Acta Mathematica Sinica, English Series
April, 2003, Vol.19, No.2, pp. 405–412
Convergence of Newton’s Method and Uniqueness of
the Solution of Equations in Banach Spaces II
Xing Hua WANG
Department of Mathematics, Zhejiang University, Hangzhou 310028, P. R. China
and
Department of Mathematics, Wenzhou University, Whenzhou 325027, P. R. China
E-mail: wangxh@mail.hz.zj.cn
Chong LI
Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
and
Academy of Mathematics and System Sciences, Chinese Academy of Sciences,
Beijing 100080, P. R. China
E-mail: cli@seu.edu.cn
Abstract Some results on convergence of Newton’s method in Banach spaces are established under
the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition
with a weak L average.
Keywords Nonlinear operator equation, Newton’s method, Lipschitz condition with L average,
Convergence ball
MR(2000) Subject Classification 65H10
Let f be an operator mapping from some domain D in a real or complex Banach space X to
another space Y . We continue to investigate the convergence problem of Newton’s method
xn+1 = xn − f (xn )−1 f (xn ),
n = 0, 1, 2, . . . ,
(1)
for finding the approximation solution of the equation
f (x) = 0.
(2)
In the previous paper [1], under the assumption that the derivative of f satisfies “the center
Lipschitz condition with L average”, the uniqueness theorem of the solution of Equation (2) was
established, where L is a positive integrable function. Furthermore, if L is nondecreasing and
Received May 30, 2000, Accepted September 6, 2000
This project is supported by the Special Funds for Major State Basic Research Projects (Grant No. G19990328)
and the National Natural Science Foundation of China (Grant No. 19971013, 10271025), also supported partly
by Zhejiang Provincial (Grant No. 100002, 101027) Natural Science Foundation of China
Wang X. H. and Li C.
406
the derivative of f satisfies “the radius Lipschitz condition with L average”, the convergence
theorem of Newton’s method was also established in the same paper.
As shown in paper [1], the center Lipschitz condition with L average is much weaker than the
radius Lipschitz condition with L average, see also the example given at the end of the present
paper. Thus one interesting question is that whether the radius Lipschitz condition with L
average and the nondecreasing of L are necessary for the convergence of Newton’s method. The
main purpose is, under the assumption that the derivative of f satisfies “the center Lipschitz
condition with L average”, to give the convergence theorems while L is assumed to belong to
some family of positive integrable functions that are not necessarily nondecreasing.
1
Main Theorems
For convenience, we write ρ(x) = x − x∗ and the derivative means the Frechet derivative in
the whole paper. The following two theorems were given in paper [1], which give, respectively,
the exact radius of the convergence ball of Newton’s method (1) and the uniqueness ball for
the solution of Equation (2).
Theorem A Suppose that f (x∗ ) = 0, f has a continuous derivative f in the ball B(x∗ , r),
f (x∗ )−1 exists and f (x∗ )−1 f satisfies the radius Lipschitz condition with the L average:
∗ −1
f (x )
τ
(f (x) − f (x )) ≤
ρ(x)
τ ρ(x)
L(u)du,
∀x ∈ B(x∗ , r), 0 ≤ τ ≤ 1,
where xτ = x∗ + τ (x − x∗ ) and L is a positive nondecreasing function. Let r satisfy
1 r
(r + u)L(u)du ≤ 1.
r 0
(3)
(4)
Then Newton’s method (1) is convergent for all x0 ∈ B(x∗ , r) and
xn − x∗ ≤ q 2
where
Theorem B
n
−1
x0 − x∗ , n = 0, 1, 2, . . . ,
ρ(x0 )
uL(u)du
0
.
q=
ρ(x0 )
ρ(x0 ) 1 − 0
L(u)du
(5)
(6)
Suppose that f (x∗ ) = 0, f has a continuous derivative f in the ball B(x∗ , r),
f (x∗ )−1 exists and f (x∗ )−1 f satisfies the center Lipschitz condition with the L average:
f (x∗ )−1 f (x) − I ≤
0
ρ(x)
L(u)du,
where L is a positive integrable function. Let r satisfy
1 r
(r − u)L(u)du ≤ 1.
r 0
Then Equation (2) has a unique solution in B(x∗ , r).
∀x ∈ B(x∗ , r),
(7)
(8)
Convergence of Newton’s Method and Uniqueness of the Solution of Equations in Banach Spaces II
407
It is noted that the result of Theorem A is completely new, which improves the result of [2]
under the assumption of Smale [2]. Theorem B includes the corresponding results of [4,5] and
[3,4] under the common Lipschitz condition and Smale’s condition, respectively.
The first theorem of the present paper gives the convergence result of Newton’s method
under the assumptions of Theorem A but L need not be nondecreasing. In this case, of course,
the convergence order will decrease.
Theorem 1.1 Suppose that f (x∗ ) = 0, f has a continuous derivative f in the ball B(x∗ , r),
f (x∗ )−1 exists and f (x∗ )−1 f satisfies the radius Lipschitz condition with the L average (3),
where L is positive integrable. Let r satisfy
r
1
L(u)du ≤ .
2
0
(9)
Then Newton’s method (1) is convergent for all x0 ∈ B(x∗ , r) and
xn − x∗ ≤ q n x0 − x∗ , n = 0, 1, 2, . . . ,
where
(10)
ρ(x0 )
q=
L(u)du
.
ρ(x0 )
1− 0
L(u)du
0
(11)
Furthermore, suppose that the function Lα defined by
Lα (t) = t1−α L(t)
is nondecreasing for some α with 0 ≤ α ≤ 1. Let r satisfy
1 r
(r + u)L(u)du ≤ 1.
r 0
Then Newton’s method (1) is convergent for all x0 ∈ B(x∗ , r) and
n−1
i
xn − x∗ ≤ q i=0 (1+α) x0 − x∗ , n = 0, 1, 2, . . . ,
(12)
(13)
(14)
where q is determined by (6).
The second theorem of the present paper gives a similar result to Theorem 1.1 under the
assumptions of Theorem B. In this case, of course, the radius of the convergence ball will
decrease even in the case when L is nondecreasing.
Theorem 1.2
Suppose that f (x∗ ) = 0, f has a continuous derivative f in the ball B(x∗ , r),
f (x∗ )−1 exists and f (x∗ )−1 f satisfies the center Lipschitz condition with the L average (7),
where L is a positive integrable function. Let r satisfy
r
1
L(u)du ≤ .
3
0
(15)
Then Newton’s method (1) is convergent for all x0 ∈ B(x∗ , r) and the inequality (10) holds for
ρ(x )
2 0 0 L(u)du
q=
.
(16)
ρ(x )
1 − 0 0 L(u)du
Wang X. H. and Li C.
408
Furthermore, suppose that the function Lα defined by (12) is nondecreasing for some α with
0 < α ≤ 1. Then
xn − x∗ ≤ q
2
(1+α)n −1
α
x0 − x∗ , n = 0, 1, 2, . . . .
(14)
Proof of Main Theorems
In order to prove the main theorems, we need two lemmas.
Lemma 2.1 [1]
Suppose that f has a continuous derivative f in the ball B(x∗ , r), f (x∗ )−1
exists and f (x∗ )−1 f satisfies the center Lipschitz condition (7). Then for r satisfying
r
L(u)du ≤ 1,
(17)
0
f is invertible in the ball B(x∗ , r) and
f (x)−1 f (x∗ ) ≤
Lemma 2.2
1−
1
ρ(x)
0
L(u)du
∀x ∈ B(x∗ , r).
,
(18)
Suppose that L is positive integrable. Assume that the function Lα defined by
(12) is nondecreasing for some α with 0 ≤ α ≤ 1. Then, for each β ≥ 0, the function ϕβ,α
defined by
1
ϕβ,α (t) =
tα+β
t
uβ L(u)du
0
is also nondecreasing.
Proof
We may assume that α + β > 0, without loss of generality. Now for any t2 > t1 > 0, we
have that
ϕβ,α (t2 ) − ϕβ,α (t1 ) =
=
1
tβ+α
2
1
tβ+α
2
t2
0
t2
t1
≥ Lα (t1 )
= Lα (t1 )
−
−
1
tβ+α
2
1
tβ+α
2
1
tβ+α
1
1
t1
t2
0
uβ L(u)du
0
tβ+α
1
t2
t1
−
−
−
1
tβ+α
2
1
tβ+α
1
1
tβ+α
1
0
0
−
t1
t1
1
tβ+α
2
uβ+α−1 Lα (u)du
t1
0
uβ+α−1 du
uβ+α−1 du
= 0.
This completes the proof.
Proof of Theorem 1.1
Let x0 ∈ B(x∗ , r) and {xn } be the sequence generated by Newton’s
method (1) and r satisfies (17). Then, exactly as in the proof of Theorem A in [1], we have
that
∗
xn+1 − x ≤
ρ(xn )
uL(u)du
.
ρ(xn )
1− 0
L(u)du
0
(19)
Convergence of Newton’s Method and Uniqueness of the Solution of Equations in Banach Spaces II
409
Thus, if x0 ∈ B(x∗ , r) while r satisfies (9), it follows from (19) that
ρ(xn )
L(u)du
∗
0
xn+1 − x ≤
ρ(xn ) ≤ qxn − x∗ ,
ρ(xn )
1− 0
L(u)du
where q < 1 is determined by (11). This implies that (10) holds so that xn converges to x∗ .
Furthermore, if the function Lα defined by (12) is nondecreasing for some α with 0 ≤ α ≤ 1
and r is determined by (13), it follows from (19) and Lemma 2.2 that
xn+1 − x∗ ≤
ϕ1,α (ρ(xn ))
ρ(xn )1+α
ρ(xn )
1− 0
L(u)du
ϕ1,α (ρ(x0 ))
ρ(xn )1+α
ρ(xn )
1− 0
L(u)du
q
=
xn − x∗ 1+α ,
ρ(x0 )α
≤
where q < 1 is determined by (6). Hence (14) holds and xn converges to x∗ . The proof is
complete.
Proof of Theorem 1.2
Similarly, let x0 ∈ B(x∗ , r), and let {xn } be the sequence generated by
Newton’s method (1). Let r and q be determined by (15) and (16), respectively. Assume that
xn ∈ B(x∗ , r). Then
xn+1 − x∗ = f (xn )−1 {f (x∗ ) − f (xn ) + f (xn )(xn − x∗ )}
1
= f (xn )−1 f (x∗ )
f (x∗ )−1 {f (xn ) − f (xτ )}(xn − x∗ )dτ,
0
τ
∗
∗
where x = x + τ (xn − x ). Therefore,
∗
xn+1 − x ≤ f (xn )
−1 ∗
f (x )
0
1
f (x∗ )−1 {f (xn ) − f (xτ )}xn − x∗ dτ
1
∗ −1 f (x ) {f (xn ) − f (x∗ )}
0
+ f (x∗ )−1 {f (xτ ) − f (x∗ )} xn − x∗ dτ.
≤ f (xn )−1 f (x∗ )
Using Lemma 2.1 and condition (7), we have that
τ ρ(xn )
1 ρ(xn )
L(u)du
+
L(u)du
xn − x∗ 0
0
0
∗
xn+1 − x ≤
ρ(x )
1 − 0 n L(u)du
ρ(xn )
(2ρ(xn ) − u)L(u)du
= 0
ρ(x )
1 − 0 n L(u)du
ρ(x )
2 0 n L(u)du
≤
ρ(xn ).
ρ(x )
1 − 0 n L(u)du
Taking n = 0, we have that
x1 − x∗ ≤ qx0 − x∗ < x0 − x∗ < r.
(20)
Wang X. H. and Li C.
410
Now assume that
xn − x∗ ≤ qxn−1 − x∗ < xn−1 − x∗ < x0 − x∗ .
Then, it follows from (15) and (20) that
xn+1 − x∗ ≤
2
ρ(x0 )
L(u)du
ρ(xn ) = qxn − x∗ .
ρ(x0 )
1− 0
L(u)du
0
Hence, by mathematical induction, (10) holds and xn converges to x∗ .
Furthermore, if the function Lα , defined by (12), is nondecreasing for some α with 0 < α ≤ 1,
then, from Lemma 2.2, we have
xn+1 − x∗ ≤
2ϕ0,α (ρ(xn ))
ρ(xn )1+α
ρ(xn )
1− 0
L(u)du
2ϕ0,α (ρ(x0 ))
ρ(xn )1+α
ρ(x0 )
1− 0
L(u)du
q
=
xn − x∗ 1+α .
ρ(x0 )α
≤
This implies that (14) holds and proves the theorem.
3
Corollaries of Main Theorem and Example
From Theorem 1.1 and Theorem 1.2, we immediately have:
Corollary 3.1
∗ −1
f (x )
Suppose that f (x∗ ) = 0, f has a continuous derivative f in the ball B(x∗ , r),
exists and f (x∗ )−1 f satisfies the radius Lipschitz condition with L(u) = cαuα−1 :
f (x∗ )−1 (f (x) − f (xτ )) ≤ c(1 − τ α )x − x∗ ,
∀x ∈ B(x∗ , r), 0 ≤ τ ≤ 1,
(3a)
where xτ = x∗ + τ (x − x∗ ), 0 < α ≤ 1 and c > 0. Let
r=
1+α
c(1 + 2α)
α1
.
(4a)
Then Newton’s method (1) is convergent for all x0 ∈ B(x∗ , r) and (14) holds for
q=
Corollary 3.2
cαx0 − x∗ α
.
(1 + α) (1 − cx0 − x∗ α )
(6a)
Suppose that f (x∗ ) = 0, f has a continuous derivative f in the ball B(x∗ , r),
f (x∗ )−1 exists and f (x∗ )−1 f satisfies the center Lipschitz condition with L(u) = cαuα−1 :
f (x∗ )−1 f (x) − I ≤ cx − x∗ α ,
where 0 < α ≤ 1 and c > 0. Let
r=
1
3c
∀x ∈ B(x∗ , r),
(7a)
α1
.
(15a)
Convergence of Newton’s Method and Uniqueness of the Solution of Equations in Banach Spaces II
411
Then, Newton’s method (1) is convergent for all x0 ∈ B(x∗ , r) and (14) holds for
q=
2cx0 − x∗ α
.
1 − cx0 − x∗ α
(16a)
Corollary 3.3 Suppose that f (x∗ ) = 0, f has a continuous derivative f in the ball B(x∗ , r),
2γc
f (x∗ )−1 exists and f (x∗ )−1 f satisfies the center Lipschitz condition with L(u) = (1−γu)
3:
f (x∗ )−1 f (x) − I ≤
where c > 0, γ > 0. Let
c
− c,
(1 − γx − x∗ )2
∀x ∈ B(x∗ , r),
3c + 1 − 3c(3c + 1)
r=
.
(3c + 1)γ
(7b)
(15b)
Then, Newton’s method (1) is convergent for all x0 ∈ B(x∗ , r) and (14) holds for
q=
2c − 2c(1 − γx0 − x∗ )2
.
(1 + c)(1 − γx0 − x∗ )2 − c
Example 3.1 Let X = Y = R, the reals. Define
x
π
f (x) =
1 + x sin
dx,
x
0
Then
f (x) =
1 + x sin
π
,
x
1,
(16b)
∀x ∈ R.
x = 0,
x = 0.
Obviously, x∗ = 0 is a zero of f and f satisfies that
π f (x∗ )−1 f (x) − I = x sin ≤ |x − x∗ |,
x
∀x ∈ R.
It follows from Theorem 1.2 that for any x0 ∈ B(x∗ , 1/3)
∗
xn − x ≤
2|x0 |
1 − |x0 |
2n
x0 − x∗ ,
n = 1, 2, . . . .
However, there is no positive integrable function L such that (3) is satisfied. In fact, observe
that
π 2
π
,
f (x∗ )−1 (f (x) − f (xτ )) = x sin − τ x sin
=
x
τx
2k + 1
for x = 1/k, τ =
2k
2k+1
and k = 1, 2, . . . . Thus, if there was a positive integrable function L
such that (3) holds on B(x∗ , r) for some r > 0, it follows that there exists some k0 > 1 such
that
0
r
L(u)du ≥
+∞ k=k0
1
k
2
2k+1
L(u)du ≥
+∞
k=k0
2
= +∞,
2k + 1
which is a contradiction. This example shows that Theorem 1.2 is an essential improvement of
Theorem 1.2 if the radius of the convergence ball is ignored.
412
Wang X. H. and Li C.
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