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Strong convergence of a Halpern-type algorithm for common solutions of fixed point and equilibrium problems

Abstract

In this article, fixed points of nonexpansive mappings and equilibrium problems based on a Halpern-type algorithm are investigated. Strong convergence theorems for common solutions of the two problems are obtained in the framework of real Hilbert spaces.

1 Introduction

The study of equilibrium problems is an important branch of optimization theory and nonlinear functional analysis. Numerous problems in physics, optimization, transportation, signal processing, and economics are reduced to find a solution to equilibrium problems, which cover fixed point problems, variational inequalities, saddle problems, inclusion problems, and so on. A closely related subject of current interest is the problem of finding common elements in the fixed point set of nonlinear operators and in the solution set of monotone variational inequalities; see [115] and the references therein. The motivation for this subject is mainly due to its possible applications to mathematical modeling of concrete complex problems. The aim of this paper is to investigate a common element problem based on a Halpern-type algorithm. Strong convergence of the algorithm is obtained in the framework of real Hilbert spaces. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a Halpern-type algorithm is proposed and analyzed. Strong convergence theorems for common solutions of two problems are established in the framework of Hilbert spaces. In Section 4, applications of the main results are provided.

2 Preliminaries

Let H be a real Hilbert space with inner product , and norm . Let C be a nonempty, closed, and convex subset of H and let Proj C be the metric projection from H onto C.

Let T:CC be a mapping. In this paper, we use F(T) to denote the fixed point set of T. Recall that T is said to be contractive iff there exists a constant α(0,1) such that

TxTyαxy,x,yC.

For such a case, T is also said to be α-contractive. Recall that T is said to be nonexpansive iff

TxTyxy,x,yC.

It is well known that the fixed point set of nonexpansive mappings is nonempty provided that the subset C is bounded, convex, and closed.

Let A:CH be a mapping. Recall that A is said to be monotone iff

AxAy,xy0,x,yC.

Recall that A is said to be inverse-strongly monotone iff there exists a constant α>0 such that

AxAy,xyα A x A y 2 ,x,yC.

For such a case, A is also said to be α-inverse-strongly monotone.

Recall that the classical variational inequality is to find an xC such that

Ax,yx0,yC.
(2.1)

In this paper, we use VI(C,A) to denote the solution set of (2.1). It is well known that xC is a solution of the variational inequality (2.1) iff x is a solution of the fixed point equation P C (IrA)x=x, where r>0 is a constant.

Recall that a set-valued mapping M:HH is said to be monotone iff, for all x,yH, fMx and gMy imply xy,fg>0. M is maximal iff the graph Graph(M) of R is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping M is maximal if and only if, for any (x,f)H×H, xy,fg0, for all (y,g)Graph(M) implies fRx.

For a maximal monotone operator M on H, and r>0, we may define the single-valued resolvent J r :HD(M), where D(M) denotes the domain of M. It is known that J r is firmly nonexpansive, and M 1 (0)=F( J r ), where F( J r ):={xD(M):x= J r x}, and M 1 (0):={xH:0Mx}.

Let A:CH be a monotone mapping, and let F be a bifunction of C×C into , where denotes the set of real numbers. We consider the following generalized equilibrium problem:

Find xC such that F(x,y)+Ax,yx0,yC.
(2.2)

In this paper, we use EP(F,A) to denote the solution set of the generalized equilibrium problem (2.2).

Next, we give some special cases of the generalized equilibrium problem (2.2).

  1. (I)

    If F0, then problem (2.2) is reduced to the classical variational inequality (2.1).

  2. (II)

    If A0, the zero mapping, then problem (2.2) is reduced to the following equilibrium problem:

    Find xC such that F(x,y)0,yC.
    (2.3)

In this paper, we use EP(F) to denote the solution set of the equilibrium problem (2.3).

To study the equilibrium problems, we may assume that F satisfies the following conditions:

  1. (A1)

    F(x,x)=0 for all xC;

  2. (A2)

    F is monotone, i.e., F(x,y)+F(y,x)0 for all x,yC;

  3. (A3)

    for each x,y,zC,

    lim sup t 0 F ( t z + ( 1 t ) x , y ) F(x,y);
  4. (A4)

    for each xC, yF(x,y) is convex and weakly lower semi-continuous.

Recently, many authors have studied fixed point problems of nonexpansive mappings and solution problems of the equilibrium problems (2.2) and (2.3); for more details, see [1625] and the references therein. In this paper, motivated and inspired by the research going on in this direction, we consider common element problems based on a mean iterative process. Strong convergence of the iterative process is obtained in the framework of real Hilbert spaces. The results presented in this paper improve and extend the corresponding results in Hao [1], Qin et al. [24], Chang et al. [25].

In order to prove our main results, we need the following lemmas.

Lemma 2.1 [26]

Assume that { α n } is a sequence of nonnegative real numbers such that

α n + 1 (1 γ n ) α n + δ n ,

where { γ n } is a sequence in (0,1) and { δ n } is a sequence such that

  1. (1)

    n = 1 γ n =;

  2. (2)

    lim sup n δ n / γ n 0 or n = 1 | δ n |<.

Then lim n α n =0.

Lemma 2.2 [27]

Let F:C×CR be a bifunction satisfying (A1)-(A4). Then, for any r>0 and xH, there exists zC such that

F(z,y)+ 1 r yz,zx0,yC.

Define a mapping T r :HC as follows:

T r x= { z C : F ( z , y ) + 1 r y z , z x 0 , y C } ,xH,

then the following conclusions hold.

  1. (1)

    T r is single-valued;

  2. (2)

    T r is firmly nonexpansive, i.e., for any x,yH,

    T r x T r y 2 T r x T r y,xy;
  3. (3)

    F( T r )=EP(F);

  4. (4)

    EP(F) is closed and convex.

Let { S i :CC} be a family of infinitely nonexpansive mappings and { γ i } be a nonnegative real sequence with 0 γ i <1, i1. For n1 define a mapping W n :CC as follows:

U n , n + 1 = I , U n , n = γ n S n U n , n + 1 + ( 1 γ n ) I , U n , n 1 = γ n 1 S n 1 U n , n + ( 1 γ n 1 ) I , U n , k = γ k S k U n , k + 1 + ( 1 γ k ) I , U n , k 1 = γ k 1 S k 1 U n , k + ( 1 γ k 1 ) I , U n , 2 = γ 2 S 2 U n , 3 + ( 1 γ 2 ) I , W n = U n , 1 = γ 1 S 1 U n , 2 + ( 1 γ 1 ) I .
(2.4)

Such a mapping W n is nonexpansive from C to C and it is called a W-mapping generated by S n , S n 1 ,, S 1 and γ n , γ n 1 ,, γ 1 ; see [28] and the references therein.

Lemma 2.3 [28]

Let { S i :CC} be a family of infinitely nonexpansive mappings with a nonempty common fixed point set and let { γ i } be a real sequence such that 0< γ i l<1, where l is some real number, i1. Then

  1. (1)

    W n is nonexpansive and F( W n )= i = 1 F( S i ), for each n1;

  2. (2)

    for each xC and for each positive integer k, the limit lim n U n , k exists;

  3. (3)

    the mapping W:CC defined by

    Wx:= lim n W n x= lim n U n , 1 x,xC,
    (2.5)

is a nonexpansive mapping satisfying F(W)= i = 1 F( S i ) and it is called the W-mapping generated by S 1 , S 2 , and γ 1 , γ 2 , .

Lemma 2.4 [29]

Let B:CH be a mapping and let M:HH be a maximal monotone operator. Then F( J r (IrB))= ( B + M ) 1 (0), where r is some positive real number.

Lemma 2.5 [25]

Let { S i :CC} be a family of infinitely nonexpansive mappings with a nonempty common fixed point set and let { γ i } be a real sequence such that 0< γ i l<1, i1. If K is any bounded subset of C, then

lim n sup x K Wx W n x=0.

Throughout this paper, we always assume that 0< γ i l<1, i1.

Lemma 2.6 [30]

Let A:CH a Lipschitz monotone mapping and let N C x be the normal cone to C at xC; that is, N C x={yH:xu,y,uC}. Define

Wx={ A x + N C x , x C , x C .

Then W is maximal monotone and 0Wx if and only if xVI(C,A).

Lemma 2.7 [31]

Let { x n } and { y n } be bounded sequences in H and let { β n } be a sequence in (0,1) with 0< lim inf n β n lim sup n β n <1. Suppose that x n + 1 =(1 β n ) y n + β n x n for all n0 and

lim sup n ( y n + 1 y n x n + 1 x n ) 0.

Then lim n y n x n =0.

3 Main results

Theorem 3.1 Let C be a nonempty closed convex subset of a Hilbert space H and let F be a bifunction from C×C to which satisfies (A1)-(A4). Let A:CH be an α-inverse-strongly monotone mapping and let B:CH be a β-inverse-strongly monotone mapping. Let M:HH be a maximal monotone operator. Let { S i :CC} be a family of infinitely nonexpansive mappings. Assume that F:= i = 1 F( S i )EP(F,B) ( A + M ) 1 (0). Let f:CC be a κ-contraction. Let { x n } be a sequence generated by the process: x 1 C and

{ F ( u n , y ) + B x n , y u n + 1 s n y u n , u n x n 0 , y C , x n + 1 = α n f ( x n ) + β n x n + γ n W n Proj C J r n ( u n r n A u n ) , n 1 ,

where { W n :CC} is the sequence generated in (2.4), { α n }, { β n }, and { γ n } are sequences in (0,1) such that α n + β n + γ n =1 and { r n } and { s n } are positive number sequences. Assume that the above control sequences satisfy the following restrictions:

  1. (a)

    0<r r n r <2α, 0< r s n r <2β;

  2. (b)

    lim n α n =0 and n = 1 α n =;

  3. (c)

    0< lim inf n β n lim sup n β n <1;

  4. (d)

    lim n | s n s n + 1 |= lim n | r n r n + 1 |=0.

Then the sequence { x n } converges strongly to x ¯ = Proj F f( x ¯ ).

Proof First, we show that I r n A is nonexpansive. For x,yC, we have

( I r n A ) x ( I r n A ) y 2 = x y 2 2 r n x y , A x A y + r n 2 A x A y 2 x y 2 2 r n α A x A y 2 + r n 2 A x A y 2 = x y 2 + r n ( r n 2 α ) A x A y 2 .

Using restriction (a), we have I r n A is nonexpansive, so is I s n B. Fix x F. It follows that u n x (I s n B) x n (I s n B) x x n x . Putting y n = J r n ( u n r n A u n ), one finds that y n x u n x x n x . It follows that

x n + 1 x α n f ( x n ) x + β n x n x + γ n W n Proj C y n x α n f ( x n ) f ( x ) + α n f ( x ) x + β n x n x + γ n y n x ( 1 α n ( 1 κ ) ) x n x + α n f ( x ) x .

Hence, we have x n x max{ x 1 x , f ( x ) x 1 α }. This yields the result that { x n } is bounded. Therefore, both { y n } and { u n } are also bounded. Next, without loss of generality, we assume that there exists a bounded set KC such that x n , y n , u n K. Notice that F( u n + 1 ,y)+ 1 s n + 1 y u n + 1 , u n + 1 (I s n + 1 B) x n + 1 0, yC, and F( u n ,y)+ 1 s n y u n , u n (I s n B) x n 0, yC. It follows that

u n + 1 u n , u n ( I s n B ) x n s n u n + 1 ( I s n + 1 B ) x n + 1 s n + 1 0.

Hence, we have

u n + 1 u n 2 u n + 1 u n , ( I s n + 1 B ) x n + 1 ( I s n B ) x n + ( 1 s n s n + 1 ) ( u n + 1 ( I s n + 1 B ) x n + 1 ) u n + 1 u n ( ( I s n + 1 B ) x n + 1 ( I s n B ) x n + | 1 s n s n + 1 | u n + 1 ( I s n + 1 B ) x n + 1 ) .

This yields the result that

u n + 1 u n ( I s n + 1 B ) x n + 1 ( I s n B ) x n + | s n + 1 s n | s n + 1 u n + 1 ( I s n + 1 B ) x n + 1 = ( I s n + 1 B ) x n + 1 ( I s n + 1 B ) x n + ( I s n + 1 B ) x n ( I s n B ) x n + | s n + 1 s n | s n + 1 u n + 1 ( I s n + 1 B ) x n + 1 x n + 1 x n + | s n + 1 s n | M 1 ,
(3.1)

where M 1 is an appropriate constant such that

M 1 = sup n 1 { B x n + u n + 1 ( I s n + 1 B ) x n + 1 a ¯ } .

Since J r n is firmly nonexpansive, one sees that

y n + 1 y n = J r n ( u n + 1 r n + 1 A u n + 1 ) J r n ( u n r n A u n ) u n + 1 r n + 1 A u n + 1 ( u n r n A u n ) = ( I r n + 1 A ) u n + 1 ( I r n + 1 A ) u n + ( r n r n + 1 ) A u n u n + 1 u n + | r n r n + 1 | A u n .
(3.2)

Substituting (3.1) into (3.2), one finds that

y n + 1 y n x n + 1 x n + ( | s n + 1 s n | + | r n r n + 1 | ) M 2 ,
(3.3)

where M 2 is an appropriate constant such that M 2 =max{ sup n 1 {A u n }, M 1 }. On the other hand, one has

W n + 1 Proj C y n + 1 W n Proj C y n W n + 1 y n + 1 W n y n W n + 1 y n + 1 W y n + 1 + W y n + 1 W y n + W y n W n y n sup x K { W n + 1 x W x + W x W n x } + y n + 1 y n ,
(3.4)

where K is the bounded subset of C defined above. Combining (3.3) with (3.4), one finds

W n + 1 Proj C y n + 1 W n Proj C y n sup x K { W n + 1 x W x + W x W n x } + x n + 1 x n + ( | r n + 1 r n | + | s n s n + 1 | ) M 2 .
(3.5)

Letting x n + 1 =(1 β n ) z n + β n x n we see that

z n + 1 z n α n + 1 1 β n + 1 f ( x n + 1 ) W n + 1 y n + 1 + α n 1 β n f ( x n ) W n y n + W n + 1 Proj C y n + 1 W n Proj C y n .
(3.6)

Substituting (3.5) into (3.6), we see that

z n + 1 z n x n + 1 x n α n + 1 1 β n + 1 f ( x n + 1 ) W n + 1 y n + 1 + α n 1 β n f ( x n ) W n y n + sup x K { W n + 1 x W x + W x W n x } + ( | r n + 1 r n | + | s n s n + 1 | ) M 2 .

It follows from restrictions (a), (c), and (d) that lim sup n ( z n + 1 z n x n + 1 x n )0. Using Lemma 2.7, we find that lim n z n x n =0. It follows that

lim n x n + 1 x n =0.
(3.7)

For any x F, we see that

x n + 1 x 2 α n f ( x n ) x 2 + β n x n x 2 + γ n W n Proj C y n x 2 α n f ( x n ) x 2 + β n x n x 2 + γ n y n x 2 .
(3.8)

Since

y n x 2 = J r n ( u n r n A u n ) x 2 ( I r n A ) u n ( I r n A ) x 2 = u n x 2 2 r n u n x , A u n A x + r n 2 A u n A x 2 x n x 2 + r n ( r n 2 α ) A u n A x 2 ,

we find from (3.8) that

lim n A u n A x =0.
(3.9)

It also follows from (3.8) that

x n + 1 x 2 α n f ( x n ) x 2 + β n x n x 2 + γ n x n x s n ( B x n B x ) 2 α n f ( x n ) x 2 + β n x n x 2 + γ n ( x n x 2 + s n 2 B x n B x 2 2 s n B x n B x , x n x ) α n f ( x n ) x 2 + β n x n x 2 + γ n x n x 2 s n γ n ( 2 β s n ) B x n B x 2 .

Using (3.7), one arrives at

lim n B x n B x =0.
(3.10)

Since T s n is firmly nonexpansive, we find that

u n x 2 ( I s n B ) x n ( I s n B ) x , u n x 1 2 ( x n x 2 + u n x 2 x n u n 2 s n 2 B x n B x 2 + 2 s n B x n B x , x n u n ) ,

which implies that u n x 2 x n x 2 x n u n 2 +2 s n B x n B x x n u n . Hence

γ n x n u n 2 α n f ( x n ) x 2 + ( x n x + x n + 1 x ) x n x n + 1 + 2 s n B x n B x x n u n .

Using (3.7) and (3.10), one has

lim n x n u n =0.
(3.11)

Similarly, one also has

lim n y n u n =0.
(3.12)

Since

W n y n y n y n u n + u n x n + x n W n y n ,

we find from (3.11) and (3.12) that

lim n W n y n y n =0.
(3.13)

Now, we are in a position to show lim sup n f( x ¯ ) x ¯ , x n z0, where x ¯ = Proj F f( x ¯ ). To prove this, we choose a subsequence { x n i } of { x n } such that

lim sup n f ( x ¯ ) x ¯ , x n x ¯ = lim i f ( x ¯ ) x ¯ , x n i x ¯ .
(3.14)

Since { x n i } is bounded, without loss of generality, we may assume that x n i q. Using (3.11) and (3.12), we have lim n x n y n =0. Therefore, we see that y n i q. Now, we are in a position to prove q ( A + M ) 1 (0). Notice that u n y n r n A u n M y n . Let μMν. Since M is monotone, we find that u n y n r n A u n μ, y n ν0. This implies that Aqμ,qν0. This implies that AqMq, that is, q ( A + M ) 1 (0). Next, we show that qEP(F,B). Since u n = T s n (I s n B) x n , we find from (A2) that

B x n i ,y u n i + y u n i , u n i x n i s n i F(y, u n i ),yC.
(3.15)

Putting y t =ty+(1t)q for any t(0,1] and yC, we see that y t C. Using (3.15), we find that

y t u n i , B y t y t u n i , B y t B x n i , y t u n i y t u n i , u n i x n i s n i + F ( y t , u n i ) = y t u n i , B y t B u n i + y t u n i , B u n i B x n i y t u n i , u n i x n i s n i + F ( y t , u n i ) .

Since B is monotone, we obtain from (A4) that y t w,B y t F( y t ,w). Using (A1) and (A4), we find that

0 = F ( y t , y t ) t F ( y t , y ) + ( 1 t ) F ( y t , w ) t F ( y t , y ) + ( 1 t ) y t w , B y t = t F ( y t , y ) + ( 1 t ) t y w , B y t .

Hence, 0F( y t ,y)+(1t)yw,B y t , yC. It follows from (A3) that wEP(F,B). Next, we prove that q i = 1 F( S i ). Suppose to the contrary, q i = 1 F( S i ), i.e., Wqq. Since y n i q and the space satisfies Opial’s condition, one has

lim inf i y n i q < lim inf i y n i W q lim inf i { y n i W y n i + W y n i W q } lim inf i { y n i W y n i + y n i q } .
(3.16)

Since W y n y n sup x K Wx W n x+ W n y n y n , we find from Lemma 2.5 that lim n W y n y n =0. It follows that lim inf i y n i q< lim inf i y n i q. This leads to a contradiction. Thus, we have q i = 1 F( S i ). This proves that qF. Therefore, one has

lim sup n f ( x ¯ ) x ¯ , x n x ¯ 0.

Finally, we show that x n x ¯ , as n. Note that

x n + 1 x ¯ 2 α n f ( x n ) f ( x ¯ ) , x n + 1 x ¯ + α n f ( x ¯ ) x ¯ , x n + 1 x ¯ + β n x n x ¯ x n + 1 x ¯ + γ n y n x ¯ x n + 1 x ¯ κ 2 α n ( x n x ¯ 2 + x n + 1 x ¯ 2 ) + α n f ( x ¯ ) x ¯ , x n + 1 x ¯ + ( 1 α n ) x n x ¯ x n + 1 x ¯ 1 α n ( 1 κ ) 2 x n x ¯ 2 + 1 2 x n + 1 x ¯ 2 + α n f ( x ¯ ) x ¯ , x n + 1 x ¯ ,

which implies that

x n + 1 x ¯ 2 ( 1 α n ( 1 κ ) ) x n x ¯ 2 +2 α n f ( x ¯ ) x ¯ , x n + 1 x ¯ .

Using Lemma 2.1, we find that lim n x n x ¯ =0. This completes the proof. □

4 Applications

Recall that a mapping T:CC is said to be a k-strict pseudo-contraction if there exists a constant k[0,1) such that

T x T y 2 x y 2 +k ( I T ) x ( I T ) y 2

for all x,yC. Note that the class of k-strict pseudo-contractions strictly includes the class of nonexpansive mappings. Put A=IT, where T:CC is a k-strict pseudo-contraction. Then A is 1 k 2 -inverse-strongly monotone. Now, we are in a position to state a results on fixed points of strict pseudo-contractions.

Theorem 4.1 Let C be a nonempty closed convex subset of a Hilbert space H and let F be a bifunction from C×C to which satisfies (A1)-(A4). Let T:CH be a k-strict pseudo-contraction, B:CH be a β-inverse-strongly monotone mapping, and { S i :CC} be a family of infinitely nonexpansive mappings. Assume that F:= i = 1 F( S i )EP(F,B)F(T). Let f:CC be a κ-contraction. Let { x n } be a sequence generated by x 1 C and

{ F ( u n , y ) + B x n , y u n + 1 s n y u n , u n x n 0 , y C , y n = ( 1 r n ) u n + r n T u n , x n + 1 = α n f ( x n ) + β n x n + γ n W n y n , n 1 ,

where { W n :CC} is the sequence generated in (2.4), { α n }, { β n }, and { γ n } are sequences in (0,1) such that α n + β n + γ n =1 and { r n }, and { s n } are positive number sequences. Assume that the above control sequences satisfy the following restrictions:

  1. (a)

    0<r s n r <2β, 0< r r n r <1k;

  2. (b)

    lim n α n =0 and n = 1 α n =;

  3. (c)

    0< lim inf n β n lim sup n β n <1;

  4. (d)

    lim n | s n s n + 1 |= lim n | r n r n + 1 |=0.

Then the sequence { x n } converges strongly to x ¯ = Proj F f( x ¯ ).

Proof Taking A=IT, wee see that A:CH is a α-strict pseudo-contraction with α= 1 k 2 and F(T)=VI(C,A). Using Theorem 3.1, we find the desired conclusion immediately. □

Let g:H(,+] be a proper convex lower semi-continuous function. Then the subdifferential ∂g of g is defined as follows:

fg(x)= { y H : g ( z ) g ( x ) + z x , y , z H } ,xH.

From Rockafellar [30], we know that ∂g is maximal monotone. It is not hard to verify that 0g(x) if and only if g(x)= min y H g(y).

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

I C (x)={ 0 , x C , + , x C .

Since I C is a proper lower semi-continuous convex function on H, we see that the subdifferential I C of I C is a maximal monotone operator. It is clear that J r x= P C x, xH. Notice that ( A + I C ) 1 (0)=VI(C, A 1 ). Now, we are in a position to state the result on variational inequalities.

Theorem 4.2 Let C be a nonempty closed convex subset of a Hilbert space H and let F be a bifunction from C×C to which satisfies (A1)-(A4). Let A:CH be an α-inverse-strongly monotone mapping, B:CH be a β-inverse-strongly monotone mapping, and { S i :CC} be a family of infinitely nonexpansive mappings. Assume that F:= i = 1 F( S i )EP(F,B)VI(C,A). Let f:CC be a κ-contraction. Let { x n } be a sequence generated by x 1 C and

{ F ( u n , y ) + B x n , y u n + 1 s n y u n , u n x n 0 , y C , x n + 1 = α n f ( x n ) + β n x n + γ n W n P C ( u n s n A u n ) , n 1 ,

where { W n :CC} is the sequence generated in (2.4), { α n }, { β n }, and { γ n } are sequences in (0,1) such that α n + β n + γ n =1 and { r n }, and { s n } are positive number sequences. Assume that the above control sequences satisfy the following restrictions:

  1. (a)

    0<r s n r <2β, 0< r r n r <2α;

  2. (b)

    lim n α n =0 and n = 1 α n =;

  3. (c)

    0< lim inf n β n lim sup n β n <1;

  4. (d)

    lim n | s n s n + 1 |= lim n | r n r n + 1 |=0.

Then the sequence { x n } converges strongly to x ¯ = Proj F f( x ¯ ).

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Acknowledgements

The author is grateful to Prof. Tian Cheng and the reviewers’ suggestions which improved the contents of the article.

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Zhang, Q. Strong convergence of a Halpern-type algorithm for common solutions of fixed point and equilibrium problems. J Inequal Appl 2014, 313 (2014). https://doi.org/10.1186/1029-242X-2014-313

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Keywords

  • nonexpansive mapping
  • equilibrium problem
  • fixed point
  • variational inequality