Volume 2008, Article ID 242916,7pages doi:10.1155/2008/242916
Research Article
Data Dependence for Ishikawa Iteration When
Dealing with Contractive-Like Operators
S¸. M. S¸oltuz1, 2and Teodor Grosan3
1Departamento de Matematicas, Universidad de los Andes, Carrera 1 No. 18A-10, Bogota, Colombia 2The Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca, Romania
3Department of Applied Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Correspondence should be addressed to Teodor Grosan,[email protected]
Received 13 February 2008; Accepted 27 May 2008
Recommended by Hichem Ben-El-Mechaiekh
We prove a convergence result and a data dependence for Ishikawa iteration when applied to contraction-like operators. An example is given, in which instead of computing the fixed point of an operator, we approximate the operator with a contractive-like one. For which it is possible to compute the fixed point, and therefore to approximate the fixed point of the initial operator.
Copyrightq2008 S¸. M. S¸oltuz and T. Grosan. 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
Let Xbe a real Banach space; let B ⊂ Xbe a nonempty convex closed and bounded set. Let
T, S:B → Bbe two maps. For a givenx0, u0 ∈B,we consider the Ishikawa iterationsee1
forT andS:
xn11−αnxnαnTyn, yn1−βnxnβnTxn, 1.1
un11−αnunαnSvn, vn 1−βnunβnSun, 1.2
where{αn} ⊂0,1,{βn} ⊂0,1, and
lim
n→∞αnnlim→∞βn0, ∞
n1
αn∞. 1.3
Setβn0, ∀n∈N,to obtain the Mann iteration, see2.
The mapT is called Kannan mappings, see3, if there existsb ∈ 0,1/2such that for allx, y∈B,
Similar mappings are Chatterjea mappings, see4, for which there existsc ∈ 0,1/2 such that for allx, y∈B,
Tx−Ty ≤cx−Tyy−Tx. 1.5
Zamfirescu collected these classes. He introduced the following definition, see5.
Definition 1.1see5,6. The operatorT : X→Xsatisfies conditionZZamfirescu condition if and only if there exist the real numbersa, b, c satisfying 0 < a < 1,0 < b, c < 1/2 such that for each pairx, yinX,at least one condition is true:
i z1Tx−Ty ≤ax−y,
ii z2Tx−Ty ≤bx−Txy−Ty, iii z3Tx−Ty ≤cx−Tyy−Tx.
It is known, see Rhoades 7, that z1, z2, and z3 are independent conditions.
Considerx, y ∈ B.SinceT satisfies condition Z,at least one of the conditions fromz1,z2,
andz3is satisfied. Ifz2holds, then
Tx−Ty ≤bx−Txy−Ty≤bx−Txy−xx−TxTx−Ty.
1.6
Thus
1−bTx−Ty ≤bx−y2bx−Tx. 1.7
From 0≤b <1 one obtains,
Tx−Ty ≤ b
1−bx−y 2b
1−bx−Tx. 1.8
Ifz3holds, then one gets
Tx−Ty ≤cx−Tyy−Tx ≤cx−TxTx−Tyx−yx−Tx 1.9
Hence,
1−cTx−Ty ≤cx−y2cx−Tx, 1.10
that is,
Tx−Ty ≤ c
1−cx−y 2c
1−cx−Tx. 1.11
Denote
δ :max
a, b
1−b,
c
1−c
to obtain
0≤δ <1. 1.13
Finally, we get
Tx−Ty ≤δx−y2δx−Tx, ∀x, y∈B. 1.14
Formula1.14was obtained as in8.
Osilike and Udomene introduced in9a more general definition of a quasicontractive operator; they considered the operator for which there existsL≥0 andq∈0,1such that
Tx−Ty ≤qx−yLx−Tx, ∀x, y∈B. 1.15
Imoru and Olatinwo considered in10, the following general definition. Because they failed to name them, we will call them here contractive-like operators.
Definition 1.2. One callscontractive-likethe operator T if there exist a constantq ∈ 0,1and a strictly increasing and continuous functionφ :0,∞→0,∞withφ0 0 such that for each
x, y ∈X,
Tx−Ty ≤qx−yφx−Tx. 1.16
In both papers9,10, theT-stability of Picard and Mann iterations was studied.
2. Preliminaries
The data dependence abounds in literature of fixed point theory when dealing with Picard-Banach iteration, but is quasi-inexistent when dealing with Mann-Ishikawa iteration. As far as we know, the only data-dependence result concerning Mann-Ishikawa iteration is in 11. There, the data dependence of Ishikawa iteration was proven when applied to contractions. In this note, we will prove data-dependence results for Ishikawa iteration when applied to the above contractive-like operators. Usually, Ishikawa iteration is more complicated but nevertheless more stable as Mann iteration. There is a classic example, see 12, in which Mann iteration does not converge while Ishikawa iteration does. This is the main reason for considering Ishikawa iteration inTheorem 3.2.
The following remark is obvious by using the inequality1−x≤expx, ∀x≥0.
Remark 2.1. Let{θn}be a nonnegative sequence such thatθn ∈0,1, ∀n∈N.If ∞n1θn ∞,then
∞
n11−θn 0.
The following is similar to lemma from 13.Note that another proof for this lemma
13can be found in11.
Lemma 2.2. Let{an}be a nonnegative sequence for which one supposes there existsn0∈N,such that
for alln≥n0one has satisfied the following inequality:
an1 ≤1−λnanλnσn, 2.1
whereλn ∈0,1, ∀n∈N,∞n1λn∞,andσn ≥0 ∀n∈N. Then,
0≤ lim
Proof. There existsn1 ∈ Nsuch thatσn ≤ lim supσn, ∀n ≥ n1. Setn2 max{n0, n1}such that
the following inequality holds, for alln≥n2:
an1 ≤1−λn1−λn−1
· · ·1−λn1
an1nlim→∞supσn. 2.3
Using the above Remark 2.1 with θn λn, we get the conclusion. In order to prove 2.3,
consider2.1and the induction step:
an2≤1−λn1an1λn1σn1≤1−λn11−λn1−λn−1
· · ·1−λn1
an1
1−λn1nlim→∞supσnλn1σn1
1−λn11−λn1−λn−1
· · ·1−λn1
an1nlim→∞supσn.
2.4
3. Main results
Theorem 3.1. LetXbe a real Banach space,B⊂Xa nonempty convex and closed set, andT :B→Ba contractive-like map withx∗being the fixed point. Then for allx0∈B,the iteration1.1converges to
the unique fixed point ofT.
Proof. The uniqueness comes from1.16; supposing we have two fixed pointsx∗andy∗,we get
x∗−y∗Tx∗−Ty∗ ≤qx∗−y∗φx∗−Tx∗qx∗−y∗, 3.1
that is,1−qx∗−y∗0.From1.1and1.16we obtain
xn1−x∗ ≤1−α
n xn−x∗ αn Tyn−Tx∗
≤1−αn xn−x∗ αnq yn−x∗
≤1−αn xn−x∗ αnq1−βn xn−x∗ qαnβn Txn−Tx∗
≤1−αn1−q1−1−qβn xn−x∗
≤1−αn1−q xn−x∗ ≤ · · · ≤
n
k1
1−αkq x0−x∗ .
3.2
UseRemark 2.1withθk αkqto obtain the conclusion.
This result allows us to formulate the following data dependence theorem.
Theorem 3.2. LetXbe a real Banach space, letB ⊂ Xbe a nonempty convex and closed set, and let ε >0be a fixed number. IfT :B→Bis a contractive-like operator with the fixed pointx∗andS:B→B
is an operator with a fixed pointu∗,(supposed nearest tox∗),and if the following relation is satisfied:
then
x∗−u∗ ≤ ε
1−q. 3.4
Proof. From1.1and1.2, we have
xn1−un11−αnxn−unαnTyn−Svn. 3.5
Thus
xn1−un1 1−αn
xn−unαnSvn−Tyn
≤1−αn xn−un αn Svn−TvnTvn−Tyn
≤1−αn xn−un αn Tvn−Svn αn Tvn−Tyn
≤1−αn xn−un αnεqαn yn−vn αnφ yn−Tyn
≤1−αn xn−un αnεqαn1−βn xn−un qαnβn Txn−Sun αnφ yn−Tyn
≤1−αn xn−un αnεqαn1−βn xn−un
αnβnq Txn−Tun Tun−Sun αnφ yn−Tyn
≤1−αn xn−un αnεqαn1−βn xn−un
q2α
nβnxn−unqαnβnφ xn−Txn qαnβnεαnφ yn−Tyn 1−αn1−q1−βn−βnq2 xn−un αnεqαnβnε
qαnβnφ xn−Txn αnφ yn−Tyn
1−αn1−q1qβn xn−un αnqβnφ xn−Txn φ yn−Tyn qβnεε
≤1−αn1−q xn−un αn1−qqβnφ
xn−Txn φ yn−Tyn qβnεε
1−q .
3.6
Note that limn→∞φxn−Txn limn→∞φyn−Tyn 0 becauseφis a continuous map and
both{xn},{yn}converge to the fixed point ofT.Set
λn:αn1−q,
σn: qβnφ
xn−Txn φ yn−Tyn qβnεε
1−q ,
and useLemma 2.2to obtain the conclusion
x∗−u∗ ≤ ε
1−q. 3.8
Remark 3.3. iSetβn0, ∀n∈N,to obtain the data dependence for Mann iteration.
iiThe Zamfirescu operators and implicitlyChatterjea and Kannanare contractive-like operators, therefore ourTheorem 3.2remains true for these classes.
4. Numerical example
The following example follows the example from8.
Example 4.1. LetT :R→Rbe given by
Tx0, if x∈−∞,2
−0.5, if x∈2,∞. 4.1
ThenT is contractive-like operator withq0.2 andφidentity.
Note the unique fixed point is 0.Consider now the mapS:R→R,
Sx1, ifx∈−∞,2
−1.5, ifx∈2,∞ 4.2
with the unique fixed point 1.Takeεto be the distance between the two maps as follows:
Sx−Tx ≤1, ∀x∈R. 4.3
Setu0 x0 0, αn βn 1/n1.Independently of above theory, the Ishikawa iteration
applied toS,leads to
Iteration step Ishikawa iteration 1 0.5
10 0.9 100 0.99
4.4
Note that forn1,
0.5 n1 10
1
n1S
1 2
, 4.5
since y1 1/n10 1/n11 1/2. The above computations can be obtained also
by using a Matlab program.This leads us to “conclude” that Ishikawa iteration applied toS converges to fixed point,x∗ 1. Eventually, one can see that the distance between the two fixed points is one. Actually, without knowing the fixed point ofSand without computing it, viaTheorem 3.2, we can do the following estimate for it:
x∗−u∗ ≤ 1
1−q 1 1−0.2
10
Acknowledgments
The authors are indebted to referee for carefully reading the paper and for making useful suggestions. This work was supported by CEEX ET 90/2006-2008.
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