On Convergence Results for Lipschitz Pseudocontractive Mappings

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Volume 2012, Article ID 902601,8pages doi:10.1155/2012/902601

Research Article

On Convergence Results for Lipschitz Pseudocontractive Mappings

Shin Min Kang

¹

and Arif Rafiq

²

1Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of Korea

2Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore, Pakistan

Correspondence should be addressed to Shin Min Kang,[email protected] Received 4 June 2012; Revised 3 September 2012; Accepted 3 September 2012 Academic Editor: Alicia Cordero

Copyrightq2012 S. M. Kang and A. Rafiq. 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.

We establish the strong convergence for the Ishikawa iteration scheme associated with Lipschitz pseudocontractive mappings in real Banach spaces. Moreover, our technique of proofs is of independent interest.

1. Introduction and Preliminaries

Let Ebe a real Banach space and K be a nonempty convex subset ofE. Let J denote the normalized duality mapping fromEto 2^E^∗defined by

Jx

f^∗∈E^∗: x, f^∗

x², f^∗x

, ∀x∈E, 1.1

whereE^∗denotes the dual space ofEand·,·denotes the generalized duality pairing. We will denote the single-valued duality mapping byj.

LetT :DT⊂E → Ebe a mapping with domainDTinE.

Definition 1.1. Tis said to be Lipschitz if there exists a constantL >1 such that

Tx−Ty≤Lx−y, ∀x, y∈DT. 1.2

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Definition 1.2. Tis said to be nonexpansive if

Tx−Ty≤x−y, ∀x, y∈DT. 1.3 Definition 1.3. T is said to be pseudocontractive if for allx, y ∈ DT, there existsjx−y ∈ Jx−ysuch that

Tx−Ty, j x−y

≤x−y². 1.4

Remark 1.4. It is well known that every nonexpansive mapping is pseudocontractive. Indeed, ifT is nonexpansive mapping, then for allx,y ∈DT, there existsjx−y ∈Jx−ysuch that

Tx−Ty, j x−y

≤Tx−Tyx−y≤x−y². 1.5 Rhoades1showed that the class of pseudocontractive mappings properly contains the class of nonexpansive mappings.

The class of pseudocontractions is, perhaps, the most important generalization of the class of nonexpansive mappings because of its strong relationship with the class of accretive mappings. A mappingA:E → Eis accretive if and only ifI−Ais pseudocontractive.

For a nonempty convex subsetKof a normed spaceEand a mappingT :K → K.

The Mann iteration scheme2: the sequence{x_n}is defined by x1∈K,

x_n1 1−a_nx_na_nTx_n, n≥1, 1.6

where{an}is a sequence in0,1.

The Ishikawa iteration scheme3: the sequence{x_n}is defined by x1∈K,

x_n1 1−a_nx_na_nTy_n, yn 1−bnxnbnTxn, n≥1,

1.7

where{an}and{bn}are sequences in0,1.

In the last few years or so, numerous papers have been published on the iterative approximation of fixed points of Lipschitz strongly pseudocontractive mappings using the Ishikawa iteration schemee.g.,3. Results which had been known only in Hilbert spaces and only for Lipschitz mappings have been extended to more general Banach spacese.g., 4–6and the references cited therein.

In 1974, Ishikawa3introduced an iteration scheme which, in some sense, is more general than that of Mann and which converges, under this setting, to a fixed point ofT. He proved the following result.

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Theorem 1.5. Let K be a compact convex subset of a Hilbert spaceH and let T : K → K be a Lipschitz pseudocontractive mapping. For arbitrary x1 ∈ K, let {x_n}^∞_n1 be a sequence defined iteratively by

x_n1 1−α_nx_nα_nTy_n, yn

1−βn

xnβnTxn, n≥1, 1.8

where{α_n}^∞_n1and{β_n}^∞_n1are sequences satisfying conditions i0≤αn≤βn<1;

iilim_n_{→ ∞}β_n0;

iii ^∞_n1αnβn∞.

Then{x_n}^∞_n1converges strongly to a fixed point of T.

In4, Chidume extended the results of Schu 7 from Hilbert spaces to the much more general class of real Banach spaces and approximated the fixed points of strongly pseudocontractive mappings.

In this paper, we establish the strong convergence for the Ishikawa iteration scheme associated with Lipschitz pseudocontractive mappings in real Banach spaces. Moreover, our technique of proofs is of independent interest.

2. Main Results

We will need the following results.

Lemma 2.1see8. LetJ :E → 2^Ebe the normalized duality mapping. Then, for anyx, y∈E, one has

xy² ≤ x²2 y, j

xy , ∀j

xy

∈J xy

. 2.1

Lemma 2.2see9. If there exists a positive integerNsuch that for alln≥N,n∈N(the set of all positive integers)

ρ_n1≤ 1−δ²_n

ρ_nb_n, 2.2

whereδ_n∈0,1, ^∞_n1δ²_n∞andb_noδ_n, then

nlim→ ∞ρ_n0. 2.3

We now prove our main results.

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Theorem 2.3. LetKbe a nonempty closed convex subset of a real Banach spaceEandT :K → K be a Lipschitz pseudocontractive mapping such thatp∈FT:{x∈K:Txx}. Let{α_n}^∞_n1and {βn}^∞_n1be sequences in0,1satisfying the conditions:

iv ^∞_n1α²_n∞;

vlim_n_{→ ∞}αn0;

vilim_n_{→ ∞}βn0.

For arbitraryx1∈K, let{xn}^∞_n1be defined iteratively by xn1 1−αnxnαnTyn, yn

1−βn

xnβnTxn, n≥1. 2.4 Then the following conditions are equivalent:

a{xn}^∞_n1converges strongly to the fixed pointpofT. b{Txn}^∞_n1and{Tyn}^∞_n1are bounded.

Proof. Becausepis a fixed point ofT, then the setFTof fixed points ofT is nonempty.

Suppose that lim_{n→ ∞}xnp, then sinceT is Lipschitz, so

nlim→ ∞Tx_np,

nlim→ ∞yn lim

n→ ∞

1−βn

xnβnTxn

p, 2.5 which implies that lim_{n→ ∞}Ty_np. Therefore{Tx_n}^∞_n1and{Ty_n}^∞_n1are bounded.

Set

M1x0−psup

n≥1

Txn−psup

n≥1

Tyn−p. 2.6

ObviouslyM1 <∞.

It is clear thatx0−p ≤M1. Letxn−p ≤M1. Next we will prove thatxn1−p ≤M1. Consider

xn1−p1−αnxnαnTyn−p 1−α_n

x_n−p α_n

Ty_n−p

≤ 1−αnxn−pαnTyn−p

≤M1.

2.7

So, from the above discussion, we can conclude that the sequence{xn−p}^∞_n1is bounded. Let M2sup_n≥1xn−p.

DenoteMM1M2. ObviouslyM <∞.

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Now fromLemma 2.1we obtain for alln≥1 xn1−p²1−αnxnαnTyn−p²

1−α_n x_n−p

α_n

Ty_n−p²

≤1−αn²xn−p²2αn

Tyn−p, j

xn1−p 1−α_n²x_n−p²2α_n

Tx_n1−p, j

x_n1−p 2αn

Tyn−Txn1, j

xn1−p

≤1−α_n²x_n−p²2α_nx_n1−p² 2αnTyn−Txn1xn1−p

≤1−α_n²x_n−p²2α_nx_n1−p²2α_nλ_n,

2.8

where

λnMTyn−Txn1. 2.9

Using2.4we have

yn−xn1≤yn−xnxn−xn1 βnxn−Txnαnxn−Tyn

≤2M

α_nβ_n .

2.10

From the conditions lim_n_{→ ∞}α_n0lim_n_{→ ∞}β_nand2.10, we obtain

nlim→ ∞yn−xn10, 2.11

and sinceTis Lipschitz,

nlim→ ∞Ty_n−Tx_n10, 2.12

thus, we have

nlim→ ∞λ_n0. 2.13

The real functionf : 0,∞ → 0,∞defined byft t²is increasing and convex. For all λ∈0,1andt1,t2>0 we have

1−λt1λt2²≤1−λt²₁λt²₂. 2.14

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Consider

xn1−p²1−αnxnαnTyn−p² 1−α_nx_n−p α_nTy_n−p²

≤

1−αnxn−pαnTyn−p²

≤1−αnxn−p²αnTyn−p²

≤1−α_nx_n−p²M²α_n.

2.15

Substituting2.15in2.8, we get xn1−p²≤

1−αn²2αn1−αnxn−p² 2α_n

M²α_nλ_n

1−α²_nx_n−p²ε_nα_n,

2.16

whereεn 2M²αnλn. Now, with the help of ^∞_n1α²_n ∞, limn→ ∞αn 0,2.13, and Lemma 2.2, we obtain from2.16that

nlim→ ∞x_n−p0. 2.17

This completes the proof.

Remark 2.4. Our technique of proofs is of independent interest.

Corollary 2.5. LetKbe a nonempty closed convex subset of a real Hilbert spaceEand letT :K → K be a Lipschitz pseudocontractive mapping such thatp∈FT. Let{α_n}^∞_n1and{β_n}^∞_n1be sequences in0,1satisfying the conditions (iv), (v), and (vi).

For arbitrary x1 ∈ K, let {xn}^∞_n1 be the sequence defined iteratively by 2.4. Then the following conditions are equivalent:

a{xn}^∞_n1converges strongly to the fixed point pofT. b{Txn}^∞_n1and{Tyn}^∞_n1are bounded.

The proof of the following result runs on the lines of proof of theTheorem 2.3, so is omitted.

Theorem 2.6. LetKbe a nonempty closed convex subset of a real Banach spaceEand letT, S:K → Kbe two Lipschitz pseudocontractive mappings such thatp ∈FT∩FS:{x∈K :Tx x

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Sx}. Let{αn}^∞_n1and{βn}^∞_n1be sequences in0,1satisfying the conditions (iv), (v), and (vi). For arbitraryx1∈K, let{x_n}^∞_n1be a sequence defined iteratively by

xn1 1−αnxnαnTyn, y_n

1−β_n

x_nβ_nSx_n, n≥1. 2.18 Then the following conditions are equivalent:

a{x_n}^∞_n0converges strongly to the common fixed pointpofT andS.

b{Txn}^∞_n0and{Syn}^∞_n0are bounded.

Corollary 2.7. LetKbe a nonempty closed convex subset of a real Hilbert spaceEand letT, S:K → Kbe two Lipschitz pseudocontractive mappings such thatp∈FT∩FS. Let{α_n}^∞_n1and{β_n}^∞_n1 be sequences in0,1satisfying conditions (iv), (v), and (vi).

For arbitrary x1 ∈ K, let {xn}^∞_n1 be the sequence defined iteratively by 2.18. Then the following conditions are equivalent:

a{x_n}^∞_n0converges strongly to the common fixed pointpofT andS.

b{Txn}^∞_n0and{Syn}^∞_n0are bounded.

Remark 2.8. It is worth to mentioning that we have the following.

1The results of Chidume4and Zhou and Jia10depend on the geometry of the Banach space, whereas in our case we do not need such geometry.

2We remove the boundedness assumption onKintroduced in4,10.

Acknowledgments

The authors are grateful to the referees for their valuable comments and suggestions.

References

1 B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of the American Mathematical Society, vol. 226, pp. 257–290, 1977.

2 W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol.

4, pp. 506–510, 1953.

3 S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974.

4 C. E. Chidume, “Approximation of fixed points of strongly pseudocontractive mappings,” Proceedings of the American Mathematical Society, vol. 120, no. 2, pp. 545–551, 1994.

5 C. E. Chidume and A. Udomene, “Strong convergence theorems for uniformly continuous pseudo- contractive maps,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 88–99, 2006.

6 X. Weng, “Fixed point iteration for local strictly pseudo-contractive mapping,” Proceedings of the American Mathematical Society, vol. 113, no. 3, pp. 727–731, 1991.

7 J. Schu, “Approximating fixed points of Lipschitzian pseudocontractive mappings,” Houston Journal of Mathematics, vol. 19, no. 1, pp. 107–115, 1993.

8 H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis A, vol. 16, no. 12, pp.

1127–1138, 1991.

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9 A. Rafiq, “Some results on asymptotically pseudocontractive mappings,” Fixed Point Theory, vol. 11, no. 2, pp. 355–360, 2010.

10 H. Zhou and Y. Jia, “Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption,” Proceedings of the American Mathematical Society, vol. 125, no. 6, pp. 1705–1709, 1997.

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