Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 841643, 11 pages
doi:10.1155/2010/841643
Research Article
Approximation of Solution of Some m-Point
Boundary Value Problems on Time Scales
Rahmat Ali Khan1 and Mohammad Rafique2
1
Centre for Advanced Mathematics and Physics, National University of Sciences and Technology (NUST),
H-12, Islamabad 46000, Pakistan
2
Department of Basic Sciences, College of Electrical and Mechanical Engineering, National University of
Sciences and Technology (NUST), Peshawar Road, Rawalpindi 46000, Pakistan
Correspondence should be addressed to Mohammad Rafique, mrdhillon@yahoo.com
Received 24 August 2009; Revised 13 May 2010; Accepted 2 June 2010
Academic Editor: Ondřej Došlý
Copyright q 2010 R. A. Khan and M. Rafique. 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.
The method of upper and lower solutions and the generalized quasilinearization technique for
second-order nonlinear m-point dynamic equations on time scales of the type xΔΔ t ft, xσ ,
m−1
t ∈ 0, 1T 0, 1 ∩ T, x0 0, xσ 2 1 m−1
i1 αi xηi , ηi ∈ 0, 1T ,
i1 αi ≤ 1, are developed.
A monotone sequence of solutions of linear problems converging uniformly and quadratically to
a solution of the problem is obtained.
1. Introduction
Many dynamical processes contain both continuous and discrete elements simultaneously.
Thus, traditional mathematical modeling techniques, such as differential equations or
difference equations, provide a limited understanding of these types of models. A simple
example of this hybrid continuous-discrete behavior appears in many natural populations:
for example, insects that lay their eggs at the end of the season just before the generation dies
out, with the eggs laying dormant, hatching at the start of the next season giving rise to a
new generation. For more examples of species which follow this type of behavior, we refer
the readers to 1.
Hilger 2 introduced the notion of time scales in order to unify the theory of
continuous and discrete calculus. The field of dynamical equations on time scales contain,
links and extends the classical theory of differential and difference equations, besides many
others. There are more time scales than just R corresponding to the continuous case and N
corresponding to the discrete case and hence many more classes of dynamic equations. An
excellent resource with an extensive bibliography on time scales was produced by Bohner
and Peterson 3, 4.
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Advances in Difference Equations
Recently, existence theory for positive solutions of boundary value problems BVPs
on time scales has attracted the attention of many authors; see, for example, 5–12 and the
references therein for the existence theory of some two-point BVPs, and 13–16 for threepoint BVPs on time scales. For the existence of solutions of m-point BVPs on time scales, we
refer the readers to 17.
However, the method of upper and lower solutions and the quasilinearization
technique for BVPs on time scales are still in the developing stage and few papers are devoted
to the results on upper and lower solutions technique and the method of quasilinearization on
time scales 18–21. The pioneering paper on multipoint BVPs on time scales has been the one
in 21 where lower and upper solutions were combined with degree theory to obtain very
wide-ranging existence results. Further, the authors of 21 studied existence results for more
general three-point boundary conditions which involve first delta derivatives and they also
developed some compatibility conditions. We are very grateful to the reviewer for directing
us towards this important work.
Recently, existence results via upper and lower solutions method and approximation
of solutions via generalized quasilinearization method for some three-point boundary value
problems on time scales have been studied in 16. Motivated by the work in 16, 17, in this
paper, we extend the results studied in 16 to a class of m-point BVPs of the type
xΔΔ t ft, xσ t,
t ∈ 0, 1T ,
m−1
x0 0, x σ 2 1 αi x ηi ,
1.1
i1
where ηi ∈ 0, 1T , m−1
i1 αi ≤ 1, and t is from a so-called time scale T which is an arbitrary
closed subset of R. Existence of at least one solution for 1.1 has already been studied in 17
by the Krasnosel’skii and Zabreiko fixed point theorems. We obtain existence and uniqueness
results and develop a method to approximate the solutions.
Assume that T has a topology that it inherits from the standard topology on R and
define the time scale interval 0, 1T {t ∈ T : 0 ≤ t ≤ 1}. For t ∈ T, define the forward jump
operator σ : T → T by σt inf{s ∈ T : s > t} and the backward jump operator ρ : T → T
by ρt sup{s ∈ T : s < t}. If σt > t, t is said to be right scattered, and if σt t, t is said to
be right dense. If ρt < t, t is said to be left scattered, and if ρt t, t is said to be left dense.
A function f : T → R is said to be rd-continuous provided it is continuous at all rightdense points of T and its left-sided limit exists at left-dense points of T. A function f : T → R
is said to be ld-continuous provided it is continuous at all left-dense points of T and its rightsided limit exists at right-dense points of T. Define Tk T − {m} if T has a left-scattered
maximum at m; otherwise Tk T. For f : T → R and t ∈ Tk , the delta derivative f Δ t of f
at t if exists is defined by the following Given that > 0, there exists a neighborhood U of t
such that
1.2
f σt − fs − f Δ tσt − s ≤ |σt − s|, ∀s ∈ U.
If there exists a function F : T → R such that F Δ t ft for all t ∈ T, F is said to be the
delta antiderivative of f and the delta integral is defined by
b
a
fτΔτ Fb − Fa,
a, b ∈ T.
1.3
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3
2
0, σ 2 1T to be the set of all functions y : T → R such that
Definition 1.1. Define Crd
2
Crd
0, σ 2 1
T
y : y, yΔ ∈ C 0, σ 2 1
and yΔΔ ∈ Crd 0, 1T .
T
1.4
2
A solution of 1.1 is a function y ∈ Crd
0, σ 2 1T which satisfies 1.1 for each t ∈ 0, 1T .
Let us denote
Crd 0, 1T × R yt, x : y·, x is Crd 0, 1T for every x ∈ R and yt, ·
is continuous on R uniformly at each t ∈ 0, 1T ,
2
Crd
0, 1T × R yt, x : y·, x, yx ·, x, yx ·, x are Crd 0, 1T
1.5
for every x ∈ R and yt, ·, yx t, ·, yxx t, ·
are continuous on R uniformly at each t ∈ 0, 1T .
The purpose of this paper is to develop the method of upper and lower solutions and the
method of quasilinearization 22–26. Under suitable conditions on f, we obtain a monotone
sequence of solutions of linear problems. We show that the sequence of approximants
converges uniformly and quadratically to a unique solution of the problem.
2. Upper and Lower Solutions Method
We write the BVP 1.1 as an equivalent Δ-integral equation
xt σb
Gt, sfs, xσ sΔs,
a
2.1
t ∈ 0, σ 2 1T ,
where Gt, s is a Green’s function for the problem
yΔΔ t ,
y0 0,
t ∈ 0, 1T ,
2.2
m−1
y σ 2 1 −
αi x ηi 0,
i1
and it is given by 17
Gt, s ⎧
k
⎪
1 ⎪
⎪
, t ≤ s, σ ηk ≤ s ≤ ηk1 ,
αi σ ηi − σs − σsα
t 1
⎪
⎪
⎪
T i1
⎨
⎪
⎪
k
⎪
t ⎪
⎪
⎪
αi σ ηi − σs − σsα ,
⎩σs T
i1
where k 0, 1, 2, . . . , m − 1, η0 0, and ηk1 σ 2 1.
σs ≤ t, σ ηk ≤ s ≤ ηk1 ,
2.3
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Advances in Difference Equations
Notice that Gt, s > 0 on 0, σ 2 1T × 0, σ1T and is rd-continuous. Define an
operator N : C0, σ 2 1T → C0, σ 2 1T by
Nxt σ1
Gt, sfs, xσ sΔs,
0
t ∈ 0, σ 2 1T .
2.4
By a solution of 2.1, we mean a solution of the operator equation
I − Nx 0,
that is, a fixed point of N,
2.5
where I is the identity. If f ∈ C0, 1T × R and is bounded on 0, 1T × R, then by ArzelaAscoli theorem N is compact and Schauder’s fixed point theorem yields a fixed point of N.
We discuss the case when f is not necessarily bounded on 0, σ 2 1T × R.
2
Definition 2.1. We say that α ∈ Crd
0, σ 2 1T is a lower solution of the BVP 1.1, if
αΔΔ t ≥ ft, ασ t,
α0 ≤ 0,
α σ 2 1 ≤
t ∈ 0, 1T ,
αi α ηi .
m−1
2.6
i1
2
0, σ 2 1T is an upper solution of the BVP 1.1 if
Similarly, β ∈ Crd
βΔΔ t ≤ f t, βσ t ,
β0 ≥ 0,
β σ 2 1 ≥
t ∈ 0, 1T ,
αi β ηi .
m−1
2.7
i1
Theorem 2.2. (comparison result) Assume that α, β are lower and upper solutions of the boundary
value problem 1.1. If ft, x ∈ Crd 0, 1T × R and is strictly increasing in x for each t ∈
0, σ 2 1T , then α ≤ β on 0, σ 2 1T .
2
Proof. Define vt αt − βt, t ∈ 0, σ 2 1T . Then v ∈ Crd
0, σ 2 1T and the BCs imply that
m−1
v0 ≤ 0, v σ 2 1 ≤
αi v ηi .
2.8
i1
Assume that the conclusion of the theorem is not true. Then, v has a positive maximum at
some t0 ∈ 0, σ 2 1T . Clearly, t0 > 0. If t0 ∈ 0, σ 2 1T , then, the point t0 is not simultaneously
left dense and right scattered; see, for example, 12. Hence by Lemma 1 of 12,
vΔΔ ρt0 ≤ 0.
2.9
On the other hand, using the definitions of lower and upper solutions, we obtain
vΔΔ ρt0 αΔΔ ρt0 − βΔΔ ρt0 ≥ f ρt0 , ασ ρt0 − f ρt0 , βσ ρt0 .
2.10
Advances in Difference Equations
5
Since t0 is not simultaneously left dense and right scattered, it is left scattered and right
scattered, left dense and right dense, or left scattered and right dense. In either case σρt0 t0 . Using the increasing property of ft, x in x, we obtain
vΔΔ ρt0 > 0,
2.11
a contradiction. Hence vt has no positive local maximum.
If t0 σ 2 1, then vσ 2 1 > 0. If any one of the ηi is such that vηi vσ 2 1, then v
has a positive local maximum, a contradiction. Hence
v ηi < v σ 2 1 ,
for each i 1, 2, 3, . . . , m − 1.
2.12
Moreover, if αi 0 for each i 1, 2, 3, . . . , m − 1, then, from the BCs
m−1
αi v ηi ,
v σ 2 1 ≤
2.13
i1
0 for some i 1, 2, 3, . . . , m − 1, and
we have vσ 2 1 ≤ 0, a contradiction. Hence, αi /
consequently, in view of 2.12 and the BCs, it follows that
m−1
m−1
v σ 2 1 ≤
αi v ηi <
αi v σ 2 1 .
i1
2.14
i1
m−1
2
Hence, 1 − m−1
i1 αi vσ 1 < 0, which leads to
i1 αi > 1, a contradiction.
2
2
1.
Thus,
vt
≤
0
on
0,
σ
1
σ
Hence t0 /
T.
Corollary 2.3. Under the hypotheses of Theorem 2.2, the solutions of the BVP 1.1, if they exist, are
unique.
The following theorem establishes existence of solutions to the BVP 1.1 in the
presence of well-ordered lower and upper solutions.
Theorem 2.4. Assume that α, β are lower and upper solutions of the BVP 1.1 such that α ≤
β on 0, σ 2 1T . If ft, x ∈ Crd 0, 1T × R, then the BVP 1.1 has a solution x such that
α ≤ x ≤ β,
on 0, σ 2 1T .
2.15
The proof essentially is a minor modification of the ideas in 21 and so is omitted.
3. Generalized Approximations Technique
We develop the approximation technique and show that, under suitable conditions on
f, there exists a bounded monotone sequence of solutions of linear problems that
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Advances in Difference Equations
converges uniformly and quadratically to a solution of the nonlinear original problem. If
∂2 /∂x2 ft, x ∈ C0, 1T × R and is bounded on 0, σ 2 1T × α, β, where
α min αt, t ∈ 0, σ 2 1T ,
β max βt, t ∈ 0, σ 2 1T ,
3.1
there always exists a function Φ such that
∂2 ft, x Φt, x ≤ 0,
∂x2
on 0, 1T × α, β ,
3.2
2
0, σ 2 1T × R, and it is such that ∂2 /∂x2 Φt, x ≤ 0 on 0, σ 2 1T × α, β.
where Φ ∈ Crd
For example, let M max{|fxx t, x| : t, x ∈ 0, σ 2 1T × α, β}, then we choose Φ −t −
M/2x2 . Clearly,
∂2 ft, x Φt, x ≤ 0,
2
∂x
on 0, σ 2 1T × α, β .
3.3
2
Define F : 0, 1T × R → R by Ft, x ft, x Φt, x. Note that F ∈ Crd
0, σ 2 1T × R and
∂2
Ft, x ≤ 0,
∂x2
on 0, 1T × α, β .
3.4
Theorem 3.1. Assume that
A1 α, β are lower and upper solutions of the BVP 1.1 such that α ≤ β on 0, σ 2 1T ,
2
A2 f ∈ Crd
0, σ 2 1T × R and f is increasing in x for each t ∈ 0, σ 2 1T .
Then, there exists a monotone sequence {wn } of solutions of linear problems converging uniformly
and quadratically to a unique solution of the BVP 1.1.
Proof. Conditions A1 and A2 ensure the existence of a unique solution x of the BVP 1.1
such that
αt ≤ xt ≤ βt,
3.5
t ∈ 0, σ 2 1T .
In view of 3.4, we have
ft, x ≤ f t, y Fx t, y x − y − Φt, x − Φ t, y ,
on 0, 1T × α, β .
3.6
The mean value theorem and the fact that Φx is nonincreasing in x on α, β for each t ∈
0, σ 2 1T yield
Φt, x − Φ t, y Φx t, c x − y ≥ Φx t, β x − y ,
for x ≥ y,
3.7
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7
where x, y ∈ α, β such that y ≤ c ≤ x. Substituting in 3.6, we have
ft, x ≤ f t, y Fx t, y − Φx t, β
x−y ,
for x ≥ y,
3.8
on 0, σ 2 1T × α, β. Define g : 0, σ 2 1T × R2 → R by
g t, x, y f t, y Fx t, y − Φx t, β
x−y .
3.9
We note that gt, ., . is continuous for each t ∈ 0, 1T and g., x, y is rd-continuous for each
x, y ∈ R2 . Moreover, g satisfies the following relations on 0, 1T × α, β:
gx t, x, y Fx t, y − Φx t, β ≥ Fx t, y − Φx t, y fx t, y ≥ 0,
ft, x ≤ g t, x, y ,
for x ≥ y,
3.10
3.11
ft, x gt, x, x.
Now, we develop the iterative scheme to approximate the solution. As an initial approximation, we choose w0 α and consider the linear problem
xΔΔ t g t, xσ t, w0σ t ,
x0 0,
t ∈ 0, 1T ,
m−1
x σ 2 1 αi x ηi .
3.12
i1
Using 3.11 and the definition of lower and upper solutions, we get
g t, w0σ t, w0σ t f t, w0σ t ≤ w0ΔΔ t, t ∈ 0, 1T ,
g t, βσ t, w0σ t ≥ f t, βσ t ≥ βΔΔ t, t ∈ 0, 1T ,
3.13
which imply that w0 and β are lower and upper solutions of 3.12, respectively. Hence by
2
0, σ 2 1T of 3.12
Theorem 2.4 and Corollary 2.3, there exists a unique solution w1 ∈ Crd
such that
w0 t ≤ w1 t ≤ βt,
on 0, σ 2 1 .
T
3.14
Using 3.11 and the fact that w1 is a solution of 3.12, we obtain
w1ΔΔ t g t, w1σ t, w0σ t ≥ f t, w1σ t ,
w1 0 0,
t ∈ 0, 1T ,
m−1
αi w1 ηi ,
w1 σ 2 1 i1
3.15
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Advances in Difference Equations
which implies that w1 is a lower solution of the problem 1.1. Similarly, in view of A1 ,
3.11, and 3.15, we can show that w1 and β are lower and upper solutions of the problem
xΔΔ t g t, xσ t, w1σ t ,
x0 0,
t ∈ 0, 1T ,
3.16
m−1
x σ 2 1 αi x ηi .
i1
2
Hence by Theorem 2.4 and Corollary 2.3, there exists a unique solution w2 ∈ Crd
0, σ 2 1T of
the problem 3.16 such that
w1 t ≤ w2 t ≤ βt,
on 0, σ 2 1 .
3.17
T
Continuing in the above fashion, we obtain a bounded monotone sequence {wn } of
solutions of linear problems satisfying
w0 t ≤ w1 t ≤ w2 t ≤ w3 t ≤ · · · ≤ wn t ≤ βt,
on 0, σ 2 1T ,
3.18
where the element wn of the sequence is a solution of the linear problem
σ
xΔΔ t g t, xσ t, wn−1
t ,
x0 0,
t ∈ 0, 1T ,
3.19
m−1
x σ 2 1 αi x ηi ,
i1
and is given by
wn t σ1
0
σ
Gt, sg s, wnσ s, wn−1
s Δs,
t ∈ 0, σ 2 1 .
T
3.20
By standard arguments as in 19, the sequence converges to a solution of 1.1.
Now, we show that the convergence is quadratic. Set vn1 t xt − wn1 t, t ∈
0, σ 2 1T , where x is a solution of 1.1. Then, vn1 t ≥ 0 on 0, σ 2 1T and the boundary
conditions imply that
m−1
vn1 0 0, vn1 σ 2 1 αi vn1 ηi .
i1
3.21
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9
Now, in view of the definitions of F and g, we obtain
σ
vnΔΔ t ft, xσ t − g s, wnσ t, wn−1
t
Ft, xσ t − Φt, xσ t
σ
σ
σ
− f t, wn−1
wnσ t − wn−1
t Fx t, wn−1
t − Φx t, β
t
3.22
σ
σ
σ
Ft, xσ t − F t, wn−1
t − Fx t, wn−1
t wnσ t − wn−1
t
σ
σ
− Φt, xσ t − Φ t, wn−1
t − Φx t, β wnσ t − wn−1
t ,
t ∈ 0, 1T .
Using the mean value theorem repeatedly and the fact that Φxx ≤ 0 on 0, 1T × α, β, we
obtain
σ
σ
σ
Φt, xσ t − Φ t, wn−1
t ≤ Φx t, wn−1
t xσ t − wn−1
t ,
σ
σ
σ
Ft, xσ t − F t, wn−1
t − Fx t, wn−1
t wnσ t − wn−1
t
Fxx t, ξ σ
2
σ
σ
σ
Fx t, wn−1
x t − wn−1
t xσ t − wn−1
t t
2
σ
σ
− Fx t, wn−1
t wnσ t − wn−1
t
3.23
2
Fxx t, ξ σ
σ
σ
Fx t, wn−1
x t − wn−1
t xσ t − wnσ t t
2
σ
≥ Fx t, wn−1
t xσ t − wnσ t − dvn−1 2 ,
σ
t ≤ ξ ≤ xσ t, d max{|Fxx t, x|/2 : t, x ∈ 0, σ 2 1T × α, β}, and v where wn−1
max{vt : t ∈ 0, σ 2 1T }. Hence 3.22 can be rewritten as
σ
vnΔΔ t ≥ Fx t, wn−1
t xσ t − wnσ t − dvn−1 2
σ
σ
σ
− Φx t, wn−1
t xσ t − wn−1
t Φx t, β wnσ t − wn−1
t
σ
fx t, wn−1
t xσ t − wnσ t − dvn−1 2
σ
σ
Φx t, β − Φx t, wn−1
t wnσ t − wn−1
t
σ
σ
σ
σ
fx t, wn−1
t wnσ t − wn−1
t
x t − wnσ t − dvn−1 2 Φxx t, ξ1 β − wn−1
σ
σ
σ
σ
≥ fx t, wn−1
t xnσ t − wn−1
t
x t − wnσ t − dvn−1 2 Φxx t, ξ1 β − wn−1
σ
σ
≥ −dvn−1 2 − d1 β − wn−1
txσ t − wn−1
t,
t ∈ 0, 1T ,
3.24
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σ
where wn−1
t ≤ ξ1 ≤ wnσ t, d1 max{|Φxx | : t, x ∈ 0, 1T × α, β}, and we used the fact that
fx ≥ 0 on 0, 1T × α, β. Choose r > 1 such that
σ
β t − wσ t ≤ r xσ t − wσ t,
n−1
n−1
on 0, 1T .
3.25
Therefore, we obtain
vnΔΔ t ≥ −d2 vn−1 2 ,
t ∈ 0, 1T ,
3.26
where d2 d rd1 .
By comparison result, vn t ≤ zt, t ∈ 0, 1T , where zt is the unique solution of the
linear BVP
zΔΔ t d2 vn−1 2 ,
t ∈ a, bT ,
m−1
z0 0, z σ 2 1 αi z ηi .
3.27
i1
Hence,
vn t ≤ zt d2
σ1
Gt, svn−1 2 Δs ≤ d3 vn−1 2 ,
3.28
0
σ1
where d3 d2 max{ 0 |Gt, s|Δs : t ∈ 0, σ 2 1T }. Taking the maximum over 0, 1T , we
obtain
vn ≤ d3 vn−1 2 ,
3.29
which shows the quadratic convergence.
Acknowledgement
The authors are thankful to reviewers for their valuable comments and suggestions.
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