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Research | Open | Published:

Common fixed points for some generalized nonexpansive mappings and nonspreading-type mappings in uniformly convex Banach spaces

Abstract

In this article, we study the fixed point theorems for nonspreading mappings, defined by Kohsaka and Takahashi, in Banach spaces but using the sense of norm instead of using the function ϕ. Furthermore, we prove a weak convergence theorem for finding a common fixed point of two quasi-nonexpansive mappings having demiclosed property in a uniformly convex Banach space. Consequently, such theorem can be deduced to the case of the nonspreading type mappings and some generalized nonexpansive mappings.

MSC:49J40, 47J20.

1 Introduction

Let T be a mapping on a nonempty subset E of a Banach space X. The mapping T is said to be quasi-nonexpansive[1] if F(T) and Txyxy for all xE and for all yF(T), where F(T) denoted the set of all fixed points of T.

In 2008, Suzuki [2] introduced a condition on T which is weaker than nonexpansiveness and stronger than quasi-nonexpansiveness, called condition (C) and obtained some fixed point theorems for such mappings.

Since then, Dhompongsa et al.[3] extended Suzuki’s main theorems to a wider class of Banach spaces. Furthermore, the fixed point theorems of such mappings have been studied by the authors of [46], etc.

During the same period, Kohsaka and Takahashi [7] introduced a nonlinear mapping called nonspreading mapping in a smooth, strictly convex, and reflexive Banach space X as follows:

Let E be a nonempty closed and convex subset of X. Then, a mapping S:EE is said to be nonspreading if

ϕ(Sx,Sy)+ϕ(Sy,Sx)ϕ(Sx,y)+ϕ(Sy,x)
(1.1)

for all x,yE, where ϕ(x,y)= x 2 2x,Jy+ y 2 for all x,yX and J is the duality mapping on E. When X is a Hilbert space, we know that ϕ(x,y)= x y 2 for all x,yX so a mapping S:EE is said to be nonspreading if

2 S x S y 2 S x y 2 + x S y 2
(1.2)

for all x,yE.

Since then, some fixed point theorems of such mapping has been studied by many researchers such as [810].

To discuss about weak convergence theorems for two nonexpansive mappings T 1 T 2 on E to itself, Takahashi and Tamura [11] constructed the following iterative scheme:

In 2011, Dhompongsa et al.[12] showed, by giving examples, that the class of nonspreading mappings is different from the class of mappings satisfying condition (C) and proved weak convergence theorems for a common fixed point of such two mappings in Hilbert spaces by using Takahashi and Tamura’s iterative scheme.

In this article, motivated by Dhompongsa et al.[12], we prove some fixed point theorems for nonspreading mappings for a general Banach space, i.e., nonspreading mappings satisfying (1.2) instead of (1.1). Furthermore, we prove a weak convergence theorem for a common fixed point of any two quasi-nonexpansive mappings having demiclosed property in a uniformly convex Banach space. Consequently, such theorem can be deduced to the case of the nonspreading type mappings and some generalized nonexpansive mappings.

2 Preliminaries

Let E be a nonempty closed and convex subset of a Banach space X and { x n } be a bounded sequence in X. For xX, define the asymptotic radius of { x n } at x as the number

r ( x , { x n } ) = lim sup n x n x.

Let

rr ( E , { x n } ) :=inf { r ( x , { x n } ) : x E }

and

AA ( E , { x n } ) := { x E : r ( x , { x n } ) = r } .

The number r and the set A are, respectively, called the asymptotic radius and asymptotic center of { x n } relative to E. It is known that A(E,{ x n }) is nonempty, weakly compact and convex as E is [13].

Definition 2.1[14]

A Banach space X is said to have the Opial property if for each sequence { x n }X weakly converging to a point xX (denote as x n x) and for any yX such that yx there holds

lim inf n x n x< lim inf n x n y

or equivalently

lim sup n x n x< lim sup n x n y.

Definition 2.2 The modulus of convexity of a Banach space X is the function δ X :[0,2][0,1] defined by

δ X (ε)=inf { 1 x + y 2 : x 1 , y 1 , x y ε } ,

for all ε[0,2]. A Banach space X is said to be uniformly convex if δ X (0)=0 and δ X (ε)>0 for all 0<ε2.

In 2008, the following condition was defined by Suzuki [2]:

Definition 2.3[2]

Let T be a mapping on a subset E of Banach space X. Then T is said to be a satisfy condition (C) if

1 2 xTxxyimpliesTxTyxy

for all x,yE.

We further have the following from [2].

Theorem 2.4[2]

Let E be a weakly compact convex subset of a uniformly convex Banach space X. Let T be a mapping on E. Assume that T satisfies condition (C). Then T has a fixed point.

Proposition 2.5[2]

Assume that a mapping T satisfies condition (C) and has a fixed point. Then T is a quasi-nonexpansive mapping.

Lemma 2.6[2]

Let T be a mapping on a closed subset E of a Banach space X. Assume that T satisfies condition (C). ThenF(T)is closed. Moreover, if X is strictly convex and E is convex, thenF(T)is also convex.

Proposition 2.7[2]

Let T be a mapping on subset E of Banach space X with the Opial property. Assume that T satisfies condition (C). If{ x n }converges weakly to z and lim n T x n x n =0, thenTz=z. That is(IT)is demiclosed at 0.

In 2008, Kohsaka and Takahashi [7] introduced the following nonlinear mapping.

Definition 2.8[7]

Let X be a smooth, strictly convex, and reflexive Banach space, J be the duality mapping of X and let E be a nonempty closed convex subset of X. Then, a mapping S:EE is said to be nonspreading if

ϕ(Sx,Sy)+ϕ(Sy,Sx)ϕ(Sx,y)+ϕ(Sy,x)

for all x,yE, where ϕ(x,y)= x 2 2x,Jy+ y 2 for all x,yX. In the case when X is a Hilbert space, S is said to be nonspreading if 2 S x S y 2 S x y 2 + x S y 2 for all x,yE.

Theorem 2.9[7]

Let X be a smooth, strictly convex, and reflexive Banach space, E be a nonempty closed convex subset of X and let S be a nonspreading mapping of E into itself. Then the following are equivalent:

  • there existsxEsuch that{ S n x}is bounded;

  • F(S)is nonempty.

In 2011, Dhompongsa et al.[12] proved that, by giving the following examples, in Banach spaces, the class of nonspreading mappings for a general Banach space and the class of mappings satisfying condition (C) are different. For the sake of completeness, we give the proof.

Example 1[12]

Define a mapping T on [0,3] by

Tx={ 0 , if x 3 ; 2 , if x = 3 .

From [2], T does not satisfy condition (C). But T is nonspreading. Indeed if x=3 and y3, we have

2 T x T y 2 =8<9= T y x 2 .

It is easy to see in the other cases that 2 T x T y 2 x T y 2 + y T x 2 .

Example 2[12] Define a mapping T on [0,1] by

Tx=1xfor allx[0,1].

Thus, T is nonexpansive mapping and hence it satisfies condition (C). But T is not nonspreading. In fact, if x=0 and y=1, we have

2 T x T y 2 =2>0= x T y 2 + y T x 2 .

The authors also studied the iterative scheme of Takahashi and Tamura [11] for approximation a common fixed point of nonspreading mappings and Suzuki’s mappings in Hilbert spaces as follows:

Theorem 2.10[12]

Let E be a nonempty closed convex subset of a Hilbert space H, let S be a nonspreading mapping of E into itself and let T be a condition (C) mapping of E into itself such thatF(S)F(T). Define a sequence{ x n }and{ z n }as follows:

for allnN, where{ α n }(0,1]and{ β n }[0,1]. Then, the following hold.

  • if lim inf n α n (1 α n )>0and n = 1 β n <, then{ x n }generated by (A) and{ z n }generated by (B) converge weakly tovF(S)anduF(T), respectively;

  • if lim inf n α n (1 α n )>0and lim inf n β n (1 β n )>0, then{ x n }generated by (A) and{ z n }generated by (B) converge weakly touF(S)F(T)andvF(S)F(T), respectively, whereu= lim n P F ( S ) F ( T ) x n andv= lim n P F ( S ) F ( T ) z n .

Since our purpose is to study fixed point theorems of mappings defined on uniformly convex Banach spaces, we need the following result.

Lemma 2.11[15]

Let E be a uniformly convex Banach space andr>0. Then there exists a strictly increasing, continuous, and convex functiong:[0,2r]Rsuch thatg(0)=0and

t x + ( 1 t ) y 2 t x 2 +(1t) y 2 t(1t)g ( x y )

for allx,y B r andt[0,1], where B r ={zE:zr}.

3 Fixed point theorems for nonspreading mappings for a general Banach space

We recall that S:EE is a nonspreading mapping for a general Banach space if

2 S x S y 2 S x y 2 + x S y 2 for allx,yE.

First, we consider the existence of a fixed point for such mappings in Banach spaces.

Theorem 3.1 Let X be a Banach space and E be a nonempty weakly compact convex subset of X such thatA(E,{ x n })is singleton for all bounded sequence{ x n }in X. IfS:EEis a nonspreading mapping for a general Banach space, thenF(S)is nonempty.

Proof Let xE. Since E is weakly compact, E is bounded and hence { S n x} is bounded nN. Let yA(E,{ S n x}). By the definition of S, we have

S n x S y 2 1 2 S n x y 2 + 1 2 S n 1 x S y 2 .

Therefore,

lim sup n S n x S y 2 lim sup n ( 1 2 S n x y 2 + 1 2 S n 1 x S y 2 ) , lim sup n 1 2 S n x S y 2 lim sup n 1 2 S n x y 2

thus, we have lim sup n S n x S y 2 lim sup n S n x y 2 . This implies that SyA(E,{ S n x}). By the uniqueness of A(E,{ S n x}), we have Sy=y and hence F(S) is nonempty. □

It follows from the fact that, in a uniformly convex Banach space, the asymptotic center of a bounded sequence with respect to a bounded closed convex subset is singleton. So, we have the following.

Theorem 3.2 Let X be a uniformly convex Banach space and E be a nonempty weakly compact convex subset of X. IfS:EEis a nonspreading mapping for a general Banach space, thenF(S)is nonempty.

Proposition 3.3 Let X be a Banach space and E be a nonempty subset of X. IfS:EEis a nonspreading mapping for a general Banach space andF(S). Then S is a quasi-nonexpansive mapping.

Proof Let xE and yF(S). By the definition of S, we have

2 S x y 2 =2 S x S y 2 S x y 2 + x S y 2 .

Therefore, S x y 2 x S y 2 = x y 2 and hence the proof is complete. □

Theorem 3.4 Let X be a uniformly convex Banach space and E be a nonempty weakly compact convex subset of X. Assume thatS:EEis a nonspreading mapping for a general Banach space andT:EEsatisfies condition (C). If S and T are commutative, thenF(S)F(T).

Proof By Theorem 2.4 and Lemma 2.6, we have F(T) is nonempty, closed, and convex. By the commutative of S and T, we have Sx=S(Tx)=T(Sx), and hence SxF(T) for all xF(T). Therefore, S:F(T)F(T). Since E is weakly compact convex and F(T) is a closed subset of E, F(T) is weakly compact convex. By Theorem 3.2, we have F(S). So there exists yF(S) such that y=SyF(T) which implies that yF(S)F(T). □

Open problem Can Theorem 3.4 be improved to a commutative family F of nonspreading mappings for a general Banach space when F generates a left reversible semigroup (i.e., any two right ideals have nonvoid intersection) (see [16, 17])?

We show the demiclosedness of a nonspreading mapping for a general Banach space as follows:

Theorem 3.5 Let X be a Banach space having Opial property and E be a nonempty closed convex subset of X. Assume thatS:EEis a nonspreading mapping for a general Banach space. If{ x n }is a sequence in E such that x n xand lim n S x n x n =0, thenxF(S).

Proof Let x n x and lim n S x n x n =0. Assume that Sxx. By Opial property of X, we have

lim sup n x n x 2 < lim sup n x n S x 2 .

By the definition of S, we have

x n S x 2 ( x n S x n + S x n S x ) 2 x n S x n 2 + 2 x n S x n S x n S x + 1 2 S x n x 2 + 1 2 x n S x 2 .

Since x n x, { x n } is bounded and hence {S x n Sx} is bounded. Thus lim n S x n x n =0 implies that

lim sup n x n S x 2 lim sup n S x n x 2 lim sup n ( S x n x n + x n x ) 2 .

By the boundedness of { x n x} and lim n S x n x n =0, we have

lim sup n x n S x 2 lim sup n x n x 2

which is a contradiction. Thus we have xF(S). □

Lemma 3.6 Let X be a Banach space. Let E be a nonempty closed convex subset of X. IfS:EEandT:EEare quasi-nonexpansive mappings such thatF(S)F(T). Let{ x n }be defined as

{ x 1 = x E , x n + 1 = α n S { β n T x n + ( 1 β n ) x n } + ( 1 α n ) x n

for allnN, where{ α n }(0,1)and{ β n }(0,1).

Then lim n x n wexists for allwF(T)F(S)and{ x n }is bounded.

Proof Let wF(S)F(T) and y n = β n T x n +(1 β n ) x n . By the quasi-nonexpansiveness of S and T, we have

S y n w y n w = β n T x n + ( 1 β n ) x n w β n x n w + ( 1 β n ) x n w = x n w .
(3.1)

By (3.1) we have,

x n + 1 w = α n S y n + ( 1 α n ) x n w α n S y n w + ( 1 α n ) x n w α n x n w + ( 1 α n ) x n w = x n w .

We can conclude by induction that x n wxw for all nN. This imply that { x n w} is a decreasing and bounded sequence and hence lim n x n w exists. Furthermore, { x n } is bounded since x n x n w+w. □

Now, we are in a position to prove our main result.

Theorem 3.7 Let X be a uniformly convex Banach space having Opial property. Let E be a nonempty closed convex subset of X. IfS:EEandT:EEare quasi-nonexpansive mappings having demiclosed property. Assume thatF(S)F(T). Let{ x n }be defined as

{ x 1 = x E , x n + 1 = α n S { β n T x n + ( 1 β n ) x n } + ( 1 α n ) x n

for allnN, where{ α n }(0,1)and β n (0,1).

Then lim inf n α n (1 α n )>0and lim inf n β n (1 β n )>0imply that x n vF(S)F(T).

Proof Let wF(S)F(T). As in the proof in Lemma 3.6, we have x n wxw for all nN. Using Lemma 2.11, we put r=xw so that there exists a strictly increasing, continuous, and convex function g:[0,2r]R such that g(0)=0 and

y n w 2 = β n ( T x n w ) + ( 1 β n ) ( x n w ) 2 β n T x n w 2 + ( 1 β n ) x n w 2 β n ( 1 β n ) g ( T x n x n ) .

Hence, by the quasi-nonexpansiveness of T, we obtain

y n w 2 x n w 2 β n (1 β n )g ( T x n x n )
(3.2)
x n w 2 .
(3.3)

From x n + 1 w 2 = α n ( S y n w ) + ( 1 α n ) ( x n w ) 2 and (3.3), we put r=xw in Lemma 2.11 again to get a strictly increasing, continuous, and convex function g:[0,2r]R such that g(0)=0 and

x n + 1 w 2 α n S y n w 2 +(1 α n ) x n w 2 α n (1 α n )g ( S y n x n ) .

By the quasi-nonexpansiveness of S and from (3.3), we obtain

x n + 1 w 2 α n y n w 2 +(1 α n ) x n w 2 α n (1 α n )g ( S y n x n )
(3.4)
α n x n w 2 + ( 1 α n ) x n w 2 α n ( 1 α n ) g ( S y n x n ) = x n w 2 α n ( 1 α n ) g ( S y n x n ) .
(3.5)

Hence

α n (1 α n )g ( S y n x n ) x n w 2 x n + 1 w 2 .

Since lim inf n α n (1 α n )>0, there exist k 1 >0 and NN such that

α n (1 α n ) k 1 ,nN.

By Lemma 3.6, we have

0= lim sup n ( x n w 2 x n + 1 w 2 )

and hence

0 lim sup n α n (1 α n )g ( S y n x n ) k 1 lim sup n g ( S y n x n ) .

Since k 1 >0, we have lim sup n g(S y n x n )=0 and hence lim n g(S y n x n )=0.

Since S y n x n 2xw for all nN, {S y n x n } is bounded and hence we can put M= lim sup n S y n x n . So there exists {S y n k x n k }{S y n x n } such that

lim k S y n k x n k =M.

Since g is a continuous function, we have

0= lim k g ( S y n k x n k ) =g(M).

Since g(0)=0 and g is strictly increasing, M=0.

Therefore, lim sup n S y n x n =0 and hence lim n S y n x n =0.

From (3.4), we have

0 α n ( 1 α n ) g ( S y n x n ) x n w 2 x n + 1 w 2 + α n ( y n w 2 x n w 2 ) .

Hence,

α n ( x n w 2 y n w 2 ) x n w 2 x n + 1 w 2 .
(3.6)

Since α n (1 α n )< α n for all nN, 0< lim inf n α n (1 α n )< lim inf n α n .

Therefore, there exist k 2 >0 and NN such that

α n k 2 ,nN.

Then from (3.6) and lim n x n w exists, we have lim n ( x n w 2 y n w 2 )=0.

On the other hand, we have from (3.2) that

β n (1 β n )g ( T x n x n ) x n w 2 y n w 2 .

Since lim inf n β n (1 β n )>0 so there exist k 3 >0 and NN such that

α n k 3 ,nN.

Therefore, we can conclude that lim n g(T x n x n )=0.

Similarly, the continuity and strictly convexity of g imply that lim n T x n x n =0.

Since { x n } is bounded, there exists { x n i }{ x n } such that x n i v. From demiclosedness of T, we have vF(T). Since

lim sup n y n x n = lim sup n β n T x n + ( 1 β n ) x n x n = lim sup n β n T x n x n ,

where { β n }(0,1) and lim n T x n x n =0, we have lim n y n x n =0.

Using lim n y n x n =0 and x n i v, by passing through subsequences, if necessary, we can assume that there exists a weakly convergent subsequence { y n i } of { y n } such that y n i v.

Furthermore, consider

S y n y n = S y n x n + x n y n S y n x n + x n y n .

Since lim n S y n x n =0 and lim n y n x n =0, lim n S y n y n =0.

By the demiclosedness of S, we have vF(S) and hence vF(S)F(T).

Finally, we show that x n v. Let { x n k } be arbitrary subsequence of { x n }. Since { x n k } is bounded, there exists { x n k i }{ x n k } that x n k i u. The same proof as v above, there exists { y n k i }{ y n k } such that y n k i u and uF(S)F(T).

Suppose that vu. Using Lemma 3.6 to guarantee that lim n x n v and lim n x n u exist and hence we have from the Opial property that

lim n x n v = lim i x n i v < lim i x n i u = lim n x n u = lim i x n k i u < lim i x n k i v = lim n x n v .

This is a contradiction. So x n vF(T)F(S). □

Since the class of nonspreading mappings for a general Banach space is different from the class of mappings satisfying condition (C), we can apply Proposition 2.5 and Proposition 3.3 to deduce Theorem 3.7 as follows:

Corollary 3.8 Let X be a uniformly convex Banach space having Opial property. Let E be a nonempty closed convex subset of X. Assume thatS:EEis a nonspreading mapping for a general Banach space andT:EEsatisfies condition (C) such thatF(S)F(T). Let{ x n }and{ z n }be defined as

for allnN, where{ α n }(0,1)and β n (0,1).

If lim inf n α n (1 α n )>0and lim inf n β n (1 β n )>0, then{ x n }generated by (A) and{ z n }generated by (B) converge weakly touF(S)F(T)andvF(S)F(T), respectively.

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Acknowledgements

This article is dedicated to Professor Anthony To-Ming Lau for celebrating his great achievements in the development of fixed point theory and applications. The authors are indebted to the anonymous referee(s) for comments which lead to the improvement and for the kindness in providing us the open problem in the article. This research was (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

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Correspondence to Warunun Inthakon.

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Keywords

  • fixed point
  • common fixed point
  • generalized nonexpansive mapping
  • nonspreading mapping
  • uniformly convex Banach space