Volume 2011, Article ID 973028,23pages doi:10.1155/2011/973028
Research Article
Hybrid Proximal-Type Algorithms for Generalized
Equilibrium Problems, Maximal Monotone
Operators, and Relatively Nonexpansive Mappings
Lu-Chuan Zeng,
1, 2Q. H. Ansari,
3and S. Al-Homidan
31Department of Mathematics, Shanghai Normal University, Shanghai 200234, China 2Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
3Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals (KFUPM), P.O. Box 119, Dhahran 31261, Saudi Arabia
Correspondence should be addressed to S. Al-Homidan,[email protected]
Received 24 September 2010; Accepted 18 October 2010
Academic Editor: Jen Chih Yao
Copyrightq2011 Lu-Chuan Zeng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The purpose of this paper is to introduce and consider new hybrid proximal-type algorithms for finding a common element of the set EP of solutions of a generalized equilibrium problem, the setFSof fixed points of a relatively nonexpansive mappingS, and the setT−10 of zeros of a
maximal monotone operatorTin a uniformly smooth and uniformly convex Banach space. Strong convergence theorems for these hybrid proximal-type algorithms are established; that is, under appropriate conditions, the sequences generated by these various algorithms converge strongly to the same point in EP∩FS∩T−10. These new results represent the improvement, generalization,
and development of the previously known ones in the literature.
1. Introduction
LetEbe a real Banach space with the dualE∗andCbe a nonempty closed convex subset of
E. We denote byNandRthe sets of positive integers and real numbers, respectively. Also, we denote byJthe normalized duality mapping fromEto 2E∗defined by
Jxx∗∈E∗:x, x∗x2x∗2, ∀x∈E, 1.1
f : C×C → Rbe a bifunction andA: C → E∗be a nonlinear mapping. We consider the following generalized equilibrium problem:
findu∈Csuch thatfu, yAu, y−u≥0, ∀y∈C. 1.2
The set of suchu∈Cis denoted by EP, that is,
EPu∈C:fu, yAu, y−u≥0, ∀y∈C . 1.3
WheneverEHa Hilbert space, problem1.2was introduced and studied by S. Takahashi and W. Takahashi 1. Similar problems have been studied extensively recently. See, for example,2–11. In the case ofA ≡ 0,EP is denoted by EPf. In the case of f ≡ 0, EP is also denoted by VIC, A. The problem1.2is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games and others; see, for example, 12–14. A mappingS:C → Eis called nonexpansive ifSx−Sy ≤ x−yfor allx, y∈C. Denote by
FSthe set of fixed points ofS, that is,FS {x∈C:Sxx}. A mappingA:C → E∗is calledα-inverse-strongly monotone, if there exists anα >0 such that
Ax−Ay, x−y≥αAx−Ay2, ∀x, y∈C. 1.4
It is easy to see that ifA : C → E∗is anα-inverse-strongly monotone mapping, then it is 1/α-Lipschitzian.
LetEbe a real Banach space with the dualE∗. A multivalued operatorT :E → 2E∗ with domainDT {z ∈ E : Tz /∅}is called monotone ifx1−x2, y1 −y2 ≥0 for each
xi ∈ DT and yi ∈ Txi, i 1,2. A monotone operator T is called maximal if its graph
GT {x, y : y ∈ Tx} is not properly contained in the graph of any other monotone operator. A method for solving the inclusion 0∈Txis the proximal point algorithm. Denote byIthe identity operator onEHa Hilbert space. The proximal point algorithm generates, for any initial pointx0x∈H, a sequence{xn}inH, by the iterative scheme
xn1 IrnT−1xn, n0,1,2, . . . , 1.5
where{rn}is a sequence in the interval0,∞. Note that this iteration is equivalent to
0∈Txn1 1
rnxn1−xn, n0,1,2, . . . . 1.6
This algorithm was first introduced by Martinet12and generally studied by Rockafellar
15in the framework of a Hilbert space. Later many authors studied its convergence in a Hilbert space or a Banach space. See, for instance,16–21and the references therein.
LetEbe a reflexive, strictly convex, and smooth Banach space with the dualE∗andC be a nonempty closed convex subset ofE. LetT :E → 2E∗ be a maximal monotone operator
with domainDT CandS :C → Cbe a relatively nonexpansive mapping. LetA:C →
A1–A4:A1 fx, x 0, ∀x ∈ C; A2 f is monotone, that is, fx, y fy, x ≤ 0,
∀x, y ∈ C;A3limsupt↓0fxtz−x, y ≤ fx, y,∀x, y, z ∈ C;A4 the functiony →
fx, yis convex and lower semicontinuous. The purpose of this paper is to introduce and investigate two new hybrid proximal-type Algorithms1.1and1.2for finding an element of EP∩ FS∩ T−10.
Algorithm 1.1.
x0∈Carbitrarily chosen,
0vn 1
rnJxn−Jxn, vn∈Txn,
znJ−1βnJxn1−βnJSxn,
ynJ−1αnJxn 1−αnJSzn,
un∈Csuch that
fun, yAun, y−un r1 n
y−un, Jun−Jyn≥0, ∀y∈C,
Hnv∈C:φv, un≤αnφv,xn 1−αnφv, zn, v−xn, vn ≤0 ,
Wn{v∈C:v−xn, Jx0−Jxn ≤0},
xn1 ΠHn∩Wnx0, n0,1,2, . . . ,
1.7
where{rn}∞n0is a sequence in0,∞and{αn}∞n0,{βn}n∞0are sequences in0,1.
Algorithm 1.2.
x0∈Carbitrarily chosen,
0vn r1
nJxn−Jxn, vn∈Txn,
ynJ−1αnJx0 1−αnJSxn,
un∈Csuch that
fun, yAun, y−un r1 n
y−un, Jun−Jyn≥0, ∀y∈C,
Hnv∈C:φv, un≤αnφv, x0 1−αnφv,xn, v−xn, vn ≤0 ,
Wn{v∈C:v−xn, Jx0−Jxn ≤0},
xn1 ΠHn∩Wnx0, n0,1,2, . . . ,
1.8
In this paper, strong convergence results on these two hybrid proximal-type algorithms are established; that is, under appropriate conditions, the sequence{xn}generated byAlgorithm 1.1and the sequence{xn}generated byAlgorithm 1.2, converge strongly to the
same pointΠEP∩FS∩T−10x0. These new results represent the improvement, generalization and
development of the previously known ones in the literature including Solodov and Svaiter
22, Kamimura and Takahashi23, Qin and Su24, and Ceng et al.25.
Throughout this paper the symbolstands for weak convergence and → stands for strong convergence.
2. Preliminaries
Let E be a real Banach space with the dual E∗. We denote by J the normalized duality mapping fromEto 2E∗defined by
Jxx∗∈E∗:x, x∗x2x∗2, ∀x∈X, 2.1
where·,·denotes the generalized duality pairing. A Banach spaceEis called strictly convex ifxy/2 < 1 for allx, y ∈ Ewithx y 1 andx /y. It is said to be uniformly convex ifxn−yn → 0 for any two sequences{xn},{yn} ⊂Esuch thatxn yn 1 and limn→ ∞xnyn/21. LetU{x∈E:x1}be a unit sphere ofE, then the Banach
spaceEis called smooth if
lim
t→0
xty− x
t 2.2
exists for each x, y ∈ U. IfEis smooth, thenJ is single valued. We still denote the single valued duality mapping byJ.
It is also said to be uniformly smooth if the limit is attained uniformly forx, y ∈ U. Recall also that ifEis uniformly smooth, then Jis uniformly norm-to-norm continuous on bounded subsets ofE. A Banach spaceEis said to have the Kadec-Klee property if for any sequence{xn} ⊂E, wheneverxn x∈Eandxn → x, we havexn → x. It is known that
ifEis uniformly convex, thenEhas the Kadec-Klee property; see26,27for more details. LetCbe a nonempty closed convex subset of a real Hilbert spaceHandPC :H → C be the metric projection ofHonto C, thenPCis nonexpansive. This fact actually characterizes
Hilbert spaces and hence, it is not available in more general Banach spaces. Nevertheless, Alber 28 recently introduced a generalized projection operator ΠC in a Banach space E which is an analogue of the metric projection in Hilbert spaces.
Next, we assume thatEis a smooth Banach space. Consider the functional defined as in28,29by
φx, yx2−2x, Jyy2, ∀x, y∈E. 2.3
The generalized projectionΠC:E → Cis a mapping that assigns to an arbitrary point
x∈Ethe minimum point of the functionalφy, x; that is,ΠCxx, wherexis the solution to the minimization problem
φx, x min
y∈C φ
y, x. 2.4
The existence and uniqueness of the operatorΠCfollows from the properties of the functional
φx, yand strict monotonicity of the mappingJsee, e.g.,30. In a Hilbert spaceH,ΠC
PC. From28, in uniformly smooth and uniformly convex Banach spaces, we have
x −y2≤φx, y≤xy2, ∀x, y∈E. 2.5
LetCbe a nonempty closed convex subset ofE, and letSbe a mapping fromCinto itself. A pointp∈Cis called an asymptotically fixed point ofS31ifCcontains a sequence{xn}
which converges weakly topsuch thatSxn−xn → 0. The set of asymptotical fixed points of
Swill be denoted byFS. A mappingSfromCinto itself is called relatively nonexpansive
32–34ifFS FSandφp, Sx≤φp, xfor allx∈Candp∈FS.
We remark that ifEis a reflexive, strictly convex and smooth Banach space, then for anyx, y ∈E, φx, y 0 if and only ifx y. It is sufficient to show that ifφx, y 0 then
x y. From2.5, we havex y. This implies that x, Jy x2 y2. From the definition ofJ, we haveJxJy. Therefore, we havexy; see26,27for more details.
We need the following lemmas for the proof of our main results.
Lemma 2.1see23. LetEbe a smooth and uniformly convex Banach space and let{xn}and{yn} be two sequences ofE. Ifφxn, yn → 0and either{xn}or{yn}is bounded, thenxn−yn → 0.
Lemma 2.2see23,28. LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE, letx∈Eand letz∈C, then
z ΠCx⇐⇒y−z, Jx−Jz≤0, ∀y∈C. 2.6
Lemma 2.3see23,28. LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE, then
φx,ΠCyφΠCy, y≤φx, y, ∀x∈C, y∈E. 2.7
Lemma 2.4see35. LetCbe a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach spaceE, and letS:C → Cbe a relatively nonexpansive mapping, thenFSis closed and convex.
Lemma 2.5see36. LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE, letf be a bifunction fromC×CtoRsatisfying (A1)–(A4), and letr >0
andx∈E, then, there existsz∈Csuch that
fz, y1ry−z, Jz−Jx≥0, ∀y∈C. 2.8
Motivated by Combettes and Hirstoaga 37 in a Hilbert space, Takahashi and
Zembayashi38established the following lemma.
Lemma 2.6 see38. LetCbe a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach spaceE, and letfbe a bifunction fromC×CtoRsatisfying (A1)–(A4). Forr >0andx∈E, define a mappingTr :E → Cas follows:
Trx
z∈C:fz, y1
r
y−z, Jz−Jx≥0, ∀y∈C
2.9
for allx∈E, then, the following hold:
iTr is single valued;
iiTr is a firmly nonexpansive-type mapping, that is, for allx, y∈E,
Trx−Try, JTrx−JTry≤Trx−Try, Jx−Jy; 2.10
iiiFTr FTr EPf;
ivEPfis closed and convex.
UsingLemma 2.6, one has the following result.
Lemma 2.7see38. LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE, letfbe a bifunction fromC×CtoRsatisfying (A1)–(A4), and letr >0, then, forx∈Eandq∈FTr,
φq, TrxφTrx, x≤φq, x. 2.11
Utilizing Lemmas2.5,2.6and2.7as above, Chang39derived the following result.
Iforx∈E, there existsu∈Csuch that
fu, yAu, y−u1ry−u, Ju−Jx≥0, ∀y∈C; 2.12
IIifEis additionally uniformly smooth andKr :E → Cis defined as
Krx
u∈C:fu, yAu, y−u1
r
y−u, Ju−Jx≥0, ∀y∈C
, ∀x∈E, 2.13
then the mappingKrhas the following properties:
iKris single valued,
iiKris a firmly nonexpansive-type mapping, that is,
Krx−Kry, JKrx−JKry≤Krx−Kry, Jx−Jy, ∀x, y∈E, 2.14
iiiFKr FKr EP,
ivEPis a closed convex subset ofC,
vφp, Krx φKrx, x≤φp, x, for allp∈FKr.
Proof. Define a bifunctionF:C×C → Ras follows:
Fx, yfx, yAx, y−x, ∀x, y∈C. 2.15
Then it is easy to verify thatFsatisfies the conditionsA1–A4. Therefore, The conclusions
IandIIofProposition 2.8follow immediately from Lemmas2.5,2.6and2.7.
Lemma 2.9see13,14. LetE be a reflexive, strictly convex and smooth Banach space, and let
T :E → 2E∗be a maximal monotone operator withT−10/∅, then,
φz, Jrx φJrx, x≤φz, x, ∀r >0, z∈T−10, x∈E. 2.16
3. Main Results
Throughout this section, unless otherwise stated, we assume thatT :E → 2E∗is a maximal monotone operator with domainDT C,S:C → Cis a relatively nonexpansive mapping,
Algorithm 3.1.
x0∈Carbitrarily chosen,
0vn r1
nJxn−Jxn, vn∈Txn,
znJ−1βnJxn1−βnJSxn,
ynJ−1αnJxn 1−αnJSzn,
un∈Csuch that
fun, yAun, y−un r1 n
y−un, Jun−Jyn≥0, ∀y∈C,
Hnv∈C:φv, un≤αnφv,xn 1−αnφv, zn, v−xn, vn ≤0 ,
Wn{v∈C:v−xn, Jx0−Jxn ≤0},
xn1 ΠHn∩Wnx0, n0,1,2, . . . ,
3.1
where{rn}∞n0is a sequence in0,∞and{αn}∞n0,{βn}n∞0are sequences in0,1.
First we investigate the condition under which the Algorithm 3.1 is well defined. Rockafellar40proved the following result.
Lemma 3.2Rockafellar40. LetEbe a reflexive, strictly convex, and smooth Banach space and letT:E → 2E∗be a multivalued operator, then there hold the following:
iT−10is closed and convex ifTis maximal monotone such thatT−10/∅;
iiT is maximal monotone if and only ifTis monotone withRJrT E∗for allr >0.
Utilizing this result, we can show the following lemma.
Lemma 3.3. LetEbe a reflexive, strictly convex, and smooth Banach space. IfEP∩FS∩T−10/∅, then the sequence{xn}generated byAlgorithm 3.1is well defined.
Proof. For eachn≥0, define two setsCnandDnas follows:
Cn v∈C:φv, un≤αnφv,xn 1−αnφv, zn ,
Dn{v∈C:v−xn, vn ≤0}.
3.2
It is obvious thatCnis closed andDn, Wnare closed convex sets for eachn≥0. Let us show
thatCnis convex. Forv1, v2∈Cnandt∈0,1, putvtv1 1−tv2. It is sufficient to show thatv∈Cn. Indeed, observe that
is equivalent to
2αnv, Jxn21−αnv, Jzn −2v, Jun ≤αnxn2 1−αnzn2− un2. 3.4
Note that there hold the following:
φv, un v2−2v, Junun2,
φv,xn v2−2v, Jxnxn2,
φv, zn v2−2v, Jznzn2,
3.5
Thus we have
2αnv, Jxn21−αnv, Jzn −2v, Jun
2αntv1 1−tv2, Jxn21−αntv1 1−tv2, Jzn
−2tv1 1−tv2, Jun
2tαnv1, Jxn21−tαnv2, Jxn21−αntv1, Jzn
21−αn1−tv2, Jzn −2tv1, Jun −21−tv2, Jun
≤αnxn2 1−αnzn2− un2.
3.6
This implies thatv ∈ Cn. Therefore, Cn is convex and henceHn Cn ∩ Dn is closed and
convex.
On the other hand, letw ∈EP∩ FS∩ T−10 be arbitrarily chosen, thenw ∈EP, w∈
FSandw∈T−10. FromAlgorithm 3.1, it follows that
φw, un φw, Krnyn
≤φw, yn
φw, J−1α
nJxn 1−αnJSzn
w2−2w, α
nJxn 1−αnJSznαnJxn 1−αnJSzn2
≤ w2−2α
nw, Jxn −21−αnw, JSznαnxn2 1−αnSzn2
≤αnφw,xn 1−αnφw, Szn
≤αnφw,xn 1−αnφw, zn.
3.7
Sow∈Cnfor alln≥0. Now, fromLemma 3.2it follows that there existsx0, v0∈E×E∗such that 0v01/r0Jx0−Jx0andv0∈Tx0. SinceTis monotone, it follows thatx0−w, v0 ≥0, which implies thatw∈D0and hencew∈H0. Furthermore, it is clear thatw∈W0C, then
w ∈H0∩ W0, and thereforex1 ΠH0∩W0x0is well defined. Suppose thatw ∈Hn−1∩ Wn−1
andxnis well defined for somen≥1. Again byLemma 3.2, we deduce thatxn, vn∈E×E∗
conclude thatxn−w, vn ≥0, which implies thatw∈Dnand hencew∈Hn. It follows from
Lemma 2.4that
w−xn, Jx0−Jxnw−ΠHn−1∩Wn−1x0, Jx0−JΠHn−1∩Wn−1x0 ≤0, 3.8
which implies thatw∈Wn. Consequently,w∈Hn∩Wnand so EP∩FS∩T−10⊂Hn∩Wn.
Thereforexn1 ΠHn∩Wnx0 is well defined, then, by induction, the sequence{xn}generated
byAlgorithm 3.1, is well defined for each integern≥0.
Remark 3.4. From the above proof, we obtain that
EP∩ FS∩ T−10⊂Hn∩ Wn 3.9
for each integern≥0.
We are now in a position to prove the main theorem.
Theorem 3.5. LetEbe a uniformly smooth and uniformly convex Banach space. Let{rn}∞n0 be a sequence in0,∞and{αn}∞n0,{βn}∞n0be sequences in0,1such that
lim inf
n→ ∞ rn>0, lim supn→ ∞ αn<1, nlim→ ∞βn1. 3.10
Let EP∩ FS∩ T−10/∅. If S is uniformly continuous, then the sequence {xn} generated by Algorithm 3.1converges strongly toΠEP∩FS∩T−10x0.
Proof. First of all, if follows from the definition of Wn that xn ΠWnx0. Since xn1
ΠHn∩Wnx0∈Wn, we have
φxn, x0≤φxn1, x0, ∀n≥0. 3.11
Thus{φxn, x0}is nondecreasing. Also fromxn ΠWnx0andLemma 2.3, we have that
φxn, x0 φΠWnx0, x0≤φw, x0−φw, xn≤φw, x0 3.12
for eachw∈EP∩FS∩T−10⊂W
nand for eachn≥0. Consequently,{φxn, x0}is bounded. Moreover, according to the inequality
we conclude that {xn} is bounded. Thus, we have that limn→ ∞φxn, x0 exists. From
Lemma 2.3, we derive the following:
φxn1, xn φxn1,ΠWnx0
≤φxn1, x0−φΠWnx0, x0
φxn1, x0−φxn, x0,
3.14
for alln≥0. This implies thatφxn1, xn → 0. So it follows fromLemma 2.1thatxn1−xn →
0. Sincexn1 ΠHn∩Wnx0∈Hn, from the definition ofHn, we also have
φxn1, un≤αnφxn1,xn 1−αnφxn1, zn, xn1−xn, vn ≤0. 3.15
Observe that
φxn1, zn φ
xn1, J−1βnJxn1−βnJSxn
xn12−2xn1, βnJxn1−βnJSxnβnJxn1−βnJSxn2
≤ xn12−2βnxn1, Jxn −21−βnxn1, JSxnβnxn21−βnSxn2
βnφxn1,xn 1−βnφxn1, Sxn.
3.16
At the same time,
φΠHnxn, xn−φxn, xn ΠHnxn2− xn22xn−ΠHnxn, Jxn
≥2ΠHnxn−xn, Jxn2xn−ΠHnxn, Jxn
2xn−ΠHnxn, Jxn−Jxn.
3.17
SinceΠHnxn∈Hnandvn 1/rnJxn−Jxn, it follows that
xn−ΠHnxn, Jxn−Jxnrnxn−ΠHnxn, vn ≥0 3.18
and hence that φΠHnxn, xn ≥ φxn, xn. Further, from xn1 ∈ Hn, we have φxn1, xn ≥
φΠHnxn, xn, which yields
Then it follows fromφxn1, xn → 0 thatφxn, xn → 0. Hence it follows fromLemma 2.1
thatxn−xn → 0. Since from3.15we derive
φxn1,xn−φxn, xn
xn12−2xn1, Jxnxn2−
xn2−2xn, Jxnxn2
xn12− xn2−2xn1, Jxn2xn, Jxn
xn12− xn2−2xn1−xn, Jxn−Jxn
−2xn1−xn, Jxn2xn, Jxn−Jxn
xn1 − xnxn1xn 2rnxn1−xn, vn −2xn1−xn, Jxn
2xn, Jxn−Jxn
≤ xn1−xnxn1xn 2xn1−xnxn2xnJxn−Jxn
≤ xn1−xnxn1xn 2xn1−xnxn−xnxn2xnJxn−Jxn,
3.20
we have
φxn1,xn≤φxn, xn xn1−xnxn1xn
2xn1−xnxn−xnxn2xnJxn−Jxn.
3.21
Thus, fromφxn, xn → 0,xn−xn → 0, andxn1−xn → 0, we know thatφxn1,xn → 0. Consequently from3.16,φxn1,xn → 0, andβn → 1 it follows that
φxn1, zn−→0. 3.22
So it follows from3.15,φxn1,xn → 0, andφxn1, zn → 0 thatφxn1, un → 0. Utilizing
Lemma 2.1we deduce that
lim
Furthermore, foru∈EP∩FS∩ T−10 arbitrarily fixed, it follows fromProposition 2.8that
φun, ynφKrnyn, yn
≤φu, yn−φu, Krnyn
φu, J−1α
nJxn 1−αnJSzn
−φu, un
u2−2u, α
nJxn 1−αnJSznαnJxn 1−αnJSzn2−φu, un
≤ u2−2α
nu, Jxn −21−αnu, JSznαnxn2 1−αnSzn2−φu, un
αnφu,xn 1−αnφu, Szn−φu, un
≤1−αnφu, zn αnφu,xn−φu, un
1−αnφ
u, J−1β
nJxn1−βnJSxnαnφu,xn−φu, un
1−αn
u2−2u, β
nJxn1−βnJSxnβnJxn1−βnJSxn2
αnφu,xn−φu, un
≤1−αn
u2−2β
nu, Jxn −21−βnu, JSxnβnxn21−βnSxn2
αnφu,xn−φu, un
1−αnβnφu,xn 1−βnφu, Sxnαnφu,xn−φu, un
≤1−αnβnφu,xn 1−βnφu,xnαnφu,xn−φu, un
1−αnφu,xn αnφu,xn−φu, un
φu,xn−φu, un
φu,xn−φu, xn1 φu, xn1−φu, un
xn2− xn122u, Jxn1−Jxnxn12− un22u, Jun−Jxn1
≤ xn−xn1xnxn1 2uJxn1−Jxn
xn1−unxn1un 2uJun−Jxn1.
3.24
SinceJis uniformly norm-to-norm continuous on bounded subsets ofE, it follows from3.23 thatJxn1−Jxn → 0 andJun−Jxn1 → 0, which hence yieldφun, yn → 0. Utilizing
Lemma 2.1, we getun−yn → 0. Observe that
xn1−yn≤ xn1−unun−yn−→0, 3.25
due to3.23. Since J is uniformly norm-to-norm continuous on bounded subsets ofE, we have that
lim
On the other hand, we have
xn−zn ≤ xn−xn1xn1−zn −→0. 3.27
Noticing that
Jxn1−JynJxn1−αnJxn 1−αnJSzn
αnJxn1−Jxn 1−αnJxn1−JSzn
1−αnJxn1−JSzn−αnJxn−Jxn1
≥1−αnJxn1−JSzn −αnJxn−Jxn1,
3.28
we have
Jxn1−JSzn ≤ 1−1α n
Jxn1−JynαnJxn−Jxn1
. 3.29
From3.26and lim supn→ ∞αn<1, we obtain
lim
n→ ∞Jxn1−JSzn0. 3.30
SinceJ−1is also uniformly norm-to-norm continuous on bounded subsets ofE∗, we obtain
lim
n→ ∞xn1−Szn0. 3.31
Observe that
xn−Sxn ≤ xn−xn1xn1−SznSzn−Sxn. 3.32
SinceS is uniformly continuous, it follows from3.27,3.31andxn1−xn → 0 thatxn−
Sxn → 0.
Now let us show thatωw{xn}⊂EP∩FS∩ T−10, where
ωw{xn}:x∈C:xnk xfor some subsequence{nk} ⊂ {n}withnk↑ ∞ . 3.33
Indeed, since {xn} is bounded and X is reflexive, we know that ωw{xn}/∅. Take x ∈
ωw{xn}arbitrarily, then there exists a subsequence{xnk}of{xn}such thatxnk x. Hence
x∈FS. Let us show thatx∈T−10. Sincex
n−xn → 0, we have thatxnk x. Moreover, since
Jis uniformly norm-to-norm continuous on bounded subsets ofEand lim infn→ ∞rn>0, we
obtain
vn r1
It follows fromvn∈Txnand the monotonicity ofTthat
z−xn, z−vn≥0 3.35
for allz∈DTandz∈Tz. This implies that
z−x, z ≥0 3.36
for allz∈DTandz∈Tz. Thus from the maximality ofT, we infer thatx∈T−10. Therefore,
x∈FS∩ T−10. Further, let us show thatx∈EP. Sinceu
n−yn → 0 andxn−un → 0, from
xnk xwe obtain thatynk xandunk x.
SinceJ is uniformly norm-to-norm continuous on bounded subsets ofE, fromun −
yn → 0 we derive
lim
n→ ∞Jun−Jyn0. 3.37
From lim infn→ ∞rn>0, it follows that
lim
n→ ∞
Jun−Jyn
rn 0. 3.38
By the definition ofun :Krnyn, we have
Fun, yr1 n
y−un, Jun−Jyn≥0, ∀y∈C, 3.39
where
Fun, yfun, yAun, y−un. 3.40
Replacingnbynk, we have fromA2that
1
rnk
y−unk, Junk −Jynk
≥ −Funk, y
≥Fy, unk
, ∀y∈C. 3.41
Sincey→fx, y Ax, y−xis convex and lower semicontinuous, it is also weakly lower semicontinuous. Lettingnk → ∞in the last inequality, from3.38andA4we have
Fy,x≤0, ∀y∈C. 3.42
Fort, with 0< t≤1, andy∈C, letytty 1−tx. Sincey∈Candx∈C, thenyt∈Cand henceFyt,x≤0. So, fromA1we have
Dividing byt, we have
Fyt, y≥0, ∀y∈C. 3.44
Lettingt↓0, fromA3it follows that
Fx, y ≥0, ∀y∈C. 3.45
So,x∈EP. Therefore, we obtain thatωw{xn}⊂EP∩ FS∩ T−10 by the arbitrariness ofx. Next, let us show thatωw{xn} {ΠEP∩FS∩T−10x0}andxn → ΠEP∩FS∩T−10x0.
Indeed, putx ΠEP∩FS∩T−10x0. Fromxn1 ΠHn∩Wnx0andx∈EP∩ FS∩ T−10 ⊂ Hn∩Wn, we haveφxn1, x0≤φx, x0. Now from weakly lower semicontinuity of the norm, we derive for eachx∈ωw{xn}
φx, x 0 x2−2x, x 0x02
≤lim inf
k→ ∞
xnk2−2xnk, x0x02
lim inf
k→ ∞ φxnk, x0
≤lim sup
k→ ∞ φxnk, x0
≤φx, x0.
3.46
It follows from the definition ofΠEP∩FS∩T−10x0thatxxand hence
lim
k→ ∞φxnk, x0 φx, x0. 3.47
So we have limk→ ∞xnk x. Utilizing the Kadec-Klee property ofE, we conclude that
{xnk} converges strongly toΠEP∩FS∩T−10x0. Since{xnk} is an arbitrary weakly convergent
subsequence of {xn}, we know that {xn} converges strongly to ΠEP∩FS∩T−10x0. This
completes the proof.
Theorem 3.5 covers 25, Theorem 3.1 by taking C E, f ≡ 0 and A ≡ 0. Also
Theorem 3.5covers24, Theorem 2.1by takingf ≡0,A≡0 andT≡0.
Theorem 3.6. LetCbe a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach spaceE. LetT :E → 2E∗be a maximal monotone operator with domainDT C,S:C →
Cbe a relatively nonexpansive mapping,A :C → E∗be anα-inverse-strongly monotone mapping andf:C×C → Rbe a bifunction satisfying (A1)–(A4). Assume that{rn}∞n0is a sequence in0,∞
satisfyinglim infn→ ∞rn >0and that{αn}∞n0is a sequences in0,1satisfyinglimn→ ∞αn0.
Algorithm 3.7.
x0∈Carbitrarily chosen,
0vn r1
nJxn−Jxn, vn∈Txn,
ynJ−1αnJx0 1−αnJSxn,
un∈Csuch that
fun, yAun, y−un r1 n
y−un, Jun−Jyn≥0, ∀y∈C,
Hnv∈C:φv, un≤αnφv, x0 1−αnφv,xn, v−xn, vn ≤0 ,
Wn{v∈C:v−xn, Jx0−Jxn ≤0},
xn1 ΠHn∩Wnx0, n0,1,2, . . . ,
3.48
whereJis the single valued duality mapping onE. Let EP∩FS∩T−10/∅. IfSis uniformly continuous, then{xn}converges strongly toΠEP∩FS∩T−10x0.
Proof. For eachn≥0, define two setsCnandDnas follows:
Cnv∈C:φv, un≤αnφv, x0 1−αnφv,xn ,
Dn{v∈C:v−xn, vn ≤0}.
3.49
It is obvious that Cn is closed and Dn, Wn are closed convex sets for each n ≥ 0. Let us
show thatCn is convex and so Hn Cn∩ Dn is closed and convex. Similarly to the proof ofLemma 3.3, since
φv, un≤αnφv, x0 1−αnφv,xn 3.50
is equivalent to
we know thatCnis convex and so isHnCn∩ Dn. Next, let us show that EP∩FS∩T−10⊂
Cnfor eachn≥0. Indeed, utilizingProposition 2.8, we have, for eachw∈EP∩ FS∩ T−10,
φw, un φw, Krnyn
≤φw, yn
φw, J−1α
nJx0 1−αnJSxn
w2−2w, α
nJx0 1−αnJSxnαnJx0 1−αnJSxn2
≤ w2−2α
nw, Jx0 −21−αnw, JSxnαnx02 1−αnSxn2
αnφw, x0 1−αnφw, Sxn
≤αnφw, x0 1−αnφw,xn.
3.52
Sow ∈ Cnfor alln≥ 0 and EP∩ FS∩ T−10 ⊂C
n. As in the proof ofLemma 3.3, we can
obtainw∈Dnand hencew∈Hn. It follows fromLemma 2.4that
w−xn, Jx0−Jxnw−ΠHn−1∩Wn−1x0, Jx0−JΠHn−1∩Wn−1x0 ≤0, 3.53
which implies thatw∈Wn. Consequently,w∈Hn∩Wnand so EP∩FS∩T−10⊂Hn∩Wn
for alln≥0. Therefore, the sequence{xn}generated byAlgorithm 3.7is well defined. As in
the proof ofTheorem 3.5, we can obtainφxn1, xn → 0. Sincexn1 ΠHn∩Wnx0∈Hn, from
the definition ofHnwe also have
φxn1, un≤αnφxn1, x0 1−αnφxn1,xn, xn1−xn, vn ≤0. 3.54
As in the proof of Theorem 3.5, we can deduce not only from φxn1, xn → 0 that
φxn, xn → 0 but also fromφxn, xn → 0,xn−xn → 0 andxn1−xn → 0 that
lim
n→ ∞φxn1,xn 0. 3.55
Sincexn1 ΠHn∩Wnx0∈Hn, from the definition ofHn, we also have
φxn1, un≤αnφxn1, x0 1−αnφxn1,xn. 3.56
It follows from3.55andαn → 0 that
lim
n→ ∞φxn1, un 0. 3.57
UtilizingLemma 2.1we have
lim
Furthermore, foru∈EP∩FS∩ T−10 arbitrarily fixed, it follows fromProposition 2.8that
φun, ynφKrnyn, yn
≤φu, yn−φu, Krnyn
φu, J−1α
nJx0 1−αnJSxn
−φu, un
u2−2u, α
nJx0 1−αnJSxnαnJx0 1−αnJSxn2−φu, un
≤ u2−2α
nu, Jx0 −21−αnu, JSxnαnx02 1−αnSxn2−φu, un
αnφu, x0 1−αnφu, Sxn−φu, un
≤αnφu, x0 φu,xn−φu, un
αnφu, x0 φu,xn−φu, xn1 φu, xn1−φu, un
αnφu, x0 xn2− xn122u, Jxn1−Jxnxn12
− un22u, Jun−Jxn1
≤αnφu, x0 xn−xn1xnxn1 2uJxn1−Jxn
xn1−unxn1un 2uJun−Jxn1.
3.59
SinceJ is uniformly norm-to-norm continuous on bounded subsets ofE, it follows from3.58thatJxn1−Jxn → 0 andJun−Jxn1 → 0, which together withαn → 0,
yieldφun, yn → 0. UtilizingLemma 2.1, we getun−yn → 0. Observe that
xn1−yn≤ xn1−unun−yn−→0, 3.60
due to3.58. Since J is uniformly norm-to-norm continuous on bounded subsets ofE, we have
lim
n→ ∞Jxn1−Jynnlim→ ∞Jxn1−Jxnnlim→ ∞Jxn1−Jxn0. 3.61
Note that
JSxn−JynJSxn−αnJx0 1−αnJSxnαnJx0−JSxn. 3.62
Therefore, fromαn → 0 we get
lim
n→ ∞JSxn−Jyn0. 3.63
SinceJ−1is also uniformly norm-to-norm continuous on bounded subsets ofE∗, we obtain
lim
It follows that
xn−Sxn ≤ xn−xn1xn1−ynyn−SxnSxn−Sxn. 3.65
SinceSis uniformly continuous, it follows from3.58and3.64thatxn−Sxn → 0.
Finally, we prove that xn → ΠEP∩FS∩T−10x0. Indeed, for x ∈ EP∩ FS∩ T−10
arbitrarily fixed, there exists a subsequence {xnk} of {xn} such that xnk x ∈ C, then
x ∈ FS. Now let us show that x ∈ T−10. Since x
n −xn → 0, we have thatxnk x. Moreover, since J is uniformly norm-to-norm continuous on bounded subsets of E, and lim infn→ ∞rn > 0, we obtain that vn 1/rnJxn −Jxn → 0. It follows from vn ∈ Txn
and the monotonicity ofT thatz−xn, z−vn ≥0 for allz∈DTandz∈Tz. This implies thatz−x, z ≥0 for allz∈DTandz∈Tz. Thus from the maximality ofT, we infer that
x∈T−10. Further, let us show thatx∈EP. Sinceu
n−yn → 0 andxn−un → 0, fromxnk x we obtain thatynk xandunk x.
SinceJ is uniformly norm-to-norm continuous on bounded subsets ofE, fromun −
yn → 0 we derive limn→ ∞Jun−Jyn0. From lim infn→ ∞rn>0 it follows that
lim
n→ ∞
Jun−Jyn
rn 0. 3.66
By the definition ofun :Krnyn, we have
Fun, yr1 n
y−un, Jun−Jyn≥0, ∀y∈C, 3.67
whereFun, y fun, y Aun, y−un. Replacingnbynk, we have fromA2that
1
rnk
y−unk, Junk −Jynk
≥ −Funk, y
≥Fy, unk
, ∀y∈C. 3.68
Since y → fx, y Ax, y − x is convex and lower semicontinuous, it is also weakly
lower semicontinuous. Lettingnk → ∞in the last inequality, from3.66andA4we have
Fy,x ≤0, for ally∈C. Fort, with 0< t≤ 1, andy∈C, letyt ty 1−tx. Sincey ∈C andx∈C, thenyt∈Cand henceFyt,x≤0. So, fromA1we have
0Fyt, yt≤tFyt, y 1−tFyt,x≤tFyt, y. 3.69
Dividing byt, we haveFyt, y ≥ 0, for ally ∈ C. Lettingt ↓ 0, fromA3 it follows that
Fx, y ≥0, for ally∈C. So,x∈EP. Therefore, we obtain thatωw{xn}⊂EP∩ FS∩ T−10
by the arbitrariness ofx.
Indeed, putx ΠEP∩FS∩T−10x0. Fromxn1 ΠHn∩Wnx0andx∈EP∩ FS∩ T−10 ⊂ Hn∩Wn, we haveφxn1, x0≤φx, x0. Now from weakly lower semicontinuity of the norm, we derive for eachx∈ωw{xn}
φx, x 0 x2−2x, x 0x02
≤lim inf
k→ ∞
xnk2−2xnk, x0x02
lim inf
k→ ∞ φxnk, x0
≤lim sup
k→ ∞ φxnk, x0
≤φx, x0.
3.70
It follows from the definition ofΠEP∩FS∩T−10x0 thatx xand hence limk→ ∞φxnk, x0 φx, x0. So we have limk→ ∞xnk x. Utilizing the Kadec-Klee property ofE, we know thatxnk → ΠEP∩FS∩T−10x0. Since{xnk} is an arbitrary weakly convergent subsequence of {xn}, we know thatxn → ΠEP∩FS∩T−10x0. This completes the proof.
Theorem 3.6 covers 25, Theorem 3.2 by taking C E, f ≡ 0 and A ≡ 0. Also
Theorem 3.6covers24, Theorem 2.2by takingf ≡0, A≡0 andT ≡0.
Acknowledgments
This research was partially supported by the Leading Academic Discipline Project of
Shanghai Normal University no. DZL707, Innovation Program of Shanghai Municipal
Education Commission Grant no. 09ZZ133, National Science Foundation of China no. 11071169, Ph.D. Program Foundation of Ministry of Education of Chinano. 20070270004, Science and Technology Commission of Shanghai Municipality Grantno. 075105118, and Shanghai Leading Academic Discipline Project no. S30405. Al-Homidan is grateful to KFUPM for providing research facilities.
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