NON-SELF-MAPPINGS
SOMYOT PLUBTIENG AND RATTANAPORN PUNPAENG
Received 14 February 2005 and in revised form 14 June 2005
SupposeCis a nonempty closed convex subset of real Hilbert spaceH. LetT:C→H be a nonexpansive non-self-mapping and P is the nearest point projection ofH onto C. In this paper, we study the convergence of the sequences{xn},{yn},{zn}satisfying xn=(1−αn)u+αnT[(1−βn)xn+βnTxn],yn=(1−αn)u+αnPT[(1−βn)yn+βnPT yn], andzn=P[(1−αn)u+αnTP[(1−βn)zn+βnTzn]], where{αn} ⊆(0, 1), 0≤βn≤β <1 andαn→1 asn→ ∞. Our results extend and improve the recent ones announced by Xu and Yin, and many others.
1. Introduction
LetCbe a nonempty closed convex subset of a Banach spaceE. Then a non-self-mapping TfromCintoEis callednonexpansiveifTx−T y ≤ x−yfor allx,y∈C. Givenu∈ Cand{αn}is a sequence such that 0< αn<1, we can define a contractionTn:C→Eby
Tnx=1−αnu+αnTx, x∈C. (1.1)
IfTis a self-mapping (i.e.,T(C)⊂C), thenTnmapsCinto itself, and hence, by Banach’s contraction principle,Tnhas a unique fixed pointxninC, that is, we have
xn=1−αnu+αnTxn, ∀n≥1 (1.2)
(such a sequence{xn}is said to be an approximating fixed point ofTsince it possesses the property that if{xn}is bounded, then limn→∞Txn−xn =0) whenever limn→∞αn=1. The strong convergence of {xn}as αn→1 for a self-mapping T of a bounded C was proved in a Hilbert space independently by Browder [1] and Halpern [3] and in a uni-formly smooth Banach space by Reich [7]. Thereafter, Singh and Watson [8] extended the result of Browder and Halpern to nonexpansive non-self-mappingT satisfying Rothe’s boundary conditionT(∂C)⊂C(here∂Cdenotes the boundary ofC). Recently, Xu and Yin [11] proved that ifCis a nonempty closed convex (not necessarily bounded) subset of Hilbert spaceH, ifT:C→H is a nonexpansive non-self-mapping, and if{xn}is the sequence defined by (1.2) which is bounded, then{xn}converges strongly asαn→1 to a
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fixed point ofT. Marino and Trombetta [5] defined contractionsSnandUnfromCinto itself by
Snx=1−αnu+αnPTx, ∀x∈C, (1.3)
Unx=P1−αnu+αnTx, ∀x∈C, (1.4)
whereP is the nearest point projection ofH ontoC. Then by the Banach contraction principle, there exists a unique fixed pointyn(resp.,zn) ofSn(resp.,Un) inC, that is,
yn=1−αnu+αnPT yn, (1.5)
zn=P1−αnu+αnTzn. (1.6)
Xu and Yin [11] also proved that ifCis a nonempty closed convex subset of a Hilbert spaceH, ifT:C→His a nonexpansive non-self-mapping satisfying the weak inwardness condition, and{xn}is bounded, then{yn}(resp.,{zn}) defined by (1.5) (resp., (1.6)) converges strongly asαn→1 to a fixed point ofT.
LetCbe a nonempty convex subset of Banach spaceE. Then forx∈C, we define the inward setIc(x) as follows:
Ic(x)=y∈E:y=x+a(z−x) for somez∈C a≥0. (1.7)
A mappingT:C→Eis said to beinwardifTx∈Ic(x) for allx∈C.Tis also said to be weakly inwardif for eachx∈C,Txbelongs to the closure ofIc(x).
In this paper, we extend Xu and Yin’s results [11] to study the contraction mappings Tn,Sn, andUndefine by
Tnx=1−αnu+αnT1−βnx+βnTx, (1.8) Snx=1−αnu+αnPT1−βnx+βnPTx, (1.9) Unx=P1−αnu+αnTP1−βnx+βnTx, (1.10)
where{αn} ⊆(0, 1), 0≤βn≤β <1, andP is the nearest point projection ofH ontoC. Moreover, we also prove the strong convergence of the sequences{xn},{yn}, and{zn} satisfying
xn=1−αnu+αnT1−βnxn+βnTxn, (1.11) yn=1−αnu+αnPT1−βnyn+βnPT yn, (1.12) zn=P1−αnu+αnTP1−βnzn+βnTzn, (1.13)
whereαn→1 asn→ ∞. We note that ifβn≡0, then (1.11), (1.12), (1.13) reduce to (1.2), (1.5), and (1.6), respectively. The results presented in this paper extend and improve the corresponding onces announced by Xu and Yin [11], and others.
2. Main results
Theorem2.1. LetH be a real Hilbert space, letCbe a nonempty closed convex subset of H, and letT:C→H be a nonexpansive non-self-mapping. Suppose that for someu∈C, {αn} ⊆(0, 1), and0≤βn≤β <1, the mappingTndefined by (1.8) has a (unique) fixed pointxn∈Cfor alln≥1. ThenThas a fixed point if and only if{xn}remains bounded as αn→1. In this case,{xn}converges strongly asαn→1to a fixed point ofT.
Proof. We denote byF(T) the fixed point set ofT. Suppose thatF(T) is nonempty. Let w∈F(T). Then for eachn≥1, we have
w−xn=w−
1−αnu−αnT1−βnxn+βnTxn ≤1−αnw−u+αnw−T1−βnxn+βnTxn ≤1−αnw−u+αnw−1−βnxn−βnTxn ≤1−αnw−u+αn1−βnw−xn+αnβnw−xn =1−αnw−u+αnw−xn,
(2.1)
and hence (1−αn)w−xn≤(1−αn)w−u, for alln≥1. This implies thatw−xn ≤ w−ufor alln≥1. Then{xn}is a bounded sequence. Conversely, suppose that{xn} is bounded,zis a weak cluster point of{xn}, andαn→1 asn→ ∞. Then we show that F(T)= ∅and{xn}converges strongly to a fixed point ofT. We choose a subsequence {xni}of the sequence{xn}withαni→1 such thatxni →zweakly, we can define a real-valued functiongonHgiven by
g(x)=lim sup i→∞
xni−x2
, for everyx∈H, (2.2)
observing thatxni−x2= xni−z2+ 2xni−z,z−x+z−x2. Sincexni→zweakly,
we immediately get
g(x)=g(z) +x−z2, ∀x∈H, (2.3)
in particular,
g(Tz)=g(z) +Tz−z2. (2.4)
On the other hand, we have
xni−Txni≤
1−αniu−Txni+αniT1−βnixni+βniTxni−Txni ≤1−αniu−Txni+αni1−βnixni+βniTxni−xni ≤1−αniu−Txni+βniTxni−xni,
(2.5)
for alli≥1. This implies that (1−βni)xni−Txni ≤(1−αni)u−Txni, and hence
xni−Txni=1−αni
1−βniu−Txni ≤
1−αni
(1−β) u−Txni−→0 asi−→ ∞.
Note that
xni−Tz2
=xni−Txni+Txni−Tz2
≤xni−Txni+Txni−Tz2 =xni−Txni2
+ 2xni−TxniTxni−Tz+Txni−Tz2
(2.7)
for alln∈N. Hence,
g(Tz)=lim sup i→∞
xni−Tz2
≤lim sup i→∞
Txni−Tz2
≤lim sup i→∞
xni−z2
=g(z).
(2.8)
This, together with (2.4), implies thatTz=zandzis a fixed point ofT. Now sinceF(T) is nonempty, closed, and convex, there exists a uniquev∈F(T) that is closest tou; namely, vis the nearest point projection ofuontoF(T). For anyy∈F(T), we have
xn−u
+αn(u−y)2=1−αnu+αnT1−βnxn+βnTxn−u +αn(u−y)2 =α2
nT1−βnxn+βnTxn−y2 ≤α2
n1−βnxn+βnTxn−y2 =α2
n1−βnxn−y+βnTxn−y2 ≤α2
n1−βnxn−y+βnxn−y
2
=α2
nxn−y2 =α2
nxn−u+u−y2,
(2.9)
and so
xn−u2
+α2nu−y2+ 2αnxn−u,u−y
≤α2
nxn−u2+u−y2+ 2xn−u,u−y ≤αnxn−u2
+αnu−y2+ 2αnxn−u,u−y
(2.10)
for alln≥1. It follows that
xn−u2
≤αny−u2≤ y−u2, ∀y∈F(T), {αn} ⊆(0, 1)∀n∈N. (2.11)
Since the norm ofHis weakly lower semicontinuous (w-l.s.c.), we get
z−u ≤lim inf
Therefore, we must havez=vforvis the unique element inF(T) that is closest tou. This shows thatvis the only weak cluster point of{xn}withαn→1. It remains to verify that the convergence is strong. In fact, it follows that
xn−v2
=xn−u2− u
−v2−2xn−v,v−u
≤ −2xn−v,v−u−→0 asn−→ ∞. (2.13)
This completes the proof.
Corollary2.2. LetH,C,Tbe as inTheorem 2.1. Suppose in addition thatCis bounded and that the weak inwardness condition is satisfied. Then for eachu∈C, the sequence{xn} satisfying (1.11) converges strongly asαn→1to a fixed point ofT.
Theorem2.3. LetHbe a Hilbert space, letCbe a nonempty closed convex subset ofH, let T:C→H be a nonexpansive non-self-mapping satisfying the weak inwardness condition, and letP:H→Cbe the nearest point projection. Suppose that for someu∈C, each{αn} ⊆ (0, 1)and0≤βn≤β <1. Then, a mappingSn defined by (1.9) has a unique fixed point yn∈C. Further,T has a fixed point if and only if{yn}remains bounded asαn→1. In this case,{yn}converges strongly asαn→1to a fixed point ofT.
Proof. It is straightforward thatSn:C→Cis a contraction for everyn≥1. Therefore by the Banach contraction principle, there exists a unique fixed pointynofSninCsatisfying (1.12). Letwbe a fixed point ofT. Then as in the proof ofTheorem 2.1,{yn}is bounded. Conversely, suppose that{yn}is bounded. ApplyingTheorem 2.1, we obtain that{yn} converges strongly to a fixed pointzofPT. Next, let us show thatz∈F(T). Sincez=PTz andPis the nearest point projection ofHontoC, it follows by [9] that
Tz−z,J(z−v)≥0, ∀v∈C. (2.14)
On the other hand,Tzbelongs to the closure ofIc(z) by the weak inwardness conditions. Hence for each integern≥1, there existzn∈Candan≥0 such that the sequence
rn:=z+anzn−z−→Tz. (2.15)
Thus it follows that
0≤anTz−z,z−zn =Tz−z,an(z−zn)
=Tz−z,z−rn−→ Tz−z,z−Tz = −Tz−z2.
(2.16)
Hence we haveTz=z.
Theorem2.5. LetH,C,T,P,u,{αn}, and{βn}be as inTheorem 2.3. Then a mappingUn defined by (1.10) has a unique fixed pointzn∈C. Further,Thas a fixed point if and only if {zn}remains bounded asαn→1andβn→0. In this case,{zn}converges strongly asαn→1 andβn→0to a fixed point ofT.
Proof. It follows by the Banach contraction principle that there exists a unique fixed point znofUnsuch that
zn=P1−αnu+αnTP1−βnzn+βnTzn. (2.17)
Letw∈F(T). Then for eachn≥1, we have
w−zn= Pw−P
1−αnu+αnTP1−βnzn+βnTzn ≤w−1−αnu−αnTP1−βnzn+βnTzn ≤1−αnw−u+αnw−TP1−βnzn+βnTzn ≤1−αnw−u+αn1−βnw−zn+αnβnw−Tzn ≤1−αnw−u+αn1−βnw−zn+αnβnw−zn =1−αnw−u+αnw−zn,
(2.18)
and hence (1−αn)w−zn ≤(1−αn)w−u, for alln >1. This implies thatw−zn ≤ w−u, for alln >1. Then{zn}is bounded. Conversely, suppose that{zn}is bounded, αn→1, andβn→0. We show thatF(T)= ∅. For any subsequence{zni}of the sequence {zn}converging weakly to ¯zsuch thatαni→1, we can define a real-valued functiong on Hgiven by
g(z)=lim sup i→∞
zni−z2
, for everyz∈H, (2.19)
observing thatzni−z2= zni−z¯ 2+ 2zni−z¯, ¯z−z+z¯−z2. Sincezni→z¯weakly,
we get
g(z)=g( ¯z) +z¯−z2, ∀z∈H, (2.20)
in particular,
g(PTz¯)=g( ¯z) +PTz¯−z¯ 2. (2.21)
For instance, the straightforward verification gives
zni−PTzni=P
1−αniu+αniTP1−βnizni+βniTzni−PTzni
and this implies thatzni−PTzni ≤(1−αni)u−Tzni+αniβniTzni−zni →0 asi→ ∞. Moreover, we note that
zni−PTz¯2
=zni−PTzni+PTzni−PTz¯2 ≤zni−PTzni+PTzni−PTz¯2 =zni−PTzni2
+ 2zni−PTzniPTzni−PTz¯+PTzni−PTz¯2 (2.23)
for alli∈N. It follows that
gPTz¯=lim sup i→∞
zni−PTz¯2
≤lim sup i→∞
PTzni−PTz¯2
≤lim sup i→∞
zni−z¯2
=g(z)
(2.24)
which in turn, together with (2.21), implies thatPT( ¯z)=z¯. Since T satisfies the weak inwardness condition, by the same argument as in the proof ofTheorem 2.3, we see that
¯
zis a fixed point ofT. For anyw∈F(T), we have
αnTP1−βnw+βnw−u+u=αn(w−u) +u =αnw+1−αnu =Pαnw+1−αnu
(2.25)
for alln∈N. By following as in the proof ofTheorem 2.1, we have
zn−u2
≤αnw−u2≤ w−u2, ∀w∈F(T), {αn} ⊆(0, 1)∀n∈N. (2.26)
From (2.26) and the w-l.s.c. of the norm ofH, it follows that
z¯−u ≤lim infn
→∞ zn−u≤ w−u (2.27)
for allw∈F(T). Hence ¯zis the nearest point projectionzinF(T) ofuontoF(T) which exists uniquely sinceF(T) is nonempty, closed, and convex. Moreover,
zn−z2
=zn−u2
− u−z2−2zn−z,z−u
≤ −2zn−z,z−u−→0 asn−→ ∞. (2.28)
This completes the proof.
Acknowledgment
The authors would like to thank the Thailand Research Fund for financial support.
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Somyot Plubtieng: Department of Mathematics, Faculty of Science, Naresuan University, Phitsan-ulok 65000, Thailand
E-mail address:somyotp@nu.ac.th
Rattanaporn Punpaeng: Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
