Volume 2010, Article ID 518243,10pages doi:10.1155/2010/518243
Research Article
Convergence of Inexact Iterative Schemes for
Nonexpansive Set-Valued Mappings
Simeon Reich and Alexander J. Zaslavski
Department of Mathematics, The Technion-Israel Institute of Technology, 32000 Haifa, Israel
Correspondence should be addressed to Alexander J. Zaslavski,[email protected]
Received 23 October 2009; Accepted 10 January 2010
Academic Editor: Mohamed A. Khamsi
Copyrightq2010 S. Reich and A. J. Zaslavski. 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.
Taking into account possibly inexact data, we study iterative schemes for approximating fixed points and attractors of contractive and nonexpansive set-valued mappings, respectively. More precisely, we are concerned with the existence of convergent trajectories of nonstationary dynamical systems induced by approximations of a given set-valued mapping.
1. Introduction
Tis the given mapping, which approximates the best approximation ofxtinTxt. SinceTis
assumed to act on a general complete metric space, we cannot, in general, choosext1to be the
best approximation ofxtby elements ofTxt. Instead, we choosext1so that it provides an
approximation up to a positive numbert, such that the sequence{t}∞t0is summable. This
method allowed Nadler9to obtain the existence of a fixed point of a strictly contractive set-valued mapping and the authors of10to obtain more general results.
In view of the above discussion, it is obviously important to study convergence properties of the iterates ofset-valuednonexpansive mappings in the presence of errors and possibly inaccurate data. The present paper is a contribution in this direction. More precisely, we are concerned with the existence of convergent trajectories of nonstationary dynamical systems induced by approximations of a given set-valued mapping. In the second section of the paper, we consider an iterative scheme for approximating fixed points of closed-valued strict contractions in metric spaces and prove our first convergence theoremseeTheorem 2.1 below. Our second convergence theorem Theorem 3.1 is established in the third section of our paper. We show there that if for any initial point, there exists a trajectory of the dynamical system induced by a nonexpansive set-valued mappingT, which converges to a given invariant setF, then a convergent trajectory also exists for a nonstationary dynamical system induced by approximations ofT.
2. Convergence to a Fixed Point of a Contractive Mapping
In this section we consider iterative schemes for approximating fixed points of closed-valued strict contractions in metric spaces.
We begin with a few notations.
Throughout this paper,X, ρis a complete metric space. Forx∈Xand a nonempty subsetAofX, set
ρx, A infρx, y:y∈A. 2.1
For each pair of nonemptyA, B⊂X, put
HA, B max
sup
x∈Aρx, B,supy∈Bρ
y, A
. 2.2
LetT :X → 2X\ {∅}be such thatTxis a closed subset ofXfor eachx∈Xand
HTx, Ty≤cρx, y, ∀x, y∈X, 2.3
wherec∈0,1is a constant.
Theorem 2.1. Let{i}∞i0⊂0,∞and{δi}∞i0⊂0,∞satisfy
∞
i0
i<∞,
∞
i0
LetTi:X → 2X\ {∅}satisfy, for each integeri≥0,
HTx, Tix≤i, ∀x∈X. 2.5
Assume thatx0∈Xand that for each integeri≥0,
xi1 ∈Tixi, ρxi, xi1≤ρxi, Tixi δi. 2.6
Then{xi}∞i0converges to a fixed point ofT.
Proof. We first show that{xi}∞i0is a Cauchy sequence. To this end, leti≥0 be an integer. Then by2.6and2.5,
ρxi1, xi2≤ρxi1, Ti1xi1 δi1
≤ρxi1, Txi1 sup
ρz, Ti1xi1:z∈Txi1
δi1
≤ρxi1, Txi1 i1δi1
≤HTixi, Txi1 i1δi1
≤HTxi, Txi1 ii1δi1
≤cρxi, xi1 ii1δi1.
2.7
By2.7,
ρx1, x2≤cρx0, x1 1δ10, 2.8
ρx2, x3≤cρx1, x2 12δ2 2.9
≤c2ρx
0, x1 c10δ1 12δ2. 2.10
Now we show by induction that for each integern≥1,
ρxn, xn1≤cnρx0, x1 n−1
i0
ci
n−iδn−in−i−1. 2.11
Assume thatk≥1 is an integer and that2.11holds fornk. When combined with 2.7, this implies that
ρxk1, xk2≤cρxk, xk1 k1δk1k
≤ck1ρx 0, x1
k−1
i0
ci1
k−iδk−ik−1−i k1δk1k
ck1ρx 0, x1
k
i0
ci
k1−iδk1−ik−i.
2.12
Thus2.11holds fornk1. Therefore, we have shown by induction that2.11holds for all integersn≥1. By2.11,
∞
n1
ρxn, xn1≤
∞
n1
cnρx 0, x1
n
i1
cn−i
iδii−1
≤ρx0, x1
∞
n1
cn∞ i1 ⎛ ⎝∞ j0 cj ⎞
⎠iδii−1
≤ ∞
n0
cn
ρx0, x1
∞
n1
nδnn−1
<∞.
2.13
Thus{xn}∞n0is a Cauchy sequence and there exists
x∗nlim→ ∞xn. 2.14
We claim that
x∗∈Tx∗. 2.15
Indeed, by2.14, there is an integern0≥1 such that for each integern≥n0,
Letn≥n0be an integer. By2.3,2.16and2.5,
ρx∗, Tx∗≤ρx∗, xn1 ρxn1, Tx∗
≤ρx∗, xn1 ρxn1, Txn HTxn, Tx∗
≤ρx∗, xn1 ρxn1, Txn ρxn, x∗
≤ρxn1, Txn 4
≤ρxn1, Tnxn HTnxn, Txn 4
≤n4 −→4
2.17
asn → ∞. Sinceis an arbitrary positive number, we conclude that
x∗∈Tx∗, 2.18
as claimed.Theorem 2.1is proved.
3. Convergence to an Attractor of a Nonexpansive Mapping
In this section we show that if for any initial point, there exists a trajectory of the dynamical system induced by a nonexpansive set-valued mappingT, which converges to an invariant setF, then a convergent trajectory also exists for a nonstationary dynamical system induced by approximations ofT.
LetT :X → 2X\ {∅}be such thatTxis a closed set for eachx∈Xand
HTx, Ty≤ρx, y, ∀x, y∈X. 3.1
Theorem 3.1. Let{i}∞i0⊂0,∞,∞i0i<∞,Fa nonempty closed subset ofX,
TF⊂F, 3.2
and for each integeri≥0, letTi:X → 2X\ {∅}satisfy
HTx, Tix≤i, ∀x∈X. 3.3
Assume that for eachx∈X, there exists a sequence{xi}∞i0⊂Xsuch that
x0x, xi1∈Txi, i0,1, . . . ,
lim
i→ ∞ρxi, F 0.
Then for eachx∈X, there is a sequence{xi}∞i0⊂Xsuch that
x0x, xi1 ∈Tixi, i0,1, . . . ,
lim
i→ ∞ρxi, F 0.
3.5
We begin the proof ofTheorem 3.1with two lemmata.
Lemma 3.2. Letx∈X,pa natural number,{xi}pi0⊂X,
x0x, xi1∈Tixi, i0, . . . , p−1, 3.6
and letδ >0. Then there is a natural numberq > pand a sequence{xi}qip⊂Xsuch that
xi1∈Tixi, ip, . . . , q−1,
ρxq, F≤δ.
3.7
Proof. Choose a natural numberp1 > psuch that
∞
ip1
i < δ8 3.8
and a sequence{xi}pip1 ⊂Xsuch that
xi1∈Tixi, ip, . . . , p1−1. 3.9
There is a sequence{yi}∞ip1⊂Xsuch that
yp1xp1, ilim→ ∞ρ
yi, F0, 3.10
yi1∈T
yi, for all integersi≥p1. 3.11
We are now going to define by induction a sequence{xi}∞ip1 ⊂X.
To this end, assume thatk≥p1is an integer and that we have already definedxi ∈X,
ip1, . . . , k, such that
xi1 ∈Tixi, ip1, . . . , k−1, 3.12
ρxk, yk≤3 ⎛ ⎝k
ip1
i−k ⎞
⎠. 3.13
By3.11and3.1,
yk1 ∈T
yk, 3.14
HTyk, Txk≤ρxk, yk. 3.15
By3.15, there isyk1∈Xsuch that
yk1∈Txk, ρyk1,yk1
≤ρxk, ykk. 3.16
Together with3.3, this implies that
ρyk1, Tkxk≤k, 3.17
and there is
xk1∈Tkxk 3.18
such that
ρyk1, xk1
≤2k. 3.19
When combined with3.16and3.13, this implies that
ρxk1, yk1
≤ρxk1,yk1
ρyk1, yk1
≤ρxk, yk3k≤3 k
ip1
i. 3.20
Thus, by3.18and3.20, the assumption we have made concerningkalso holds fork1. Therefore, we have indeed defined by induction a sequence{xi}∞ip1such that
xi1∈Tixi, ip1, . . . , 3.21
and3.13holds for all integersk≥p1. By3.11, there is an integerq > p12 such that
ρyq, F< δ4. 3.22
Together with3.8and3.13, this inequality implies that
ρxq, F≤ρxq, yqρyq, F≤
∞
ip1
iδ4 < δ2 δ4. 3.23
Lemma 3.3. Let{xi}∞i0⊂X,
xi1∈Tixi, i0,1, . . . , 3.24
δ >0,pa natural number,
ρxp, F≤δ, 3.25
∞
ip
i < δ. 3.26
Thenρxi, F≤3δfor all integersi≥p.
Proof. We intend to show by induction that for all integersn≥p,
ρxn, F≤δ n
ip
2i−2n. 3.27
Clearly, fornpinequality3.27does hold. Assume now thatn≥pis an integer and3.27 holds. Then there is
yn∈F 3.28
such that
ρxn, yn≤δ n
ip
2i−n. 3.29
By3.24and3.3, there is
xn1∈Txn 3.30
such that
ρxn1, xn1≤2n. 3.31
By3.29and3.1,
HTxn, Tyn≤ρxn, yn, 3.32
and, in view of3.30, there is
yn1∈T
such that
ρyn1,xn1
≤ρxn, ynn. 3.34
By3.33,3.28, and3.2,
yn1∈F. 3.35
By3.35,3.31,3.34, and3.27,
ρxn1, F≤ρ
xn1, yn1
≤ρxn1,xn1 ρ
xn1, yn1
≤2nnρxn, yn≤δ2n n
ip
2i.
3.36
Thus, the assumption we have made concerningnalso holds forn1. Therefore, we may conclude that inequality3.27indeed holds for all integersn≥p. Together with3.26, this implies that for all integersn≥p,
ρxn, F≤δ2δ3δ. 3.37
Lemma 3.3is proved.
Completion of the Proof of
Theorem 3.1
Letx ∈ X. Since∞i0i < ∞, it follows fromLemma 3.2that there exist a sequence{xi}∞i0
and a strictly increasing sequence of natural numbers{nk}∞k1, constructed by induction, such that
xi1∈Tixi, i0,1, . . . , 3.38
and for each integerk≥1,
ρxnk, F≤2−k,
∞
ink
i<2−k. 3.39
It now follows from3.39andLemma 3.3that
lim
n→ ∞ρxn, F 0. 3.40
Acknowledgment
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.
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