A Newton type method and its application

A NEWTON-TYPE METHOD AND ITS APPLICATION

Received 6 March 2006; Accepted 26 March 2006

We prove an existence and uniqueness theorem for solving the operator equationF(x) + G(x)=0, whereFis a continuous and Gˆateaux differentiable operator and the operator Gsatisfies Lipschitz condition on an open convex subset of a Banach space. As corollaries, a recent theorem of Argyros (2003) and the classical convergence theorem for modified Newton iterates are deduced. We further obtain an existence theorem for a class of non-linear functional integral equations involving the Urysohn operator.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

1. Introduction

This paper considers the problem of approximating a locally unique solutionx∗of the equationF(x) +G(x)=0, whereFandGare continuous operators defined on an open convex subsetDof a Banach spaceXwith values in a Banach spaceY. The solution is obtained as the limit of a sequence of Newton-type iterates given by

xn+1=xn−

F_xn−1Fxn

+Gxn

. (1.1)

It may be noted that forG≡0, (1.1) reduces to Newton’s method of iterates whose con-vergence is proved under the usual hypotheses thatFis Fr´echet differentiable andF_xis in-vertible. Recently, Argyros [3] obtained an interesting generalization of Newton’s method which is further extended in this paper along with an earlier result of Vijesh and Subrah-manyam [1]. More specifically, we prove the convergence of a sequence of Newton-type iterates under mild conditions onF. In particular, the Gˆateaux differentiable operatorF is assumed to satisfy the inequalityF_x0−1F_x−F_x0 ≤in a certain neighbourhood Nx0

ofx0whileGis required to be a contraction inN

x0

. Using the main theorem, the existence of a unique continuous solution to a nonlinear functional integral equation involving the Urysohn operator is deduced from a corollary to the main theorem. This seems to be the first application of Newton-type algorithm to such nonlinear equations. Illustrative examples are also discussed.

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 23674, Pages1–9

2. Convergence analysis

Theorem 2.1is a general theorem on the convergence of Newton-type iterates, proved under mild assumptions. It generalizes the theorems in [1,3].

Theorem 2.1. LetFandGbe continuous operators defined on an open convex subsetDof a Banach spaceXwith values in a Banach spaceY. Suppose thatFis Gˆateaux differentiable at each point in some neighbourhood ofx0∈DandGis Lipschitz onD. Assume further that

(i) (Fxo )−1∈L(Y,X), the space of bounded linear operators fromYtoX;

(ii) for someη≥0,(F_xo )−1[F(x0) +G(x0)] ≤η; (iii) for somer >0,(F_x0)−1_[F

x−Fxo] ≤(>0) wheneverx∈U(x0,r), whereU(x0, r)= {x∈B:x−x0< r}andG(x)−G(y) ≤kx−yforx,y∈Dsuch that 3+∗<1 and (1 +c0/(1−c))η < r, where∗=k(Fx0)

−1_,c

0=(+∗)/(1−), andc=(2+∗_)/(1−);

(iv)F_xis piecewise hemicontinuous for eachx∈U(x0,r) andU(x0,r)⊂D.

Then the sequencexn(n≥0) generated recursively by (1.1) is well defined, remains in

U(x0,r) for alln≥0, and converges to a unique solution x∗∈U(x0,r) of the equation F(x) +G(x)=0. Moreover, the following error bounds hold for alln≥2:

_{xn+1−xn≤c}n−1_c 0η,

_xn−x∗_≤ cn−1 1−cc0η.

(2.1)

Proof. Clearly by (ii) x1−x0 = (Fx0)

−1_{[F(x0) +G(x0)] ≤η < r, and hence x} 1∈ U(x0,r). It follows from the choice of and the well-known Banach lemma (see Rall [4, page 50]) that (F_x1)−

1 _{exists and(F}

x1)−

1_F

x0 ≤1/(1−). Moreover (1.1) gives for

n=1 that

x2−x1= −

F_x1−1Fx1

+Gx1

= −F_x1−1F_x0F_x0−1Fx1

+Gx1

−Fx0

−Gx0

+Fx0

+Gx0

= −F_x1−1F_x0F_x0−1

×

1

0

F_θx1+(1−θ)xo−F

x0

x1−x0dθ+Gx1−Gx0 (using (iv))

= −F_x1−1F_x0

×

1

0

F_x0−1F_{θx1+(1−θ)xo}−F_x0x1−x0

dθ+F_x0−1Gx1

−Gx0 . (2.2)

So

_x2−x1≤F

x1

−1 F_x0

×

1

0

_F

x0

−1

F_{θx 1+(1−θ)xo}−F

x0

x1−x0

dθ+F_x0−1Gx1

−Gx0

≤ 1 1−

x1−x0+Fx0

−1

= 1 1−

x1−x0+∗x1−x0=+ ∗

1− x1−x0,

_{x2−x0≤x2−x1+x1−x0≤c0η+η < r.}

(2.3)

Thusx2∈U(x0,r). Again by Banach’s lemma (see [4]) (Fx2)

−1_{exists and(F}

x2)

−1_F

x0 ≤

1/(1−). Assume that

xk∈U

x0,r, xk+1−xk≤ck−1c0η fork=2, 3,. . .,n−1. (2.4)

In view of hypotheses (iii) and (iv), it follows as before from Banach’s lemma that (F_xn)−1 exists and

_F

xn −1

F_x0≤ 1

1−. (2.5)

By hypotheses (iii) and (iv) and (2.5), we obtain

xn+1−xn

= −Fxn −1

Fxn

+Gxn

= −F_xn−1F_x0F_x0−1Fxn+Gxn

= −F_xn−1F_x0F_x0−1 ×Fxn

+Gxn

−Fxn−1

−Gxn−1

+Fxn−1

+Gxn−1

= −F_xn−1F_x0F_x0−1

×1 0

F_{θxn+(1−θ)xn−1}−F_x0+F_x0−F_xn−1xn−xn−1dθ+Gxn

−Gxn−1

= −F_xn−1F_x0

1

0

F_x0−1F_{θxn+(1−θ)xn−1}−F_x0+F_x0−F_xn−1xn−xn−1dθ

+F_x0−1Gxn

−Gxn−1 ,

_{xn+1−xn≤F}

xn −1

F_x0

1

0

_F

x0

−1

F_{θx n+(1−θ)xn−1}−F_x0 xn−xn−1dθ

+

1

0

_F

x0

−1 F_x0−F

xn−1 xn−xn−1dθ

+F_x0−1kxn−xn−1 ,

_xn+1−xn≤2+∗

1− xn−xn−1.

Thus by induction hypothesis (2.4),xn+1−xn ≤cn−1c0η. Since

_{xn+1−x0≤xn+1−xn+xn−xn−1+···+x1−x0} ≤cn−1c0η+cn−2c0η+···+cc0η+c0η+η,

_{xn+1−x0≤η1 +c0}1−cn

1−c

≤η

1 + c0 1−c

< r.

(2.7)

Hencexn∈U(x0,r) for alln≥0. Fork≥2,m≥0.

_{xk+m−xk≤xk+m−xk+m−1+xk+m−1−xk+m−2+···+xk+1−xk} ≤ck+m−2c0η+ck+m−3c0η+. . .+ck−1c0η

≤ηc0ck−1

1 +c+···+cm−1

≤1−cm 1−c c

k−1_c 0η≤ c

k−1

1−cc0η.

(2.8)

As 0< c <1,xnis a Cauchy sequence in the closed subsetU(x0,r) of the Banach space

X; it hence converges to an elementx∗inU(x0,r). From hypothesis (iii) using triangle inequality, it follows thatF

xn ≤M, whereM=(/(F

x0)

−1_+(F

x0)

−1_{). Since}

xn+1=xn−

F_xn−1Fxn

+Gxn

, Fxn

+Gxn

= −F

xn

xn+1−xn

. (2.9)

So

_F xn

+Gxn≤Fxnxn+1−xn≤Mxn+1−xn. (2.10)

Proceeding to the limit in (2.10) asntends to infinity and using the continuity ofF andGit follows from the convergence of (xn) tox∗thatF(x∗) +G(x∗)=0. Ifx∗andy∗

are two solutions ofF(x) +G(x)=0 inU(x0,r), then

_x∗_−y∗_=x∗_−y∗_−F

x0

−1

Fx∗+Gx∗−Fy∗−Gy∗

≤ 1

0

I−F_x0−1F_θx ∗_+(1−θ)y∗

x∗−y∗dθ

+Fx0

−1

Gx∗−Gy∗ ≤x∗−y∗+∗_x∗_−y∗

=+∗_x∗_−y∗_<x∗_−y∗ _{as 0≤+}∗_≤3+∗_<1.

(2.11)

This contradiction implies thatx∗=y∗. Hence the theorem holds. Corollary 2.2 (see Argyros [3, Theorem 1]). LetFbe a continuous operator defined on an open convex subsetDof a Banach spaceXwith values in a Banach spaceYand continuously Fr´echet differentiable at somex0∈D. Assume that

(i) (F_xo)−1_∈L(Y,X);

(ii) there exists a parameterηsuch that 0≤ (Fxo)

(iii) for some∈(0, 1/3), there existsδ >0 such that(F_xo)−1_(F

x−Fxo) ≤whenever x∈U(x0,δ)= {x∈X:x−x0< δ, with (c2/(1−c) +c0+ 1)η < δ, wherec0=

/(1−) andc=2c0.

Then the Newton iterates (xn) generated by (1.1) are well defined, remain inU(x0,δ) for

alln≥0, and converge to a solutionx∗∈U(x0,δ) of equationF(x)=0. Moreover, for all n≥2, the following error bounds hold:

_{xn+1−xn≤c}n_x

1−x0,

_xn−x∗_≤ cn

1−cx1−x0.

(2.12)

Proof. SinceFis Fr´echet differentiable, it is Gˆateaux differentiable. TakeG≡0 in Theorem 2.1. Letδ=rand note that

δ >

c2

1−c+c0+ 1

η≥

1 + c0 1−c

η. (2.13)

SinceFis continuously Fr´echet differentiable,F_xis hemicontinuous at eachx∈U(x0,δ) and thus all the conditions ofTheorem 2.1are satisfied. SoFhas a unique zero inU(x0,δ). Corollary 2.3. LetF be a continuous operator defined on an open convex subsetDof a Banach spaceXwith values in a Banach spaceY having a Gˆateaux derivative at each point in some neighbourhood ofx0∈D. Assume further that

(i) the inverse ofF_xoexists and (F_xo)−1_∈L(Y,X); (ii) for someη≥0(Fxo)

−1_{F(x0) ≤η;}

(iii) for some r >0,(F_x0)−1[F_x−F_xo] ≤whenever x∈U(x0,r). Suppose (1 +c0/ (1−c))η < rand 0<3<1, wherec0=/(1−) andc=2c0;

(iv)Fxis piecewise hemi continuous at eachx∈U(x0,r).

Then the modified Newton iterates (xn),n≥0, generated by

xn+1=xn−

F_xo−1Fxn

(2.14)

are well defined, remain inU(x0,r) for alln≥0, and converge to a solutionx∗∈U(x0,r) of F(x)=0.

To proveCorollary 2.3, setG≡0 inTheorem 2.1and proceed as inTheorem 2.1.

3. Solutions of a class of nonlinear functional integral equations

In this section, some existence theorems for an operator equation involving the Urysohn operator are proved. More specifically, given a compact subset ΩofR,g ∈C(Ω×R) andK∈C(Ω2_{×R),Theorem 3.2gives sufficient conditions for the existence of a unique} solution of the nonlinear functional integral equation of the form

x(t) +

ΩK

t,s,x(s)ds+gt,x(t)=0 ∀t∈Ω,x∈C(Ω). (3.1)

Lemma 3.1. ForK∈C(Ω2_{×R), suppose thatK3(t,s,u)=K/u∈C(Ω}2_{×R) satisfies} the Lipschitz condition with respect to the third variable. Then the Fr´echet derivative ofF exists forx∈U(x0,r) andx0∈C(Ω) and the derivative atxis given by

Fx(h)t=h(t) +

ΩK3

t,s,x(s)h(s)ds. (3.2)

For the proof, see [2].

Theorem 3.2. LetK(t,s,u)∈C(Ω2_{×R), letK3(t,s,u)∈C(Ω}2_{×R), letg ∈C(Ω×R),} and letΩbe a compact subset ofRwhose Lebesgue measure is equal tod >0. Suppose that

(i) for somem∈(0, 1),K3(t,s,u1)−K3(t,s,u2)≤mu1−u2holds for all (t,s,ui)∈ Ω2_{×R,i=1, 2;}

(ii)|g(t,y1)−g(t,y2)| ≤∗_{|y1−y2|for ally1,y2∈R,t∈Ω, whereβ+}∗_<1β= sup_t∈Ω|_ΩK3(t,s,x0(s))ds|. Further let η≥0 be chosen such that η≥sup_t∈Ω| Fx0(t) +g(t,x0(t))|;

(iii) letα1andα2be the positive roots ofp(x)=4m2d2x2−4md(1−β)(2−2β−∗)x+ [(1−β−∗_)(1−β)]2_{and letr}

1andr2withr1≤r2be the positive roots ofq(x)= 3md(1−β)x2_{−[(1−β−}∗_{)(1−β) + 2mdη]x+ (1−β)η;}

(iv) suppose further thatη≤min{α1,α2}andη/(1−β)< r1<(1−β−∗)/3md. Then the functional integral equationFx(t) +g(t,x(t))=0 for allt∈Ωhas a unique so-lution inU(x0,r0) for allr0∈

r1, minr2, (1−β−∗_{)/3mdand the sequence of iterates} given in (1.1) converges to this solution.

Proof. Clearly F maps C(Ω) into itself. Let G:C(Ω)→C(Ω) be defined by Gx(t)= g(t,x(t))∀t∈Ω. ThenGmapsC(Ω) into itself,

_{Gx1(t)−Gx2(t)=g}

t,x1(t)−gt,x2(t) ≤∗_{x1(t)−x2(t) (by (ii))} ≤∗_x

1−x2.

(3.3)

ThusGx1−Gx2 ≤∗x1−x2. Assumption (i) together withLemma 3.1implies that F_xexists. Now,

_I−F

x0

h(t)=h(t)−h(t)−

ΩK3

t,s,x0(s)h(s)ds

≤

Ω

_K3

t,s,x0(s)h(s)ds≤ h

Ω

_K3

t,s,x0(s)ds ≤ hβ.

(3.4)

This implies thatI−Fx0 ≤β <1, and henceFx0 is invertible and by Banach’s lemma,

(F_x0)−1 ≤1/(1−β). Since the rootsα1andα2ofp(x) are positive and

x−α1

x−α2

=x2₋

(1−β)2−2β−∗ md

x+ 1−β−∗

(1−β) 2md

2 ,

by assumption (iv), (η−α1)(η−α2)≥0. This implies that

η2₋

(1−β)2−2β−∗ md

η+

1−β−∗_(1−β) 2md

2

≥0. (3.6)

Thus

1−β−∗_(1−β)2

−4md(1−β)2−2β−∗_{η+ 4m}2_d2_η2_{≥0. (3.7)}

In other words, the discriminant of the quadratic equationq(x)=0 being

1−β−∗_{(1−β) + 2mdη}2

−12md(1−β)2η (3.8)

is nonnegative by (3.6). Soq(x) always has positive rootsr1 andr2. Since by assump-tion (iv)η/(1−β)< r1<(1−β−∗)/3md, chooser0withr1< r0<min{r2, (1−β−∗)/ 3md}. Then

qr0

=3md(1−β)r02−r0

1−β−∗_{(1−β) + 2mdη+ (1−β)η <0. (3.9)}

So

1−β−∗_{(1−β) + 2mdηr}

0−3md(1−β)r20>(1−β)η. (3.10)

Hence

_{1−β−2 mr0d}

1−β−∗_{−3 mr0d}

_η

1−β< r0. (3.11)

Now fort∈Ω,

_F

x0

−1 Fx−Fx0

h(t)=Fx0

−1

Ω

K3

t,s,x(s)−K3

t,s,x0(s)h(s)ds

≤Fx0

−1

Ω

_K3

t,s,x(s)−K3

t,s,x0(s)h(s)ds

≤mr0d

1−βh (using (i)).

(3.12)

This implies that(Fx0)

−1_(F

x−Fx0) ≤mr0d/(1−β). Setting =mr0d/(1−β),

∗ 1 =

∗_{/(1−β), η}

functionx∗satisfying

x∗(t) +

ΩK

t,s,x∗(s)ds+gt,x∗(t)=0, ∀t∈Ω, (3.13)

and this solution can obtained as the limit of the iterates in (1.1).

4. Illustrative examples

The example below illustratesTheorem 3.2.

Example 4.1. Consider the problem of solving the functional integral equation

x(t)− 1 1000

1

0tcos(st) sin

x(s)−1

1000

ds−sin x(t) 1000

=0 inC[0, 1]. (4.1)

For the choice F(x)=x(t)−1/100001tcos(st)sin((x(s)−1)/1000)ds and g(t,x(t))= −sin(|x(t)|/1000), where Ω=[0, 1] andx0≡1/1000, β=supt∈Ω|Fx0(t) +g(t,x0(t))|

=sup_t∈Ω|1/1000−0.17×10−7sint−0.17×10−7|<1/1000,Fxh(t)=h(t)−1/10002 1

0t cos(st) cos((x(s)−1)/1000)h(s)ds. For η=1/1000, η/(1−β)=0.001 000 001. Setting m=10−6_, ∗₌₁₀−3_{, and d =1, we have p(x)=4×10}−12_x2_{−0.000 007 993x+} 0.997 997 006 andα1 the smaller positive root of p(x) is 0.000 007 993−0.000 006 923/ 8×10−12=133750> η. For q(x)=0.000 002 999x2_{−0.998 999 002x+ 0.000 999 99, r}

1 the smaller positive root of q(x) is (0.998 999 002−0.998 998 996)/0.000 005 998= 0.001 000 333> η/(1−β). As the other positive rootr2=338 110.7032>332 999.667= (1−β−∗_{)/3md,Theorem 3.2(4) is satisfied. Forr}

0∈(0.001 000 333, 332 999.667), all the conditions of Theorem 3.2 are fulfilled. So the functional integral equation has a unique solution inU(x0,r0) and this solution can be obtained as the limit of the sequence of iterates in (1.1).

Example 4.2. The next example shows thatTheorem 2.1is more general thanCorollary 2.2, the main theorem obtained by Argyros [3].

Let f :R→Rbe the map defined by

f(x)= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

_x−0.01

25

20 +x2_sin1 x

, x=0,

− 1

125, x=0,

(4.2)

letg:R→Rbe the map defined byg(x)= |x| −0.01/1000. Choosex0=0,=0.25, andr=0.4. Clearly fory∈R,

f_x(y)= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

x−0.01 25

2xsin1 x−cos

1 x

+

20 +x2_sin(1/x) 25

y, forx=0,

0.8y, forx=0,

(f_x0)−

1 _{exists and (f}

x0)−

1 _{=1/0.8. As G satisfies the Lipschitz condition with M=} 1/1000,∗_=(M(f

x0)

−1_{)=0.001 25. Forx∈(−0.4, 0.4),}

_f

x−fxo=supf

x−fxo

(y):y ≤1

=sup

x−0.01 25

2xsin1 x−cos

1 x

+

x2_sin(1/x) 25

y:y ≤1

≤sup

_{0.4 + 0.01}

25 (2×0.4 + 1) + 0.16

25

|y|:y ≤1

=0.03592.≤0.25×0.8= F

x0

−1,

(4.4)

and (F_x0)−

1_(F

x−Fx0)<=0.25. Also c0=0.335, c=0.668 33, η=0.010 0125, 0≤

(fx0)

−1_{(f(x0) +g(x0)) ≤ 0.010 0125, and (c}

0/(1−c) + 1)= 2.009 789 1587. Since (c0/(1−c) + 1)η < r, all the conditions ofTheorem 2.1are verified. Thus f(x) +g(x)=0 has a unique solution inU(0, 0.4). However,F_x is not continuous at zero but piecewise hemicontinuous inU(0, 0.4). It may be noted thatCorollary 2.2(due to Argyros) cannot be applied to prove that this equation has a solution, whereasTheorem 2.1insures this.

Acknowledgment

V. Antony Vijesh acknowledges the Council of Scientific and Industrial Research (India) for the financial support provided in the form of a Junior Research Fellowship to carry out this research work.

References

[1] V. Antony Vijesh and P. V. Subrahmanyam, An extension of Newton’s method, Proceedings of the International Conference on Analysis and Applications, Allied Publishers Pvt., New Delhi, 2004, pp. 120–126.

[2] J. Appell, E. De Pascale, and P. P. Zabrejko, On the application of the Newton-Kantorovich method to nonlinear integral equations of Uryson type, Numerical Functional Analysis and Optimization 12 (1991), no. 3-4, 271–283.

[3] I. K. Argyros, On a theorem of L. V. Kantorovich concerning Newton’s method, Journal of Compu-tational and Applied Mathematics 155 (2003), no. 2, 223–230.

[4] L. B. Rall, Computational Solution of Nonlinear Operator Equations, John Wiley & Sons, New York, 1969.

V. Antony Vijesh: Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India

E-mail address:[email protected]

P. V. Subrahmanyam: Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India