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Convergence and certain control conditions for hybrid iterative algorithms
Applied Mathematics and Computation 219 (2013) 10325–10332
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Convergence and certain control conditions for hybrid iterative algorithms q Shuang Wang School of Mathematical Sciences, Yancheng Teachers University, Yancheng, 224051 Jiangsu, PR China
a r t i c l e
i n f o
a b s t r a c t Very recently, Yao et al. (2010) [4] proposed a hybrid iterative algorithm. Under the parameter sequences satisfying some quite restrictive conditions, they derived a strong convergence theorem in a Hilbert space. In this paper, under the weaker conditions, we prove the strong convergence of the sequence generated by their iterative algorithm to a common fixed point of an infinite family of nonexpansive mappings, which solves a variational inequality. An appropriate example, such that all conditions of this result that are satisfied and that other conditions are not satisfied, is provided. Moreover, the method of our paper is different from the method in Yao et al. (2010) [4]. Ó 2013 Elsevier Inc. All rights reserved.
Keywords: Strong convergence Variational inequality Fixed points k-Lipschitzian g-Strongly monotone Hilbert spaces
1. Introduction Let H be a real Hilbert space. Recall that a mapping T : H ! H is said to be nonexpansive if kTx Tyk 6 kx yk; 8x; y 2 H. We use FðTÞ to denote the set of fixed points of T, that is FðTÞ ¼ fx 2 H : Tx ¼ xg. Yamada [1] introduced the following hybrid iterative method for solving the variational inequality
xnþ1 ¼ Txn lkn FðTxn Þ;
n P 0;
ð1:1Þ 2
where F is a k-Lipschitzian and g-strongly monotone operator with k > 0; g > 0; 0 < l < 2g=k . Under a sequence fkn g of P 2 real numbers in (0, 1) satisfying the conditions: (C1) limn!1 kn ¼ 0, (C2) 1 n¼0 kn ¼ 1 and (C3) limn!1 ðkn knþ1 Þ=knþ1 ¼ 0, he proved that fxn g generated by (1.1) converges strongly to the unique solution of variational inequality
hF ~x; x ~xi P 0;
8x 2 FðTÞ:
An example of sequence fkn g which satisfies conditions (C1)-(C3) is given by kn ¼ 1=nr , where 0 < r < 1. We note that the condition (C3) was first used by Lions [2]. It was observed that Lion’s conditions on the sequence fkn g excluded the canonical choice kn ¼ 1=n. This was overcome in 2003 by Xu and Kim [3], if fkn g satisfies conditions (C1), (C2) and (C4)
ðC4Þ : lim kn =knþ1 ¼ 1; n!1
or equivalently;
lim ðkn knþ1 Þ=knþ1 ¼ 0:
n!1
It is easy to see the condition (C4) is strictly weaker than condition (C3), coupled with conditions (C1) and (C2). Moreover, (C4) includes the important and natural choice f1=ng of fkn g. Very recently, motivated by the above work, Yao et al. [4] considered the following algorithm: for an arbitrary point x0 2 C,
q
Supported by the Natural Science Foundation of Yancheng Teachers University under Grant (12YCKL001). E-mail addresses: [email protected], [email protected]
0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.04.004
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S. Wang / Applied Mathematics and Computation 219 (2013) 10325–10332
yn ¼ xn kn Fðxn Þ; xnþ1 ¼ ð1 an Þyn þ an W n yn ;
n P 0;
ð1:2Þ
where F is a k-Lipschitzian and monotone operator on H; W n is a W-mapping defined by below (2.4). They proved g-strongly if k 2 ½1; 1Þ; g 2 ð0; 1Þ, an 2 c; 12 for some c > 0 and fkn g satisfying appropriate conditions, the sequence fxn g and fyn g defined by (1.2) converge strongly to x 2 \1 n¼1 FðT n Þ, which solves the variational inequality
hFx ; x x i P 0;
8x 2 \ 1 n¼1 FðT n Þ:
We remind the reader of the following fact: in order to guarantee the strong convergence of the iterative sequence fxn g, we need k 2 ½1; 1Þ; g 2 ð0; 1Þ and an 2 ½c; 12 for some constant c > 0 in Yao et al. [4]. The assumptions are quite restrictive. The purpose of this paper is twofold. Firstly, we use contractions to regularize the nonexpansive mapping T to propose a scheme that generates a net fxt gt2ð0;g=k2 Þ and prove the strong convergence of fxt g to a fixed point x of T which solves the variational inequality hFx ; x ui 6 0; 8u 2 FðTÞ. Secondly, under the weaker conditions, we prove the sequence generated by the iterative algorithm (1.2) converges to a common fixed point of an infinite family of nonexpansive mappings, which solves variational inequality hFx ; x x i P 0; 8x 2 \1 n¼1 FðT n Þ. Moreover, the method of our paper is different from the method in Yao et al. [4]. An appropriate example, such that all conditions of this result that are satisfied and that other conditions are not satisfied, is provided. The advantages of our results in this paper are that weaker and fewer restrictions are imposed on parameters. 2. Preliminaries Let H be a real Hilbert space with inner product h; i and norm kk. For the sequence fxn g in H, we write xn * x to indicate that the sequence fxn g converges weakly to x. xn ! x means that fxn g converges strongly to x. In a real Hilbert space H, we have
kx yk2 ¼ kxk2 þ kyk2 2hx; yi;
8x; y 2 H:
ð2:1Þ
A mapping F : H ! H is called k-Lipschitzian if there exists a positive constant k such that
kFx Fyk 6 kkx yk;
8x; y 2 H:
ð2:2Þ
F is said to be g-strongly monotone if there exists a positive constant g such that
hFx Fy; x yi P gkx yk2 ;
8x; y 2 H:
ð2:3Þ
~ > 0 such that Let A be a strongly positive bounded linear operator on H, that is, there exists a constant c
~kxk2 ; hAx; xi P c
8x 2 H:
A typical problem is that of minimizing a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H:
min
x2FðTÞ
1 hAx; xi hx; bi; 2
where b is a given point in H. Remark 2.1. From the definition of A, we note that a strongly positive bounded linear operator A is a kAk-Lipschitzian and c~-strongly monotone operator. In order to prove our main results, we need the following Lemmas. Lemma 2.2 ([5]). Let H be a Hilbert space, C a closed convex subset of H, and T : C ! C a nonexpansive mapping with FðTÞ – ;, if fxn g is a sequence in C weakly converging to x and if fðI TÞxn g converges strongly to y, then ðI TÞx ¼ y. Lemma 2.3 ([6]). Let fxn g and fzn g be bounded sequences in Banach space E and fcn g be a sequence in ½0; 1 which satisfies the following condition:
0 < lim inf cn 6 lim sup cn < 1: n!1
n!1
Suppose that xnþ1 ¼ cn xn þ ð1 cn Þzn ; n P 0, and lim supn!1 ðkznþ1 zn k kxnþ1 xn kÞ 6 0. Then limn!1 kzn xn k ¼ 0. Lemma 2.4 ([7,8]). Let fsn g be a sequence of non-negative real numbers satisfying
snþ1 6 ð1 kn Þsn þ kn dn þ cn ;
n P 0;
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P1 where fkn g; fdn g and fcn g satisfy the following conditions: (i) fkn g ½0; 1 and n¼0 kn ¼ 1, (ii) lim supn!1 dn 6 0 or P1 P1 n¼0 kn dn < 1, (iii) cn P 0ðn P 0Þ; n¼0 cn < 1. Then limn!1 sn ¼ 0. Let fT i : H ! Hg be a family of infinitely nonexpansive mappings, fni g be a real sequence such that 0 6 ni 6 b < 1; 8i P 1. For any n P 1 define a mapping W n : H ! H as follows:
8 U n;nþ1 ¼ I; > > > > U n;n ¼ nn T n U n;nþ1 þ ð1 nn ÞI; > > > >U > n;n1 ¼ nn1 T n1 U n;n þ ð1 nn1 ÞI; > > > > < U n;k ¼ nk T k U n;kþ1 þ ð1 nk ÞI > > > U n;k1 ¼ nk1 T k1 U n;k þ ð1 nk1 ÞI > > > > > > > > > > U n;2 ¼ n2 T 2 U n;3 þ ð1 n2 ÞI; : W n ¼ U n;1 ¼ n1 T 1 U n;2 þ ð1 n1 ÞI:
ð2:4Þ
Such a mapping W n : H ! H is called a W-mapping generated by T n ; T n1 ; . . . ; T 1 and nn ; nn1 ; . . . ; n1 . We have the following crucial conclusion concerning W n . We can find them in [9–12]. Now we only need the following similar version in Hilbert spaces. Lemma 2.5. Let H be a real Hilbert space, fT i : H ! Hg be a family of infinitely nonexpansive mappings with \1 i¼1 FðT i Þ – ;; fni g be a real sequence such that 0 < ni 6 b < 1; 8i P 1. Then: (1) (2) (3) (4)
W n is a nonexpansive and FðW n Þ ¼ \1 i¼1 FðT i Þ for each n P 1; For every x 2 H and k 2 N, the limit limn!1 U n;k x exists; If we define a mapping W : H ! H as: Wx ¼ limn!1 W n x ¼ limn!1 U n;1 x, for every x 2 H. Then, FðWÞ ¼ \1 i¼1 FðT i Þ; For any bounded sequence fxn g H, we have limn!1 kWxn W n xn k ¼ 0. 2
Lemma 2.6. Let F be a k-Lipschitzian and g-strongly monotone operator on a Hilbert space H with 0 < g 6 k and 0 < t < g=k . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Then S ¼ ðI tFÞ : H ! H is a contraction with contraction coefficient st ¼ 1 tð2g tk Þ. Proof. From (2.1)–(2.3), we have
kSx Syk2 ¼ kðx yÞ tðFx FyÞk2 ¼ kx yk2 þ t2 kFx Fyk2 2thFx Fy; x yi 2
2
6 kx yk2 þ t 2 k kx yk2 2tgkx yk2 ¼ ½1 tð2g tk Þkx yk2 2
2
for all x; y 2 H. From 0 < g 6 k and 0 < t < g=k , we have 0 < 1 tð2g tk Þ < 1 and
kSx Syk 6 st kx yk; where
st ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 tð2g tk Þ 2 ð0; 1Þ. Hence S is a contraction with contraction coefficient
st . h
3. Main results Let F be a k-Lipschitzian and g-strongly monotone operator on H with 0 < g 6 k; T : H ! H be a nonexpansive mapping. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Let t 2 ð0; g=k Þ and st ¼ 1 tð2g tk Þ, consider a mapping St on H defined by
St x ¼ T½ðI tFÞx;
x 2 H:
It is easy to see that St is a contraction. Indeed, from Lemma 2.6, we have
kSt x St yk 6 kT½ðI tFÞx T½ðI tFÞyk 6 kðI tFÞx ðI tFÞyk 6 st kx yk for all x; y 2 H. Hence it has a unique fixed point, denoted xt , which uniquely solves the fixed point equation
xt ¼ T½ðI tFÞxt ;
xt 2 H:
ð3:1Þ
Theorem 3.1. Let H be a real Hilbert space. Let T : H ! H be a nonexpansive mapping such that FðTÞ–;. Let F be a k-Lipschitzian 2 and g-strongly monotone operator on H with 0 < g 6 k. For each t 2 ð0; g=k Þ, let the net fxt g be generated by (3.1). Then, as t ! 0, the net fxt g converges strongly to a fixed point x of T which solves the variational inequality:
hFx ; x ui 6 0;
8u 2 FðTÞ:
ð3:2Þ
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S. Wang / Applied Mathematics and Computation 219 (2013) 10325–10332
Proof. We first show the uniqueness of a solution of the variational inequality (3.2), which is indeed a consequence of the strong monotonicity of F. Suppose x 2 FðTÞ and ~ x 2 FðTÞ both are solutions to (3.2), then
hFx ; x ~xi 6 0
ð3:3Þ
hF ~x; ~x x i 6 0:
ð3:4Þ
and
Adding (3.3) to (3.4), we gets
hFx F ~x; x ~xi 6 0: The strong monotonicity of F implies that x ¼ ~ x and the uniqueness is proved. Below we use x 2 FðTÞ to denote the unique solution of (3.2). Next, we prove that fxt g is bounded. Taking u 2 FðTÞ, from (3.1) and using Lemma 2.6, we have
kxt uk ¼ kT½ðI tFÞxt Tuk 6 kðI tFÞxt uk 6 kðI tFÞxt ðI tFÞu tFuk 6 kðI tFÞxt ðI tFÞuk þ tkFuk 6 st kxt uk þ t kFuk; that is,
kxt uk 6
t kFuk: 1 st
ð3:5Þ
Observe that
lim
t!0þ
t 1 ¼ : 1 st g
From t ! 0, we may assume, without loss of generality, that t 6 kg2 , where is an arbitrary small positive number. Thus, we have 1t st is continuous, 8t 2 ½0; kg2 . Therefore, we obtain
sup
t g < þ1: : t 2 0; 2 1 st k
ð3:6Þ
From (3.5) and (3.6), we have fxt g is bounded and so is fFxt g. On the other hand, from (3.1), we obtain
kxt Txt k ¼ kT½ðI tFÞxt Txt k 6 kðI tFÞxt xt k ¼ tkFxt k ! 0ðt ! 0Þ:
ð3:7Þ
To prove that xt ! x . For a given u 2 FðTÞ, by (2.1) and using Lemma 2.6, we have 2 2 kxt uk2 ¼ kT½ðI tFÞxt Tuk 6 kðI tFÞxt ðI tFÞu tFuk 6 s2t kxt uk2 þ t 2 kFuk2 þ 2thðI tFÞu ðI tFÞxt ; Fui
6 st kxt uk2 þ t 2 kFuk2 þ 2thu xt ; Fui þ 2t 2 hFxt Fu; Fui 6 st kxt uk2 þ t 2 kFuk2 þ 2thu xt ; Fui þ 2t 2 kkxt ukkFuk: Therefore,
kxt uk2 6
t2 2t 2t2 k kFuk2 þ hu xt ; Fui þ kxt ukkFuk: 1 st 1 st 1 st
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 From st ¼ 1 tð2g tk Þ, 2t limt!0 1st hu xt ; Fui ¼ 0.
we
have
2
limt!0 1t st ¼ 0
ð3:8Þ 2
2t k limt!0 1 st ¼ 0.
and
Observe
that,
if
xt * u,
we
have
x, then by (3.8), we see Since fxt g is bounded, we see that if ft n g is a sequence in ð0; g2 such that tn ! 0 and xtn * ~ k
x. Moreover, by (3.7) and using Lemma 2.2, we have ~ x 2 FðTÞ. We next prove that ~ x solves the variational inequality xtn ! ~ (3.2). From (3.1) and u 2 FðTÞ, we have 2 kxt uk2 6 kðI tFÞxt uk ¼ kxt uk2 þ t 2 kFxt k2 2thFxt ; xt ui;
that is,
hFxt ; xt ui 6
t kFxt k2 : 2
Now replacing t in (3.9) with tn and letting n ! 1, we have
hF ~x; ~x ui 6 0:
ð3:9Þ
S. Wang / Applied Mathematics and Computation 219 (2013) 10325–10332
10329
That is ~ x 2 FðTÞ is a solution of (3.2), hence ~ x ¼ x by uniqueness. In a summary, we have shown that each cluster point of fxt g (at t ! 0) equals x . Therefore, xt ! x as t ! 0. Setting F ¼ A in Theorem 3.1, we can obtain the following result. Corollary 3.2. Let H be a real Hilbert space. Let T : H ! H be a nonexpansive mapping such that FðTÞ – ;. Let A be a strongly ~ 6 kAk. For each t 2 ð0; c~ 2 Þ, let the net fxt g be generated by positive bounded linear operator with coefficient 0 < c kAk
xt ¼ T½ðI tAÞxt . Then, as t ! 0, the net fxt g converges strongly to a fixed point x of T which solves the variational inequality:
hAx ; x ui 6 0;
8u 2 FðTÞ:
Setting F ¼ I, the identity mapping, in Theorem 3.1, we can obtain the following result. Corollary 3.3. Let H be a real Hilbert space. Let T : H ! H be a nonexpansive mapping such that FðTÞ–;. For each t 2 ð0; 1Þ, let the net fxt g be generated by xt ¼ T½ð1 tÞxt . Then, as t ! 0, the net fxt g converges strongly to a fixed point x of T which solves the variational inequality:
hx ; x ui 6 0;
8u 2 FðTÞ:
Theorem 3.4. Let H be a real Hilbert space. Let F be a k-Lipschitzian and g-strongly monotone operator on H with 0 < g 6 k. Let 1 fT n g1 n¼1 : H ! H be an infinite family of nonexpansive mappings such that \n¼1 FðT n Þ–; and W n be a W-mapping defined by (2.4). Let fkn g be a sequence in ½0; 1Þ and fan g be a sequences in ½0; 1. Assume that P (A1) limn!1 kn ¼ 0 and 1 n¼0 kn ¼ 1; (A2) 0 < c < lim inf n!1 an 6 lim supn!1 an < 1 for some c 2 ð0; 1Þ. For given x0 2 H arbitrarily, let the sequence fxn g be generated by (1.2). Then the sequence fxn g and fyn g converge strongly to a x 2 \1 n¼1 FðT n Þ which solves the variational inequality
hFx ; x ui 6 0;
8u 2 \1 n¼1 FðT n Þ:
Proof. We proceed with the following steps. Step 1. We claim that fxn g is bounded. From (A1), we may assume, without loss of generality, that 0 < kn 6 g2 for all n, k where is an arbitrary small positive number. In fact, let u 2 \1 n¼1 FðT n Þ, from (1.2) and using Lemma 2.6, we have
kyn uk ¼ kxn kn Fðxn Þ uk 6 kðI kn FÞxn ðI kn FÞu kn Fuk 6 skn kxn uk þ kn kFuk; where
skn
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ¼ 1 kn ð2g kn k Þ 2 ð0; 1Þ. Then from (1.2) and (3.10), we obtain
ð3:10Þ
kxnþ1 uk ¼ kð1 an Þðyn uÞ þ an ðW n yn uÞk 6 kyn uk 6 ½1 ð1 skn Þkxn uk þ kn kFuk kn kFuk : 6 max kxn uk; 1 skn By induction, we have
kxn uk 6 maxfkx0 uk; M1 kFukg;
n o where M 1 ¼ sup 1ksnk : 0 < kn 6 kg2 < þ1. Therefore, fxn g is bounded. We also obtain that fyn g, fW n yn g and fFxn g are n bounded. Step 2. We claim that limn!1 kxnþ1 xn k ¼ 0. To this end, define xnþ1 ¼ ð1 an Þxn þ an zn . We observe that
xnþ2 ð1 anþ1 Þxnþ1 xnþ1 ð1 an Þxn kznþ1 zn k ¼ anþ1 an ð1 anþ1 Þynþ1 þ anþ1 W nþ1 ynþ1 ð1 anþ1 Þxnþ1 ð1 an Þyn þ an W n yn ð1 an Þxn 6 anþ1 an anþ1 W nþ1 ynþ1 ð1 anþ1 Þknþ1 Fðxnþ1 Þ an W n yn ð1 an Þkn Fðxn Þ 6 a a nþ1
6
ð1 anþ1 Þknþ1
kFðx
Þk þ
n
ð1 an Þkn
kFðxn Þk þ W nþ1 ynþ1 W n yn
nþ1 anþ1 an ð1 cÞknþ1 ð1 cÞkn kFðxnþ1 Þk þ kFðxn Þk þ W nþ1 ynþ1 W n yn : 6 c c
ð3:11Þ
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S. Wang / Applied Mathematics and Computation 219 (2013) 10325–10332
From (2.4), for u 2 \1 n¼1 FðT n Þ, we have
kW nþ1 yn W n yn k ¼ n1 T 1 U nþ1;2 yn n1 T 1 U n;2 yn 6 n1 U nþ1;2 yn U n;2 yn ¼ n1 n2 T 2 U nþ1;3 yn n2 T 2 U n;3 yn 6 n1 n2 U nþ1;3 yn U n;3 yn 6 6 n1 n2 nn U nþ1;nþ1 yn U n;nþ1 yn ! nþ1 Y ¼ n1 n2 nn knnþ1 T nþ1 yn þ ð1 nnþ1 Þyn yn k 6 ni ðkT nþ1 yn uk þ ku yn kÞ nþ1 Y
6
! ni
i¼1 nþ1 Y ð2kyn ukÞ 6 M 2 ni ;
i¼1
ð3:12Þ
i¼1
where M 2 ¼ supf2kyn uk; n P 0g. By (1.2) and (3.12), we have
W nþ1 ynþ1 W n yn 6 W nþ1 ynþ1 W nþ1 yn þ kW nþ1 yn W n yn k 6 ynþ1 yn þ kW nþ1 yn W n yn k 6 kxnþ1 knþ1 Fðxnþ1 Þ xn þ kn Fðxn Þk þ M2
nþ1 Y ni i¼1
6 kxnþ1 xn k þ knþ1 kFðxnþ1 Þk þ kn kFðxn Þk þ M2
nþ1 Y ni :
ð3:13Þ
i¼1
Substituting (3.13) into (3.11), we have
kznþ1 zn k 6 ¼
ð1 cÞknþ1
c knþ1
c
kFðxnþ1 Þk þ
kFðxnþ1 Þk þ
kn
c
ð1 cÞkn
c
kFðxn Þk þ kxnþ1 xn k þ knþ1 kFðxnþ1 Þk þ kn kFðxn Þk þ M 2
nþ1 Y
ni
i¼1
kFðxn Þk þ kxnþ1 xn k þ M 2
nþ1 Y
ni ;
i¼1
that is,
kznþ1 zn k kxnþ1 xn k 6
knþ1
c
kFðxnþ1 Þk þ
kn
c
kFðxn Þk þ M 2
nþ1 Y ni : i¼1
Observing limn!1 kn ¼ 0 and 0 < ni 6 b < 1, it follows that
lim sup ðkznþ1 zn k kxnþ1 xn kÞ 6 0:
ð3:14Þ
n!1
By (A2) and using Lemma 2.3, we have limn!1 kzn xn k ¼ 0. Therefore
lim kxnþ1 xn k ¼ lim an kzn xn k ¼ 0:
n!1
n!1
Step 3. We claim that limn!1 kxn W n xn k ¼ 0. Observe that
kxn W n xn k 6 kxn xnþ1 k þ kxnþ1 W n xn k 6 kxn xnþ1 k þ ð1 an Þkyn W n xn k þ an kW n yn W n xn k 6 kxn xnþ1 k þ ð1 an Þkyn xn k þ ð1 an Þkxn W n xn k þ an kyn xn k ¼ kxn xnþ1 k þ kyn xn k þ ð1 an Þkxn W n xn k; that is,
kxn W n xn k 6
1
an
ðkxnþ1 xn k þ kyn xn kÞ 6
1
c
ðkxnþ1 xn k þ kn kFðxn ÞkÞ ! 0ðn ! 1Þ:
ð3:15Þ
Step 4. We claim that limn!1 kxn Wxn k ¼ 0. Observe that
kxn Wxn k 6 kxn W n xn k þ kW n xn Wxn k:
ð3:16Þ
By (3.15) and (3.16) and using Lemma 2.5, we obtain
lim kxn Wxn k ¼ 0:
n!1
Step 5. We claim that lim supn!1 hFx ; x xn i 6 0, where x ¼ limt!0 xt and xt defined by xt ¼ W½ð1 tFÞxt . Since xn is bounded, there exists a subsequence fxnk g of fxn g which converges weakly to x. From Step 4, we obtain Wxnk * x. From Lemma 2.2, we have x 2 FðWÞ. Hence, by Theorem 3.1, we have
lim suphFx ; x xn i ¼ lim hFx ; x xnk i ¼ hFx ; x xi 6 0: n!1
k!1
Step 6. We claim that fxn g and fyn g converge strongly to x 2 \1 n¼1 FðT n Þ. From (1.2), we have
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S. Wang / Applied Mathematics and Computation 219 (2013) 10325–10332 2 kxnþ1 x k2 6 ð1 an Þkyn x k2 þ an kW n yn x k2 6 kyn x k2 ¼ kxn kn Fðxn Þ x k 2
2
6 kðI kn FÞxn ðI kn FÞx kn Fx k 6 s2kn kxn x k2 þ k2n kFx k þ 2kn hðI kn FÞx ðI kn FÞxn ; Fx i 2
6 skn kxn x k2 þ k2n kFx k þ 2kn hx xn ; Fx i þ 2k2n hFxn Fx ; Fx i 2
6 ½1 ð1 skn Þkxn x k2 þ k2n kFx k þ 2kn hx xn ; Fx i þ 2kk2n kxn x kkFx k " # k2n 2kn 2k2n 2 6 ½1 ð1 skn Þkxn x k þ ð1 skn Þ M3 þ hx xn ; Fx i þ M3 1 s kn 1 s kn 1 skn " # 3k2n 6 ½1 ð1 skn Þkxn x k2 þ ð1 skn Þ M 3 þ 2M 1 hx xn ; Fx i ¼ ð1 ln Þkxn x k2 þ ln dn ; 1 s kn 2
2
n M3 where M 3 ¼ supfkFx k ; kkxn x kkFx kg, ln ¼ 1 skn , dn ¼ 3k þ 2M 1 hx xn ; Fx i. It is easy to see that 1skn lim supn!1 dn 6 0. Hence, by Lemma 2.4, the sequence fxn g converges strongly to x 2 \1 n¼1 FðT n Þ. On the other hand, we have
P1
n¼1
ln ¼ 1 and
kyn xn k 6 kn kFðxn Þk ! 0 ðn ! 1Þ: It follows that the sequence fyn g converges strongly to x 2 \1 n¼1 FðT n Þ. From x ¼ limt!0 xt and Theorem 3.1, we have x is the unique solution of the variational inequality: hFx ; x ui 6 0; u 2 \1 FðT Þ. h n n¼1 From Theorem 3.4, we can obtain the following result immediately.
Corollary 3.5 (Yao et al. [4, Theorem 3.2]). Let H be a real Hilbert space. Let F : H ! H be k-Lipschitzian and g-strongly monotone 1 operator with k 2 ½1; 1Þ and g 2 ð0; 1Þ. Let fT n g1 n¼1 : H ! H be an infinite family of nonexpansive mappings such that \n¼1 FðT n Þ – ; and fW n g be W-mapping defined by (2.4). Let fkn g be a sequence in ½0; 1Þ and fan g be a sequences in ½0; 1. Assume that (i) limn!1 kn ¼ 0; P1 (ii) n¼0kn ¼ 1; (iii) an 2 c; 12 for some c > 0. Then the sequence fxn g and fyn g generated by (1.2) converge strongly to x 2 \1 n¼1 FðT n Þ which solves the following variational inequality hFx ; x xi 6 0; x 2 \1 n¼1 FðT n Þ. Remark 3.6. Theorem 3.4 include [4, Theorem 3.2] as a special case. The following example shows that all conditions of Theorem 3.4 are satisfied. However, the conditions g 2 ð0; 1Þ and an 2 c; 12 for some c > 0 in [4, Theorem 3.2] are not satisfied. Example 3.7. Let H ¼ R the set of real numbers and T n T. Define a nonexpansive mapping T : H ! H and an operator F : H ! H as follows:
Tx ¼ 0 and FðxÞ ¼ x;
8x 2 R:
1 1 It is easy to see that FðTÞ ¼ f0g; \1 n¼1 FðT n Þ ¼ f0g and W n x ¼ ð1 n1 Þx; 8x 2 R. Take n1 ¼ 2, We have W n x ¼ 2 x; 8x 2 R. Given sequences fan g and fkn g : an ¼ 23 ; kn ¼ 1n for all n P 0. For an arbitrary x0 2 H, let fxn g defined as follows
yn ¼ xn kn Fðxn Þ; xnþ1 ¼ ð1 an Þyn þ an W n yn ;
that is,
1 2 y þ W n yn 3 n 3 2 1 ¼ xn Fðxn Þ 3 n 2 1 xn ; n P 0: 1 ¼ 3 n
xnþ1 ¼
Observe that for all n P 0,
2 1 x 1 0 kxnþ1 0k ¼ n 3 n 2 1 ¼ kxn 0k 1 3 n 2 6 kxn 0k: 3
n P 0;
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nþ1 Hence we have kxnþ1 0k 6 23 kx0 0k for all n P 0. This implies that fxn g converges strongly to 0 2 \1 n¼1 FðT n Þ. On one other hand, we have
kyn 0k ¼
1 1 kxn k 6 kxn k: n
Therefore fyn g converges strongly to 0 2 \1 n¼1 FðT n Þ. On the other hand, we have hFð0Þ; 0 ui 6 0; u 2 \1 n¼1 FðT n Þ, that is, 0 is the solution of the variational inequality hFx ; x ui 6 0; u 2 \1 FðT Þ. n n¼1 By Fx ¼ x, we have g ¼ k ¼ 1. Furthermore, it is easy to see that there hold the following: P (B1) limn!1 kn ¼ 0 and 1 n¼0 kn ¼ 1; (B2) 0 < 12 < lim inf n!1 an ¼ 23 ¼ lim supn!1 an < 1 for some constant c ¼ 12. Hence there is no doubt that all conditions of Theorem 3.4 are satisfied. Since an ¼ 23 R c; 12 , the conditions an 2 c; 12 for some c > 0 and g 2 ð0; 1Þ of Yao et al. [4, Theorem 3.2] are not satisfied. Acknowledgments The author thanks the editor and the referees for their useful comments and suggestions. References [1] I. Yamada, The hybrid steepest descent for the variational inequality problems over the intersection of fixed points sets of nonexpansive mapping, in: D. Butnariu, Y. Censor, S. Reich (Eds.), Inherently Parallel Algorithms in Feasibility and Optimization and Their Application, Elservier, New York, 2001, pp. 473–504. [2] P.L. Lions, Approximation de points fixes de contractions, C.R. Acad. Sci. Paris 284 (1977) 1357–1359. [3] H.K. Xu, T.H. Kim, Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim. Theory Appl. 119 (2003) 185–201. [4] Y.H. Yao, M.N. Noor, Y.C. Liou, A new hybrid iterative algorithm for variational inequalities, Appl. Math. Comput. 216 (2010) 822–829. [5] K. Geobel, W.A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge Univ. Press, 1990. pp. 473– 504. [6] T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one parameter nonexpansive semigroups without Bochner integral, J. Math. Anal. Appl. 35 (2005) 227–239. [7] L.S. Liu, Iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995) 114–125. [8] H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002) 240–256. [9] W. Takahashi, K. Shimoji, Convergence theorems for nonexpansive mappings and feasibility problems, Math. Comput. Model. 32 (2000) 1463–1471. [10] K. Shimoji, W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math. 5 (2001) 387–404. [11] S.-S. Chang, A new method for solving equilibrium problem and variational inequality problem with application to optimization, Nonlinear Anal. 70 (2009) 3307–3319. [12] Y.H. Yao, Y.C. Liou, J.C. Yao, Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings, Fixed Point Theory Appl. 2007 (2007) (Article ID 64363, 12 p.).
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Convergence and certain control conditions for hybrid iterative algorithms q Shuang Wang School of Mathematical Sciences, Yancheng Teachers University, Yancheng, 224051 Jiangsu, PR China
a r t i c l e
i n f o
a b s t r a c t Very recently, Yao et al. (2010) [4] proposed a hybrid iterative algorithm. Under the parameter sequences satisfying some quite restrictive conditions, they derived a strong convergence theorem in a Hilbert space. In this paper, under the weaker conditions, we prove the strong convergence of the sequence generated by their iterative algorithm to a common fixed point of an infinite family of nonexpansive mappings, which solves a variational inequality. An appropriate example, such that all conditions of this result that are satisfied and that other conditions are not satisfied, is provided. Moreover, the method of our paper is different from the method in Yao et al. (2010) [4]. Ó 2013 Elsevier Inc. All rights reserved.
Keywords: Strong convergence Variational inequality Fixed points k-Lipschitzian g-Strongly monotone Hilbert spaces
1. Introduction Let H be a real Hilbert space. Recall that a mapping T : H ! H is said to be nonexpansive if kTx Tyk 6 kx yk; 8x; y 2 H. We use FðTÞ to denote the set of fixed points of T, that is FðTÞ ¼ fx 2 H : Tx ¼ xg. Yamada [1] introduced the following hybrid iterative method for solving the variational inequality
xnþ1 ¼ Txn lkn FðTxn Þ;
n P 0;
ð1:1Þ 2
where F is a k-Lipschitzian and g-strongly monotone operator with k > 0; g > 0; 0 < l < 2g=k . Under a sequence fkn g of P 2 real numbers in (0, 1) satisfying the conditions: (C1) limn!1 kn ¼ 0, (C2) 1 n¼0 kn ¼ 1 and (C3) limn!1 ðkn knþ1 Þ=knþ1 ¼ 0, he proved that fxn g generated by (1.1) converges strongly to the unique solution of variational inequality
hF ~x; x ~xi P 0;
8x 2 FðTÞ:
An example of sequence fkn g which satisfies conditions (C1)-(C3) is given by kn ¼ 1=nr , where 0 < r < 1. We note that the condition (C3) was first used by Lions [2]. It was observed that Lion’s conditions on the sequence fkn g excluded the canonical choice kn ¼ 1=n. This was overcome in 2003 by Xu and Kim [3], if fkn g satisfies conditions (C1), (C2) and (C4)
ðC4Þ : lim kn =knþ1 ¼ 1; n!1
or equivalently;
lim ðkn knþ1 Þ=knþ1 ¼ 0:
n!1
It is easy to see the condition (C4) is strictly weaker than condition (C3), coupled with conditions (C1) and (C2). Moreover, (C4) includes the important and natural choice f1=ng of fkn g. Very recently, motivated by the above work, Yao et al. [4] considered the following algorithm: for an arbitrary point x0 2 C,
q
Supported by the Natural Science Foundation of Yancheng Teachers University under Grant (12YCKL001). E-mail addresses: [email protected], [email protected]
0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.04.004
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S. Wang / Applied Mathematics and Computation 219 (2013) 10325–10332
yn ¼ xn kn Fðxn Þ; xnþ1 ¼ ð1 an Þyn þ an W n yn ;
n P 0;
ð1:2Þ
where F is a k-Lipschitzian and monotone operator on H; W n is a W-mapping defined by below (2.4). They proved g-strongly if k 2 ½1; 1Þ; g 2 ð0; 1Þ, an 2 c; 12 for some c > 0 and fkn g satisfying appropriate conditions, the sequence fxn g and fyn g defined by (1.2) converge strongly to x 2 \1 n¼1 FðT n Þ, which solves the variational inequality
hFx ; x x i P 0;
8x 2 \ 1 n¼1 FðT n Þ:
We remind the reader of the following fact: in order to guarantee the strong convergence of the iterative sequence fxn g, we need k 2 ½1; 1Þ; g 2 ð0; 1Þ and an 2 ½c; 12 for some constant c > 0 in Yao et al. [4]. The assumptions are quite restrictive. The purpose of this paper is twofold. Firstly, we use contractions to regularize the nonexpansive mapping T to propose a scheme that generates a net fxt gt2ð0;g=k2 Þ and prove the strong convergence of fxt g to a fixed point x of T which solves the variational inequality hFx ; x ui 6 0; 8u 2 FðTÞ. Secondly, under the weaker conditions, we prove the sequence generated by the iterative algorithm (1.2) converges to a common fixed point of an infinite family of nonexpansive mappings, which solves variational inequality hFx ; x x i P 0; 8x 2 \1 n¼1 FðT n Þ. Moreover, the method of our paper is different from the method in Yao et al. [4]. An appropriate example, such that all conditions of this result that are satisfied and that other conditions are not satisfied, is provided. The advantages of our results in this paper are that weaker and fewer restrictions are imposed on parameters. 2. Preliminaries Let H be a real Hilbert space with inner product h; i and norm kk. For the sequence fxn g in H, we write xn * x to indicate that the sequence fxn g converges weakly to x. xn ! x means that fxn g converges strongly to x. In a real Hilbert space H, we have
kx yk2 ¼ kxk2 þ kyk2 2hx; yi;
8x; y 2 H:
ð2:1Þ
A mapping F : H ! H is called k-Lipschitzian if there exists a positive constant k such that
kFx Fyk 6 kkx yk;
8x; y 2 H:
ð2:2Þ
F is said to be g-strongly monotone if there exists a positive constant g such that
hFx Fy; x yi P gkx yk2 ;
8x; y 2 H:
ð2:3Þ
~ > 0 such that Let A be a strongly positive bounded linear operator on H, that is, there exists a constant c
~kxk2 ; hAx; xi P c
8x 2 H:
A typical problem is that of minimizing a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H:
min
x2FðTÞ
1 hAx; xi hx; bi; 2
where b is a given point in H. Remark 2.1. From the definition of A, we note that a strongly positive bounded linear operator A is a kAk-Lipschitzian and c~-strongly monotone operator. In order to prove our main results, we need the following Lemmas. Lemma 2.2 ([5]). Let H be a Hilbert space, C a closed convex subset of H, and T : C ! C a nonexpansive mapping with FðTÞ – ;, if fxn g is a sequence in C weakly converging to x and if fðI TÞxn g converges strongly to y, then ðI TÞx ¼ y. Lemma 2.3 ([6]). Let fxn g and fzn g be bounded sequences in Banach space E and fcn g be a sequence in ½0; 1 which satisfies the following condition:
0 < lim inf cn 6 lim sup cn < 1: n!1
n!1
Suppose that xnþ1 ¼ cn xn þ ð1 cn Þzn ; n P 0, and lim supn!1 ðkznþ1 zn k kxnþ1 xn kÞ 6 0. Then limn!1 kzn xn k ¼ 0. Lemma 2.4 ([7,8]). Let fsn g be a sequence of non-negative real numbers satisfying
snþ1 6 ð1 kn Þsn þ kn dn þ cn ;
n P 0;
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S. Wang / Applied Mathematics and Computation 219 (2013) 10325–10332
P1 where fkn g; fdn g and fcn g satisfy the following conditions: (i) fkn g ½0; 1 and n¼0 kn ¼ 1, (ii) lim supn!1 dn 6 0 or P1 P1 n¼0 kn dn < 1, (iii) cn P 0ðn P 0Þ; n¼0 cn < 1. Then limn!1 sn ¼ 0. Let fT i : H ! Hg be a family of infinitely nonexpansive mappings, fni g be a real sequence such that 0 6 ni 6 b < 1; 8i P 1. For any n P 1 define a mapping W n : H ! H as follows:
8 U n;nþ1 ¼ I; > > > > U n;n ¼ nn T n U n;nþ1 þ ð1 nn ÞI; > > > >U > n;n1 ¼ nn1 T n1 U n;n þ ð1 nn1 ÞI; > > > > < U n;k ¼ nk T k U n;kþ1 þ ð1 nk ÞI > > > U n;k1 ¼ nk1 T k1 U n;k þ ð1 nk1 ÞI > > > > > > > > > > U n;2 ¼ n2 T 2 U n;3 þ ð1 n2 ÞI; : W n ¼ U n;1 ¼ n1 T 1 U n;2 þ ð1 n1 ÞI:
ð2:4Þ
Such a mapping W n : H ! H is called a W-mapping generated by T n ; T n1 ; . . . ; T 1 and nn ; nn1 ; . . . ; n1 . We have the following crucial conclusion concerning W n . We can find them in [9–12]. Now we only need the following similar version in Hilbert spaces. Lemma 2.5. Let H be a real Hilbert space, fT i : H ! Hg be a family of infinitely nonexpansive mappings with \1 i¼1 FðT i Þ – ;; fni g be a real sequence such that 0 < ni 6 b < 1; 8i P 1. Then: (1) (2) (3) (4)
W n is a nonexpansive and FðW n Þ ¼ \1 i¼1 FðT i Þ for each n P 1; For every x 2 H and k 2 N, the limit limn!1 U n;k x exists; If we define a mapping W : H ! H as: Wx ¼ limn!1 W n x ¼ limn!1 U n;1 x, for every x 2 H. Then, FðWÞ ¼ \1 i¼1 FðT i Þ; For any bounded sequence fxn g H, we have limn!1 kWxn W n xn k ¼ 0. 2
Lemma 2.6. Let F be a k-Lipschitzian and g-strongly monotone operator on a Hilbert space H with 0 < g 6 k and 0 < t < g=k . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Then S ¼ ðI tFÞ : H ! H is a contraction with contraction coefficient st ¼ 1 tð2g tk Þ. Proof. From (2.1)–(2.3), we have
kSx Syk2 ¼ kðx yÞ tðFx FyÞk2 ¼ kx yk2 þ t2 kFx Fyk2 2thFx Fy; x yi 2
2
6 kx yk2 þ t 2 k kx yk2 2tgkx yk2 ¼ ½1 tð2g tk Þkx yk2 2
2
for all x; y 2 H. From 0 < g 6 k and 0 < t < g=k , we have 0 < 1 tð2g tk Þ < 1 and
kSx Syk 6 st kx yk; where
st ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 tð2g tk Þ 2 ð0; 1Þ. Hence S is a contraction with contraction coefficient
st . h
3. Main results Let F be a k-Lipschitzian and g-strongly monotone operator on H with 0 < g 6 k; T : H ! H be a nonexpansive mapping. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Let t 2 ð0; g=k Þ and st ¼ 1 tð2g tk Þ, consider a mapping St on H defined by
St x ¼ T½ðI tFÞx;
x 2 H:
It is easy to see that St is a contraction. Indeed, from Lemma 2.6, we have
kSt x St yk 6 kT½ðI tFÞx T½ðI tFÞyk 6 kðI tFÞx ðI tFÞyk 6 st kx yk for all x; y 2 H. Hence it has a unique fixed point, denoted xt , which uniquely solves the fixed point equation
xt ¼ T½ðI tFÞxt ;
xt 2 H:
ð3:1Þ
Theorem 3.1. Let H be a real Hilbert space. Let T : H ! H be a nonexpansive mapping such that FðTÞ–;. Let F be a k-Lipschitzian 2 and g-strongly monotone operator on H with 0 < g 6 k. For each t 2 ð0; g=k Þ, let the net fxt g be generated by (3.1). Then, as t ! 0, the net fxt g converges strongly to a fixed point x of T which solves the variational inequality:
hFx ; x ui 6 0;
8u 2 FðTÞ:
ð3:2Þ
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Proof. We first show the uniqueness of a solution of the variational inequality (3.2), which is indeed a consequence of the strong monotonicity of F. Suppose x 2 FðTÞ and ~ x 2 FðTÞ both are solutions to (3.2), then
hFx ; x ~xi 6 0
ð3:3Þ
hF ~x; ~x x i 6 0:
ð3:4Þ
and
Adding (3.3) to (3.4), we gets
hFx F ~x; x ~xi 6 0: The strong monotonicity of F implies that x ¼ ~ x and the uniqueness is proved. Below we use x 2 FðTÞ to denote the unique solution of (3.2). Next, we prove that fxt g is bounded. Taking u 2 FðTÞ, from (3.1) and using Lemma 2.6, we have
kxt uk ¼ kT½ðI tFÞxt Tuk 6 kðI tFÞxt uk 6 kðI tFÞxt ðI tFÞu tFuk 6 kðI tFÞxt ðI tFÞuk þ tkFuk 6 st kxt uk þ t kFuk; that is,
kxt uk 6
t kFuk: 1 st
ð3:5Þ
Observe that
lim
t!0þ
t 1 ¼ : 1 st g
From t ! 0, we may assume, without loss of generality, that t 6 kg2 , where is an arbitrary small positive number. Thus, we have 1t st is continuous, 8t 2 ½0; kg2 . Therefore, we obtain
sup
t g < þ1: : t 2 0; 2 1 st k
ð3:6Þ
From (3.5) and (3.6), we have fxt g is bounded and so is fFxt g. On the other hand, from (3.1), we obtain
kxt Txt k ¼ kT½ðI tFÞxt Txt k 6 kðI tFÞxt xt k ¼ tkFxt k ! 0ðt ! 0Þ:
ð3:7Þ
To prove that xt ! x . For a given u 2 FðTÞ, by (2.1) and using Lemma 2.6, we have 2 2 kxt uk2 ¼ kT½ðI tFÞxt Tuk 6 kðI tFÞxt ðI tFÞu tFuk 6 s2t kxt uk2 þ t 2 kFuk2 þ 2thðI tFÞu ðI tFÞxt ; Fui
6 st kxt uk2 þ t 2 kFuk2 þ 2thu xt ; Fui þ 2t 2 hFxt Fu; Fui 6 st kxt uk2 þ t 2 kFuk2 þ 2thu xt ; Fui þ 2t 2 kkxt ukkFuk: Therefore,
kxt uk2 6
t2 2t 2t2 k kFuk2 þ hu xt ; Fui þ kxt ukkFuk: 1 st 1 st 1 st
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 From st ¼ 1 tð2g tk Þ, 2t limt!0 1st hu xt ; Fui ¼ 0.
we
have
2
limt!0 1t st ¼ 0
ð3:8Þ 2
2t k limt!0 1 st ¼ 0.
and
Observe
that,
if
xt * u,
we
have
x, then by (3.8), we see Since fxt g is bounded, we see that if ft n g is a sequence in ð0; g2 such that tn ! 0 and xtn * ~ k
x. Moreover, by (3.7) and using Lemma 2.2, we have ~ x 2 FðTÞ. We next prove that ~ x solves the variational inequality xtn ! ~ (3.2). From (3.1) and u 2 FðTÞ, we have 2 kxt uk2 6 kðI tFÞxt uk ¼ kxt uk2 þ t 2 kFxt k2 2thFxt ; xt ui;
that is,
hFxt ; xt ui 6
t kFxt k2 : 2
Now replacing t in (3.9) with tn and letting n ! 1, we have
hF ~x; ~x ui 6 0:
ð3:9Þ
S. Wang / Applied Mathematics and Computation 219 (2013) 10325–10332
10329
That is ~ x 2 FðTÞ is a solution of (3.2), hence ~ x ¼ x by uniqueness. In a summary, we have shown that each cluster point of fxt g (at t ! 0) equals x . Therefore, xt ! x as t ! 0. Setting F ¼ A in Theorem 3.1, we can obtain the following result. Corollary 3.2. Let H be a real Hilbert space. Let T : H ! H be a nonexpansive mapping such that FðTÞ – ;. Let A be a strongly ~ 6 kAk. For each t 2 ð0; c~ 2 Þ, let the net fxt g be generated by positive bounded linear operator with coefficient 0 < c kAk
xt ¼ T½ðI tAÞxt . Then, as t ! 0, the net fxt g converges strongly to a fixed point x of T which solves the variational inequality:
hAx ; x ui 6 0;
8u 2 FðTÞ:
Setting F ¼ I, the identity mapping, in Theorem 3.1, we can obtain the following result. Corollary 3.3. Let H be a real Hilbert space. Let T : H ! H be a nonexpansive mapping such that FðTÞ–;. For each t 2 ð0; 1Þ, let the net fxt g be generated by xt ¼ T½ð1 tÞxt . Then, as t ! 0, the net fxt g converges strongly to a fixed point x of T which solves the variational inequality:
hx ; x ui 6 0;
8u 2 FðTÞ:
Theorem 3.4. Let H be a real Hilbert space. Let F be a k-Lipschitzian and g-strongly monotone operator on H with 0 < g 6 k. Let 1 fT n g1 n¼1 : H ! H be an infinite family of nonexpansive mappings such that \n¼1 FðT n Þ–; and W n be a W-mapping defined by (2.4). Let fkn g be a sequence in ½0; 1Þ and fan g be a sequences in ½0; 1. Assume that P (A1) limn!1 kn ¼ 0 and 1 n¼0 kn ¼ 1; (A2) 0 < c < lim inf n!1 an 6 lim supn!1 an < 1 for some c 2 ð0; 1Þ. For given x0 2 H arbitrarily, let the sequence fxn g be generated by (1.2). Then the sequence fxn g and fyn g converge strongly to a x 2 \1 n¼1 FðT n Þ which solves the variational inequality
hFx ; x ui 6 0;
8u 2 \1 n¼1 FðT n Þ:
Proof. We proceed with the following steps. Step 1. We claim that fxn g is bounded. From (A1), we may assume, without loss of generality, that 0 < kn 6 g2 for all n, k where is an arbitrary small positive number. In fact, let u 2 \1 n¼1 FðT n Þ, from (1.2) and using Lemma 2.6, we have
kyn uk ¼ kxn kn Fðxn Þ uk 6 kðI kn FÞxn ðI kn FÞu kn Fuk 6 skn kxn uk þ kn kFuk; where
skn
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ¼ 1 kn ð2g kn k Þ 2 ð0; 1Þ. Then from (1.2) and (3.10), we obtain
ð3:10Þ
kxnþ1 uk ¼ kð1 an Þðyn uÞ þ an ðW n yn uÞk 6 kyn uk 6 ½1 ð1 skn Þkxn uk þ kn kFuk kn kFuk : 6 max kxn uk; 1 skn By induction, we have
kxn uk 6 maxfkx0 uk; M1 kFukg;
n o where M 1 ¼ sup 1ksnk : 0 < kn 6 kg2 < þ1. Therefore, fxn g is bounded. We also obtain that fyn g, fW n yn g and fFxn g are n bounded. Step 2. We claim that limn!1 kxnþ1 xn k ¼ 0. To this end, define xnþ1 ¼ ð1 an Þxn þ an zn . We observe that
xnþ2 ð1 anþ1 Þxnþ1 xnþ1 ð1 an Þxn kznþ1 zn k ¼ anþ1 an ð1 anþ1 Þynþ1 þ anþ1 W nþ1 ynþ1 ð1 anþ1 Þxnþ1 ð1 an Þyn þ an W n yn ð1 an Þxn 6 anþ1 an anþ1 W nþ1 ynþ1 ð1 anþ1 Þknþ1 Fðxnþ1 Þ an W n yn ð1 an Þkn Fðxn Þ 6 a a nþ1
6
ð1 anþ1 Þknþ1
kFðx
Þk þ
n
ð1 an Þkn
kFðxn Þk þ W nþ1 ynþ1 W n yn
nþ1 anþ1 an ð1 cÞknþ1 ð1 cÞkn kFðxnþ1 Þk þ kFðxn Þk þ W nþ1 ynþ1 W n yn : 6 c c
ð3:11Þ
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From (2.4), for u 2 \1 n¼1 FðT n Þ, we have
kW nþ1 yn W n yn k ¼ n1 T 1 U nþ1;2 yn n1 T 1 U n;2 yn 6 n1 U nþ1;2 yn U n;2 yn ¼ n1 n2 T 2 U nþ1;3 yn n2 T 2 U n;3 yn 6 n1 n2 U nþ1;3 yn U n;3 yn 6 6 n1 n2 nn U nþ1;nþ1 yn U n;nþ1 yn ! nþ1 Y ¼ n1 n2 nn knnþ1 T nþ1 yn þ ð1 nnþ1 Þyn yn k 6 ni ðkT nþ1 yn uk þ ku yn kÞ nþ1 Y
6
! ni
i¼1 nþ1 Y ð2kyn ukÞ 6 M 2 ni ;
i¼1
ð3:12Þ
i¼1
where M 2 ¼ supf2kyn uk; n P 0g. By (1.2) and (3.12), we have
W nþ1 ynþ1 W n yn 6 W nþ1 ynþ1 W nþ1 yn þ kW nþ1 yn W n yn k 6 ynþ1 yn þ kW nþ1 yn W n yn k 6 kxnþ1 knþ1 Fðxnþ1 Þ xn þ kn Fðxn Þk þ M2
nþ1 Y ni i¼1
6 kxnþ1 xn k þ knþ1 kFðxnþ1 Þk þ kn kFðxn Þk þ M2
nþ1 Y ni :
ð3:13Þ
i¼1
Substituting (3.13) into (3.11), we have
kznþ1 zn k 6 ¼
ð1 cÞknþ1
c knþ1
c
kFðxnþ1 Þk þ
kFðxnþ1 Þk þ
kn
c
ð1 cÞkn
c
kFðxn Þk þ kxnþ1 xn k þ knþ1 kFðxnþ1 Þk þ kn kFðxn Þk þ M 2
nþ1 Y
ni
i¼1
kFðxn Þk þ kxnþ1 xn k þ M 2
nþ1 Y
ni ;
i¼1
that is,
kznþ1 zn k kxnþ1 xn k 6
knþ1
c
kFðxnþ1 Þk þ
kn
c
kFðxn Þk þ M 2
nþ1 Y ni : i¼1
Observing limn!1 kn ¼ 0 and 0 < ni 6 b < 1, it follows that
lim sup ðkznþ1 zn k kxnþ1 xn kÞ 6 0:
ð3:14Þ
n!1
By (A2) and using Lemma 2.3, we have limn!1 kzn xn k ¼ 0. Therefore
lim kxnþ1 xn k ¼ lim an kzn xn k ¼ 0:
n!1
n!1
Step 3. We claim that limn!1 kxn W n xn k ¼ 0. Observe that
kxn W n xn k 6 kxn xnþ1 k þ kxnþ1 W n xn k 6 kxn xnþ1 k þ ð1 an Þkyn W n xn k þ an kW n yn W n xn k 6 kxn xnþ1 k þ ð1 an Þkyn xn k þ ð1 an Þkxn W n xn k þ an kyn xn k ¼ kxn xnþ1 k þ kyn xn k þ ð1 an Þkxn W n xn k; that is,
kxn W n xn k 6
1
an
ðkxnþ1 xn k þ kyn xn kÞ 6
1
c
ðkxnþ1 xn k þ kn kFðxn ÞkÞ ! 0ðn ! 1Þ:
ð3:15Þ
Step 4. We claim that limn!1 kxn Wxn k ¼ 0. Observe that
kxn Wxn k 6 kxn W n xn k þ kW n xn Wxn k:
ð3:16Þ
By (3.15) and (3.16) and using Lemma 2.5, we obtain
lim kxn Wxn k ¼ 0:
n!1
Step 5. We claim that lim supn!1 hFx ; x xn i 6 0, where x ¼ limt!0 xt and xt defined by xt ¼ W½ð1 tFÞxt . Since xn is bounded, there exists a subsequence fxnk g of fxn g which converges weakly to x. From Step 4, we obtain Wxnk * x. From Lemma 2.2, we have x 2 FðWÞ. Hence, by Theorem 3.1, we have
lim suphFx ; x xn i ¼ lim hFx ; x xnk i ¼ hFx ; x xi 6 0: n!1
k!1
Step 6. We claim that fxn g and fyn g converge strongly to x 2 \1 n¼1 FðT n Þ. From (1.2), we have
10331
S. Wang / Applied Mathematics and Computation 219 (2013) 10325–10332 2 kxnþ1 x k2 6 ð1 an Þkyn x k2 þ an kW n yn x k2 6 kyn x k2 ¼ kxn kn Fðxn Þ x k 2
2
6 kðI kn FÞxn ðI kn FÞx kn Fx k 6 s2kn kxn x k2 þ k2n kFx k þ 2kn hðI kn FÞx ðI kn FÞxn ; Fx i 2
6 skn kxn x k2 þ k2n kFx k þ 2kn hx xn ; Fx i þ 2k2n hFxn Fx ; Fx i 2
6 ½1 ð1 skn Þkxn x k2 þ k2n kFx k þ 2kn hx xn ; Fx i þ 2kk2n kxn x kkFx k " # k2n 2kn 2k2n 2 6 ½1 ð1 skn Þkxn x k þ ð1 skn Þ M3 þ hx xn ; Fx i þ M3 1 s kn 1 s kn 1 skn " # 3k2n 6 ½1 ð1 skn Þkxn x k2 þ ð1 skn Þ M 3 þ 2M 1 hx xn ; Fx i ¼ ð1 ln Þkxn x k2 þ ln dn ; 1 s kn 2
2
n M3 where M 3 ¼ supfkFx k ; kkxn x kkFx kg, ln ¼ 1 skn , dn ¼ 3k þ 2M 1 hx xn ; Fx i. It is easy to see that 1skn lim supn!1 dn 6 0. Hence, by Lemma 2.4, the sequence fxn g converges strongly to x 2 \1 n¼1 FðT n Þ. On the other hand, we have
P1
n¼1
ln ¼ 1 and
kyn xn k 6 kn kFðxn Þk ! 0 ðn ! 1Þ: It follows that the sequence fyn g converges strongly to x 2 \1 n¼1 FðT n Þ. From x ¼ limt!0 xt and Theorem 3.1, we have x is the unique solution of the variational inequality: hFx ; x ui 6 0; u 2 \1 FðT Þ. h n n¼1 From Theorem 3.4, we can obtain the following result immediately.
Corollary 3.5 (Yao et al. [4, Theorem 3.2]). Let H be a real Hilbert space. Let F : H ! H be k-Lipschitzian and g-strongly monotone 1 operator with k 2 ½1; 1Þ and g 2 ð0; 1Þ. Let fT n g1 n¼1 : H ! H be an infinite family of nonexpansive mappings such that \n¼1 FðT n Þ – ; and fW n g be W-mapping defined by (2.4). Let fkn g be a sequence in ½0; 1Þ and fan g be a sequences in ½0; 1. Assume that (i) limn!1 kn ¼ 0; P1 (ii) n¼0kn ¼ 1; (iii) an 2 c; 12 for some c > 0. Then the sequence fxn g and fyn g generated by (1.2) converge strongly to x 2 \1 n¼1 FðT n Þ which solves the following variational inequality hFx ; x xi 6 0; x 2 \1 n¼1 FðT n Þ. Remark 3.6. Theorem 3.4 include [4, Theorem 3.2] as a special case. The following example shows that all conditions of Theorem 3.4 are satisfied. However, the conditions g 2 ð0; 1Þ and an 2 c; 12 for some c > 0 in [4, Theorem 3.2] are not satisfied. Example 3.7. Let H ¼ R the set of real numbers and T n T. Define a nonexpansive mapping T : H ! H and an operator F : H ! H as follows:
Tx ¼ 0 and FðxÞ ¼ x;
8x 2 R:
1 1 It is easy to see that FðTÞ ¼ f0g; \1 n¼1 FðT n Þ ¼ f0g and W n x ¼ ð1 n1 Þx; 8x 2 R. Take n1 ¼ 2, We have W n x ¼ 2 x; 8x 2 R. Given sequences fan g and fkn g : an ¼ 23 ; kn ¼ 1n for all n P 0. For an arbitrary x0 2 H, let fxn g defined as follows
yn ¼ xn kn Fðxn Þ; xnþ1 ¼ ð1 an Þyn þ an W n yn ;
that is,
1 2 y þ W n yn 3 n 3 2 1 ¼ xn Fðxn Þ 3 n 2 1 xn ; n P 0: 1 ¼ 3 n
xnþ1 ¼
Observe that for all n P 0,
2 1 x 1 0 kxnþ1 0k ¼ n 3 n 2 1 ¼ kxn 0k 1 3 n 2 6 kxn 0k: 3
n P 0;
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S. Wang / Applied Mathematics and Computation 219 (2013) 10325–10332
nþ1 Hence we have kxnþ1 0k 6 23 kx0 0k for all n P 0. This implies that fxn g converges strongly to 0 2 \1 n¼1 FðT n Þ. On one other hand, we have
kyn 0k ¼
1 1 kxn k 6 kxn k: n
Therefore fyn g converges strongly to 0 2 \1 n¼1 FðT n Þ. On the other hand, we have hFð0Þ; 0 ui 6 0; u 2 \1 n¼1 FðT n Þ, that is, 0 is the solution of the variational inequality hFx ; x ui 6 0; u 2 \1 FðT Þ. n n¼1 By Fx ¼ x, we have g ¼ k ¼ 1. Furthermore, it is easy to see that there hold the following: P (B1) limn!1 kn ¼ 0 and 1 n¼0 kn ¼ 1; (B2) 0 < 12 < lim inf n!1 an ¼ 23 ¼ lim supn!1 an < 1 for some constant c ¼ 12. Hence there is no doubt that all conditions of Theorem 3.4 are satisfied. Since an ¼ 23 R c; 12 , the conditions an 2 c; 12 for some c > 0 and g 2 ð0; 1Þ of Yao et al. [4, Theorem 3.2] are not satisfied. Acknowledgments The author thanks the editor and the referees for their useful comments and suggestions. References [1] I. Yamada, The hybrid steepest descent for the variational inequality problems over the intersection of fixed points sets of nonexpansive mapping, in: D. Butnariu, Y. Censor, S. Reich (Eds.), Inherently Parallel Algorithms in Feasibility and Optimization and Their Application, Elservier, New York, 2001, pp. 473–504. [2] P.L. Lions, Approximation de points fixes de contractions, C.R. Acad. Sci. Paris 284 (1977) 1357–1359. [3] H.K. Xu, T.H. Kim, Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim. Theory Appl. 119 (2003) 185–201. [4] Y.H. Yao, M.N. Noor, Y.C. Liou, A new hybrid iterative algorithm for variational inequalities, Appl. Math. Comput. 216 (2010) 822–829. [5] K. Geobel, W.A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge Univ. Press, 1990. pp. 473– 504. [6] T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one parameter nonexpansive semigroups without Bochner integral, J. Math. Anal. Appl. 35 (2005) 227–239. [7] L.S. Liu, Iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995) 114–125. [8] H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002) 240–256. [9] W. Takahashi, K. Shimoji, Convergence theorems for nonexpansive mappings and feasibility problems, Math. Comput. Model. 32 (2000) 1463–1471. [10] K. Shimoji, W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math. 5 (2001) 387–404. [11] S.-S. Chang, A new method for solving equilibrium problem and variational inequality problem with application to optimization, Nonlinear Anal. 70 (2009) 3307–3319. [12] Y.H. Yao, Y.C. Liou, J.C. Yao, Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings, Fixed Point Theory Appl. 2007 (2007) (Article ID 64363, 12 p.).