Skip to main content

Approximation of solutions to an equilibrium problem in a nonuniformly smooth Banach space

Abstract

An equilibrium problem based on a projection algorithm is investigated. A strong convergence theorem for solutions of the equilibrium problems is established in a nonuniformly smooth Banach space.

1 Introduction

Recently, equilibrium problems have been studied as an effective and powerful tool for studying a wide class of real world problems which arise in economics, finance, image reconstruction, ecology, transportation, and networks; see [116] and the references therein. It is well known that equilibrium problems include many important problems in nonlinear analysis and optimization such as the Nash equilibrium problem, variational inequalities, complementarity problems, vector optimization problems, fixed point problems, saddle point problems, and game theory. For the solutions of equilibrium problems, there are several algorithms to solve the problem; see [1728] and the references therein. However, most of these results are obtained in the framework of Hilbert spaces or uniformly smooth Banach spaces.

The purpose of this paper is to study solution problems of an equilibrium problem based on a projection algorithm in a nonuniformly smooth Banach space. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a projection algorithm is introduced and the convergence analysis is given. A strong convergence theorem is established in a nonuniformly smooth Banach space. Applications of the main results are also discussed in this section.

2 Preliminaries

Let E be a real Banach space, and let E be the dual space of E. We denote by J the normalized duality mapping from E to 2 E defined by

Jx= { f E : x , f = x 2 = f 2 } ,

where , denotes the generalized duality pairing. A Banach space E is said to be strictly convex if x + y 2 <1 for all x,yE with x=y=1 and xy. It is said to be uniformly convex if lim n x n y n =0 for any two sequences { x n } and { y n } in E such that x n = y n =1 and lim n x n + y n 2 =1. Let U E ={xE:x=1} be the unit sphere of E. Then the Banach space E is said to be smooth provided

lim t 0 x + t y x t

exists for each x,y U E . It is also said to be uniformly smooth if the above limit is attained uniformly for x,y U E . It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. It is also well known that E is uniformly smooth if and only if E is uniformly convex.

Recall that a Banach space E enjoys the Kadec-Klee property if for any sequence { x n }E and xE with x n x and x n x, then x n x0 as n. For more details on the Kadec-Klee property, the readers can refer to [29] and the references therein. It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property. In this paper, we use → and to denote the strong convergence and weak convergence, respectively.

Let C be a nonempty subset of E. Recall that a mapping Q:C E is said to be monotone iff

xy,QxQy0,x,yC.

Q:C E is said to be α-inverse-strongly monotone iff there exists a positive real number α such that

xy,QxQyα Q x Q y 2 ,x,yC.

Recall also that a monotone mapping Q is said to be maximal iff its graph Graph(Q)={(x,f):fQx} is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping Q is maximal iff for (x,f)E× E , xy,fg0 for every (y,g)Graph(Q) implies fQx. An operator Q from C into E is said to be hemi-continuous if for all x,yC, the mapping f of [0,1] into E defined by f(t)=Q(tx+(1t)y) is continuous with respect to the weak topology of E . Let f be a bifunction from C×C to , where denotes the set of real numbers. In this paper, we investigate the following equilibrium problem. Find pC such that

f(p,y)0,yC.
(2.1)

We use EP(f) to denote the solution set of equilibrium problem (2.1). That is,

EP(f)= { p C : f ( p , y ) 0 , y C } .

Given a mapping Q:C E , let

f(x,y)=Qx,yx,x,yC.

Then pEP(f) iff p is a solution of the following variational inequality. Find p such that

Qp,yp0,yC.
(2.2)

In order to study the solution problem of equilibrium problem (2.1), we assume that f satisfies the following conditions:

(A1) f(x,x)=0, xC;

(A2) f is monotone, i.e., f(x,y)+f(y,x)0, x,yC;

(A3)

lim sup t 0 f ( t z + ( 1 t ) x , y ) f(x,y),x,y,zC;

(A4) for each xC, yf(x,y) is convex and weakly lower semi-continuous.

As we all know, if C is a nonempty closed convex subset of a Hilbert space H and P C :HC is the metric projection of H onto C, then P C is nonexpansive. This fact actually characterizes Hilbert spaces and consequently it is not available in more general Banach spaces. In this connection, Alber [30] recently introduced a generalized projection operator Π C in a Banach space E which is an analogue of the metric projection P C in Hilbert spaces.

Next, we assume that E is a smooth Banach space. Consider the functional defined by

ϕ(x,y)= x 2 2x,Jy+ y 2 ,x,yE.

Observe that in a Hilbert space H, the equality is reduced to ϕ(x,y)= x y 2 , x,yH. The generalized projection Π C :EC is a map that assigns to an arbitrary point xE the minimum point of the functional ϕ(x,y), that is, Π C x= x ¯ , where x ¯ is the solution to the minimization problem

ϕ( x ¯ ,x)= min y C ϕ(y,x).

The existence and uniqueness of the operator Π C follow from the properties of the functional ϕ(x,y) and strict monotonicity of the mapping J; see, for example, [29] and [30]. In Hilbert spaces, Π C = P C . It is obvious from the definition of a function ϕ that

( x y ) 2 ϕ(x,y) ( y + x ) 2 ,x,yE,
(2.3)

and

ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2xz,JzJy,x,y,zE.
(2.4)

Let T:CC be a mapping. In this paper, we use F(T) to denote the fixed point set of T. A point p in C is said to be an asymptotic fixed point of T iff C contains a sequence { x n } which converges weakly to p such that lim n x n T x n =0. The set of asymptotic fixed points of T will be denoted by F ˜ (T). T is said to be relatively nonexpansive iff F ˜ (T)=F(T) and ϕ(p,Tx)ϕ(p,x) for all xC and pF(T). T is said to be quasi-ϕ-nonexpansive iff F(T) and ϕ(p,Tx)ϕ(p,x) for all xC and pF(T).

Remark 2.1 The class of quasi-ϕ-nonexpansive mappings is more general than the class of relatively nonexpansive mappings which requires the restriction: F(T)= F ˜ (T).

Remark 2.2 The class of quasi-ϕ-nonexpansive mappings is a generalization of quasi-nonexpansive mappings in Hilbert spaces.

In this paper, we investigate the solution problem of equilibrium problem (2.1) based on a projection algorithm. A strong convergence theorem for solutions of the equilibrium problems is established in a reflexive, strictly convex, and smooth Banach space such that both E and E have the Kadec-Klee property.

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

Lemma 2.3 [30]

LetCbe a nonempty, closed, and convex subset of a smooth Banach spaceE, andxE. Then x 0 = Π C xif and only if

x 0 y,JxJ x 0 0,yC.

Lemma 2.4 [30]

LetEbe a reflexive, strictly convex, and smooth Banach space, letCbe a nonempty, closed, and convex subset ofE, andxE. Then

ϕ(y, Π C x)+ϕ( Π C x,x)ϕ(y,x),yC.

Lemma 2.5 [30]

LetEbe a reflexive, strictly convex, and smooth Banach space. Then

ϕ(x,y)=0x=y.

The following lemma can be obtained from [15] and [31].

Lemma 2.6LetCbe a closed convex subset of a smooth, strictly convex, and reflexive Banach spaceE. Letfbe a bifunction fromC×Cto satisfying (A1)-(A4). Letr>0andxE. Then

  1. (a)

    There exists zC such that

    f(z,y)+ 1 r yz,JzJx0,yC.
  2. (b)

    Define a mapping T r :EC by

    S r x= { z C : f ( z , y ) + 1 r y z , J z J x , y C } .

Then the following conclusions hold:

  1. (1)

    F( S r )=EP(f);

  2. (2)

    S r is a firmly nonexpansive-type mapping, i.e., for allx,yE,

    S r x S r y,J S r xJ S r y S r x S r y,JxJy;
  3. (3)

    S r is single-valued;

  4. (4)
    ϕ(q, S r x)+ϕ( S r x,x)ϕ(q,x),qF( S r );
  5. (5)

    EP(f)is closed and convex;

  6. (6)

    S r is quasi-ϕ-nonexpansive.

3 Main results

Theorem 3.1LetEbe a reflexive, strictly convex, and smooth Banach space such that bothEand E have the Kadec-Klee property. LetCbe a nonempty, closed, and convex subset of E. Letfbe a bifunction fromC×Cto satisfying (A1)-(A4) such thatEF(f)is nonempty. Let{ x n }be a sequence generated in the following manner:

{ x 0 E chosen arbitrarily , C 1 = C , x 1 = Π C 1 x 0 , y n C  such that  f ( y n , y ) + 1 r n y y n , J y n J x n 0 , y C , C n + 1 = { z C n : ϕ ( z , y n ) ϕ ( z , x n ) } , x n + 1 = Π C n + 1 x 1 ,

where{ r n }is a real sequence such that lim inf n r n >0. Then the sequence{ x n }converges strongly to Π EP ( f ) x 1 , where Π EP ( f ) is the generalized projection fromEontoEP(f).

Proof First, we show that C n is closed and convex, that the projection on it is well defined. It is obvious that C 1 =C is closed and convex. Suppose that C m is closed and convex for some mN. We next prove that C m + 1 is also closed and convex for the same m. Let For z 1 , z 2 C m + 1 , we see that z 1 , z 2 C m . It follows that z=t z 1 +(1t) z 2 C m , where t(0,1). Notice that

ϕ( z 1 , y m )ϕ( z 1 , x m )andϕ( z 1 , y h )ϕ( z 1 , x m ).

The above inequalities are equivalent to

2 z 1 ,J x m J y m x m 2 y m 2
(3.1)

and

2 z 2 ,J x m J y m x m 2 y m 2 .
(3.2)

Multiplying t and (1t) on both sides of (3.1) and (3.2), respectively, yields that

2z,J x m J y m x m 2 y m 2 .

That is,

ϕ(z, y m )ϕ(z, x m ).

This gives that C m + 1 is closed and convex. Then C n is closed and convex. Now, we are in a position to prove that EP(f) C n . EP(f) C 1 =C is obvious. Suppose that EP(f) C m for some mN. Fix pEP(f) C m . It follows that

ϕ(p, y m )=ϕ(p, S r m x m )ϕ(p, x m ),

which implies that p C m + 1 . This proves that EP(f) C n . In the light of x n = Π C n x 1 , from Lemma 2.3, we find that x n z,J x 1 J x n 0 for any z C n . It follows from EP(f) C n that

x n w,J x 1 J x n 0,wEP(f).
(3.3)

It follows from Lemma 2.4 that

ϕ ( x n , x 1 ) = ϕ ( Π C n x 1 , x 1 ) ϕ ( Π F ( T ) x 1 , x 1 ) ϕ ( Π F ( T ) x 1 , x n ) ϕ ( Π F ( T ) x 1 , x 1 ) .

This implies that the sequence {ϕ( x n , x 1 )} is bounded. It follows from (2.3) that the sequence { x n } is also bounded. Since the space is reflexive, we may assume that x n x ¯ . Since C n is closed and convex, we find that x ¯ C n . On the other hand, we see from the weak lower semicontinuity of the norm that

ϕ ( x ¯ , x 1 ) = x ¯ 2 2 x ¯ , J x 1 + x 1 2 lim inf n ( x n 2 2 x n , J x 1 + x 1 2 ) = lim inf n ϕ ( x n , x 1 ) lim sup n ϕ ( x n , x 1 ) ϕ ( x ¯ , x 1 ) ,

which implies that lim n ϕ( x n , x 1 )=ϕ( x ¯ , x 1 ). Hence, we have lim n x n = x ¯ . In view of the Kadec-Klee property of E, we find that x n x ¯ as n. Next, we show that pEF(f). In view of construction of x n + 1 = Π EP ( f ) x 1 C n + 1 C n , we find that

ϕ ( x n + 1 , x n ) = ϕ ( x n + 1 , Π C n x 1 ) ϕ ( x n + 1 , x 1 ) ϕ ( Π C n x 1 , x 1 ) = ϕ ( x n + 1 , x 1 ) ϕ ( x n , x 1 ) .
(3.4)

Since x n = Π C n x 1 and x n + 1 = Π C n + 1 x 1 C n + 1 C n , we arrive at ϕ( x n , x 1 )ϕ( x n + 1 , x 1 ). This shows that {ϕ( x n , x 1 )} is nondecreasing. It follows from the boundedness that lim n ϕ(x, x 1 ) exists. This implies from (3.4) that

lim n ϕ( x n + 1 , x n )=0.
(3.5)

Since x n + 1 = Π C n + 1 x 1 C n + 1 , we arrive at ϕ( x n + 1 , y n )ϕ( x n + 1 , x n ). It follows that

lim n ϕ( x n + 1 , y n )=0.

In view of (2.3), we see that lim n ( x n + 1 y n )=0. This in turn implies that lim n y n = x ¯ . It follows that lim n J y n =J x ¯ . This implies that {J y n } is bounded. Note that both E and E are reflexive. We may assume that J y n y E . In view of the reflexivity of E, we see that J(E)= E . This shows that there exists an element yE such that Jy= y . It follows that

ϕ ( x n + 1 , y n ) = x n + 1 2 2 x n + 1 , J y n + y n 2 = x n + 1 2 2 x n + 1 , J y n + J y n 2 .

Taking lim inf n on both sides of the equality above yields that

0 x ¯ 2 2 x ¯ , y + y 2 = x ¯ 2 2 x ¯ , J y + J y 2 = x ¯ 2 2 x ¯ , J y + y 2 = ϕ ( x ¯ , y ) .

That is, x ¯ =y, which in turn implies that y =J x ¯ . It follows that J y n J x ¯ E . Since E enjoys the Kadec-Klee property, we obtain that lim n J y n =J x ¯ . Notice that

J x n J y n J x n J x ¯ +J x ¯ J y n .

It follows that lim n J x n J y n =0. From the restriction on the sequence { r n }, we find that

lim n J x n J y n r n =0.
(3.6)

In view of y n = S r n x n , we see that

f( y n ,y)+ 1 r n y y n ,J y n J x n 0,yC.

It follows from (A2) that

y y n J y n J x n r n 1 r n y y n ,J y n J x n f( y n ,y)f(y, y n ),yC.

By taking the limit as n in the above inequality, from (A4) we obtain that

f(y, x ¯ )0,yC.

For 0<t<1 and yC, define y t =ty+(1t) x ¯ . It follows that y t C, which yields that f( y t , x ¯ )0. It follows from (A1) and (A4) that

0=f( y t , y t )tf( y t ,y)+(1t)f( y t , x ¯ )tf( y t ,y).

That is,

f( y t ,y)0.

Letting t0, we obtain from (A3) that f( x ¯ ,y)0, yC. This implies that x ¯ EP(f).

Finally, we prove that x ¯ = Π EP ( f ) x 1 . Letting n in (3.3), we see that

x ¯ w,J x 1 J x ¯ 0,wEP(f).

In the light of Lemma 2.3, we find that x ¯ = Π EP ( f ) x 1 . This completes the proof. □

We remark that L p , where p>1 is a space which satisfies the restriction in Theorem 3.1. Since every uniformly convex and uniformly smooth Banach space is a reflexive, strictly convex, and smooth Banach space such that both E and E have the Kadec-Klee property, we find from Theorem 3.1 the following result.

Corollary 3.2LetEbe a uniformly convex and uniformly smooth Banach space. LetCbe a nonempty, closed, and convex subset ofE. Letfbe a bifunction fromC×Cto satisfying (A1)-(A4) such thatEF(f)is nonempty. Let{ x n }be a sequence generated in the following manner:

{ x 0 E chosen arbitrarily , C 1 = C , x 1 = Π C 1 x 0 , y n C  such that  f ( y n , y ) + 1 r n y y n , J y n J x n 0 , y C , C n + 1 = { z C n : ϕ ( z , y n ) ϕ ( z , x n ) } , x n + 1 = Π C n + 1 x 1 ,

where{ r n }is a real sequence such that lim inf n r n >0. Then the sequence{ x n }converges strongly to Π EP ( f ) x 1 , where Π EP ( f ) is the generalized projection fromEontoEP(f).

In the framework of Hilbert spaces, we find from Theorem 3.1 the following result.

Corollary 3.3LetEbe a Hilbert space. LetCbe a nonempty, closed, and convex subset of E. Letfbe a bifunction fromC×Cto satisfying (A1)-(A4) such thatEF(f)is nonempty. Let{ x n }be a sequence generated in the following manner:

{ x 0 E chosen arbitrarily , C 1 = C , x 1 = P C 1 x 0 , y n C  such that  f ( y n , y ) + 1 r n y y n , y n x n 0 , y C , C n + 1 = { z C n : z y n 2 z x n 2 } , x n + 1 = P C n + 1 x 1 ,

where{ r n }is a real sequence such that lim inf n r n >0. Then the sequence{ x n }converges strongly to P EP ( f ) x 1 , where P EP ( f ) is the metric projection fromEontoEP(f).

Proof Notice that ϕ(x,y)= x y 2 . The generalized metric projection is reduced to the metric projection and the normalized duality mapping J is reduced to the identity mapping I in Hilbert spaces. The result can be obtained from Theorem 3.1 immediately. □

References

  1. Censor Y, Cohen N, Kutscher T, Shamir J: Summed squared distance error reduction by simultaneous multiprojections and applications. Appl. Math. Comput. 2002, 126: 157–179. 10.1016/S0096-3003(00)00144-2

    MathSciNet  Article  MATH  Google Scholar 

  2. Iiduka H: Fixed point optimization algorithm and its application to network bandwidth allocation. J. Comput. Appl. Math. 2012, 236: 1733–1742. 10.1016/j.cam.2011.10.004

    MathSciNet  Article  MATH  Google Scholar 

  3. Qin X, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 2010, 11: 2963–2972. 10.1016/j.nonrwa.2009.10.017

    MathSciNet  Article  MATH  Google Scholar 

  4. Dhage BC, Jadhav NS: Differential inequalities and comparison theorems for first order hybrid integro-differential equations. Adv. Inequal. Appl. 2013, 2: 61–80.

    Google Scholar 

  5. Abdel-Salam HS, Al-Khaled K: Variational iteration method for solving optimization problems. J. Math. Comput. Sci. 2012, 2: 1475–1497.

    MathSciNet  Google Scholar 

  6. Qin X, Cho SY, Kang SM: Strong convergence of shrinking projection methods for quasi- ϕ -nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math. 2010, 234: 750–760. 10.1016/j.cam.2010.01.015

    MathSciNet  Article  MATH  Google Scholar 

  7. Iiduka H: Strong convergence for an iterative method for the triple-hierarchical constrained optimization problem. Nonlinear Anal. 2009, 71: e1292-e1297. 10.1016/j.na.2009.01.133

    MathSciNet  Article  MATH  Google Scholar 

  8. Noor MA, Noor KI, Waseem M: Decomposition method for solving system of linear equations. Eng. Math. Lett. 2013, 2: 34–41.

    Google Scholar 

  9. Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224–228. 10.1016/j.aml.2010.09.008

    MathSciNet  Article  MATH  Google Scholar 

  10. Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2012, 2: 374–397.

    MathSciNet  MATH  Google Scholar 

  11. Qin X, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi- ϕ -nonexpansive mappings. Appl. Math. Comput. 2010, 215: 3874–3883. 10.1016/j.amc.2009.11.031

    MathSciNet  Article  MATH  Google Scholar 

  12. Al-Bayati AY, Al-Kawaz RZ: A new hybrid WC-FR conjugate gradient-algorithm with modified secant condition for unconstrained optimization. J. Math. Comput. Sci. 2012, 2: 937–966.

    MathSciNet  Google Scholar 

  13. Shen J, Pang LP: An approximate bundle method for solving variational inequalities. Commun. Optim. Theory 2012, 1: 1–18.

    Google Scholar 

  14. Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011

    MathSciNet  Article  MATH  Google Scholar 

  15. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.

    MathSciNet  MATH  Google Scholar 

  16. He RH: Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities in FC-spaces. Adv. Fixed Point Theory 2012, 2: 47–57.

    Google Scholar 

  17. Zegeye H: A hybrid iteration scheme for equilibrium problems, variational inequality problems and common fixed point problems in Banach spaces. Nonlinear Anal. 2010, 72: 2136–2146. 10.1016/j.na.2009.10.014

    MathSciNet  Article  MATH  Google Scholar 

  18. Ye J, Huang J: Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. J. Math. Comput. Sci. 2011, 1: 1–18.

    MathSciNet  Article  Google Scholar 

  19. Qin X, Cho SY, Kang SM: Iterative algorithms for variational inequality and equilibrium problems with applications. J. Glob. Optim. 2010, 48: 423–445. 10.1007/s10898-009-9498-8

    MathSciNet  Article  MATH  Google Scholar 

  20. Gu F, Lu J: Projection methods of iterative solutions in Hilbert spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 162

    Google Scholar 

  21. Kim JK: Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings. J. Inequal. Appl. 2010., 2010: Article ID 312602

    Google Scholar 

  22. Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 10

    Google Scholar 

  23. Kim JK, Anh PN, Nam YM: Strong convergence of an extended extragradient method for equilibrium problems and fixed point problems. J. Korean Math. Sci. 2012, 49: 187–200. 10.4134/JKMS.2012.49.1.187

    MathSciNet  Article  MATH  Google Scholar 

  24. Yang S, Li W: Iterative solutions of a system of equilibrium problems in Hilbert spaces. Adv. Fixed Point Theory 2011, 1: 15–26.

    Google Scholar 

  25. Qin X, Shang M, Su Y: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Comput. Model. 2008, 48: 1033–1046. 10.1016/j.mcm.2007.12.008

    MathSciNet  Article  MATH  Google Scholar 

  26. Luo H, Wang Y: Iterative approximation for the common solutions of a infinite variational inequality system for inverse-strongly accretive mappings. J. Math. Comput. Sci. 2012, 2: 1660–1670.

    MathSciNet  Google Scholar 

  27. Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618.

    MathSciNet  Article  MATH  Google Scholar 

  28. Kang SM, Cho SY, Liu Z: Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings. J. Inequal. Appl. 2010., 2010: Article ID 827082

    Google Scholar 

  29. Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht; 1990.

    Book  MATH  Google Scholar 

  30. Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Dekker, New York; 1996.

    Google Scholar 

  31. Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 2009, 70: 45–57. 10.1016/j.na.2007.11.031

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The author is grateful to the editor and the reviewers for their suggestions which improved the contents of the article.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Juan Zhao.

Additional information

Competing interests

The author declares that they have no competing interests.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Zhao, J. Approximation of solutions to an equilibrium problem in a nonuniformly smooth Banach space. J Inequal Appl 2013, 387 (2013). https://doi.org/10.1186/1029-242X-2013-387

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-387

Keywords

  • equilibrium problem
  • fixed point
  • quasi-ϕ-nonexpansive mapping
  • projection