# Some Existence and Convergence Theorems for Nonexpansive Type Mappings

Authors: S. N. Mishra; Rajendra Pant; R. Panicker
Journal: International Journal of Analysis
 (2013)

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## Abstract

Some existence and convergence theorems for a class of nonexpansive type mappings are obtained in a normed space. The results obtained herein generalize certain known results.


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## Body

## 1. Introduction

LetC be a nonempty subset of a normed space X. A mapping T:C→C is said to be nonexpansive if
(1)∥Tx-Ty∥≤∥x-y∥,
for all x,y∈C. Suppose S:C→C is another mapping on C. Then the mapping T is said to be S-nonexpansive if
(2)∥Tx-Ty∥≤∥Sx-Sy∥,
for all x,y∈C. The class of S-nonexpansive mappings is more general than nonexpansive mappings [1–3].We extend the above notion ofS-nonexpansive mappings to a more general class of nonexpansive mappings.Definition 1.
LetX be a normed space, C a nonempty subset of X and S,T:C→C. We say that T is a generalized S-nonexpansive type mapping if
(3)∥Tx-Ty∥≤M(x,y),
for all x,y∈C, where
(4)M(x,y)=max{∥Sx-Sy∥,∥Sx-Tx∥+∥Sy-Ty∥2,∥Sx-Ty∥+∥Sy-Tx∥2}.
Further T will be called a generalized nonexpansive type mappings if
(5)∥Tx-Ty∥≤m(x,y),
for all x,y∈C, where
(6)m(x,y)=max{∥x-y∥,∥x-Tx∥+∥y-Ty∥2,∥x-Ty∥+∥y-Tx∥2}.In this paper we obtain some common fixed point theorems for generalized nonexpansive type mappings in normed spaces. Specifically, in Section3, we obtain some coincidence and common fixed point theorems while in Section 4, the weak convergence of Mann iterations (cf. Mann [4]) to a common fixed point for the above class of mappings is discussed. The results obtained herein generalize certain results of Kim et al. [1], Rhoades and Temir [3], and Shazad [5] among others.


## 2. Preliminaries

LetC be a nonempty subset of a normed space X and S,T:C→X. A point z∈C is called a coincidence point of S and T if Sz=Tz and a common fixed point if Sz=Tz=z.Now onwardsℕ will denote the set of naturals while F(S) and F(T) the set of fixed points of S and T, respectively.Definition 2.
LetX be a normed space, C a nonempty subset of X, and S,T:C→C. The pair of mappings (S,T) is called(i)
commuting ifTSx=STx for all x∈C.(ii)
weakly commuting if for allx∈C, ∥STx-TSx∥≤∥Sx-Tx∥ (see [6]).(iii)
R-weakly commuting if for all x∈C, there exists R>0 such that  ∥STx-TSx∥≤R∥Sx-Tx∥ (see [7]).
The following example illustrates that weakly commuting mappings areR-weakly commuting but the converse is not true in general.Example 3.
LetX=C=[1,∞) be endowed with usual norm ∥x∥=|x|. Let S,T:C→C be mappings defined by
(7)Sx=x2,Tx=2x2-1,∀x∈X.
Then
(8)∥STx-TSx∥=∥2x2-4x+2∥=2|x2-2x+1|,∥Sx-Tx∥=|x2-2x+1|.
Therefore ∥STx-TSx∥=2|x2-2x+1|=2∥Sx-Tx∥ and the pair (S,T) is R-weakly commuting with R=2 but not weakly commuting.
In general, commuting⇒ weakly commuting ⇒R-weakly commuting.Definition 4 (cf. [1]).
LetX be a normed space and C a nonempty subset of X. The set C is called q-starshaped with q∈C if for all x∈C, the segment [q,x]={(1-t)q+tx} joining q to x, is contained in C; where 0≤t≤1.
Further ifC is a nonempty q-starshaped subset of a normed space X, then the mapping S:C→C is said to be q-affine if
(9)tSx+(1-t)Sq∈C
for all x∈C and 0≤t≤1.Definition 5 (cf. [1]).
LetX be a normed space, C a nonempty subset of X, and S,T:C→C such that F(S)≠∅. Suppose q∈F(S) and C is q-starshaped. Then the pair of mappings (S,T) is called R-subweakly commuting on C if for all x∈C, there exists a real number R>0 such that
(10)∥STx-TSx∥≤Rdist(Sx,[q,Tx]),
where dist(Sx,[q,Tx])=inf{∥Sx-y∥:y∈[q,Txq]}.
We note thatR-subweakly commuting mappings are R-weakly commuting but the converse is not true in general.Example 6.
LetX=ℝ (set of reals) with norm ∥x∥=|x| and C=[0,10]. Define S,T:C→C by
(11)Tx=x+12,Sx=x2.
Then
(12)∥TSx-STx∥=|x+24-x+14|=14,∥Tx-Sx∥=|x+12-x2|=12,∀x∈C.
Therefore
(13)∥TSx-STx∥=14=12(12)=R∥Tx-Sx∥
holds for R=1/2 and T and S are R-weakly commuting on C.
On the other hand,q=0∈F(S), and for all x∈C,
(14)∥Sx-[Tx,q]∥=|x2-[x+12,0]|=0.
So, there does not exist any R>0 such that for all x∈C,
(15)∥TSx-STx∥=14≤R∥Sx-[Tx,q]∥=0
holds. Thus T and S are not R-subweakly commuting on C.Definition 7.
LetC be a nonempty subset of a normed space X and T:C→C. Let {xn} be a sequence in X. We denote the weak and strong convergence of {xn} to x by xn⇀x and xn→x, respectively. The mapping T is said to be demicontinuous if {xn} is a sequence in X such that xn→x, then Txn⇀Tx.Definition 8.
A Banach spaceX is said to satisfy the Opial's condition (see [8]), if whenever a sequence {xn} in X, converges weakly to x  (  xn⇀x), then
(16)limn→∞inf∥xn-x∥<limn→∞inf∥xn-y∥,
for all y∈X, y≠x.
We note that theLp spaces, p≠2 do not satisfy Opial' condition while all lp spaces (1<p<∞) do (see for details Goebel and Kirk  [9]).


## 3. Existence Results

The following common fixed point theorem is due to Shahzad [5, Theorem 2.1]. For related results we refer to [2, 10–12].Theorem 9.
Let(X,d) be a metric space and C a nonempty subset of X. Let S,T:C→C be a pair of mappings such that(i)
T(C)⊆S(C);(ii)
d(Tx,Ty)≤kmax{d(Sx,Sy),d(Sx,Tx), d(Sy,Ty),(d(Sx,Ty)+d(Sy,Tx))/2},  k∈(0,1);(iii)
the pair(S,T) is R-weakly commuting on C.
Ifcl(T(C)) (closure of T(C)), is complete and T is continuous, then F(S)∩F(T)∩C is a singleton.Now we obtain a more general version of the above theorem, where the continuity condition onT has been dispensed with and the completeness of cl(T(C)) has been replaced by completeness of T(C).Theorem 10.
Let(X,d) be a metric space and C a nonempty subset of X. Let S,T:C→C be a pair of mappings such that(i)
T(C)⊆S(C);(ii)
d(Tx,Ty)≤kmax{d(Sx,Sy),d(Sx,Tx),d(Sy,Ty),(d(Sx,Ty)+d(Sy,Tx))/2},  k∈(0,1);(iii)
the pair(S,T) is R-weakly commuting on C.
Then we have the following:(a)
F(S)∩F(T)∩T(C) is a singleton if T(C) is complete,(b)
F(S)∩F(T)∩S(C) is a singleton if S(C) is complete.Proof .
Pickx0∈C. Since T(C)⊆S(C), we can construct a sequence {xn} in C such that Sxn=Txn-1 for all n∈ℕ (see [12]). By (ii), we have
(17)d(Sxn+1,Sxn)=d(Txn,Txn-1)≤kmax{d(Sxn,Txn-1)+d(Sxn-1,Txn)2d(Sxn,Sxn-1),d(Sxn,Txn),d(Sxn-1Txn-1),d(Sxn,Txn-1)+d(Sxn-1,Txn)2}=kmax{d(Sxn,Sxn)+d(Sxn-1,Sxn+1)2d(Sxn,Sxn-1),d(Sxn,Sxn+1),d(Sxn-1,Sxn),d(Sxn,Sxn)+d(Sxn-1,Sxn+1)2}=kmax{d(Sxn-1,Sxn+1)2d(Sxn,Sxn-1),d(Sxn,Sxn+1),d(Sxn-1,Sxn),d(Sxn-1,Sxn+1)2}≤kmax{d(Sxn-1,Sxn)+d(Sxn,Sxn+1)2d(Sxn,Sxn-1),d(Sxn,Sxn+1),d(Sxn-1,Sxn),d(Sxn-1,Sxn)+d(Sxn,Sxn+1)2},=kmax{d(Sxn,Sxn+1)+d(Sxn-1,Sxn)2d(Sxn,Sxn-1),d(Sxn,Sxn+1)+d(Sxn-1,Sxn)2}.
Now if
(18)max{d(Sxn,Sxn-1),d(Sxn,Sxn+1)+d(Sxn-1,Sxn)2}=d(Sxn,Sxn+1)+d(Sxn-1,Sxn)2,
then
(19)d(Sxn+1,Sxn)≤k2[d(Sxn,Sxn-1)+d(Sxn+1,Sxn)].
Therefore
(20)d(Sxn+1,Sxn)≤kd(Sxn,Sxn-1).
Since k<1, {Sxn} is a Cauchy sequence in C (see [1, 5]).
(a) Suppose thatT(C) is complete. Then there exists a point z∈T(C) such that Txn→z∈T(C). Thus, Sxn→z. Since z∈T(C)⊆S(C), there exists u∈C such that z=Su. Again by (ii), we have
(21)d(Tu,Txn)≤kmax{d(Su,Txn)+d(Sxn,Tu)2d(Su,Sxn),d(Su,Tu),d(Sxn,Txn),d(Su,Txn)+d(Sxn,Tu)2}.
Making n→∞, yields
(22)d(Tu,Su)≤kmax{0,d(Su,Tu),d(Su,Tu)2}=kd(Su,Tu)<d(Su,Tu),
a contradiction. Therefore d(Tu,Su)=0 and Su=Tu=z. Since the pair (S,T) is R-weakly commuting on C, it follows that
(23)d(STu,TSu)≤Rd(Su,Tu)=Rd(z,z)=0.
Therefore d(STu,TSu)=0 and Sz=Tz. Again by (ii), we have
(24)d(Tz,Txn)≤kmax{d(Sz,Txn)+d(Sxn,Tz)2d(Sz,Sxn),d(Sz,Tz),d(Sxn,Txn),d(Sz,Txn)+d(Sxn,Tz)2}.
Making n→∞, yields
(25)d(Tz,z)≤kmax{d(Sz,z)+d(z,Tz)2d(Sz,z),d(Sz,Tz),d(z,z),d(Sz,z)+d(z,Tz)2}=kmax{d(Sz,z),0,d(Sz,z)}=kd(Sz,z)=kd(Tz,z),
which implies z=Tz=Sz. Using (ii), uniqueness of z can be proved. Since z∈T(C), we conclude that F(S)∩F(T)∩T(C)={z}.
(b) SupposeS(C) is complete. Then Sxn→z for some z∈S(C) and there exist u∈C such that z=Su. As in part (a), we can show that Sz=Tz=z. Thus F(S)∩F(T)∩S(C)={z}.Recently Kim et al. [1] obtained the following result for S-nonexpansive type mappings in a normed space.Theorem 11.
LetC be a nonempty q-star shaped subset of a normed space X and S,T:C→C two mappings satisfying the following conditions:(i)
the mappingT is S-nonexpansive and S is q-affine with q∈F(S);(ii)
T(C)⊆S(C);(iii)
the pair(S,T) is R-subweakly commuting.SupposeS(C) is compact. Then we have the following:(a)
S and T have a coincidence point y∈S(C).(b)
IfS or T is demicontinuous, then y∈F(S)∩F(T).We extend the above theorem to a more general class of nonexpansive type mappings. In the sequel we will need the following Lemma12 and Proposition 13.Lemma 12.
Let(X,d) be a metric space and C a nonempty subset of X. Let S,T:C→C be a pair of mappings such that  (i)
T(C)⊆S(C);(ii)
d(Tx,Ty)≤kmax{d(Sx,Sy),(d(Sx,Tx)+d(Sy,Ty))/2,(d(Sx,Ty)+d(Sy,Tx))/2}, k∈(0,1);(iii)
the pair(S,T) is R-weakly commuting on C.
Then we have the following:(a)
F(S)∩F(T)∩T(C) is a singleton if  T(C) is complete;(b)
F(S)∩F(T)∩S(C) is a singleton if  S(C) is complete.Proof.
The proof can be completed on the lines of the proof of Theorem10, where
(26)max{d(Sx,Ty)+d(Sy,Tx)2d(Sx,Sy),d(Sx,Tx),d(Sy,Ty),d(Sx,Ty)+d(Sy,Tx)2}
is replaced by
(27)max{d(Sx,Sy),d(Sx,Tx)+d(Sy,Ty)2,d(Sx,Ty)+d(Sy,Tx)2}.Proposition 13.
LetC be a nonempty q-star shaped subset of a normed space X and S,T:C→C two mappings such that(i)
Tis a generalizedS-nonexpansive type mapping andSisq-affine withq∈F(S);(ii)
T(C)⊆S(C);(iii)
the pair(S,T)isR-subweakly commuting;(iv)
S(C)is complete.
Then there exist exactly one pointxλ such that
(28)xλ=Sxλ=(1-λ)q+λTxλ,
for all λ∈(0,1).Proof.
DefineTλ:C→C by Tλx=(1-λ)q+λTx, for all x∈C and for each λ∈(0,1).
Since(S,T) is R-subweakly commuting and S is q-affine, we have
(29)∥STλx-TλSx∥=∥[(1-λ)q+λSTx]-[(1-λ)q+λTSx]∥=λ∥TSx-STx∥≤λR∥Sx-Tλx∥,
for all x∈C. Thus, the pair (S,Tλ) is R-weakly commuting on C.
Also(30)∥Tλx-Tλy∥=λ∥Tx-Ty∥≤λmax{∥Sx-Sy∥,∥Sx-Tx∥+∥Sy-Ty∥2,∥Sx-Ty∥+∥Sy-Tx∥2},
for all x,y∈C. For x∈C, we have Tx∈T(C)⊆S(C), that is, there exist a point y∈C such that Tx=Sy∈S(C).
Observe that(31)Tλx=(1-λ)q+λTx=(1-λ)q+λSy∈S(C).
It follows that Tλ(C)⊆S(C) for all λ∈(0,1). Now for each λ∈(0,1), we conclude that (i)*Tλ(C)⊆S(C),(ii)*∥Tλx-Tλy∥≤λmax{∥Sx-Sy∥,(∥Sx-Tx∥+∥Sy-Ty∥)/2,(∥Sx-Ty∥+∥Sy-Tx∥)/2},(iii)*S(C) is complete,(iv)*(S,Tλ) is R-weakly commuting on C.
Therefore by Lemma12, there exist exactly one point xλ∈S(C) such that
(32)xλ=Sxλ=Tλxλ,
which implies that xλ=Sxλ=(1-λ)q+λTxλ.Now we obtain a common fixed point theorem for generalizedS-nonexpansive type mappings.Theorem 14.
LetC be a nonempty subset of a normed space X. Let S,T:C→C be two mappings satisfying conditions (i)–(iii) of Proposition 13. Suppose S(C) is compact. Then we have the following:(a)
S and T have a coincidence point y∈S(C),(b)
IfS or T is demicontinuous, then y∈F(S)∩F(T).Proof.
Let{λn} be a sequence in (0,1) such that λn→1. By Proposition 13, there exists exactly one point xλn∈S(C) such that
(33)xλn=Sxλn=(1-λn)q+λnTxλn,
for all n∈ℕ.
Setxλn:=yn. Since S(C) is compact, there exist a subsequence {ynj} of {yn} such that
(34)limj→∞Synj=y∈S(C).
Thus y=Su for some u∈C.
The assumption(ii) implies that {Tynj} is bounded. It follows that
(35)∥ynj-Tynj∥=∥(1-λm)q+λmTynj-Tynj∥=(1-λm)∥q-Tynj∥→0,asj→∞.
Thus limj→∞Tynj=limj→∞ynj=y. By the condition (3), we have
(36)∥Tynj-Tu∥≤max{∥Synj-Su∥,∥Synj-Tynj∥+∥TynjSu-Tu∥2,∥Synj-Tu∥+∥Su-Tynj∥2}=max{∥Synj-y∥,∥Synj-Tynj∥+∥Tynjy-Tu∥2,∥Synj-Tu∥+∥y-Tynj∥2}.
Making j→∞, we get
(37)∥y-Tu∥≤max{0,12∥y-Tu∥,12∥y-Tu∥}=12∥y-Tu∥,
a contradiction. Therefore ∥y-Tu∥=0 and Tu=y.(a)
Since the pair(S,T) is R-subweakly commuting, we have
(38)∥STu-TSu∥≤Rdist(Su,[Tu,q])=R∥Su-[(1-λn)q+λnTu∥=R∥Su-Su∥=0.
Thus Sy=Ty.(b)
SupposeS is demicontinuous. Since limm→∞xm=limm→∞Sxm=y, it follows from the demicontinuity of S that Sy=y. But Sy=Ty. Thus we conclude that y∈F(S)∩F(T). Similarly we can prove that y∈F(S)∩F(T) when T is demicontinuous.The following example shows the generality of Theorem14 over Theorem 11.Example 15.
LetX=ℝ (set of reals) with norm ∥x∥=|x| and C=[0,4]. Define S,T:C→C by
(39)Tx={1,ifx∈{1,3,4},2,ifx=2,32,otherwise,Sx={1,ifx=1,4,ifx∈{2,3},2,ifx=4,12,otherwise.
For x=2 and y=3, we have
(40)∥Tx-Ty∥=1>0=∥Sx-Sy∥,
and the condition (2) of Theorem 11 is not satisfied. Further, it can be easily verified that S and T satisfy all the hypotheses of Theorem 14 and S1=T1=1, is a common fixed point of S and T.


## 4. Convergence Results

Recently, Rhoades and Temir [3] obtained the following theorem.Theorem 16.
LetX be a Banach space and C a closed convex subset of X which satisfies Opial's condition. Let S,T:C→C be mappings such that(i)
T is S-nonexpansive;(ii)
S is nonexpansive. Suppose {kn} is a real sequence in (0,1). Then the sequence of Mann iterates defined for an arbitrary x0∈C by
(41)xn+1=(1-kn)xn+knTxn,n∈{0}∪ℕ
converges weakly to a common fixed point of S and T.The following theorem extends Theorem16 to generalized S-nonexpansive type mappings.Theorem 17.
LetX be a Banach space and C a closed convex subset of X which satisfies Opial's condition. Let S,T:C→C be such that(i)
T is generalized S-nonexpansive type;(ii)
S is nonexpansive.
Suppose{kn} is a real sequence in (0,1). Then the sequence of Mann iterates defined for an arbitrary x0∈C by
(42)xn+1=(1-kn)xn+knTxn,n∈{0}∪ℕ
converges weakly to a common fixed point of S and T.Proof.
IfF(T)∩F(S) is nonempty and singleton, then the proof is complete. Assume that F(T)∩F(S) is nonempty and that F(T)∩F(S) is not a singleton. Let z∈F(T)∩F(S). Then
(43)∥xn+1-z∥=∥(1-kn)xn+knTxn-z∥=∥(1-kn)(xn-z)+kn(Txn-z)∥.
Since T is generalized S-nonexpansive type, we have
(44)∥xn+1-z∥≤(1-kn)∥xn-z∥+knM(xn,z).
Now the following cases arise.
Case
1. (M(xn,z)=∥Sxn-Sz∥). Then
(45)∥Txn-z∥≤∥Sxn-Sz∥.
Since S is nonexpansive on C, the above inequality reduces to
(46)∥Txn-z∥≤∥xn-z∥.
Case
2.
(
M
(
x
n
,
z
)
=
(
∥
S
x
n
-
T
x
n
∥
+
∥
S
z
-
T
z
∥
)
/
2
). Then
(47)∥Txn-z∥≤∥Sxn-Txn∥+∥Sz-Tz∥2=∥Sxn-Txn∥2≤∥Sxn-z∥+∥z-Txn∥2=∥Sxn-Sz∥+∥z-Txn∥2,∥Txn-z∥≤∥Sxn-Sz∥.
Nonexpansiveness ofS on C implies
(48)∥Txn-z∥≤∥xn-z∥.
Case
3.
(
M
(
x
n
,
z
)
=
(
∥
S
x
n
-
T
z
∥
+
∥
S
z
-
T
x
n
∥
)
/
2
). Then
(49)∥Txn-z∥≤∥Sxn-Tz∥+∥Sz-Txn∥2=∥Sxn-Tz∥+∥Txn-z∥2=∥Sxn-Sz∥+∥Txn-z∥2,∥Txn-z∥≤∥Sxn-Sz∥.
Again, sinceS is nonexpansive on C, it follows that
(50)∥Txn-z∥≤∥xn-z∥.
Therefore in all the cases, we get
(51)∥Txn-z∥≤∥xn-z∥.
By (44) and (51), we get
(52)∥xn+1-z∥≤(1-kn)∥xn-z∥+kn∥xn-z∥=∥xn-z∥.
Thus, for kn≠0, {∥xn-z∥} is a nonincreasing sequence. Hence, limn→∞∥xn-z∥ exists. Now we show that {xn} converges weakly to a common fixed point of S and T. Let {xnk} and {xmk} be two subsequences of {xn} which converge weakly to z and z~, respectively. We will show that z=z~. Suppose the contrary. Since X satisfies Opial's condition and limn→∞∥xn-z∥ exists for any z∈F(T)∩F(S), we get
(53)limn→∞∥xn-z∥=limk→∞∥xnk-z∥<limk→∞∥xnk-z~∥=limn→∞∥xn-z~∥=limj→∞∥xmj-z~∥<limj→∞∥xmj-z∥=limn→∞∥xn-z∥,
a contradiction. Hence z=z~.Corollary 18.
Theorem16.Proof.
It comes from Theorem17, when M(x,y)=∥Sx-Sy∥.Corollary 19.
LetX be a Banach space and C a closed convex subset of X which satisfies Opial's condition. Let T:C→C be generalized nonexpansive type mapping.
Suppose{kn} is a real sequence in (0,1). Then the sequence of Mann iterates defined for an arbitrary x0∈C by
(54)xn+1=(1-kn)xn+knTxn,n∈{0}∪ℕ
converges weakly to a common fixed point of S and T.Proof.
It comes from Theorem17, when S is an identity mapping on X.Corollary 20.
LetX be a Banach space and C a closed convex subset of X which satisfies Opial's condition. Let T:C→C be a nonexpansive mapping.
Suppose{kn} is a real sequence in (0,1). Then the sequence of Mann iterates defined for arbitrary x0∈C defined
(55)xn+1=(1-kn)xn+knTxn,n∈{0}∪ℕ
converges weakly to a common fixed point of S and T.Proof.
It comes from Corollary19 when m(x,y)=∥x-y∥.



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