Monotone convergence of point to set-mapping

Chapter 11

Monotone convergence of point to set-mapping In this chapter we discuss the monotonicity of the iteration sequence generated by the algorithmic model Xk+l e f k ( X O , X l , . . . , X k )

(k _> l - 1).

(11.0.1)

Here, we assume again that for k >_ 1 - 1, f k " x k + l --+ 2 x , where X is a subset of a partially ordered space L, and let < denote the order in L, (see also Chapter 2, [97]-[101])

11.1

General

Theorems

Our main results will be based on a certain type of isotony of the algorithmic mappings.

Definition 11.1.1 The sequence of point-to-set m a p p i n g gk " X k+l ~ 2 L is called increasingly isotone f r o m the left on X , if f o r any sequence {x (k) } such that x (k) C X (k > O) and x (~ < x (1) < x (2) < ..., y~ < y2 f o r all y~ E g k ( x ( ~ Y2 C gk+l(X(O),X(1),...,X(k+l)) and k > l - 1.

Consider the special case of single variable mappings gk. The above properties are now reduced to the following. Assume that x(~ (1) C X, x (~ _< x (1), yl E gk(x(O)), and Y2 C gk+l (X(1)), then Yl < Y2. Similarly, the sequence of point-to-set mappings gk " x k + l ~ 2L is called increasingly isotone from the right on X, if for any sequence {x (k) } such that x (k) E X (k > 0) and x (~ < x (1) _< x (2) < ..., yl > y2 for all yl E g k ( x ( ~ Y2 C g k + l ( X ( O ) , x ( 1 ) , . . . , x ( k + l ) ) and k > l - 1. Our main result is the following: 409

11. Monotone convergence of point to set-mapping

410

T h e o r e m 11.1.2 Assume that the sequence of mappings fk is increasingly isotone from the left; futhermore, xo <_ Xl ~ ... ~ Xl-1 ~ Xl (Xk C X , 0 ~ k ~ l - 1). Then for all k >_ 0, Xk+l >__xk. P r o o f . By induction, assume that for all indices i (i < k), Xi+l ~_ Xi. The relations Xk C fk-l(XO ~ Xl ~ ... ~ Xk-1)

and

Xk+l C fk(XO ~ Xl ~ ... ~ Xk)

and Definition 11.1.1 imply that Xk+l >_ Xk. Because this inequality holds for k _< 1 - 1, the proof is complete. 9 Consider next the modified algorithmic model (11.1.1)

Yk+l E fk(Yk, Y k - l , . . . , Yl, Y0),

which differs from process (11.0.1) only in the order of the variables in the righthand side. Using finite induction again similarly to Theorem 11.1.2, one may easily prove the following theorem. T h e o r e m 11.1.3 Assume that the sequence of mappings fk is increasingly isotone from the right; furthermore, yo >_ yl >_ ... >_ yl-1 >_ yl (Yk C X, 0 <_ k <_ l - 1). The for all k >_ 0, yk+l ~_ Yk. C o r o l l a r y 11.1.4 Assume that X C_ I~n; furthermore, xk ---* s* and Yk ---* s* as k ~ cx~. It is also assumed that " ~ " is the usual partial order of vectors; i.e., a - (a (i)) < b - (b (~)) if and only if for all i, a (~) < b (~). Under the conditions of Theorems 11.1.2 and 11.1.3,

Xk<_S*<_yk. This relation is very useful in constructing an error estimator and a stopping rule for methods (11.0.1) and (11.1.1), since for all coordinates of vectors xk, Yk and s*, 0 _< s *(~) - x~~) _< Yk(i) _ x~ i) and . (i)

0 _ < y ~ i ) - s *(i) < y k

-xk

(i)

9

Hence, if each element of yk - Xk is less than an error tolerance e, then all elements of Xk and Yk are closer than e to the corresponding elements of the limit vector. 11.2

Convergence

in linear

spaces

Assume now that X is a subset of a partially ordered linear space L. In addition, the partial order satisfies the following conditions:

411

11.2. Convergence in linear spaces

(F1). x <_ y (x, y E L) implies that - x >_ y; (F2). If x(1),x(2),y(1),y (2) y(1) + y(2).

E

L, x (i) _< y(1) and x (2) _< y(2) , then x (1) + x (2) <

Assume that for k _> l - 1 and p - 1,2, mapping K (p), Hs p) are point-to-set mapping defined on X k+l, and for all (x(1), x (2), ...,x (k+~)) E X k+~, K~P)(x(1),x (2) ,...,x(k+l)), and H(kP)(x(1),x(2),...,x (k+l)) are nonempty subsets of L; i.e., Kk(p), and H~ p) 9 X k+l ~ 2 L. Consider next the algorithmic model: Xk+l _ §~ k + l - - ~ e(1)

and

Yk+l _ §~ k + l -- ~~(2) k+l

( 1 1 . 2 "1)

where §~ k + l C K

1 ) ( x 0 ~Xl ~ ""~ xk) ~ ~~(1) k+l

E H~ 1)(yk ~Yk-1 ~ "'" Yo)

and

t(2) k+l

CK

(2)

~(1)

(Yk Yk-1 ~ ""~ YO), ~ k + l E

U(2)

(Xo, Xl~ ...~ Xk) 9

In the formulation of our next theorem we will need what follows. 11.2.1 A point-to-set mapping g " X k + l ~ 2 L is called increasingly isotone on X , if for all x (1) <_ x (2) <_ ... < x (k+2) (x (i) E X , i - 1,2,...,k + 2), relations yl E g ( x ( i ) , x (2), ...,x (k+l)) and y2 E g(x(2),x (3), ...,x (k+2) imply that yl <

Definition

Y2.

Increasingly isotone functions have the following property. L e m m a 11.2.2 A s s u m e that mapping g 9 X k+l ~ 2 L is increasingly isotone. Then, for all x (~) and y(i) E X (i - 1, 2, ..., k + 1) such that x(1) ~_ x(2) _~ ...,x(k+l) ~ y(1) _~ y(2) ~ ... ~ y ( k + l )

and for any x E g(x (1), x (2), ..., x (k+l)) and y E g(y(1), y(2), ..., y(k+l)), x _< y.

Proof. Then

Let Yi E g(x(i+l),...,x(k+l),y(1),...,y (i)) be arbitrary for i -

1,2,...,k.

x ~ Yl ~_ Y2 <_ ... <_ Yk <_ Y,

which completes the proof. 9 R e m a r k 11.2.1 In the literature a point-to-set mapping g " X k+l --~ 2 L is called isotone on X , if for all x (~) and y(i) such that x (~) < y(i) (i - 1, 2, ..., k + 1) and for all x E g(x(1),...,x (k+l)) and y E g(y(1),...,y(k+i)), x < y. It is obvious that an isotone mapping is increasingly isotone, but the reverse is not necessarily true as the following example illustrates.

11. Monotone convergence of point to set-mapping

412 Example

11.2.1

Define L -

IR~, X - [ 0 , 1], k -

1, _< to be the usual order of real

numbers, and

{x(2) g(X(1)'X(2))

--

X (1)'- X (2) + l,

if x(1) >_ 2x(2) _ 1 if x (1) < 2x (2) - 1.

We will now verify that g is increasingly isotone, but not isotone on X. Select arbitrary x (1) <__ x (2) <_ x (3) from the unit interval. Note first that g(x(1),x (2)) <_ x (2), since if x (1) _> 2x (2) - 1, then g(x(1),x (2)) - x (2), and if x (1) < 2x (2) - 1, then g(x(1),x (2)) - x (1) - x (2) + 1 < 2x (2) - 1 - x (2) + 1 - x (2). Note next that g(x(2),x (3)) >_ x (2), since if x (2) > 2x (3) - 1, then g(x(2),x (3)) - x (3) _> x (2), and if x (2) < 2x (3) - 1, then 9(x(2),x (3)) - x (2) - x (3) + 1 >_ x (2). Consequently, g(x(~),x (2)) <_ x (2) <_ 9(x(2),x(3)). Hence, g is increasingly isotone; however, it is not isotone on X, since for any points (t, 1) and (t, 1 - e ) (where t, e > 0 and t+2e<

1),

g(t, 1) - t -

1 + 1 - t < g(t, 1 - e )

- t-(1

-e)+

1 - t + e,

but i > 1 - e. C o n s i d e r now t h e following a s s u m p t i o n s : (G1) S e q u e n c e s of m a p p i n g K~ 1) a n d H (2) are i n c r e a s i n g l y i s o t o n e f r o m t h e left. P u t h e r m o r e , Kk(1) a n d H (2) are i n c r e a s i n g l y i s o t o n e f r o m t h e right. (G2) For all k a n d t (~ < t (1) _< ... <_ t (k) <_ t-(k) <_ ... <_ ~-(1) ~

~-(0) (t(i)

a n d t-(i) E X ,

= o,l,...,k),

<_ K(:)

..., t o))

and H~l)(t--(k), ..., t--(~ _> H~2)(t(~

(G3) T h e initial p o i n t s are selected so t h a t

x0
<_ ... <_ xz-x <_xl_
(G4) (xo,yo) -- {X IX E L, Xo <_ x <_ yo} C X. 11.2.3 Under assumtions (F1), (/'2) and (G1) through (G4), xk and yk are in X for all k > O. Furthermore,

Theorem

xk <_ xk+~ <_ yk+~ <_ yk.

413

11.2. Convergence in linear spaces

P r o o f . By induction, assume that y0 > xi+l > xi and xo <=yi+l <_ yi for all i < k. Therefore, points xi and yi (i - 0, 1, ..., k) are in X. Note next that assumption (G1) implies that

t(1) ~ t k(1),(2)
(1) s(k2)'

~ 8k+1-

and hence, §

Xk+l

-- ~k+l

(1) -- 8k+l

> t(1)

--

k

(1)

-- 8 k

-~ X k

and Yk+l -- t(2) k+l

~

< t}r)c9 _ S~()(o --'-- Y k . --

-- ~

Assume next that xi < yi (i <_ k). Then, assumption (G2) implies that

t(1) < +(2) and _(1)> _(2) k+l

-- ~k+l

~

-- ~

o(1) -- ~

.< t ( 2 ) -- ~ k + l

Therefore, §

Xk+l

-- ~k+l

e(2) -- ~

-- Yk+l.

Thus, the proof is complete. 9 Introduce next the point-to-set mappings f(P)(x

(1),...,x

(k+l))

-- {t--8

It C K(kP)(x(1),...,X(k+I)),

8 E H(kP)(x(1),...,x(k+I))}.

Assume that z is a common fixed point of mappings f~l) and f~2); i.e., for p - 1,2, z C f ( P ) ( z , z , ..., z), furthermore (G5). Mappings KO)K(2)H(kl)r-r(2)..k are increasingly isotone. T h e o r e m 11.2.4 A s s u m e that Xl <_ z < yl. Then, under assumptions (F1), (F2) and (Gz) through (Gs), x k < z < Yk for all k > O.

P r o o f . By induction, assume that xi < z < yi, for i < k. Then, for p = 1, 2 and for all t (p) C K(P)(z, z, ..., z ) a n d s (p) E H (p)k (z, z, ..., z), §

~k+l

_(2) < t(1) t(2) < +(2) and o(1) > s(1) s(2) > ~k+l --

'

-- ~k+l

~

--

~

--

"

Therefore, +(1) ~(1) < t(p) (p) +(2) _(2) Xk+l = ~ k + l - - ~ k + l _ --S --<~k+l--~k+l =Yk+l,

and since z - t - s with some t and s, the proof is completed.

(p=l,2) 9

11. Monotone convergence of point to set-mapping

414 Remark

11.2.2 Two important special cases of algorithm (11.2.1) can be given:

1 If K (1) - K ( 2 ) - Kk and H ( 1 ) - H ( 2 ) - Hk, then f ( 1 ) _ f ( 2 ) _ fk, and s o z is a common fixed point of the mapping fk. 2 Assume next that Kk and Hk are 1-variable functions. Then one may select K~l)(t(o),...,t(k)) - Kk(t(k-l+l) ...,t(k)), K (2) (t(0), ..., t (k)) - Kk(t(~

..., t(/-1)),

H~ 1) (t (0) ' ..., t (k)) - Hk(t(~

t (1-1))

and H (2) (t(0), ..., t (k)) - Hk(t(k-l+l), ..., t(k)). Note that the resulting process for 1 - i was discussed in Ortega and Rheinboldt ([263], Section 13.2). C o r o l l a r y 11.2.5 Assume that in addition L is a Hausdorff topological space which satisfies the first axiom of countabi;ity, and

(F3) If xk C L (k >_ O) such that Xk < x with some x C L and xk <_ Xk+l for all k _> 0, then the sequence {xk } converges to a limit point x* c L; furthermore, X* < X .

(F4) If xk C L (k >_ O) and xk ~ x, t h e n - x k

~ -x.

Note first that assumptions (F1) through (F4) imply that any sequence {Yk} with the properties yk E L, Yk ~_ Yk+l, and Yk _~ Y (k _> 0) is convergent to a limit point y* C L; furthermore, y* > y. Since under the conditions of the theorem {Xk } is increasing with an upper bound y0 and {yk} decreases with a lower bound x0, both sequences are convergent. If x* and y* denote the limit points and z is a common fixed point of mappings fk such that xz _< z _< Yz then x* _< z _< y*. Consider the s p e c i a l / - s t e p algorithm given as special case 2 above. Assume that for all k _> l - 1, mappings Kk and Hk are closed. Then, x* is the smallest fixed point and y* is the largest fixed point in (xz, yl}. As a further special case we mention the following lemma. L e m m a 11.2.6 Let B be a partially ordered topological space and let x. y be two points of B such that x <_ y. If f " (x, y) ~ B is a continuous isotone mappings having the property that x < f (x) and y > f(y), then a point z E (x, y) exists such that z - f ( z ) . P r o o f . Select X - (x, y}, 1 - 1, Hk = 0 and Kk = f, and consider the iteration sequences {xk} and {Yk} starting from x0 -- x and Y0 - Y. Then, both sequences are convergent, and the limits are fixed points. 9 Remark

11.2.3 This result is known as Theorem 2.2.2, (see, Chapter 2).

11.3. Applications

11.3

415

Applications

Consider the simple iteration method

x k + ! - f(xk),

(11.3.1)

where f ' [ a , b] -~ R with the additional properties:

f(a)>a,

f(b)
for all x c[a,b],

where q is a given constant. Consider the iteration sequences Xk+l

--

f(Xk),

X0 -

a

and

yk+l - f(yk),

Y0 - b

(11.3.2)

Since f is increasingly, Theorems 11.1.2 and 11.1.3 apply to both sequences {xk} and {Yk} that converge to the unique fixed point s*; furthermore, sequence {xk} is increasing and {Yk} decreases. This situation is illustrated in Figure 11.3.1. Next we drop the assumption f'(x) <_ q < 1, but retain all other conditions. In other words, f is continuous, increasing in [a, b],

f (a) > a

f (b) < b.

Notice that Theorems 11.1.2 and 11.1.3 still can be applied for the iteration sequence (11.3.2). However, the uniqueness of the fixed point is no longer true. This situation is shown in Figure 11.3.2. Becausef is continuous, there is at least one fixed point in the interval (a, b). Alternatively, Theorems 11.2.3 and 11.2.4 may also be applied with K (1) - K (2) f and H~ 1) - H (2) - 0, which has the following additional consequence. Let s* and s** denote the smallest and the largest fixed points between a and b. Because a < s* and b > s**, f(a) <_ f ( s * ) = s* and f(b) >__f ( s * * ) - s**, thus, s* and s** are between Xl and yl. Therefore, for all k _> 1, xk -< s* = s** -<_ yk; futhermore, xk ~ s* and Yk --~ s**. If the fixed point is unique, the uniqueness of the fixed point can be estabilished by showing that sequences {xk} and {Yk} have the same limit. Finally, we mention that the above facts can be easily extended to the more general case when f " IRn ~ ~n. E x a m p l e 11.3.1 We now illustrate procedure (11.3.2) in the case of the single

variable nonlinear equation x-we

1

10

x +1.

416

11. Monotone convergence of point to set-mapping f(x)

y=x

y=f(x)

f

.............

/ ,j

.

x0

x

.

.

.

.

x2 s

Y2

.

.

.

.

.

.

.

.

Yl

~

YO

Figure 11.1" Monotone fixed point iterations.

f(x)

y=x

=

J~x o

Xl

s*

s**

y.

Yo

)

"~

Figure 11.2: The case of multiple fixed points

It is easy to see that

le1+1> 10

1

and

1 e2 + 1 < 2 .

Further, d

x(le x

_~ex e (o, o.75).

11.~. Exercises

417

Therefore, all conditions are satisfied. Select xo - 1 and yo - 2. Then,

Xl -- ~1 el -~- 1 ~ 1.27183;

y l - ] - ~ 1 e2 + 1 ~ 1 . 7 3 8 9 1 ;

1 el.27183 x2 - 1-0 + 1 ~ 1.35674;

Y 2 - 1----e1"73891 -+-1 ~ 1.56911; 10 "

y3 = 1 e1"56911 -+- 1 ~ 1.48024; _0

1 el.38835 x4 - 1-0 + 1 ~ 1.40082;

Y4 = ~ e 1"48024 -t- 1 ~ 1.43940;

1 ela~176 x3 - 1-0 + 1 ~ 1.40585;

Y5 -- ~1 e143940 -+- 1 ~ 1.42182,

-

and so on. visible. 11.4

4

1 el.35674 x3 -- 1-0 + 1 ~ 1.38835;

4

10

The monotonicity of both iteration sequences, as expected, is clearly

Exercises

1. Consider the p r o b l e m of a p p r o x i m a t i n g a locally unique zero s* of the equation

g(x)-0, where g is a given real function defined on a closed interval [a, b]. T h e N e w t o n iterates {x~}, {Yn}, n _> 0 are defined as g ( X n ) + g t ( X n ) ( X n + l -- Xn) -- 0

xo=a

g ( Y n ) Jr- g t ( Y n ) ( Y n + l -- Yn) -- 0

yo - b.

and

Use T h e o r e m s 11.1.2 and 11.1.3 to find sufficient conditions for the m o n o t o n e convergence of the above sequences to s* C [a, b]. 2. E x a m i n e the above problem, b u t using the secant iterations {xn}, {y~}, n >_ - 1 given by

g(Xn) + g(Y~) +

g(Xn) --g(Xn--1)

(Xn+l -- Xn) -- 0

Xl -- a, X0 -- given

(Yn+l - Yn) = 0

Yl = b, Y0 = given.

Xn -- X n - 1 g(Yn)--g(Yn-1) Yn -- Y n - 1

3. E x a m i n e the above problem, b u t using the Newton-like iteration, where {xn}, {y~}, n _> 0 are given by g(x

) +

-

-

0

xo-a

418

11. M o n o t o n e

convergence

o f p o i n t to s e t - m a p p i n g

and g(Yn) + h(yn)(Yn+l

-

Yn) -- 0

yo - b

with a suitably chosen function h. 4. Examine the above problem, but using the Newton-like iteration, where {xn }, {yn}, n _> 0 are given by g(Xn)+Cn(Xn+l - - X n ) - - 0

x0 - a

g(Yn) + dn(Yn+l - Yn) - 0

y0 - b

and

with a suitably chosen real sequences {cn }, {dn }. 5. Generalize Problems 1 to 4 to R n. 6. Generalize Problems 1 to 4 to I~n by choosing g to be defined on a given subset of a partially ordered topological space. (For a definition, see Chapter

2). 7. Compare the answers given in Problem 1 to 6 with the results of Chapter 2. 8. To find a zero s* for g ( x ) - O, where g is a real function defined on [a, b], rewrite the equation as x -

x + cg(x) -

f(x)

for some constant c r 0. Consider the sequences Xn+l

--

f(xn)

xo -

a

and Y n + l -- f ( Y n )

YO -- b.

Use Theorems 11.1.2 and 11.1.3 to find sufficient conditions for the monotone convergence of the above sequences to a locally unique fixed point s* E [a, b] ot the equation x - f ( x ) .