ AαΔ , AβΔ ≥ βΔ on a, σb.
Let Aαt wt. Thus
2
2
2
w − αΔ t − qtw − αΔ σt
2
≥ f t, ασt, αΔ σt
2
−f t, ασt, αΔ σt 0,
4
2
2
t ∈ a, b,
3.14
w − α σ 2 b λw − αΔ σ 2 b ≥ 0,
w − αa ≥ 0,
ζw − αΔ ξ1 − ηw − αΔ ξ1 ≤ 0,
2
γw − αΔ ξ2 δw − αΔ ξ2 ≤ 0.
3
2
3
Using Lemma 2.6, we obtain that α ≤ Aα, αΔ ≥ AαΔ on a, σ 2 b and a, σb, respectively. In case that σ 2 b is right scattered, the assumptions λ ≥ σ 3 b − σ 2 b and w −
ασ 2 b w − αΔ σ 2 b ≥ 0 imply that ασ 3 b ≤ Aασ 3 b. Similarly, we show that
2
2
Aβt ≤ βt, AβΔ t ≥ βΔ t on a, σ 3 b and a, σb, respectively.
2
2
Step 2. If y1 , y2 ∈ α, β; y1Δ , y2Δ ∈ βΔ , αΔ , y1 ≤ y2 and y2Δ ≤ y1Δ , then we have Ay1 ≤ Ay2
2
2
and Ay1 Δ ≥ Ay2 Δ .
Let Ay1 w1 and Ay2 w2 . Thus
2
2
2
2
2
2
w2 − w1 Δ t − qtw2 − w1 Δ σt
2
f t, y2 σt, y2Δ σt
2
−f t, y1 σt, y1Δ σt ≥ 0, t ∈ a, b,
4
w2 − w1 a 0,
2
w2 − w1 σ 3 b λw2 − w1 Δ σ 3 b 0,
ζw2 − w1 Δ ξ1 − ηw2 − w1 Δ ξ1 0,
2
γw2 − w1 Δ ξ2 δw2 − w1 Δ ξ2 0.
3.15
3
2
3
Hence we get that Ay1 t ≤ Ay2 t and Ay1 Δ t ≥ Ay2 Δ t on a, σ 3 b and a, σb,
respectively.
Now, we define the sequences {αn t} and {βn t} by
2
α0 t αt,
αn1 t Aαn t,
β0 t βt,
2
βn1 t Aβn t,
for n ≥ 0.
3.16
Abstract and Applied Analysis
13
From the properties of A, we have
α α0 ≤ α1 ≤ · · · ≤ βn ≤ · · · ≤ β1 ≤ β0 β,
3.17
Δ
Δ
Δ
Δ
Δ
αΔ αΔ
0 ≤ α1 ≤ · · · ≤ βn ≤ · · · ≤ β1 ≤ β0 β .
2
2
2
2
2
2
2
But then α∗ and β∗ defined by
α∗ lim αn ,
β∗ lim βn ,
n→∞
n→∞
α∗
Δ2
lim αΔ
n ,
β∗
2
n→∞
Δ2
lim βn
n→∞
3.18
are extremal solutions of 1.1.
Example 3.5. Consider the BVP
3
y t − t y σt −y σt, t ∈
,4 ,
2
3
y
0,
y σ 2 4 yΔ σ 2 4 0,
2
Δ4
2 Δ2
yΔ 2 − 3yΔ 2 0,
2
3
Δ2
3.19
7yΔ 3 8yΔ 3 0,
2
3
where σ4 ≤ 29/7 and σ 3 4 − σ 2 4 ≤ 1. It is easy to check that αt 0, βt t are lower
and upper solutions of the BVP 3.19, respectively, and that all assumptions of Theorem 3.3
are fulfilled. So the BVP 3.19 has a solution yt satisfying 0 ≤ yt ≤ t for t ∈ 3/2, σ 3 4,
2
yΔ t 0 for t ∈ 3/2, σ4.
References
1 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Application,
Birkhäuser Boston, Boston, Mass, USA, 2001.
2 M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston, Boston,
Mass, USA, 2003.
3 E. Akin, “Boundary value problems for a differential equation on a measure chain,” Panamerican
Mathematical Journal, vol. 10, no. 3, pp. 17–30, 2000.
4 F. M. Atici and A. Cabada, “Existence and uniqueness results for discrete second-order periodic
boundary value problems,” Computers & Mathematics with Applications, vol. 45, no. 6-9, pp. 1417–1427,
2003.
5 P. W. Eloe and Q. Sheng, “Approximating crossed symmetric solutions of nonlinear dynamic equations
via quasilinearization,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 56, no. 2, pp. 253–272,
2004.
6 B. Kaymakçalan, “Monotone iterative method for dynamic systems on time scales,” Dynamic Systems
and Applications, vol. 2, no. 2, pp. 213–220, 1993.
7 S. Leela and S. Sivasundaram, “Dynamic systems on time scales and superlinear convergence of
iterative process,” WSSIAA, vol. 3, pp. 431–436, 1994.
14
Abstract and Applied Analysis
8 Y. Pang and Z. Bai, “Upper and lower solution method for a fourth-order four-point boundary value
problem on time scales,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2243–2247, 2009.
9 F. M. Atici and G. S. Guseinov, “On Green’s functions and positive solutions for boundary value
problems on time scales,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 75–
99, 2002.
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