Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces

Fixed Point Theory and Applications Volume 2008, Article ID 528614,10pages doi:10.1155/2008/528614

Research Article

Convergence to Compact Sets of

Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces

Evgeniy Pustylnik, Simeon Reich, and Alexander J. Zaslavski Department of Mathematics, The Technion-Israel Institute of Technology, 32000 Haifa, Israel

Correspondence should be addressed to Simeon Reich,[email protected] Received 27 September 2008; Accepted 17 November 2008

Recommended by Brailey Sims

We study the influence of computational errors on the convergence to compact sets of orbits of nonexpansive mappings in Banach and metric spaces. We first establish a convergence theorem assuming that the computational errors are summable and then provide examples which show that the summability of errors is necessary for convergence.

Copyrightq2008 Evgeniy Pustylnik et al. 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.

1. Introduction

Convergence analysis of iterations of nonexpansive mappings in Banach and metric spaces is a central topic in nonlinear functional analysis. It began with the classical Banach theorem 1on the existence of a unique fixed point for a strict contraction. Banach’s celebrated result also yields convergence of iterates to the unique fixed point. There are several generalizations of Banach’s fixed point theorem which show that the convergence of iterates holds for larger classes of nonexpansive mappings. For instance, Rakotch2introduced the class of contractive mappings and showed that their iterates also converged to their unique fixed point.

In view of these results and their numerous applications, it is natural to ask if convergence of the iterates of nonexpansive mappings will be preserved in the presence of computational errors. In 3, we provide affirmative answers to this question. Related results can be found, for example, in4,5. More precisely, in3we show that if all exact iterates of a given nonexpansive mapping convergeto fixed points, then this convergence continues to hold for inexact orbits with summable errors. In6, we continued to study the influence of computational errors on the convergence of iterates of nonexpansive mappings in both Banach and metric spaces. We show there that if all the orbits of a nonexpansive

self-mapping of a metric spaceX converge to some closed subsetF of X, then all inexact orbits with summable errors also converge to this attractor setF. On the other hand, we also construct examples which show that inexact orbits may fail to converge if the errors are not summable.

Our purpose in the present paper is to consider the case where different exact orbits converge to possibly different compact subsets ofX. InSection 2, we obtain a convergence result see Theorem 2.1 below under the assumption that the computational errors are summable. This result is an extension of3, Theorem 4.2. In Sections3 and4, we provide examples which show that the summability of errors is necessary for convergence see Proposition 3.1andTheorem 4.1.

2. Convergence to compact sets

LetX, ρbe a complete metric space. For eachx∈X, and each nonempty and closed subset A⊂X, put

ρx, A inf

ρx, y: y∈A

. 2.1

For each mappingT :X → X, setT⁰xxfor allx∈X.

Theorem 2.1. LetT :X → Xsatisfy

ρTx, Ty≤ρx, y ∀x, y∈X. 2.2

Suppose that for eachx∈X, there exists a nonempty compact setEx⊂X, such that

ilim→ ∞ρ

Tⁱx, Ex

0. 2.3

Assume that{γn}^∞_n0⊂0,∞,_∞

n0γn<∞, x_n∞

n0⊂X, ρ

x_{n 1}, Tx_n

≤γ_n, n0,1, . . . . 2.4

Then, there exists a nonempty compact subsetFofXsuch that

n→ ∞limρ xn, F

0. 2.5

Proof. In order to prove the theorem, it is sufficient to show that any subsequence of{xn}^∞_n0 has a convergent subsequence.

To see this, it is sufficient to show that for any >0, the following assertion holds:

P1 any subsequence of{xn}^∞_n0 has a subsequence which is contained in a ball of radius.

Indeed, there is an integerk≥1 such that ∞

ik

γ_i<

8. 2.6

Define a sequence{yi}^∞_ikby

y_kx_k, 2.7

y_{i 1}Ty_i for all integersi≥k. 2.8

There exists a nonempty compact setE⊂Xsuch that

ilim→ ∞ρ yi, E

0. 2.9

By2.4,2.7, and2.8,

ρ

x_{k 1}, y_{k 1}

≤γ_k. 2.10

Assume thatq≥k 1 is an integer and that forik 1, . . . , q,

ρ xi, yi

≤ⁱ⁻¹

jk

γj. 2.11

Note that in view of2.10, inequality2.11is valid whenqk 1.

By2.2and2.11,

ρ

Tyq, Txq

≤ρ yq, xq

≤^q−1

jk

γj. 2.12

When combined with2.4, this implies that

ρ

x_{q 1}, y_{q 1}

≤ρ

x_{q 1}, Tx_q ρ

Tx_q, Ty_q

≤γ_q ^q−1

jk

γ_j^q

jk

γ_j, 2.13

so that2.11also holds foriq 1. Thus, we have shown that for all integersq≥k 1,

ρ yq, xq

≤^q−1

jk

γj<^∞

jk

γj<

8 2.14

by2.6. In view of2.9, we have for all large enough natural numbersq, ρ

xq, E

<

4. 2.15

By2.15, there exist an integerq0> kand a sequence{z_i}^∞_iq0 ⊂Esuch that ρ

xi, zi

<

3 for all integersi≥q0. 2.16

Consider any subsequence{xqi}^∞_i1of{xn}^∞_n0. Since the setEis compact, the sequence{zqi}^∞_i1 has a convergent subsequence{z_qij}^∞

j1.

We may assume without loss of generality that all elements of this convergent subsequence belong toBu, /16for someu∈X.

In view of2.16, x_qij ∈B

u,

2 for all sufficiently large natural numbersj. 2.17 Thus,P1holds and this completes the proof ofTheorem 2.1.

Note thatTheorem 2.1is an extension of the following result established in3.

Theorem 2.2. LetX, ρbe a complete metric space and letT :X → Xbe such that

ρTx, Ty≤ρx, y ∀x, y∈X, 2.18

and for eachx∈X, the sequence{Tⁿx}^∞_n1converges inX, ρ.

Assume that{γ_n}^∞_n0 ⊂0,∞satisfies_∞

n0γ_n<∞, and that a sequence{x_n}^∞_n0 ⊂Xsatisfies ρxn 1, Txn≤γn, n0,1, . . . .Then, the sequence{xn}^∞_n1converges to a fixed point ofTinX, ρ.

3. First example of nonconvergence to compact sets

In this section, we show that both Theorems2.1and2.2cannot, in general, be improvedcf.

6, Proposition 3.1.

Proposition 3.1. For any normed spaceX, there exists an operatorT : X → X such that Tx− Ty ≤ x−y for allx, y ∈ X, the sequence{Tⁿx}^∞_n1 converges for eachx ∈ X and, for any sequence of positive numbers{γn}^∞_n0, there exists a sequence{xn}^∞_n0 ⊂X with xn 1−Txn ≤γn

for all nonnegative integersn, which converges to a compact set if and only if the sequence{γ_n}^∞_n0is summable, that is,_∞

n0γ_n<∞.

Proof. This is a simple fact because we may takeT to be the identity operator:Tx x,∀x.

Then, we may takex0to be an arbitrary element ofXwith x0 1 and define by induction xn 1Txn γnx0, n0,1,2, . . . . 3.1

Evidently, xn 1 −Txn γn andxn 1 x0 1 _n

i0γifor all integers n ≥ 0, so that the convergence of {x_n}^∞_n0 to a compact set is equivalent to the summability of the sequence {γn}^∞_n0.Proposition 3.1is proved.

4. Second example of nonconvergence to compact sets

In Section 3, we have shown that Theorems 2.1 and 2.2 cannot, in general, be improved.

However, inProposition 3.1every point of the space is a fixed point of the operatorT, and the inexact orbits tend to infinity. In this section, we construct an operatorTon a certain complete metric spaceXa bounded, closed, and convex subset of a Banach spacesuch that all of its

orbits converge to its unique fixed point, and for any nonsummable sequence of errors and any initial point, there exists an inexact orbit which does not converge to any compact setcf.

6, Theorem 4.1.

Let X be the set of all sequences x {xi}^∞_i1 of nonnegative numbers such that _∞

i1x_i≤1. Forx{x_i}^∞_i1andy{y_i}^∞_i1inX, set ρ

x_i∞

i1, y_i∞

i1

^∞

i1

x_i−y_i. 4.1

Clearly,X, ρis a complete metric space.

Define a mappingT :X → Xas follows:

T

xi_∞

i1

x2, x3, . . . , xi, . . .

,

xi_∞

i1∈X. 4.2

In other words, for any{xi}^∞_i1∈X,

T

xi_∞

i1

yi_∞

i1, whereyixi 1 for all integersi≥1. 4.3 SetT⁰xxfor allx∈X. Clearly,

ρTx, Ty≤ρx, y ∀x, y∈X,

Tⁿxconverges to0,0, . . . , . . .asn−→ ∞ 4.4 for allx∈X.

Theorem 4.1. Let{ri}^∞_i0⊂0,∞,

∞ i0

r_i∞, 4.5

andx{xi}^∞_i1∈X. Then, there exists a sequence{yⁱ}^∞_i0⊂Xsuch that

y⁰x, ρ

Tyⁱ, y^{i 1}

≤r_i, i0,1, . . . , 4.6

and the following property holds.

There is no nonempty compact setE⊂Xsuch that

ilim→ ∞ρ yⁱ, E

0. 4.7

In the proof of this theorem, we may assume without loss of generality that

ri≤16⁻¹ for all integersi≥0. 4.8 We precede the proof ofTheorem 4.1with the following lemma.

Lemma 4.2. Letz⁰ {z⁰_i }^∞_i1 ∈X, letk ≥0 be an integer, and letj0be a natural number. Then, there exist an integern≥4 and a sequence{zⁱ}ⁿ_i0⊂Xsuch that

ρ

z^{i 1}, Tzⁱ

≤rk i, i0, . . . , n−1, zⁿ

zⁿ₁ , . . . , zⁿ_i , . . .

zⁿ_{i ∞}

i1

4.9

withzⁿ_{j0 1}≥4⁻¹.

Proof. There is a natural numberm >4 such that

m > j0 4, 4.10

∞ im

z⁰_i <16⁻¹. 4.11

Set

z^{i 1}Tzⁱ, i0, . . . , m−1. 4.12

Then,

z^m

z⁰_{m 1}, z⁰_{m 2}, . . . , z⁰_i , . . .

. 4.13

By4.5, there is a natural numbern > msuch that

k n

jk m

rj≥2⁻¹. 4.14

By4.14and4.8,

n≥m 7, 4.15

and we may assume without loss of generality that

k n−1

jk m

rj < 1

2. 4.16

In view of4.14and4.8,

k n−1

jk m

rj ^{k n}

jk m

rj−rk n≥2⁻¹−16⁻¹. 4.17

Forim 1, . . . , n, definezⁱ{zⁱ_j }^∞_j1as follows:

zⁱ_j z⁰_{j i}, j∈ {1,2, . . .} \

n 1 j0−i

, 4.18

zⁱ_{n 1 j}

0−iz⁰_{n 1 j}

0

k i−1

jk m

r_j. 4.19

Clearly, forim 1, . . . , n,zⁱis well defined and by4.18,4.19,4.10, and4.16, ∞

j1

zⁱ_j ^∞

ji 1

z⁰_j ^{k i−1}

jk m

r_j≤^∞

jm

z⁰_j ^{k n−1}

jk m

r_j ≤16⁻¹ 2⁻¹<1. 4.20

Thuszⁱ ∈X,im 1, . . . , n.

Leti ∈ {m, . . . , n−1}. We now estimateρz^{i 1}, Tzⁱ. Ifi m, then by4.2,4.3, 4.13, and4.18,

ρ

z^{i 1}, Tzⁱ

≤r_{k i}. 4.21

Leti > m. We first set

zj_∞

j1Tzⁱ. 4.22

In view of4.14,4.2, and4.3,zj zⁱ_{j 1}for all integersj≥1. When combined with4.18, this implies that

zjz⁰_{j 1 i} ∀j∈ {1,2, . . .} \

n−i j0

, 4.23

zn j0−izⁱ_{n 1 j0−i}z⁰_{n 1 j0} ^{k i−1}

jk m

rj. 4.24

By4.18and4.23,

zjz^{i 1}_j 4.25

for allj ∈ {1,2, . . .} \ {n j0−i}. It now follows from4.22,4.25,4.18,4.19, and4.23 that

ρ

z^{i 1}, Tzⁱ ρ

z^{i 1}, z_j∞

j1

z^{i 1}_{n j0−i}−z_{n j0−i}

z⁰_{n 1 j0} ^{k i}

jk m

rj−

z⁰_{n 1 j0} ^{k i−1}

jk m

rj

< rk i. 4.26

When combined with4.12, this implies that ρ

z^{i 1}, Tzⁱ

≤r_{k i}, i0, . . . , n−1. 4.27

By4.17and4.18,

zⁿ_{j0 1}zⁿ_{n 1 j0−n}≥^{k n−1}

jk m

rj ≥4⁻¹. 4.28

This completes the proof ofLemma 4.2.

Proof ofTheorem 4.1. In order to prove the theorem, we construct by induction, using Lemma 4.2, a sequence of nonnegative integers{sk}^∞_k0 and a sequence{yⁱ}^∞_i0 ⊂ X such that

y⁰x, 4.29

ρ

y^{i 1}, Tyⁱ

≤ri for all integersi≥0, 4.30 s0 0, s_k< s_{k 1} for all integersk≥0, 4.31

and for all integersk≥1,

y_{k 1}^sk≥ 1

4. 4.32

In the sequel, we use the notationyⁱ{yⁱ_j }^∞_j1,i0,1, . . . . Set

y⁰x, s00. 4.33

Assume that q ≥ 0 is an integer and that we have already defined afinite sequence of nonnegative numbers{sk}^q_k0and afinitesequence of points{yⁱ}^s_i0^q ⊂Xsuch that4.33 is valid,4.30holds for all integersisatisfying 0≤i < s_q,

si< si 1 for all integersisatisfying 0≤i < q, 4.34

and that4.32holds for all integersksatisfying 0< k≤q.Note that forq0 this assumption does hold.

Now, we show that this assumption also holds forq 1.

Indeed, applyingLemma 4.2with

z⁰y^sq, j0q 1, ksq, 4.35

we obtain that there exist an integersq 1≥4 sqand a sequence{yⁱ}^s_is^{q 1}_q⊂Xsuch that ρ

y^{i 1}, Tyⁱ

≤ri, isq, . . . , sq 1−1, y_{q 2}^{sq 1}≥ 1

4. 4.36

Thus, the assumption made forq also holds forq 1. Therefore, we have constructed by induction a sequence of points{yⁱ}^∞_i0 ⊂X and a sequence of nonnegative integers{s_k}^∞_k0 which satisfy4.30and4.31for all integersi, k≥0, respectively, and4.32for all integers k≥1.

Finally, we show that there is no nonempty compact setE⊂Xsuch that

ilim→ ∞ρ yⁱ, E

0. 4.37

Assume the contrary. Then, there does exist a nonempty compact setE⊂Xsuch that

ilim→ ∞ρ yⁱ, E

0. 4.38

This implies that any subsequence of{y^k}^∞_k0has a convergent subsequence.

Consider such a subsequence{y^sq}^∞_q1. This subsequence has a convergent subse- quence{y^sqp}^∞_p1. There are therefore a pointz{z_i}^∞_i0∈Xsuch that

z lim

p→ ∞y^sqp 4.39

and a natural numberp0such that ρ

z, y^sqp

≤16⁻¹ for all integersp≥p0. 4.40

Hence we have, for all integersp≥p0,

z_{qp 1}−y^s_qp^qp ₁≤ρ

z, y^sqp

≤16⁻¹,

z_{qp 1}≥y_q^s_p^qp ₁−16⁻¹>8⁻¹. 4.41

This, of course, contradicts the inequality_∞

i1z_i ≤ 1. The contradiction we have reached completes the proof ofTheorem 4.1.

Acknowledgments

This research was supported by the Israel Science FoundationGrant no. 647/07, the Fund for the Promotion of Research at the Technion, and by the Technion President’s Research Fund.

References

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