Projection methods of iterative solutions in Hilbert spaces - Springer Link

Gu and Lu Fixed Point Theory and Applications 2012, 2012:162 http://www.fixedpointtheoryandapplications.com/content/2012/1/162

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Projection methods of iterative solutions in Hilbert spaces Feng Gu and Jing Lu* *

Correspondence: [email protected]; [email protected] Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, China

Abstract In this paper, zero points of the sum of two monotone mappings, solutions of a monotone variational inequality, and fixed points of a nonexpansive mapping are investigated based on a hybrid projection iterative algorithm. Strong convergence of the purposed iterative algorithm is obtained in the framework of real Hilbert spaces without any compact assumptions. MSC: 47H05; 47H09; 47J25; 90C33 Keywords: fixed point; monotone operator; nonexpansive mapping; variational inequality; zero point

1 Introduction and preliminaries Throughout this paper, we always assume that H is a real Hilbert space with an inner product ·, · and a norm · . Let C be a nonempty, closed, and convex subset of H. Let S : C → C be a nonlinear mapping. F(S) stands for the fixed point set of S; that is, F(S) := {x ∈ C : x = Tx}. Recall that S is said to be nonexpansive iff Sx – Sy ≤ x – y,

∀x, y ∈ C.

If C is a bounded, closed, and convex subset of H, then F(S) is not empty, closed, and convex; see []. Let A : C → H be a mapping. Recall that A is said to be inverse-strongly monotone iff there exists a constant α >  such that Ax – Ay, x – y ≥ αAx – Ay ,

∀x, y ∈ C.

For such a case, A is also said to be α-inverse-strongly monotone. A is said to be monotone iff Ax – Ay, x – y ≥ ,

∀x, y ∈ C.

Recall that the classical variational inequality is to find an x ∈ C such that Ax, y – x ≥ ,

∀y ∈ C.

(.)

© 2012 Gu and Lu; 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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In this paper, we use VI(C, A) to denote the solution set of (.). It is known that x ∈ C is a solution of (.) if and only if x is a fixed point of the mapping ProjC (I – rA), where r >  is a constant, I stands for the identity mapping, and ProjC stands for the metric projection from H onto C. If A is α-inverse-strongly monotone and r ∈ (, α], then the mapping ProjC (I – rA) is nonexpansive; see [] for more details. It follows that VI(C, A) is closed and convex. Monotone variational inequality theory has emerged as a fascinating branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, ecology, social, regional, pure, and applied sciences. In recent years, much attention has been given to developing efficient numerical methods for treating solution problems of monotone variational inequality. The gradient-projection method is a powerful tool for solving constrained convex optimization problems and has extensively been studied; see [–] and the references therein. It has recently been applied to solving split feasibility problems which find applications in image reconstructions and the intensity modulated radiation theory; see [–] and the references therein. However, the gradientprojection method requires the operator to be strongly monotone and Lipschitz continuous. These strong conditions rule out many applications. Extra gradient-projection method which was first introduce by Korpelevich [] in the finite dimensional Euclidean space has been studied recently for relaxing the strong monotonicity of operators; see [– ] and the references therein. Recall that a set-valued mapping M : H ⇒ H is said to be monotone iff, for all x, y ∈ H, f ∈ Mx and g ∈ My imply x – y, f – g > . A monotone mapping M : H ⇒ H is maximal iff the graph Graph(M) of R is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if, for any (x, f ) ∈ H × H, x – y, f – g ≥ , for all (y, g) ∈ Graph(M) implies f ∈ Rx. For a maximal monotone operator M on H and r > , we may define the single-valued resolvent Jr : H → D(M), where D(M) denotes the domain of M. It is known that Jr is firmly nonexpansive, and M– () = F(Jr ), where F(Jr ) := {x ∈ D(M) : x = Jr x} and M– () : {x ∈ H :  ∈ Mx}. Recently, variational inequalities, fixed point problems, and zero point problems have been investigated by many authors based on iterative methods; see, for example, [–] and the references therein. In this paper, zero point problems of the sum of a maximal monotone operator and an inverse-strongly monotone mapping, solution problems of a monotone variational inequality, and fixed point problems of a nonexpansive mapping are investigated. A hybrid iterative algorithm is considered for analyzing the convergence of iterative sequences. Strong convergence theorems are established in the framework of real Hilbert spaces without any compact assumptions. In order to prove our main results, we also need the following definitions and lemmas. Lemma . Let C be a nonempty, closed, and convex subset of H. Then the following inequality holds: x – ProjC x + y – ProjC  ≤ x – y ,

∀x ∈ H, y ∈ C.

Lemma . [] Let C be a nonempty, closed, and convex subset of H. Let S : C → C be a nonexpansive mapping. Then the mapping I – S is demiclosed at zero, that is, if {xn } is a sequence in C such that xn x¯ and xn – Sxn → , then x¯ ∈ F(S).


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Lemma . Let C be a nonempty, closed, and convex subset of H, B : C → H be a mapping, and M : H ⇒ H be a maximal monotone operator. Then F(Jr (I – sB)) = (B + M)– (). Proof Notice that p ∈ F Jr (I – sB) ⇐⇒

⇐⇒

p = Jr (I – sB)p

 ∈ (B + M)– ()

⇐⇒

⇐⇒

p – sBp ∈ p + sMp

p ∈ (B + M)– ().

This completes the proof.

Lemma . [] Let C be a nonempty, closed, and convex subset of H, A : C → H be a Lipschitz monotone mapping, and NC x be the normal cone to C at x ∈ C; that is, NC x = {y ∈ H : x – u, y, ∀u ∈ C}. Define ⎧ ⎨Ax + N x, x ∈ C, C Wx = ⎩∅, x ∈/ C. Then W is maximal monotone and  ∈ Wx if and only if x ∈ VI(C, A).

2 Main results Now, we are in a position to give our main results. Theorem . Let C be a nonempty, closed, and convex subset of H. Let S : C → C be a nonexpansive mapping with a nonempty fixed point set, A : C → H be an α-Lipschitz continuous and monotone mapping, and B : C → H be a β-inverse-strongly monotone mapping. Let M : H ⇒ H be a maximal monotone operator such that D(M) ⊂ C. Assume that F := F(S) ∩ (B + M)– () ∩ VI(C, A) is not empty. Let {xn } be a sequence generated by the following iterative process: ⎧ ⎪ x ∈ C, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C = C, ⎪ ⎪ ⎪ ⎪ ⎨z = Proj (J (x – s Bx ) – r AJ (x – s Bx )), n n n n sn n n n C sn n ⎪ ⎪yn = αn xn + ( – αn )S ProjC (Jsn (xn – sn Bxn ) – rn Azn ), ⎪ ⎪ ⎪ ⎪ ⎪ Cn+ = {v ∈ Cn : yn – v ≤ xn – v}, ⎪ ⎪ ⎪ ⎪ ⎩x = Proj n ≥ , n+ Cn+ x ,

(.)

where Jsn = (I + sn M)– , {rn } is a sequence in (, α ), {sn } is a sequence in (, β), and {αn } is a sequence in (, ). Assume that the following restrictions are satisfied: (a)  < a ≤ rn ≤ b < α ; (b)  < c ≤ sn ≤ d < β; (c)  ≤ αn ≤ e < , where a, b, c, d, and e are real constants. Then the sequence {xn } converges strongly to ProjF x . Proof First, we show that Cn is closed and convex for each n ≥ . From the assumption, we see that C = C is closed and convex. Suppose that Cm is closed and convex for some


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m ≥ . We show that Cm+ is closed and convex for the same m. Let v , v ∈ Cm+ and v = tv + ( – t)v , where t ∈ (, ). Notice that ym – v ≤ xm – v is equivalent to ym  – xm  – v, ym – xm ≥ . It is clear that v ∈ Cm+ . This shows that Cn is closed and convex for each n ≥ . Next, we show that F ⊂ Cn for each n ≥ . Put un = ProjC (vn – rn Azn ), where vn = Jsn (xn – sn Bxn ). From the assumption, we see that F ⊂ C = C . Suppose that F ⊂ Cm for some m ≥ . For any p ∈ F ⊂ Cm , we see from Lemma . that um – p ≤ vm – rm Azm – p – vm – rm Azm – um  = vm – p – vm – um  + rm Azm , p – um = vm – p – vm – um  + rm Azm – Ap, p – zm + Ap, p – zm + Azm , zm – um ≤ vm – p – vm – zm + zm – um  + rm Azm , zm – um = vm – p – vm – zm  – zm – um  + vm – zm – rm Azm , um – zm .

(.)

Notice that A is Lipschitz continuous. In view of zm = ProjC (vm – rm Avm ), we find that vm – zm – rm Azm , um – zm = vm – zm – rm Avm , um – zm + rm Avm – rm Azm , um – zm ≤ rm αvm – zm um – zm .

(.)

Substituting (.) into (.), we obtain that um – p ≤ vm – p – vm – zm  – zm – um  + rm αvm – zm um – zm   α vm – zm  . (.) ≤ vm – p –  – rm This in turn implies from restriction (a) that ym – p ≤ αm xm – p + ( – αm )Sum – p ≤ αm xm – p + ( – αm )um – p   ≤ αm xm – p + ( – αm ) vm – p –  – rm α vm – zm    α vm – zm  ≤ xm – p – ( – αm )  – rm ≤ xm – p .

(.)


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This shows that p ∈ Cm+ . This proves that F ⊂ Cn for each n ≥ . Note xn = ProjCn x . For each p ∈ F ⊂ Cn , we have x – xn ≤ x – p. Since B is inverse-strongly monotone, we see from Lemma . that (B + M)– () is closed and convex. Since A is Lipschitz continuous, we find that VI(C, A) is close and convex. This proves that F is closed and convex. It follows that x – xn ≤ x – ProjF x .

(.)

This implies that {xn } is bounded. Since xn = ProjCn x and xn+ = ProjCn+ x ∈ Cn+ ⊂ Cn , we have  ≤ x – xn , xn – xn+ = x – xn , xn – x + x – xn+ ≤ –x – xn  + x – xn x – xn+ . It follows that xn – x ≤ xn+ – x . This proves that limn→∞ xn – x exists. Notice that xn – xn+  = xn – x  + xn – x , x – xn+ + x – xn+  = xn – x  – xn – x  + xn – x , xn – xn+ + x – xn+  ≤ x – xn+  – xn – x  . It follows that lim xn – xn+ = .

n→∞

(.)

In view of xn+ = ProjCn+ x ∈ Cn+ , we see that yn – xn+ ≤ xn – xn+ . This implies that yn – xn ≤ yn – xn+ + xn – xn+ ≤ xn – xn+ . From (.), we find that lim xn – yn = .

n→∞

(.)


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Since B is β-inverse-strongly monotone, we see from restriction (b) that (I – sn B)x – (I – sn B)y = x – y – sn x – y, Bx – By + s Bx – By n ≤ x – y – sn (β – sn )Bx – By ≤ x – y ,

∀x, y ∈ C.

This implies from (.) that yn – p ≤ αn xn – p + ( – αn )vn – p  = αn xn – p + ( – αn )Jsn (xn – sn Bxn ) – Jsn (p – sn Bp) ≤ xn – p – ( – αn )sn (β – sn )Bxn – Bp . It follows that ( – αn )sn (β – sn )Bxn – Bp ≤ xn – p – yn – p ≤ xn – yn xn – p + yn – p . In view of restrictions (b) and (c), we find from (.) that lim Bxn – Bp = .

n→∞

(.)

Since Jsn is firmly nonexpansive, we find that  vn – p = Jsn (xn – sn Bxn ) – Jsn (p – sn Bp)

≤ vn – p, (xn – sn Bxn ) – (p – sn Bp) =

  vn – p + (xn – sn Bxn ) – (p – sn Bp)   – (vn – p) – (xn – sn Bxn ) – (p – sn Bp)

  vn – p + xn – p – vn – xn + sn (Bxn – Bp)   = vn – p + xn – p – vn – xn  – sn Bxn – Bp  – sn vn – xn , Bxn – Bp

≤

≤

 vn – p + xn – p – vn – xn  + sn vn – xn Bxn – Bp . 

This in turn implies that vn – p ≤ xn – p – vn – xn  + sn vn – xn Bxn – Bp. Combining (.) with (.), we arrive at yn – p ≤ αn xn – p + ( – αn )vn – p ≤ xn – p – ( – αn )vn – xn  + sn vn – xn Bxn – Bp.

(.)


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It follows that ( – αn )vn – xn  ≤ xn – p – yn – p + sn vn – xn Bxn – Bp ≤ xn – yn xn – p + yn – p + sn vn – xn Bxn – Bp. In view of (.) and (.), we see from restriction (c) that lim vn – xn = .

n→∞

(.)

On the other hand, we find from (.) that ( – αn )  – rn α  vn – zn  ≤ xn – p – yn – p ≤ xn – yn xn – p + yn – p . In view of restrictions (a) and (c), we obtain from (.) that lim vn – zn = .

n→∞

(.)

Notice that  un – zn  = PC (vn – rn Azn ) – PC (vn – rn Avn )  ≤ (vn – rn Azn ) – (vn – rn Avn ) ≤ rn α  zn – vn  . Thanks to (.), we arrive at lim un – zn = .

n→∞

(.)

Notice that xn – Sxn ≤ xn – Sun + Sun – Sxn ≤

xn – yn + un – xn  – αn

≤

xn – yn + un – zn + zn – vn + vn – xn .  – αn

In view of (.), (.), (.), and (.), we find from restriction (c) that lim xn – Sxn = .

n→∞

Since {xn } is bounded, we find that there exists a subsequence {xni } of {xn } such that xni q ∈ C. From Lemma ., we easily conclude that q ∈ F(S). Now, we are in a position to show that x ∈ VI(C, A). Define ⎧ ⎨Ax + N x, x ∈ C, C Wx = ⎩∅, x ∈/ C.


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For any given (x, y) ∈ G(W ), we have y – Ax ∈ NC x. Since un ∈ C, we see from the definition of NC x – un , y – Ax ≥ .

(.)

In view of un = PC (vn – rn Azn ), we obtain that x – un , un + rn Azn – vn ≥  and hence

un – vn x – un , + Azn ≥ . rn

(.)

In view of (.) and (.), we find that x – uni , y ≥ x – uni , Ax

un – vni ≥ x – uni , Ax – x – uni , i + Azni rni

= x – uni , Ax – Auni + x – uni , Auni – Azni un – vni – x – uni , i rni un – vni . ≥ x – uni , Auni – Azni – x – uni , i rni

(.)

Notice that xn – un ≤ xn – vn + vn – zn + zn – un . It follows from (.), (.), and (.) that lim xn – un = .

n→∞

Since A is Lipschitz continuous, we find from (.) that x – q, y ≥ . Since W is maximal monotone, we conclude that q ∈ W – (). This proves that q ∈ VI(C, A). Finally, we prove that q ∈ (B + M)– (). Notice that xn – sn Bxn ∈ vn + sn Mvn ; that is, xn – vn – Bxn ∈ Mvn . sn

(.)


Let μ ∈ Mν. Since M is monotone, we find from (.)

xn – vn – Bxn – μ, vn – ν ≥ . sn

In view of the restriction (b), we see that –Bq – μ, q – ν ≥ . This implies that –Bq ∈ Mq, that is, q ∈ (B + M)– (). This completes q ∈ F . Assume that there exists another subsequence {xnj } of {xn } which converges weakly to q ∈ F . We can easily conclude from Opial’s condition that q = q . Finally, we show that q = ProjF x and {xn } converges strongly to q. This completes the proof of Theorem .. In view of the weak lower semicontinuity of the norm, we obtain from (.) that x – ProjF x ≤ x – q ≤ lim inf x – xn n→∞

≤ lim sup x – xn ≤ x – ProjF x , n→∞

which yields that limn→∞ x – xn = x – ProjF x = x – q. It follows that {xn } con verges strongly to ProjF x . This completes the proof. If B = , then Theorem . is reduced to the following. Corollary . Let C be a nonempty, closed, and convex subset of H. Let S : C → C be a nonexpansive mapping with a nonempty fixed point set and A : C → H be an α-Lipschitz continuous and monotone mapping. Let M : H ⇒ H be a maximal monotone operator such that D(M) ⊂ C. Assume that F := F(S) ∩ M– () ∩ VI(C, A) is not empty. Let {xn } be a sequence generated by the following iterative process: ⎧ ⎪ x ∈ C, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C = C, ⎪ ⎪ ⎪ ⎪ ⎨z = Proj (J x – r AJ x ), n n sn n C sn n ⎪ ⎪ yn = αn xn + ( – αn )S ProjC (Jsn xn – rn Azn ), ⎪ ⎪ ⎪ ⎪ ⎪ C = {v ∈ Cn : yn – v ≤ xn – v}, ⎪ ⎪ ⎪ n+ ⎪ ⎩x = Proj n ≥ , n+ Cn+ x , where Jsn = (I + sn M)– , {rn } is a sequence in (, α ), {sn } is a sequence in (, +∞), and {αn } is a sequence in (, ). Assume that the following restrictions are satisfied: (a)  < a ≤ rn ≤ b < α ; (b)  < c ≤ sn < ∞; (c)  ≤ αn ≤ d < , where a, b, c, and d are real constants. Then the sequence {xn } converges strongly to ProjF x . If M = , then Jsn = I. Corollary . is reduced to the following.

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Corollary . Let C be a nonempty, closed, and convex subset of H. Let S : C → C be a nonexpansive mapping with a nonempty fixed point set and A : C → H be an α-Lipschitz continuous and monotone mapping. Assume that F := F(S) ∩ VI(C, A) is not empty. Let {xn } be a sequence generated by the following iterative process: ⎧ ⎪ ⎪x ∈ C, ⎪ ⎪ ⎪ ⎪C = C, ⎪  ⎪ ⎪ ⎪ ⎪ ⎨z = Proj (x – r Ax ), n n n C n ⎪ ⎪ yn = αn xn + ( – αn )S ProjC (xn – rn Azn ), ⎪ ⎪ ⎪ ⎪ ⎪ Cn+ = {v ∈ Cn : yn – v ≤ xn – v}, ⎪ ⎪ ⎪ ⎪ ⎩x = Proj n ≥ , n+ Cn+ x , where {rn } is a sequence in (, α ), and {αn } is a sequence in (, ). Assume that the following restrictions are satisfied: (a)  < a ≤ rn ≤ b < α ; (b)  ≤ αn ≤ c < , where a, b, and c are real constants. Then the sequence {xn } converges strongly to ProjF x . If A = , then Theorem . is reduced to the following. Corollary . Let C be a nonempty, closed, and convex subset of H. Let S : C → C be a nonexpansive mapping with a nonempty fixed point set and B : C → H be a β-inverse-strongly monotone mapping. Let M : H ⇒ H be a maximal monotone operator such that D(M) ⊂ C. Assume that F := F(S) ∩ (B + M)– () is not empty. Let {xn } be a sequence generated by the following iterative process: ⎧ ⎪ x ∈ C, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪C = C, ⎪ ⎨ yn = αn xn + ( – αn )SJsn (xn – sn Bxn ), ⎪ ⎪ ⎪ ⎪Cn+ = {v ∈ Cn : yn – v ≤ xn – v}, ⎪ ⎪ ⎪ ⎪ ⎩ xn+ = ProjCn+ x , n ≥ , where Jsn = (I + sn M)– , {sn } is a sequence in (, β), and {αn } is a sequence in (, ). Assume that the following restrictions are satisfied: (a)  < a ≤ sn ≤ b < β; (b)  ≤ αn ≤ c < , where a, b, and c are real constants. Then the sequence {xn } converges strongly to ProjF x . Let f : H → (–∞, +∞] be a proper convex lower semicontinuous function. Then the subdifferential ∂f of f is defined as follows:

∂f (x) = y ∈ H : f (z) ≥ f (x) + z – x, y, z ∈ H ,

∀x ∈ H.

From Rockafellar [], we know that ∂f is maximal monotone. It is not hard to verify that  ∈ ∂f (x) if and only if f (x) = miny∈H f (y).


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Let IC be the indicator function of C, i.e.,

IC (x) =

⎧ ⎨,

x ∈ C, ⎩+∞, x ∈/ C.

Since IC is a proper lower semicontinuous convex function on H, we see that the subdifferential ∂IC of IC is a maximal monotone operator. It is clear that Js x = PC x, ∀x ∈ H. Notice that (B + ∂IC )– () = VI(C, B). Indeed, x ∈ (B + ∂IC )– ()

⇐⇒

⇐⇒

–Bx ∈ ∂IC x

⇐⇒

Bx, y – x ≥ 

⇐⇒

x ∈ VI(C, B).

 ∈ Bx + ∂IC x

(.)

In the light of the above, the following is not hard to derive from Corollary .. Corollary . Let C be a nonempty, closed, and convex subset of H. Let S : C → C be a nonexpansive mapping with a nonempty fixed point set and B : C → H be a β-inversestrongly monotone mapping. Assume that F := F(S) ∩ VI(C, B) is not empty. Let {xn } be a sequence generated by the following iterative process: ⎧ ⎪ x ∈ C, ⎪ ⎪ ⎪ ⎪ ⎪C = C, ⎪ ⎪ ⎨  yn = αn xn + ( – αn )SPC (xn – sn Bxn ), ⎪ ⎪ ⎪ ⎪ Cn+ = {v ∈ Cn : yn – v ≤ xn – v}, ⎪ ⎪ ⎪ ⎪ ⎩ xn+ = ProjCn+ x , n ≥ , where {sn } is a sequence in (, β), and {αn } is a sequence in (, ). Assume that the following restrictions are satisfied: (a)  < a ≤ sn ≤ b < β; (b)  ≤ αn ≤ c < , where a, b, and c are real constants. Then the sequence {xn } converges strongly to ProjF x .

3 Applications First, we consider the problem of finding a minimizer of a proper convex lower semicontinuous function. Theorem . Let f : H → (–∞, +∞] be a proper convex lower semicontinuous function such that (∂f )– () is not empty. Let {xn } be a sequence generated by the following iterative


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process: ⎧ ⎪ x ∈ H, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨C = H,



n yn = arg mins∈H {f (z) + z–x }, sn ⎪ ⎪ ⎪ ⎪ ⎪ ⎪Cn+ = {v ∈ Cn : yn – v ≤ xn – v}, ⎪ ⎪ ⎩ xn+ = ProjCn+ x , n ≥ ,

where {sn } is a positive sequence such that  < a ≤ sn , where a is a real constant. Then the sequence {xn } converges strongly to Proj(∂f )– () x . Proof Putting A = B = , S = I, and αn ≡ , we can immediately draw the desired conclusion from Theorem .. Second, we consider the problem of approximating a common fixed point of a pair of nonexpansive mappings. Theorem . Let C be a nonempty, closed, and convex subset of H. Let S : C → C and T : C → C be a pair of nonexpansive mappings with a nonempty common fixed point set. Let {xn } be a sequence generated by the following iterative process: ⎧ ⎪ ⎪x ∈ C, ⎪ ⎪ ⎪ ⎪ ⎪ C = C, ⎪ ⎪ ⎪ ⎪ ⎨z = ( – s )x + s Tx , n n n n n ⎪ ⎪ yn = αn xn + ( – αn )Szn , ⎪ ⎪ ⎪ ⎪ ⎪ Cn+ = {v ∈ Cn : yn – v ≤ xn – v}, ⎪ ⎪ ⎪ ⎪ ⎩x = Proj n ≥ , n+ Cn+ x , where {sn }, and {αn } are sequences in (, ). Assume that the following restrictions are satisfied: (a)  < a ≤ sn ≤ b < ; (b)  ≤ αn ≤ c < , where a, b, and c are real constants. Then the sequence {xn } converges strongly to ProjF(S)∩F(T) x . Proof Putting A = , M = ∂IC , and B = I – T, we see that B is  -inverse-strongly monotone. We also have F(T)=VI(C, B) and PC (xn – sn Bxn ) = ( – sn )xn + sn Txn . In view of (.), we can immediately obtain the desired result. Let F be a bifunction of C × C into R, where R denotes the set of real numbers. Recall the following equilibrium problem in the terminology of Blum and Oettli [] (see also Fan []): Find x ∈ C such that F(x, y) ≥ ,

∀y ∈ C.

(.)


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To study the equilibrium problem (.), we may assume that F satisfies the following conditions: (A) F(x, x) =  for all x ∈ C; (A) F is monotone, i.e., F(x, y) + F(y, x) ≤  for all x, y ∈ C; (A) for each x, y, z ∈ C, lim sup F tz + ( – t)x, y ≤ F(x, y); t↓

(A) for each x ∈ C, y → F(x, y) is convex and lower semi-continuous. Putting F(x, y) = Ax, y – x for every x, y ∈ C, we see that the equilibrium problem (.) is reduced to the variational inequality (.). The following lemma can be found in [] and []. Lemma . Let C be a nonempty, closed, and convex subset of H and F : C × C → R be a bifunction satisfying (A)-(A). Then, for any s >  and x ∈ H, there exists z ∈ C such that  F(z, y) + y – z, z – x ≥ , s

∀y ∈ C.

Further, define  Ts x = z ∈ C : F(z, y) + y – z, z – x ≥ , ∀y ∈ C s

(.)

for all s >  and x ∈ H. Then, the following hold: (a) Ts is single-valued; (b) Ts is firmly nonexpansive; that is, Ts x – Ts y ≤ Ts x – Ts y, x – y,

∀x, y ∈ H;

(c) F(Ts ) = EP(F); (d) EP(F) is closed and convex. Lemma . [] Let C be a nonempty, closed, and convex subset of H, F be a bifunction from C × C to R which satisfies (A)-(A), and AF be a multivalued mapping of H into itself defined by

AF x =

⎧ ⎨{z ∈ H : F(x, y) ≥ y – x, z, ∀y ∈ C},

x ∈ C,

⎩∅,

x ∈/ C.

(.)

Then AF is a maximal monotone operator with the domain D(AF ) ⊂ C, EP(F) = A– F (), where FP(F) stands for the solution set of (.), and Ts x = (I + sAF )– x,

∀x ∈ H, r > ,

where Ts is defined as in (.). Finally, we consider finding a solution of the equilibrium problem.


Theorem . Let C be a nonempty, closed, and convex subset of H. Let F : C × C → R be a bifunction satisfying (A)-(A) such that EP(F) = ∅. Let {xn } be a sequence generated by the following iterative process: ⎧ ⎪ x ∈ C, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨C = C, yn = (I + sn AF )– xn , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪Cn+ = {v ∈ Cn : yn – v ≤ xn – v}, ⎪ ⎪ ⎩ xn+ = ProjCn+ x , n ≥ , where AF is defined as (.), and {sn } is a positive sequence such that  < a ≤ sn < ∞, where a is a real constant Then the sequence {xn } converges strongly to ProjFP(F) x . Proof Putting A = B = , S = I and αn ≡ , we immediately reach the desired conclusion from Lemma ..

Competing interests The authors declare that they have no competing interests. Authors’ contributions Both authors contributed equally to this work. Both authors read and approved the final manuscript. Author details Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, China. Authors’ information Author’s information Acknowledgements The first author was supported by the National Natural Science Foundation of China (11071169, 11271105), the Natural Science Foundation of Zhejiang Province (Y6110287, Y12A010095). Received: 27 June 2012 Accepted: 6 September 2012 Published: 25 September 2012 References 1. Browder, FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math. 18, 78-81 (1976) 2. Takahshi, W, Toyoda, M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417-428 (2003) 3. Polyak, BT: Introduction to Optimization. Optimization Software, New York (1987) 4. Calamai, PH, Moré, JJ: Projected gradient methods for linearly constrained problems. Math. Program. 39, 93-116 (1987) 5. Zhang, M: Iterative algorithms for common elements in fixed point sets and zero point sets with applications. Fixed Point Theory Appl. 2012, 21 (2012) 6. Byrne, C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103-120 (2008) 7. Censor, Y, Elfving, T, Kopf, N, Bortfeld, T: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071-2084 (2005) 8. Censor, Y, Bortfeld, T, Martin, B, Trofimov, A: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353-2365 (2006) 9. Lopez, G, Martin, V, Xu, HK: Perturbation techniques for nonexpansive mappings with applications. Nonlinear Anal. 10, 2369-2383 (2009) 10. Korpelevich, GM: An extragradient method for finding saddle points and for other problems. Ekonom. i Mat. Metody 12, 747-756 (1976) 11. Qin, X, Cho, SY, Kang, SM: An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings. J. Glob. Optim. 49, 679-693 (2011) 12. Anh, PN, Kim, JK, Muu, LD: An extragradient algorithm for solving bilevel pseudomonotone variational inequalities. J. Glob. Optim. 52, 627-639 (2012) 13. Nadezhkina, N, Takahashi, W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191-201 (2006)

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doi:10.1186/1687-1812-2012-162 Cite this article as: Gu and Lu: Projection methods of iterative solutions in Hilbert spaces. Fixed Point Theory and Applications 2012 2012:162.

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