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Volume 2011, Article ID 929061, 17 pagesdoi:10.1155/2011/929061 Research Article Approximation of Solutions for Second-Order m-Point Nonlocal Boundary Value Problems via the Method of Ge

Volume 2011, Article ID 929061, 17 pages

doi:10.1155/2011/929061

Research Article

Approximation of Solutions for Second-Order

m-Point Nonlocal Boundary Value Problems via

the Method of Generalized Quasilinearization

Ahmed Alsaedi

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O Box 80203,

Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Ahmed Alsaedi,aalsaedi@hotmail.com

Received 11 May 2010; Revised 29 July 2010; Accepted 2 October 2010

Academic Editor: Gennaro Infante

Copyrightq 2011 Ahmed Alsaedi 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 discuss the existence and uniqueness of the solutions of a second-order m-point nonlocal

boundary value problem by applying a generalized quasilinearization technique A monotone sequence of solutions converging uniformly and quadratically to a unique solution of the problem

is presented

1 Introduction

The monotone iterative technique coupled with the method of upper and lower solutions

1 7 manifests itself as an effective and flexible mechanism that offers theoretical as well as constructive existence results in a closed set, generated by the lower and upper solutions In general, the convergence of the sequence of approximate solutions given by the monotone iterative technique is at most linear 8, 9 To obtain a sequence of approximate solutions converging quadratically, we use the method of quasilinearization10 This method has been developed for a variety of problems11–20 In view of its diverse applications, this approach

is quite an elegant and easier for application algorithms

The subject of multipoint nonlocal boundary conditions, initiated by Bicadze and Samarski˘ı 21, has been addressed by many authors, for instance, 22–32 The multipoint boundary conditions appear in certain problems of thermodynamics, elasticity and wave propagation, see 23 and the references therein The multipoint boundary conditions may

be understood in the sense that the controllers at the endpoints dissipate or add energy according to censors located at intermediate positions

In this paper, we develop the method of generalized quasilinearization to obtain a sequence of approximate solutions converging monotonically and quadratically to a unique

solution of the following second-order m−point nonlocal boundary value problem

−xt  ft, x t, xt, t ∈ 0, 1, 1.1

px 0 − qx0 m−2

i1 τix

ηi

, px 1  qx1 m−2

i1 σix

ηi

, ηi ∈ 0, 1, 1.2

where f : 0, 1 × R × R → R is continuous and τ i, σi i  1, 2, , m − 2 are nonnegative real

constants such thatm−2

i1 τi < 1,m−2

i1 σi < 1, and p, q > 0 with p > 1.

Here we remark that26 studies 1.1 with the boundary conditions of the form

δx 0 − γx0  0, x1  m−2

i1 αix

ηi

A perturbed integral equation equivalent to the problem1.1 and 1.3 considered in 26 is

x t 

1

0

k t, sfs, x s, xsds 

m−2

i1 αix

ηi

where

k t, s   1

δ  γ

⎧

⎨

⎩



γ  δt

1 − s, 0 ≤ t ≤ s,



δ  γs

It can readily be verified that the solution given by1.4 does not satisfy 1.1 On the other hand, by Green’s function method, a unique solution of the problem1.1 and 1.3 is

x t 

1

0

k t, sfs, x s, xsds 

m−2



i1 αix

ηiγ  δt

δ  γ , 1.6

where kt, s is given by 1.5 Thus, 1.6 represents the correct form of the solution for the problem1.1 and 1.3

2 Preliminaries

For x ∈ C10, 1, we define x1 x  x, where x  max{|xt| : t ∈ 0, 1} It can easily

be verified that the homogeneous problem associated with1.1-1.2 has only the trivial solution Therefore, by Green’s function method, the solution of1.1-1.2 can be written as

x t 

1

0

G t, sfs, x s, xsds 

m−2

i1 τix

ηi −t 2q  p q  p

p

2q  p





m−2



i1 σix

ηi t

p

2q  p



,

2.1

where Gt, s is the Green’s function and is given by

G t, s  1

p

p  2q

⎧

⎨

⎩



q  pt

q  p 1 − s, 0 ≤ t ≤ s,



q  ps

q  p 1 − t, s ≤ t ≤ 1. 2.2

Note that Gt, s > 0 on 0, 1 × 0, 1.

We say that α ∈ C20, 1 is a lower solution of the boundary value problem 1.1 and

1.2 if

−αt ≤ ft, α t, αt, t ∈ 0, 1,

pα 0 − qα0 ≤m−2

i1 τiα

ηi

, pα 1  qα1 ≤m−2

i1 σiα

ηi

,

2.3

and β ∈ C20, 1 is an upper solution of 1.1 and 1.2 if

−βt ≥ ft, β t, βt, t ∈ 0, 1,

pβ 0 − qβ0 ≥m−2

i1 τiβ

ηi

, pβ 1  qβ1 ≥m−2

i1 σiβ

ηi

Definition 2.1 A continuous function h : 0, ∞ → 0, ∞ is called a Nagumo function if

∞

λ

sds

for λ ≥ 0 We say that f ∈ C0, 1 × R × R satisfies a Nagumo condition on 0, 1 relative

to α, β if for every t ∈ 0, 1 and x ∈ min t∈0,1αt, maxt∈0,1βt, there exists a Nagumo

function h such that |ft, x, x| ≤ h|x|.

We need the following result33 to establish the main result

Theorem 2.2 Let f : 0, 1 × R2 → R be a continuous function satisfying the Nagumo condition

on E  {t, x, y ∈ 0, 1 × R2 : α ≤ x ≤ β} where α, β : 0, 1 → R are continuous functions such

that αt ≤ βt for all t ∈ 0, 1 Then there exists a constant M > 0 (depending only on α, β, the Nagumo function h) such that every solution x of 1.1-1.2 with αt ≤ xt ≤ βt, t ∈ 0, 1

satisfies |x| ≤ M.

If α, β ∈ C20, 1 are assumed to be lower and upper solutions of 1.1-1.2, respectively, in the statement of Theorem 2.2, then there exists a solution, xt of 1.1 and

1.2 such that αt ≤ xt ≤ βt, t ∈ 0, 1.

Theorem 2.3 Assume that α, β ∈ C20, 1 are, respectively, lower and upper solutions of 1.1-1.2.

If ft, x, y ∈ C0, 1 × R × R is decreasing in x for each t, y ∈ 0, 1 × R, then α ≤ β on 0, 1 Proof Let us define ut  αt − βt so that u ∈ C20, 1 and satisfies the boundary

conditions

pu 0 − qu0 ≤m−2

i1 τiu

ηi

, pu 1  qu1 ≤m−2

i1 σiu

ηi

For the sake of contradiction, let u have a positive maximum at some t0∈ 0, 1 If t0 ∈ 0, 1, then ut0  0 and ut0 ≤ 0 On the other hand, in view of the decreasing property of

ft, x, y in x, we have

ut0  αt0 − βt0 ≥ −ft0, α t0, αt0 ft0, β t0, βt0> 0, 2.7

which is a contradiction If we suppose that u has a positive maximum at t0  0, then it follows from the first of boundary conditions2.6 that

pu 0 − qu0 ≤m−2

i1 τiu

ηi

which implies thatp − 1u0 ≤ qu0 Now as p > 1, q > 0, u0 > 0, u0 ≤ 0, therefore

we obtain a contradiction We have a similar contradiction at t0  1 Thus, we conclude that

αt ≤ βt, t ∈ 0, 1.

3 Main Results

Theorem 3.1 Assume that

A1 the functions α, β ∈ C20, 1 are, respectively, lower and upper solutions of 1.1-1.2

such that α ≤ β on 0, 1;

A2 the function f ∈ C20, 1 × R × R satisfies a Nagumo condition relative to α, β

and fx ≤ 0 on 0, 1 × min t∈0,1αt, maxt∈0,1βt × −M, M, where M is a positive constant depending on α, β, and the Nagumo function h Further, there exists

a function φ ∈ C20, 1 × R2 such that Ψf  φ ≥ 0 with Ψφ ≥ 0 on 0, 1 ×

mint∈0,1αt, maxt∈0,1βt × −M, M, where

Ψ x − y2 ∂2

∂x2  2x − y

x− y ∂2

∂x∂xx− y2 ∂2

Then, there exists a monotone sequence {α n } of approximate solutions converging uniformly to a

unique solution of the problems1.1-1.2.

Proof For y ∈ R, we define ωy  max{−M, min{y, M}} and consider the following

modified m-point BVP

−xt  ft, x t, ωxt, t ∈ 0, 1,

px 0 − qx0 m−2

i1 τix

ηi

, px 1  qx1 m−2

i1 σix

ηi

.

3.2

We note that α, β are, respectively, lower and upper solutions of 3.2 and for every t, x ∈

0, 1 × min t∈0,1αt, maxt∈0,1βt, we have

∞

0

sds 

M

0

sds

h s 

∞

M

sds

Nagumo function h such that

M

0

sds ≥

N

0

sds

h s >

 max

β t : t ∈ 0, 1− min{αt : t ∈ 0, 1}, 3.5

where M > max{N, α, β} Thus, any solution x of 3.2 with αt ≤ xt ≤ βt, t ∈ 0, 1

satisfies|x| ≤ M on 0, 1 and hence it is a solution of 1.1-1.2

Let us define a function F : 0, 1 × R2 → R by

F

t, x, x

 ft, x, x

 φt, x, x− ωx

In view of the assumptionA2, it follows that F ∈ C20, 1 × R2 and satisfies ΨF ≥ 0 on

0, 1 × min t∈0,1αt, maxt∈0,1βt × −M, M Therefore, by Taylor’s theorem, we obtain

f

t, x, ω

x

≥ ft, y, ω

y

 F x



t, y, ω

y

x − y

 F x

t, y, ω

y

ω

x

− ωy

−φ t, x, 0 − φt, y, 0

≥ ft, y, ω

y

Fx

t, y, ω

y

− φ x



t, β, 0

x − y

 F x

t, y, ω

y

ω

x

− ωy

.

3.7

We set

H

t, x, x; y, y

 ft, y, ω

y

Fx

t, y, ω

y

− φ x



t, β, 0

x − y

 F x

t, y, ω

y

ω

x

− ωy

and observe that

f

t, x, ω

x

≥ Ht, x, x; y, y

,

f

t, x, ω

x

 Ht, x, x; x, x

By the mean value theorem, we can find α ≤ c1 ≤ y and α ≤ c2 ≤ yc1, c2depend on y, y, resp., such that

f

t, y, ω

y

− ft, α t, αt f x t, c1, c2y − α t f xt, c1, c2ω

y

− αt 3.10

Letting

H1



t, x, x; y, y

 ft, α t, αt f x t, c1, c2x − αt  f xt, c1, c2ω

x

− αt,

3.11

we note that

f

t, y, ω

y

 H1



t, y, y; y, y

,

f

t, α t, αt H1



t, α t, αt; y, y. 3.12 Let us define H as



H 

⎧

⎨

⎩

H

t, x, x; y, y

, for x ≥ y,

H1



t, x, x; y, y

, for x ≤ y. 3.13

Clearly H is continuous and bounded on 0, 1×mint∈0,1αt, maxt∈0,1βt×R and satisfies

a Nagumo condition relative to α, β For every αt ≤ y ≤ βt and y ∈ R, we consider the

m-point BVP

−x H

t, x, x; y, y

, t ∈ 0, 1,

px 0 − qx0 m−2

i1 τix

ηi

, px 1  qx1 m−2

i1 σix

ηi

.

3.14

Using3.9, 3.12 and 3.13, we have



H

t, α t, αt; y, y

 H1



t, α t, αt; y, y

 ft, α t, αt≥ −αt,

pα 0 − qα0 ≤m−2

i1 τiα

ηi

, pα 1  qα1 ≤m−2

i1 σiα

ηi

,



H

t, β t, βt; y, y

 Ht, β t, βt; y, y

≤ ft, β t, βt≤ −βt,

pβ 0 − qβ0 ≥m−2

i1 τiβ

ηi

, pβ 1  qβ1 ≥m−2

i1 σiβ

ηi

.

3.15

Thus, α, β are lower and upper solutions of 3.14, respectively Since H satisfies a Nagumo

condition, there exists a constant M1 > max{α, β} depending on α, β and a Nagumo

function such that any solution x of 3.14 with αt ≤ xt ≤ βt satisfies |x| < M1on0, 1 Now, we choose α0 α and consider the problem

−x H

t, x, x; α0, α0

, t ∈ 0, 1,

px 0 − qx0 m−2

i1 τix

ηi

, px 1  qx1 m−2

i1 σix

ηi

.

3.16

UsingA1, 3.9, 3.12 and 3.13, we obtain



H

t, α0, α0; α0, α0



 ft, α0, α0

≥ −α

0t,

pα00 − qα

00 ≤m−2

i1 τiα0



ηi

, pα01  qα

01 ≤m−2

i1 σiα0



ηi

,



H

t, β t, βt; α0, α0

 Ht, β t, βt; α0, α0

≤ ft, β t, βt≤ −βt,

pβ 0 − qβ0 ≥m−2

i1 τiβ

ηi

, pβ 1  qβ1 ≥m−2

i1 σiβ

ηi

,

3.17

which imply that α0 and β are lower and upper solutions of 3.16 Hence by Theorems2.2

and2.3, there exists a unique solution α1of3.16 such that

α0 ≤ α1 ≤ βt, α

Note that the uniqueness of the solution follows by Theorem 2.3 Using 3.9 and 3.13

together with the fact that α1is solution of3.16, we find that α1is a lower solution of3.2, that is,

−α

1  H

t, α1, α1; α0, α0

≤ ft, α1, ω

α1

, t ∈ 0, 1,

pα10 − qα

10 m−2

i1 τiα1



ηi

, pα11  qα

11 m−2

i1 σiα1



ηi

.

3.19

In a similar manner, it can be shown by usingA1, 3.12, 3.13, and 3.19 that α1and β are lower and upper solutions of the following m-point BVP

−x H

t, x, x; α1, α1

, t ∈ 0, 1,

px 0 − qx0 m−2

i1 τix

ηi

, px 1  qx1 m−2

i1 σix

ηi

.

3.20

Again, by Theorems2.2and2.3, there exists a unique solution α2of3.20 such that

α1t ≤ α2t ≤ βt, α

Continuing this process successively, we obtain a bounded monotone sequence {α n} of solutions satisfying

α1t ≤ α2t ≤ α3t ≤ · · · ≤ α n t ≤ βt, t ∈ 0, 1, 3.22

where α nis a solution of the problem

−x H

t, x, x; α n−1, αn−1

, t ∈ 0, 1,

px 0 − qx0 m−2

i1 τix

ηi

, px 1  qx1 m−2

i1 σix

ηi

,

3.23

and is given by

x t 

1

0

G t, s H

s, αn, αn ; α n−1, αn−1

ds 

m−2

i1 τix

ηi −t 2q  p q  p

p

2q  p





m−2



i1

σix

ηi t

p

2q  p



.

3.24

Since H is bounded on 0, 1 × mint∈0,1αt, maxt∈0,1βt × R × mint∈0,1αt,

maxt∈0,1βt × R, therefore it follows that the sequences {α j n }j  0, 1 are uniformly

bounded and equicontinuous on 0, 1 Hence, by Ascoli-Arzela theorem, there exist the subsequences and a function x ∈ C10, 1 such that α j n → x j uniformly on 0, 1 as

n → ∞ Taking the limit n → ∞, we find that  Ht, αn, αn ; α n−1, αn−1  → ft, x, ωx which consequently yields

x t 

1

0

G t, sfs, x s, ωxsds 

m−2



i1 τix

ηi −t 2q  p q  p

p

2q  p





m−2

i1 σix

ηi t

p

2q  p



.

3.25

This proves that x is a solution of 3.2

Theorem 3.2 Assume that A1 and A2 hold Further, one assumes that

A3 the function F ∈ C20, 1 × R × R satisfies y∂/∂xFt, x, y  my2 ≤ 0 for |y| ≥

M, where m  max{|Fxxt, x, y| : t, x, y ∈ 0, 1 × min t∈0,1αt, maxt∈0,1βt ×

−M, M}, and F  f  φ.

Then, the convergence of the sequence {α n } of approximate solutions (obtained in Theorem 3.1 ) is quadratic.

Proof Let us set en1 t  xt − α n1 t ≥ 0 so that e n1satisfies the boundary conditions

pen1 0 − qe

n10 m−2

i1 τien1

ηi

, pen1 1  qe

n11 m−2

i1 σien1

ηi

In view of the assumption A3, for every t, x ∈ 0, 1 × min t∈0,1αt, maxt∈0,1βt, it

follows that

Fxt, x, M  2mM ≤ 0, Fxt, x, −M − 2mM ≥ 0. 3.27 Now, by Taylor’s theorem, we have

−e

n1 t F

t, x, x

− φt, x, 0

−f

t, αn, ω

αn

 F x



t, α, ω

αn

α n1 − α n

− φ x



t, β, 0

α n1 − α n   F x



t, αn, ω

αn

ω

αn1

− ωαn

 F x



t, αn, ω

αn

x − α n1   F x

t, αn, ω

αn

x− ωαn1

1

2



x − α n2Fxx t, z1, z2  2x − α nx− ωαn

Fxxt, z1, z2

x− ωαn2

Fxxt, z1, z2

−φ t, x, 0 − φt, α n, 0  − φ x



t, β, 0

α n1 − α n

≤ F x



t, αn, ω

αn

x− ωαn1





M2

2



|x − α n|  x− ωαn 2 ρ1x − α n2,

3.28

where α n ≤ z1 ≤ x, ωα

n  ≤ z2 ≤ x, α n ≤ ξ ≤ β, M2  max{|F xx |, |F xx |, |F xx|} on

0, 1 × min t∈0,1αt, maxt∈0,1βt × −M, M and ρ1  ρ max{φ xx t, x, 0 : t, x, 0 ∈

0, 1 × min t∈0,1αt, maxt∈0,1βt} with ρ > 1 satisfying β − αn ≤ ρx − α n  on 0, 1 Also,

in view of3.13, we have

−e

n1 t  ft, x, x

− H

t, αn1, αn1 ; α n, αn

≥ ft, x, x

− ft, αn1, ω

αn1

 f x t, c3, c4e n1  f xt, c3, c4x− ωαn1

≥ −γe n1  f xt, c3, c4x− ωαn1

,

3.29

where α n1 ≤ c3 ≤ x, ωα

n1  ≤ c4 ≤ x and γ  max{|f x t, x, y| : t, x, y ∈ 0, 1 ×

mint∈0,1αt, maxt∈0,1βt × −M, M}.

Now we show that ωαn1 t  α

n1 t By the mean value theorem, for every y1 ∈

−M, M and ωα

n1 t ≤ c5≤ y1, we obtain

Fx

t, αn t, y1



 F x



t, αn t, ωαn1 t F xxt, α n t, c5y1− ωαn1 t. 3.30

Let αn1 > M for some t ∈ 0, 1 Then ωαn1 t  M and 3.30 becomes

Fx

t, αn t, y1



 F xt, α n t, M  F xxt, α n t, c5y1− M

In particular, taking y1 −M and using 3.27, we have

Fxt, α n t, −M ≤ F xt, α n t, M  2mM ≤ 0, 3.32

which contradicts that F xt, α n t, −M ≥ 2mM > 0 Similarly, letting α

n1 < −M for some

t ∈ 0, 1, we get a contradiction Thus, it follows that |αn1 t| ≤ M for every t ∈ 0, 1, which implies that ωαn1 t  α

n1 t and consequently, 3.28 and 3.29 take the form

−e

n1 t ≤ F x

t, αn, ω

αn ten1 t  M3e n2

where M3 ρ1 M2/2 and

−e

n1 t ≥ −γe n1 t  f xt, c3, c4e

Now, by a comparison principle, we can obtain e n1 t ≤ rt on 0, 1, where rt is a solution

of the problem

−rt  F x

t, αn, ω

αn trt  M3e n2

1,

pr 0 − qr0 m−2

i1 τien1

ηi

, pr 1  qr1 m−2

i1 σien1

ηi

.

3.35

n1 t By the mean value theorem, for every y1 ∈

−M, M and ωα

n1... t. 3.30

Let αn1 > M for some t ∈ 0, 1 Then ωαn1 t  M and 3.30 becomes

Fx...

n1 < −M for some

t ∈ 0, 1, we get a contradiction Thus, it follows that |αn1 t| ≤ M for every t ∈ 0, 1, which implies