Báo cáo sinh học: " Research Article Approximation of Solution of Some m-Point Boundary Value Problems on Time Scales Rahmat Ali Khan1 and Mohammad Rafique2 1" pot

Volume 2010, Article ID 841643, 11 pagesdoi:10.1155/2010/841643 Research Article Approximation of Solution of Some m-Point Boundary Value Problems on Time Scales 1 Centre for Advanced Ma

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

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: Ondˇrej Doˇsl ´y

Copyrightq 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 xΔΔt  ft, x σ,

t ∈ 0, 1T 0, 1 ∩ T, x0  0, xσ21 m−1

i1 α i x η i , η i ∈ 0, 1T,m−1

i1 α i≤ 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 to1

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 justR 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 Peterson3,4

Recently, existence theory for positive solutions of boundary value problemsBVPs

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, and13–16 for

three-point BVPs on time scales For the existence of solutions of m-three-point BVPs on time scales, we

refer the readers to17

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 scales18–21 The pioneering paper on multipoint BVPs on time scales has been the one

wide-ranging existence results Further, the authors of21 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 in16 Motivated by the work in 16,17, in this paper, we extend the results studied in16 to a class of m-point BVPs of the type

xΔΔt  ft, x σ t, t ∈ 0, 1T,

x 0  0, xσ21m−1

i1

α i x

η i



,

1.1

where η i ∈ 0, 1T,m−1

i1 α i ≤ 1, and t is from a so-called time scale T which is an arbitrary

closed subset ofR 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

define the time scale interval0, 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 right-dense points ofT 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 ofT and its right-sided limit exists at right-dense points ofT Define Tk  T − {m} if T has a left-scattered maximum at m; otherwiseTk  T For f : T → R and t ∈ T k , the delta derivative fΔt of f

at t if exists is defined by the following Given that  > 0, there exists a neighborhood U of t

such that



 f

If there exists a function F : T → R such that FΔt  ft for all t ∈ T, F is said to be the delta antiderivative of f and the delta integral is defined by

b

a

Definition 1.1 Define C2rd0, σ21T to be the set of all functions y : T → R such that

C2rd

0, σ21

T



y : y, yΔ∈ C0, σ21

T



and yΔΔ∈ Crd0, 1T. 1.4

A solution of1.1 is a function y ∈ C2

rd0, σ21T which satisfies 1.1 for each t ∈ 0, 1T Let us denote

Crd0, 1T× R y t, x : y·, x is Crd0, 1T for every x ∈ R and yt, ·

is continuous on R uniformly at each t ∈ 0, 1T,

C2rd0, 1T× R y t, x : y·, x, y x ·, x, y x ·, x are Crd0, 1T

for every x ∈ R and yt, ·, y x t, ·, y xx t, ·

are continuous onR uniformly at each t ∈ 0, 1T.

1.5

The purpose of this paper is to develop the method of upper and lower solutions and the method of quasilinearization22–26 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 BVP1.1 as an equivalent Δ-integral equation

x t 

σb

a

where Gt, s is a Green’s function for the problem

yΔΔt , t ∈ 0, 1T,

σ21−m−1

i1

α i x

η i

and it is given by17

G t, s 

⎧

⎪

⎪

⎨

⎪

⎪

⎩

t



1 1

T

k

i1

α i

σ

η i

− σs− σsα



, t ≤ s, ση k

≤ s ≤ η k 1 ,

σ s t

T

 k



i1

α i

σ

η i

− σs− σsα



, σ s ≤ t, ση k

≤ s ≤ η k 1 ,

2.3

where k  0, 1, 2, , m − 1, η0 0, and η k 1  σ21

Notice that Gt, s > 0 on 0, σ21T × 0, σ1T and is rd-continuous Define an

operator N : C0, σ21T → C0, σ21Tby

Nxt 

σ1

0

By a solution of2.1, we mean a solution of the operator equation

where I is the identity If f ∈ C0, 1T× R and is bounded on 0, 1T× R, then by

Arzela-Ascoli 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, σ21T× R

Definition 2.1 We say that α ∈ C2

rd0, σ21Tis a lower solution of the BVP1.1, if

αΔΔt ≥ ft, α σ t, t ∈ 0, 1T,

α 0 ≤ 0, ασ21≤m−1

i1 α i α

η i

Similarly, β ∈ C2

rd0, σ21Tis an upper solution of the BVP1.1 if

βΔΔt ≤ ft, β σ t, t ∈ 0, 1T,

β 0 ≥ 0, βσ21≥m−1

i1 α i β

η i

Theorem 2.2 (comparison result) Assume that α, β are lower and upper solutions of the boundary

value problem 1.1 If ft, x ∈ Crd0, 1T × R and is strictly increasing in x for each t ∈

0, σ21T, then α ≤ β on 0, σ21T.

Proof Define v t  αt − βt, t ∈ 0, σ21T Then v ∈ C2

rd0, σ21Tand the BCs imply that

v 0 ≤ 0, vσ21≤m−1

i1

α i v

η i

Assume that the conclusion of the theorem is not true Then, v has a positive maximum at some t0∈ 0, σ21T Clearly, t0 > 0 If t0∈ 0, σ21T, then, the point t0is not simultaneously left dense and right scattered; see, for example,12 Hence by Lemma 1 of 12,

vΔΔ

On the other hand, using the definitions of lower and upper solutions, we obtain

vΔΔ

ρ t0 αΔΔ

ρ t0− βΔΔ

ρ t0≥ fρ t0, α σ

ρ t0− fρ t0, β σ

ρ t0. 2.10

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

a contradiction Hence vt has no positive local maximum.

If t0  σ21, then vσ21 > 0 If any one of the η i is such that vη i   vσ21, then v

has a positive local maximum, a contradiction Hence

v

η i

< v

Moreover, if α i  0 for each i  1, 2, 3, , m − 1, then, from the BCs

v

σ21≤m−1

i1

α i v

η i



we have vσ21 ≤ 0, a contradiction Hence, α i /  0 for some i  1, 2, 3, , m − 1, and

consequently, in view of2.12 and the BCs, it follows that

v

σ21≤m−1

i1

α i v

η i

<

m−1

i1

α i v

Hence,1 −m−1 i1 α i vσ21 < 0, which leads tom−1 i1 α i > 1, a contradiction.

Hence t0/  σ21 Thus, vt ≤ 0 on 0, σ21T

Corollary 2.3 Under the hypotheses of Theorem 2.2 , the solutions of the BVP1.1, if they exist, are

unique.

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, σ21T If f t, x ∈ Crd0, 1T× R, then the BVP 1.1 has a solution x such that

The proof essentially is a minor modification of the ideas in21 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

converges uniformly and quadratically to a solution of the nonlinear original problem If

∂2/∂x2ft, x ∈ C0, 1T× R and is bounded on 0, σ21T× α, β, where

α minα t, t ∈ 0, σ21T, β maxβ t, t ∈ 0, σ21T, 3.1 there always exists a functionΦ such that

∂2

∂x2

whereΦ ∈ C2

rd0, σ21T× R, and it is such that ∂2/∂x2Φt, x ≤ 0 on 0, σ21T× α, β For example, let M  max{|f xx t, x| : t, x ∈ 0, σ21T× α, β}, then we choose Φ  −t −

M/2x2 Clearly,

∂2

∂x2

Define F : 0, 1T× R → R by Ft, x  ft, x Φt, x Note that F ∈ C2

rd0, σ21T× R and

∂2

Theorem 3.1 Assume that

A1 α, β are lower and upper solutions of the BVP 1.1 such that α ≤ β on 0, σ21T,

A2 f ∈ C2

rd 0, σ21T× R and f is increasing in x for each t ∈ 0, σ21T Then, there exists a monotone sequence {w n } of solutions of linear problems converging uniformly

and quadratically to a unique solution of the BVP1.1

Proof Conditions A1 and A2 ensure the existence of a unique solution x of the BVP 1.1

such that

In view of3.4, we have

f t, x ≤ ft, y

F x



t, y

x − y− Φt, x − Φt, y , on0, 1T×α, β 3.6

0, σ21Tyield

Φt, x − Φt, y

 Φx t, cx − y≥ Φx



t, β

where x, y ∈ α, β such that y ≤ c ≤ x Substituting in 3.6, we have

f t, x ≤ ft, y

F x



t, y

− Φx



on0, σ21T× α, β Define g : 0, σ21T× R2 → R by

g

t, x, y

 ft, y

F x

t, y

− Φx



We note that gt, ,  is continuous for each t ∈ 0, 1Tand g., x, y is rd-continuous for each

x, y ∈ R2 Moreover, g satisfies the following relations on 0, 1T× α, β:

g x



t, x, y

 F x



t, y

− Φx



t, β

≥ F x



t, y

− Φx



t, y

 f x



t, y

f t, x ≤ gt, x, y

, for x ≥ y,

Now, we develop the iterative scheme to approximate the solution As an initial

approxima-tion, we choose w0 α and consider the linear problem

xΔΔt  gt, x σ t, w σ

0t, t ∈ 0, 1T,

σ21m−1

i1

α i x

η i

Using3.11 and the definition of lower and upper solutions, we get

g

t, w σ0t, w σ

0t ft, w σ0t≤ wΔΔ

0 t, t ∈ 0, 1T,

g

t, β σ t, w σ

0t≥ ft, β σ t≥ βΔΔt, t ∈ 0, 1T,

3.13

which imply that w0 and β are lower and upper solutions of3.12, respectively Hence by Theorem 2.4andCorollary 2.3, there exists a unique solution w1 ∈ C2

rd0, σ21T of 3.12 such that

w0t ≤ w1t ≤ βt, on0, σ21

Using3.11 and the fact that w1is a solution of3.12, we obtain

wΔΔ1 t  gt, w σ1t, w σ

0t≥ ft, w σ1t, t ∈ 0, 1T,



σ21m−1

i1

α i w1



η i



,

3.15

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 w1and β are lower and upper solutions of the problem

xΔΔt  gt, x σ t, w σ

1t, t ∈ 0, 1T,

σ21m−1

i1

α i x

η i



.

3.16

Hence byTheorem 2.4andCorollary 2.3, there exists a unique solution w2∈ C2

rd0, σ21Tof the problem3.16 such that

w1t ≤ w2t ≤ βt, on0, σ21

solutions of linear problems satisfying

w0t ≤ w1t ≤ w2t ≤ w3t ≤ · · · ≤ w n t ≤ βt, on 0, σ21T, 3.18

where the element w nof the sequence is a solution of the linear problem

xΔΔt  gt, x σ t, w σ

n−1 t, t ∈ 0, 1T,

σ21m−1

i1

α i x

η i

and is given by

w n t 

σ1

0

G t, sgs, w σ n s, w σ

n−1 sΔs, t ∈0, σ21

By standard arguments as in19, the sequence converges to a solution of 1.1

Now, we show that the convergence is quadratic Set v n 1 t  xt − w n 1 t, t ∈

0, σ21T, where x is a solution of1.1 Then, vn 1 t ≥ 0 on 0, σ21Tand the boundary conditions imply that

v n 1 0  0, v n 1



σ21m−1

i1

α i v n 1

η i



Now, in view of the definitions of F and g, we obtain

v nΔΔt  ft, x σ t − gs, w n σ t, w σ

n−1 t

 Ft, x σ t − Φt, x σ t

−f

t, w σ n−1 t F x

t, w σ n−1 t− Φx



t, β

w σ n t − w σ

n−1 t

 F t, x σ t − Ft, w n−1 σ t− F x



t, w σ n−1 tw n σ t − w σ

n−1 t

−Φt, x σ t − Φt, w n−1 σ t− Φx



t, β

w n σ t − w σ

n−1 t , t ∈ 0, 1T.

3.22

Using the mean value theorem repeatedly and the fact thatΦxx ≤ 0 on 0, 1T× α, β, we

obtain

Φt, x σ t − Φt, w σ n−1 t≤ Φx



t, w σ n−1 tx σ t − w σ

n−1 t,

F t, x σ t − Ft, w n−1 σ t− F x



t, w σ n−1 tw n σ t − w σ

n−1 t

 F x



t, w n−1 σ tx σ t − w σ

n−1 t F xx t, ξ

2



x σ t − w σ

n−1 t2

− F x



t, w σ n−1 tw n σ t − w σ

n−1 t

 F x



t, w n−1 σ tx σ t − w σ

n t F xx t, ξ

2



x σ t − w σ

n−1 t2

≥ F x



t, w n−1 σ tx σ t − w σ

n t − dv n−12,

3.23

n−1 t ≤ ξ ≤ x σ t, d  max{|F xx t, x|/2 : t, x ∈ 0, σ21T× α, β}, and v 

max{vt : t ∈ 0, σ21T} Hence 3.22 can be rewritten as

vΔΔn t ≥ F x



t, w σ n−1 tx σ t − w σ

n t − dv n−12

− Φx



t, w σ n−1 tx σ t − w σ

n−1 t Φx



t, β

w n σ t − w σ

n−1 t

 f x



t, w n−1 σ tx σ t − w σ

n t − dv n−12

Φx



t, β

− Φx



t, w σ n−1 t w n σ t − w σ

n−1 t

 f x



t, w n−1 σ 

x σ t − w σ

n t − dv n−12 Φxx t, ξ1β − w σ

n−1 t

w σ n t − w σ

n−1 t

≥ f x



t, w n−1 σ 

x σ t − w σ

n t − dv n−12 Φxx t, ξ1β − w σ

n−1 t

x σ n t − w σ

n−1 t

≥ −dv n−12− d1β − w σ

n−1 tx σ t − w σ

where w σ n−1 t ≤ ξ1≤ w σ t, d1 max{|Φxx | : t, x ∈ 0, 1T× α, β}, and we used the fact that

f x ≥ 0 on 0, 1T× α, β Choose r > 1 such that

β σ t − w σ

n−1 t ≤ rx σ t − w σ

Therefore, we obtain

where d2 d rd1

By comparison result, v n t ≤ zt, t ∈ 0, 1T, where zt is the unique solution of the

linear BVP

zΔΔt  d2v n−12, t ∈ a, bT,

z 0  0, zσ21m−1

i1

α i z

η i

.

3.27

Hence,

v n t ≤ zt  d2

σ1

0

where d3  d2max{0σ1 |Gt, s|Δs : t ∈ 0, σ21T} Taking the maximum over 0, 1T, we obtain

which shows the quadratic convergence

Acknowledgement

The authors are thankful to reviewers for their valuable comments and suggestions

References

1 F B Christiansen and T M Fenchel, “Theories of Populations in Biological Communities, Ecological Studies,” vol 20, Springer, Berlin, Germany, 1977

2 S Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,”

Results in Mathematics, vol 18, no 1-2, pp 18–56, 1990.

3 M Bohner and A Peterson, Dynamic Equations on Time Scales: An Introduction with Applications,

Birkh¨auser, Boston, Mass, USA, 2001

4 M Bohner and A Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston,

Mass, USA, 2003

5 R P Agarwal and D O’Regan, “Nonlinear boundary value problems on time scales,” Nonlinear

Analysis: Theory, Methods & Applications, vol 44, no 4, pp 527–535, 2001.

f, there exists a bounded monotone sequence of solutions of linear problems that