Shrinking Projection Method for Fixed Point Problems of an Infinite

International Scholarly Research Network ISRN Mathematical Analysis Volume 2011, Article ID 795379, 16 pages doi:10.5402/2011/795379

Research Article Shrinking Projection Method for Fixed Point Problems of an Infinite Family of Strictly Pseudocontractive Mappings and the System of Cocoercive Quasivariational Inclusions Problems in Hilbert Spaces Pattanapong Tianchai Faculty of Science, Maejo University, Chiangmai 50290, Thailand Correspondence should be addressed to Pattanapong Tianchai, [email protected] Received 12 April 2011; Accepted 3 May 2011 Academic Editor: C. Zhu Copyright q 2011 Pattanapong Tianchai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper is concerned with a common element of the set of common fixed points for an infinite family of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method, and the applicability of the results is shown to extend and improve some well-known results existing in the current literature.

1. Introduction Throughout this paper, we always assume that C is a nonempty closed convex subset of a real Hilbert space H with inner product and norm denoted by ·, · and  · , respectively, 2H denoting the family of all the nonempty subsets of H. Let B : H → H be a single-valued nonlinear mapping and M : H → 2H a set-valued mapping. We consider the following quasivariational inclusion problem, which is to find a point x ∈ H, θ ∈ Bx  Mx,

1.1

where θ is the zero vector in H. The set of solutions of problem 1.1 is denoted by VIH, B, M.

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Recall that PC is the metric projection of H onto C; that is, for each x ∈ H, there exists the unique point in PC x ∈ C such that x − PC x  miny∈C x − y. A mapping T : C → C is called nonexpansive if T x − T y ≤ x − y for all x, y ∈ C. A point x ∈ C is a fixed point of T provided T x  x. We denote by FT  the set of fixed points of T ; that is, FT   {x ∈ C : T x  x}. If C is nonempty bounded closed convex subset of H and T is a nonexpansive mapping of C into itself, then FT  is nonempty see 1. Recall that a mapping A : C → C is said to be i monotone if   Ax − Ay, x − y ≥ 0,

∀x, y ∈ C,

1.2

ii k-Lipschitz continuous if there exists a constant k > 0 such that     Ax − Ay ≤ kx − y,

∀x, y ∈ C,

1.3

if k  1, then A is a nonexpansive, iii pseudocontractive if       Ax − Ay2 ≤ x − y2  I − Ax − I − Ay2 ,

∀x, y ∈ C,

1.4

iv k-strictly pseudocontractive if there exists a constant k ∈ 0, 1 such that       Ax − Ay2 ≤ x − y2  kI − Ax − I − Ay2 ,

∀x, y ∈ C,

1.5

it is obvious that A is a nonexpansive if and only if A is 0-strictly pseudocontractive, v α-strongly monotone if there exists a constant α > 0 such that 2    Ax − Ay, x − y ≥ αx − y ,

∀x, y ∈ C,

1.6

vi α-inverse-strongly monotone (or α-cocoercive) if there exists a constant α > 0 such that 2    Ax − Ay, x − y ≥ αAx − Ay ,

∀x, y ∈ C,

1.7

if α  1, then A is said to be firmly nonexpansive; it is obvious that any α-inversestrongly monotone mapping A is monotone and 1/α-Lipschitz continuous. The existence of common fixed points for a finite family of nonexpansive mappings has been considered by many authors see 2–5 and the references therein.

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In this paper, we study the mapping Wn defined by Un,n1  I,

  Un,n  μn Sn Un,n1  1 − μn I,   Un,n−1  μn−1 Sn−1 Un,n  1 − μn−1 I, .. .   Un,k  μk Sk Un,k1  1 − μk I,   Un,k−1  μk−1 Sk−1 Un,k  1 − μk−1 I,

1.8

.. .   Un,2  μ2 S2 Un,3  1 − μ2 I,   Wn  Un,1  μ1 S1 Un,2  1 − μ1 I, where {μi } is nonnegative real sequence in 0, 1, for all i ∈ N, S1 , S2 , . . . from a family of infinitely nonexpansive mappings of C into itself. It is obvious that Wn is a nonexpansive mapping of C into itself; such a mapping Wn is called a W-mapping generated by S1 , S2 , . . . , Sn and μ1 , μ2 , . . . , μn . Definition 1.1 see 6. Let M : H → 2H be a multivalued maximal monotone mapping. Then, the single-valued mapping JM,λ : H → H defined by JM,λ u  I  λM−1 u, for all u ∈ H, is called the resolvent operator associated with M, where λ is any positive number and I is the identity mapping. Recently, Zhang et al. 6 considered the problem 1.1 and the problem of a fixed point of nonexpansive mapping. To be more precise, they proved the following theorem. Theorem ZLC. Let H be a real Hilbert space, B : H → H an α-inverse-strongly monotone mapping, M : H → 2H a maximal monotone mapping, and T : H → H a nonexpansive mapping. Suppose that the set FT  ∩ VIH, B, M  / ∅, where VIH, B, M is the set of solutions of quasivariational inclusion 1.1. Suppose that x1  x ∈ H and {xn } is the sequence defined by yn  JM,λ xn − λBxn , xn1  αn x  1 − αn T yn ,

1.9

for all n ∈ N, where λ ∈ 0, 2α and {αn } ⊂ 0, 1 satisfying the following conditions:  (C1) limn → ∞ αn  0 and ∞ n1 αn  ∞, ∞ (C2) n1 |αn1 − αn | < ∞. Then, {xn } converges strongly to PFT ∩VIH,B,M x. Nakajo and Takahashi 7 introduced an iterative scheme for finding a fixed point of a nonexpansive mapping by a hybrid method which is called that shrinking projection method or CQ method as in the following theorem.

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Theorem NT. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of C into itself such that FT  /  ∅. Suppose that x1  x ∈ C and {xn } is the sequence defined by yn  αn xn  1 − αn T xn ,    Cn  z ∈ C : yn − z ≤ xn − z ,

1.10

Qn  {z ∈ C : xn − z, x1 − xn  ≥ 0}, xn1  PCn ∩Qn x1 ,

∀n ∈ N,

where 0 ≤ αn ≤ α < 1. Then, {xn } converges strongly to PFT  x1 . In the same way, Kikkawa and Takahashi 8 introduced an iterative scheme for finding a common fixed point of an infinite family of nonexpansive mappings as follows: yn  Wn xn ,    Cn  z ∈ C : yn − z ≤ xn − z ,

1.11

Qn  {z ∈ C : xn − z, x1 − xn  ≥ 0}, xn1  PCn ∩Qn x1 ,

∀n ∈ N,

where x1  x ∈ C and Wn is a W-mapping of C into itself generated by {Tn : C → C}

 ∅, then the sequence {xn } generated by 1.11 and {μn }. They prove that, if Ω  ∞ n1 FTn  / converges strongly to PΩ x1 . Recently, Su and Qin 9 modified the shrinking projection method for finding a fixed point of a nonexpansive mapping, for which the convergence rate of the iterative scheme is faster than that of the iterative scheme of Nakajo and Takahashi 7 as follows: yn  αn xn  1 − αn T xn ,   Cn  z ∈ Cn−1 ∩ Qn−1 : yn − z ≤ xn − z , 

Qn  {z ∈ Cn−1 ∩ Qn−1 : xn − z, x0 − xn  ≥ 0},    C0  z ∈ C : y0 − z ≤ x0 − z ,

n ≥ 1, n ≥ 1,

1.12

Q0  C, xn1  PCn ∩Qn x0 ,

∀n ∈ N ∪ {0},

where x0  x ∈ C and T is a nonexpansive mapping of C into itself. They prove that, under the  ∅, then the sequence {xn } generated by 1.12 converges parameter 0 ≤ αn ≤ α < 1, if FT  / strongly to PFT  x0 .

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On the other hand, Tada and Takahashi 10 introduced an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of solutions of a fixed point problem of a nonexpansive mapping as follows: un ∈ C

  1  such that F un , y  y − un , un − xn ≥ 0, rn

∀y ∈ C,

yn  1 − αn xn  αn T un ,    Cn  z ∈ H : yn − z ≤ xn − z ,

1.13

Qn  {z ∈ H : xn − z, x1 − xn  ≥ 0}, xn1  PCn ∩Qn x1 ,

∀n ∈ N,

where x1  x ∈ H, T is a nonexpansive mapping of C into H and F is a bifunction from C × C into R. They prove that, under the sequences {αn } ⊂ α, 1 for some α ∈ 0, 1 and {rn } ⊂ r, ∞ for some r > 0, if Ω  FT  ∩ EPF /  ∅, then the sequence {xn } generated by 1.13 converges strongly to PΩ x1  such that EPF is the set of solutions of equilibrium problem defined by    EPF  x ∈ C : F x, y ≥ 0, ∀y ∈ C .

1.14

In this paper, we introduce an iterative scheme 1.15 for finding a common element of the set of common fixed points for an infinite family of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems by the shrinking projection method in Hilbert spaces as follows: N  yn  αn Wn xn  1 − αn  βi JMi ,λi xn − λi Bi xn , i1

 2 N      n  αn 1 − αn Wn xn − βi JMi ,λi xn − λi Bi xn  ,   i1

 2 Cn1  z ∈ Cn ∩ Qn : yn − z ≤ xn − z2 − n ,

1.15

Qn1  {z ∈ Cn ∩ Qn : xn − z, x1 − xn  ≥ 0}, C1  Q1  H, xn1  PCn1 ∩Qn1 x1 ,

∀n ∈ N,

where x1  u ∈ H chosen arbitrarily, Mi : H → 2H is a maximal monotone mapping, Bi : H → H is a ξi -cocoercive mapping for each i  1, 2, . . . , N, and Wn is a W-mapping on H generated by {Sn } and {μn } such that the mapping Sn : H → H defined by Sn x  αx  1 − α Tn x for all x ∈ H, where {Tn : H → H} is an infinite family of k-strictly pseudocontractive mappings with a fixed point. It is well known that the class of strictly pseudocontractive mappings contains the class of nonexpansive mappings, and it follows that, if k  0, then the iterative scheme 1.15 is

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reduced to find a common element of the set of common fixed points for an infinite family of nonexpansive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces.  Furthermore, if Mi ≡ Bi ≡ 0 for all i  1, 2, . . . , N and N i1 βi  1, then the iterative scheme 1.15 is reduced to extend and improve the results of Kikkawa and Takahashi 8 for finding a common fixed point of an infinite family of k-strictly pseudocontractive mappings as follows: x1  u ∈ C chosen arbitrarily, yn  αn Wn xn  1 − αn xn , n  αn 1 − αn Wn xn − xn 2 ,

 2 Cn1  z ∈ Cn ∩ Qn : yn − z ≤ xn − z2 − n ,

1.16

Qn1  {z ∈ Cn ∩ Qn : xn − z, x1 − xn  ≥ 0}, C1  Q1  C, xn1  PCn1 ∩Qn1 x1 ,

∀n ∈ N,

and if k  α  0 and setting T1 ≡ T , Tn ≡ I for all n  2, 3, . . ., then the iterative scheme 1.16 is reduced to find a fixed point of a nonexpansive mapping, for which the convergence rate of the iterative scheme is faster than that of the iterative scheme of Su and Qin 9 as follows: x1  u ∈ C chosen arbitrarily, yn  σn T xn  1 − σn xn , δn  σn 1 − σn T xn − xn 2 ,

 2 Dn1  z ∈ Dn ∩ Qn : yn − z ≤ xn − z2 − δn ,

1.17

Qn1  {z ∈ Dn ∩ Qn : xn − z, x1 − xn  ≥ 0}, D1  Q1  C, xn1  PDn1 ∩Qn1 x1 ,

∀n ∈ N.

We suggest and analyze the iterative scheme 1.15 under some appropriate conditions imposed on the parameters. The strong convergence theorem for the above two sets is obtained, and the applicability of the results is shown to extend and improve some well-known results existing in the current literature.

2. Preliminaries We collect the following lemmas which will be used in the proof of the main results in the next section.

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Lemma 2.1 see 11. Let H be a Hilbert space. For any x, y ∈ H and λ ∈ R, one has      λx  1 − λy2  λx2  1 − λ y2 − λ1 − λx − y2 .

2.1

Lemma 2.2 see 1. Let C be a nonempty closed convex subset of a Hilbert space H. Then the following inequality holds: 

 x − PC x, PC x − y ≥ 0,

∀x ∈ H, y ∈ C.

2.2

Lemma 2.3 see 5. Let C be a nonempty closed convex subset of a Hilbert space H, define mapping

 ∅, and Wn as 1.8, let Si : C → C be a family of infinitely nonexpansive mappings with ∞ i1 FSi  / let {μi } be a sequence such that 0 < μi ≤ μ < 1, for all i ≥ 1. Then 1 Wn is nonexpansive and FWn  

n

i1

FSi  for each n ≥ 1,

2 for each x ∈ C and for each positive integer k, limn → ∞ Un,k x exists, 3 the mapping W : C → C defined by Wx : lim Wn x  lim Un,1 x, n→∞

n→∞

is a nonexpansive mapping satisfying FW  generated by S1 , S2 , . . . and μ1 , μ2 , . . ..

∞

i1

x ∈ C,

2.3

FSi  and it is called the W-mapping

Lemma 2.4 see 6. The resolvent operator JM,λ associated with M is single valued and nonexpansive for all λ > 0. Lemma 2.5 see 6. u ∈ H is a solution of quasivariational inclusion 1.1 if and only if u  JM,λ u − λBu, for all λ > 0, that is, VIH, B, M  FJM,λ I − λB,

∀λ > 0.

2.4

Lemma 2.6 see 12. Let C be a nonempty closed convex subset of a strictly convex Banach space

 ∅. Let X. Let {Tn : n ∈ N} be a sequence of nonexpansive mappings on C. Suppose that ∞ n1 FTn  / ∞ {αn } be a sequence of positive real numbers such that n1 αn  1. Then a mapping S on C defined by Sx 

∞  αn Tn x, n1

for x ∈ C, is well defined, nonexpansive and FS 

∞

n1

2.5

FTn  holds.

Lemma 2.7 see 13. Let C be a nonempty closed convex subset of a Hilbert space H and S : C → C a nonexpansive mapping. Then I − S is demiclosed at zero. That is, whenever {xn } is a sequence in C weakly converging to some x ∈ C and the sequence {I − Sxn } strongly converges to some y, it follows that I − Sx  y.

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Lemma 2.8 see 14. Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → C a k-strict pseudocontraction. Define S : C → C by Sx  αx  1 − αT x for each x ∈ C. Then, as α ∈ k, 1, S is nonexpansive such that FS  FT . Lemma 2.9 see 1. Every Hilbert space H has Radon-Riesz property or Kadec-Klee property, that is, for a sequence {xn } ⊂ H with xn x and xn  → x then xn → x.

3. Main Results Theorem 3.1. Let H be a real Hilbert space, Mi : H → 2H a maximal monotone mapping, and Bi : H → H a ξi -cocoercive mapping for each i  1, 2, . . . , N. Let {Tn : H → H} be an infinite family of k-strictly pseudocontractive mappings with a fixed point such that k ∈ 0, 1. Define a mapping Sn : H → H by Sn x  αx  1 − αTn x,

∀x ∈ H,

3.1

for all n ∈ N, where α ∈ k, 1. Let Wn : H → H be a W-mapping generated by {Sn } and {μn } such

N  ∅. that {μn } ⊂ 0, μ, for some μ ∈ 0, 1. Assume that Ω :  ∞ n1 FTn  ∩  i1 VIH, Bi , Mi  / For x1  u ∈ H chosen arbitrarily, suppose that {xn } is generated iteratively by N  yn  αn Wn xn  1 − αn  βi JMi ,λi xn − λi Bi xn , i1

 2 N      n  αn 1 − αn Wn xn − βi JMi ,λi xn − λi Bi xn  ,   i1

 2 Cn1  z ∈ Cn ∩ Qn : yn − z ≤ xn − z2 − n , Qn1  {z ∈ Cn ∩ Qn : xn − z, x1 − xn  ≥ 0}, C1  Q1  H, xn1  PCn1 ∩Qn1 x1 ,

∀n ∈ N,

where (C1) {αn } ⊂ a, b such that 0 < a < b < 1, (C2) βi ∈ 0, 1 and λi ∈ 0, 2ξi  for each i  1, 2, . . . , N,  (C3) N i1 βi  1. Then, the sequences {xn } and {yn } converge strongly to w  PΩ x1 . Proof. For any x, y ∈ H and for each i  1, 2, . . . , N, by the ξi -cocoercivity of Bi , we have       I − λi Bi x − I − λi Bi y2   x − y − λi Bi x − Bi y 2 2     x − y − 2λi x − y, Bi x − Bi y

3.2

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9 2   λ2i Bi x − Bi y 2  2  ≤ x − y − 2ξi − λi λi Bi x − Bi y 2  ≤ x − y  , 3.3

which implies that I − λi Bi is nonexpansive. Pick p ∈ Ω. Therefore, by Lemma 2.5, we have p  JMi ,λi I − λi Bi p,

3.4

for each i  1, 2, . . . , N. Since Sn x  αx  1 − αTn x, where α ∈ k, 1 and {Tn } is a family of k-strict pseudocontraction, therefore, by Lemma 2.8, we have that Sn is nonexpansive and



FSn   FTn . It follows from Lemma 2.31 that FWn   ni1 FSi   ni1 FTi , which implies that Wn p  p. Therefore, by C3, 3.4, Lemma 2.1, and the nonexpansiveness of Wn , JMi ,λi , and I − λi Bi , we have  2 N       yn − p2   αn Wn xn  1 − αn  βi JMi ,λi xn − λi Bi xn  − p   i1  2 N          αn Wn xn − p  1 − αn  βi JMi ,λi xn − λi Bi xn  − p    i1 2   αn Wn xn − Wn p 2  N        1 − αn  βi JMi ,λi xn − λi Bi xn  − JMi ,λi p − λi Bi p    i1  2 N      − αn 1 − αn Wn xn − βi JMi ,λi xn − λi Bi xn    i1

3.5

2  ≤ αn xn − p 2  N        1 − αn  βi xn − λi Bi xn  − p − λi Bi p − n i1

2 2   ≤ αn xn − p  1 − αn xn − p − n  2  xn − p − n , for all n ∈ N. Firstly, we prove that Cn ∩ Qn is closed and convex for all n ∈ N. It is obvious that C1 ∩ Q1 is closed and, by mathematical induction, that Cn ∩ Qn is closed for all n ≥ 2, that is Cn ∩ Qn is closed for all n ∈ N. Since yn − z2 ≤ xn − z2 − n is equivalent to     yn − xn 2  2 yn − xn , xn − z  n ≤ 0,

3.6

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for all n ∈ N, therefore, for any z1 , z2 ∈ Cn1 ∩ Qn1 ⊂ Cn ∩ Qn and  ∈ 0, 1, we have     yn − xn 2  2 yn − xn , xn − z1  1 − z2   n   2     yn − xn   2 yn − xn , xn − z1  n   2    1 −   yn − xn   2 yn − xn , xn − z2  n

3.7

≤ 0, for all n ∈ N, and we have xn − z1  1 − z2 , x1 − xn   xn − z1 , x1 − xn   1 − xn − z2 , x1 − xn  ≥ 0,

3.8

for all n ∈ N. Since C1 ∩ Q1 is convex, and by putting n  1 in 3.6, 3.7, and 3.8, we have that C2 ∩ Q2 is convex. Suppose that xk is given and Ck ∩ Qk is convex for some k ≥ 2. It follows by putting n  k in 3.6, 3.7, and 3.8 that Ck1 ∩ Qk1 is convex. Therefore, by mathematical induction, we have that Cn ∩ Qn is convex for all n ≥ 2, that is, Cn ∩ Qn is convex for all n ∈ N. Hence, we obtain that Cn ∩ Qn is closed and convex for all n ∈ N. Next, we prove that Ω ⊂ Cn ∩ Qn for all n ∈ N. It is obvious that p ∈ Ω ⊂ H  C1 ∩ Q1 . Therefore, by 3.2 and 3.5, we have p ∈ C2 and note that p ∈ H  Q2 , and so p ∈ C2 ∩ Q2 . Hence, we have Ω ⊂ C2 ∩ Q2 . Since C2 ∩ Q2 is a nonempty closed convex subset of H, there exists a unique element x2 ∈ C2 ∩ Q2 such that x2  PC2 ∩Q2 x1 . Suppose that xk ∈ Ck ∩ Qk is given such that xk  PCk ∩Qk x1 , and p ∈ Ω ⊂ Ck ∩ Qk for some k ≥ 2. Therefore, by 3.2 and 3.5, we have p ∈ Ck1 . Since xk  PCk ∩Qk x1 , therefore, by Lemma 2.2, we have xk − z, x1 − xk  ≥ 0

3.9

for all z ∈ Ck ∩ Qk . Thus, by 3.2, we have p ∈ Qk1 , and so p ∈ Ck1 ∩ Qk1 . Hence, we have Ω ⊂ Ck1 ∩ Qk1 . Since Ck1 ∩ Qk1 is a nonempty closed convex subset of H, there exists a unique element xk1 ∈ Ck1 ∩Qk1 such that xk1  PCk1 ∩Qk1 x1 . Therefore, by mathematical induction, we obtain Ω ⊂ Cn ∩ Qn for all n ≥ 2, and so Ω ⊂ Cn ∩ Qn for all n ∈ N, and we can define xn1  PCn1 ∩Qn1 x1  for all n ∈ N. Hence, we obtain that the iteration 3.2 is well defined. Next, we prove that {xn } is bounded. Since xn  PCn ∩Qn x1  for all n ∈ N, we have xn − x1  ≤ z − x1 ,

3.10

for all z ∈ Cn ∩ Qn . It follows by p ∈ Ω ⊂ Cn ∩ Qn that xn − x1  ≤ p − x1  for all n ∈ N. This implies that {xn } is bounded, and so is {yn }. Next, we prove that yn − xn  → 0 as n → ∞. Since xn1  PCn1 ∩ Qn1 x1  ∈ Cn1 ∩ Qn1 ⊂ Cn ∩ Qn , therefore, by 3.10, we have xn − x1  ≤ xn1 − x1  for all n ∈ N. This implies that {xn − x1 } is a bounded nondecreasing sequence and there exists the limit of xn − x1 , that is, lim xn − x1   m,

n→∞

3.11

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for some m ≥ 0. Since xn1 ∈ Qn1 , therefore, by 3.2, we have xn − xn1 , x1 − xn  ≥ 0.

3.12

It follows by 3.12 that xn − xn1 2  xn − x1   x1 − xn1 2  xn − x1 2  2xn − x1 , x1 − xn   2xn − x1 , xn − xn1   xn1 − x1 2

3.13

≤ xn1 − x1 2 − xn − x1 2 . Therefore, by 3.11, we have xn − xn1  −→ 0 as n −→ ∞.

3.14

Since xn1 ∈ Cn1 , therefore, by 3.2, we have   yn − xn1 2 ≤ xn − xn1 2 − n

3.15

≤ xn − xn1 2 . It follows by 3.15 that     yn − xn  ≤ yn − xn1   xn1 − xn  ≤ xn − xn1   xn1 − xn 

3.16

 2xn − xn1 . Therefore, by 3.14, we obtain   yn − xn  −→ 0 as n −→ ∞.

3.17

Since {xn } is bounded, there exists a subsequence {xni } of {xn } which converges weakly to w. Next, we prove that w ∈ Ω. Define the sequence of mappings {Qn : H → H} and the mapping Q : H → H by Qn x  αn Wn x  1 − αn 

N  βi JMi ,λi I − λi Bi x, i1

Qx  lim Qn x, n→∞

for all n ∈ N. Therefore, by C1 and Lemma 2.33, we have

∀x ∈ H,

3.18

12

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∀x ∈ H,

3.19

i1

where a ≤ c  limn → ∞ αn ≤ b. From C3 and Lemma 2.33, we have that W and N i1 βi JMi ,λi I − λi Bi  are nonexpansive. Therefore, by C2, C3, Lemmas 2.33, 2.5, 2.6, and 2.8, we have FQ  FW ∩ F 

 ∞ 





i1

βi JMi ,λi I − λi Bi 

∩

 FTi 





FSi 

i1

 ∞ 

 N 

 ∩

i1

 N  FJMi ,λi I − λi Bi  i1

N 

3.20

 VIH, Bi , Mi  ,

i1

that is, FQ  Ω. From 3.17, we have yni − xni  → 0 as i → ∞. Thus, from 3.2 and 3.18, we get Qxni − xni  → 0 as i → ∞. It follows from xni w and Lemma 2.7 that w ∈ FQ, that is, w ∈ Ω. Since Ω is a nonempty closed convex subset of H, there exists a unique w ∈ Ω such that w  PΩ x1 . Next, we prove that xn → w as n → ∞. Since w  PΩ x1 , we have x1 − w ≤ x1 − z for all z ∈ Ω, and it follows that x1 − w ≤ x1 − w.

3.21

Since w ∈ Ω ⊂ Cn ∩ Qn , therefore, by 3.10, we have x1 − xn  ≤ x1 − w.

3.22

Therefore, by 3.21, 3.22, and the weak lower semicontinuity of norm, we have x1 − w ≤ x1 − w ≤ lim infx1 − xni  i→∞

≤ lim supx1 − xni 

3.23

i→∞

≤ x1 − w. It follows that x1 − w  lim x1 − xni   x1 − w. i→∞

3.24

Since xni w as i → ∞, therefore, we have x1 − xni  x1 − w as i −→ ∞.

3.25

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Hence, from 3.24, 3.25, the Kadec-Klee property, and the uniqueness of w  PΩ x1 , we obtain xni −→ w  w

as i −→ ∞.

3.26

It follows that {xn } converges strongly to w, and so is {yn }. This completes the proof. Remark 3.2. The iteration 3.2 is the difference with some well known results as the following. 1 The sequence {xn } is the projection sequence of x1 onto Cn ∩ Qn for all n ∈ N such that C1 ∩ Q1 ⊃ C2 ∩ Q2 ⊃ · · · ⊃ Cn ∩ Qn ⊃ · · · ⊃ Ω.

3.27

2 The proof of w ∈ Ω is simple by the demiclosedness principle because the sequence {yn } is a linear nonexpansive mapping form of the mappings Wn and JMi ,λi I−λi Bi . 3 Solving a common fixed point for an infinite family of strictly pseudocontractive mappings and a system of cocoercive quasivariational inclusions problems by iteration is obtained.

4. Applications Theorem 4.1. Let H be a real Hilbert space, Mi : H → 2H a maximal monotone mapping, and Bi : H → H a ξi -cocoercive mapping for each i  1, 2, . . . , N. Let {Tn : H → H} be an infinite family of nonexpansive mappings. Define a mapping Sn : H → H by Sn x  αx  1 − αTn x,

∀x ∈ H,

4.1

for all n ∈ N, where α ∈ 0, 1. Let Wn : H → H be a W-mapping generated by {Sn } and {μn } such

N  ∅. that {μn } ⊂ 0, μ, for some μ ∈ 0, 1. Assume that Ω :  ∞ n1 FTn  ∩  i1 VIH, Bi , Mi  / For x1  u ∈ H chosen arbitrarily, suppose that {xn } is generated iteratively by yn  αn Wn xn  1 − αn 

N  i1

βi JMi ,λi xn − λi Bi xn ,

 2 N      n  αn 1 − αn Wn xn − βi JMi ,λi xn − λi Bi xn  ,   i1

 2 Cn1  z ∈ Cn ∩ Qn : yn − z ≤ xn − z2 − n , Qn1  {z ∈ Cn ∩ Qn : xn − z, x1 − xn  ≥ 0}, C1  Q1  H, xn1  PCn1 ∩Qn1 x1 ,

∀n ∈ N, 4.2

14

ISRN Mathematical Analysis

where (C1) {αn } ⊂ a, b such that 0 < a < b < 1, (C2) βi ∈ 0, 1 and λi ∈ 0, 2ξi  for each i  1, 2, . . . , N,  (C3) N i1 βi  1. Then the sequences {xn } and {yn } converge strongly to w  PΩ x1 . Proof. It is concluded from Theorem 3.1 immediately, by putting k  0. Theorem 4.2. Let C be a nonempty closed convex subset of a real Hilbert space H and {Tn : C → C} an infinite family of k-strictly pseudocontractive mappings with a fixed point such that k ∈ 0, 1. Define a mapping Sn : C → C by Sn x  αx  1 − αTn x,

∀x ∈ C,

4.3

for all n ∈ N, where α ∈ k, 1. Let Wn : C → C be a W-mapping generated by {Sn } and {μn } such

 ∅. For x1  u ∈ C chosen that {μn } ⊂ 0, μ, for some μ ∈ 0, 1. Assume that Ω : ∞ n1 FTn  / arbitrarily, suppose that {xn } is generated iteratively by yn  αn Wn xn  1 − αn xn , n  αn 1 − αn Wn xn − xn 2 ,

 2 Cn1  z ∈ Cn ∩ Qn : yn − z ≤ xn − z2 − n ,

4.4

Qn1  {z ∈ Cn ∩ Qn : xn − z, x1 − xn  ≥ 0}, C1  Q1  C, xn1  PCn1 ∩Qn1 x1 ,

∀n ∈ N,

where {αn } ⊂ a, b such that 0 < a < b < 1. Then the sequences {xn } and {yn } converge strongly to w  PΩ x1 . Proof. It is concluded from Theorem 3.1 immediately, by putting Mi ≡ Bi ≡ 0 for all i  1, 2, . . . , N. Theorem 4.3. Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → C a nonexpansive mapping. Assume that FT   / ∅. For x1  u ∈ C chosen arbitrarily, suppose that {xn } is generated iteratively by yn  σn T xn  1 − σn xn δn  σn 1 − σn T xn − xn 2 ,

 2 Dn1  z ∈ Dn ∩ Qn : yn − z ≤ xn − z2 − δn , Qn1  {z ∈ Dn ∩ Qn : xn − z, x1 − xn  ≥ 0},

ISRN Mathematical Analysis

15 D1  Q1  C, xn1  PDn1 ∩Qn1 x1 ,

∀n ∈ N, 4.5

where {σn } ⊂ a, b such that 0 < a < b < 1. Then the sequences {xn } and {yn } converge strongly to w  PFT  x1 . Proof. It is concluded from Theorem 4.2, by putting α  0. Setting T1 ≡ T , Tn ≡ I for all n  2, 3, . . . and leting μn ⊂ 0, μ for some μ ∈ 0, 1, therefore, from the definition of Sn in Theorem 4.2, we have S1  T1  T and Sn  I for all n  2, 3, . . .. Since Wn is a W-mapping generated by {Sn } and {μn }, therefore, by the definition of Un,i and Wn in 1.8, we have Un,i  I for all i  2, 3, . . . and Wn  Un,1  μ1 S1 Un,2  1 − μ1 I  μ1 T  1 − μ1 I. Hence, by Theorem 4.2, we obtain yn  αn Wn xn  1 − αn xn      αn μ1 T xn  1 − μ1 xn  1 − αn xn

4.6

 σn T xn  1 − σn xn , where σn : αn μ1 . Since, the same as in the proof of Theorem 3.1, we have that Dn ∩ Qn is a nonempty closed convex subset of C for all n ∈ N and by Theorem 4.2, we have n  αn 1 − αn Wn xn − xn 2  2  αn 1 − αn μ1 T xn  1 − μ1 xn − xn      αn μ1 μ1 − μ1 αn T xn − xn 2    σn μ1 − σn T xn − xn 2

4.7

≤ σn 1 − σn T xn − xn 2  δn , for all n ∈ N. It follows that Dn ⊂ Cn for all n ∈ N, where Cn is defined as in Theorem 4.2. Hence, by Theorem 4.2, we obtain the desired result. This completes the proof.

References 1 W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and its Application, Yokohama Publishers, Yokohama, Japan, 2000. 2 K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis. Theory, Methods & Applications Series A, vol. 67, no. 8, pp. 2350–2360, 2007. 3 H. H. Bauschke, “The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 202, no. 1, pp. 150–159, 1996. 4 M. Shang, Y. Su, and X. Qin, “Strong convergence theorems for a finite family of nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2007, Article ID 76971, 9 pages, 2007.

16

ISRN Mathematical Analysis

5 K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansive mappings and applications,” Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387–404, 2001. 6 S.-S. Zhang, J. H. W. Lee, and C. K. Chan, “Algorithms of common solutions to quasi variational inclusion and fixed point problems,” Applied Mathematics and Mechanics, vol. 29, no. 5, pp. 571–581, 2008. 7 K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379, 2003. 8 M. Kikkawa and W. Takahashi, “Approximating fixed points of infinite nonexpansive mappings by the hybrid method,” Journal of Optimization Theory and Applications, vol. 117, no. 1, pp. 93–101, 2003. 9 Y. Su and X. Qin, “Strong convergence of monotone hybrid method for fixed point iteration processes,” Journal of Systems Science & Complexity, vol. 21, no. 3, pp. 474–482, 2008. 10 A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,” Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370, 2007. 11 W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, Japan, 2009. 12 R. E. Bruck, Jr., “Properties of fixed-point sets of nonexpansive mappings in Banach spaces,” Transactions of the American Mathematical Society, vol. 179, pp. 251–262, 1973. 13 F. E. Browder, “Nonlinear monotone operators and convex sets in Banach spaces,” Bulletin of the American Mathematical Society, vol. 71, pp. 780–785, 1965. 14 F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.

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