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Strong convergence theorems for infinite family of relatively quasi nonexpansive mappings and systems of equilibrium problems
Applied Mathematics and Computation 218 (2012) 5146–5156
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Strong convergence theorems for infinite family of relatively quasi nonexpansive mappings and systems of equilibrium problems Yekini Shehu Mathematics Institute, African University of Science and Technology, Abuja, Nigeria
a r t i c l e
i n f o
a b s t r a c t
Keywords: Relatively quasi nonexpansive mappings Equilibrium problems Hybrid method Generalized f-projection operator Banach spaces
In this paper, we construct a new iterative scheme by hybrid method for approximation of common element of set of common fixed points of countably infinite family of relatively quasi-nonexpansive mappings and set of common solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. Then, we prove strong convergence of the scheme to a common element of the two sets. Furthermore, we apply our results to solve convex minimization problem. Our results extend important recent results. 2011 Elsevier Inc. All rights reserved.
1. Introduction Let E be a real Banach space and C be nonempty closed convex subset of E. A mapping T : C ? C is called nonexpansive if
kTx Tyk 6 kx yk;
8x; y 2 C:
ð1:1Þ
A point x 2 C is called a fixed point of T if Tx = x. The set of fixed points of T is defined as F(T) :¼ {x 2 C : Tx = x}. A mapping T : C ? C is called quasi-nonexpansive if
kTx x k 6 kx x k;
8x 2 C; x 2 FðTÞ:
It is clear that every nonexpansive mapping with nonempty set of fixed points is quasi-nonexpansive. Let {Tn} be a sequence of mappings from C into E. For a subset B of C, we say that (i) ({Tn}, B) satisfies condition AKTT (see, for example [3,24]) if 1 X
supfkT nþ1 x T n xk : x 2 Bg < 1;
n¼1
(ii) ({Tn}, B) satisfies condition ⁄AKTT (see, for example [24]) if 1 X
supfkJT nþ1 x JT n xk : x 2 Bg < 1:
n¼1
Recently, Nilsrakoo and Saejung [24] proved strong convergence theorems for approximation of common fixed point of countably infinite family of relatively quasi-nonexpansive mappings in a uniformly smooth real Banach space which is also uniformly convex. In particular, they proved the following theorem. E-mail address: [email protected] 0096-3003/$ - see front matter 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.001
Y. Shehu / Applied Mathematics and Computation 218 (2012) 5146–5156
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Theorem 1.1. Let E be a real uniformly convex and uniformly smooth Banach space and let C be a nonempty closed convex subset T1 of E. Let fT n g1 n¼1 be a sequence of relatively quasi-nonexpansive mappings of C into itself such that F :¼ n¼1 FðT n Þ – ; and let 1 fxn gn¼0 be a sequence in C defined as follows:
8 x0 2 KC 1 ¼ Q 1 ¼ K; > > > > 1 > > < yn ¼ J ðan Jxn þ ð1 an ÞJT n xn Þ; n P 1; C n ¼ fw 2 C n1 \ Q n1 : /ðyn ; wÞ 6 /ðxn ; wÞg; n P 1; > > > > Q n ¼ fw 2 C n1 \ Q n1 : hxn w; Jx0 Jxn i P 0g; n P 1; > > : xnþ1 ¼ PC n \Q n x0 ; n P 0;
ð1:2Þ
where fan g1 n¼1 is a sequence in [0, 1) with lim supn?1an < 1. Suppose that for each bounded subset B of C, the ordered pair ({Tn}, B) satisfies either condition AKTT or condition ⁄ AKTT. Let T be the mapping from C into E defined by Tx :¼ limn?1Tnx for all x 2 C and T 1 suppose that T is closed and FðTÞ ¼ 1 n¼0 FðT n Þ. Then, fxn gn¼0 converges strongly to PF(T)x0. Let F be a bifunction of C C into R. The equilibrium problem (see, for example [12,21,23,29,32,33,38,39]) is to find x⁄ 2 C such that
Fðx ; yÞ P 0
ð1:3Þ
for all y 2 C. We shall denote the solutions set of (1.3) by EP(F). Numerous problems in Physics, optimization and economics reduce to find a solution of problem (1.3). Some methods have been proposed to solve the equilibrium problem, see for example [4,10,12,16,17,20,27]. In [22], Matsushita and Takahashi introduced a hybrid iterative scheme for approximation of fixed points of relatively nonexpansive mapping in a uniformly convex real Banach space which is also uniformly smooth: x0 2 C,
8 yn ¼ J 1 ðan Jxn þ ð1 an ÞJTxn Þ; > > > < H ¼ fw 2 C : /ðw; y Þ 6 /ðw; x Þg; n n n > W ¼ fw 2 C : hx w; Jx Jx i; > n n 0 n > : xnþ1 ¼ PHn \W n x0 ; n P 0: They proved that fxn g1 n¼0 converges strongly to PF(T)x0, where F(T) – ;. In [28], Plubtieng and Ungchittrakool introduced the following hybrid projection algorithm for a pair of relatively nonexpansive mappings: x0 2 C,
8 > zn ¼ J 1 bð1Þ Jxn þ bð2Þ JTxn þ bnð3Þ JSxn > n n > > > > > < yn ¼ J 1 ðan Jx0 þ ð1 an ÞJzn Þ
C n ¼ fz 2 C : /ðz; yn Þ 6 /ðz; xn Þ þ an ðkx0 k2 þ 2hw; Jxn Jx0 iÞg > > > > > > Q n ¼ fz 2 C : hxn z; Jxn Jx0 i 6 0g > : xnþ1 ¼ PC n \Q n x0 ;
ð1:4Þ
n o n o n o ð2Þ ð3Þ ; bð2Þ and bð3Þ are sequences in (0, 1) satisfying bð1Þ where fan g; bð1Þ n n n n þ bn þ bn ¼ 1 and T and S are relatively nonexpansive mappings and J is the single-valued duality mapping on E. They proved under the appropriate conditions on the parameters that the sequence {xn} generated by (1.4) converges strongly to a common fixed point of T and S. In [37], Takahashi and Zembayashi introduced the following hybrid iterative scheme for approximation of fixed point of relatively nonexpansive mapping which is also a solution to an equilibrium problem in a uniformly convex real Banach space which is also uniformly smooth: x0 2 C; C 1 ¼ C; x1 ¼ PC 1 x0 ;
8 > yn ¼ J 1 ðan Jxn þ ð1 an ÞJTxn Þ; > > > < Fðu ; yÞ þ 1 hy u ; Ju Jy i P 0; 8y 2 C n n n n rn > C nþ1 ¼ fw 2 C n : /ðw; un Þ 6 /ðw; xn Þg; > > > : xnþ1 ¼ PC nþ1 x0 ; n P 1;
where J is the duality mapping on E. Then, they proved that fxn g1 n¼0 converges strongly to PFx0, where F = EP(F) \ F(T) – ;. Recently, Li et al. [19] introduced the following hybrid iterative scheme for approximation of fixed points of a relatively nonexpansive mapping using the properties of generalized f-projection operator in a uniformly smooth real Banach space which is also uniformly convex: x0 2 C,
8 1 > > < yn ¼ J ðan Jxn þ ð1 an ÞJTxn Þ; C nþ1 ¼ fw 2 C n : Gðw; Jyn Þ 6 Gðw; Jxn Þg; > > : xnþ1 ¼ Pf x0 ; n P 1; C nþ1
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They proved a strong convergence theorem for finding an element in the fixed points set of T. We remark here that the results of Li et al. [19] extended and improved on the results of Matsushita and Takahashi [22]. Motivated by the above mentioned results and the on-going research, it is our purpose in this paper to prove a strong convergence theorem for countable family of relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. Furthermore, we apply our results to solve convex minimization problem. Our results extend the results of Li et al. [19] and Takahashi and Zembayashi [37] and many other recent known results in the literature. 2. Preliminaries Let E be a real Banach space. The modulus of smoothness of E is the function qE : [0, 1) ? [0, 1) defined by
qE ðsÞ :¼ sup
1 ðkx þ yk þ kx ykÞ 1 : kxk 6 1; kyk 6 s : 2
E is uniformly smooth if and only if
lim s!0
qE ðsÞ ¼ 0: s
Let dimE P 2. The modulus of convexity of E is the function dE : (0, 2] ? [0, 1] defined by
x þ y n o dE ðÞ :¼ inf 1 : kxk ¼ kyk ¼ 1; ¼ kx yk : 2 E is uniformly convex if for any 2 (0, 2], there exists a d = d() > 0 such that if x, y 2 E with kxk 6 1, kyk 6 1 and kx yk P , then 12 ðx þ yÞ 6 1 d. Equivalently, E is uniformly convex if and only if dE() > 0 for all 2 (0, 2]. A normed space E is called strictly convex if for all x, y 2 E, x – y, kxk = kyk = 1, we have kkx + (1 k)yk < 1, "k 2 (0, 1). Let E⁄ be the dual space of E. We denote by J the normalized duality mapping from E to 2E defined by
JðxÞ ¼ ff 2 E : hx; f i ¼ kxk2 ¼ kf k2 g: The following properties of J are well known (the reader can consult [9,34,35] for more details): (1) (2) (3) (4)
If E is uniformly smooth, then J is norm-to-norm uniformly continuous on each bounded subset of E. J(x) – ;, x 2 E. If E is reflexive, then J is a mapping from E onto E⁄. If E is smooth, then J is single valued.
Throughout this paper, we denote by /, the functional on E E defined by
/ðx; yÞ :¼ kxk2 2hx; JðyÞi þ kyk2 ;
8x; y 2 E:
ð2:1Þ
Let C be a nonempty subset of E and let T be a mapping from C into E. A point p 2 C is said to be an asymptotic fixed point of T if C contains a sequence fxn g1 n¼0 which converges weakly to p and limn?1kxn Txnk = 0. The set of asymptotic fixed points of T is denoted by e F ðTÞ. We say that a mapping T is relatively nonexpansive (see, for example [5–7,22]) if the following conditions are satisfied: (R1) F(T) – ;; (R2) /(p, Tx) 6 /(p, x), "x 2 C, p 2 F(T); (R3) FðTÞ ¼ e F ðTÞ. If T satisfies (R1) and (R2), then T is said to be relatively quasi-nonexpansive. It is easy to see that the class of relatively quasi-nonexpansive mappings contains the class of relatively nonexpansive mappings. Many authors have studied the methods of approximating the fixed points of relatively quasi-nonexpansive mappings (see, for example [8,14,24–26,30,31,40,43] the references contained therein). Clearly, in Hilbert space H, relatively quasi-nonexpansive mappings and quasi-nonexpansive mappings are the same, for /(x, y) = kx yk2, "x, y 2 H and this implies that
/ðp; TxÞ 6 /ðp; xÞ () kTx pk 6 kx pk;
8x 2 C; p 2 FðTÞ:
The examples of relatively quasi-nonexpansive mappings are given in [30]. Let E be a smooth, strictly convex and reflexive real Banach space and let C be a nonempty closed convex subset of E. Following Alber [2], the generalized projection PC from E onto C is defined by
PC ðxÞ :¼ argmin /ðy; xÞ; y2C
8x 2 E:
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The existence and uniqueness of PC follows from the property of the functional /(x, y) and strict monotonicity of the mapping J (see, for example [1,2,11,18,35]). If E is a Hilbert space, then PC is the metric projection of H onto C. From [18], in uniformly convex and uniformly smooth Banach spaces, we have
ðkxk kykÞ2 6 /ðx; yÞ 6 ðkxk þ kykÞ2 ;
8x; y 2 E:
ð2:2Þ
Next, we recall the concept of generalized f-projector operator, together with its properties. Let G : C E ! R [ fþ1g be a functional defined as follows:
Gðn; uÞ ¼ knk2 2hn; ui þ kuk2 þ 2qf ðnÞ;
ð2:3Þ
⁄
where n 2 C, u 2 E , q is a positive number and f : C ! R [ fþ1g is proper, convex and lower semi-continuous. From the definitions of G and f, it is easy to see the following properties: (i) G(n, u) is convex and continuous with respect to u when n is fixed; (ii) G(n, u) is convex and lower semi-continuous with respect to n when u is fixed. Definition 2.1 [41]. Let E be a real Banach space with its dual E⁄. Let C be a nonempty, closed and convex subset of E. We say that PfC : E ! 2C is a generalized f-projection operator if
PfC u ¼ fu 2 C : Gðu; uÞ ¼ inf Gðn; uÞg;
8u 2 E :
n2C
For the generalized f-projection operator, Wu and Huang [41] proved the following theorem basic properties: Lemma 2.2 [41]. Let E be a real reflexive Banach space with its dual E⁄. Let C be a nonempty, closed and convex subset of E. Then the following statements hold: (i) (ii)
PfC is a nonempty closed convex subset of C for all u 2 E⁄. If E is smooth, then for all u 2 E ; x 2 PfC if and only if hx y; u Jxi þ qf ðyÞ qf ðxÞ P 0;
(iii)
8y 2 C:
If E is strictly convex and f : C ! R [ fþ1g is positive homogeneous (i.e., f(tx) = tf(x) for all t > 0 such that tx 2 C where x 2 C), then PfC is a single valued mapping.
Fan et al. [15] showed that the condition f is positive homogeneous which appeared in Lemma 2.2 can be removed. Lemma 2.3 [15]. Let E be a real reflexive Banach space with its dual E⁄ and C a nonempty, closed and convex subset of E. Then if E is strictly convex, then PfC is a single valued mapping. Recall that J is a single valued mapping when E is a smooth Banach space. There exists a unique element u 2 E⁄ such that u = Jx for each x 2 E. This substitution in (2.3) gives
Gðn; JxÞ ¼ knk2 2hn; Jxi þ kxk2 þ 2qf ðnÞ:
ð2:4Þ
Now, we consider the second generalized f-projection operator in a Banach space. Definition 2.4. Let E be a real Banach space and C a nonempty, closed and convex subset of E. We say that PfC : E ! 2C is a generalized f-projection operator if
PfC x ¼ fu 2 C : Gðu; JxÞ ¼ inf Gðn; JxÞg; n2C
8x 2 E:
Obviously, the definition of relatively quasi-nonexpansive mapping T is equivalent to (R0 1)F(T) – ;; (R0 2)G(p, JTx) 6 G(p, Jx), "x 2 C, p 2 F(T). Lemma 2.5 [13]. Let E be a Banach space and f : E ! R [ fþ1g be a lower semi-continuous convex functional. Then there exists x⁄ 2 E⁄ and a 2 R such that
f ðxÞ P hx; x i þ a;
8x 2 E:
We know that the following lemmas hold for operator PfC . Lemma 2.6 [19]. Let C be a nonempty, closed and convex subset of a smooth and reflexive Banach space E. Then the following statements hold:
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(i) PfC x is a nonempty closed and convex subset of C for all x 2 E; (ii) for all x 2 E; ^ x 2 PfC x if and only if
h^x y; Jx J^xi þ qf ðyÞ qf ðxÞ P 0; (iii) if E is strictly convex, then P
f Cx
8y 2 C;
is a single valued mapping.
Lemma 2.7 [19]. Let C be a nonempty, closed and convex subset of a smooth and reflexive Banach space E. Let x 2 E and ^ x 2 PfC . Then
/ðy; ^xÞ þ Gð^x; JxÞ 6 Gðy; JxÞ;
8y 2 C:
The fixed points set F(T) of a relatively quasi-nonexpansive mapping is closed convex as given in the following lemma. Lemma 2.8 ([30,24]). Let C be a nonempty closed convex subset of a smooth, uniformly convex Banach space E. Let T be a closed relatively quasi- nonexpansive mapping of C into itself. Then F(T) is closed and convex. Also, this following lemma will be used in the sequel. Lemma 2.9 [18]. Let C be a nonempty closed convex subset of a smooth, uniformly convex Banach space E. Let fxn g1 n¼1 and 1 1 fyn g1 n¼1 be sequences in E such that either fxn gn¼1 or fyn gn¼1 is bounded. If limn?1/(xn, yn) = 0, then limn?1kxn ynk = 0. Lemma 2.10 [42]. Let E be a uniformly convex real Banach space. For arbitrary r > 0, let Br(0) :¼ {x 2 E : kxk 6 r} and k 2 [0, 1]. Then, there exists a continuous strictly increasing convex function
g : ½0; 2r ! R;
gð0Þ ¼ 0
such that for every x, y 2 Br(0), the following inequality holds:
kkx þ ð1 kÞyk2 6 kkxk2 þ ð1 kÞkyk2 kð1 kÞgðkx ykÞ: Lemma 2.11 ([24,3]). Let C be a nonempty closed subset of a Banach space E and let fT n g1 n¼1 be a sequence of mappings of K into E. Let B be a subset of C with ({Tn}, B) satisfying AKTT. Then there exists a mapping T : B ? E such that Tx :¼ limn?1Tnx, for all x 2 B. and limn?1sup{kTnx Txk : x 2 B} = 0. Lemma 2.12 [24]. Let E be reflexive and strictly convex Banach space whose norm is Fréchet differentiable, let C be a nonempty ⁄ closed subset of E which and let fT n g1 n¼1 be a sequence of mappings of K into E. Let B be a subset of C with ({Tn}, B) satisfying AKTT. Then there exists a mapping T : B ? E such that Tx :¼ limn?1Tnx, for all x 2 B and limn?1sup{kJTnx JTxk : x 2 B} = 0. For solving the equilibrium problem for a bifunction F : C C ! R, let us assume that F satisfies the following conditions: (A1) (A2) (A3) (A4)
F(x, x) = 0 for all x 2 C; F is monotone, i.e., F(x, y) + F(y, x) 6 0 for all x, y, 2C; for each x, y 2 C, limt?0F(tz + (1 t)x, y) 6 F(x, y); for each x 2 C, y ´ F(x, y) is convex and lower semicontinuous.
Lemma 2.13 [4]. Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E and let F be a bifunction of C C into R satisfying (A1)–(A4). Let r > 0 and x 2 E. Then, there exists z 2 C such that
1 Fðz; yÞ þ hy z; Jz Jxi P 0 for all y 2 K: r Lemma 2.14 [36]. Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E. Assume that F : C C ! R satisfies (A1)–(A4). For r > 0 and x 2 E, define a mapping T Fr : E ! C as follows:
T Fr ðxÞ ¼
1 z 2 C : Fðz; yÞ þ hy z; Jz Jxi P 0; 8y 2 C r
for all z 2 E. Then, the following hold: 1. T Fr is single-valued; 2. T Fr is firmly nonexpansive-type mapping, i.e., for any x, y 2 E,
D E D E T Fr x T Fr y; JT Fr x JT Fr y 6 T Fr x T Fr y; Jx Jy ;
3. FðT Fr Þ ¼ EPðFÞ; 4. EP(F) is closed and convex.
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Lemma 2.15 [36]. Let C be a nonempty closed convex subset of a smooth, strictly and reflexive Banach space E. Assume convex that F : C C ! R satisfies (A1)–(A4) and let r > 0. Then for each x 2 E and q 2 F T Fr ,
/ q; T Fr x þ / T Fr x; x 6 /ðq; xÞ:
For the rest of this paper, the sequence fxn g1 n¼1 converges strongly to p shall be denoted by xn ? p, n ? 1. Observe that an operator T in a Banach space E is said to be closed if xn ? x and Txn ? y, then Tx = y. 3. Main results We now state and prove the following theorem. Theorem 3.1. Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty closed convex 1 subset of E. For each k = 1, 2, . . . , m, let Fk be a bifunction from C C satisfying (A1)–(A4) 1 m and let fT n gn¼1 be an infinite family of relatively-quasi nonexpansive mappings of C into itself such that F :¼ \n¼1 FðT n Þ \ \k¼1 EPðF k Þ – ;. Let f : E ! R be a convex f and lower semicontinuous mapping with C int(D(f)) and suppose fxn g1 n¼0 is iteratively generated by x0 2 C; C 1 ¼ C; x1 ¼ PC 1 x0 ;
8 yn ¼ J 1 ðan Jxn þ ð1 an ÞJT n xn Þ; > > > > < un ¼ T F m T F m1 . . . T F 2 T F 1 yn r m;n r m1;n r 2;n r1;n
n P 1;
> C nþ1 ¼ fw 2 C n : Gðw; Jun Þ 6 Gðw; Jxn Þg; > > > : xnþ1 ¼ PfC nþ1 x0 ; n P 1;
ð3:1Þ
n P 1;
where J is the duality mapping on E. Suppose fan g1 n¼1 is a sequence in (0, 1) such that lim infn?1an(1 an) > 0 and fr k;n g1 n¼1 ð0; 1Þ; ðk ¼ 1; 2; . . . ; mÞ satisfying lim infn?1rk,n > 0, (k = 1, 2, . . . , m). Suppose that for each bounded subset B of C, the ordered pair ({Tn} , B) satisfies either condition AKTT or condition ⁄AKTT. Let T be the mapping from C into E defined by 1 f Tx :¼ limn?1Tnx for all x 2 C and suppose that T is closed and FðTÞ ¼ \1 n¼1 FðT n Þ. Then, fxn gn¼0 converges strongly to PF x0 . Proof. We first show that Cn, "n P 1 is closed and convex. It is obvious that C1 = C is closed and convex. Suppose that Cn is closed convex for some n > 1. From the definition of Cn+1, we have that z 2 Cn+1 implies G(z, Jun) 6 G(z, Jxn). This is equivalent to
2ðhz; Jxn i hz; Jun iÞ 6 kxn k2 kun k2 : This implies that Cn+1 is closed convex for the same n > 1. Hence, Cn is closed and convex "n P 1. This shows that PC nþ1 x0 is well defined for all n P 0. 2 1 k k1 By taking hkn ¼ T Frk;n T Frk1;n . . . T Fr2;n T Fr1;n ; k ¼ 1; 2; . . . ; m and h0n ¼ I for all n P 1, we obtain un ¼ hm n yn . k We next show that F Cn, "n P 1. From Lemma 2.14, one has that T Frk;n ; k ¼ 1; 2; . . . ; m is relatively nonexpansive mapping. For n = 1, we have F C = C1. Now, assume that F Cn1 for some n P 2. Then for each x⁄ 2 F, we obtain
Gðx ; Jun Þ ¼ G x ; Jhm n yn 6 Gðx ; Jyn Þ ¼ Gðx ; ðan Jxn þ ð1 an ÞJT n xn ÞÞ
¼ kx k2 2an hx ; Jxn i 2ð1 an Þhx ; JT n xn i þ kan Jxn þ ð1 an ÞJT n xn k2 þ 2qf ðx Þ 6 kx k2 2an hx ; Jxn i 2ð1 an Þhx ; JT n xn i þ an kJxn k2 þ ð1 an ÞkJT n xn k2 þ 2qf ðx Þ ¼ an Gðx ; Jxn Þ þ ð1 an ÞGðx ; JT n xn Þ 6 Gðx ; Jxn Þ:
ð3:2Þ
fxn g1 n¼0
⁄
So, x 2 Cn. This implies that F Cn, "n P 1 and the sequence generated by (3.1) is well defined. We now show that limn?1G(xn, Jx0) exists. Since f : E ! R is a convex and lower semi-continuous, applying Lemma 2.5, we see that there exists u⁄ 2 E⁄ and a 2 R such that
f ðyÞ P hy; u i þ a;
8y 2 E:
It follows that
Gðxn ; Jx0 Þ ¼ kxn k2 2hxn ; Jx0 i þ kx0 k2 þ 2qf ðxn Þ P kxn k2 2hxn ; Jx0 i þ kx0 k2 þ 2qhxn ; u i þ 2qa ¼ kxn k2 2hxn ; Jx0 qu i þ kx0 k2 þ 2qa P kxn k2 2kxn kkJx0 qu k þ kx0 k2 þ 2qa ¼ ðkxn k kJx0 qu kÞ2 þ kx0 k2 kJx0 qu k2 þ 2qa: f C n x0 ,
Since xn ¼ P
it follows from (3.3) that
Gðx ; Jx0 Þ P Gðxn ; Jx0 Þ P ðkxn k kJx0 qu kÞ2 þ kx0 k2 kJx0 qu k2 þ 2qa
ð3:3Þ
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1 for each x⁄ 2 F(T). This implies that fxn g1 n¼1 is bounded and so is fGðxn ; Jx0 Þgn¼0 . By the construction of Cn, we have that Cm Cn and xm ¼ PfC m x0 2 C n for any positive integer m P n. It then follows Lemma 2.7 that
/ðxm ; xn Þ þ Gðxn ; Jx0 Þ 6 Gðxm ; Jx0 Þ:
ð3:4Þ
It is obvious that
/ðxm ; xn Þ P ðkxm k kxn kÞ2 P 0: In particular,
/ðxnþ1 ; xn Þ þ Gðxn ; Jx0 Þ 6 Gðxnþ1 ; Jx0 Þ and
/ðxnþ1 ; xn Þ P ðkxnþ1 k kxn kÞ2 P 0; 1 and so fGðxn ; Jx0 Þg1 n¼0 is nondecreasing. It follows that the limit of fGðxn ; Jx0 Þgn¼0 exists. f By the fact that Cm Cn and xm ¼ PC m x0 2 C n for any positive integer m P n, we obtain
/ðxm ; un Þ 6 /ðxm ; xn Þ: Now, (3.4) implies that
/ðxm ; un Þ 6 /ðxm ; xn Þ 6 Gðxm ; Jx0 Þ Gðxn ; Jx0 Þ:
ð3:5Þ
Taking the limit as m, n ? 1 in (3.5), we obtain
lim /ðxm ; xn Þ ¼ 0:
n!1
It then follows from Lemma 2.9 that kxm xnk ? 0 as m, n ? 1. Hence, fxn g1 n¼0 is Cauchy. Since E is a Banach space and C is closed and convex, then there exists p 2 C such that xn ? p as n ? 1. Now since /(xm, xn) ? 0 as m, n ? 1 we have in particular that /(xn+1, xn) ? 0 as n ? 1 and this further implies that limn?1kxn+1 xnk = 0. Since xnþ1 ¼ PfC nþ1 x0 2 C nþ1 ; we have
/ðxnþ1 ; un Þ 6 /ðxnþ1 ; xn Þ;
8n P 0:
Then, we obtain
lim /ðxnþ1 ; un Þ ¼ 0:
n!1
Since E is uniformly convex and smooth, we have from Lemma 2.9 that
lim kxnþ1 xn k ¼ 0 ¼ lim kxnþ1 un k:
n!1
n!1
So,
kxn un k 6 kxnþ1 xn k þ kxnþ1 un k: Hence,
lim kxn un k ¼ 0:
n!1
ð3:6Þ
Since J is uniformly norm-to-norm continuous on bounded sets and limn?1kxn unk = 0, we obtain
lim kJxn Jun k ¼ 0:
n!1
ð3:7Þ
Let r :¼ supnP1{kxnk, kTnxnk}. Since E is uniformly smooth, we know that E⁄ is uniformly convex. Then from Lemma 2.10, we have Gðx ; Jun Þ ¼ Gðx ; Jhm n yn Þ 6 Gðx ; Jyn Þ ¼ Gðx ; ðan Jxn þ ð1 an ÞJT n xn ÞÞ
¼ kx k2 2an hx ; Jxn i 2ð1 an Þhx ; JT n xn i þ kan Jxn þ ð1 an ÞJT n xn k2 þ 2qf ðx Þ 6 kx k2 2an hx ; Jxn i 2ð1 an Þhx ; JT n xn i þ an kJxn k2 þ ð1 an ÞkJT n xn k2 an ð1 an ÞgðkJxn JT n xn kÞ þ 2qf ðx Þ ¼ an Gðx ; Jxn Þ þ ð1 an ÞGðx ; JT n xn Þ an ð1 an ÞgðkJxn JT n xn kÞ 6 Gðx ; Jxn Þ an ð1 an ÞgðkJxn JT n xn kÞ: It then follows that
an ð1 an ÞgðkJxn JT n xn kÞ 6 Gðx ; Jxn Þ Gðx ; Jun Þ:
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But
Gðx ; Jxn Þ Gðx ; Jun Þ ¼ kxn k2 kun k2 2hx ; Jxn Jun i 6 jkxn k2 kun k2 j þ 2jhx ; Jxn Jun ij 6 jkxn k kun kjðkxn k þ kun kÞ þ 2kx kkJxn Jun k 6 kxn un kðkxn k þ kun kÞ þ 2kx kkJxn Jun k: From (3.6) and (3.7), we obtain
Gðx ; Jxn Þ Gðx ; Jun Þ ! 0;
n ! 1:
Using the condition lim infn?1 an(1 an) > 0, we have
lim gðkJxn JT n xn kÞ ¼ 0:
n!1
By property of g, we have limn?1 kJxn JTnxnk = 0. Since J1 is also uniformly norm-to-norm continuous on bounded sets, we have
lim kxn T n xn k ¼ 0:
n!1
Case 1: ({Tn}, {xn}) satisfies condition AKTT. We apply Lemma 2.11 to get
kxn Txn k 6 kxn T n xn k þ kT n xn Txn k 6 kxn T n xn k þ supfkT n x Txk : x 2 fxn gg ! 0: Case 2: ({Tn}, {xn}) satisfies condition ⁄AKTT. We apply Lemma 2.12 to get
kJxn JTxn k 6 kJxn JT n xn k þ kJT n xn JTxn k 6 kJxn JT n xn k þ supfkJT n x JTxk : x 2 fxn gg ! 0: Hence,
lim kxn Txn k ¼ lim kJ 1 ðJxn Þ J 1 ðJTxn Þk ¼ 0:
n!1
n!1
From both cases, we obtain
lim kxn Txn k ¼ 0:
n!1
T Since T is closed and xn ? p, n ? 1, we have p 2 FðTÞ ¼ 1 n¼1 FðT n Þ: m Next, we show that p 2 \k¼1 EPðF k Þ. From (3.2), we obtain
F m m1 /ðx ; un Þ ¼ / x ; hm yn 6 / x ; hm1 yn 6 6 /ðx ; xn Þ: n yn ¼ / x ; T r m;n hn n
ð3:8Þ
m for all n P 1, it follows from (3.8) and Lemma 2.15 that Since x 2 EPðF m Þ ¼ F T Frm;n
m / un ; hm1 yn ¼ / T Frm;n hm1 yn ; hm1 yn 6 / x ; hm1 yn /ðx ; un Þ 6 /ðx ; xn Þ /ðx ; un Þ: n n n n
m1 From (3.6) and (3.7), we obtain limn!1 / hm yn ¼ limn!1 / un ; hm1 yn ¼ 0: From Lemma 2.9, we have n yn ; hn n
m1 lim hm yn ¼ lim un hm1 yn ¼ 0: n yn hn n
n!1
n!1
ð3:9Þ
Hence, we have from (3.9) that
m1 lim Jhm yn ¼ 0: n yn Jhn
n!1
ð3:10Þ
m1 Again, since x 2 EPðF m1 Þ ¼ F T Frm1;n
for all n P 1, it follows from (3.8) and Lemma 2.15 that
m1 / hm1 yn ; hm2 yn ¼ / T Frm1;n hm2 yn ; hm2 yn 6 / x ; hm2 yn / x ; hm1 yn 6 /ðx ; xn Þ /ðx ; un Þ: n n n n n n
Again, from (3.6) and (3.7), we obtain limn!1 / hm1 yn ; hm2 yn ¼ 0: From Lemma 2.9, we have n n
lim hm1 yn hm2 yn ¼ 0 n n
ð3:11Þ
n!1
and hence,
lim Jhm1 yn Jhm2 yn ¼ 0: n n
ð3:12Þ
n!1
In a similar way, we can verify that
lim hm2 yn hm3 yn ¼ ¼ lim h1n yn yn ¼ 0: n n
n!1
n!1
ð3:13Þ
From (3.9), (3.11) and (3.13), we can conclude that
lim hkn yn hk1 n yn ¼ 0;
n!1
k ¼ 1; 2; . . . ; m:
ð3:14Þ
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Y. Shehu / Applied Mathematics and Computation 218 (2012) 5146–5156
Since xn ? p, n ? 1, we obtain from (3.6) that un ? p, n ? 1. Again, from (3.9), (3.11), (3.13) and un ? p, n ? 1, we have that hkn yn ! p; n ! 1, for each k = 1, 2, . . . , m. Also, using (3.14), we obtain
lim Jhkn yn Jhk1 n yn ¼ 0;
n!1
k ¼ 1; 2; . . . ; m:
Since lim infn?1rk,n > 0, k = 1, 2, . . . , m,
lim
n!1
k Jhn yn Jhk1 n yn r k;n
¼ 0:
ð3:15Þ
By Lemma 2.14, we have that for each k = 1, 2, . . . , m
1 F k hkn yn ; y þ hy hkn yn ; Jhkn yn Jhk1 n yn i P 0; r k;n
8y 2 C:
Furthermore, using (A2) we obtain
E 1 D k y hkn yn ; Jhkn yn Jhk1 n yn P F k y; hn yn : rk;n
ð3:16Þ
By (A4), (3.15) and hkn yn ! p, we have for each k = 1, 2, . . . , m
F k ðy; pÞ 6 0;
8y 2 C:
For fixed y 2 C, let zt,y :¼ ty + (1 t)p for all t 2 (0, 1]. This implies that zt 2 C. This yields that Fk(zt, p) 6 0. It follows from (A1) and (A4) that
0 ¼ F k ðzt ; zt Þ 6 tF k ðzt ; yÞ þ ð1 tÞF k ðzt ; pÞ 6 tF k ðzt ; yÞ and hence
0 6 F k ðzt ; yÞ: From condition (A3), we obtain
F k ðp; yÞ P 0;
8y 2 C:
1 m This implies that p 2 EP(Fk), k = 1, 2, . . . , m. Thus, p 2 \m k Þ. Hence, we have p 2 F ¼ \k¼1 EPðF k Þ \ \n¼1 FðT n Þ . k¼1 EPðF f 1 Finally, we show that p ¼ PF x0 . Since F ¼ \m k¼1 EPðF k Þ \ \n¼1 FðT n Þ is a closed and convex set, from Lemma 2.6, we know that PfF x0 is single valued and denote w ¼ PfF x0 . Since xn ¼ PfC n x0 and w 2 F Cn, we have
Gðxn ; Jx0 Þ 6 Gðw; Jx0 Þ;
8n P 0:
We know that G(n, Ju) is convex and lower semi-continuous with respect to n when u is fixed. This implies that
Gðp; Jx0 Þ 6 lim inf Gðxn ; Jx0 Þ 6 lim sup Gðxn ; Jx0 Þ 6 Gðw; Jx0 Þ: n!1
n!1
f F x0
From the definition of P
and p 2 F, we see that p = w. This completes the proof.
h
Corollary 3.2. Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty, closed and convex subset of E. For each k = 1, 2, . . . , m, let Fk be a bifunction from C C satisfying (A1)–(A4) and let fT n g1 n¼1 be an infinite family m 1 of relatively-quasi nonexpansive mappings of C into itself such that F :¼ \1 n¼1 FðT n Þ \ \k¼1 EPðF k Þ – ;. Suppose fxn gn¼0 is iteratively generated by x0 2 C; C 1 ¼ C; x1 ¼ PC 1 x0 ;
8 > yn ¼ J 1 ðan Jxn þ ð1 an ÞJT n xn Þ; n P 1; > > > < u ¼ T F m T F m1 . . . T F 2 T F 1 y n r m;n r m1;n r 2;n r 1;n n > C ¼ fw 2 C : /ðw; un Þ 6 /ðw; xn Þg; n P 1; > nþ1 n > > : xnþ1 ¼ PC nþ1 x0 ; n P 1;
where J is the duality mapping on E. Suppose fan g1 n¼1 is a sequence in (0, 1) such that lim infn?1an(1 an) > 0 and frk;n g1 ð0; 1Þ; ðk ¼ 1; 2; . . . ; mÞ satisfying lim inf rk,n > 0, (k = 1, 2, . . . , m). Suppose that for each bounded subset B of C, n?1 n¼1 the ordered pair ({Tn} , B) satisfies either condition AKTT or condition ⁄AKTT. Let T be the mapping from C into E defined by 1 Tx :¼ limn?1Tnx for all x 2 C and suppose that T is closed and FðTÞ ¼ \1 n¼1 FðT n Þ. Then, fxn gn¼0 converges strongly to PFx0.
Proof. Take f(x) = 0 for all x 2 E in Theorem 3.1, then G(n, Jx) = /(n, x) and PfC x0 ¼ PC x0 . Then, the desired conclusion follows. h
Y. Shehu / Applied Mathematics and Computation 218 (2012) 5146–5156
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Corollary 3.3. Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty closed convex 1 subset that 1of E. Let fT n gn¼1 be an infinite family of relatively-quasi nonexpansive mappings of C into itself such F :¼ \n¼1 FðT n Þ – ;. Let f : E ! R be a convex and lower semicontinuous mapping with C int(D(f)) and suppose fxn g1 n¼0 is iteratively generated by x0 2 C; C 1 ¼ C; x1 ¼ PfC 1 x0 ;
8 1 > > < yn ¼ J ðan Jxn þ ð1 an ÞJT n xn Þ; n P 1; C nþ1 ¼ fw 2 C n : Gðw; Jyn Þ 6 Gðw; Jxn Þg; n P 1; > > : xnþ1 ¼ Pf x0 ; n P 1;
ð3:17Þ
C nþ1
where J is the duality mapping on E. Suppose fan g1 n¼1 is a sequence in (0, 1) such that lim infn?1an(1 an) > 0 and that for each bounded subset B of C, the ordered pair ({Tn}, B) satisfies either condition AKTT or condition ⁄AKTT. Let T be the mapping from C into 1 f E defined by Tx :¼ limn?1Tnx for all x 2 C and suppose that T is closed and FðTÞ ¼ \1 n¼1 FðT n Þ. Then, fxn gn¼0 converges strongly to PF x0 . Remark 3.4. Our Corollary 3.3 extends and improves on Theorem 1.1. In fact, the iterative procedure (3.17) is simpler than (1.2) in the following two aspects: (a) the process of computing Qn = {w 2 Cn1 \ Qn1 : hxn w, Jx0 Jxni P 0} is removed; (b) the process of computing PC n \Q n is replaced by computing PfC n . 4. Application Let u : C ! R be a real valued-function. The convex minimization problem is to find x⁄ 2 C such that
uðx Þ 6 uðyÞ;
ð4:1Þ
"y 2 C. The set of solutions of (4.1) is denoted by CMP(u). For each r > 0 and x 2 E, define the mapping
Tu r ðxÞ :¼
1 z 2 C : uðyÞ þ hy z; Jz Jxi P uðzÞ; 8y 2 C : r
Theorem 4.1. Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty closed and convex subset of E. For each k = 1, 2, . . . , m, let uk be a lower semi-continuous and convex function from C to R and let fT n g1 be an m n¼1 infinite family of relatively-quasi nonexpansive mappings of C into itself such that F :¼ \1 n¼1 FðT n Þ \ \k¼1 CMPðuk Þ – ;. Let f : E ! R be a convex and lower semicontinuous mapping with C int(D(f)) and suppose fxn g1 n¼0 is iteratively generated by x0 2 C; C 1 ¼ C; x1 ¼ PfC 1 x0 ;
8 y ¼ J 1 ðan Jxn þ ð1 an ÞJT n xn Þ; > > > n > < un ¼ T um T um1 . . . T u2 T u1 yn r m;n r m1;n r 2;n r1;n
n P 1;
> C nþ1 ¼ fw 2 C n : Gðw; Jun Þ 6 Gðw; Jxn Þg; > > > : xnþ1 ¼ PfC nþ1 x0 ; n P 1;
n P 1;
ð4:2Þ
where J is the duality mapping on E. Suppose fan g1 n¼1 is a sequence in (0, 1) such that lim infn?1an(1 an) > 0 and fr k;n g1 n¼1 ð0; 1Þ; ðk ¼ 1; 2; . . . ; mÞ satisfying lim infn?1rk,n > 0, (k = 1, 2, . . . , m). Suppose that for each bounded subset B of C, the ordered pair ({Tn} , B) satisfies either condition AKTT or condition ⁄AKTT. Let T be the mapping from C into E defined by 1 f Tx :¼ limn?1Tnx for all x 2 C and suppose that T is closed and FðTÞ ¼ \1 n¼1 FðT n Þ. Then, fxn gn¼0 converges strongly to PF x0 . k for each Proof. Define Fk(x, y) :¼ uk(y) uk(x), x, y 2 C and k = 1, 2, . . . , m. Then F T Frkk ¼ EPðF k Þ ¼ CMPðuk Þ ¼ F T u rk m k = 1, 2, . . . , m, and therefore fF k gm k¼1 satisfies conditions (A1) and (A2). Furthermore, one can easily show that fF k gk¼1 satisfies conditions (A3) and (A4). Therefore, from Theorem 3.1, we obtain the desired result. h Acknowledgement The author would like to express his sincere thanks to Professor Melvin Scott, the Editor and the referees for their valuable suggestions and comments on improving the manuscript. References [1] Y.I. Alber, S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces, PanAmer. Math. J 4 (1994) 39–54. [2] Y.I. Alber, Metric and generalized projection operator in Banach spaces: properties and applications, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics, vol. 178, Dekker, New York, NY, USA, 1996, pp. 15–50. [3] K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2006) 2350–2360.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Strong convergence theorems for infinite family of relatively quasi nonexpansive mappings and systems of equilibrium problems Yekini Shehu Mathematics Institute, African University of Science and Technology, Abuja, Nigeria
a r t i c l e
i n f o
a b s t r a c t
Keywords: Relatively quasi nonexpansive mappings Equilibrium problems Hybrid method Generalized f-projection operator Banach spaces
In this paper, we construct a new iterative scheme by hybrid method for approximation of common element of set of common fixed points of countably infinite family of relatively quasi-nonexpansive mappings and set of common solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. Then, we prove strong convergence of the scheme to a common element of the two sets. Furthermore, we apply our results to solve convex minimization problem. Our results extend important recent results. 2011 Elsevier Inc. All rights reserved.
1. Introduction Let E be a real Banach space and C be nonempty closed convex subset of E. A mapping T : C ? C is called nonexpansive if
kTx Tyk 6 kx yk;
8x; y 2 C:
ð1:1Þ
A point x 2 C is called a fixed point of T if Tx = x. The set of fixed points of T is defined as F(T) :¼ {x 2 C : Tx = x}. A mapping T : C ? C is called quasi-nonexpansive if
kTx x k 6 kx x k;
8x 2 C; x 2 FðTÞ:
It is clear that every nonexpansive mapping with nonempty set of fixed points is quasi-nonexpansive. Let {Tn} be a sequence of mappings from C into E. For a subset B of C, we say that (i) ({Tn}, B) satisfies condition AKTT (see, for example [3,24]) if 1 X
supfkT nþ1 x T n xk : x 2 Bg < 1;
n¼1
(ii) ({Tn}, B) satisfies condition ⁄AKTT (see, for example [24]) if 1 X
supfkJT nþ1 x JT n xk : x 2 Bg < 1:
n¼1
Recently, Nilsrakoo and Saejung [24] proved strong convergence theorems for approximation of common fixed point of countably infinite family of relatively quasi-nonexpansive mappings in a uniformly smooth real Banach space which is also uniformly convex. In particular, they proved the following theorem. E-mail address: [email protected] 0096-3003/$ - see front matter 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.001
Y. Shehu / Applied Mathematics and Computation 218 (2012) 5146–5156
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Theorem 1.1. Let E be a real uniformly convex and uniformly smooth Banach space and let C be a nonempty closed convex subset T1 of E. Let fT n g1 n¼1 be a sequence of relatively quasi-nonexpansive mappings of C into itself such that F :¼ n¼1 FðT n Þ – ; and let 1 fxn gn¼0 be a sequence in C defined as follows:
8 x0 2 KC 1 ¼ Q 1 ¼ K; > > > > 1 > > < yn ¼ J ðan Jxn þ ð1 an ÞJT n xn Þ; n P 1; C n ¼ fw 2 C n1 \ Q n1 : /ðyn ; wÞ 6 /ðxn ; wÞg; n P 1; > > > > Q n ¼ fw 2 C n1 \ Q n1 : hxn w; Jx0 Jxn i P 0g; n P 1; > > : xnþ1 ¼ PC n \Q n x0 ; n P 0;
ð1:2Þ
where fan g1 n¼1 is a sequence in [0, 1) with lim supn?1an < 1. Suppose that for each bounded subset B of C, the ordered pair ({Tn}, B) satisfies either condition AKTT or condition ⁄ AKTT. Let T be the mapping from C into E defined by Tx :¼ limn?1Tnx for all x 2 C and T 1 suppose that T is closed and FðTÞ ¼ 1 n¼0 FðT n Þ. Then, fxn gn¼0 converges strongly to PF(T)x0. Let F be a bifunction of C C into R. The equilibrium problem (see, for example [12,21,23,29,32,33,38,39]) is to find x⁄ 2 C such that
Fðx ; yÞ P 0
ð1:3Þ
for all y 2 C. We shall denote the solutions set of (1.3) by EP(F). Numerous problems in Physics, optimization and economics reduce to find a solution of problem (1.3). Some methods have been proposed to solve the equilibrium problem, see for example [4,10,12,16,17,20,27]. In [22], Matsushita and Takahashi introduced a hybrid iterative scheme for approximation of fixed points of relatively nonexpansive mapping in a uniformly convex real Banach space which is also uniformly smooth: x0 2 C,
8 yn ¼ J 1 ðan Jxn þ ð1 an ÞJTxn Þ; > > > < H ¼ fw 2 C : /ðw; y Þ 6 /ðw; x Þg; n n n > W ¼ fw 2 C : hx w; Jx Jx i; > n n 0 n > : xnþ1 ¼ PHn \W n x0 ; n P 0: They proved that fxn g1 n¼0 converges strongly to PF(T)x0, where F(T) – ;. In [28], Plubtieng and Ungchittrakool introduced the following hybrid projection algorithm for a pair of relatively nonexpansive mappings: x0 2 C,
8 > zn ¼ J 1 bð1Þ Jxn þ bð2Þ JTxn þ bnð3Þ JSxn > n n > > > > > < yn ¼ J 1 ðan Jx0 þ ð1 an ÞJzn Þ
C n ¼ fz 2 C : /ðz; yn Þ 6 /ðz; xn Þ þ an ðkx0 k2 þ 2hw; Jxn Jx0 iÞg > > > > > > Q n ¼ fz 2 C : hxn z; Jxn Jx0 i 6 0g > : xnþ1 ¼ PC n \Q n x0 ;
ð1:4Þ
n o n o n o ð2Þ ð3Þ ; bð2Þ and bð3Þ are sequences in (0, 1) satisfying bð1Þ where fan g; bð1Þ n n n n þ bn þ bn ¼ 1 and T and S are relatively nonexpansive mappings and J is the single-valued duality mapping on E. They proved under the appropriate conditions on the parameters that the sequence {xn} generated by (1.4) converges strongly to a common fixed point of T and S. In [37], Takahashi and Zembayashi introduced the following hybrid iterative scheme for approximation of fixed point of relatively nonexpansive mapping which is also a solution to an equilibrium problem in a uniformly convex real Banach space which is also uniformly smooth: x0 2 C; C 1 ¼ C; x1 ¼ PC 1 x0 ;
8 > yn ¼ J 1 ðan Jxn þ ð1 an ÞJTxn Þ; > > > < Fðu ; yÞ þ 1 hy u ; Ju Jy i P 0; 8y 2 C n n n n rn > C nþ1 ¼ fw 2 C n : /ðw; un Þ 6 /ðw; xn Þg; > > > : xnþ1 ¼ PC nþ1 x0 ; n P 1;
where J is the duality mapping on E. Then, they proved that fxn g1 n¼0 converges strongly to PFx0, where F = EP(F) \ F(T) – ;. Recently, Li et al. [19] introduced the following hybrid iterative scheme for approximation of fixed points of a relatively nonexpansive mapping using the properties of generalized f-projection operator in a uniformly smooth real Banach space which is also uniformly convex: x0 2 C,
8 1 > > < yn ¼ J ðan Jxn þ ð1 an ÞJTxn Þ; C nþ1 ¼ fw 2 C n : Gðw; Jyn Þ 6 Gðw; Jxn Þg; > > : xnþ1 ¼ Pf x0 ; n P 1; C nþ1
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They proved a strong convergence theorem for finding an element in the fixed points set of T. We remark here that the results of Li et al. [19] extended and improved on the results of Matsushita and Takahashi [22]. Motivated by the above mentioned results and the on-going research, it is our purpose in this paper to prove a strong convergence theorem for countable family of relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. Furthermore, we apply our results to solve convex minimization problem. Our results extend the results of Li et al. [19] and Takahashi and Zembayashi [37] and many other recent known results in the literature. 2. Preliminaries Let E be a real Banach space. The modulus of smoothness of E is the function qE : [0, 1) ? [0, 1) defined by
qE ðsÞ :¼ sup
1 ðkx þ yk þ kx ykÞ 1 : kxk 6 1; kyk 6 s : 2
E is uniformly smooth if and only if
lim s!0
qE ðsÞ ¼ 0: s
Let dimE P 2. The modulus of convexity of E is the function dE : (0, 2] ? [0, 1] defined by
x þ y n o dE ðÞ :¼ inf 1 : kxk ¼ kyk ¼ 1; ¼ kx yk : 2 E is uniformly convex if for any 2 (0, 2], there exists a d = d() > 0 such that if x, y 2 E with kxk 6 1, kyk 6 1 and kx yk P , then 12 ðx þ yÞ 6 1 d. Equivalently, E is uniformly convex if and only if dE() > 0 for all 2 (0, 2]. A normed space E is called strictly convex if for all x, y 2 E, x – y, kxk = kyk = 1, we have kkx + (1 k)yk < 1, "k 2 (0, 1). Let E⁄ be the dual space of E. We denote by J the normalized duality mapping from E to 2E defined by
JðxÞ ¼ ff 2 E : hx; f i ¼ kxk2 ¼ kf k2 g: The following properties of J are well known (the reader can consult [9,34,35] for more details): (1) (2) (3) (4)
If E is uniformly smooth, then J is norm-to-norm uniformly continuous on each bounded subset of E. J(x) – ;, x 2 E. If E is reflexive, then J is a mapping from E onto E⁄. If E is smooth, then J is single valued.
Throughout this paper, we denote by /, the functional on E E defined by
/ðx; yÞ :¼ kxk2 2hx; JðyÞi þ kyk2 ;
8x; y 2 E:
ð2:1Þ
Let C be a nonempty subset of E and let T be a mapping from C into E. A point p 2 C is said to be an asymptotic fixed point of T if C contains a sequence fxn g1 n¼0 which converges weakly to p and limn?1kxn Txnk = 0. The set of asymptotic fixed points of T is denoted by e F ðTÞ. We say that a mapping T is relatively nonexpansive (see, for example [5–7,22]) if the following conditions are satisfied: (R1) F(T) – ;; (R2) /(p, Tx) 6 /(p, x), "x 2 C, p 2 F(T); (R3) FðTÞ ¼ e F ðTÞ. If T satisfies (R1) and (R2), then T is said to be relatively quasi-nonexpansive. It is easy to see that the class of relatively quasi-nonexpansive mappings contains the class of relatively nonexpansive mappings. Many authors have studied the methods of approximating the fixed points of relatively quasi-nonexpansive mappings (see, for example [8,14,24–26,30,31,40,43] the references contained therein). Clearly, in Hilbert space H, relatively quasi-nonexpansive mappings and quasi-nonexpansive mappings are the same, for /(x, y) = kx yk2, "x, y 2 H and this implies that
/ðp; TxÞ 6 /ðp; xÞ () kTx pk 6 kx pk;
8x 2 C; p 2 FðTÞ:
The examples of relatively quasi-nonexpansive mappings are given in [30]. Let E be a smooth, strictly convex and reflexive real Banach space and let C be a nonempty closed convex subset of E. Following Alber [2], the generalized projection PC from E onto C is defined by
PC ðxÞ :¼ argmin /ðy; xÞ; y2C
8x 2 E:
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The existence and uniqueness of PC follows from the property of the functional /(x, y) and strict monotonicity of the mapping J (see, for example [1,2,11,18,35]). If E is a Hilbert space, then PC is the metric projection of H onto C. From [18], in uniformly convex and uniformly smooth Banach spaces, we have
ðkxk kykÞ2 6 /ðx; yÞ 6 ðkxk þ kykÞ2 ;
8x; y 2 E:
ð2:2Þ
Next, we recall the concept of generalized f-projector operator, together with its properties. Let G : C E ! R [ fþ1g be a functional defined as follows:
Gðn; uÞ ¼ knk2 2hn; ui þ kuk2 þ 2qf ðnÞ;
ð2:3Þ
⁄
where n 2 C, u 2 E , q is a positive number and f : C ! R [ fþ1g is proper, convex and lower semi-continuous. From the definitions of G and f, it is easy to see the following properties: (i) G(n, u) is convex and continuous with respect to u when n is fixed; (ii) G(n, u) is convex and lower semi-continuous with respect to n when u is fixed. Definition 2.1 [41]. Let E be a real Banach space with its dual E⁄. Let C be a nonempty, closed and convex subset of E. We say that PfC : E ! 2C is a generalized f-projection operator if
PfC u ¼ fu 2 C : Gðu; uÞ ¼ inf Gðn; uÞg;
8u 2 E :
n2C
For the generalized f-projection operator, Wu and Huang [41] proved the following theorem basic properties: Lemma 2.2 [41]. Let E be a real reflexive Banach space with its dual E⁄. Let C be a nonempty, closed and convex subset of E. Then the following statements hold: (i) (ii)
PfC is a nonempty closed convex subset of C for all u 2 E⁄. If E is smooth, then for all u 2 E ; x 2 PfC if and only if hx y; u Jxi þ qf ðyÞ qf ðxÞ P 0;
(iii)
8y 2 C:
If E is strictly convex and f : C ! R [ fþ1g is positive homogeneous (i.e., f(tx) = tf(x) for all t > 0 such that tx 2 C where x 2 C), then PfC is a single valued mapping.
Fan et al. [15] showed that the condition f is positive homogeneous which appeared in Lemma 2.2 can be removed. Lemma 2.3 [15]. Let E be a real reflexive Banach space with its dual E⁄ and C a nonempty, closed and convex subset of E. Then if E is strictly convex, then PfC is a single valued mapping. Recall that J is a single valued mapping when E is a smooth Banach space. There exists a unique element u 2 E⁄ such that u = Jx for each x 2 E. This substitution in (2.3) gives
Gðn; JxÞ ¼ knk2 2hn; Jxi þ kxk2 þ 2qf ðnÞ:
ð2:4Þ
Now, we consider the second generalized f-projection operator in a Banach space. Definition 2.4. Let E be a real Banach space and C a nonempty, closed and convex subset of E. We say that PfC : E ! 2C is a generalized f-projection operator if
PfC x ¼ fu 2 C : Gðu; JxÞ ¼ inf Gðn; JxÞg; n2C
8x 2 E:
Obviously, the definition of relatively quasi-nonexpansive mapping T is equivalent to (R0 1)F(T) – ;; (R0 2)G(p, JTx) 6 G(p, Jx), "x 2 C, p 2 F(T). Lemma 2.5 [13]. Let E be a Banach space and f : E ! R [ fþ1g be a lower semi-continuous convex functional. Then there exists x⁄ 2 E⁄ and a 2 R such that
f ðxÞ P hx; x i þ a;
8x 2 E:
We know that the following lemmas hold for operator PfC . Lemma 2.6 [19]. Let C be a nonempty, closed and convex subset of a smooth and reflexive Banach space E. Then the following statements hold:
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(i) PfC x is a nonempty closed and convex subset of C for all x 2 E; (ii) for all x 2 E; ^ x 2 PfC x if and only if
h^x y; Jx J^xi þ qf ðyÞ qf ðxÞ P 0; (iii) if E is strictly convex, then P
f Cx
8y 2 C;
is a single valued mapping.
Lemma 2.7 [19]. Let C be a nonempty, closed and convex subset of a smooth and reflexive Banach space E. Let x 2 E and ^ x 2 PfC . Then
/ðy; ^xÞ þ Gð^x; JxÞ 6 Gðy; JxÞ;
8y 2 C:
The fixed points set F(T) of a relatively quasi-nonexpansive mapping is closed convex as given in the following lemma. Lemma 2.8 ([30,24]). Let C be a nonempty closed convex subset of a smooth, uniformly convex Banach space E. Let T be a closed relatively quasi- nonexpansive mapping of C into itself. Then F(T) is closed and convex. Also, this following lemma will be used in the sequel. Lemma 2.9 [18]. Let C be a nonempty closed convex subset of a smooth, uniformly convex Banach space E. Let fxn g1 n¼1 and 1 1 fyn g1 n¼1 be sequences in E such that either fxn gn¼1 or fyn gn¼1 is bounded. If limn?1/(xn, yn) = 0, then limn?1kxn ynk = 0. Lemma 2.10 [42]. Let E be a uniformly convex real Banach space. For arbitrary r > 0, let Br(0) :¼ {x 2 E : kxk 6 r} and k 2 [0, 1]. Then, there exists a continuous strictly increasing convex function
g : ½0; 2r ! R;
gð0Þ ¼ 0
such that for every x, y 2 Br(0), the following inequality holds:
kkx þ ð1 kÞyk2 6 kkxk2 þ ð1 kÞkyk2 kð1 kÞgðkx ykÞ: Lemma 2.11 ([24,3]). Let C be a nonempty closed subset of a Banach space E and let fT n g1 n¼1 be a sequence of mappings of K into E. Let B be a subset of C with ({Tn}, B) satisfying AKTT. Then there exists a mapping T : B ? E such that Tx :¼ limn?1Tnx, for all x 2 B. and limn?1sup{kTnx Txk : x 2 B} = 0. Lemma 2.12 [24]. Let E be reflexive and strictly convex Banach space whose norm is Fréchet differentiable, let C be a nonempty ⁄ closed subset of E which and let fT n g1 n¼1 be a sequence of mappings of K into E. Let B be a subset of C with ({Tn}, B) satisfying AKTT. Then there exists a mapping T : B ? E such that Tx :¼ limn?1Tnx, for all x 2 B and limn?1sup{kJTnx JTxk : x 2 B} = 0. For solving the equilibrium problem for a bifunction F : C C ! R, let us assume that F satisfies the following conditions: (A1) (A2) (A3) (A4)
F(x, x) = 0 for all x 2 C; F is monotone, i.e., F(x, y) + F(y, x) 6 0 for all x, y, 2C; for each x, y 2 C, limt?0F(tz + (1 t)x, y) 6 F(x, y); for each x 2 C, y ´ F(x, y) is convex and lower semicontinuous.
Lemma 2.13 [4]. Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E and let F be a bifunction of C C into R satisfying (A1)–(A4). Let r > 0 and x 2 E. Then, there exists z 2 C such that
1 Fðz; yÞ þ hy z; Jz Jxi P 0 for all y 2 K: r Lemma 2.14 [36]. Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E. Assume that F : C C ! R satisfies (A1)–(A4). For r > 0 and x 2 E, define a mapping T Fr : E ! C as follows:
T Fr ðxÞ ¼
1 z 2 C : Fðz; yÞ þ hy z; Jz Jxi P 0; 8y 2 C r
for all z 2 E. Then, the following hold: 1. T Fr is single-valued; 2. T Fr is firmly nonexpansive-type mapping, i.e., for any x, y 2 E,
D E D E T Fr x T Fr y; JT Fr x JT Fr y 6 T Fr x T Fr y; Jx Jy ;
3. FðT Fr Þ ¼ EPðFÞ; 4. EP(F) is closed and convex.
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Lemma 2.15 [36]. Let C be a nonempty closed convex subset of a smooth, strictly and reflexive Banach space E. Assume convex that F : C C ! R satisfies (A1)–(A4) and let r > 0. Then for each x 2 E and q 2 F T Fr ,
/ q; T Fr x þ / T Fr x; x 6 /ðq; xÞ:
For the rest of this paper, the sequence fxn g1 n¼1 converges strongly to p shall be denoted by xn ? p, n ? 1. Observe that an operator T in a Banach space E is said to be closed if xn ? x and Txn ? y, then Tx = y. 3. Main results We now state and prove the following theorem. Theorem 3.1. Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty closed convex 1 subset of E. For each k = 1, 2, . . . , m, let Fk be a bifunction from C C satisfying (A1)–(A4) 1 m and let fT n gn¼1 be an infinite family of relatively-quasi nonexpansive mappings of C into itself such that F :¼ \n¼1 FðT n Þ \ \k¼1 EPðF k Þ – ;. Let f : E ! R be a convex f and lower semicontinuous mapping with C int(D(f)) and suppose fxn g1 n¼0 is iteratively generated by x0 2 C; C 1 ¼ C; x1 ¼ PC 1 x0 ;
8 yn ¼ J 1 ðan Jxn þ ð1 an ÞJT n xn Þ; > > > > < un ¼ T F m T F m1 . . . T F 2 T F 1 yn r m;n r m1;n r 2;n r1;n
n P 1;
> C nþ1 ¼ fw 2 C n : Gðw; Jun Þ 6 Gðw; Jxn Þg; > > > : xnþ1 ¼ PfC nþ1 x0 ; n P 1;
ð3:1Þ
n P 1;
where J is the duality mapping on E. Suppose fan g1 n¼1 is a sequence in (0, 1) such that lim infn?1an(1 an) > 0 and fr k;n g1 n¼1 ð0; 1Þ; ðk ¼ 1; 2; . . . ; mÞ satisfying lim infn?1rk,n > 0, (k = 1, 2, . . . , m). Suppose that for each bounded subset B of C, the ordered pair ({Tn} , B) satisfies either condition AKTT or condition ⁄AKTT. Let T be the mapping from C into E defined by 1 f Tx :¼ limn?1Tnx for all x 2 C and suppose that T is closed and FðTÞ ¼ \1 n¼1 FðT n Þ. Then, fxn gn¼0 converges strongly to PF x0 . Proof. We first show that Cn, "n P 1 is closed and convex. It is obvious that C1 = C is closed and convex. Suppose that Cn is closed convex for some n > 1. From the definition of Cn+1, we have that z 2 Cn+1 implies G(z, Jun) 6 G(z, Jxn). This is equivalent to
2ðhz; Jxn i hz; Jun iÞ 6 kxn k2 kun k2 : This implies that Cn+1 is closed convex for the same n > 1. Hence, Cn is closed and convex "n P 1. This shows that PC nþ1 x0 is well defined for all n P 0. 2 1 k k1 By taking hkn ¼ T Frk;n T Frk1;n . . . T Fr2;n T Fr1;n ; k ¼ 1; 2; . . . ; m and h0n ¼ I for all n P 1, we obtain un ¼ hm n yn . k We next show that F Cn, "n P 1. From Lemma 2.14, one has that T Frk;n ; k ¼ 1; 2; . . . ; m is relatively nonexpansive mapping. For n = 1, we have F C = C1. Now, assume that F Cn1 for some n P 2. Then for each x⁄ 2 F, we obtain
Gðx ; Jun Þ ¼ G x ; Jhm n yn 6 Gðx ; Jyn Þ ¼ Gðx ; ðan Jxn þ ð1 an ÞJT n xn ÞÞ
¼ kx k2 2an hx ; Jxn i 2ð1 an Þhx ; JT n xn i þ kan Jxn þ ð1 an ÞJT n xn k2 þ 2qf ðx Þ 6 kx k2 2an hx ; Jxn i 2ð1 an Þhx ; JT n xn i þ an kJxn k2 þ ð1 an ÞkJT n xn k2 þ 2qf ðx Þ ¼ an Gðx ; Jxn Þ þ ð1 an ÞGðx ; JT n xn Þ 6 Gðx ; Jxn Þ:
ð3:2Þ
fxn g1 n¼0
⁄
So, x 2 Cn. This implies that F Cn, "n P 1 and the sequence generated by (3.1) is well defined. We now show that limn?1G(xn, Jx0) exists. Since f : E ! R is a convex and lower semi-continuous, applying Lemma 2.5, we see that there exists u⁄ 2 E⁄ and a 2 R such that
f ðyÞ P hy; u i þ a;
8y 2 E:
It follows that
Gðxn ; Jx0 Þ ¼ kxn k2 2hxn ; Jx0 i þ kx0 k2 þ 2qf ðxn Þ P kxn k2 2hxn ; Jx0 i þ kx0 k2 þ 2qhxn ; u i þ 2qa ¼ kxn k2 2hxn ; Jx0 qu i þ kx0 k2 þ 2qa P kxn k2 2kxn kkJx0 qu k þ kx0 k2 þ 2qa ¼ ðkxn k kJx0 qu kÞ2 þ kx0 k2 kJx0 qu k2 þ 2qa: f C n x0 ,
Since xn ¼ P
it follows from (3.3) that
Gðx ; Jx0 Þ P Gðxn ; Jx0 Þ P ðkxn k kJx0 qu kÞ2 þ kx0 k2 kJx0 qu k2 þ 2qa
ð3:3Þ
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1 for each x⁄ 2 F(T). This implies that fxn g1 n¼1 is bounded and so is fGðxn ; Jx0 Þgn¼0 . By the construction of Cn, we have that Cm Cn and xm ¼ PfC m x0 2 C n for any positive integer m P n. It then follows Lemma 2.7 that
/ðxm ; xn Þ þ Gðxn ; Jx0 Þ 6 Gðxm ; Jx0 Þ:
ð3:4Þ
It is obvious that
/ðxm ; xn Þ P ðkxm k kxn kÞ2 P 0: In particular,
/ðxnþ1 ; xn Þ þ Gðxn ; Jx0 Þ 6 Gðxnþ1 ; Jx0 Þ and
/ðxnþ1 ; xn Þ P ðkxnþ1 k kxn kÞ2 P 0; 1 and so fGðxn ; Jx0 Þg1 n¼0 is nondecreasing. It follows that the limit of fGðxn ; Jx0 Þgn¼0 exists. f By the fact that Cm Cn and xm ¼ PC m x0 2 C n for any positive integer m P n, we obtain
/ðxm ; un Þ 6 /ðxm ; xn Þ: Now, (3.4) implies that
/ðxm ; un Þ 6 /ðxm ; xn Þ 6 Gðxm ; Jx0 Þ Gðxn ; Jx0 Þ:
ð3:5Þ
Taking the limit as m, n ? 1 in (3.5), we obtain
lim /ðxm ; xn Þ ¼ 0:
n!1
It then follows from Lemma 2.9 that kxm xnk ? 0 as m, n ? 1. Hence, fxn g1 n¼0 is Cauchy. Since E is a Banach space and C is closed and convex, then there exists p 2 C such that xn ? p as n ? 1. Now since /(xm, xn) ? 0 as m, n ? 1 we have in particular that /(xn+1, xn) ? 0 as n ? 1 and this further implies that limn?1kxn+1 xnk = 0. Since xnþ1 ¼ PfC nþ1 x0 2 C nþ1 ; we have
/ðxnþ1 ; un Þ 6 /ðxnþ1 ; xn Þ;
8n P 0:
Then, we obtain
lim /ðxnþ1 ; un Þ ¼ 0:
n!1
Since E is uniformly convex and smooth, we have from Lemma 2.9 that
lim kxnþ1 xn k ¼ 0 ¼ lim kxnþ1 un k:
n!1
n!1
So,
kxn un k 6 kxnþ1 xn k þ kxnþ1 un k: Hence,
lim kxn un k ¼ 0:
n!1
ð3:6Þ
Since J is uniformly norm-to-norm continuous on bounded sets and limn?1kxn unk = 0, we obtain
lim kJxn Jun k ¼ 0:
n!1
ð3:7Þ
Let r :¼ supnP1{kxnk, kTnxnk}. Since E is uniformly smooth, we know that E⁄ is uniformly convex. Then from Lemma 2.10, we have Gðx ; Jun Þ ¼ Gðx ; Jhm n yn Þ 6 Gðx ; Jyn Þ ¼ Gðx ; ðan Jxn þ ð1 an ÞJT n xn ÞÞ
¼ kx k2 2an hx ; Jxn i 2ð1 an Þhx ; JT n xn i þ kan Jxn þ ð1 an ÞJT n xn k2 þ 2qf ðx Þ 6 kx k2 2an hx ; Jxn i 2ð1 an Þhx ; JT n xn i þ an kJxn k2 þ ð1 an ÞkJT n xn k2 an ð1 an ÞgðkJxn JT n xn kÞ þ 2qf ðx Þ ¼ an Gðx ; Jxn Þ þ ð1 an ÞGðx ; JT n xn Þ an ð1 an ÞgðkJxn JT n xn kÞ 6 Gðx ; Jxn Þ an ð1 an ÞgðkJxn JT n xn kÞ: It then follows that
an ð1 an ÞgðkJxn JT n xn kÞ 6 Gðx ; Jxn Þ Gðx ; Jun Þ:
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But
Gðx ; Jxn Þ Gðx ; Jun Þ ¼ kxn k2 kun k2 2hx ; Jxn Jun i 6 jkxn k2 kun k2 j þ 2jhx ; Jxn Jun ij 6 jkxn k kun kjðkxn k þ kun kÞ þ 2kx kkJxn Jun k 6 kxn un kðkxn k þ kun kÞ þ 2kx kkJxn Jun k: From (3.6) and (3.7), we obtain
Gðx ; Jxn Þ Gðx ; Jun Þ ! 0;
n ! 1:
Using the condition lim infn?1 an(1 an) > 0, we have
lim gðkJxn JT n xn kÞ ¼ 0:
n!1
By property of g, we have limn?1 kJxn JTnxnk = 0. Since J1 is also uniformly norm-to-norm continuous on bounded sets, we have
lim kxn T n xn k ¼ 0:
n!1
Case 1: ({Tn}, {xn}) satisfies condition AKTT. We apply Lemma 2.11 to get
kxn Txn k 6 kxn T n xn k þ kT n xn Txn k 6 kxn T n xn k þ supfkT n x Txk : x 2 fxn gg ! 0: Case 2: ({Tn}, {xn}) satisfies condition ⁄AKTT. We apply Lemma 2.12 to get
kJxn JTxn k 6 kJxn JT n xn k þ kJT n xn JTxn k 6 kJxn JT n xn k þ supfkJT n x JTxk : x 2 fxn gg ! 0: Hence,
lim kxn Txn k ¼ lim kJ 1 ðJxn Þ J 1 ðJTxn Þk ¼ 0:
n!1
n!1
From both cases, we obtain
lim kxn Txn k ¼ 0:
n!1
T Since T is closed and xn ? p, n ? 1, we have p 2 FðTÞ ¼ 1 n¼1 FðT n Þ: m Next, we show that p 2 \k¼1 EPðF k Þ. From (3.2), we obtain
F m m1 /ðx ; un Þ ¼ / x ; hm yn 6 / x ; hm1 yn 6 6 /ðx ; xn Þ: n yn ¼ / x ; T r m;n hn n
ð3:8Þ
m for all n P 1, it follows from (3.8) and Lemma 2.15 that Since x 2 EPðF m Þ ¼ F T Frm;n
m / un ; hm1 yn ¼ / T Frm;n hm1 yn ; hm1 yn 6 / x ; hm1 yn /ðx ; un Þ 6 /ðx ; xn Þ /ðx ; un Þ: n n n n
m1 From (3.6) and (3.7), we obtain limn!1 / hm yn ¼ limn!1 / un ; hm1 yn ¼ 0: From Lemma 2.9, we have n yn ; hn n
m1 lim hm yn ¼ lim un hm1 yn ¼ 0: n yn hn n
n!1
n!1
ð3:9Þ
Hence, we have from (3.9) that
m1 lim Jhm yn ¼ 0: n yn Jhn
n!1
ð3:10Þ
m1 Again, since x 2 EPðF m1 Þ ¼ F T Frm1;n
for all n P 1, it follows from (3.8) and Lemma 2.15 that
m1 / hm1 yn ; hm2 yn ¼ / T Frm1;n hm2 yn ; hm2 yn 6 / x ; hm2 yn / x ; hm1 yn 6 /ðx ; xn Þ /ðx ; un Þ: n n n n n n
Again, from (3.6) and (3.7), we obtain limn!1 / hm1 yn ; hm2 yn ¼ 0: From Lemma 2.9, we have n n
lim hm1 yn hm2 yn ¼ 0 n n
ð3:11Þ
n!1
and hence,
lim Jhm1 yn Jhm2 yn ¼ 0: n n
ð3:12Þ
n!1
In a similar way, we can verify that
lim hm2 yn hm3 yn ¼ ¼ lim h1n yn yn ¼ 0: n n
n!1
n!1
ð3:13Þ
From (3.9), (3.11) and (3.13), we can conclude that
lim hkn yn hk1 n yn ¼ 0;
n!1
k ¼ 1; 2; . . . ; m:
ð3:14Þ
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Since xn ? p, n ? 1, we obtain from (3.6) that un ? p, n ? 1. Again, from (3.9), (3.11), (3.13) and un ? p, n ? 1, we have that hkn yn ! p; n ! 1, for each k = 1, 2, . . . , m. Also, using (3.14), we obtain
lim Jhkn yn Jhk1 n yn ¼ 0;
n!1
k ¼ 1; 2; . . . ; m:
Since lim infn?1rk,n > 0, k = 1, 2, . . . , m,
lim
n!1
k Jhn yn Jhk1 n yn r k;n
¼ 0:
ð3:15Þ
By Lemma 2.14, we have that for each k = 1, 2, . . . , m
1 F k hkn yn ; y þ hy hkn yn ; Jhkn yn Jhk1 n yn i P 0; r k;n
8y 2 C:
Furthermore, using (A2) we obtain
E 1 D k y hkn yn ; Jhkn yn Jhk1 n yn P F k y; hn yn : rk;n
ð3:16Þ
By (A4), (3.15) and hkn yn ! p, we have for each k = 1, 2, . . . , m
F k ðy; pÞ 6 0;
8y 2 C:
For fixed y 2 C, let zt,y :¼ ty + (1 t)p for all t 2 (0, 1]. This implies that zt 2 C. This yields that Fk(zt, p) 6 0. It follows from (A1) and (A4) that
0 ¼ F k ðzt ; zt Þ 6 tF k ðzt ; yÞ þ ð1 tÞF k ðzt ; pÞ 6 tF k ðzt ; yÞ and hence
0 6 F k ðzt ; yÞ: From condition (A3), we obtain
F k ðp; yÞ P 0;
8y 2 C:
1 m This implies that p 2 EP(Fk), k = 1, 2, . . . , m. Thus, p 2 \m k Þ. Hence, we have p 2 F ¼ \k¼1 EPðF k Þ \ \n¼1 FðT n Þ . k¼1 EPðF f 1 Finally, we show that p ¼ PF x0 . Since F ¼ \m k¼1 EPðF k Þ \ \n¼1 FðT n Þ is a closed and convex set, from Lemma 2.6, we know that PfF x0 is single valued and denote w ¼ PfF x0 . Since xn ¼ PfC n x0 and w 2 F Cn, we have
Gðxn ; Jx0 Þ 6 Gðw; Jx0 Þ;
8n P 0:
We know that G(n, Ju) is convex and lower semi-continuous with respect to n when u is fixed. This implies that
Gðp; Jx0 Þ 6 lim inf Gðxn ; Jx0 Þ 6 lim sup Gðxn ; Jx0 Þ 6 Gðw; Jx0 Þ: n!1
n!1
f F x0
From the definition of P
and p 2 F, we see that p = w. This completes the proof.
h
Corollary 3.2. Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty, closed and convex subset of E. For each k = 1, 2, . . . , m, let Fk be a bifunction from C C satisfying (A1)–(A4) and let fT n g1 n¼1 be an infinite family m 1 of relatively-quasi nonexpansive mappings of C into itself such that F :¼ \1 n¼1 FðT n Þ \ \k¼1 EPðF k Þ – ;. Suppose fxn gn¼0 is iteratively generated by x0 2 C; C 1 ¼ C; x1 ¼ PC 1 x0 ;
8 > yn ¼ J 1 ðan Jxn þ ð1 an ÞJT n xn Þ; n P 1; > > > < u ¼ T F m T F m1 . . . T F 2 T F 1 y n r m;n r m1;n r 2;n r 1;n n > C ¼ fw 2 C : /ðw; un Þ 6 /ðw; xn Þg; n P 1; > nþ1 n > > : xnþ1 ¼ PC nþ1 x0 ; n P 1;
where J is the duality mapping on E. Suppose fan g1 n¼1 is a sequence in (0, 1) such that lim infn?1an(1 an) > 0 and frk;n g1 ð0; 1Þ; ðk ¼ 1; 2; . . . ; mÞ satisfying lim inf rk,n > 0, (k = 1, 2, . . . , m). Suppose that for each bounded subset B of C, n?1 n¼1 the ordered pair ({Tn} , B) satisfies either condition AKTT or condition ⁄AKTT. Let T be the mapping from C into E defined by 1 Tx :¼ limn?1Tnx for all x 2 C and suppose that T is closed and FðTÞ ¼ \1 n¼1 FðT n Þ. Then, fxn gn¼0 converges strongly to PFx0.
Proof. Take f(x) = 0 for all x 2 E in Theorem 3.1, then G(n, Jx) = /(n, x) and PfC x0 ¼ PC x0 . Then, the desired conclusion follows. h
Y. Shehu / Applied Mathematics and Computation 218 (2012) 5146–5156
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Corollary 3.3. Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty closed convex 1 subset that 1of E. Let fT n gn¼1 be an infinite family of relatively-quasi nonexpansive mappings of C into itself such F :¼ \n¼1 FðT n Þ – ;. Let f : E ! R be a convex and lower semicontinuous mapping with C int(D(f)) and suppose fxn g1 n¼0 is iteratively generated by x0 2 C; C 1 ¼ C; x1 ¼ PfC 1 x0 ;
8 1 > > < yn ¼ J ðan Jxn þ ð1 an ÞJT n xn Þ; n P 1; C nþ1 ¼ fw 2 C n : Gðw; Jyn Þ 6 Gðw; Jxn Þg; n P 1; > > : xnþ1 ¼ Pf x0 ; n P 1;
ð3:17Þ
C nþ1
where J is the duality mapping on E. Suppose fan g1 n¼1 is a sequence in (0, 1) such that lim infn?1an(1 an) > 0 and that for each bounded subset B of C, the ordered pair ({Tn}, B) satisfies either condition AKTT or condition ⁄AKTT. Let T be the mapping from C into 1 f E defined by Tx :¼ limn?1Tnx for all x 2 C and suppose that T is closed and FðTÞ ¼ \1 n¼1 FðT n Þ. Then, fxn gn¼0 converges strongly to PF x0 . Remark 3.4. Our Corollary 3.3 extends and improves on Theorem 1.1. In fact, the iterative procedure (3.17) is simpler than (1.2) in the following two aspects: (a) the process of computing Qn = {w 2 Cn1 \ Qn1 : hxn w, Jx0 Jxni P 0} is removed; (b) the process of computing PC n \Q n is replaced by computing PfC n . 4. Application Let u : C ! R be a real valued-function. The convex minimization problem is to find x⁄ 2 C such that
uðx Þ 6 uðyÞ;
ð4:1Þ
"y 2 C. The set of solutions of (4.1) is denoted by CMP(u). For each r > 0 and x 2 E, define the mapping
Tu r ðxÞ :¼
1 z 2 C : uðyÞ þ hy z; Jz Jxi P uðzÞ; 8y 2 C : r
Theorem 4.1. Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty closed and convex subset of E. For each k = 1, 2, . . . , m, let uk be a lower semi-continuous and convex function from C to R and let fT n g1 be an m n¼1 infinite family of relatively-quasi nonexpansive mappings of C into itself such that F :¼ \1 n¼1 FðT n Þ \ \k¼1 CMPðuk Þ – ;. Let f : E ! R be a convex and lower semicontinuous mapping with C int(D(f)) and suppose fxn g1 n¼0 is iteratively generated by x0 2 C; C 1 ¼ C; x1 ¼ PfC 1 x0 ;
8 y ¼ J 1 ðan Jxn þ ð1 an ÞJT n xn Þ; > > > n > < un ¼ T um T um1 . . . T u2 T u1 yn r m;n r m1;n r 2;n r1;n
n P 1;
> C nþ1 ¼ fw 2 C n : Gðw; Jun Þ 6 Gðw; Jxn Þg; > > > : xnþ1 ¼ PfC nþ1 x0 ; n P 1;
n P 1;
ð4:2Þ
where J is the duality mapping on E. Suppose fan g1 n¼1 is a sequence in (0, 1) such that lim infn?1an(1 an) > 0 and fr k;n g1 n¼1 ð0; 1Þ; ðk ¼ 1; 2; . . . ; mÞ satisfying lim infn?1rk,n > 0, (k = 1, 2, . . . , m). Suppose that for each bounded subset B of C, the ordered pair ({Tn} , B) satisfies either condition AKTT or condition ⁄AKTT. Let T be the mapping from C into E defined by 1 f Tx :¼ limn?1Tnx for all x 2 C and suppose that T is closed and FðTÞ ¼ \1 n¼1 FðT n Þ. Then, fxn gn¼0 converges strongly to PF x0 . k for each Proof. Define Fk(x, y) :¼ uk(y) uk(x), x, y 2 C and k = 1, 2, . . . , m. Then F T Frkk ¼ EPðF k Þ ¼ CMPðuk Þ ¼ F T u rk m k = 1, 2, . . . , m, and therefore fF k gm k¼1 satisfies conditions (A1) and (A2). Furthermore, one can easily show that fF k gk¼1 satisfies conditions (A3) and (A4). Therefore, from Theorem 3.1, we obtain the desired result. h Acknowledgement The author would like to express his sincere thanks to Professor Melvin Scott, the Editor and the referees for their valuable suggestions and comments on improving the manuscript. References [1] Y.I. Alber, S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces, PanAmer. Math. J 4 (1994) 39–54. [2] Y.I. Alber, Metric and generalized projection operator in Banach spaces: properties and applications, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics, vol. 178, Dekker, New York, NY, USA, 1996, pp. 15–50. [3] K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2006) 2350–2360.
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