Approximating fixed points of nonexpansive type mappings

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APPROXIMATING FIXED POINTS OF NONEXPANSIVE

TYPE MAPPINGS

HEMANT K. PATHAK and MOHAMMAD S. KHAN

(Received 1 March 1999)

Abstract.In a uniformly convex Banach space, the convergence of Ishikawa iterates to a unique fixed point is proved for nonexpansive type mappings under certain conditions.

2000 Mathematics Subject Classification. 47H10, 54H25.

1. Introduction. LetDbe a nonempty, closed, and convex subset of a uniformly convex Banach spaceB, and T :D→D with fixed point set F(T ). Recently, Ghosh and Debnath [1] introduced the generalized versions of the conditions of Senter and Dotson [6] as: the mappingTwithF(t)= ∅is said to satisfy the following conditions. Condition1.1. If there exists a nondecreasing functionf :[0,∞)→[0,∞)with f (0)=0 andf (r ) >0 for allr∈(0,∞)such that

_{1−T Tµ}

x≥fd(x,F) ∀x∈D, (1.1)

whereTµx=(1−µ)x+µT xwith 0≤µ≤β <1 andd(x,F)=infz∈Fx−z.

Condition1.2. If there exists a positive real numberksuch that _{1−T Tµ}

x≥k dx,F(T ) ∀x∈D. (1.2)

Whenµ=0, both conditions reduce to those of Senter and Dotson [6]. It may be noted that the mapping which satisfiesCondition 1.2also satisfiesCondition 1.1.

In this paper, we wish to use Conditions1.1and1.2to prove the convergence of Ishikawa iterates [3] of certain nonexpansive type mappings.

2. Ishikawa’s iterative process. LetDbe a convex subset of a Banach spaceBand T:D→D. Forx1∈D, Ishikawa [3] defined a sequence{xn}such that

xn+1=1−αnxn+αnT1−βnxn+βnT xn, (2.1)

where{αn}∞n=1and {βn}∞n=1 are sequences of nonnegative numbers with 0≤αn≤

βn≤1,limn→∞βn=0, and∞n=1αnβn= ∞.

Using notation forTµxofSection 1, (2.1) may be written as

In this paper, we assume thatαnandβnsatisfy

(i) 0< a≤αn< b <1,

(ii) 0≤βn≤β <1.

We denote the sequence (2.1) byM(x1,αn,βn,T ), whereαn andβn satisfy (i) and

(ii). We also assume thatαn=λandβn=µfor allnin the Ishikawa iterates defined

above, that is,

xn+1=Tλ,µn x1,Tλ,µ=(1−λ)I+λT(1−µ)I+µT. (2.3)

3. Nonexpansive type mappings and convergence theorems. Before we state and prove our main results we need to recall several definitions.

Definition3.1. A mappingT:D→Dis called nonexpansive if for allx,y∈D,

_{T x−T y≤x−y. (3.1)}

Definition 3.2. A mappingT :D→D is called generalized nonexpansive if it satisfies the condition, for allX,Y∈D,

_{T x−T y≤ax−y+bx−T x+y−T y+cx−T y+y−T x, (3.2)}

wherea,c≥0,b >0, anda+2b+2c≤1. This type of mapping was introduced by Hardy and Rogers [2] in metric spaces.

Definition3.3. A mapping T :D→D is said to satisfyCondition 1.1 if for all x,y∈D,

_{T x−T y≤max βx−y,}x−T x+y−T y

2 ,

_{x−T y+y−T x} 2

, (3.3) andT is said to satisfyCondition 1.2if for allx,y∈D,

_{T x−T y≤max βx−y,} x−T x+y−T y

2 ,x−T y, βy−T x

, (3.4) where 0≤µ≤β <1.

Remark3.4. It is to be noted that

(i) a nonexpansive mapping is generalized nonexpansive,

(ii) generalized nonexpansive mappings and mappings satisfyingCondition 1.1 also satisfyCondition 1.2, but the converse is not true as can be seen from the following example.

Example3.5. LetB=Rwith the usual norm and letD=D1D2where

D1=m_n, m=0,1,3,9,...; n=1,4,...,3k+1,

D2=m_n, m=1,3,9,27,...;n=2,5,...,3k+2.

DefineT:D→Dby

T x=

         3x

4 , x∈D1, x

2, x∈D2.

(3.6)

ThenTsatisfiesCondition 1.2, but it does not satisfyCondition 1.1and coincidentally thatT is not a generalized nonexpansive mapping; for instance, takex=1,y=3/5. Then

_{T x−T y=} 9 20≥max

₂ 5β, 11 40, 17 40

=max 2₅β,1_{2 1}

4+ 3 10

,1₂

₇

10+ 3 20

=max βx−y,x−T x+₂y−T y,x−T y+₂y−T x

. (3.7)

We now show that a mappingT satisfyingCondition 1.2is a quasi-nonexpansive mapping. Supposepis a fixed point ofT. Then puttingy=pin (3.4) and forT x=p, we obtain

0<T x−p=T x−T p

≤max

βT x−p, 1₂x−T x,x−p, βp−T x

≤max

βT x−p, 1₂x−p+p−T x,x−p,βp−T x

.

(3.8)

SinceT x−p ≤βp−T xis not possible, we have _{T x−p≤max}1

2x−p+p−T x,x−p

(3.9)

which implies that

_{T x−p≤x−p. (3.10)}

Therefore, T is quasi-nonexpansive. Next we show that

F(T )=FTλ,µ=FT Tµ. (3.11)

ObviouslyF(T )⊂F(Tλ,µ).

Letp∈F(Tλ,µ). ThenTλ,µp=pimplies thatTλ,µp=(1−λ)Ip+λT [(1−µ)Ip+µT p]=

(1−λ)p+λT Tµpand soT Tµp=p.

It follows from (3.4) that _{T p−p=T p−T Tµp}

≤max

βp−Tµp,1₂p−T p+Tµp−p,0,βTµp−T p

=max

βµp−T p,1₂(1+µ)p−T p,0,β(1−µ)p−T p

,

whence we obtainT p=p, since max{βµ,(1/2)(1+µ),β(1−µ)}<1. Thus,F(Tλ,µ)⊂

F(T )leading to the result (3.11).

Now, we show that the mappingT satisfiesCondition 1.2. We have from (3.4) _{T Tµx−p=T Tµx−T p}

≤max

βTµX−P,1₂Tµx−T Tµx,Tµx−p,βp−T Tµx

=maxTµx−p,1₂Tµx−T Tµx,βp−T Tµx

≤maxTµx−p,1₂Tµx−T Tµx,βp−Tµx

=maxTµx−p,1₂Tµx−T Tµx

≤max(1−µ)x−p+µT x−P,1

2x−Tµx+x−T Tµx

≤max(1−µ)x−p+µx−p,1 2

µx−T x+x−T Tµx

≤maxx−p,1 2

µx−p+p−T x+x−T Tµx

≤maxx−p,1₂2µx−p+x−T Tµx

.

(3.13)

Also,we know that

_{T Tµx−p≥x−p−x−T Tµx. (3.14)} From (3.13) and (3.14), we deduce that

maxx−p,1₂2µx−p+x−T Tµx

≥x−p−x−T Tµx (3.15)

which impliesx−T Tµx ≥(2(1−µ)/3)x−p. Then we may write

_{x−T Tµx≥kx−p, (3.16)}

where

0< k=2(1₃−µ)<1. (3.17) ThusT satisfiesCondition 1.2with 0< k <1. Consequently, by Maiti and Ghosh [4, Theorem 1, page 114], we have the following.

Theorem3.6. LetDbe a closed convex Banach space B, and letT :D→Dbe a mapping which satisfies (3.4) and has a fixed point inD. ThenT satisfiesCondition 1.2 and, for anyx1∈D,M(x1,αn,βn,T )converges to the fixed point ofD.

Theorem 3.7. Let D be a closed convex bounded subset ofa uniformly convex Banach space B, and let T : D→D be a mapping satisfying (3.3). Then T satisfies Condition 1.1, and for anyx1∈D,M(x1,αn,βn,T )converges to the unique fixed point

ofT.

Proof. The mappingT satisfying (3.3) also satisfies Naimpally and Singh [5, Con-dition II(D)], and so T has a unique fixed point. We now show that T is quasinon-expansive. Letp∈F(T ). Then, for anyx∈D, we have from (3.3),

_{T x−p=T x−T p≤maxβx−p,}1

2x−T x, 1

2x−p+p−T x

≤maxβx−p,1

2x−p+p−T x

(3.18)

implying

_{T x−p≤x−p. (3.19)}

Next, we show thatT satisfiesCondition 1.1. Letp∈F(T ). Then we have from (3.1),

_{T Tµx−p=T Tµx−T}

≤max

βTµx−p,1₂Tµx−T Tµx,1₂Tµx−p+p−T Tµx

≤max

βTµx−p,1₂Tµx−p+p−T Tµx

≤maxβTµx−p,Tµx−p

=Tµx−p=x−p.

(3.20)

From (3.14) and (3.20), we derive

_{x−p≥x−p−x−T Tµx (3.21)}

which implies that

_{x−T Tµx≥0=f (0). (3.22)}

ThusT satisfies all conditions which ensure the convergence ofM(x1,αn,βn,T ).

References

[1] M. K. Ghosh and L. Debnath,Approximation ofthe fixed points ofquasi-nonexpansive map-pings in a uniformly convex Banach space, Appl. Math. Lett.5(1992), no. 3, 47–50. MR 93b:47117. Zbl 760.47026.

[2] G. E. Hardy and T. D. Rogers,A generalization ofa fixed point theorem ofReich, Canad. Math. Bull.16(1973), 201–206.MR 48 #2847. Zbl 266.54015.

[4] M. Maiti and M. K. Ghosh,Approximating fixed points by Ishikawa iterates, Bull. Austral. Math. Soc.40(1989), no. 1, 113–117.MR 90j:47076. Zbl 667.47030.

[5] S. A. Naimpally and K. L. Singh,Extensions ofsome fixed point theorems ofRhoades, J. Math. Anal. Appl.96(1983), no. 2, 437–446.MR 85h:47069. Zbl 524.47033.

[6] H. F. Senter and W. G. Dotson, Jr.,Approximating fixed points ofnonexpansive mappings, Proc. Amer. Math. Soc.44(1974), 375–380.MR 49 #11333. Zbl 299.47032.

Hemant K. Pathak: Department of Mathematics, Kalyan Mahavidyalaya, Bhilai Nagar, CG490006, India

E-mail address:[email protected]

Mohammad S. Khan: Department of Mathematics and Statistics, Sultan Qaboos Uni-versity, P.O. Box36, P. C ode123, Al-Khod Muscat, Oman