On solving split equilibrium problems and fixed

Suantai et al. Fixed Point Theory and Applications (2016) 2016:35 DOI 10.1186/s13663-016-0509-4

RESEARCH

Open Access

On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces Suthep Suantai1* , Prasit Cholamjiak2 , Yeol Je Cho3,4 and Watcharaporn Cholamjiak2 *

Correspondence: [email protected] 1 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand Full list of author information is available at the end of the article

Abstract In this paper, we introduce and study iterative schemes for solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces and prove that the modified Mann iteration converges weakly to a common solution of the considered problems. Moreover, we present some examples and numerical results for the main results. MSC: 47H10; 54H25 Keywords: nonspreading multi-valued mapping; split equilibrium problem; fixed point problem; weak convergence; Hilbert space

1 Introduction In the following, let H and H be real Hilbert spaces with the inner product ·, · and the norm · . Let C be a nonempty subset of H . The equilibrium problem is to find a point xˆ ∈ C such that F (ˆx, y) ≥ 

(.)

for all y ∈ C. Since its inception by Blum and Oettli [] in , the equilibrium problem (.) has received much attention due to its applications in a large variety of problems arising in numerous problems in physics, optimizations, and economics. Some methods have been rapidly established for solving this problem (see [–]). Very recently, Kazmi and Rizvi [] introduced and studied the following split equilibrium problem: Let C ⊆ H and Q ⊆ H . Let F : C × C → R and F : Q × Q → R be two bifunctions. Let A : H → H be a bounded linear operator. The split equilibrium problem is to find xˆ ∈ C such that F (ˆx, x) ≥  for all x ∈ C

(.)

and such that yˆ = Aˆx ∈ Q solves F (ˆy, y) ≥  for all y ∈ Q.

(.)

© 2016 Suantai et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Suantai et al. Fixed Point Theory and Applications (2016) 2016:35

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Note that the problem (.) is the classical equilibrium problem and we denote its solution set by EP(F ). The inequalities (.) and (.) constitute a pair of equilibrium problems which have to find the image yˆ = Aˆx, under a given bounded linear operator A, of the solution xˆ of (.) in H is the solution of (.) in H . We denote the solution set of (.) by EP(F ). The solution set of the split equilibrium problem (.) and (.) is denoted by = {z ∈ EP(F ) : Az ∈ EP(F )}. A subset C ⊂ H is said to be proximinal if, for each x ∈ H , x – y = d(x, C) = inf x – z : z ∈ C . Let CB(C), K(C), and P(C) denote the families of nonempty closed bounded subsets, nonempty compact subsets and nonempty proximinal bounded subset of C, respectively. The Hausdorff metric on CB(C) is defined by H(A, B) = max sup d(x, B), sup d(y, A) x∈A

y∈B

for all A, B ∈ CB(C) where d(x, B) = infb∈B x – b. An element p ∈ C is called a fixed point of T : C → CB(C) if p ∈ Tp. The set of fixed points of T is denoted by F(T). We say that T : C → CB(C) is: () nonexpansive if H(Tx, Ty) ≤ x – y for all x, y ∈ C; () quasi-nonexpansive if H(Tx, Tp) ≤ x – p for all x ∈ C and p ∈ F(T). Recently, the existence of fixed points and the convergence theorems of multi-valued mappings have been studied by many authors (see [–]). Hussain and Khan [] presented the fixed point theorems of a *-nonexpansive multivalued mapping and the strong convergence of its iterates to a fixed point defined on a closed and convex subset of a Hilbert space by using the best approximation operator PT x, which is defined by PT x = {y ∈ Tx : y – x = d(x, Tx)}. The convergence theorems and its applications in this direction have been established by many authors (for instance, see [, , ]). In , Song and Cho [] gave the example of a multi-valued mapping T which is not necessary nonexpansive, but PT is nonexpansive. This is an important tool for studying the fixed point theory for multi-valued mappings. Kohsaka and Takahashi [] introduced a class of mappings which is called nonspreading mapping. Let C be a subset of Hilbert spaces H . A mapping T : C → C is said to be nonspreading if Tx – Ty ≤ Tx – y + Ty – x


Page 3 of 16

for all x, y ∈ C. Subsequently, Iemoto and Takahashi [] showed that T : C → C is nonspreading if and only if Tx – Ty ≤ x – y + x – Tx, y – Ty for all x, y ∈ C. Very recently, Liu [] introduced the following class of multi-valued mappings: a multivalued mapping T : C → CB(C) is said to be nonspreading if ux – uy  ≤ ux – y + uy – x for some ux ∈ Tx and uy ∈ Ty for all x, y ∈ C. He proved a weak convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points. In this paper, we introduce, by using Hausdorff metric, the class of nonspreading multivalued mappings. We say that a mapping T : C → CB(C) is a k-nonspreading multi-valued mapping if there exists k >  such that H(Tx, Ty) ≤ k d(Tx, y) + d(x, Ty)

(.)

for all x, y ∈ C. It is easy to see that, if T is  -nonspreading, then T is nonspreading in the case of singlevalued mappings (see [, ]). Moreover, if T is a  -nonspreading and F(T) = ∅, then T is quasi-nonexpansive. Indeed, for all x ∈ C and p ∈ F(T), we have H(Tx, Tp) ≤ d(Tx, p) + d(x, Tp) ≤ H(Tx, Tp) + x – p . It follows that H(Tx, Tp) ≤ x – p.

(.)

We now give an example of a  -nonspreading multi-valued mapping which is not nonexpansive. Example . Consider C = [–, ] with the usual norm. Define T : C → CB(C) by {}, x ∈ [–, ]; Tx = |x| [– |x|+ , ], x ∈ [–, –). Now, we show that T is  -nonspreading. In fact, we have the following cases: Case : If x, y ∈ [–, ], then H(Tx, Tx) = . |y| Case : If x ∈ [–, ] and y ∈ [–, –), then Tx = {} and Ty = [– |y|+ , ]. This implies that H(Tx, Ty) = 

|y| |y| + 

 <  < y ≤ d(Tx, y) + d(x, Ty) .


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|y| |x| Case : If x, y ∈ [–, –), then Tx = [– |x|+ , ] and Ty = [– |y|+ , ]. This implies that

|y| |x| H(Tx, Ty) =  – |x| +  |y| + 





<  < d(Tx, y) + d(x, Ty) .

On the other hand, T is not nonexpansive since for x = – and y = –  , we have Tx = {} and Ty = [–  , ]. This shows that H(Tx, Ty) =  >  = | –  – (–  )| = x – y. In , Mann [] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping T in a Hilbert space H: xn+ = αn xn + ( – αn )Txn

(.)

for each n ∈ N, where the initial point x is taken in C arbitrarily and {αn } is a sequence in [, ]. Motivated by the previous results, in this paper, we introduce and study the Mann-type iteration to approximate a common solution of the split equilibrium problem and the fixed point problem for a  -nonspreading multi-valued mapping and prove some weak convergence theorems in Hilbert spaces. Finally, we give some examples and numerical results to illustrate our main results.

2 Preliminaries We now provide some results for the main results. In a Hilbert space H , let C be a nonempty closed convex subset of H . For every point x ∈ H , there exists a unique nearest point of C, denoted by PC x, such that x – PC x ≤ x – y for all y ∈ C. Such a PC is called the metric projection from H onto C. We know that PC is a firmly nonexpansive mapping from H onto C, i.e., PC x – PC y ≤ PC x – PC y, x – y,

∀x, y ∈ H .

Further, for any x ∈ H and z ∈ C, z = PC x if and only if x – z, z – y ≥ ,

∀y ∈ C.

A mapping A : C → H is called α-inverse strongly monotone if there exists α >  such that x – y, Ax – Ay ≥ αAx – Ay ,

∀x, y ∈ C.

Lemma . Let H be a real Hilbert space. Then the following equations hold: () x – y = x – y – x – y, y for all x, y ∈ H ; () x + y ≤ x + y, x + y for all x, y ∈ H ; () tx + ( – t)y = tx + ( – t)y – t( – t)x – y for all t ∈ [, ] and x, y ∈ H ; () If {xn }∞ n= is a sequence in H which converges weakly to z ∈ H , then lim sup xn – y = lim sup xn – z + z – y n→∞

for all y ∈ H .

n→∞


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A space X is said to satisfy Opial’s condition if, for any sequence xn with xn x, then lim inf xn – x < lim inf xn – y n→∞

n→∞

for all y ∈ X with y = x. It is well known that every Hilbert space satisfies Opial’s condition. Lemma . [] Let X be a Banach space which satisfies Opial’s condition and {xn } be a sequence in X. Let u, v ∈ X be such that limn→∞ xn – u and limn→∞ xn – v exist. If {xnk } and {xmk } are subsequences of {xn } which converge weakly to u and v, respectively, then u = v. Assumption . [] Let F : C × C → R be a bifunction satisfying the following assumptions: () F (x, x) =  for all x ∈ C; () F is monotone, i.e., F (x, y) + F (y, x) ≤  for all x ∈ C; () for each x, y, z ∈ C, lim supt→+ F (tz + ( – t)x, y) ≤ F (x, y); () for each x ∈ C, y → F (x, y) is convex and lower semi-continuous. Lemma . [] Assume that F : C × C → R satisfies Assumption .. For any r >  and x ∈ H , define a mapping TrF : H → C as follows: TrF (x) =

 z ∈ C : F (z, y) + y – z, z – x ≥ , ∀y ∈ C . r

Then the following hold: () TrF is nonempty and single-valued; () TrF is firmly nonexpansive, i.e., for any x, y ∈ H ,

F

T  x – T F y  ≤ T F x – T F y, x – y ; r r r r () F(TrF ) = EP(F ); () EP(F ) is closed and convex. Further, assume that F : Q × Q → R satisfying Assumption .. For each s >  and w ∈ H , define a mapping TsF : H → Q as follows:  TsF (w) = d ∈ Q : F (d, e) + e – d, d – w ≥ , ∀e ∈ Q . s Then we have the following: () TsF is nonempty and single-valued; () TsF is firmly nonexpansive; () F(TsF ) = EP(F , Q); () EP(F , Q) is closed and convex. Condition (A) Let H be a Hilbert space and C be a subset of H . A multi-valued mapping T : C → CB(C) is said to satisfy Condition (A) if x – p = d(x, Tp) for all x ∈ H and p ∈ F(T).


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Remark . We see that T satisfies Condition (A) if and only if Tp = {p} for all p ∈ F(T). It is well known that the best approximation operator PT , which is defined by PT x = {y ∈ Tx : y – x = d(x, Tx)}, also satisfies Condition (A).

3 Main results Now, we are ready to prove some weak convergence theorem for  -nonspreading multivalued mappings in Hilbert spaces. To this end, we need the following crucial results. Lemma . Let C be a closed and convex subset of a real Hilbert space H and T : C → K(C) be a k-nonspreading multi-valued mapping such that k ∈ (,  ]. If x, y ∈ C and a ∈ Tx, then there exists b ∈ Ty such that a – b ≤ H(Tx, Ty) ≤

k x – y + x – a, y – b . –k

Proof Let x, y ∈ C and a ∈ Tx. By Nadler’s theorem (see []), there exists b ∈ Ty such that a – b ≤ H(Tx, Ty) . It follows that  H(Tx, Ty) k ≤ d(Tx, y) + d(x, Ty) ≤ a – y + x – b ≤ a – x + a – x, x – y + x – y + x – a + x – a, a – b + a – b = a – x + x – y + a – b +  a – x, x – a – (y – b) ≤ a – x + x – y + H(Tx, Ty) +  a – x, x – a – (y – b) . This implies that H(Tx, Ty) ≤

k x – y + x – a, y – b . –k

This completes the proof.

Lemma . Let C be a closed and convex subset of a real Hilbert space H and T : C → K(C) be a k-nonspreading multi-valued mapping such that k ∈ (,  ]. Let {xn } be a sequence in C such that xn p and limn→∞ xn – yn =  for some yn ∈ Txn . Then p ∈ Tp. Proof Let {xn } be a sequence in C which converges weakly to p and let yn ∈ Txn be such that xn – yn → . Now, we show that p ∈ F(T). By Lemma ., there exists zn ∈ Tp such that yn – zn  ≤

k xn – p + xn – yn , p – zn . –k


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Since Tp is compact and zn ∈ Tp, there exists {zni } ⊂ {zn } such that zni → z ∈ Tp. Since {xn } converges weakly, it is bounded. For each x ∈ H , define a function f : H → [, ∞) by f (x) := lim sup i→∞

k xni – x . –k

Then, by Lemma .(), we obtain f (x) = lim sup i→∞

k xni – p + p – x –k

for all x ∈ H . Thus f (x) = f (p) + f (z) = f (p) +

k p – x –k

for all x ∈ H . It follows that

k p – z . –k

(.)

We observe that f (z) = lim sup i→∞

≤ lim sup i→∞

k k xni – z = lim sup xni – yni + yni – z –k  – k i→∞ k yn – z . –k i

This implies that f (z) ≤ lim sup i→∞

= lim sup i→∞

≤ lim sup i→∞

≤ lim sup i→∞

k yn – z –k i  k yni – zni + zni – z –k k xni – p + xni – yni , p – zni –k k xni – p –k

= f (p). Hence it follows from (.) and (.) that p – z = . This completes the proof.

(.)

Theorem . Let H , H be two real Hilbert space and C ⊂ H , Q ⊂ H be nonempty closed convex subsets of Hilbert spaces H and H , respectively. Let A : H → H be a bounded linear operator and T : C → K(C) a  -nonspreading multi-valued mapping. Let F : C × C → R, F : Q × Q → R be bifunctions satisfying Assumption . and F is upper semicontinuous in the first argument. Assume that T satisfies Condition (A) and = F(T) ∩ = ∅, where = {z ∈ C : z ∈ EP(F ) and Az ∈ EP(F )}. Let {xn } be a sequence defined by ⎧ ⎪ ⎨x ∈ C arbitrarily, un = TrFn (I – γ A∗ (I – TrFn )A)xn , ⎪ ⎩ xn+ ∈ αn xn + ( – αn )Tun ,

(.)

for all n ≥ , where {αn } ⊂ (, ), rn ⊂ (, ∞), and γ ∈ (, /L) such that L is the spectral radius of A∗ A and A∗ is the adjoint of A. Assume that the following conditions hold:


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()  < lim infn→∞ αn ≤ lim supn→∞ αn < ; () lim infn→∞ rn > . Then the sequence {xn } defined by (.) converges weakly to p ∈ . Proof We first show that A∗ (I – TrFn )A is a L -inverse strongly monotone mapping. Since TrFn is firmly nonexpansive and I – TrFn is -inverse strongly monotone, we see that

∗

A I – T F Ax – A∗ I – T F Ay  = A∗ I – T F (Ax – Ay), A∗ I – T F (Ax – Ay) rn rn rn rn = I – TrFn (Ax – Ay), AA∗ I – TrFn (Ax – Ay) ≤ L I – TrFn (Ax – Ay), I – TrFn (Ax – Ay)



= L I – T F (Ax – Ay) rn

≤ L Ax – Ay, I – TrFn (Ax – Ay) = L x – y, A∗ I – TrFn Ax – A∗ I – TrFn Ay for all x, y ∈ H . This implies that A∗ (I – TrFn )A is a L -inverse strongly monotone mapping. Since γ ∈ (, L ), it follows that I – γ A∗ (I – TrFn )A is nonexpansive. Now, we divide the proof into six steps as follows: Step . Show that {xn } is bounded. Let p ∈ . Then p = TrFn p and (I – γ A∗ (I – TrFn )A)p = p. Thus we have

un – p = TrFn I – γ A∗ I – TrFn A xn – TrFn I – γ A∗ I – TrFn A p

≤ I – γ A∗ I – TrFn A xn – I – γ A∗ I – TrFn A p ≤ xn – p.

(.)

It follows that xn+ – p ≤ αn xn – p + ( – αn )zn – p

for some zn ∈ Tun

≤ αn xn – p + ( – αn )d(zn , Tp) ≤ αn xn – p + ( – αn )H(Tun , Tp) ≤ xn – p. Hence limn→∞ xn – p exists. Step . Show that zn – xn →  as n → ∞ for all zn ∈ Tun . From Lemma . and T satisfying Condition (A), we have xn+ – p ≤ αn xn – p + ( – αn )zn – p – αn ( – αn )xn – zn  = αn xn – p + ( – αn )d(zn , Tp) – αn ( – αn )xn – zn  ≤ αn xn – p + ( – αn )H(Tun , Tp) – αn ( – αn )xn – zn  ≤ αn xn – p + ( – αn )un – p – αn ( – αn )xn – zn ≤ xn – p – αn ( – αn )xn – zn  .


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This implies that αn ( – αn )xn – zn  ≤ xn – p – xn+ – p . From Condition () and the existence of limn→∞ xn – p, we have lim xn – zn = .

n→∞

(.)

Step . Show that un – zn →  as n → ∞ for all zn ∈ Tun . For any p ∈ , we estimate

 un – p = TrFn I – γ A∗ I – TrFn A xn – p

 = TrFn I – γ A∗ I – TrFn A xn – TrFn p

 ≤ xn – γ A∗ I – TrFn Axn – p

 ≤ xn – p + γ  A∗ I – TrFn Axn + γ p – xn , A∗ I – TrFn Axn . Thus we have un – p ≤ xn – p + γ  Axn – TrFn Axn , AA∗ I – TrFn Axn + γ p – xn , A∗ I – TrFn Axn .

(.)

On the other hand, we have γ  Axn – TrFn Axn , AA∗ I – TrFn Axn ≤ Lγ  Axn – TrFn Axn , Axn – TrFn Axn

 = Lγ  Axn – TrFn Axn

(.)

γ p – xn , A∗ I – TrFn Axn = γ A(p – xn ), Axn – TrFn Axn = γ A(p – xn ) + Axn – TrFn Axn – Axn – TrFn Axn , Axn – TrFn Axn

 = γ Ap – TrFn Axn , Axn – TrFn Axn – Axn – TrFn Axn



  F F

≤ γ Axn – Trn Axn – Axn – Trn Axn 

 = –γ Axn – TrFn Axn .

(.)

and

Using (.), (.), and (.), we have



 un – p ≤ xn – p + Lγ  Axn – TrFn Axn – γ Axn – TrFn Axn



= xn – p + γ (Lγ – ) Axn – TrFn Axn .

(.)


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It follows that, for all zn ∈ Tun ,

 xn+ – p = αn xn + ( – αn )zn – p ≤ αn xn – p + ( – αn )zn – p = αn xn – p + ( – αn )d(zn , Tp) ≤ αn xn – p + ( – αn )H(Tun , Tp) ≤ αn xn – p + ( – αn )un – p



≤ αn xn – p + ( – αn ) xn – p + γ (Lγ – ) Axn – TrFn Axn



≤ xn – p + γ (Lγ – ) Axn – TrFn Axn . Therefore, we have

 –γ (Lγ – ) Axn – TrFn Axn ≤ xn – p – xn+ – p . Since γ (Lγ – ) <  and limn→∞ xn – p exists, by (.), we obtain

lim Axn – TrFn Axn = .

(.)

n→∞

Since TrFn is firmly nonexpansive and I – γ A∗ (TrFn – I)A is nonexpansive, it follows that un – p

 = TrFn xn – γ A∗ I – TrFn Axn – TrFn p ≤ TrFn xn – γ A∗ I – TrFn Axn – TrFn p, xn – γ A∗ I – TrFn Axn – p = un – p, xn – γ A∗ I – TrFn Axn – p



  un – p + xn – γ A∗ I – TrFn Axn – p – un – xn – γ A∗ I – TrFn Axn 



 ≤ un – p + xn – p – un – xn – γ A∗ I – TrFn Axn 

  = un – p + xn – p – un – xn  + γ  A∗ I – TrFn Axn  – γ un – xn , A∗ I – TrFn – I Axn , =

which implies that un – p ≤ xn – p – un – xn  + γ un – xn , A∗ I – TrFn Axn

≤ xn – p – un – xn  + γ un – xn A∗ I – TrFn Axn . It follows from (.) that xn+ – p ≤ αn xn – p + ( – αn )zn – p

for all zn ∈ Tun

(.)


Page 11 of 16

≤ αn xn – p + ( – αn )d(zn , Tp) ≤ αn xn – p + ( – αn )H(Tun , Tp) ≤ αn xn – p + ( – αn )un – p

≤ αn xn – p + ( – αn ) xn – p – un – xn  + γ un – xn A∗ I – TrFn Axn . Therefore, we have

( – αn )un – xn  ≤ γ un – xn A∗ I – TrFn Axn + xn – p – xn+ – p . From Condition () and (.), we have lim un – xn = .

n→∞

(.)

From (.) and (.), we have un – zn ≤ un – xn + xn – zn → 

(.)

as n → ∞. Step . Show that xn+ – xn →  as n → ∞. From (.) and (.), we have

xn+ – un = αn xn + ( – αn )zn – un ≤ αn xn – un + ( – αn )zn – un → 

(.)

as n → ∞. From (.) and (.), we also have un – un+ ≤ un – xn+ + xn+ – un+ → 

(.)

as n → ∞. It follows from (.) and (.) that xn+ – xn ≤ xn+ – un + un – xn → 

(.)

as n → ∞. Step . Show that ωw (xn ) ⊂ , where ωw (xn ) = {x ∈ H : xni x, {xni } ⊂ {xn }}. Since {xn } is bounded and H is reflexive, ωw (xn ) is nonempty. Let q ∈ ωw (xn ) be an arbitrary element. Then there exists a subsequence {xni } ⊂ {xn } converging weakly to q. From (.), it follows that uni q as i → ∞. By Lemma . and (.), we obtain q ∈ F(T). Next, we show that q ∈ EP(F ). From un = TrFn (I + γ A∗ (I – TrFn )A)xn , we have F (un , y) +

 y – un , un – xn – γ A∗ I – TrFn Axn ≥  rn

for all y ∈ C, which implies that F (un , y) +

  y – un , γ A∗ I – TrFn Axn ≥  y – un , un – xn – rn rn


Page 12 of 16

for all y ∈ C. By Assumption .(), we have   y – uni , γ A∗ I – TrFn Axni ≥ F (y, uni ) y – uni , uni – xni – i rni rni for all y ∈ C. From lim infn→∞ rn > , from (.), (.), and Assumption .(), we obtain F (y, q) ≤  for all y ∈ C. For any  < t ≤  and y ∈ C, let yt = ty + ( – t)q. Since y ∈ C and q ∈ C, yt ∈ C, and hence F (yt , q) ≤ . So, by Assumption .() and (), we have  = F (yt , yt ) ≤ tF (yt , y) + ( – t)F (yt , q) ≤ tF (yt , y) and hence F (yt , y) ≥ . So F (q, y) ≥  for all y ∈ C by (.) and hence q ∈ EP(F ). Since A is a bounded linear operator, Axni Aq. Then it follows from (.) that TrFn Axni Aq i

(.)

as i → ∞. By the definition of TrFni Axni , we have  y – TrFn Axni , TrFn Axni – Axni ≥  F TrFn Axni , y + i i i rni for all y ∈ C. Since F is upper semi-continuous in the first argument and (.), it follows that F (Aq, y) ≥  for all y ∈ C. This shows that Aq ∈ EP(F ). Hence q ∈ . Step . Show that {xn } and {un } converge weakly to an element of . It is sufficient to show that ωw (xn ) is single point set. Let p, q ∈ ωw (xn ) and {xnk }, {xnm } ⊂ {xn } be such that xnk p and xnm q. From (.), we also have unk p and unm q. By Lemma . and (.), it follows that p, q ∈ F(T). Applying Lemma ., we obtain p = q. This completes the proof. If Tp = {p} for all p ∈ F(T), then T satisfies Condition (A) and so we can obtain the following result. Theorem . Let H , H be two real Hilbert spaces and C ⊂ H , Q ⊂ H be nonempty closed convex subsets of Hilbert spaces H and H , respectively. Let A : H → H be a bounded linear operator and T : C → K(C) a  -nonspreading multi-valued mapping. Let F : C × C → R, F : Q × Q → R be bifunctions satisfying Assumption . and F is upper semi-continuous in the first argument. Assume that = F(T) ∩ = ∅ and Tp = {p} for all p ∈ F(T), where = {z ∈ C : z ∈ EP(F ) and Az ∈ EP(F )}. Let {xn } be a sequence defined by ⎧ ⎪ ⎨x ∈ C arbitrarily, un = TrFn (I – γ A∗ (I – TrFn )A)xn , ⎪ ⎩ xn+ ∈ αn xn + ( – αn )Tun ,

(.)


Page 13 of 16

for all n ≥ , where {αn } ⊂ (, ), rn ⊂ (, ∞), and γ ∈ (, /L) such that L is the spectral radius of A∗ A and A∗ is the adjoint of A. Assume that the following conditions hold: ()  < lim infn→∞ αn ≤ lim supn→∞ αn < ; () lim infn→∞ rn > . Then the sequence {xn } defined by (.) converges weakly to p ∈ . Since PT satisfies Condition (A), we also obtain the following results. Theorem . Let H , H be two real Hilbert spaces and C ⊂ H , Q ⊂ H be nonempty closed convex subsets of Hilbert spaces H and H , respectively. Let A : H → H be a bounded linear operator and T : C → P(C) a multi-valued mapping. Let F : C × C → R, F : Q × Q → R be bifunctions satisfying Assumption . and F is upper semi-continuous in the first argument. Assume that PT is  -nonspreading multi-valued mapping and I – T is demiclosed at  with = F(T) ∩ = ∅, where = {z ∈ C : z ∈ EP(F ) and Az ∈ EP(F )}. Let {xn } be a sequence defined by ⎧ ⎪ ⎨x ∈ C arbitrarily, un = TrFn (I – γ A∗ (I – TrFn )A)xn , ⎪ ⎩ xn+ ∈ αn xn + ( – αn )PT un ,

(.)

for all n ≥ , where {αn } ⊂ (, ), rn ⊂ (, ∞), and γ ∈ (, /L) such that L is the spectral radius of A∗ A and A∗ is the adjoint of A. Assume that the following conditions hold: ()  < lim infn→∞ αn ≤ lim supn→∞ αn < ; () lim infn→∞ rn > . Then the sequence {xn } defined by (.) converges weakly to p ∈ . Proof By the same proof as in Theorem ., we have un → zn ∈ PT un . This implies that d(un , Tun ) ≤ d(un , PT un ) ≤ un – zn →  as n → ∞. Since I – T is demiclosed at , we obtain this result.

(.)

4 Examples and numerical results In this section, we give examples and numerical results for supporting our main theorem. Example . Let H = H = R, C = [–, ], and Q = (–∞, ]. Let F (u, v) = (u – )(v – u) for all u, v ∈ C and F (x, y) = (x + )(y – x) for all x, y ∈ Q. Define two mappings A : R → R and T : C → K(C) by Ax = x for all x ∈ R and {}, x ∈ [–, ]; Tx = |x| , ], x ∈ [–, –). [– |x|+ n n Choose αn = n+ , rn = n+ , and γ =  . It is easy to check that F and F satisfy all conditions in Theorem . and T satisfies Condition (A) such that F(T) = {}. For each r >  and x ∈ C, we divide the process of our iteration into five steps as follows:


Page 14 of 16

Table 1 Numerical results of Example 1.1 being randomized in the first time n

zn

xn

|xn+1 – xn |

1 2 3 4 5 6 7 8 9 10 .. . 50

–5.74683E–01 0 0 0 0 0 0 0 0 0 .. . 0

–3.00000E+00 –1.38312E+00 –5.53249E–01 –2.37107E–01 –1.05381E–01 –4.79003E–02 –2.21078E–02 –1.03170E–02 –4.85506E–03 –2.29976E–03 .. . –9.26099E–16

1.61688E+00 8.29873E–01 3.16142E–01 1.31726E–01 5.74804E–02 2.57925E–02 1.17909E–02 5.46194E–03 2.55529E–03 1.20464E–03 .. . 4.67634E–16

Step . Find z ∈ Q such that F (z, y) + r y – z, z – Ax ≥  for all y ∈ Q. Noting that Ax = x, we have  F (z, y) + y – z, z – Ax ≥  r

 (z + )(y – z) + y – z, z – x ≥  r r(z + )(y – z) + (y – z)(z – x) ≥  (y – z) ( + r)z – (x – r) ≥ .

⇐⇒ ⇐⇒ ⇐⇒

By Lemma ., we know that TrF Ax is single-valued. Hence z = x–r . +r F ∗ Step . Find s ∈ C such that s = x – γ A (I – Tr )Ax. From Step , we have s = x – γ A∗ I – TrF Ax = x – γ A∗ Ax – TrF Ax

(x – r) = x – γ x – +r = ( – γ )x +

γ (x – r). +r

Step . Find u ∈ C such that F (u, v) + r v – u, u – s ≥  for all v ∈ C. From Step , we have  F (u, v) + v – u, u – s ≥  r

⇐⇒ ⇐⇒ ⇐⇒

(–γ )x+r (x–r) + γ (+r) .  +r (–γ )xn +rn n –rn ) xn+ ∈ αn xn + ( – αn )Tun , where un = + γ (x . +rn (+rn )

Similarly, by Lemma ., we obtain u = Step . Find

 (u – )(v – u) + v – u, u – s ≥  r r(u – )(v – u) + (v – u)(u – s) ≥  (v – u) ( + r)u – (s + r) ≥ .

s+r +r

=

From

{}, x ∈ [–, ]; Tx = |x| [– |x|+ , ], x ∈ [–, –), and αn =

n ,r n+ n

xn+ =

=

n , n+

and γ =

 , 

we have

n n xn +  – zn , n +  n + 

(.)


Page 15 of 16

Table 2 Numerical results of Example 1.1 being randomized in the second time n

zn

xn

|xn+1 – xn |

1 2 3 4 5 6 7 8 9 10 .. . 50

–1.26779E–00 0 0 0 0 0 0 0 0 0 .. . 0

–3.00000E+00 –1.08452E+00 –4.33808E–01 –1.85918E–01 –8.26300E–02 –3.75591E–02 –1.73350E–02 –8.08965E–03 –3.80690E–03 –1.80327E–03 .. . –7.26163E–16

1.91548E+00 6.50711E–01 2.47890E–01 1.03288E–01 4.50709E–02 2.02241E–02 9.24532E–03 4.28276E–03 2.00363E–03 9.44568E–04 .. . 3.66677E–16

where zn ∈

{}, un ∈ [–, ]; |un | [– |un |+ , ], un ∈ [–, –).

Step . Compute the numerical results. Choosing x = – and taking randomly zn in the above interval, we obtain Tables  and . From Table  and Table , we see that  is the solution in Example ..

Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand. 2 School of Science, University of Phayao, Phayao, 56000, Thailand. 3 Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju, 660-701, Republic of Korea. 4 Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia. Acknowledgements The first author would like to thank the Thailand Research Fund under the project RTA5780007 and Chiang Mai University. The third author thanks the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (Grant Number: 2014R1A2A2A01002100). W Cholamjiak and P Cholamjiak would like to thank University of Phayao. Received: 13 August 2015 Accepted: 27 February 2016 References 1. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123-145 (1994) 2. Ceng, LC, Yao, JC: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 214, 186-201 (2008) 3. Combettes, PL, Hirstoaga, SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117-136 (2005) 4. Peng, JW, Liou, YC, Yao, JC: An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions. Fixed Point Theory Appl. 2009, Article ID 794178 (2009) 5. Tada, A, Takahashi, W: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. In: Takahashi, W, Tanaka, T (eds.) Nonlinear Analysis and Convex Analysis. Yokohama Publishers, Yokohama (2005) 6. Kazmi, KR, Rizvi, SH: Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem. J. Egypt. Math. Soc. 21, 44-51 (2013) 7. Assad, NA, Kirk, WA: Fixed point theorems for set-valued mappings of contractive type. Pac. J. Math. 43, 553-562 (1972) 8. Nadler, SB Jr.: Multi-valued contraction mappings. Pac. J. Math. 30, 475-488 (1969) 9. Pietramala, P: Convergence of approximating fixed points sets for multivalued nonexpansive mappings. Comment. Math. Univ. Carol. 32, 697-701 (1991)


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