Viemam J. Math, (2016) 44:351-374 DOI 10.1007/S10013-015-0129-Z
Parallel Hybrid Iterative Methods for Variational Inequalities, Equilibrium Problems, and Common Fixed Point Problems
Pham Ky Anh • Dang Van Hieu
Received. 15 October 2014/Accepted: 16 November 2014/Published online 4March20l5
© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015
Abstract In this paper, we propose two strongly convergent parallel hybrid iterative meth- ods for finding a common element of the set of fixed points of a family of asymptotically quasi 0-nonexpansive mappings, the set of solutions of variational inequalities and the set of solutions of equilibrium problems in uniformly smooth and 2-uniformly convex Banach spaces. A numerical experiment is given to verify the efficiency of the proposed parallel algorithms.
Keywords Asymptotically quasi 0-nonexpansive mapping • Variational inequality • Equilibrium problem • Hybnd method • Parallel computadon
Mathematics Subject Classification (2010) 47H05 • 47H09 • 47H10 • 47J25 • 65J15 • 65Y05
I Introduction
Let C be a nonempty closed convex subset of a Banach space E. The vanalional inequality for a possibly nonlinear mapping A : C -»• £*, consists of finding p* G Csuchthat
{Ap*,p-p*]>Q Vp^C. (1) The set of solutions of (1) is denoted by Vl(A.C). Takahashi and Toyoda [19] proposed
a weakly convergent method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an a-inverse strongly monotone mapping in a Hilbert space.
Dedicated to Professor Nguyen Khoa Son's 65th birthday, R K. Anh m) D. V. Hieu
Hanoi University of Science, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
e-mail: anhpk@vnu edu vn D, V Hieu e-mail: dv,[email protected]
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352 P K, Anh. D V Hieu Theorem 1 [19] Let K be a closed convex subset of a real Hilberi space H. Let cr > 0, Let A be an a-inverse strongly monotone mapping ofK into H, and let Shea nonexpansive mapping of K into itself such that F(S) f l VliK. A) ^ 0. Let {jr„l be a sequence generated by
\x^eK,
1 Jr„+l = dnX,, + ( t - a„)SPKiXn - K^AXn)
foreveryn = 0, 1,2,.., where k„ e [a, b] for some a, b e (0,2a)anda„ e [c.d]
forsomec.d € (0,l).Then, lx„) converges weakly to z € F(S) f) VI{K.A),when Z— lifn„-tooPF(S)f\VHK.A)Xn-
In 2008, liduka and Takahashi [8] considered problem (1) in a 2-unifoniily convex, uniformly smooth Banach space under the fodowing assumptions.
(VI) A IS«-inverse-strongly monotone.
(V2) VliA.C) / 0.
(V3) \\Ay\\ < \\Ay - AM||forally € C and u e VI(A.C).
Theorem 2 [8] Let E be a 2-uniformly convex, uniformly smooth Banach space whose duality mapping J is weakly sequentially continuous, and let Cbea nonempty, closed convex subset of E. Assume that A is a mapping of C into E* satisfying conditions (VI)-(V3).
Suppose that xi — x ^ C and [xn] is given by
Xn+i = Tic J'Hjx„-/.nAXn)
for every n = 1,2,..., where lk„ ] is a sequence of positive numbers. If X„ is cliosen so thatXn e la,b]forsomea,bwithO < a < b <: -^, then the sequence [x,,] converges weakly to some element z in VI(C, A). Here, l/c is the 2-uniform convexity constant of E, andz = \im„ ^ CO f^ V HA.C)Xn-
In 2009. Zegeye and Shahzad [22] smdied the following hybrid iterative algorithm in a 2- uniformly convex and uniformly smooth Banach space for finding a common element of the set of fixed points of a weakly relatively nonexpansive mapping T and the set of solutioos of a variational inequality involving an a-inverse strongly monotone mapping A:
yn^nc(J-HJx„-^„Axn)),
Zn - Tyn.
Ho = {veC: <]>(v, zo) < <t>iv. yo) < (fiv. XQ)} . H„ = {v^ H„.i n ^n-l • <i>(V, Zn) < 'f>(V, yn) < <P(v.Xn)] , Wo = C,
)Vn^{v& Hn-l n Wn-l ' {Xn - V. Jxo - J Xn) > O) , Xn+\ = PH„C]w„xo,n > I,
where J is the normalized duality mapping on E The strong convergence of {x„] lo n F ( r ) n v/|/i,c)-»^o has been estabhshed.
Kang et al. [9] extended this algoridim to a weakly relatively nonexpansive mapping, a variational inequality and an equilibrium problem. Recently, Saewan and Kumam [14] have constmcted a sequential hybrid block iterative algoridim for an infinite family of closed and uniformly asymptotically quasi 0-nonexpansive mappings, a variational inequality for an or-mverse-strongly monotone mapping, and a system of equilibnum problems.
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Parallel Hybrid lieracive Methods for Vis, EPs, and FPPs
Qin et al. [12] considered the following sequential hybnd method for a pair of inverse strongly monotone and a quasi ^-nonexpansive mappings in a 2-uniformly convex and uniformly smooth Banach space:
.ro = E,Ci = C, xi = Uc^xo, u„ = nc(J-HJx„-q„Bx„)), Zn = Tic {j'HjUn - KAlIn)) , .V,i = TZn.
C„+i = {veCn-<i>iV,yn)<<PiV,Zn):
Xn+l ^nc„^jXo,n > 0 .
4>iV.Un)<(l>iv,Xn)], They proved the strong convergence of the sequence {x„] to TIFXQ, where F = FiT) n VliA.C) n VI(B,C)
Let/beabifunctionfrom C x C to a set of real numbers R. The equilibnum problem for/consists of finding an element JF e C s u c h t h a t
/ ( ? , y ) > 0 V V G C . (2) The set of solutions of the equilibrium problem (2) is denoted by EP(f). Equilibrium problems include several problems such as: variational inequalities, optimizadon problems, fixed point problems, etc. In recent years, equilibrium problems have been studied widely and several solution methods have been proposed (see [3, 9, 14, 15, 18]). On the other hand, for finding a common element in F ( r ) P | £ P ( / ) , Takahashi and Zembayashi [20]
introduced the following algorithm in a uniformly smooth and uniformly convex Banach space:
jroeC,
= J-^anJxn -I- (1 - a„)JTyn),
u„ e C , s.t. f(u„,y)-\-}-^(y-Un.Ju„- Jyn) > 0 Vy e C, Hn ={v€C:^iv,u„) <<l>iv,xn)),
W„ = {v€C \ {x„ - V, J.XO - Jx„) > 0}, Xn+l = /'//^nii'.^O.n > 1-
The strong convergence of the sequences {x„] and [u„} to Uf^J-,(-^£p^f'fXo has been established.
Recently, the above-mentioned algorithms have been generalized and modified for find- ing a common point of the set of solutions of variational inequalities, the set of fixed points of (asymptotically) quasi 0-nonexpansive mappings, and the set of solutions of equilibrium problems by several authors, such as Takahashi and Zembayashi [20], Wang et al. [21], and others.
Very recently, Anh and Chung [4] have considered the following parallel hybrid method for a finite family of relatively nonexpansive mappings {T,}^^^:
xo e y'n - in =
c„ =
Xn+
a.
C, J-\a„Jx„ + H argmaxi<,<jv {||j
= iveC:4,(.v.%
= {v eC : {Jxo -
= nc„ne.-'^o,«
- « , ) j r , . t , ) .
; - « . l l l .
< * ( " , * • ) ) Jx„ ,Xn — v) i O .
i = l .
••» : = y'n
i O ) .
This algorithm was extended, modified and generalized by Anh and Hieu [5] for a finite family of asymptotically quasi 0-nonexpansive mappings in Banach spaces. Note dial die proposed parallel hybrid methods m [4, 5] can be used for solving simultaneous systems
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P K Anh, D V Hieu of maximal monotone mappings. Other parallel methods for solving accretive operator equations can be found in [3].
In this paper, motivated and inspired by the above-mentioned results, we propose two novel parallel iterative methods for finding a common element of the set of fixed poinis of a family of asymptotically quasi 0-nonexpansive mappings |f"(Sj))J'^,, the set of solutions of variational inequalities (V/(A,, C)]fLi, and the set of solutions of equihbrium problems [EP(fk)]^^^ in uniformly smooth and 2-uniformly convex Banach spaces, namely;
Method A
where, TrX :
jco e C IS chosen arbitrarily,
y'„ = Uc {J~HJX„ - knA,Xn)), i = l,2,...M, in = argmax {||y^ — XnW : i = I,- . , M } , y„ = yj,", zi = 7 " ' (anJx„-\-(l -a„)JS"jyn). j = l...N, jn = argmax I ||ii - Xn\\ : j - I N\,Z„=ZJ",
"n = Tl^^Zn.k = 1 K,
fcn = argmax {||u*-.K„|| : it = 1,2. .. K] . u„=H*", C„+] = {; e C„ : 4>(z,Un) < 4>(z.Zn) < cl>iz.Xn)-i-e„], Xn + l = ric„+,Xo, W > 0,
: is a unique solution to a regulanzed equilibrium problem fiz,y)-\--{y-z,Jz-Jx)>0 Vy e C. 1, Furdier, the control parameter sequences [k„], (a„}, and (r^) satisfy the conditions
0 < a„ < 1, lim sup a„ < 1, A.„ e [a, b], rn > d. (4) for some a,b e (0,ac^/2),d > 0, where l / c is the 2-uniform convexity constant of E. Concerning the sequence {€„}, we consider two cases If the mappings [S,] are asymptotically quasi 0-nonexpansive, we assume that the solution set F is hounded, le, there exists a positive number w, such that F c Q — [u & C : \\u\\ < w] and put
€„ .— (k„ — l)(o) + ||J:M||)^. If the mappings (5,) are quasi 0-nonexapansive, then4„ = 1, and we put e„ = 0
Method B
xo € C IS chosen arbiti^arily,
y'^ = H e {J~HJX„ - \nA,X„)) , I - 1 , . . , , M, . = a r g m a x { | | y i - . x „ | | : i = 1 , , . . , M } , y n = y ; " ,
Zn = / ' • {a„.0JXn + EU "n.j JS"jy„) , (jj ' =T,%,k = l,...,K,
/:„ = argmaxj||H^-Jr„|| : * = 1, .., K] .Hn = u'„", Cn+l =lzeC„ :4>iZ.Un)<^iZ,Xn)-i-en], x„+i = Tlc„^fXo. rt > 0,
where the control parameter sequences [Xn], {oinj], and lr„} satisfy the conditions N
0 < an.] < 1, X)«H„/ ^ 1. lim infan.oa„.j > 0, A„ e [a, b], rn > d. (6)
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Parallel Hybnd Iterative Methods for Vis, EPs, and FPPs 355 In Method A (3). knowing x„ we find the intermediate approximations y'^.i = I. ....M
in parallel Using the farthest element among y^ from J:„, we compute Zn.y = 1, - - . , /V in parallel. Further, among zi, we choose the farthest element from x,, and determine solutions of regularized equilibnum problems u^,k = l , . . . , A ' i n parallel. Then the farthest from x„ element among «*, denoted by Un is chosen. Based on iin, a closed convex subset C„+i is constmcted. Finally, the next approximation Xn+i is defined as the generalized projection of xoontoCn+i.
A sunilar ideaof parallehsm is employed in Method B (5). However, the subset C„+i in Method B is simpler than that in Method A,
The results obtained in this paper extend and modify the corresponding results of Zegeye and Shahzad [22], Takahashi and Zembayashi [20], Anh and Chung [4], Anh and Hieu [5], and others.
The paper is organized as follows. In Section 2, we collect some definitions and results needed for further investigation. Section 3 deals with the convergence analysis of the methods (3) and (5). In Section 4, a novel parallel hybnd iterative method for vanalional inequalities and closed, quasi 0-nonexpansive mappings is studied. Finally, a numerical experiment is considered in Section 5 to verify the efficiency of the proposed parallel hybrid mediods.
2 Preliminaries
In this section, we recall some definitions and results which will be used later. The reader is referred to [2] for more details
Definition 1 A Banach space E is called
1) strictly convex if the unit sphere 5i (0) = [x ^ X : \\x\\ 1} is strictly convex, i.e., die inequality IIJC -|- y|| < 2holdsfor all J:, y e S|(0),x j ^ y;
2) umformly convex if for any given e > 0 there exists 5 = Sie) > 0 such diat for all jc.y e EwithlUII < 1, llyll < 1, ||J: - y | | = c die inequality ||J:-|-y|| < 2(1 - S) holds;
3) smooth if the hmit
hm (7) f^-O t
exists for all J:, y e Si(0);
4) uniformly smooth if die limit (7) exists uniformly for all jr.y e 5i(0).
The modulus of convexity of £ IS die function ^E , [0,2] -^ [0, 1] defined by SE(€) = inf j l - - i i i ^ : 11.^11 = llyll = 1- \\x 'y\\=A for all e e [0,2] Note diat E is uniformly convex if and only if 5£(e) > 0 for all 0 < e < 2and5£(0) = 0. Let p > 1, £ is said to be p-imiformly convex if there exists some constant c > 0 such that ^E(e) > « ' ' . It is well-known diat die spaces L'',;P, and W^ are p-uniformly convex if p > 2 and 2-umformly convex if I < p < 2 and a Hilbert space H is unifomdy smooth and 2-uniformly convex.
Let E be a real Banach space widi its dual £*. The dual product of / e £*and;c € E is denoted by {v. f)ov {f.x}. Forthesakeof simplicity, the norms of £ and £* are denoted
PK Anh. D.V Hieu by die same symbol |i • ||. The nonnalized duality mapping J E ^ 2^' is defined by , ,
J(x) = \f^E^:(fx) = \\xf = \\ff\.
The following properties can be found in [7]:
i) If £ IS a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping J : £ - * 2^ is single-valued, one-to-one, and onto;
ii) If £ is a reflexive and stiictly convex Banach space, dien 7 " ' is norm to weak' continuous;
iii) If £ is a uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E,
iv) A Banach space £ is uniformly smooth if and only if £ ' is uniformly convex;
v) Each uniformly convex Banach space £ has the Kadec-Klee property, i.e,, for any sequence{j:„) c £.ifj:n -^ x e £and||j:„|| -> ||.t||, dien Jn -> x.
Lemma 1 [22] IfE is a 2-uniformly convex Banach space, then
\\x-y\\<~\\Jx-Jy\\ Vx,yEE, where J is the normalized duality mapping on E and 0 < c < 1.
The best constant ~ is called the 2-uniform convexity constant of £ Next, we assume tha
£ IS a smooth, stncdy convex, and reflexive Banach space. In the sequel, we always use 0 : £ X £ - • [0, oo) to denote the Lyapunov functional defined by
<P(x, y) = l\xf - 2{x. Jy) + \\yf V.r. y e £ From the definition of 0, we have
i\\x\\ - \\y\\)^ < 4>ix,y) < (t|:r[| -i- \\y\\)^. (8) Moreover, Ihe Lyapunov functional satisfies the identity
^(x, y)=<l>(x.z) + ^(z, y) -\- 2{z - x, Jy - Jz) (9) forall.v.y.z € £.
The generalized projection r i c " £ -*• C i s defmed by Tlc(x) =argmin0(jc, y),
ysC
In what follows, we need the following properties of the functional 0 and the generahzed projection Ylc-
Lemma 2 [I] Let E be a smooth, strictly convex, and reflexive Banach space and C a nonempty closed convex subset ofE. Then the following conclusions hold.
i) •*(j:,nc(y)) + 0 ( n c ( y ) , y ) < 0(;c, y) Vjr e C. v e £ ;
11) ifx € £ . z e C.thenz = Udx) if and only if {z - y.Jx - Jz) > OVy e C;
111) (p(x,y) = 0 ifandonly ifx = y.
Lemma 3 [10] Let C be a nonempty closed convex subset of a smooth Banach E.
x.y.z € EandX e [H, l]. For a given real number a, the set T>-={veC :^(v.z)<X4>{v.x) + (l-X)(t>iv.y)+a]
is closed and convex.
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Parallel Hybrid IteraHve Methods for Vis. EPs, and FPPs
Lemma 4 [1] Let [xn) and [y„] be two sequences in a uniformly convex and uniformly smooth real Banach space E. If4>(x„, y„) -^ 0 and either {Xn} or {y„] is bounded, then
\\xn - ynW ^^ Oasn -¥• oo.
Lemma 5 [6] Let E be a uniformly convex Banach space, r be a positive number, and Br (0) C E be the closed ball with center at origin and radius r. Then, for any given subset [xi,X2,...,Xf:j}C Br(0) and for any positive numbers Xi, X2,.... X/v withY,^^^ Xi = 1, there exists a continuous, strictly increasing, and convex function g : [0, 2r) -^ [0. oo) withgiO) = 0 such that, for any i.j e {1,2 N] with i < j .
II ^ f ^
U2^kxkl <Y.Xk\\xkf -X,kjg(\\x, -xjW).
Definition 2 A mapping A : £ - * £ ' is called I) monotone, if
(A(x) - Aiy),x - y} > 0 Vx.y e E;
I) uniformly monotone, if there exists a strictly increasing function ^ [0, oo) -^
[0, 00), Vr(0) = 0, such that
(A(x) - Aiy),x - y) > ilr(\\x - y\\) Vx.yeE: (10) i) rj-strongly monotone, if there exists a positive constant rj, such diat in (10),
Ht) = nth
4) a-inverse strongly monotone, if there exists a positive constant a. such that (A(x) - A{y).x - y) > a\\A(x) - Aiy)t Vj:,y e £ ; 5) £-Lipschitz continuous if there exists a positive constant L, such that
\\A(x) - A(y)\\ < L\\x - y\\ Wx.y e £ .
If A is a-mverse strongly monotone then it is ^-Lipschitz continuous. If A is jj-strongly monotone and Z,-Lipschitz continuous then it is -A-inverse strongly monotone Lemma 6 [17] Let C be a nonempty, closed convex subset of a Banach space E and A bea monotone, hemicontinuous mapping ofC into E*. Then
VI(C,A) ^ {u € C : (V - u,A(v)) > 0 Vy e C } . Let C be a nonempty closed convex subset of a smooth, stncdy convex, and reflexive Banach space £, 7" : C - * C be a mapping. The set
F(T) = {x € C : Tx ^ x]
is called the set of fixed points of T. A point /J e C is said to be an asymptotic fixed point of T if there exists a sequence {x„] C C such that x„ -^ p and \\x„ - Tx„|| - • 0 as n -^ + oo. The set of all asymptotic fixed points of T will be denoted by FiT).
Definition 3 A mapping 7" : C -> C is called
i) relatively nonexpansive mapping if £(?") ^ ^. F{T) = £ ( r ) , and HP.TX) < HP.X) Vp e F(T). Wx e C;
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358 '.K. Anh, D.V. Hieu \-j^
ii) closed if for any sequence {jc„! C C,Xn -> x and rAr„ - * y, then Tx: = y;
iii) quasi 0-nonexpansive mapping (or hemi-relatively nonexpansive mapping) if
£(7") ^ 13 and
HP<7X) < HP.X) Vp e F(T), Vx e C.
iv) asymptotically quasi 0-nonexpansive if F(T) ?t 0 and there exists a sequence {k„] C [1,-Foo) widiftn -^ l a s « -»• -Hoosuchthat
4>(p,T''x) < k„<p(p,x) Vn > 1, Vp e £ ( 7 ) , VJC G C ; v) uniformly L-Lipschitz continuous, if there exists a constant £ > 0 such that
||r":t: - T"y|| < £||;i: - y|| Vn > i, Vx.y e C.
The reader is referred to [6, 16] for examples of closed and asymptotically quasi 0-nonexpansive mappings. It has been shown that the class of asymptotically quasi 0- nonexpansive mappings contains properly the class of quasi 0-nonexpansive mappings, and ihe class of quasi 0-nonexpansive mappings contains the class of relatively nonexpansive mappings as a proper subset.
Lemma 7 [6] Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed convex subset of E. Le! T : C -* C be a closed and asymptotically quasi ^-nonexpansive mapping with a sequence {kn\ C [1,-1-00),fcn -*• I. Then F(T) is a closed convex subset of C.
Next, for solving the equilibrium problem (2), we assume that Ihe bifuncrton/ safisfies the following conditions-
(Al) f(x,x) = Oforall;r e C;
(A2) / is monotone, i,e.,/(j:,y) -|- f(y,x) < OforallA:,y e C;
(A3) Eora\\x,y,z e C,
lim^sapf(tz + (1 - t)x,y) < fix,y);
(A4) For all J: e C , / ( J : , •) is convex and lower semicontinuous.
The following results show that in a smooth (uniformly smooth), strictly convex and reflexive Banach space, die regularized equihbrium problem has a solution (unique solution, respectively).
Lemma 8 [20] Let Cbea closed and convex subset of a smooth, strictly convex and reflex- ive Banach space E,fbea bifunction from C x C ro R satisfying conditions (A1)-(A4) and letr > Q, x e E. Then, there exists z e C such that
f(z.y)-\--{y~z.Jz~Jx)>0 VysC.
Lemma 9 [20] Let Cbea closed and convex subset of a uniformly smooth, strictly convex, and reflexive Banach spaces E.fbea bifunction from C x C ro K satisfying conditions iA\)-^A4). For all r > Qandx e E. define the mapping
TrX = {z^C : f(z,y)-ir -{y^z.Jz- Jx)>0 Vy e C|.
Then the following hold:
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Parallel Hybnd Iterative Methods for Vis, EPs, and FPPs
(BI) Tr is single-valued;
(B2) Tr is a firmly nonexpansive-type mapping, i.e., for all x,y e E, {TrX - Try, JT.x - JTry) < (TrX - Try. Jx - Jy);
(B3) F(Tr) = F(T) = EP(f):
(B4) EP(f) IS closed and convex and Tr isa relatively nonexpansive mapping.
Lemma 10 [20] Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E. Letf be a bifunction from C x C to R satisfying (A\)-iA4) and letr > 0, Then, for X e Eandq e F(Tr).
4>(q.TrX)-^4>(TrX,x)S<l>iq,x).
Let E b e a real Banach space. Alber[l] studied the function y : £ x £ * -^ Kdefinedby V(x,x^) = \\xf - 2(x,x*) + \\x''f
Clearly, V(:i,:c*) - 4>(x,J-^x*)
Lemma 11 [1] Let E be a reflexive, strictly convex and smooth Banach space with E* as its dual Then
V(x,x'') -\- 2{J-^x - x*.y*) < V(x,x* -j- y*) Vx e £ , Vx'^.y" e E*.
Consider die normal cone Nc to a set C at a point x € C defined by Nc(x) - {x* e E'' : (x - y,x*) > 0 Vy e C].
We have flie following result.
Lemma 12 [ 13] Let Cbe a nonempty closed convex subset of a Banach space E and let A be a monotone and hemi-continuous mapping of C into E* with D(A) — C. Let Q be a mapping defined by:
n( i - \Ax + Nc(x) ifxeC.
^^^^ ^ I 0 ifx^C Then Q is a maximal monotone and Q 0 — VliA.C).
3 Convergence Analysis
Throughout this section, we assume that C is a nonempty closed convex subset of a real umformly smooth and 2-umformly convex Banach space £, Denote
F = (n vKAi.c)^ n (n "(^A n (n '''-(M)
and assume that the set F is nonempty
We prove convergence theorems for methods (3) and (5) widi the control parame- ter sequences satisfying conditions (4) and (6), respectively. We also propose similar parallel hybrid methods for quasi 0-nonexpansive mappmgs, vanalional inequalities and equilibrium problems,
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Theorem 3 Let {A,),^, be a finite family of mappings from C to E" satisfying condi- tions (\l)-(^3). Let {fi]^^^ C y. C ^ R be a finite family of bifiinctions satisfying conditions (A1)-(A4). Let {Sj]'J^j : C ^ C be a finite family of uniform L- Lipschitz continuous and asymptotically quasi-<f)-nonexpansive mappings with the same sequence {k^} C [1, -|-oo), k„ —> 1. Assume that there exists a positive number cu such that
£ C f i := {w e C : ||H|| < co). If the control parameter sequences [an], {Xn\. and {r„}
satisfy condition (4), then the sequence {x„) generated by (3) converges strongly to Ufxt).
Proof We divide the proof of Theorem 3 Into seven steps
Step 1 Claim that F, C„ are closed convex subsets of C, Indeed, since each mapping S, is uniformly £-Lipschitz continuous, it is closed. By Lemmas 6, 7, and 9, £(5;), VI(Aj,C) and £ £ ( / * ) are closed convex sets, therefore, H^iiEi^j))^ Cl^^i "^'iA,. C) and r\k=i EP(fk) are also closed and convex Hence, £ i s a closed and convex subsetof C. h is obvious that C„ is closed for all « > 0, We prove the convexity of C„ by induction Clearly, Co := C is closed convex. Assume that C„ is closed convex for some n > 0.
From the construction of C„+i, we find
C„ + | = Cnf]{Z e E • 4>(Z.U„) < 0(Z.Z„) < Hz.Xn) -1- C„|.
Lemma 3 ensures that Cn+\ is convex. Thus, C„ is closed convex for all n > 0. Hence, flpx^andxn+i :— nc„+|j:o are well-defined.
Step 2 Claim that £ C Cn for all n > 0. By Lemma 10 and the relative nonexpan- siveness of 7)-^, we obtain
0(H,5n) = <p(u,Tr„Zn) < 4>(u.Zn) for all u € F.
From the convexity of || • ||^ and die asymptotical quasi 0-nonexpansiveness of Sj, we find 0 ( « , ; „ ) = <l>{u,J-^(a„Jxn-\-(l-an)JS';jn^'^
= \\uf - 2a„ {u. Xn) - 2(1 - an)tu, JS''jJn^ + \\a„Jx„ -i- (1 - a„)JSly„f
< \ii'f-2a„{u,Xn)-2(\-an)(^u,JS';j„)-\-an\\xnf-\-(\-a„)\\S';j,f
= Ot„4>{u.Xn)-i-ii-an)<i>iu,S'lyn)
< 0!n<t>iu,Xn)-\-(l-a„)k„(t>(u,y„) (11) forallM e £-By the hypotheses of Theorem 3, Lemmas 1,2, and 11, and H e £,wehave
0(H,y„) = <p(u,Uc(^J-\jXn-XnA,„Xn)'\)
< <l>(u.J'\jXn-A.nAi„X„))
•= y(u.JXn-XnA,„Xn)
< V(U, JXn - X„A,„X„ + XnA,„Xn) -2{j-^ {j Xn - X„A,„X„) ~ U, X„A,„Xn)
= 0(«.^„) - 2Xn ( y - ' (Jx„ - XnA.„X„) - / - ' (Jxn) , A,„Xn) -2Xn {x„ - u, A,„Jt„ - A,^(u)) - 2Xn {xn - U. A,„u)
^ Springer
1
pM^lel H^bpd'Iterative Methods for Vis, EPs, and FPPs
^ ^ < H".Xn) + -^l\Jxn-/.„A,„Xn - /;i:„|11|A,,;r„|| - 2A„a;||A,„:r„ - A,„M|1^
4X^
< <l>(u.x„) + ^ | | A , „ J : „ | | ^ - 2X„a||A,„jc„ - A,„«||^
< ip(u.Xn) -2a{a-—\ ||A,-„x„ - A,-„M||^
<(l>(u,Xn). (1 From (11), (12) and die estimate (8), we obtain
')(u,x„) + ikn-l)4>(u,x„)-2a(\-a„){a~^] || A,„ J:„ - A,„Mf -A.„uf
< <l>iu,xn)-^en. (13) Therefore £ C C„ for all« > 0.
Step 3 Claim that the sequence {x„]. {y'„], [zi] and {«*} converge strongly to p € C.
By Lemma 2 and-tn = flc^Ji^O'we have
^(xn.xo) < 0(M,J:O) - 4'(u,x„) < <t>iu,xo)
for all H € £ . Hence, {0(jr„, xo)} is bounded. By (8), [x„] is bounded, and so are the sequences {y„j, {ii„),and(zn),Bytheconstructionof Cn,J:n-n = Hcn+i^o e C„+i C Cn.
From Lemma 2 and A:,, — nc„J:o7Wcget
(piXn.Xo) < (t>ix„+i,Xo) - (piXn+UXn) < 4>(Xn+i, XQ).
Therefore, the sequence {^(xn.xo)] is nondecreasing, hence, it has a finite hmit. Note diat for all m > n.x,,, & Cm C Cn, and by Lemma 2 we obtain
HXm.Xn) < HXm.xo) - (f>iXn,Xo) -* 0 (14) as m,« - * CO. From (14) and Lemma 4, we have ||:>:„ - x^W -» 0. This shows that
{x„} c C is a Cauchy sequence. Smce £ is complete and C is a closed convex subset of £, fjTn} converges strongly t o p £ C. From(14), 0(.Vn+i,.*:n) -*• Oas« -* oo. Taking into account that ,:(„+1 e Cn+i,wefind
(I>(X„+1,U„) <(i>ix„+l,Zn) <<P(Xn+i.X„)-\-€„. (15) Since (;K„} is bounded, we can put M - sup{||jc„|| : n = 0, 1,2,,..), hence
t„ : = ( J t „ - l ) ( w - l - | | ; r „ | | ) - < ( i „ - l)(w + M ) 2 ^ 0 . (16) By(15),(16)and0(;c„+[,j:„) -* 0,wefinddiat
lim 0(j:„+i,u„)=^lim^0(A:„-(-i,3„) = ^Ihn^0(;c„+i,j:„) = 0. (17) Therefore, from Lemma 4,
lim l|;c„+i-iinll = lim ||jrn+i-z„|l = Irni ||j:„+i-;cn|| = 0
^ Springer
362 P K' Anh, D V Hieu This together with tl-icn-i-i - JTnll - * 0 implies that
lim ||j:n—iinll— hm ||jrn — z„|| = 0, By the definitions of y„ and k„, we obtain
lim | | . r „ - H ^ | | = lim | | j t „ - z ; ; | | = 0 (18) forall 1 < k < K a n d l < j < Af. Hence,
lim .ic„ — lim u* = lim zi = p (19) for all 1 < k < A^ and 1 .< j < N By the hypotheses of Theorem 3, Lemmas 1, 2,
and 11, we also have
HXn, yn) = 0 (xn, Uc (j~UjXn " A„A,„;r„)))
< lp(x„,J-\jXn-XnA,„Xn))
= V{x„,JXn -X„Ai„Xn)
< V(Xn,JXn - XnA,„Xn+XnA,„Xn) -2(^J-^ {jXn-A„A,,,Xn)-X„,X„A,„X„\
- -2Xn(j-^ {jXn - X„A,^Xn) - J-' JX„. A,„Xn\
4^2
< -^U,„xn - Ai„uf (20) forallH e Q l i ^ ^ ( ^ . C), From(13), weobtain
2(1 - a „ ) a (a - — j ||A;„;r„ - A,„«|l^ < ( 0 ( H , ;r„) - 0(M, Zn)) + €„
= 2{u.JZn - JXn) + (||^„||^ - \\Zn\\^)+€n
<nu\\\\JZn-JXn\\ + \\Xn-Zn\\i\\x„\\ + \\-Z„\\) + €n- (21) Using die fact that ||J:„ - Z„ || ^ 0 and J is uniformly continuous on each bounded set, we can conclude diat II/j„ - Jx„\\ - * 0 as n ^ cxj. This, together with (21), and the relations lim sup„_,^ an < l a n d t ^ -> 0 imply Uiat
lim ||A,„jr„ - A,,M|| = 0 , (22) From (20) and (22), we obtain
hm 0(j:n,yn) = 0.
Therefore,lim„^^l|j:„-3ij| = O.Bydiedefinitionof (n, weget lim„^co lk« - yill = 0 Hence,
lim y; - p (23) for all 1 < / < M.
fi Springer
Parallel Hybnd Iterauve Methods for Vis, EPs, and FPPs
Step4 Clahnthatp e n " = i FiSj) The relation z;! ^ y-'(o,i-/-rn + (1 - ff„)/Sj"y„) implies diat/z;; = a„Jxn + (I - a„) 7 S^y„. Therefore,
\\Jx„ - JziW = (1 - a „ ) | | y ; r „ - JS"^yn\\. (24) Since ||jCn — z^ || - » 0 and J is uniformly continuous on each bounded subset of £,
Pxn - JziW - > O a s n - > oo. This, togedier with (24) and lim supn^^oTn < I implies that
hm \\Jx„ - JS^ynW = 0.
Therefore
lim ]\x„ - S'JynW = U. (25) Since limn^oo Ikn — ynll = 0, lim„_,oo ||yn — S"yn|| = 0, hence
lim 5"y„ = p. (26) Further,
+ ll»+i-j,ll + l|v„-s;y.|l
< (L + 1) ly,+i -M + 1^;+'j»+i - j»+i II + II j . - s; j . II -* 0.
therefore
lim^ SJ+'j„ = ^lirn^ Sj ( s j y , j = p. (27) From (26), (27) and the closedness of 5 j , we obtain p E F ( 5 j ) for all I < j < N.
Hence, p € n J L l ' ' ( • ' ; ) •
5fep 5 Claim that p e C]fLi^HAi.^'' Lemma 12 ensures that the mapping
„ , , \A,x + NcM ,( xeC,
2'W = {0 nxtc
is maximal monotone, where JVc(J:) is the normal cone to Cat j : e C. For all (j:,j')inthe graph of (2M i-e, (^, y) 6 G(!2,), we have y - A,(x) e Wc(jt). By the definition of Nc(x), we find that
{j: - Z.J - A,U)) > 0 for all z e C Since y^ e C,
(» - yi-)" - 1.W) i "•
Therefore
{x-y'„.y)>{x-yi,.A,ix)). (28) Taking into account y; = UcU'^Jx,, - ^„A,j:„)) andLemma2, weget
(x-y'„.Jyi-Jx„+>-.A,x„)>0. (29) Therefore, from (28), (29) and the monotonicity of A,-, we find that
i'-y'„.y) > (;(->',',. A,(JT))
= i«- j i . ' t . U ) - A,(y'„)} + ('- y',- A,(y',) - A,-(jr,)) + {x-y'„. A,(jr„)) / , Jx„- Jy' \
ix-y',.A,l.y'„)-A,fxM + {x-y'„. r ^ ^ 1 . (30)
^ Sprmger
364 P K Anh. D. V Hieu Since ||j:n—yj,|| —* O a n d / i s uniform continuous on each bounded set, \\Jx„ — Jy'nW —* 0.
By An > a > 0, we obtain
JXn -J y'n
^hrn^ —^—— = 0. (31) Since A, is a-inverse strongly monotone. A, is ^-Lipschitz continuous. This, together widi
lljTn — yi II -> 0 implies that
^lira^||A,(y;,)-A,(;c„)||=0. (32) From (30), (31), (32), and y^ - * p , we obtain (JT - p,y) > 0 for all (jr,y) e G(Q,).
Therefore,p € G^'O = V/(A/. C) foraU 1 < / < M.Hence,p € n , l i VHA,,C).
S'rep (5 Claim that p e Hf^i £'^(/*)-Sincelimn^(x, ||u*-Znll = O a n d / i s uniformly continuous on every bounded subset of £, we have
^^lim b w * - y i n | - 0 . This together with rn > d > 0 implies dial
^lim_ 1__=_ 'J. = 0. (33) Wehaveu* = r / z „ , a n d
A ( « J . y ) + — ( ? - " ; . , ' » , ; - . / z . ) 2 0 VyeC. (34) Frr)m (34) and condition (A2). we get
-(y-»;,ju;-jz„)i-/,(«j.).)> A(j,»;) vyec. (35)
Letung « -* 00, by (33). (35) and (A4). we obtain
A ( y , p ) i O Vyec. (36) Putting y, = ry + (1 - r)p, where 0 < r < 1 and y e C. we get y, e C. Hence, for
sufficiently small t, from (A3) and (36). we have
fkiyi.p) = A d y + (1 - r)p. p) < 0.
By the properties (Al) and (A4). we find
0 = A(y,. y,) = A(y,. ly + d - DP) < iA(y,. y) + d - <)A(y.. p) < >A(y.. y).
Dividing both sides ofthe last mequahty by r > 0. we obtain A (yi, y) > Oforally € C.
i.e..
A C y + (1 - O p . y ) > 0 Vy e C.
Passing ( ^ 0+, from (A3) we get A ( p . y) > 0 Vy e C and 1 < t < AT, i.e..
p e nLi t^nft).
Step 7 Claim that the sequence {x„] converges strongly to nf.ro.
Indeed, since»' .= Urixo) e f c C „ . i „ = He, do) from Lemma 2. we have
4'(x„.xo'l <<Hx'',xo)-<l,t.x\x,-i ^'HxKxo). (37) a Spring!,
Parallel Hybnd Iterative Methods for Vis, EPs, and FPPs Therefore *
^ ^ V - 4>(X\XQ) > ^liin^0(j:n,xo)=^lim^{||j:„|p-2(x„,yxo} + ||j:o|j-j
^ ^ = l l / ' f - 2 ( p , / j r o ) + ||jrol|^
= 0 ( P , J : O ) -
From the definition ofa:^ it follows that p = jr'^.Theproof of Theorem3iscomplete. D Remark I Assume that [ A , ] ^ , is a fimte family of ))-strongiy monotone and L- Lipschitz continuous mappings. Then each A, is ^-inverse strongly monotone and VIiA,,C) = A-'O.Hence, ||A;jc|) < ||A,-^ - A,«|| forall x e Candw € VliA,.C).
Thus, all the conditions (V1)-(V3) for the variational inequalities VI(A,. C) hold.
Theorem 4 Let {At ]fLi be a finite family of mappings from C to £* satisfying conditions (V1)-(V3). £er {/jtlf^] : C x C -»• R be a finite family of bifunctions satisfying con- ditions (A1)-(A4). Let {5j)J'^j : C -^ C be a finite family of uniform L-Lipschitz contin- uous and asymptotically quasi-cp-nonexpansive mappings with the same sequence {kn] C {1,-l-oo), An —• I. Assume that F is a subset of Q, and suppose that the con- trol parameter sequences {«„!, [Xn], [r„] satisfy condition (6). TTien, the sequence {x„]
d by method (5) converges strongly to TIFXQ.
Proof Arguing similarly as in Step 1 of the proof of Theorem 3, we conclude that £ , C„
are closed convex for all n > 0. Now, we show that F C Cn for all n > 0, For all u e £ , by Lemma 5 and the convexity of || • |p we obtain
0(M,S„) - 0 U , J ' - ' Un.oJXn + '^an.lJSfynjj
N 11 N 11^
= IIHII^ - 2a„,o iu.xn) - 2 ^ f f n , ; («, Sfy„) + a^.o^^n + J ] a n . / y 5 ; y n N
< liu 11^ - 2a„.o(«, Xn)-2J2"".' ("• •^"5^") + <^''.o\\x„ f
N . -\-J2<^n.l\\S"ynf -Cln.oan.jg []\-fx., - JS]yn\\)
(=1 N
< a„.o0(K,JCn) + Xl""''*^*"- Si'y»)-ein.Qan,jg {\\Jx„ - JS'jynWj 1=1
N
< a„,o0(M, jr„) -H ^ a „ , / i „ 0 ( M , %)-a„,oan.jg [\\Jxn - JSpnWj •
From (12), we get
<l>(u,yn)<<Pi'i.Xn)-2a(a- — \\\A,^Xn-A,„u\\-
^ Spnnger
P K. Anh, D. V. Hieu Using (38), (39), and the estimate (8), we find
0(H,Hn) = 0 ( H , T . S n ) 5 0(«,z«)
: 0(w,jr„)-l-^a„./(/:n - D'Piu.Xn)-a„,Qa„,jg (\\Jx„ - JS'JynW) -2j^a,ja (a - - j - j ||A,„.r„ - A^^uf 2b\
: (f>(u, x„) -I- ikn - 1) (ro + Iknll)^ - an,oa„jg {\\JXn - JS^ynW) -2"S\ctn.\a {a. T" 1 ll'^rn-ffi — - ^ ' , " 1 1 "
0(M.«n) <0(«.-«n) + e„
for all u e £, This implies that £ C Cnforalln > 0. Using (40) and arguing similarly as in Steps 3, 5, and 6 of Theorem 3, we obtain
lira Un = lim «„ = lim yn = lira y'^ = lira x„ = p € C, andp € (C]LiEP(fi,))f)(f]fi^VI(A„C)).
Next, we show diat p e HjLi £(-5y)-Indeed, from (40), we have
a«,0««,jS (ll/^n - -^S;y„||) < (0(H,x„) - 0 ( « , «„)) -!-«„ (41) Since lljTn - Un\\ -> O , | 0 ( H , J : „ ) - 0(M,Mn)| -i- O a s « - ^ CO. This, together with (41) and the facts that e„ -* Oand lim infn^ooOn.oan.y > Oimplythat
^ l i m ^ ^ ( | | 7 x „ - y s ; y „ [ | ) = 0 . By Lemma 5, we get
lim ||/jc„ - JS'jynW = 0 .
Since J IS uniformly continuous on each bounded subsetof E, we conclude that lim \\x„ - 5 " y n | | = 0
Using the last equality and a similar argument for proving relations (25), (26), (27), and acting as in Step 7 of the proof of Theorem 3, we obtain p s n f = i FiSj) and
p = jr'f = n/pj:o. The proof of Theorem 4 is complete, D Next, we consider two parallel hybrid methods for solving variational inequalities,
equihbrium problems and quasi 0-nonexpansive mappings, when the boundedness of die solution set F and the uniform Lipschitz continuity of 5, are not required.
Theorem 5 Assume that {A,]fi^, {fk)^^^, {an], [tn], and {k„] satisfy all conditions of Theorem 3 and {S^}^^, Is a finite family of closed and quasi <}>-nonexpansive mappings. In
S Springer
Parallel Hybnd Iterative Methods for Vis, EPs. and FPPs
addition, suppose that the solution set F is nonempty. For an initial point XQ fc C, define the sequence [xn] as follows:
y'n ^ Tic {J'HjXn - XnA,Xn)) . 1 - 1 . 2 , . .M.
in = argmax{llyj ~x„\\ : i = 1.2,..., M} , y„ = y^", zi = J-^{a„Jxn + (l ~an)JSjyn), j = l.2,...,N,
;„ = a r g m a x j | [ 2 ; ; - j ; „ [ j : j = 1,2, . , i v j , z „ = E ^ " , ^^2) ui^TJ^^Zn. k=\,2 K,
i „ - a r g m a x { | l M * - ^ „ | | . A = l , 2 , . . . , A : } , «„ = M^", C„+i = {z e C„ : 0 ( r , u„) < 0(z, E„) < 0(z, x„)], J:„+I = nc„+,J:o. « > 0.
Then the sequence {xn] converges strongly to TIFXQ.
Proof Smce S, is a closed and quasi ^-nonexpansive mapping, it is closed and asymptoti- cally qu^i 0-nonexpansive mapping with kn = 1 for all /i > 0. Hence, £„ = 0 by definition.
Arguing similarly as in the proof of Theorem 3, we come to the desired conclusion. D Theorem 6 Assume that [A,\^^^, [fk]k=i' \''n\, (Cn.;). and [X„] satisfy all conditions of Theorem 4 and [Sj |J^[ is a finite family of closed and quasi <p-nanexpanslve mappings In addition, suppose tlmt the solution set F is nonempty For an initial approximation XQ e C, let the sequence {Xn ] be defined by
y'n = nc{J~'^(JXn-XnA^Xn)), i = \,2,....M, in — argmax JHyJ, — j:n|| : < = I, 2 M\. y„ =y'n'', Zn = J " ' {an.oJXn -\- E^^l^^.jJSjVn) .
u^„ = T,lzn. k=l,2, ,K, (43) kn — argmax(llw* — x„\\ : k •= 1,2,..., K] , u„=u^,
Cn+l = {zeCn •.^(Z,Un)<<PiZ,Xn)],
{ x„+i = nc„+i^o. n > 0.
Then the sequence {x„| converges strongly to UFXO.
Proof The proof is similar to that of Theorem 4 for S, being a closed and quasi 0-
asymptotically nonexpansive mapping with k„ = 1 for all « > 0 •
4 A Parallel Iterative Method for Quasi 0-Nonexpansive Mappings and Variational InequaliHes
In 2004, using Mann's iteration, Matsushita and Takahashi [11] proposed die following scheme for finding a fixed point of a relatively nonexpansive mappmg T:
ncJ~^(anJxn -I- (1 - a„)JTx„). (44)
where jro e C is given. They proved that if the interior of £(7") is nonempty then die sequence {Xn ] generated by (44) converges strongly to some point in FiT). Recently, using Halpem's and Ishikawa's iterative processes, Zhang et al. [23] have proposed modified iterative algoridims of (44) for a relatively nonexpansive mapping.
^ Springer
p. K Anh, D V. Hieu In this section, employing the ideas of Matsushita and Takahashi [II] and Anh and Chung [4], we propose a parallel hybrid iterative algoridim for finite farmhes of closed and quasi 0-nonexpansive mappings {Sj)'J^^ and variational inequahties {VIiA,,C)]^y
XQ e C is chosen arbitrarily,
y'„ = Tlc{J'\JXn-XnA,Xn)). i = 1.2 M, i„ = argmax {||y^ - J : „ | | : ( = 1,2 M], %=/„".
zi = J-^anJx„ + il-an)JSjy„), j = 1.2. . ,N, in = argmax jljz;; -Xnll : ; = 1, 2 N\.in=zi\
Xn + \ =Tlcin. n>0.
(45)
where {on) c [0,1] such that lim„_,coari = 0,
Remark 2 One can employ method (45) for a finite farmly of relatively nonexpansive mappings without assuming their closedness.
Remark 3 Method (45) modifies the corresponding method (44) in the following aspects:
A relatively nonexpansive mapping T is replaced with a finite family of quasi 0-nonexpansive mappings, where the restriction F(Sj) = F(Sj) is not requked, A parallel hybrid method for finite families of closed and quasi 0-nonexpansive map- pings and variational inequalities is considered instead of an iterative method for a relatively nonexpansive mapping,
Theorem 7 Let Ebea real uniformly smooth and 2-uniformly convex Banach space with dual space E* and C be a nonempty closed convex subset of E. Assume that {Ai]fLi is a finite family of mappings satisfying conditions (V1)-(V3), ( 5 / } ^ , is a finite family of closed and quasi (p-nonexpansive mappings, and [an] C [0, 1] satisfies limn_K«ofii = 0, Xn € [a, b] for some a.b € (0, ac^/2). In addition, suppose that the interior of F - ( D ^ i VI (A,,C))r\(r\'!^iE(Sj)) is nonempty. Then, the sequence {Xn] generated by (45) converges strongly to some point u € F Moreover, u = limn-*(XJ ri/^,'i:n.
Proof By Lemma 7, the subset £ is closed and convex, hence the generalized projections Up, T\c are well-defined. We now show that the sequence [xn] is bounded. Indeed, for every u e F. from Lemma 2 and the convexity of || • ||^, we have
(l>(u,X„+i) ^(l>iu,ncZn)<4>iti,Zn) = \\uf-2(u,J-Z„) + l\-Znf
= \\uf-2a„{u.Jxn}-2(l-an){u.JSj„y„)-\-\\a„Jx„-\-il-a„)JSjJ„f
< \\uf - 2a„(M, Jx„) - 2(1 - a„){u. JSjJn)
= an4>(u, Xn) + (1 - (^„)0(«, Sjjn)
< a„<i>(u,Xn)-\-i\ -an)(l>(u,yn).
Arguing similarly to (12) and (13), we obtam
0(M,jr„+i) < 0 ( H , J : „ ) -2<i(l - a „ ) (a - ~ \ ||A,„jc„ - A,„H||^ < <i>iu,x„). (46)
® Springer
paraiiel Hybnd IteraLve Methods for Vis, EPs, and FPPs 369 Therefore, the sequence {^(u,x„)} is decreasing. Hence, diere exists a finite limit of
{(piu.Xn)]. This, togedier witii (8) and (46) rniply that die sequences |jCni is bounded and
Um |lA,„:r„-A,„«|| = 0 . (47) Next, we show that {jtn} converges strongly to some element umC Since the interior of F
is nonempty, there exist p ^ F and r > 0 such that p -[• rh e t
for all/i G Eand||ft|| < I. Since the sequence {0(M,J:„)} is decreasing for all u e £ , we have
0 ( p -H rh,x„+i) < 0 ( p + rh.x„). (48) From (9), we find that
(l>(u,Xn) = ^iu,Xn+l) + lf>(x„ + l,X„) + 2{Xn + l - U, J Xn - JXn+l) for all u e £. Therefore
<l>(p + rh,Xn) = <p(p + rh,Xn+\) + 4>iXn+\.Xn)
+ 2 (Xn + l - (p + rh). JXn - J Xn + l) - (49) From (48), (49), we obtain
'l>(Xn+\.X„)-\-2iXn+i -(p + rh),JXn-Jx„+l) > 0.
This inequality is equivalent to
(h,JXn-Jx„+:) < :-{<l>(Xr^+uXn) + 2{Xn+l - p. JXn - JXn+l)] • (50) From (9), we also have
4>(p,Xn) = <p(p.Xn+]) + (l>iX„+\.Xn) + 2(Xn+l ~ p. JXn ~ JXn + l) . (51) From (50), (51), we obtain
I - J Xn+l) <
forall ||AII < 1. Hence,
sup (h,Jx„ - JXn + l) < :^{4>iP'Xn) - <l>(p,X„+l)].
The last relation is equivalent to
\\Jx„ - Jxn+l\\ < —{(pip.Xn) - 0(p,j:n-H))- Therefore, for all/I, m e Naxidn > m, we have
WJXn - JXn, II = WJXn ' Jx„-l + Jx„-l - Jx„^2 + ' " ' + JXn,+l " Jx„, \\
n-l
< ^ | i y x , + i - / j : , l l
< - L y (0(p,jr,-)-*(p,J:,-i-i))
^ _(^(p,j(„)-0(p,J:n)).
2r
^ Springer
P K. Anh, D. V, Hieu Letting OT,H —* oo, we obtain
lim \\Jx„ - Jx„\\ = 0 .
Since £ is a uniformly convex and uniformly smooth Banach space, / " ' is uniformly continuous on every bounded subset ofE. From the last relation we have
lim \\x„ - jc„ II = 0.
Therefore, [Xn] is a Cauchy sequence. Since £ is complete and C is closed and convex, {x„]
converges strongly to some element u in C By arguing similarly to (20), we obtain 46^
<t>iXn,y„)<'^\\A,^Xn-A.„uf.
Thisrelationtogether with (47)implies that0(x„,yn) - * 0.Therefore, ||ji:„ - ynll -^ 0, By the definition of in, we conclude that ||j:n — y^|| -*• Oforalll < i < A/. Hence,
Hm y'n=u eC (52) for all 1 < i < M Fromj:„+i = HcJn and Lemma 2, we have
4>iSjJ„.X„+i) + <l>iXn+l.Z„)^<l3(Sj„yn.ncin)+'l>(^cin,Zn)<(t>iSjJn.Zn). (53) Using the convexity of || • ||^ we have
<l>(SjJn.Zn) = \\SjJ„f-2{SjJn.JZn)+\\Znf
= \\SjJ„ f - 2«„ {SjJn, JX„) - 2(1 - an){SjJn. JSjJn)
•\-\\anJXn + (l-an)JSjJn\\-
< \\Sj„yn f - IxXn [SjJn. JXn) " 2(1 - a„) {SjJn. JSjJn) +an\\Xnf + il~an)\\SjJ„f
= OnHSjJn.Xn) + (1 - a„)^(Sj„y„, SjJn)
= an(f>(SjJn.X.,).
The last inequality together widi (53) imphes diat
<t>iXn + UZn) <an<piSjJ„,X„).
Therefore,fromtiieboundednessof {0(5j„y„,j:„)| andlimn_^oo«„ = 0, we get lim (pix„+i,Zn) = 0 .
Hence, |U„+| - z„|| -^ 0. Smce ||;c„ - Xn+i\\ -^ 0,wefind||jrn - Znli -*• 0,andby diedefinitionof Jn, weobtain ||jc„ - ;;i|| -^ Oforalll < j < W. Thus,
lim zi = p. (54) Arguing similarly to Steps 4,5 in die proof of Theorem 3, we obtain
pe t = ({yi(A,,C)'^^\X\nsA.
The proof of Theorem 7 is complete D fi Springer
PuruHel Hybnd Iterauve Methods for Vis. EPs, and FPPs 5 A Numerical Example
Let £ = M ' be a Hilbert space with the standard inner product (x, y) := xy and the norm IIJII := \x\fota\]x,y e £ . Let C ;= [0,1] C £ , The normalized dual mapping 7 = / and the Lyapunov functional 0 ( J : , y) = \x - y p . It is well known that, the modulus of convexity of Hilbert space E is 5£(e) = 1 - v ' r ^ - ^ V 4 > ^e^.
Therefore, £ is 2-uniformly convex. Moreover, the best constant ^ satisfying relations
\x - y\ < ^\Jx - Jy\ ^\x - y| with 0 < c < 1 is 1, This implies dial c = l.Define the mappings A, (J:) := X — ^ , x € C,i = 1 , . . , , M, and consider the variational inequahties
{A,(p*).p - p*) > 0 Vp G C,
fori 1 , . . , M, Clearly, V/(A,, C) = {0},i 1 , . , . , A/, Since each mapping i^iix) '•= j+i is nonexpansive, the mapping A,- - I — U,,i = I , . ., M, is ~- inverse stirongly monotone. Besides, |A, (y)| = |A,(j) - A,(0)|, hence all the assumptions (V1)-(V3) for the variational inequalities are satisfied.
Further, let {t, ]^_^ and {s, ]^^.^ be two sequences of positive numbers, such that 0 < (! < ••• < tN < l a n d j , G ( L ^ r r ] ' ' = 1- •-, A', Define die mappings Si : C -^ C.i = I Af, by putting S, (A-) ='0forA: e [0,t,],andS,(x) = s,ix-t,) i(x s [It. I].
his easy to verify diat £(5,) = {O},0(5,(;c), 0) = |5,(^)1^ < \x\^ ^ 0 (J: , 0) for every X e C, and |5,(1) — 5,(/,)| = s,{l — n) > |1 — r,|. Hence, the mappings 5, are quasi
^-nonexpansive but not nonexpansive,
Finally,letO < ^i < • • < ?K < 1 and i?* G (0,1^),A = 1 , . . , K^ be two given sequences Consider AT bifunctions/i(A:,y) := Bi!(x)(y — x),k — I,. ., AT, where Bk(x) = l^jfifO < ;r < ^k.and Btix) = r/kif^k < :«: < 1.
It is easy to verify that all the assumptions (Ai)-(A4) for die bifunctions fkix, y) are fulfilled. Besides, £ £ ( / * ) ^ {0}. Thus, the solution set
(n"w..c))n(n''<5^))n(n^
eorem 6, the iteration sequence {x„] gei According to Theorem 6, the iteration sequence {x„ ] generated by
argmax jy^ — Xn\ • i — 1,2, a„Xn + il-a„)Sjy„, ; = 1 argmax jlziJ - - x „ | : y ^ I, 2 , . r / i n , ; t - l , 2 , . . . , A : , arg max l\Un - x„\ ik—1,2,.
. I\
,2.
C„+i = {zeC„ :(t>iz.iin)<<Piz.Zn)S<l>(z,x„)].
Xn+l = nc„+i^o, " 2 : 0
fi Springer
Table 1 Experimeot with
" " ^ log(loE(«+IO))
P K, Anh. D. V, Hieu
strongly converges to x' : = y'n = (I - Xn)Xn - X„^^fff-,i - 1, is a solution of the following inequality
0. A sQ-aightforward calculation yields . . . Af. Further, the element H := u* = 7"*^
(y-u)[Bk(u)+u-Zn]>0 V y e t O , I]. (55) From (55), we find that w — 0 if and only if z„ = 0, Therefore, if Zn = 0, then ihe algorithm stops and J:''' = O.lfzn ?^ 0 then inequaUty (55) is equivalent to the system of inequalities
l-u(Bk(u) + u-
\a-u)iBkiu)- Zn) > 0, U-Zn) 2 0.
The last system yields Bkiu) -\- u = Zn- Hence, if 0 < Zn S H* :— M = ^ ^ ^ Z n . Otherwise, if ^i + 1* < 2n < UthenK* :— u ••
Using the fact that £ = {0} c C„+] we can conclude diat 0 < Wn <
From the definition of C„+i we find
t + fi dien Zn - m-
Cn + l
c^n[o^H^]
I closed convex subset, hence ,Xn, it implies that According to Step I of the proof of Theorem 3, C„ is i
[0.x„] c Cn because 0,ATn e Cn- Further, i [ 0 , 5 ^ 4 ^ ] C [0,;tn] C C„.Thus,C„+| = [0,^^].
For the sake of comparison between the computing times in the parallel and sequential modes, we choose sufficiently large numbers A', K. and M and a slowly convergent to zero sequence {«„).
The numerical experiment is performed on a LINUX cluster 1350 with eight computing nodes. Each node contains two Intel Xeon dual core 3,2 GHz and 2 GB Ram. All the pro- grams are written in C. For given tolerances we compare the execution time of algoridim (55) in both parallel and sequential modes. We denote by TOL-the tolerance \\xk - x*\\:
Tp— the execution time in parallel mode using two CPUs (in seconds), and T",—the execu- tion time in sequential mode (in seconds). The computing times in both modes are given in Tables 1 and 2
According to Tables 1 and 2, in the most favorable cases, the speed-up and the efficiency of die parallel algonthm are 5p = T^/Tp ss 2; £ p = Sp/2 ^ 1, respectively.
l^ble 2 Experiment with
fi Springer
faraUel Hybnd Iterative Methods for Vis. EPs, and FPPs
6 C o n c l u s i o n s
In this p a p e r , w e p r o p o s e d t w o s t r o n g l y c o n v e r g e n t p a r a l l e l h y b r i d i t e r a t i v e m e t h o d s for finding a c o m m o n e l e m e n t o f d i e set o f fixed p o i n t s o f q u a s i 0 - a s y m p t o t i c a l l y n o n e x p a n s i v e m a p p i n g s , t h e s e t o f s o l u t i o n s o f v a r i a t i o n a l i n e q u a l i t i e s , a n d d i e set o f s o l u t i o n s o f e q u i l i b - rium p r o b l e m s in u n i f o r m l y s m o o t h a n d 2 - u n i f o r m l y c o n v e x B a n a c h s p a c e s . A n u m e r i c a l e x a m p l e w a s g i v e n t o d e m o n s t r a t e t h e e f f i c i e n c y o f t h e p r o p o s e d p a r a l l e l a l g o r i d i m s
Acknowledgments The research of the first author was partially supported by Vietnam Insdtuie for Advanced Smdy in Malhemahcs (VIASM) and Vietnam National Foundation for Science and Technology Development (NAFOSTED).
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