International Journal of Mathematics and Mathematical Sciences Volume 2008, Article ID 630589,5pages
doi:10.1155/2008/630589
Research Article
Characterization for the Convergence of
Krasnoselskij Iteration for Non-Lipschitzian Operators
S¸tefan M. S¸oltuz1, 2and B. E. Rhoades3
1Departamento de Matematicas, Universidad de Los Andes, Carrera 1 no. 18A-10, Bogota, Colombia
2“Tiberiu Popoviciu” Institute of Numerical Analysis, 400110 Cluj-Napoca, Romania
3Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA
Correspondence should be addressed to S¸tefan M. S¸oltuz,[email protected] Received 13 August 2007; Revised 2 February 2008; Accepted 24 February 2008 Recommended by Enrico Obrecht
We establish the convergence of Krasnoselskij iteration for various classes of non-Lipschitzian op- erators.
Copyrightq2008 S¸. M. S¸oltuz and B. E. Rhoades. 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
LetXbe a real Banach space;Ba nonempty, convex subset ofX; andT :B →Ban operator.
Letx0∈B.The following iteration is known as Krasnoselskij iterationsee1:
xn1 1−λxnλTxn. 1.1
The mapJ :X →2X∗given byJx:{f∈X∗:x, fx2, fx}, for allx∈X,is called the normalized duality mapping. It is easy to see that we have
y, jx
≤ xy, ∀x, y∈X, ∀jx∈Jx. 1.2 Denote
Ψ:
ψ | ψ:0,∞−→0,∞ is astrictly increasing map with ψ0 0
. 1.3
Definition 1.1. LetXbe a real Banach space, and letBbe a nonempty subset ofX. A mapT : B→Bis called uniformly pseudocontractive if there exists a mapψ ∈Ψandjx−y∈Jx−y such that
Tx−Ty, jx−y
≤ x−y2−ψ
x−y
, ∀x, y∈B. 1.4
A mapS:X→Xis called uniformly accretive if there exists a mapψ∈Ψandjx−y∈ Jx−ysuch that
Sx−Sy, jx−y
≥ψ
x−y
, ∀x, y∈X. 1.5 Takingψa:ψa·a,for alla∈0,∞,ψ ∈Ψ, reduces to the usual definitions of ψ-strongly pseudocontractive andψ-strongly accretive. Takingψa:γ ·a2,γ ∈0,1, for all a∈0,∞,ψ ∈Ψ,we get the usual definitions of strongly pseudocontractive and strongly accretive. Therefore, the class of strongly pseudocontractive maps is included stricly in the class ofψ-strongly pseudocontractive maps. The example from2shows that this inclusion is proper. Remark, further, that the class ofψ-strongly pseudocontractive maps is also included strictly in the class of uniformly pseudocontractive mapssee also3.
We will give a characterization for the convergence of1.1when applied to uniformly pseudocontractive operators. For this purpose, we need the following lemma similar to 4, Lemma 1. Next,Ndenotes the set of all natural numbers.
Lemma 1.2. Let{an}be a positive bounded sequence and assume that there existsn0∈N such that
an1≤1−λanλan1−λψ an1
an1 λεn, ∀n≥n0, 1.6 whereλ∈0,1, εn≥0, for alln∈Nand limn→∞εn0.Then limn→∞an0.
Proof. There exists anM > 0 such thatan ≤M, for alln∈ N. Denotea :lim infan.We will prove thata0.Suppose on the contrary thata >0.Then there exists anN1∈N such that
an≥ a
2, ∀n≥N1. 1.7
From limn→∞εn0,we know that there exists anN2∈Nsuch that εn≤ ψa/2
2M , ∀n≥N2. 1.8
SetN0:max{N1, N2}.Using the fact that−1/M≥ −1/an1,we get the following:
an1≤1−λanλan1−λψ an1 an1 λεn
≤1−λanλan1−λψa/2
M λψa/2 2M
≤1−λanλan1−λψa/2 2M ,
1.9
which implies that1−λan1≤1−λan−λψa/2/2M,or an1≤an− λ
1−λ ψa/2
2M ≤an−λψa/2
2M , 1.10
since−λ/1−λ ≤ −λ.Thusλψa/2/2M ≤ an−an1,which implies that
λ < ∞,in contradiction to
λ ∞.Therefore, lim infan 0.Hence there exists a subsequence{anj} ⊂ {an}such that limj→∞anj 0.Fixε >0. Then there exists ann3∈Nsuch that
anj< ε
4, ∀j≥n3. 1.11
Also there exists ann4∈Nsuch that
εn< ψε/4
2M , ∀n≥n4. 1.12
Definen0:max{n3, n4, N0}.We claim thatanjk< ε/4 for eachj > n0and eachk >0.Suppose not. Then there exists ann0and ak >0 such that
anjk≥ ε
4. 1.13
For thisnj,letkdenote the smallest positive integer for which1.13is true. Thenanjk−1≤ε/4.
From1.6,
anjk≤1−λanjk−1λanjk−λψ anjk
anjk λεnjk−1
≤1−λanjk−1λanjk−λψε/4
anjk λψε/4 2M
≤1−λanjk−1λanjk−λψε/4 2M ,
1.14
which implies thatanjk≤ε/4−λ/1−λψε/4/2M.This leads to the contradiction:
ε
4 ≤anjk≤ ε 4 − λ
1−λ ψε/4
2M < ε
4. 1.15
Therefore,anjk< ε/4,for allk∈N, and eachj > n0,hence limn→∞an0.
2. Main result
Theorem 2.1. LetXbe a real Banach space,Ba nonempty, closed, convex, bounded subset ofX. LetT: B →Bbe a uniformly pseudocontractive and uniformly continuous operator withFT/∅. Then for x0∈B, the Krasnoselskij iteration1.1converges to the fixed point ofTif and only if limn→∞xn1− xn0.
Proof. SinceTis a self-map ofB,which is bounded and convex, then, from1.1, eachxn ∈B, so{xn}is bounded for eachn ∈ N.Uniqueness of the fixed point follows from1.4. If{xn} converges to the fixed point ofT,that is, limn→∞xnx∗,then, obviously, limn→∞xn1−xn0.
Conversely, we will prove that if limn→∞xn1−xn 0, then limn→∞xn x∗.Suppose that
xn x∗ for somen ∈ N.Then from1.1, it follows thatxm x∗ for each m > n,and the theorem is proved. Now suppose thatxn/x∗for eachn∈N.Using1.1and1.2,
xn1−x∗ 2
xn1−x∗, j
xn1−x∗
1−λ
xn−x∗ λ
Txn−Tx∗ , j
xn1−x∗ 1−λ
xn−x∗ , j
xn1−x∗ λ
Txn−Tx∗, j
xn1−x∗
≤1−λ xn−x∗ xn1−x∗ λ
Txn1−Tx∗, j
xn1−x∗ λ
Txn−Txn1, j
xn1−x∗
≤1−λ xn−x∗ xn1−x∗ λ xn1−x∗ 2−λψ xn1−x∗ λ Txn−Txn1 xn1−x∗
≤ xn1−x∗
1−λ xn−x∗ λxn1−x∗ −λψ
xn1−x∗
xn1−x∗ λTxn−Txn1
. 2.1 Hence
xn1−x∗ ≤1−λ xn−x∗ λ xn1−x∗ −λψ xn1−x∗
xn1−x∗ λ Txn−Txn1 . 2.2
Since limn→∞xn1−xn0 andTis uniformly continuous, it follows that
n→∞lim Txn−Txn1 0. 2.3
Setanxn−x∗,εnTxn−Txn1and useLemma 1.2to obtain the conlcusion.
Remark 2.2. 1IfBis not bounded, thenTheorem 2.1holds under the assumption that{xn}is bounded.
2IfTBis bounded, then{xn}is bounded.
3IfT is strongly pseudocontractive, then automaticallyFT/∅.
3. Further results
LetI denote the identity map. A mapT : B → B is called pseudocontractive if there exists jx−y∈Jx−ysuch thatTx−Ty, jx−y ≤ x−y2.
Remark 3.1. The operator T is a uniformly, stronglypseudocontractive map if and only if I−Tis auniformly, stronglyaccretive map.
Remark 3.2. 1LetT, S : X → X,and let f ∈ X be given. A fixed point for the map Tx f I−Sx, for allx∈X, is a solution forSxf.
2Letf ∈ X be a given point. IfSis an accretive map, thenT f −Sis a strongly pseudocontractive map.
Consider Krasnoselskij iteration withTxf I−Sx, xn1 1−λxnλ
f I−Sxn
. 3.1
Remarks3.1and3.2andTheorem 2.1lead to the following result.
Corollary 3.3. LetXbe a real Banach space and letS:X→Xbe a uniformly accretive and uniformly continuous operator, withI−SXbounded. Suppose thatSxfhas a solution. Then for anyx0∈ X, the Krasnoselskij iteration3.1converges to the solution ofSxf if and only if limn→∞xn1− xn0.
LetSbe an accretive operator. The operatorTxf−Sxis strongly pseudocontractive for a givenf∈X.A solution forTxxbecomes a solution forxSxf.Consider Krasnoselskij iteration withTx:f−Sx,
xn1 1−λxnλ
f−Sxn
. 3.2
Again, using Remarks3.1and3.2andTheorem 2.1, we obtain the following result.
Corollary 3.4. LetXbe a real Banach space and letS:X →Xbe an accretive and uniformly contin- uous operator, withI−SXbounded. Suppose thatxSxfhas a solution. Then forx0∈X,the Krasnoselskij iteration3.2converges to the solution ofxSxfif and only if limn→∞xn1−xn0.
Remark 3.5. If1.4holds for allx∈Bandy:x∗∈FT,then such a map is called uniformly hemicontractive. It is trivial to see that our results hold for the uniformly hemicontractive maps.
Acknowledgment
The authors are indebted to referee for carefully reading the paper and for making useful sug- gestions.
References
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