A new iterative method for a common solution of fixed points for ...

Chamnarnpan and Kumam Fixed Point Theory and Applications 2012, 2012:67 http://www.fixedpointtheoryandapplications.com/content/2012/1/67

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A new iterative method for a common solution of fixed points for pseudo-contractive mappings and variational inequalities Tanom Chamnarnpan and Poom Kumam* * Correspondence: poom. [email protected] Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand

Abstract In this article, we introduce a new iterative scheme for finding a common element of the set of fixed points for a continuous pseudo-contractive mapping and the solution set of a variational inequality problem governed by continuous monotone mappings. Strong convergence for the proposed iterative scheme is proved. Our results improve and extend some recent results in the literature. 2000 Mathematics Subject Classification: 46C05; 47H09; 47H10. Keywords: monotone mapping, nonexpansive mapping, pseudo-contractive mappings, variational inequality

1. Introduction The theory of variational inequalities represents, in fact, a very natural generalization of the theory of boundary value problems and allows us to consider new problems arising from many fields of applied mathematics, such as mechanics, physics, engineering, the theory of convex programming, and the theory of control. While the variational theory of boundary value problems has its starting point in the method of orthogonal projection, the theory of variational inequalities has its starting point in the projection on a convex set. Let C be a nonempty closed and convex subset of a real Hilbert space H. The classical variational inequality problem is to find a u Î C such that 〈v-u, Au〉 ≥ 0 for all v Î C, where A is a nonlinear mapping. The set of solutions of the variational inequality is denoted by VI(C, A). The variational inequality problem has been extensively studied in the literature, see [1-5] and the reference therein. In the context of the variational inequality problem, this implies that u Î VI(C, A) ⇔ u = PC(u - lAu), ∀l > 0, where PC is a metric projection of H into C. Let A be a mapping from C to H, then A is called monotone if and only if for each x, y Î C,   x − y, Ax − Ay ≥ 0.

(1:1)

An operator A is said to be strongly positive on H if there exists a constant γ¯ > 0 such that

© 2012 Chamnarnpan and Kumam; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Ax, x ≥ γ¯ x2 , ∀x ∈ H.

A mapping A of C into itself is called L-Lipschitz continuous if there exits a positive and number L such that     Ax − Ay ≤ L x − y , ∀x, y ∈ C. A mapping A of C into H is called a-inverse-strongly monotone if there exists a positive real number a such that  2   x − y, Ax − Ay ≥ α Ax − Ay , for all x, y Î C; see [2,6-10]. If A is an a-inverse strongly monotone mapping of C 1 -Lipschitz continuous, that is, into H, then it is obvious that A is α    1 Ax − Ay ≤ x − y for all x, y Î C. Clearly, the class of monotone mappings α include the class of a-inverse strongly monotone mappings. Recall that a mapping T of C into H is called pseudo-contractive if for each x, y Î C, we have 2    (1:2) Tx − Ty, x − y ≤ x − y .

T is said to be a k-strict pseudo-contractive mapping if there exists a constant 0 ≤ k ≤ 1 such that 2  2    x − y, Tx − Ty ≤ x − y − k(I − T) x − (I − T) y , for all x, y ∈ D(T). A mapping T of C into itself is called nonexpansive if ∥Tx - Ty∥ ≤ ∥x - y∥, for all x, y Î C. We denote by F(T) the set of fixed points of T. Clearly, the class of pseudo-contractive mappings include the class of nonexpansive and strict pseudo-contractive mappings. For finding an element of F(T), where T is a nonexpansive mapping of C into itself, Halpern [11] was the first to study the convergence of the following scheme: xn+1 = αn+1 u + (1 − αn+1 )T(xn ), n ≥ 0,

(1:3)

where u, x0 Î C and a sequence {an} of real numbers in (0,1) in the framework of Hilbert spaces. Lions [12] improved the result of Halpern by proving strong convergence of {xn} to a fixed point of T provided that the real sequence {an} satisfies certain mild conditions. In 2000, Moudafi [13] introduced viscosity approximation method and proved that if H is a real Hilbert space, for given x0 Î C, the sequence {xn} generated by the algorithm xn+1 = αn f (xn ) + (1 − αn )T(xn ), n ≥ 0,

(1:4)

where f : C ® C is a contraction mapping with a constant b Î (0,1) and {an} ⊂ (0,1) satisfies certain conditions, converges strongly to fixed point of Moudafi [13] generalizes Halpern’s theorems in the direction of viscosity approximations. In [14,15], Zegeye and Shahzad extended Moudafi’s result to Banach spaces which more general than Hilbert spaces. For other related results, see [16-18]. Viscosity approximations are very important because they are applied to convex optimization, linear programming, monotone inclusion and elliptic differential equations. Marino and Xu [19], studied the

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viscosity approximation method for nonexpansive mappings and considered the following general iterative method: xn+1 = (I − αn A)Txn + αn γ f (xn ), n ≥ 0.

(1:5)

They proved that if the sequence {an} of parameters satisfies appropriate conditions, then the sequence {xn} generated by (1.5) converges strongly to the unique solution of the variational inequality    A − γ f x∗ , x − x∗ ≥ 0, x ∈ C, which is the optimality condition for the minimization problem min x∈C

1 Ax, x − h(x), 2

where h is a potential function for gf (i.e., h’(x) = gf(x) for x Î H). For finding an element of F(T) ∩ VI(C, A), where T is nonexpansive and A is a-inverse strongly monotone, Takahashi and Toyoda [20] introduced the following iterative scheme: xn+1 = αn xn + (1 − αn )TPC (xn − λn Axn ), n ≥ 0.

(1:6)

where x 0 Î C, {a n} is a sequence in (0,1), and {l n} is a sequence in (0, 2a), and obtained weak convergence theorem in a Hilbert space H. Iiduka and Takahashi [7] proposed a new iterative scheme x1 = x Î C and xx+1 = αn x + (1 − αn )TPC (xn − λn Axn ), n ≥ 0,

(1:7)

and obtained strong convergence theorem in a Hilbert space. Motivated and inspired by the work mentioned above which combined from Equations (1.5) and (1.6), in this article, we introduced a new iterative scheme (3.1) below which converges strongly to common element of the set of fixed points of continuous pseudo-contractive mappings which more general than nonexpansive mappings and the solution set of the variational inequality problem of continuous monotone mappings which more general than a-inverse strongly monotone mappings. As a consequence, we provide an iterative scheme which converges strongly to a common element of set of fixed points of finite family continuous pseudo-contractive mappings and the solutions set of finite family of variational inequality problems for continuous monotone mappings. Our theorems extend and unify most the results that have been proved for these important class of nonlinear operators.

2. Preliminaries Let H be a nonempty closed and convex subset of a real Hilbert space H. Let A be a mapping from C into H. For every point x Î H, there exists a unique nearest point in C, denoted by PCx, such that   x − PC x ≤ x − y , ∀y ∈ C.

PC is called the metric projection of H onto C. We know that PC is a nonexpansive mapping of H onto C.

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Lemma 2.1.Let H be a real Hilbert space. The following identity holds:     x + y2 ≤ x2 + 2 y, x + y , ∀x, y ∈ H. Lemma 2.2.Let C be a closed convex subset of a Hilbert space H. Let x Î H and x0 Î C. Then x0 = PCx if and only if z − x0 , x0 − x , ∀z ∈ C.

Lemma 2.3.[21]Let {an} be a sequence of nonnegative real numbers satisfying the following relation an+1 ≤ (1 − γn )an + σn , n ≥ 0,

where, (i) {γn } ⊂ (0, 1),

∞ 

γn = ∞;

n=1

(ii) lim supn→∞ σγnn ≤ 0 or

∞ 

|σn | < ∞.

n=1

Then, the sequence {an} ® 0 as n ® ∞. Lemma 2.4.[22]Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A : C ® H be a continuous monotone mapping. Then, for r > 0 and x Î H, there exist z Î C such that   1  y − z, Az + y − z, z − x ≥ 0, ∀y ∈ C. r

(2:1)

Moreover, by a similar argument of the proof of Lemmas 2.8 and 2.9 in[23], Zegeye [22]obtained the following lemmas: Lemma 2.5.[22]Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A : C ® H be a continuous monotone mapping. For r > 0 and x Î H, define a mapping Fr : H ® C as follows:    1  Fr x := z ∈ C : y − z, Az + y − z, z − x ≥ 0, ∀y ∈ C r for all x Î H. Then the following hold: (1) Fr is single-valued; (2) Fr is a firmly nonexpansive type mapping, i.e., for all x, y Î H,     Fr x − Fr y2 ≤ Fr x − Fr y, x − y ; (3) F(Fr) = VI(C,A); (4) VI(C, A) is closed and convex. In the sequel, we shall make use of the following lemmas: Lemma 2.6.[22]Let C be a nonempty closed and convex subset of a real Hilbert space H. Let T : C ® H be a continuous pseudo-contractive mapping. Then, for r > 0 and x Î H, there exist z Î C such that

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 1  y − z, Tz − y − z, (1 + r)z − x ≤ 0, r

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∀y ∈ C.

(2:2)

Lemma 2.7.[22]Let C be a nonempty closed and convex subset of a real Hilbert space H. Let T : C ® C be a continuous pseudo-contractive mapping. For r > 0 and x Î H, define a mapping Tr : H ® C as follows:    1  Tr x := z ∈ C : y − z, Tz + y − z, (1 + r)z − x ≤ 0, ∀y ∈ C r for all x Î H. Then the following hold: (1) Tr is single - valued; (2) Tr is a firmly nonexpansive type mapping, i.e., for all x, y Î H,     Tr x − Tr y2 ≤ Tr x − Tr y, x − y ; (3) F(Tr) = F(T); (4) F(T) is closed and convex. Lemma 2.8.[19]Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient γ¯ > 0and 0 0and let {x n } be a sequence generated by x1 Î C and 

yn = Frn xn xn+1 = αn γ f (xn ) + δn xn + [(1 − δn )I − αn B]Trn yn ,

(3:1)

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where {an} ⊂ [0,1] and {rn} ⊂ (0, ∞) such that (C1) limn→∞ αn = 0, (C2) limn→∞ δn = 0,

∞

αn = ∞;

n=1 ∞ 

|δn+1 − δn | < ∞;

n=1

(C3) lim infn→∞ rn > 0,

∞ 

|rn+1 − rn | < ∞.

n=1

Then, the sequence {xn} converges strongly to z ∈ F, which is the unique solution of the variational inequality:   (3:2) (B − γ f )z, x − z ≥ 0, ∀x ∈ F. Equivalently, z = PF (I − B + γ f )z, which is the optimality condition for the minimization problem min x∈C

1 Az, z − h(z), 2

where h is a potential function for gf (i.e.,h’(z) = gf(z) for z Î H). Remark: (1) The variational inequality (3.2) has the unique solution; (see [19]). (2) It follows from condition (C1) that (1 - δ n )I - a n B is positive and   (1 − δn )I − αn B ≤ I − δn − αn β¯ for all n ≥ 1; (see [24]). Proof. We processed the proof with following four steps: Step 1. First, we will prove that the sequence {xn} is bounded. Let v ∈ F and let un = Trn yn and yn = Frn xn. Then, from Lemmas 2.5 and 2.7 that       un − v = Trn yn − Trn v ≤ yn − v = Frn xn − Frn v ≤ xn − v . (3:3) Moreover, from (3.1) and (3.2), we compute   xn+1 − v = αn (γ (f (xn ) − Bv)) + δn (xn − v) + [(1 − δn )I − αn B]Trn − v .        ≤ αn γ f (xn ) − Bv) + δn (xn − v) + (1 − δn )I − αn B Trn − v .     ¯ Trn yn − v . ≤ αn βγ xn − v + αn γ f (v) − Bv + δn xn − v + (1 + δn − αn β)   ¯ un − v . ≤ αn βγ xn − v + αn γ f (v) − Bv + δn xn − v + (1 + δn − αn β)     ¯ xn − v . ≤ αn βγ xn − v + αn γ f (v) − Bv + δn xn − v + (1 + δn − αn β)   = αn βγ xn − v + αn γ f (v) − Bv + δn xn − v + xn − v − δn xn − v − αn β¯ xn − v .   = αn βγ xn − v + αn γ f (v) − Bv + xn − v − αn β¯ xn − v .     ≤ αn βγ + 1 − αn β¯ xn − v + αn γ f (v) − Bv .   = (1 − αn (β¯ − βγ )) xn − v + αn γ f (v) − Bv .    γ f (v) − Bv ∀n ≥ 1. ≤ max xn − v , , β¯ − βγ

Therefore, by the simple introduction, we have    γ f (v) − Bv xn − v = max x1 − v , , ∀n ≥ 1 β¯ − βγ which show that {xn} is bounded, so {yn}, {un}, and {f(xn)} are bounded.

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Step 2. We will show that ∥xn+1 - xn∥ ® 0 and ∥un - yn∥ ® 0 as n ® ∞. Notice that each Trn and Frn are firmly nonexpansive. Hence, we have       un+1 − un  = Trn yn+1 − Trn yn  ≤ yn+1 − yn  = Frn xn+1 − Frn xn  ≤ xn+1 − xn  .

From (3.1), we note that  xn+1 − xn  = αn γ f (xn ) + δn xn + [(1 − δn )I − αn B]Trn yn

 −αn−1 γ f (xn−1 ) − δn−1 xn−1 − [(1 − δn−1 )I − αn−1 B]Trn yn−1  .  = αn γ f (xn ) + δn xn + (I − δn − αn B)un  −αn−1 γ f (xn−1 ) − δn−1 xn−1 − (I − δn−1 − αn−1 B)un−1  .  ≤ αn γ f (xn ) − αn γ f (xn−1 ) + αn γ f (xn−1 ) + δn xn − δn−1 xn−1 − αn−1 γ f (xn−1 ) + (I − δn − αn B)un − (I − δn − αn B)un−1 + (I − δn − αn B)un−1  −(I − δn−1 − αn−1 B)un−1  .       ≤ αn γ f (xn ) − αn γ f (xn−1 ) + αn γ f (xn−1 ) − αn−1 γ f (xn−1 ) + δn xn − δn−1 xn−1     + (I − δn − αn B)un − (I − δn − αn B)un−1  + (I − δn − αn B)un−1  −(I − δn−1 − αn−1 B)un−1  .     = αn γ f (xn ) − f (xn−1 ) + |αn − αn−1 | γ f (xn−1 ) + δn xn − δn−1 xn−1 + δn xn−1 − δn xn−1   + (I − δn − αn B) un − un−1  + (I − δn − αn B − I + δn−1 + αn−1 B)un−1  .   = αn γβ xn − xn−1  + |αn − αn−1 | γ f (xn−1 ) + δn xn − xn−1  + |δn − δn−1 | xn−1  + |I − δn − αn B| un − un−1  + |δn−1 − δn + αn−1 B + αn B| un−1  .   ≤ αn γβ xn − xn−1  + |αn − αn−1 | γ f (xn−1 ) + δn xn − xn−1  + |δn − δn−1 | xn−1 

(3:4)

+ |I − δn − αn B| xn − xn−1  + |δn−1 − δn | un−1  + |αn−1 B + αn B| un−1  .   ≤ αn γβ xn − xn−1  + |αn − αn−1 | γ f (xn−1 ) + δn xn − xn−1  + |δn − δn−1 | xn−1  + |I − δn − αn B| xn − xn−1  + |δn−1 − δn | xn−1  − |αn−1 − αn | B xn−1  .   ≤ αn γβ xn − xn−1  + |αn − αn−1 | γ f (xn−1 ) + δn xn − xn−1  + |I − δn − αn B| xn − xn−1  − |αn−1 − αn | B xn−1  .   ≤ αn γβ xn − xn−1  + |αn − αn−1 | γ f (xn−1 ) + |δn + I − δn − αn B| xn − xn−1  − |αn−1 − αn | B xn−1  .   ≤ αn γβ xn − xn−1  + |αn − αn−1 | γ f (xn−1 ) + |I − αn B| xn − xn−1  − |αn − αn−1 | B xn−1  .   ≤ αn γβ xn − xn−1  + |αn − αn−1 | γ f (xn−1 ) − Bxn−1  + |I − αn B| xn − xn−1      ≤ αn γβ xn − xn−1  + |αn − αn−1 | γ f (xn−1 ) − Bxn−1  + |I − αn B| yn − yn−1    ≤ αn γβ xn − xn−1  + |αn − αn−1 | K + |I − αn B| yn − yn−1  ,

where K = ∥gf(xn-1) - Bxn-1∥ = 2 sup{∥f(xn) ∥ + ∥un∥:n Î N}. Moreover, since yn = Frn xn and yn+1 = Frn+1 xn+1, we get   1  y − yn , Ayn + y − yn , yn − xn ≥ 0, rn

∀y ∈ C

(3:5)

and    1  y − yn+1 , Ayn+1 + y − yn+1 , yn+1 − xn+1 ≥ 0, rn+1

∀y ∈ C.

(3:6)

Putting y = yn+1 in (3.5) and y = yn in (3.6), we obtain   1   yn+1 − yn , Ayn + yn+1 − yn , yn − xn ≥ 0 rn

(3:7)

   1  yn − yn+1 , Ayn+1 + yn − yn+1 , yn+1 − xn+1 ≥ 0. rn+1

(3:8)

and

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Adding (3.7) and (3.8), we have

   yn − x n yn+1 − xn+1 yn+1 − yn , Ayn − Ayn+1 + yn+1 − yn , − ≥0 rn rn+1 which implies that  y n − xn yn+1 − xn+1 − yn+1 − yn , Ayn+1 − Ayn + yn+1 − yn , − ≥ 0. rn rn+1 



Using the fact that A is monotone, we get

 yn − x n yn+1 − xn+1 yn+1 − yn , − ≥ 0. rn rn+1 and hence

 rn yn+1 − yn , yn − yn+1 + yn+1 − yn − (yn+1 − xn+1 ) ≥ 0. rn+1 We observe that

   yn+1 − yn 2 ≤ yn+1 − yn , xn+1 − xn (1 − rn )(yn+1 − xn+1 ) rn+1        rn   ) (yn+1 − xn+1 ) . ≤ yn+1 − yn  xn+1 − xn  + (1 − rn+1

(3:9)

Without loss of generality, let k be a real number such that rn >k > 0 for all n Î N. Then, we have     yn+1 − yn  ≤ xn+1 − xn  + 1 |rn+1 − rn | yn+1 − xn+1  rn+1 1 ≤ xn+1 − xn  + |rn+1 − rn | M, k

(3:10)

where M = sup{∥yn - xn∥: n Î N}. Furthermore, from (3.4) and (3.10), we have   1 xn+1 − xn  ≤ αn γβ xn − xn−1  + αn − αn−1  K + (1 − αn ) xn − xn−1  + |rn − ri−1 | M . k 1 = (1 − αn + αn γβ) xn − xn−1  + |αn − αn−1 | K + |rn − rn−1 | M. k M |rn − rn−1 | . = (1 − αn (1 − γβ)) xn − xn−1  + K |αn − αn−1 | + k

Using Lemma 2.3, and by the conditions (C1) and (C3), we have lim xn+1 − xn  = 0.

n→∞

Consequently, from (3.10), we obtain   lim yn+1 − yn  = 0.

(3:11)

n→∞

Since un = Trn yn and un+1 = Trn+1 yn+1, we have   1   y − un , Tun − y − un , (1 − rn )un − yn ≤ 0, rn

∀y ∈ C

(3:12)

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and 

  1  y − un+1 , Tun+1 − y − un+1 , (1 − rn+1 )un+1 − yn+1 ≤ 0, rn+1

∀y ∈ C.

(3:13)

Putting y := un+1 in (3.12) and y := un in (3.13), we get un+1 − un , Tun  −

 1 un+1 − un , (1 − rn )un − yn ≤ 0. rn

(3:14)

and un − un+1 , Tun+1  −

 1  un − un+1 , (1 − rn+1 )un+1 − yn+1 ≤ 0.

rn+1

(3:15)

Adding (3.14) and (3.15), we have

 (1 − rn )un − yn (1 − rn+1 )un+1 − yn+1 un+1 − un , Tun − Tun+1 − un+1 − un , − ≤ 0. rn rn+1

Using the fact that T is pseudo-contractive, we get

 un − yn un+1 − yn+1 un+1 − un , − ≥0 rn rn+1

and hence

 rn un+1 − un , un − un+1 + un+1 − yn − (un+1 + yn+1 ) ≥ 0. rn+1 Thus, using the methods in (3.9) and (3.10), we can obtain   1 un+1 − un  ≤ yn+1 − yn  + |rn+1 + rn | M1 , rn+1

(3:16)

where M1 = sup{∥un - yn∥: n Î N}. Therefore, from (3.11) and property of {rn}, we get lim un+1 − un  = 0.

n→∞

Furthermore, since xn = αn−1 γ f (xn+1 ) + δn−1 xn−1 + [(1 − δn−1 )I − αn−1 B]Trn yn−1, we have xn − un  ≤ xn − un−1  + un−1 − un    = αn−1 γ f (xn−1 ) + δn−1 xn−1 + [(1 − δn−1 )I − αn−1 B]Trn yn−1 − un−1  + un−1 − un   = αn−1 γ f (xn−1 ) + δn−1 xn−1 + (I − δn−1 − αn−1 B)un−1 − un−1  + un−1 − un   = αn−1 γ f (xn−1 ) + δn−1 xn−1 + un−1 − δn−1 un−1 − αn−1 Bun−1 − un−1  + un−1 − un   ≤ αn−1 γ f (xn−1 ) − αn−1 Bun−1 + δn−1 xn−1 − δn−1 un−1  + un−1 − un    ≤ αn−1 γ f (xn−1 ) − Bun−1  + δn−1 xn−1 − un−1  + un−1 − un  .

Thus, by (C1) and (C2), we obtain xn − un  → 0, n → ∞.

(3:17)

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For v ∈ F, using Lemma 2.5, we obtain     yn − v2 = Fr yn − Fr v 2 n n   ≤ Frn xn − Frn v, xn − v   ≤ yn − v, xn − v 2 2  1  = (yn − v + xn − v2 − xn − yn  ) 2 and     yn − v2 ≤ xn − v2 − xn − yn 2 .

(3:18)

Therefore, from (3.1), the convexity of ∥·∥2, (3.2) and (3.18), we get  2 xn+1 − v2 = αn γ f (xn ) + δn xn + [(1 − δn )I − αn B]Trn yn − v  2 = (1 − δn )(Trn yn − v) + δn (xn − v) + αn (γ f (xn ) − BTrn yn )  2   ≤ (1 − δn )(Trn yn − v) + δn (xn − v) + 2αn γ f (xn ) − BTrn yn , xn+1 − v   2 2 ≤ (1 − δn )(yn − v) + δn (xn − v) + 2αn L2

and hence   2 2  2 (1 − δn )(yn − v) ≤ δn (xn − v) − (xn+1 − v) + 2αn L2 .

(3:19)

So, we have ∥yn - v∥ ® 0 as n ® ∞. Consequently, from (3.16) and (3.18), we obtain     un − yn  ≤ un − xn  + xn − yn  → 0 as n → ∞. Step 3. We will show that   lim sup (γ f − B)z, xn − z ≤ 0.

(3:20)

n→∞

Let Q = PF, and since, Q(I - B + gf) is contraction on H into C (see also [[25], pp. 18]) and H is complete. Thus, by Banach Contraction Principle, then there exist a unique element z of H such that z = Q(I - B + gf)z. We choose subsequence {xni } of {xn} such that     lim sup (γ f − B)z, xn − z = lim γ fz − Bz, xni − z n→∞

n→∞

Since {xni } is bounded, there exists a sequence {xnij } of {xni } and y Î C such that {xnij } y. Without loss of generality, we may assume that xni y. Since C is closed and convex it is weakly closed and hence y Î C. Since xn - yn ® 0 as n ® ∞ we have that yni y. Now, we show that y ∈ F. Since yn = Frn, Lemma 2.5 and using (3.5), we get

   yn − x n (3:21) y − yn , Ayn + y − yn , ≥ 0, ∀y ∈ C. rn and 

y − yni , Ayni



 yni − xni ≥ 0, + y − yni , rni

∀y ∈ C.

(3:22)

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Set vt = tv + (1- t)y for all t Î (0,1] and v Î C. Consequently, we get vi Î C. From (3.22), it follows that 

      yni − xni vt − yni ≥ vt − yni , Avt − vt − yni , Avt − vt − yni , rn

   yn − xni = vt − yni , Avt − Ayni − vt − yni , i , rn u −x

n n from the fact that yni − xni → 0 as i ® ∞, we obtain that i rn i → 0 as i ® ∞. Since   A is monotone, we also have that vt − yni , Avt − Ayni ≥ 0. Thus, if follows that

  0 ≤ lim vt − yni , Avt = vt − w, Avt  , i→∞

  and hence v − y, Avt ≥ 0,

∀v ∈ C.

If t ® 0, the continuity of A gives that   v − y, Ay ≥ 0, ∀v ∈ C. This implies that y Î VI(C, A). Furthermore, since un = Trn yn, Lemma 2.5 and using (3.12), we get   1   y − uni , Tuni − y − uni , (rni + 1)uni − yn+1 ≤ 0, rn

∀y ∈ C.

(3:23)

Put zt = t(v) + (1 - t)y for all t Î (0,1] and v Î C. Then, zt Î C and from (3.23) and pseudo-contractivity of T, we get         un − zt , Tzt  = un − zt , Tzt + zt − un , Tui − 1 zt − un , (1 + rn )un − yn i i i i i i i rn      1  zt − uni , uni − yni − zt − uni , uni = − zt − uni , Tzt − rni 2      1 zt − uni , uni − yni − zt − uni , uni ≥ zt − uni  − rni 

  uni − yni = − zt − uni , zt − zt − uni , . rni

Thus, since un - yn ® 0, as n ® ∞ we obtain that

uni −yni rni

→ 0 as i ® ∞. Therefore, as

i ® ∞, it follows that     y − zt , Tzt ≥ y − zt , zt and hence     − v − y, Tzt ≥ − v − y, zt ,

∀v ∈ C.

Taking t ® 0 and since T is continuous we obtain     − v − y, Ty ≥ − v − y, y , ∀v ∈ C. Now, we get v = Ty. Then we obtain that y = Ty and hence y Î F(T). Therefore, y Î F(T) ∩ VI(C, A) and since z = PF (I − B + γ f )z, Lemma 2.2 implies that

Chamnarnpan and Kumam Fixed Point Theory and Applications 2012, 2012:67 http://www.fixedpointtheoryandapplications.com/content/2012/1/67

    lim sup (γ f − B)z, xn − z = lim (I − B + γ f )z − z, xni − z i→∞ n→∞   = (γ f − B)z, y − z ≤ 0.

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(3:24)

Step 4. Finally, we will show that xn ® z as n ® ∞, where z = PF (I − B + rf )z. From (3.1) and (3.2) we observe that   xn+1 − z2 = αn γ f (xn ) + δn xn + [(1 − δn )I − αn B]Trn yn − z, xn+1 − z   = αn γ f (xn ) − Bz, xn+1 − z + δn xn − z, xn+1 − z   + [(1 − δn )I − αn B](Trn − z), xn+1 − z     ≤ αn γ f (xn ) − f (z), xn+1 − z + αn γ f (z) − Bz, xn+1 − z ¯ zn − z xn+1 − z + δn xn − z xn+1 − z + (1 − δn − αn β)   ≤ αn γ K xn − z xn+1 − z + αn γ f (z) − Bz, xn+1 − z ¯ zn − z xn+1 − z + δn xn − z xn+1 − z + (1 − δn − αn β)   = αn γ K xn − z xn+1 − z + αn γ f (z) − Bz, xn+1 − z ¯ xn − z xn+1 − z + (1 − αn β) ≤

  γk αn (xn − z2 + xn+1 − z2 ) + αn γ f (z) − Bz, xn+1 − z 2 ¯ + (1 − αn β)(x n − z xn+1 − z)

  γk αn (xn − z2 + xn+1 − z2 ) + αn γ f (z) − Bz, xn+1 − z 2 ¯ (1 − αn β) + (xn − z2 + xn+1 − z2 ) 2   1 − αn (β¯ − kγ ) 1 xn − z2 + xn+1 − z2 + αn γ f (x) − Bz, xn+1 − z , ≤ 2 2

≤

which implies that   xn+1 + z2 ≤ [1 − αn (β¯ − kγ )]xn − z2 + 2αn γ f (z) − Bz, xn+1 − z .

By the condition (C1), (3.24) and using Lemma 2.3, we see that limn®∞ ∥xn - z∥ = 0. This complete to proof. □ If we take f(x) = u, ∀x Î H and g = 1, then by Theorem 3.1, we have the following corollary: Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C ® C be a continuous pseudo-contractive mapping and A : C ® H be a continuous monotone mapping such that F := F(T) ∩ VI(C, A)  = ∅. let B : H ® H be a strongly positive linear bounded self-adjoint operator with coefficients β¯ > 0and let {xn} be a sequence generated by x1 Î H and  yn = Frn xn (3:25) xn+1 = αn u + δn xn + [(1 − δn )I − αn B]Trn yn , where {an} ⊂ [0,1] and {rn} ⊂ (0, ∞) such that (C1) lim αn = 0, n→∞

(C2) lim δn = 0, n→∞

∞  n=1 ∞ 

|δn+1 − δn | < ∞;

n=1

(C3) lim inf rn > 0, n→∞

αn = ∞;

∞ 

n=1

|rn+1 − rn | < ∞.

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Then, the sequence {xn} converges strongly to z ∈ F, which is the unique solution of the variational inequality:   (3:26) (B − f )z, x − z ≥ 0, ∀x ∈ F. Equivalently, z = PF (I − B + f )z. If we take T ≡ 0, then Trn ≡ I (the identity map on C). So by Theorem 3.1, we obtain the following corollary. Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C ® H be a continuous monotone mapping such that F := VI(C, A)  = ∅. Let f be a contraction of H into itself and let B : H ® H be a strongly positive linear bounded self-adjoint operator with coefficients β¯ > 0and let {xn} be a sequence generated by x1 Î H and xn+1 = αn γ f (xn ) + δn xn + [(1 − δn )I − αn B]Frn xn ,

(3:27)

where {an} ⊂ [0,1] and {rn} ⊂ (0, ∞) such that (C1) lim αn = 0, n→∞

(C2) lim δn = 0, n→∞

∞  n=1 ∞ 

αn = ∞; |δn+1 − δn | < ∞;

n=1

(C3) lim inf rn > 0, n→∞

∞ 

|rn+1 − rn | < ∞.

n=1

Then, the sequence {xn} converges strongly to z ∈ F, which is the unique solution of the variational inequality:   (3:28) (B − γ f )z, x − z ≥ 0, ∀x ∈ F Equivalently, z = PF (I − B + γ f )z. If we take A ≡ 0, then Frn ≡ I (the identity map on C). So by Theorem 3.1, we obtain the following corollary. Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C ® C be a continuous pseudo-contractive mapping such that F := F(T)  = ∅. Let f be a contraction of H into itself and let B : H ® H be a strongly positive linear bounded self-adjoint operator with coefficients β¯ > 0and let {xn} be a sequence generated by x1 Î H and xn+1 = αn γ f (xn ) + δn xn + [(1 − δn )I − αn B]Trn xn ,

where {an} ⊂ [0,1] and {rn} ⊂ (0, ∞) such that (C1) lim αn = 0, n→∞

(C2) lim δn = 0, n→∞

∞  n=1 ∞ 

|δn+1 − δn | < ∞;

n=1

(C3) lim inf rn > 0, n→∞

αn = ∞;

∞ 

n=1

|rn+1 − rn | < ∞.

(3:29)

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Then, the sequence {xn}n≥1 converges strongly to z ∈ F, which is the unique solution of the variational inequality:   (3:30) (B − γ f )z, x − z ≥ 0, ∀x ∈ F. Equivalently, z = PF (I − B + γ f )z. If we take C ≡ H in Theorem 3.1, then we obtain the following corollary. Corollary 3.5. Let H be a real Hilbert space. Let T n : H ® H be a continuous pseudo-contractive mapping and A : H ® H be a continuous monotone mapping such that F := F(T) ∩ A−1 (0)  = ∅. Let f be a contraction of C into itself and let B : H ® H be a strongly positive linear bounded self-adjoint operator with coefficients β¯ > 0and let {xn} be a sequence generated by x1 Î H and  yn = Frn xn xn+1 = αn γ f (xn ) + δn xn + [(1 − δn )I − αn B]Trn yn

(3:31)

where {an} ⊂ [0,1] and {rn} ⊂ (0, ∞) such that (C1) lim αn = 0, n→∞

(C2) lim δn = 0, n→∞

∞  n=1 ∞ 

|δn+1 − δn | < ∞;

n=1

(C3) lim inf rn > 0, n→∞

αn = ∞;

∞ 

|rn+1 − rn | < ∞.

n=1

Then, the sequence {xn} converges strongly to z ∈ F, which is the unique solution of the variational inequality:   (3:32) (B − γ f )z, x − z ≥ 0, ∀x ∈ F Equivalently, z = PF (I − B + γ f )z. Proof. Since D(A) = H, we note that VI(H, A) = A -1 (0). So, by Theorem 3.1, we obtain the desired result. □ Remark 3.6. Our results extend and unify most of the results that have been proved for these important classes of nonlinear operators. In particular, Theorem 3.1 extends Theorem 3.1 of Iiduka and Takahashi [7] and Zegeye et al. [26], Corollary 3.2 of Su et al. [27] in the sense that our convergence is for the more general class of continuous pseudo-contractive and continuous monotone mappings. Corollary 3.4 also extends Theorem 4.2 of Iiduka and Takahashi [7] in the sense that our convergence is for the more general class of continuous pseudo-contractive and continuous monotone mappings. Acknowledgements This study was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC No. 54000267). Authors’ contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Competing interests The authors declare that they have no competing interests.

Chamnarnpan and Kumam Fixed Point Theory and Applications 2012, 2012:67 http://www.fixedpointtheoryandapplications.com/content/2012/1/67

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