Some Convergence Theorems of a Sequence in Complete Metric Spaces and Its Applications

(1)

Volume 2010, Article ID 647085,10pages doi:10.1155/2010/647085

Research Article

Some Convergence Theorems of a Sequence in

Complete Metric Spaces and Its Applications

M. A. Ahmed and F. M. Zeyada

Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt

Correspondence should be addressed to M. A. Ahmed,[email protected]

Received 20 June 2009; Accepted 7 September 2009

Academic Editor: Tomas Dominguez Benavides

Copyrightq2010 M. A. Ahmed and F. M. Zeyada. 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.

The concept of weakly quasi-nonexpansive mappings with respect to a sequence is introduced. This concept generalizes the concept of quasi-nonexpansive mappings with respect to a sequence due to Ahmed and Zeyada2002. Mainly, some convergence theorems are established and their applications to certain iterations are given.

1. Introduction

In 1916, Tricomi 1 introduced originally the concept of quasi-nonexpansive for real functions. Subsequently, this concept has studied for mappings in Banach and metric spaces see, e.g.,2–7. Recently, some generalized types of quasi-nonexpansive mappings in metric and Banach spaces have appeared. For example, see Ahmed and Zeyada8, Qihou9–11

and others.

Unless stated to the contrary, we assume thatX, dis a metric space. LetT :D⊆X → X be any mapping and letFTbe the set of all fixed points ofT. IfF :X → RwhereRis the set of all real numbers and ifc ∈ R, setLc : {x ∈X : Fx≤ c}. We use the symbolμ

to denote the usual Kuratowski measure of noncompactness. For some properties ofμ,see Zeidler12, pages 493–495. For a givenx0∈D, the Picard iterationxnis determined by:

IxnTxn−1 Tnx0, n∈N

whereNis the set of all positive integers.

IfX is a normed space,Dis a convex set, and T : D → D, Ishikawa13gave the following iteration:

(2)

for eachn∈N, whereα∈0,1andβ∈0,1. Whenβ0, it yields thatTα,0 1−αIαT

Tα. Therefore, the iteration schemeIIbecomes

xnTαxn−1 Tαnx0. 1.1

This iteration is called Mann iteration14.

The concepts of quasi-nonexpansive mappings, with respect to a sequence and asymptotically regular mappings at a point were given in metric spaces as follows.

Definition 1.1see6. T : D → X is said to be quasi-nonexpansive mapping if for each

x∈Dand for everyp∈FT,dTx, p≤dx, p.

Definition 1.2see8. The mapT:D → Xis said to be quasi-nonexpansive with respect to xn⊆Dif for alln∈N∪ {0}and for everyp∈FT,dxn1, p≤dxn, p.

Lemma 2.1 in 8 stated that nonexpansiveness converts to

quasi-nonexpansiveness with respect to Tn_x

0 resp., Tαnx0, T_α,βn x0 for each x0 ∈ D.

The reverse implication is not truesee,8, Example 2.1. Also, the authors8showed that the continuity ofT :D → Xleads to the closedness ofFTand the converse is not truesee, 8, Example 2.2.

Definition 1.3see15. The mapping T : X → X is called an asymptotically regular at a pointx0 ∈Xif limn→ ∞dTnx0, Tn1x0 0.

The following definition is given by Angrisani and Clavelli.

Definition 1.4see16. LetX be a topological space. The functionF :X → Ris said to be a regular-global-infr.g.iatx∈XifFx>infXFimplies that there exists >0 such that

< Fx−infXFand a neighborhoodNxofxsuch thatFy> Fx−for eachy∈Nx. If

this condition holds for eachx∈X, thenFis said to be an r.g.i onX.

Definition 1.5see17. LetDbe a convex subset of a normed spaceX. A mappingT :D → Dis called directionally nonexpansive ifTx−Tm ≤ x−mfor eachx∈Dand for all

m∈x, Txwherex, ydenotes the segment joiningxandy; that is,x, y {λx1−λy: 0≤λ≤1}.

Our objective in this paper is to introduce the concept of weakly quasi-nonexpansive mappings with respect to a sequence. Mainly, we establish some convergence theorems of a sequence in complete metric spaces. These theorems generalize and improve8, Theorems 2.1 and 2.2, of7, Theorems 1.1 and 1.1,5, Theorem 3.1, and6, Proposition 1.1.

2. Main Result

(3)

Definition 2.1. LetX, dbe a metric space and letxnbe a sequence inD⊆X. Assume that

T :D → Xis a mapping withFT/φsatisfying limn→ ∞dxn, FT 0. Thus, for a given

> 0 there is an1 ∈ N such that dxn, FT < for alln ≥ n1.T is called weakly

quasi-nonexpansive with respect toxn⊆Dif for each >0 there exists ap∈FTsuch

that for alln∈Nwithn≥n1,dxn, p< .

We state the following lemma without proof.

Lemma 2.2. LetX, dbe a metric space and,xnbe a sequence inD⊆X. Assume thatT :D → X

is a mapping withFT/φ satisfying limn→ ∞dxn, FT 0. If T is quasi-nonexpansive with

respect toxn, then

AT is weakly quasi-nonexpansive with respect toxn;

B dxn, FTis a monotonically decreasing sequence in0,∞.

The following example shows that the converse ofLemma 2.2may not be true.

Example 2.3. Let X 0,1 be endowed with the Euclidean metric d. We define the map

T :X → XbyTx 3/4x2 1/4xfor eachx ∈X. Assume thatxn 1/nfor alln ∈

N− {1,2,3}. Then

FT {0,1}, lim

n→ ∞dxn, FT nlim→ ∞d

1

n, FT

0. 2.1

Given >0, there existsn1∈N− {1,2,3}such that for alln∈N− {1,2,3}withn≥n1,

there existsp0∈FT,

dxn,0

_n1 −0< . 2.2

Thus,Tis weakly quasi-nonexpansive with respect toxn. But,Tis not quasi-nonexpansive

with respect toxn Indeed, there exists 1∈FTsuch that for alln∈N−{1,2,3},dxn1,1>

dxn,1. Furthermore, the sequence dxn, FT 1/nis monotonically decreasing in

0,∞.

Before stating the main theorem, let us introduce the following lemma without proof.

Lemma 2.4. LetX, dbe a metric space and letxnbe a sequence inD⊆X. Assume thatT:D →

Xis weakly quasi-nonexpansive with respect toxnwithFT/φsatisfyinglimn→ ∞dxn, FT

0. Then,xnis a Cauchy sequence.

Now, we give the main theorem without proof in the following way.

Theorem 2.5. Letxnbe a sequence in a subsetDof a metric spaceX, dand letT :D → Xbe a

map such thatFT/φ. Then

alimn→ ∞dxn, FT 0ifxnconverges to a point inFT;

b xnconverges to a point inFTiflimn→ ∞dxn, FT 0,FTis a closed set,T is

(4)

As corollaries ofTheorem 2.5, we have the following.

Corollary 2.6. For eachx0 ∈D, letTnx0be a sequence in a subsetDof a metric spaceX, dand

letT:D → Xbe a map such thatFT/φ. Then

alimn→ ∞dTnx0, FT 0ifTnx0converges to a point inFT; b Tn_x

0converges to a point inFTiflimn→ ∞dTnx0, FT 0,FTis a closed

set,T is weakly quasi-nonexpansive with respect toTnx0andXis complete.

Corollary 2.7. For eachx0∈D, letTαnx0be a sequence in a subsetDof a normed spaceX, ·

and letT :D → Xbe a map such thatFT/φ. Then

alimn→ ∞dTαnx0, FT 0ifTαnx0converges to a point inFT; b Tn

αx0converges to a point inFTiflimn→ ∞dTαnx0, FT 0,FTis a closed

set,T is weakly quasi-nonexpansive with respect toTn

αx0, andXis a Banach space. Corollary 2.8. For eachx0∈D, letT_α,βn x0be a sequence in a subsetDof a normed spaceX,·

and letT :D → Xbe a map such thatFT/φ. Then

alimn→ ∞dTα,βn x0, FT 0ifTα,βn x0converges to a point inFT;

b T_α,βn x0converges to a point inFTiflimn→ ∞dTα,βn x0, FT 0,FTis a closed

set,T is weakly quasi-nonexpansive with respect toTn

α,βx0, andXis a Banach space.

Remark 2.9. I Theorem 2.5generalizes and improves 8, Theorem 2.1 since T is weakly quasi-nonexpansive with respect toxninstead ofT being quasi-nonexpansive with respect

toxn.

II Corollary 2.6 generalizes and improves 7, Theorem 1.1 page 462 for some reasons. These reasons are the following:

1the closedness ofDis superfluous;

2FTis closed instead ofT being continuous;

3Xis a complete metric space instead ofXis a Banach space; 4T is weakly quasi-nonexpansive with respect toTn_x

0in lieu ofT being

quasi-nonexpansive.

IIICorollary 2.7resp.Corollary 2.8generalizes and improves7, Theorem 1.1 page 469 resp. of5, Theorem 3.1 page 98since the reasons1and2inIIhold and

1 the convexity ofDin Theorem 1.1 is superfluous; 2 Tis weakly quasi-nonexpansive with respect toTn

αx0 resp.T_α,βn x0instead of

T being quasi-nonexpansive.

IVIf we takeT :D → Xinstead ofT :X → X,FTis closed in lieu ofT :X → X

being continuous andT is weakly quasi-nonexpansive with respect toTnx0in lieu ofT

being quasi-nonexpansive, thenCorollary 2.6generalizes and improves Kirk6, Proposition 1.1.

(5)

Theorem 2.10. Letxnbe a sequence in a subsetDof a complete metric spaceX, dandT :D → X

be a map such thatFT/φis a closed set. Assume that

iT is weakly quasi-nonexpansive with respect toxn;

ii dxn, FTis a monotonically decreasing sequence in0,∞;

iiilimn→ ∞dxn, xn1 0;

ivif the sequenceynsatisfieslimn→ ∞dyn, yn1 0, then

lim inf

n d

yn, FT

0 or lim sup

n

dyn, FT

0. 2.3

Thenxnconverges to a point inFT.

Proof. From the boundedness from below by zero of the sequence dxn, FT and

ii, we obtain that limn→ ∞dxn, FT exists. So, from iii and iv, we have that

lim infndxn, FT 0 or lim supndxn, FT 0. Then limn→ ∞dxn, FT 0 see,18,

page 37. Therefore, byTheorem 2.5b, the sequencexnconverges to a point inFT.

Corollary 2.11. For eachx0∈D, letTnx0be a sequence in a subsetDof a complete metric space X, dand letT :D → Xbe a map such thatFT/φis a closed set. Assume that

iT is weakly quasi-nonexpansive with respect toTnx0; ii dTn_x

0, FTis a monotonically decreasing sequence in0,∞; iiilimn→ ∞dTnx0, Tn1x0 0;

ivif the sequenceynsatisfieslimn→ ∞dyn, yn1 0, then

lim inf

n d

yn, FT

0 or lim sup

n

dyn, FT

0. 2.4

ThenTn_x

0converges to a point inFT.

Corollary 2.12. For eachx0 ∈D,letTαnx0be a sequence in a subsetDof a Banach spaceXand

letT:D → Xbe a map such thatFT/φis a closed set. Assume that

iT is weakly quasi-nonexpansive with respect toTαnx0; ii dTn

αx0, FTis a monotonically decreasing sequence in0,∞; iiilimn→ ∞Tαnx0−Tαn1x00;

ivif the sequenceynsatisfieslimn→ ∞yn−yn10, then

lim inf

n d

yn, FT

0 or lim sup

n

dyn, FT

0. 2.5

ThenTn

αx0converges to a point inFT.

Corollary 2.13. For eachx0∈D, letT_α,βn x0be a sequence in a subsetDof a Banach spaceXand

(6)

iT is weakly quasi-nonexpansive with respect toT_α,βn x0;

ii dT_α,βn x0, FTis a monotonically decreasing sequence in0,∞;

iiilimn→ ∞Tα,βn x0−Tα,βn1x00;

ivif the sequenceynsatisfieslimn→ ∞yn−yn10, then

lim inf

n d

yn, FT

0 or lim sup

n

dyn, FT

0. 2.6

ThenT_α,βn x0converges to a point inFT.

Remark 2.14. FromLemma 2.2, we find that8, Theorem 2.2is a special case ofTheorem 2.10. Also,Corollary 2.11generalizes and improves7, Theorem 1.2 page 464for the same reasons in Remark 2.9II.

We establish another consequence ofTheorem 2.5as follows.

Theorem 2.15. Letxnbe a sequence in a subsetDof a complete metric spaceX, d. Furthermore,

letT :D → Xbe a mapping such thatFT/φis a closed set. Assume that the conditions (i) and (ii) inTheorem 2.10hold and

iii the sequencexncontains a convergent subsequencexnjconverging tox∗∈Dsuch that

there exists a continuous mappingS:D → DsatisfyingSxnj xnj1 for allj∈Nand

dSx∗, p< dx∗, pfor somep∈FT.

Thenx∗∈FTandlimn→ ∞xn x∗.

Proof. Fromii, one can deduce that limn→ ∞dxn, FTexists, say equalr∈0,∞. Suppose

thatx∗does not belong toFT. So, we have fromiii that for somep∈FT,

dx∗, p> dSx∗, pd

S

lim

j→ ∞xnj

, p d lim

j→ ∞S

xnj

, p d lim

j→ ∞xnj1, p

dx∗, p.

2.7

This contradiction implies thatx∗∈FT. Then,

r lim

n→ ∞dxn, FT jlim→ ∞d

xnj, FT

d

lim

j→ ∞xnj, FT

dx∗, FT 0. 2.8

FromTheorem 2.5b, we obtain that limn→ ∞xnx∗.

Corollary 2.16. For eachx0∈D, letTnx0be a sequence in a subsetDof a complete metric space X, d. Furthermore, letT:D → Xbe a mapping such thatFT/φis a closed set. Assume that the conditions (i) and (ii) inCorollary 2.11hold and

iii the sequenceTn_x

(7)

such that there exists a continuous mapping S : D → D satisfying STnj_x 0

Tnj1_x

0for allj ∈NanddSx∗, p< dx∗, pfor somep∈FT.

Thenx∗∈FTandlimn→ ∞Tnx0 x∗.

Corollary 2.17. For eachx0∈D, letTαnx0be a sequence in a subsetDof a complete metric space X, d. Furthermore, letT:D → Xbe a mapping such thatFT/φis a closed set. Assume that the conditions (i) and (ii) inCorollary 2.12hold and

iii the sequence Tn

αx0contains a convergent subsequenceT nj

α x0converging tox∗ ∈

D such that there exists a continuous mappingS : D → D satisfyingSTnj α x0

Tnj1

α x0for allj ∈NanddSx∗, p< dx∗, pfor somep∈FT.

Thenx∗∈FTandlimn→ ∞Tαnx0 x∗.

Corollary 2.18. For eachx0 ∈D, letT_α,βn x0be a sequence in a subsetDof a complete metric space X, d. Furthermore, letT:D → Xbe a mapping such thatFT/φis a closed set. Assume that the conditions (i) and (ii) inCorollary 2.13hold and

iii the sequenceT_α,βn x0contains a convergent subsequenceT nj

α,βx0converging tox∗ ∈

D such that there exists a continuous mapping S : D → D satisfying STnj α,βx0

Tnj1

α,β x0for allj ∈NanddSx∗, p< dx∗, pfor somep∈FT.

Thenx∗∈FTandlimn→ ∞Tα,βn x0 x∗.

Remark 2.19. Theorem 1.3 in 7 is a special case ofCorollary 2.16 for the same reasons in

Remark 2.9IIand for the generalization of the conditions1.6and1.7in7, Theorem 1.3 to the conditioniii inCorollary 2.16.

From17, Corollary 2.4andTheorem 2.5b, one can prove the following theorem.

Theorem 2.20. LetT:X → Xbe a mapping of a complete metric spaceX, dsatisfying

idTx, T2_x_≤_hd_{x, T}_x_{for some}_h_∈₀_,₁_{and for all}_x_∈_X_; iiμTLc≤kμLcfor somek <1and for allc >0;

iiiFis an r.g.i. onX;

iv xn is a sequence in X such that limn→ ∞dxn, Txn 0 and T is weakly

quasi-nonexpansive with respect toxn.

Corollary 2.21. LetT :X → Xbe a mapping of a complete metric spaceX, dsatisfying

idTx, T2_x_≤_hd_{x, T}_x_{for some}_h_∈₀_,₁_{and for all}_x_∈_X_; iiμTLc≤kμLcfor somek <1and for allc >0;

iv Tn_x

0is a sequence satisfyinglimn→ ∞dTnx0, Tn1x0 0for eachx0∈XandT

is weakly quasi-nonexpansive with respect toTnx0.

ThenTn_x

(8)

Corollary 2.22. LetT :X → Xbe a mapping of a Banach spaceX, dsatisfying

iTx−T2_x_≤_h_x₋_T_x_{for some}_h_∈₀_,₁_{and for all}_x_∈_X_; iiμTLc≤kμLcfor somek <1and for allc >0;

iv Tn

αx0is a sequence inXsuch thatlimn→ ∞Tαnx0−TTαnx0 0for eachx0 ∈ X

andT is weakly quasi-nonexpansive with respect toTn αx0.

ThenTn

Corollary 2.23. LetT :X → Xbe a mapping of a Banach spaceX, dsatisfying

iTx−T2_x_≤_h_x₋_T_x_{for some}_h_∈₀_,₁_{and for all}_x_∈_X_; iiμTLc≤kμLcfor somek <1and for allc >0;

iv Tn

α,βx0is a sequence inXsuch thatlimn→ ∞Tα,βn x0−TTα,βn x00for eachx0∈X

andT is weakly quasi-nonexpansive with respect toT_α,βn x0.

Theorem 2.24. LetDbe a bounded closed convex subset of a Banach spaceX.Suppose thatT:D → Dsatisfies

iT is directionally nonexpansive onD;

iiμTLc≤kμLcfor somek <1and for allc >0;

iiiFis an r.g.i. onD;

iv xn⊆Dsatisfieslimn→ ∞xn−Txn0andTis weakly quasi-nonexpansive with respect

toxn.

Proof. The conclusion is obtained by combining17, Theorem 3.3andTheorem 2.5b.

Corollary 2.25. LetDbe a bounded closed convex subset of a Banach spaceX.Suppose thatT:D → Dsatisfies

iT is directionally nonexpansive onD;

iv Tn_x

0 for each x0 ∈ D satisfies limn→ ∞Tnx0−Tn1x0 0 and T is weakly

quasi-nonexpansive with respect toTn_x 0.

ThenTn_x

0converges to a point inFT.

Corollary 2.26. LetDbe a bounded closed convex subset of a Banach spaceX. Suppose thatT:D → Dsatisfies

(9)

iv Tn

αx0 for each x0 ∈ D satisfies limn→ ∞Tαnx0−TTαnx0 0 and T is weakly

quasi-nonexpansive with respect toTαnx0.

ThenTn

Corollary 2.27. LetDbe a bounded closed convex subset of a Banach spaceX. Suppose thatT:D → Dsatisfies

iT is directionally nonexpansive onD;

iv T_α,βn x0for eachx0 ∈ Dsatisfieslimn→ ∞Tα,βn x0−TTα,βn x0 0andT is weakly

quasi-nonexpansive with respect toTn α,βx0.

Remark 2.28. It is worth to mention that Corollaries2.12,2.13,2.17,2.18,2.21–2.23,2.25–2.27

are new results.

References

1 F. Tricomi, “Un teorema sulla convergenza delle successioni formate della successive iterate di una funzione di una variable reale,”Giornale di Matematiche di Battaglini, vol. 54, pp. 1–9, 1916.

2 K. M. Das, S. P. Singh, and B. Watson, “A note on Mann iteration for quasinonexpansive mappings,”

Nonlinear Analysis: Theory, Methods & Applications, vol. 5, no. 6, pp. 675–676, 1981.

3 W. G. Dotson Jr., “On the Mann iterative process,”Transactions of the American Mathematical Society, vol. 149, pp. 65–73, 1970.

4 M. K. Ghosh and L. Debnath, “Approximation of the fixed points of quasi-nonexpansive mappings in a uniformly convex Banach space,”Applied Mathematics Letters, vol. 5, no. 3, pp. 47–50, 1992.

5 M. K. Ghosh and L. Debnath, “Convergence of Ishikawa iterates of quasi-nonexpansive mappings,”

Journal of Mathematical Analysis and Applications, vol. 207, no. 1, pp. 96–103, 1997.

6 W. A. Kirk, “Remarks on approximation and approximate fixed points in metric fixed point theory,”

Annales Universitatis Mariae Curie-Skłodowska. Sectio A, vol. 51, no. 2, pp. 167–178, 1997.

7 W. V. Petryshyn and T. E. Williamson Jr., “Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings,”Journal of Mathematical Analysis and Applications, vol. 43, pp. 459–497, 1973.

8 M. A. Ahmed and F. M. Zeyada, “On convergence of a sequence in complete metric spaces and its applications to some iterates of quasi-nonexpansive mappings,”Journal of Mathematical Analysis and Applications, vol. 274, no. 1, pp. 458–465, 2002.

9 L. Qihou, “Iterative sequences for asymptotically quasi-nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 1–7, 2001.

10 L. Qihou, “Iterative sequences for asymptotically quasi-nonexpansive mappings with error member,”

Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 18–24, 2001.

11 L. Qihou, “Iteration sequences for asymptotically quasi-nonexpansive mapping with an error member of uniform convex Banach space,”Journal of Mathematical Analysis and Applications, vol. 266, no. 2, pp. 468–471, 2002.

12 E. Zeidler,Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems, Springer, New York, NY, USA, 1986.

(10)

14 W. R. Mann, “Mean value methods in iteration,”Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.

15 W. A. Kirk, “Nonexpansive mappings and asymptotic regularity,”Nonlinear Analysis: Theory, Methods & Applications, vol. 40, no. 1–8, pp. 323–332, 2000.

16 M. Angrisani and M. Clavelli, “Synthetic approaches to problems of fixed points in metric space,”

Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 170, pp. 1–12, 1996.

17 W. A. Kirk and L. M. Saliga, “Some results on existence and approximation in metric fixed point theory,”Journal of Computational and Applied Mathematics, vol. 113, no. 1-2, pp. 141–152, 2000.