Approximation of Solutions for Second-Order m-Point Nonlocal Boundary Value Problems via the Method of Generalized Quasilinearization

Volume 2011, Article ID 929061,17pages 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,[email protected] Received 11 May 2010; Revised 29 July 2010; Accepted 2 October 2010

Academic Editor: Gennaro Infante

Copyrightq2011 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-orderm-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 1–7manifests 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 23and 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-orderm−point nonlocal boundary value problem

−xt f

t, xt, xt

, t∈0,1, 1.1

px0−qx0 ^m−2

i1

τix ηi

, px1 qx1 ^m−2

i1

σix ηi

, ηi∈0,1, 1.2

wheref:0,1×R×R → Ris continuous andτi, σi i1,2, . . . , m−2are nonnegative real constants such that_m−2

i1 τi<1,_m−2

i1 σi<1,andp, q >0 withp >1.

Here we remark that26studies1.1with the boundary conditions of the form

δx0−γx0 0, x1 ^m−2

i1

αix ηi

, ηi∈0,1. 1.3

A perturbed integral equation equivalent to the problem1.1and1.3considered in26is

xt ₁

0

kt, sf

s, xs, xs ds

_m−2

i1

αix ηi

t², 1.4

where

kt, s 1 δγ

⎧⎨

⎩

γδt

1−s, 0≤t≤s, δγs

1−t, s≤t≤1. 1.5

It can readily be verified that the solution given by1.4does not satisfy1.1. On the other hand, by Green’s function method, a unique solution of the problem1.1and1.3is

xt ₁

0

kt, sf

s, xs, xs ds

_m−2

i1

αix ηi

γδt

δγ, 1.6

wherekt, sis given by1.5. Thus,1.6represents the correct form of the solution for the problem1.1and1.3.

2. Preliminaries

Forx∈C¹0,1,we definex₁xx,wherexmax{|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 of1.1-1.2can be written as

xt ₁

0

Gt, sf

s, xs, xs ds

_m−2

i1

τix ηi

−t

2qp qp p

2qp

_m−2

i1

σix ηi

t

2qp q

p 2qp

,

2.1

whereGt, sis the Green’s function and is given by

Gt, s 1

p p2q

⎧⎨

⎩

qpt

qp1−s

, 0≤t≤s, qps

qp1−t

, s≤t≤1. 2.2

Note thatGt, s>0 on0,1×0,1.

We say thatα∈C²0,1is a lower solution of the boundary value problem1.1and 1.2if

−αt≤f

t, αt, αt

, t∈0,1, pα0−qα0≤^m−2

i1

τiα ηi

, pα1 qα1≤^m−2

i1

σiα ηi

,

2.3

andβ∈C²0,1is an upper solution of1.1and1.2if

−βt≥f

t, βt, βt

, t∈0,1, pβ0−qβ0≥^m−2

i1

τiβ ηi

, pβ1 qβ1≥^m−2

i1

σiβ ηi

. 2.4

Definition 2.1. A continuous functionh:0,∞ → 0,∞is called a Nagumo function if _∞

λ

sds

hs ∞, 2.5

forλ ≥ 0. We say thatf ∈C0,1 × R × Rsatisfies a Nagumo condition on0,1relative to α, βif for every t ∈ 0,1 andx ∈ min_t∈0,1αt,max_t∈0,1βt, there exists a Nagumo functionhsuch that|ft, x, x| ≤h|x|.

We need the following result33to establish the main result.

Theorem 2.2. Letf : 0,1×R² → Rbe a continuous function satisfying the Nagumo condition onE{t, x, y∈0,1×R² :α≤x≤β}whereα, β:0,1 → Rare continuous functions such thatαt ≤ βtfor allt ∈ 0,1.Then there exists a constantM >0 (depending only onα, β,the Nagumo functionh) such that every solutionxof 1.1-1.2withαt ≤ 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,xtof 1.1and 1.2such thatαt≤xt≤βt,t∈0,1.

Theorem 2.3. Assume thatα, β∈C²0,1are, respectively, lower and upper solutions of1.1-1.2.

Ifft, x, y∈C0,1×R×Ris decreasing inxfor eacht, y∈0,1×R,thenα≤βon0,1.

Proof. Let us define ut αt −βt so that u ∈ C²0,1 and satisfies the boundary conditions

pu0−qu0≤^m−2

i1

τiu ηi

, pu1 qu1≤^m−2

i1

σiu ηi

. 2.6

For the sake of contradiction, letuhave a positive maximum at somet0∈0,1. Ift0 ∈0,1, then ut0 0 and ut0 ≤ 0.On the other hand, in view of the decreasing property of ft, x, yinx,we have

ut0 αt0−βt0≥ −f

t0, αt0, αt0 f

t0, βt0, βt0

>0, 2.7 which is a contradiction. If we suppose that uhas a positive maximum at t0 0, then it follows from the first of boundary conditions2.6that

pu0−qu0≤^m−2

i1

τiu ηi

≤u0, 2.8

which implies thatp−1u0≤ qu0.Now asp > 1,q > 0,u0 > 0,u0≤ 0,therefore we obtain a contradiction. We have a similar contradiction att0 1.Thus, we conclude that αt≤βt,t∈0,1.

3. Main Results

Theorem 3.1. Assume that

A1the functions α, β ∈ C²0,1are, respectively, lower and upper solutions of 1.1-1.2 such thatα≤βon0,1;

A2the function f ∈ C²0,1 × R × R satisfies a Nagumo condition relative to α, β

and fx ≤ 0 on 0,1 × min_t∈0,1αt,max_t∈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 withΨφ ≥ 0 on 0,1× min_t∈0,1αt,max_t∈0,1βt×−M, M,where

Ψ

x−y₂ ∂²

∂x² 2 x−y

x−y ∂²

∂x∂x

x−y₂ ∂²

∂x². 3.1

Then, there exists a monotone sequence {αn} 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 modifiedm-point BVP

−xt f

t, xt, ω xt

, t∈0,1, px0−qx0 ^m−2

i1

τix ηi

, px1 qx1 ^m−2

i1

σix ηi

.

3.2

We note thatα, βare, respectively, lower and upper solutions of3.2and for everyt, x ∈ 0,1×mint∈0,1αt,max_t∈0,1βt,we have

f≤hω

xhx, 3.3

whereh· hω·.As _∞

0

sds hs

_M

0

sds

hs

_∞

M

sds

hM ∞, 3.4

soh is a Nagumo function. Furthermore, there exists a constantNdepending on α, β, and Nagumo functionhsuch that

_M

0

sds hs ≥

_N

0

sds hs >

max

βt:t∈0,1

−min{αt:t∈0,1}

, 3.5

whereM >max{N,α,β}. Thus, any solutionxof3.2withαt≤xt≤βt,t∈0,1 satisfies|x| ≤Mon0,1and hence it is a solution of1.1-1.2.

Let us define a functionF:0,1×R² → Rby F

t, x, x f

t, x, x φ

t, x, x−ω x

. 3.6

In view of the assumptionA2,it follows thatF ∈C²0,1×R²and satisfiesΨF≥0 on 0,1×mint∈0,1αt,max_t∈0,1βt×−M, M.Therefore, by Taylor’s theorem, we obtain

f t, x, ω

x

≥f t, y, ω

y Fx

t, y, ω

y

x−y Fx

t, y, ω

y

ω x

−ω y

−

φt, x,0−φ t, y,0

≥f t, y, ω

y

Fx

t, y, ω y

−φx

t, β,0 x−y Fx

t, y, ω

y

ω x

−ω y

.

3.7

We set

H

t, x, x;y, y f

t, y, ω y

Fx

t, y, ω y

−φx

t, β,0 x−y Fx

t, y, ω

y

ω x

−ω y

, 3.8

and observe that

f t, x, ω

x

≥H

t, x, x;y, y , f

t, x, ω x

H

t, x, x;x, x

. 3.9

By the mean value theorem, we can findα≤c1 ≤yandα ≤c2 ≤ yc1, c2depend ony, y, resp., such that

f t, y, ω

y

−f

t, αt, αt

fxt, c1, c2

y−αt

fxt, c1, c2 ω

y

−αt

. 3.10

Letting

H1

t, x, x;y, y f

t, αt, αt

fxt, c1, c2x−αt fxt, c1, c2 ω

x

−αt 3.11,

we note that

f t, y, ω

y H1

t, y, y;y, y , f

t, αt, αt H1

t, αt, αt;y, y

. 3.12

Let us defineHas

H

⎧⎨

⎩ H

t, x, x;y, y

, forx≥y, H1

t, x, x;y, y

, forx≤y. 3.13

ClearlyHis continuous and bounded on0,1×min_t∈0,1αt,max_t∈0,1βt×Rand satisfies a Nagumo condition relative toα, β. For everyαt ≤y ≤ βtandy ∈R, we consider the m-point BVP

−xH

t, x, x;y, y

, t∈0,1, px0−qx0 ^m−2

i1

τix ηi

, px1 qx1 ^m−2

i1

σix ηi

.

3.14

Using3.9,3.12and3.13, we have

H

t, αt, αt;y, y H1

t, αt, αt;y, y f

t, αt, αt

≥ −αt,

pα0−qα0≤^m−2

i1

τiα ηi

, pα1 qα1≤^m−2

i1

σiα ηi

, H

t, βt, βt;y, y H

t, βt, βt;y, y

≤f

t, βt, βt

≤ −βt,

pβ0−qβ0≥^m−2

i1

τiβ ηi

, pβ1 qβ1≥^m−2

i1

σiβ ηi

.

3.15

Thus,α, βare lower and upper solutions of3.14, respectively. SinceHsatisfies a Nagumo condition, there exists a constantM1 > max{α,β}depending onα, βand a Nagumo functionsuch that any solutionxof3.14withαt≤xt≤βtsatisfies|x|< M1on0,1.

Now, we chooseα0αand consider the problem

−xH

t, x, x;α0, α₀

, t∈0,1, px0−qx0 ^m−2

i1

τix ηi

, px1 qx1 ^m−2

i1

σix ηi

.

3.16

UsingA1,3.9,3.12and3.13, we obtain

H

t, α0, α₀;α0, α0

f

t, α0, α₀

≥ −α₀t,

pα00−qα₀0≤^m−2

i1

τiα0

ηi

, pα01 qα₀1≤^m−2

i1

σiα0

ηi

,

H

t, βt, βt;α0, α₀ H

t, βt, βt;α0, α₀

≤f

t, βt, βt

≤ −βt,

pβ0−qβ0≥^m−2

i1

τiβ ηi

, pβ1 qβ1≥^m−2

i1

σiβ ηi

,

3.17

which imply thatα0 andβare lower and upper solutions of3.16. Hence by Theorems2.2 and2.3, there exists a unique solutionα1of3.16such that

α0 ≤α1 ≤βt, α₁≤M1, t∈0,1. 3.18

Note that the uniqueness of the solution follows by Theorem 2.3. Using 3.9 and 3.13 together with the fact thatα1is solution of3.16, we find thatα1is a lower solution of3.2, that is,

−α₁ H

t, α1, α₁;α0, α₀

≤f

t, α1, ω α₁

, t∈0,1, pα10−qα₁0 ^m−2

i1

τiα1

ηi

, pα11 qα₁1 ^m−2

i1

σiα1

ηi

.

3.19

In a similar manner, it can be shown by usingA1,3.12,3.13, and3.19thatα1andβare lower and upper solutions of the followingm-point BVP

−xH

t, x, x;α1, α₁

, t∈0,1, px0−qx0 ^m−2

i1

τix ηi

, px1 qx1 ^m−2

i1

σix ηi

.

3.20

Again, by Theorems2.2and2.3, there exists a unique solutionα2of3.20such that

α1t≤α2t≤βt, α₂t≤M1, t∈0,1. 3.21 Continuing this process successively, we obtain a bounded monotone sequence {αn} of solutions satisfying

α1t≤α2t≤α3t≤ · · · ≤αnt≤βt, t∈0,1, 3.22 whereαnis a solution of the problem

−xH

t, x, x;α_n−1, α_n−1

, t∈0,1, px0−qx0 ^m−2

i1

τix ηi

, px1 qx1 ^m−2

i1

σix ηi

,

3.23

and is given by

xt ₁

0

Gt, sH

s, αn, α_n;αn−1, α_n−1 ds

_m−2

i1

τix ηi

−t

2qp qp p

2qp

_m−2

i1

σix ηi

t

2qp q

p 2qp

.

3.24

Since H is bounded on 0,1 × mint∈0,1αt, maxt∈0,1βt × R × mint∈0,1αt, max_t∈0,1βt × R, therefore it follows that the sequences {α^jn }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 thatα^jn → x^j uniformly on 0,1 as

n → ∞.Taking the limitn → ∞,we find thatHt, α n, α_n;αn−1, α_n−1 → ft, x, ωxwhich consequently yields

xt ₁

0

Gt, sf

s, xs, ω xs

ds _m−2

i1

τix ηi

−t

2qp qp p

2qp

_m−2

i1

σix ηi

t

2qp q

p 2qp

.

3.25

This proves thatxis a solution of3.2.

Theorem 3.2. Assume thatA1andA2hold. Further, one assumes that

A3the functionF ∈ C²0,1×R×Rsatisfiesy∂/∂xFt, x, y my² ≤ 0 for|y| ≥ M,wherem max{|Fxxt, x, y|:t, x, y∈0,1×mint∈0,1αt,max_t∈0,1βt×

−M, M},andFfφ.

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

Proof. Let us seten1t xt−αn1t≥0 so thaten1satisfies the boundary conditions

pen10−qe_n10 ^m−2

i1

τien1 ηi

, pen11 qe_n11 ^m−2

i1

σien1 ηi

. 3.26

In view of the assumption A3,for every t, x ∈ 0,1×min_t∈0,1αt,max_t∈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_n1t F

t, x, x

−φt, x,0

− f

t, αn, ω α_n

Fx

t, α, ω α_n

α_n1−αn

−φx

t, β,0

αn1−αn Fx

t, αn, ω α_n

ω α_n1

−ω α_n Fx

t, αn, ω α_n

x−αn1 Fx

t, αn, ω α_n

x−ω α_n1 1

2

x−αn²Fxxt, z1, z2 2x−αn x−ω

α_n

Fxxt, z1, z2

x−ω α_n2

Fxxt, z1, z2

−

φt, x,0−φt, αn,0−φx

t, β,0

αn1−αn

≤Fx

t, αn, ω α_n

x−ω α_n1

M2

2

|x−αn|x−ω

α_n²ρ1x−αn², 3.28

where αn ≤ z1 ≤ x, ωα_n ≤ z2 ≤ x, αn ≤ ξ ≤ β, M2 max{|Fxx|,|Fxx |,|Fxx|} on 0,1 ×min_t∈0,1αt, max_t∈0,1βt ×−M, M and ρ1 ρmax{φxxt, x,0 : t, x,0 ∈ 0,1×mint∈0,1αt, maxt∈0,1βt}withρ >1 satisfyingβ−αn ≤ρx−αnon0,1.Also, in view of3.13, we have

−e_n1t f t, x, x

−H

t, αn1, α_n1;αn, α_n

≥f t, x, x

−f

t, α_n1, ω α_n1 fxt, c3, c4en1fxt, c3, c4

x−ω α_n1

≥ −γe_n1fxt, c3, c4 x−ω

α_n1 ,

3.29

where αn1 ≤ c3 ≤ x, ωα_n1 ≤ c4 ≤ x and γ max{|fxt, x, y| : t, x, y ∈ 0,1× min_t∈0,1αt, max_t∈0,1βt×−M, M}.

Now we show thatωα_n1t α_n1t.By the mean value theorem, for everyy1 ∈

−M, Mandωα_n1t≤c5≤y1,we obtain Fx

t, αnt, y1

Fx

t, αnt, ω

α_n1t

Fxxt, αnt, c5 y1−ω

α_n1t

. 3.30

Letα_n1> Mfor somet∈0,1.Thenωα_n1t Mand3.30becomes Fx

t, αnt, y1

Fxt, αnt, M Fxxt, αnt, c5

y1−M

≤Fxt, αnt, M−m

y1−M

. 3.31

In particular, takingy1−Mand using3.27, we have

Fxt, αnt,−M≤Fxt, αnt, M 2mM≤0, 3.32 which contradicts thatFxt, αnt,−M ≥ 2mM > 0.Similarly, lettingα_n1 < −Mfor some t∈0,1,we get a contradiction. Thus, it follows that|α_n1t| ≤Mfor everyt∈0,1, which implies thatωα_n1t α_n1tand consequently,3.28and3.29take the form

−e_n1 t≤Fx

t, αn, ω α_nt

e_n1t M3en²₁, 3.33 whereM3ρ1 M2/2and

−e_n1 t≥ −γen1t fxt, c3, c4e_n1t. 3.34 Now, by a comparison principle, we can obtainen1t≤rton0,1, wherertis a solution of the problem

−rt Fx

t, αn, ω α_nt

rt M3en²₁, pr0−qr0 ^m−2

i1

τien1 ηi

, pr1 qr1 ^m−2

i1

σien1 ηi

.

3.35

SinceFx is continuous and bounded on0,1×mint∈0,1αt,max_t∈0,1βt×R, there exist ζ2, ζ1>0independent ofnsuch that−ζ1≤Fx ≤ζ2on0,1×min_t∈0,1αt,max_t∈0,1βt×

−M, M.Sinceζ2−Fxt, αn, ωα_n≥0 on0,1,so we can rewrite3.35as rt ζ2rt

ζ2−Fx

t, αn, ω α_n

rt−M3en²₁ pr0−qr0 ^m−2

i1

τien1 ηi

, pr1 qr1 ^m−2

i1

σien1 ηi

,

3.36

whose solution is given by

rt ₁

0

Gζ2t, s ζ2−Fx

t, αn, ω α_n

rs−M3en²₁ ds

_m−2

i1

τie_n1 ηi

−t

2qp qp p

2qp

_m−2

i1

σie_n1 ηi

t

2qp q

p 2qp

3.37 where

Gζ2t, s −1 ζ2

pqζ2

/p−e^−ζ2

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

1−pζ2q p e^−ζ21−s

pζ2q p −e^−ζ2t

, 0≤t≤s,

e^−ζ2t−s−pζ2q p e^−ζ21−s

pζ2q p −e^−ζ2s

, s≤t≤1, 3.38

Introducing the integrating factorμt e^{0tFxs,αns,ωαnsds}such thate^−ζ1t< μ≤e^ζ2t,3.34 takes the form

rtμt

−M3en²₁μt. 3.39

Integrating3.39from 0 totand usingr0≥−1/q_m−2

i1 τie_n1ηi,we obtain rtμt≥ −1

q

m−2

i1

τien1 ηi

−M3en²_{1 t}

0

μsds, 3.40

which can alternatively be written as

rt≥ −1 qe^ζ1t

m−2

i1

τie_n1 ηi

− M3

ζ2e^ζ1ten²₁ e^ζ2−1

≥ −1 q

m−2

i1

τien1 −M3

ζ2 en²₁ e^ζ2−1

−ρ1en1 −ρ2en²₁,

3.41

whereρ1 1/q_m−2

i1 τi,ρ2 M3/ζ2e^ζ2−1. Using the fact thatGζ2t, s≤0 together with 3.41yields

Gζ2t, sζ2−Fxrt≤Gζ2t, sζ2−Fx

ρ1en1ρ2en²₁

≤Gζ2t, sζ2ζ1

ρ1en1ρ2en²₁ ,

3.42

which, on substituting in3.37, yields

e_n1≤rt≤ ₁

0

Gζ2t, sζ2ζ1

ρ1e_n1ρ2en²₁

M3en²₁ ds

_m−2

i1

τien1 ηi

−t

2qp qp p

2qp

_m−2

i1

σien1 ηi

t

2qp q

p 2qp

≤ ₁

0

Gζ2t, sζ2ζ1

ρ1e_n1ds ₁

0

Gζ2t, sρ2ζ2ζ1 M3

en²₁

ds

_m−2

i1

τi^m−2

i1

σi

pq p

2qp

en1 ηi

≤

B _m−2

i1

τi^m−2

i1

σi

pq p

2qp

en1Aen²₁,

3.43 where

A

ρ2ζ2ζ1 M3

max ₁

0

Gζ2t, sds, B ζ2ζ1ρ1max ₁

0

Gζ2t, sds. 3.44

Taking the maximum over0,1and then solving3.43foren1,we obtain

e_n1 ≤ A

1−B−_m−2

i1 τi_m−2

i1 σi

pq/p

2qpen²₁. 3.45 Also, it follows from3.33that

e_n1 μt

≥ −M3en²₁μt≥ −M3e^ζ2ten²₁, t∈0,1. 3.46

Integrating3.46from 0 totand usingv_n1 0≥−1/q_m−2

i1 τie_n1ηi from the boundary conditionpen10−qe_n10 _m−2

i1 τien1ηi,we obtain e_n1 tμt≥ −1

q

m−2

i1

τien1 ηi

−M3

e^ζ2t−1

ζ2 en²₁, 3.47

which, in view of the facte^−ζ1t< μ≤e^ζ2tand3.45, yields

e_n1t≥e^ζ1t

⎡

⎢⎣ −1

q

m−2

i1

τi

⎛⎜⎝ A 1−B−_m−2

i1 τi_m−2

i1 σi

pq /p

2qp

⎞

⎟⎠

− M3

e^ζ2t−1 ζ2

$

en²₁≥ −δ1en²₁,

3.48

where

δ1max

⎧⎪

⎨

⎪⎩e^ζ1t

⎡

⎢⎣ 1

q

m−2

i1

τi

⎛⎜⎝ A 1−B−_m−2

i1 τi_m−2

i1 σi

pq /p

2qp

⎞

⎟⎠

M3

e^ζ2t−1 ζ2

$

, t∈0,1

% .

3.49

Asen1∈C¹0,1, there existst∈0,1such that e_n1

t

e_n11−e_n10≤e_n11

≤ 1 p

m−2

i1

σie_n1 ηi

−q

pe_n11≤ 1 p

m−2

i1

σie_n1qδ p en²₁

≤

⎡

⎢⎣ A p

1−B−_m−2

i1 σi_m−2

i1 τi

pq /p

2qp

m−2

i1

σiqδ p

⎤

⎥⎦en²₁.

3.50

Integrating3.46fromttott≤tand using3.50, we have

e_n1t≤e^ζ1t

⎡

⎢⎣ e^ζ2tA_m−2

i1 σi

p

1−B−_m−2

i1 σi_m−2

i1 τi

pq /p

2qp

qδ p M3

e^ζ2t−e^ζ2t ζ2

⎤

⎥⎦en²₁.

3.51

Using3.45in3.34, we obtain e_n1 tμ1t

≤ γAμ1t

1−B−_m−2

i1 σi_m−2

i1 τi

pq /p

2qpen²₁, 3.52

whereμ1t e^{0tfxs,c3,c4ds}. Sincefx is bounded on0,1×mint∈0,1αt, maxt∈0,1βt×

−M, M,we can chooseζ3, ζ4 > 0 such that −ζ3 ≤ fxt,c3,c4 ≤ ζ4 on0,1×mint∈0,1αt, max_t∈0,1βt×−M, Mande^−ζ3t< μ1t≤e^ζ4tso that3.52takes the form

e_n1 tμ1t

≤ γAe^ζ4t

1−B−_m−2

i1 σi_m−2

i1 τi

pq /p

2qpen²₁. 3.53

Integrating3.53fromttott≥t, and using3.51, we find that

e_n1t≤ 1 μ1t

⎡

⎢⎣e_n1 t

μ1

t

γA

e^ζ4t−e^ζ4t L2

1−B−_m−2

i1 σi_m−2

i1 τi

pq /p

2qpen²₁

⎤

⎥⎦

≤e^ζ3t

⎡

⎢⎣ Ae^ζ4t_m−2

i1 σi

p

1−B−_m−2

i1 σi_m−2

i1 τi

pq /p

2qpqδe^ζ4t p

γA

e^ζ4t−e^ζ4t ζ4

1−B−_m−2

i1 σi_m−2

i1 τi

pq /p

2qp

⎤

⎥⎦en²₁.

3.54

Letting

δ2max

⎧⎪

⎨

⎪⎩max

⎧⎪

⎨

⎪⎩e^ζ1t

⎡

⎢⎣ e^ζ2tA_m−2

i1 σi

p

1−B−_m−2

i1 σi_m−2

i1 τi

pq /p

2qp

qδ p M3

e^ζ2t−e^ζ2t ζ2

⎤

⎥⎦, t∈ 0, t⎫

⎪⎬

⎪⎭,

max

⎧⎪

⎨

⎪⎩e^ζ3t

⎡

⎢⎣ Ae^ζ4t_m−2

i1 σi

p

1−B−_m−2

i1 σi_m−2

i1 τi

pq /p

2qpqδe^ζ4t p

γA

e^ζ4t−e^ζ4t ζ4

1−B−_m−2

i1 σi_m−2

i1 τi

pq /p

2qp

⎤

⎥⎦, t∈ t,1%%

,

3.55

it follows from3.51and3.54that

e_n1t≤δ2en²₁. 3.56

Hence, from3.48and3.56, it follows that

,,en1,,≤δ3en²₁, 3.57

whereδ3max{δ1, δ2}.From3.45and3.57with

Q A

1−B−_m−2

i1 σi_m−2

i1 τi

pq /p

2qpδ3, 3.58

we obtain

en1₁en1,,v_n1,,≤Qen²₁. 3.59

This proves the quadratic convergence inC¹norm.

Example 3.3. Consider the boundary value problem

−x− 1

720te^x− 1

35x−1− tx² 16

1 x², t∈0,1, 5

4x0−11

20x0 1 7x

3 4

1

9x 4

5

, 5

4x1 11

20x1 1 3x

3 4

.

3.60

Letαt 0 andβt 1tbe, respectively, lower and upper solutions of3.60. Clearlyαt andβtare not the solutions of 3.60andαt < βt, t ∈ 0,1.Also, the assumptions of Theorem 3.1are satisfied. Thus, the conclusion ofTheorem 3.1applies to the problem3.60.

Acknowledgment

The author is grateful to the referees and professor G. Infante for their valuable suggestions and comments that led to the improvement of the original paper.