Báo cáo hóa học: " COMMON FIXED POINTS OF A FINITE FAMILY OF ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPS" potx

AKUCHU Received 3 December 2003 and in revised form 13 February 2004 Convergence theorems for approximation of common fixed points of a finite family of asymptotically pseudocontractive

ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPS

M O OSILIKE AND B G AKUCHU

Received 3 December 2003 and in revised form 13 February 2004

Convergence theorems for approximation of common fixed points of a finite family of asymptotically pseudocontractive mappings are proved in Banach spaces using an aver-aging implicit iteration process

1 Introduction

LetE be a real Banach space and let J denote the normalized duality mapping from E into

2E ∗

given byJ(x) = { f ∈ E ∗: x, f  =  x 2=  f 2}, whereE ∗denotes the dual space

ofE and ·,·denotes the generalized duality pairing IfE ∗is strictly convex, thenJ is

single-valued In the sequel, we will denote the single-valued duality mapping byj.

Let K be a nonempty subset of E A mapping T : K → K is said to be asymptoti-cally pseudocontractive (see, e.g., [3]) if there exists a sequence{ a n } ∞

n =1⊆[1,∞) such that limn →∞ a n =1 and

T n x − T n y, j(x − y)

≤ a n  x − y 2, ∀ n ≥1, (1.1) for allx, y ∈ K, j(x − y) ∈ J(x − y) In Hilbert spaces H, a self-mapping T of a nonempty

subsetK of H is asymptotically pseudocontractive if it satisfies the simpler inequality

T n x − T n y 2

≤ a n  x − y 2+x − y −

T n x − T n y 2

, ∀ n ≥1 (1.2) for allx, y ∈ K and for some sequence { a n } ∞

n =1⊆[1,∞) such that limn →∞ a n =1 The class

of asymptotically pseudocontractive mappings contains the important class of asymptot-ically nonexpansive mappings (i.e., mappings T : K → K such that

T n x − T n y  ≤ a n  x − y , ∀ n ≥1,∀ x, y ∈ K, (1.3) and for some sequence { a n } ∞

n =1⊆[1,∞) such that limn →∞ a n =1).T is called asymp-totically quasi-nonexpansive if F(T) = { x ∈ K : Tx = x } = ∅and (1.3) is satisfied for all

x ∈ K and for all y ∈ F(T) If there exists L > 0 such that  T n x − T n y  ≤ L  x − y for Copyright©2004 Hindawi Publishing Corporation

Fixed Point Theory and Applications 2004:2 (2004) 81–88

2000 Mathematics Subject Classification: 47H09, 47H10, 47J05, 65J15

URL: http://dx.doi.org/10.1155/S1687182004312027

alln ≥1 and for allx, y ∈ K, then T is said to be uniformly L-Lipschitzian A mapping

T : K → K is said to be semicompact (see, e.g., [4]) if for any sequence { x n } ∞

n =1 in K

such that limn →∞  x n − Tx n  =0, there exists a subsequence{ x n j } ∞

j =1 of { x n } ∞

n =1 such that{ x n j } ∞

j =1converges strongly to somex ∗ ∈ K.

In [5], Xu and Ori introduced an implicit iteration process and proved weak

con-vergence theorem for approximation of common fixed points of a finite family of non-expansive mappings (i.e., a subclass of asymptotically nonnon-expansive mappings for which

 Tx − T y  ≤  x − y  ∀ x, y ∈ K).

In [4], Sun modified the implicit iteration process of Xu and Ori and applied the mod-ified averaging iteration process for the approximation of fixed points of asymptotically quasi-nonexpansive maps IfK is a nonempty closed convex subset of E, and { T i } N

i =1isN

asymptotically quasi-nonexpansive self-maps ofK, then for x0∈ K and { α n } ∞

n =1⊆(0, 1), the iteration process is generated as follows:

x1= α1x0+

1− α1



T1x1,

x2= α2x1+

1− α2



T2x2,

x N = α N x N −1+

1− α N

T N x N,

x N+1 = α N+1 x N+

1− α N+1

T2x N+1,

x N+2 = α N+2 x N+1+

1− α N+2

T2x N+2,

x2N = α2N x2N −1+

1− α2N



T N2x2N,

x2N+1 = α2N+1 x2N+

1− α2N+1



T13x2N+1,

(1.4)

The iteration process can be expressed in a compact form as

x n = α n x n −1+

1− α n

wheren =(k −1)N + i, i ∈ I = {1, 2, ,N }

Assuming that the implicit iteration process is defined inK, Sun proved the following

theorem

Theorem 1.1 Let E be a Banach space and let K be a nonempty closed convex subset of E Let { T i } N

i =1be N asymptotically quasi-nonexpansive self-maps of K (i.e.,  T i n x − p i  ≤[1 +

u in] x − p i  for all n ≥ 1, for all x ∈ K, and for all p i ∈ F(T i ), i ∈ I) Let F = ∩ N

i =1F(T i)=

∅ and let∞

n =1u in < ∞ for all i ∈ I Let x0∈ K, s ∈ (0, 1), and { α n } ∞

n =1⊂(s,1 − s) Then the implicit iteration process ( 1.5 ) converges strongly to a common fixed point of the family

{ T i } N

i =1if and only if lim inf n →∞ d(x n,F) = 0, where d(x n,F) =infp ∈ F  x n − p 

Theorem 1.2 Let E be a real uniformly convex Banach space and K a nonempty closed convex bounded subset of E Let { T i } N

i =1be N uniformly Lipschitzian asymptotically quasi-nonexpansive self-maps of K such that∞

n =1u in < ∞ for all i ∈ I Let F = ∩ N

i =1F(T i)= ∅

and let one member of the family { T i } N

i =1be semicompact Let x0∈ K and let s and { α n } ∞

n =1

be as in Theorem 1.1 Then the iteration process ( 1.5 ) converges strongly to a common fixed point of the family { T i } N

i =1.

Observe that ifT : K → K is a uniformly L-Lipschitzian asymptotically

pseudocon-tractive map with sequence{ a n } ∞

n =1⊆[1,∞) such that limn →∞ a n =1, then for every fixed

u ∈ K and t ∈(L/(1 + L),1), the operator S t,n:K → K defined for all x ∈ K by

satisfies

S t,n x − S t,n y  ≤(1− t)L  x − y , ∀ x, y ∈ K. (1.7)

Since (1− t)L ∈(0, 1), it follows thatS t,n is a contraction map and hence has a unique fixed pointx t,ninK This implies that there exists a unique x t,n ∈ K such that

Thus the implicit iteration process (1.5) is defined inK for the family { T i } N

i =1ofN

uni-formly L i-Lipschitzian asymptotically pseudocontractive self-mappings of a nonempty convex subsetK of a Banach space provided that α n ∈(α,1) for all n ≥1, where α =

L/(1 + L) and L =max1≤ i ≤ N { L i }

It is our purpose in this paper to first extendTheorem 1.1to the class of uniformly

L-Lipschitzian asymptotically pseudocontractive mappings The condition∞

n =1(a in −1)<

∞for alli ∈ I = {1, 2, ,N } which is equivalent to the condition∞

n =1u in < ∞for all

i ∈ I assumed in Theorems1.1and1.2is not imposed in our theorem We do not want to make the general assumption that the iteration process is defined If one assumes that the iteration process is always defined, our result will hold for even the more general class of asymptotically hemicontractive maps (i.e., mappings for whichF(T) = ∅and (1.1) holds for allx ∈ K and y ∈ F(T)) If E = H, a Hilbert space, we obtain a strong convergence

theorem similar toTheorem 1.2for the class of uniformlyL-Lipschitzian asymptotically

pseudocontractive maps

In the sequel we will need the following lemma

Lemma 1.3 [1, page 80] Let { a n } ∞

n =1, { b n } ∞

n =1, and { δ n } ∞

n =1 be sequences of nonnegative real numbers satisfying the inequality

a n+1 ≤1 +δ n



If∞

n =1δ n < ∞ and∞

n =1b n < ∞ , then lim n →∞ a n exists If in addition { a n } ∞

n =1has a subse-quence which converges strongly to zero, then lim n →∞ a n = 0.

Throughout the remaining part of this paper,{ T i } N

i =1is a finite family of uniformly

L i-Lipschitzian asymptotically pseudocontractive self-maps of a nonempty closed convex

subsetK of a Banach space so that

T i n x − T i n y, j(x − y)

≤ a in  x − y 2, ∀ n ≥1,∀ i ∈ I = {1, 2, ,N },∀ x, y ∈ K,

(1.10)

and for some sequences{ a in } ∞

n =1,i ∈ I, with lim n →∞ a in =1, for alli ∈ I;  T i n x − T i n y  ≤

L i  x − y  for all n ≥1, for all i ∈ I, for all x, y ∈ K, and for some L i > 0, i ∈ I L =

max1≤ i ≤ N { L i }

Theorem 1.4 Let E be a real Banach space and K a nonempty closed convex subset of E Let { T i } N

i =1 be N uniformly L i -Lipschitzian asymptotically pseudocontractive self-maps of

K such that F = ∩ N

i =1F(T i)= ∅ Let x0∈ K and let { α n } ∞

n =1 be a real sequence in (α,1) satisfying the condition∞

n =1(1− α n)< ∞ , where α =(1 +L)/(2 + L) (so that 2α −1> 0) Then the implicit iteration sequence { x n } ∞

n =1generated by ( 1.5 ) exists in K and converges strongly to a common fixed point of the family { T i } N

i =1if and only if lim inf n →∞ d(x n,F) = 0, where d(x n,F) =infp ∈ F  x n − p 

Proof We will use the well-known inequality

 x + y 2≤  x 2+ 2

y, j(x + y)

(1.11)

which holds for allx, y ∈ E and for all j(x − y) ∈ J(x − y) and which was first proved in

[2]

Letp ∈ F, then using (1.1), (1.5), and (1.11), we obtain

x n − p 2

=α n

x n −1− p

+

1− α n

T i k x n − p 2

≤ α2

nx n −1− p 2

+ 2

1− α n

T i k x n − p, j

x n − p

≤ α2nx n −1− p 2

+ 2

1− α n



a ikx n − p 2

.

(1.12)

Observe that since limk →∞ a ik =1 for all i ∈ I, then there exists N0 such that for all

k > N0/N + 1 (i.e., for all n ≥ N0), we havea ik ≤1 + (2α −1)/4(1 − α) for all i ∈ I

Conse-quently, for allk > N0/N + 1 (for all n ≥ N0), we have 1−2(1− α n)a ik ≥(1/2)(2α −1)> 0.

Let a =max1≤ i ≤ N {supk ≥1{ a1k }, supk ≥1{ a2k }, ,sup k ≥1{ a Nk }} Then for all k > N0/

N + 1 (for all n ≥ N0), it follows from the last inequality in (1.12) that

x n − p 2

≤



α2

n



1−2

1− α n

a ik 

x n −1− p 2

=

1 +2



1− α n

a ik −1



1−2

1− α n

a ik+



1− α n 2



1−2

1− α n

a ik x

n −1− p 2

≤1 + 4a[2α −1]−1

1− α n



+ 2[2α −1]−1

1− α n

 2 

x n −1− p 2

= 1 +σ nx n −1− p 2

,

(1.13)

whereσ n =4a[2α −1]−1(1− α n) + 2[2α −1]−1(1− α n)2 Since∞

n =1σ n < ∞, it follows from the last equality in (1.13) andLemma 1.3that limn →∞  x n − p exists so that there

existsM > 0 such that  x n − p  ≤ M for all n ≥1 Consequently, we obtain from the last equality in (1.13) that

x

n − p 1 +σ n 1/2x

n −1− p 1 +σ nx n −1− p  ≤  x n −1− p+Mσ

n (1.14)

It follows from (1.14) that

d

x n,F

≤ 1 +σ n

d

x n −1,F

so that it again follows fromLemma 1.3that limn →∞ d(x n,F) exists.

If{ x n } ∞

n =1 converges strongly to a common fixed pointp of the family { T i } N

i =1, then limn →∞  x n − p  =0 Since

0≤ d

x n,F

≤x

we have lim infd(x n,F) =0

Conversely suppose lim infn →∞ d(x n,F) =0, then we have limn →∞ d(x n,F) =0 Thus for arbitrary
> 0, there exists a positive integer N1such that

d

x n,F

<

Furthermore, ∞

n =1σ n < ∞ implies that there exists a positive integer N2 such that

∞

j = n σ j <
/4M for all n ≥ N2 ChooseN =max{ N0,N1,N2}

Thend(x N,F) ≤
/4 and∞

j = N σ j <
/4M It follows from (1.14) that for alln,m ≥ N

and for allp ∈ F, we have

x n − x m  ≤  x n − p+x m − p

≤x

N − p+M n

j = N+1

σ j+x

N − p+M m

j = N+1

σ j

≤2x

N − p+ 2M ∞

j = N

σ j

(1.18)

Taking infinimum over allp ∈ F, we obtain

x n − x m  ≤2d

x N,F

+ 2M

∞



j = N

Thus{ x n } ∞

n =1is Cauchy Suppose limn →∞ x n = u Then u ∈ K since K is closed

Further-more, sinceF(T i) is closed for alli ∈ I, we have that F is closed Since lim n →∞ d(x n,F) =0,

Remark 1.5 Prototype for the iteration parameter { α n }inTheorem 1.4isα n = α + n2(1−

α)/(n2+ 1)n ≥1

Theorem 1.6 Let H be a real Hilbert space and let K be a nonempty closed convex subset of

H Let { T i } N

i =1be N uniformly L i -Lipschitzian asymptotically pseudocontractive self-maps of

K such that F = ∩ N

i =1F(T i)= ∅ and∞

n =1(a in −1)< ∞ for all i ∈ I Let one member of the family { T i } N

i =1be semicompact Let x0∈ K and let { α n } ∞

n =1be a sequence in (0, 1) such that

0< α ≤ α n ≤ β < 1 for all n ≥ 1, where α = L/(1 + L) Then the implicit iteration sequence

{ x n } ∞

n =1generated by ( 1.5 ) exists in K and converges strongly to a common fixed point of the family { T i } N

i =1.

Proof We will use the well-known identity

tx + (1 − t)y 2

= t  x 2+ (1− t)  y 2− t(1 − t)  x − y 2 (1.20)

which holds in Hilbert spacesH for all x, y ∈ H and for all t ∈[0, 1] Letp ∈ F, then using

(1.2) and (1.20), we obtain

x

n − p 2

=α

n



x n −1− p

+

1− α n

T k

i x n − p 2

= α nx

n −1− p 2

+

1− α nT k

i x n − p 2

− α n

1− α nx n −1− T k

i x n 2

≤ α nx n −1− p 2

+

1− α n

a ikx n − p 2

+x n − T k

i x n 2

− α n

1− α nx n −1− T k

i x n 2

= α nx

n −1− p 2

+

1− α n

a ikx

n − p 2

− α n

1− α n 2 x

n −1− T k

i x n 2

.

(1.21)

Observe that since limk →∞ a ik =1 for alli ∈ I, then there exists N0 such that for allk >

N0/N + 1 (i.e., for all n ≥ N0), we havea ik ≤1 +α2/(1 − α) for all i ∈ I Consequently,

for allk > N0/N + 1 (for all n ≥ N0), we have 1−(1− α n)a ik ≥ α(1 − α) > 0 Thus for all

k > N0/N + 1 (for all n ≥ N0), it follows from (1.21) that

x n − p 2

≤



α n



1−1− α n

a ik 

x n −1− p 2

− α n

1− α n 2 x n −1− T k

i x n 2

=

1 +



1− α n

a ik −1



1−1− α n

a ik x

n −1− p 2

− α n

1− α n 2 x

n −1− T k

i x n 2

≤ 1 + α(1 − α)−1 

1− α n



a ik −1x n −1− p 2

− α n

1− α n 2 x n −1− T k

i x n 2

≤ 1 + α(1 − α)−1 

a ik −1x n −1− p 2

− α(1 − β)2 x

n −1− T k

i x n 2

= 1 +σ ikx n −1− p 2

− α(1 − β)2 x

n −1− T k

i x n 2 ,

(1.22)

whereσ ik =[α(1 − α)] −1(a ik −1) Since∞

k =1σ ik < ∞, it follows from the last equality in (1.22) andLemma 1.3that limn →∞  x n − p exists Furthermore, there existsD > 0 such

that x n − p  ≤ D for all n ≥1 Thus from the last equality in (1.22), we obtain

x n − p 2

≤x n −1− p 2

− α(1 − β)2x n −1− T k

i x n 2 +D2σ ik, (1.23)

from which it follows that limn →∞  x n −1− T k

i x n  =0 Thus limn →∞  x n −1− T k

n x n  =0 Furthermore,

x n − T k

n x n  = α nx n −1− T k

n x n  −→0 asn −→ ∞,

x n − x n −1 = 1− α nx n −1− T k

n x n  −→0 asn −→ ∞ (1.24)

Thus limn →∞  x n − x n+i  =0 for alli ∈ I For all n > N, we have T n = T n − Nso that

x

n −1− T n x n  ≤  x n −1− T k

n x n+T k

n x n − T n x n

≤x

n −1− T k

n x n+LT k −1

n x n − x n

≤x n −1− T k

n x n+L T k −1

n x n − T k −1

n − N x n − N+T k −1

n − N x n − N − x(n − N) −1 

+x(

n − N) −1− x n

≤x n −1− T k

n x n+L2 x n − x n − N+Lx(n − N) −1− T k −1

n − N x n − N

+Lx

n − x(n − N) −1 −→0 asn −→ ∞

(1.25) Hence,

x

n − T n x n  ≤  x n − x n −1 +x

n −1− T n x n  −→0 asn −→ ∞ (1.26) Consequently, for alli ∈ I, we have

x n − T n+i x n  ≤  x n − x n+i+x n+i − T n+i x n+i+Lx n+i − x n

=(1 +L)x n+i − x n+x n+i − T n+i x n+i  −→0 asn −→ ∞ (1.27)

It follows that limn →∞  x n − T i x n  =0 for alli ∈ I Since one member of { T i } N

i =1is semi-compact, then there exists a subsequence { x n j } ∞

j =1 of the sequence { x n } ∞

n =1 such that

{ x n j } ∞

j =1converges strongly tou Since K is closed, u ∈ K, and furthermore,

u − T i u  =lim

j →∞x n

j − T i x n j  =0 ∀ i ∈ I. (1.28) Thusu ∈ F Since { x n j } ∞

j =1converges strongly tou and lim n →∞  x n − u exists, it follows fromLemma 1.3that{ x n } ∞

Remark 1.7 Prototype for the iteration parameter { α n }inTheorem 1.6isα n = α + n(1 −

α)/2(n + 1) n ≥1, for which 0< α < α + (1 − α)/4 ≤ α n < α + (1 − α)/2 < 1.

Acknowledgments

This work was completed when the first author was visiting the Abdus Salam Interna-tional Centre for Theoretical Physics, Trieste, Italy He is grateful to the Committee on Development and Exchanges (CDE) of the International Mathematical Union, and the University of Nigeria, Nsukka, for generous travel support His research is supported by a grant from TWAS (99-181 RG/MATHS/AF/AC) The authors thank the referees for their useful comments on the original manuscript and for drawing the authors attention to reference [2]

[1] M O Osilike, S C Aniagbosor, and B G Akuchu, Fixed points of asymptotically

demicontrac-tive mappings in arbitrary Banach spaces, Panamer Math J 12 (2002), no 2, 77–88.

[2] W V Petryshyn, A characterization of strict convexity of Banach spaces and other uses of duality

mappings, J Funct Anal 6 (1970), 282–291.

[3] J Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J Math.

Anal Appl 158 (1991), no 2, 407–413.

[4] Z.-H Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically

quasi-nonexpansive mappings, J Math Anal Appl 286 (2003), no 1, 351–358.

[5] H.-K Xu and R G Ori, An implicit iteration process for nonexpansive mappings, Numer Funct.

Anal Optim 22 (2001), no 5-6, 767–773.

M O Osilike: Department of Mathematics, University of Nigeria, Nsukka, Nigeria

E-mail address:osilike@yahoo.com

B G Akuchu: Department of Mathematics, University of Nigeria, Nsukka, Nigeria

E-mail address:akuchubg@yahoo.com

alln ≥1 and for allx, y ∈ K, then T is said... 8

[1] M O Osilike, S C Aniagbosor, and B G Akuchu, Fixed points of asymptotically

demicontrac-tive... i)= ∅