APPROXIMATING FIXED POINTS OF λ-FIRMLY NONEXPANSIVE MAPPINGS IN BANACH SPACES

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APPROXIMATING FIXED POINTS OF λ-FIRMLY NONEXPANSIVE MAPPINGS IN BANACH SPACES

GANG-EUN KIM (Received 2 September 1997)

Abstract.We study the convergence of the Ishikawa iteration methods to ﬁxed points for the result of Smarzewski (1991). Our theorems also improve recent theorems due to Sharma and Sahu (1996).

Keywords and phrases. λ-firmly nonexpansive mapping, fixed point, Opial’s condition, Fréchet differentiable norm.

2000 Mathematics Subject Classiﬁcation. Primary 46B20; Secondary 47H10.

1. Introduction. LetEbe a real Banach space and letCbe a nonempty closed convex subset ofE. Then a mappingT ofCinto itself is callednonexpansiveifT x−T y ≤ x−yfor allx,y∈C. A mappingT ofCinto itself is calledλ-ﬁrmly nonexpansive if there existsλ∈(0,1)such that

T x−T y ≤(1−λ)(x−y)+λ(T x−T y) ∀x,y∈C. (1.1) It is clear that everyλ-ﬁrmly nonexpansive mapping is nonexpansive. For a mapping T ofCinto itself, we consider the following iteration scheme:x1∈C,

xn+1=αnT

βnT xn+ 1−βn

xn +

1−αn

xn ∀n≥1, (1.2) where{αn}and{βn}are real sequences in[0,1]. Such an iteration scheme was intro- duced by Ishikawa [5]; see also Mann iteration scheme (corresponding to the choice βn=0 for alln∈N) [6]. Now letCbe a nonempty convex subset of a Banach spaceE, and letT,Sbe mappings ofCinto itself. Then, for anx1∈C, we consider the iterates {xn}deﬁned by

xn+1=αnT yn+ 1−αn

Sxn, yn=βnT xn+

1−βn

xn ∀n≥1, (1.3)

where αn and βn satisfy 0< a≤ αn, βn ≤b <1. If S =I, the identity mapping, the iterates (1.3) are reduced to the above special case due to Ishikawa [5]. In 1991, Smarzewski [10] proved the following result: letEbe a uniformly convex Banach space and letC=_n

i=1Cibe a union of nonempty bounded closed convex subsetsCiofE and suppose T : C →C is λ-ﬁrmly nonexpansive for some λ∈(0,1). Then T has a ﬁxed point inC. The result above is no longer true if T is merely nonexpansive, even in one-dimensional space; see [10]. Recently, Sharma and Sahu [9] studied the

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convergence of the Mann and Ishikawa iteration methods to ﬁxed points for the result of Smarzewski [10].

In this paper, we first show that the iterates{xn}and{yn}defined by (1.3) converge weakly to the same common fixed point ofT and S whenE is a uniformly convex Banach space with Opial’s condition or Fréchet differentiable norm. Next, we show that the iterates{xn}defined by (1.2) converge weakly to a fixed point ofT whenE is a uniformly convex Banach space with Opial’s condition. Finally, we show that if Eis uniform convex and if the ranges ofT are contained in a compact subset ofC, the iterates{xn}defined by (1.2) converge strongly to a fixed point ofT. This paper also improves recent theorems due to Sharma and Sahu [9] using ideas of Takahashi- Kim [12].

2. Preliminaries. Throughout this paper, we denote byE and E^∗ a real Banach space and the dual space ofE, respectively. The value ofx^∗∈E^∗atx∈Eis denoted byx,x^∗. LetC be a nonempty closed convex subset ofEand letT be a mapping fromCinto itself. Then we denote byF(T )the set of all ﬁxed points ofT, i.e.,F(T )= {x∈C:T x=x}. We also denote byNthe set of all natural numbers and byRandR⁺ the sets of all real numbers and all nonnegative real numbers, respectively. coAmeans the closure of the convex hull ofA. A Banach spaceEis calleduniformly convexif for each >0 there is aδ >0 such that forx,y∈Ewithx,y ≤1 andx−y ≥, x+y ≤2(1−δ)holds. When{xn}is a sequence inE, thenxn→x(resp.,xn x, xn x) denote strong (resp., weak,∗ weak^∗) convergence of the sequence{xn}tox.

A Banach spaceEis said to satisfyOpial’s condition[7] if for any sequence{xn}inE, xn ximplies that

limsup

n→∞ xn−x<limsup

n→∞ xn−y ∀y∈Ewithy=x. (2.1) IfI−Tis demiclosed at zero [1], i.e., for any sequence{xn}inC, the conditionsxn→x weakly andxn−T xn→0 strongly implyx−T x=0. With eachx∈E, we associate the set

Jφ(x)=

x^∗∈E^∗:

x,x^∗ = xx^∗andx^∗ =φ x

, (2.2)

whereφ:R⁺ →R⁺ is a continuous and strictly increasing function withφ(0)=0 and φ(∞)= ∞. Then Jφ:E→2Ê^∗ is said to be the duality mapping. Suppose that Jφis single-valued. ThenJφis said to be weakly sequentially continuous if for each {xn} ∈Ewith xn x, thenJφ(xn) J^∗ φ(x). For abbreviation, we setJ:=Jφ. In all our proofs we assume, without loss of generality, thatJis normalized. We know that ifEadmits a weakly sequentially continuous duality mapping, thenEsatisfies Opial’s condition; see [4]. Let S(E)= {x∈E:x =1}. Then the norm ofE is said to be Gâteaux differentiable(andEis said to besmooth) if

limt→0

x+ty−x

t (2.3)

exists for eachxandyinS(E). It is also said to beFréchet diﬀerentiableif, for each x∈S(E), the limit (2.3) is attained uniformly in y∈S(E). All Hilbert spaces and

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l^p(1< p <∞) satisfy Opial’s condition, whileL^pwith 1< p=2<∞do not. It is well known that ifE is smooth, then the duality mappingJ is single-valued and strong- weak^∗continuous; for more details, see [2] or [11].

3. Convergence theorems. We ﬁrst begin with the following.

Lemma3.1(see [8]). LetEbe a uniformly convex Banach space,0< b≤tn≤c <1 for alln≥1, anda≥0. Suppose that{xn}^∞_n=1and{yn}^∞_n=1are sequences ofEsuch thatlimsupn→∞xn ≤a,limsupn→∞yn ≤a, andlimn→∞tnxn+(1−tn)yn =a.

Thenlimn→∞xn−yn =0.

Using Lemma 3.1, we have the following.

Lemma3.2. LetC=_n

i=1Cibe a union of nonempty closed convex subsetsCiof a uniformly convex Banach spaceE and letT ,S:C→C be λ-ﬁrmly nonexpansive for someλ∈(0,1)andtT (sT x+(1−s)x)+(1−t)Sx∈Cfor allx∈C ands,t∈(0,1).

ThenF(T )∩F(S)is nonempty if and only if the iterates{xn}deﬁned by (1.3) is bounded, {xn−T xn}and{xn−Sxn}converge strongly to zero asn→ ∞.

Proof. Letw be a common ﬁxed point of T and S. SinceT and S areλ-ﬁrmly nonexpansive for someλ∈(0,1), it is easy to check thatxn+1−w ≤ xn−wfor alln≥1. So,{xn}is bounded and limn→∞xn−wexists. Putc=limn→∞xn−w.

SinceT isλ-ﬁrmly nonexpansive for someλ∈(0,1), we obtain T yn−w ≤(1−λ)(yn−w)+λ(T yn−w)

≤(1−λ)yn−w+λT yn−w, (3.1)

and thusT yn−w ≤ yn−w. Taking limsup_n→∞in both sides, we obtain limsup

n→∞ T yn−w ≤limsup

n→∞ yn−w ≤limsup

n→∞ xn−w =c. (3.2) Furthermore, since

n→∞limαn(T yn−w)+(1−αn)(Sxn−w)= lim

n→∞xn+1−w =c, (3.3) by Lemma 3.1, we have limn→∞T yn−Sxn =0. Since

xn+1−w ≤αnT yn−w+(1−αn)xn−w

≤αnyn−w+(1−αn)xn−w, (3.4) we have

xn+1−w−xn−w

αn ≤ yn−w−xn−w. (3.5)

Since{αn}is assumed to be bounded away from zero, we obtain

c≤liminf_n_→∞ yn−w. (3.6)

Sinceyn−w ≤ xn−wfor alln≥1, we have

c=_n→∞limyn−w =_n→∞limβn(T xn−w)+(1−βn)(xn−w). (3.7)

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By Lemma 3.1, we have limn→∞T xn−xn =0. Since

xn−Sxn ≤ xn−T xn+T xn−T yn+T yn−Sxn

≤(1+βn)xn−T xn+T yn−Sxn, (3.8) we havexn−Sxn→0 asn→ ∞.

Conversely, suppose that{xn}is bounded,{xn−T xn}and {xn−Sxn}converge strongly to zero asn→ ∞. Then we can consider a real-valued functiongonCgiven by

g(v)=limsup

n→∞ xn−v for eachv∈C. (3.9) By [11], we know thatg :C →Ris continuous and convex. Further, if vn → ∞, theng(vn)→ ∞. So, we have an elementv0∈Csuch thatg(v0)=r=minv∈Cg(v).

SetM= {v0∈C:r=g(v0)}. ThenMis bounded, closed, and convex. Further,Mis invariant underT. In fact, letz∈M. Then, for someλ∈(0,1), we have

limsup

n→∞ T xn−T z ≤limsup

n→∞

(1−λ)(xn−z)+λ(T xn−T z)

≤(1−λ)limsup

n→∞ xn−z+λlimsup

n→∞ T xn−T z (3.10) and thus

limsup

n→∞ xn−T z =limsup

n→∞ T xn−T z ≤limsup

n→∞ xn−z. (3.11) HenceT z∈M. Similarly,Mis invariant underS. SinceEis uniformly convex and hence Mconsists of one point, we have a common ﬁxed point ofTandSinM; see [13].

Remark3.3. In Lemma 3.2, ifF(T )∩F(S)≠∅, then we furthermore see that{yn− T yn}and{yn−Syn}converge strongly to zero asn→ ∞.

We ﬁrst consider the following weak convergence ofλ-ﬁrmly nonexpansive mappings in a Banach space.

Theorem3.4. LetEbe a uniformly convex Banach space satisfying Opial’s condition and letC=_n

i=1Cibe a union of nonempty closed convex subsetsCi ofEand letT, S:C→Cbeλ-firmly nonexpansive for someλ∈(0,1)with a common fixed point and tT (sT x+(1−s)x)+(1−t)Sx∈C for allx∈C ands,t∈(0,1). Then the iterates {xn}and{yn}defined by (1.3) converge weakly to a common fixed point ofT andS.

Further, the twow-limits of{xn}and{yn}coincide.

Proof. Letzbe a common fixed point ofT andS. Then, as in the proof of Lemma 3.2, we have limn→∞xn−zexists. Letz1andz2be two weak subsequential limits of the sequence {xn}. We claim that the conditionsxn_i z1 and xn_j z2 imply z1=z2∈F(T )∩F(S). We first show thatz1,z2∈F(T ). In fact, ifT z1≠z1, then, by Opial’s condition, we have limsupi→∞xn_i−z1<limsupi→∞xn_i−T z1. SinceT is λ-firmly nonexpansive for someλ∈(0,1), we obtain

limsup

i→∞ T xni−T z1 ≤limsup

i→∞

(1−λ)

xni−z1 +λ

T xni−T z1

≤(1−λ)limsup

i→∞ xn_i−z1+λlimsup

i→∞ T xn_i−T z1. (3.12)

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By Lemma 3.2, we have limsup

i→∞ xn_i−T z1 ≤limsup

i→∞ xn_i−z1. (3.13)

This is a contradiction. Hence we haveT z1=z1. Similarly, we havez2∈F(T ). Next, we showz1=z2. If not, by Opial’s condition,

n→∞limxn−z1 =lim

i→∞xn_i−z1<lim

i→∞xn_i−z2

=lim

n→∞xn−z2 =lim

j→∞xnj−z2

<lim

j→∞xnj−z1=lim

n→∞xn−z1.

(3.14)

This is a contradiction. Hence we havez1=z2. By using the same method as above, we havez1=z2∈F(S). This implies that{xn}converges weakly to a common ﬁxed point of T and S. As in the proof of Lemma 3.2, we have limn→∞yn−z exists.

Letyni w1 and ynj w2. Then, by using the same method as above, we obtain w1=w2∈F(T )∩F(S). Further, sincexn−yn =βnxn−T xn →0 asn→ ∞, we readily see that twow-limits of{xn}and{yn}coincide.

Theorem3.5. LetE be a uniformly convex Banach space with a Fréchet diﬀeren- tiable norm. LetC=_n

i=1Cibe a union of nonempty closed convex subsetsCiofEand letT ,S:C→Cbeλ-firmly nonexpansive for someλ∈(0,1)with a common fixed point, and letI−T,I−S be demiclosed at zero andtT (sT x+(1−s)x)+(1−t)Sx∈Cfor allx∈Cands,t∈(0,1). Then the iterates{xn}and{yn}defined by (1.3) converge weakly to a common fixed point ofT andS. Further, the twow-limits of{xn}and{yn} coincide.

Proof. SinceF(T )∩F(S)is nonempty, it follows from Lemma 3.2 that {xn}is bounded,{xn−T xn}and{xn−Sxn}converge strongly to zero asn→ ∞. There exists a subsequence{xni}of{xn}and a pointz∈Csuch thatxni z. SinceI−T andI−S are demiclosed at zero, we obtainz∈F(T )∩F(S). Fory,z∈F(T )∩F(S), as in the proof of Lemma 2 [12], we have limn→∞xn,J(y−z)exists. To prove Theorem 3.5, assumexni z1andxnj z2. Then, fory,z∈F(T )∩F(S), we have

z1,J(y−z) =lim

i→∞

xn_i,J(y−z) = lim

n→∞

xn,J(y−z)

=lim

j→∞

xn_j,J(y−z) =

z2,J(y−z) . (3.15) Settingy=z1andz=z2, we obtainz1−z2,J(z1−z2) =0 and hencez1=z2. This implies that{xn}converges weakly to a common ﬁxed point ofTandS. By using the same method as above,{yn}converges weakly to a common ﬁxed point ofT andS.

Further, sincexn−yn→0 asn→ ∞, the remaining part of the proof is trivial.

Theorem3.6. LetEbe a uniformly convex Banach space satisfying Opial’s condi- tion, and letC=_n

i=1Cibe a union of nonempty bounded closed convex subsetsCiof E and letT :C→C beλ-ﬁrmly nonexpansive for someλ∈(0,1)andtT (sT x+(1− s)x)+(1−t)x∈Cfor allx∈Cands,t∈(0,1). Then for any initial datax1inC, the iterates{xn}deﬁned by (1.2), where{αn}and{βn}are chosen so thatαn∈[a,b]and

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βn∈[0,b]orαn∈[a,1]andβn∈[a,b]for somea,bwith0< a≤b <1, converge weakly to a ﬁxed point ofT.

Proof. The existence of a ﬁxed point follows from Smarzewski [10]. Letw be a ﬁxed point ofT. Then, as in the proof of Lemma 3.2, we have limn→∞xn−wexists.

Putc=limn→∞xn−w. SinceT isλ-ﬁrmly nonexpansive for someλ∈(0,1), we obtain

T yn−w≤(1−λ)(yn−w)+λ(T yn−w)

≤(1−λ)yn−w+λT yn−w (3.16)

and thusT yn−w ≤ yn−w. Taking limsup_n→∞in both sides, we obtain limsup

n→∞ T yn−w ≤limsup

n→∞ xn−w =c. (3.17)

Further, we have

n→∞limαn(T yn−w)+(1−αn)(xn−w)=_n→∞limxn+1−w =c. (3.18) If 0< a≤αn≤b <1 and 0≤βn≤b <1, by Lemma 3.1, we have limn→∞T yn−xn = 0. Since

T xn−xn ≤ T xn−T yn+T yn−xn

≤ xn−yn+T yn−xn

≤βnT xn−xn+T yn−xn,

(3.19)

we obtain

(1−b)T xn−xn ≤(1−βn)T xn−xn ≤ T yn−xn. (3.20) Thereforexn−T xn →0 asn→ ∞. On the other hand, we have, for alln≥1,

xn+1−w ≤αnT yn−w+(1−αn)xn−w

≤αnyn−w+(1−αn)xn−w (3.21) and hence

xn+1−w−xn−w

αn ≤ yn−w−xn−w. (3.22)

If 0< a≤αn≤1 and 0< a≤βn≤b <1, we obtain

c≤liminf_n_→∞ yn−w. (3.23)

Sinceyn−w ≤ xn−wfor alln≥1, we obtain

c=_n→∞limyn−w =_n→∞limβn(T xn−w)+(1−βn)(xn−w). (3.24) By Lemma 3.1, we have limn→∞T xn−xn =0. As in the proof of Theorem 3.4, the result follows.

Corollary3.7. LetE be a uniformly convex Banach space possessing a weakly sequentially continuous duality mapping and letC=_n

i=1Ci be a union of nonempty bounded closed convex subsetsCiofEand letT:C→Cbe aλ-ﬁrmly nonexpansive for someλ∈(0,1)and letI−Tbe demiclosed at zero andtT x+(1−t)x∈Cfor allx∈C andt∈(0,1). Let{αn}be a real sequence satisfying0< a≤αn≤b <1for alln∈N.

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Pickx1∈Cand deﬁnexn+1=αnT xn+(1−αn)xnfor alln∈N. Then{xn}converges weakly to a ﬁxed point ofT.

Corollary3.8. LetE be a uniformly convex Banach space possessing a weakly sequentially continuous duality mapping and letC=_n

i=1Ci be a union of nonempty bounded closed convex subsetsCiofEand letT :C→Cbeλ-firmly nonexpansive for someλ∈(0,1)and letI−Tbe demiclosed at zero andtT (sT x+(1−s)x)+(1−t)x∈ C for all x∈C ands,t∈(0,1). Let {αn}and {βn}be two sequence real sequence satisfying0< a≤αn≤b <1and0< c≤βn≤d <1for alln∈N. Pickx1∈Cand the iterates{xn}defined by (1.2). Then{xn}converges weakly to a fixed point ofT.

Next, we consider a strong convergence ofλ-ﬁrmly nonexpansive mapping in a Banach space.

Theorem3.9. LetEbe a uniformly convex Banach space and letC=_n

i=1Cibe a union of nonempty bounded closed convex subsetsCiofEwithCi⊆Ci+1. Suppose that T:C→Cisλ-firmly nonexpansive for someλ∈(0,1)such thatT (C)is contained in a compact subset ofC. Then for any initial datax1inC, the iterates{xn}defined by (1.2), where{αn}and{βn}are chosen so thatαn∈[a,b]andβn∈[0,b]orαn∈[a,1]and βn∈[a,b]for somea,bwith0< a≤b <1, converge strongly to a fixed point ofT.

Proof. Note that{xn}is well defined. The existence of a fixed point follows from Smarzewski [10]. By Mazur’s theorem [3], co({x1} ∪T (C))is a compact subset ofC containing{xn}. There exist a subsequence{xm}of the sequence{xn}and a point z∈Csuch thatxm→z. As in the proof of Theorem 3.6,{xn−T xn}converges strongly to zero asn→ ∞. SinceT isλ-firmly nonexpansive for someλ∈(0,1), we obtain

z−T z ≤ z−xm+xm−T xm+T xm−T z

≤2z−xm+xm−T xm →0 asm → ∞. (3.25) HenceT z=z. As in the proof of Lemma 3.2, we have limn→∞xn−zexists. Hence we have limn→∞xn−z =0.

Remark3.10. In Theorem 3.9, ifT ,S:C→C areλ-ﬁrmly nonexpansive for some λ∈(0,1)such thatT (C)andS(C)are contained in a compact subset ofCandF(T )∩

F(S)≠∅, then the iterates{xn}and{yn}deﬁned by (1.3) converge strongly to the same common ﬁxed point ofT andS.

References

[1] F. E. Browder,Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) (Providence, R.I.), Amer. Math. Soc., 1976, pp. 1–308. MR 53#8982.

Zbl 327.47022.

[2] J. Diestel,Geometry of Banach Spaces—Selected Topics, Lecture Notes in Math., vol. 485, Springer-Verlag, Berlin, 1975. MR 57#1079. Zbl 307.46009.

[3] N. Dunford and J. T. Schwartz,Linear Operators. I. General Theory, Interscience Publish- ers, Inc., New York, 1958. MR 22#8302. Zbl 084.10402.

[4] J. P. Gossez and E. Lami Dozo,Some geometric properties related to the ﬁxed point the- ory for nonexpansive mappings, Paciﬁc J. Math.40(1972), 565–573. MR 46#9815.

Zbl 231.47027.

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[5] S. Ishikawa,Fixed points by a new iteration method, Proc. Amer. Math. Soc.44(1974), 147–150. MR 49#1243. Zbl 286.47036.

[6] W. R. Mann,Mean value methods in iteration, Proc. Amer. Math. Soc.4(1953), 506–510.

MR 14,988f. Zbl 050.11603.

[7] Z. Opial,Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc.73(1967), 591–597. MR 35#2183. Zbl 179.19902.

[8] J. Schu,Weak and strong convergence to ﬁxed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991), no. 1, 153–159. MR 91k:47136.

Zbl 709.47051.

[9] B. K. Sharma and D. R. Sahu, Fixed point approximation for λ-ﬁrmly nonexpansive mappings, Bull. Calcutta Math. Soc. 88 (1996), no. 4, 285–290. MR 98h:47079.

Zbl 881.47042.

[10] R. Smarzewski,On ﬁrmly nonexpansive mappings, Proc. Amer. Math. Soc.113(1991), no. 3, 723–725. MR 92b:47089. Zbl 734.47039.

[11] W. Takahashi,Nonlinear Functional Analysis, Kindaikagaku, Tokyo, 1988 (Japanese).

[12] W. Takahashi and G. E. Kim,Approximating ﬁxed points of nonexpansive mappings in Banach spaces, Math. Japon.48(1998), no. 1, 1–9. MR 2000b:47127. Zbl 913.47056.

[13] E. Zeidler, Nonlinear Functional Analysis and its Applications. I, Springer-Verlag, New York, 1986. MR 87f:47083. Zbl 583.47050.

Gang-Eun Kim: Department of Applied Mathematics, Pukyong National University, Pusan608-737, Korea