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Constructed nets with perturbations for equilibrium and fixed point problems
Journal of Inequalities and Applications volume 2014, Article number: 334 (2014)
Abstract
In this paper, an implicit net with perturbations for solving the mixed equilibrium problems and fixed point problems has been constructed and it is shown that the proposed net converges strongly to a common solution of the mixed equilibrium problems and fixed point problems. Also, as applications, some corollaries for solving the minimumnorm problems are also included.
MSC:47J05, 47J25, 47H09.
1 Introduction
The present paper is devoted to solving the following mixed equilibrium problem: Find u\in C such that
for all v\in C, where C is a nonempty closed convex subset of a real Hilbert space H, F:C\times C\to R is a bifunction and A:C\to H is a nonlinear operator. The solution set of (1.1) is denoted by S(MEP).
This problem (1.1) includes optimization problems, variational inequalities, minimax problems, and Nash equilibrium problems in noncooperative games as special cases.
Case 1. If A=0 in (1.1), then (1.1) reduces to the following equilibrium problem: Find u\in C such that
for all v\in C. The solution of (1.2) is denoted by S(EP).
Case 2. If F=0 in (1.1), then (1.1) reduces to the variational inequality problem: Find z\in C such that
for all v\in C. The solution of (1.2) is denoted by S(VI).
In the literature, there are a large number of references associated with some equilibrium problems and variational inequality problems (see, for instance, [1–32]).
The main purpose of the present paper is to construct the following implicit net with perturbations for solving the mixed equilibrium (1.1) and the fixed point problem:
for all y\in C. Also, it is shown that the proposed net \{{z}_{t}\} converges strongly to a common solution of the mixed equilibrium problems and fixed point problems. As applications, some corollaries for solving the minimumnorm problems are also included.
2 Preliminaries
Let H be a real Hilbert space with an inner product \u3008\cdot ,\cdot \u3009 and a norm \parallel \cdot \parallel, respectively, and C be a nonempty closed convex subset of a real Hilbert space H.

(1)
A mapping T:C\to C is said to be nonexpansive if
\parallel TuTv\parallel \le \parallel uv\parallel
for all u,v\in C. F(T) denotes the set of fixed points of T.

(2)
A mapping A:C\to H is said to be αinversestrongly monotone if there exists a positive real number \alpha >0 such that
\u3008AuAv,uv\u3009\ge \alpha {\parallel AuAv\parallel}^{2}
for all u,v\in C. It is clear that any αinversestrongly monotone mapping is monotone and \frac{1}{\alpha}Lipschitz continuous.
Throughout this paper, we assume that a bifunction F:C\times C\to R satisfies the following conditions:
(C1) F(u,u)=0 for all u\in C;
(C2) F is monotone, i.e., F(u,v)+F(v,u)\le 0 for all u,v\in C;
(C3) for each u,v,w\in C, {lim}_{t\downarrow 0}F(tw+(1t)u,v)\le F(u,v);
(C4) for each u\in C, v\mapsto F(u,v) is convex and lower semicontinuous.
In fact, some efforts to construct the algorithms for solving the equilibrium problems have been carried out. For instance, Moudafi [15] presented an iterative algorithm and proved a weak convergence theorem for solving the mixed equilibrium problem (1.1). Takahashi and Takahashi [24] constructed the following iterative algorithm:
for all y\in C and n\ge 0, where T:C\to C is a nonexpansive mapping. They proved that the sequence \{{x}_{n}\} generated by (2.1) converges strongly to z={Proj}_{F(T)\cap S(\mathit{MEP})}(u).
Plubtieng and Punpaeng [20] introduced and considered the following two iterative schemes for finding a common element of the set of solutions of the problem (1.2) and the set of fixed points of a nonexpansive mapping in a Hilbert space H.
Implicit iterative algorithm \{{x}_{n}\}:
for all y\in H and n\ge 1.
Explicit iterative algorithm \{{x}_{n}\}:
for all y\in H and n\ge 1.
They proved that, under certain conditions, the sequences \{{x}_{n}\} generated by (2.2) and (2.3) converge strongly to the unique solution of the variational inequality:
for all x\in F(T)\cap S(\mathit{EP}), which is the optimality condition for the minimization problem:
where h is a potential function for γf.
We know that there are perturbations always occurring in the iterative processes because the manipulations are inaccurate. Recently, Chuang et al. ([8]) constructed the following iteration process with perturbations for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points for a quasinonexpansive mapping with perturbation: {q}_{1}\in H and
for all y\in C and n\ge 0. They shown that the sequence \{{q}_{n}\} converges strongly to {Proj}_{F(T)\cap S(\mathit{EP})}.
Now, we need the following useful lemmas for our main results.
Lemma 2.1 [11]
Let C be a nonempty closed convex subset of a real Hilbert space H. Let F:C\times C\to R be a bifunction which satisfies the conditions (C1)(C4). Let r>0 and x\in H. Then there exists z\in C such that
for all y\in C. Further, if
then the following hold:

(1)
{T}_{r} is singlevalued and {T}_{r} is firmly nonexpansive, i.e., for any x,y\in H,
{\parallel {T}_{r}x{T}_{r}y\parallel}^{2}\le \u3008{T}_{r}x{T}_{r}y,xy\u3009; 
(2)
S(\mathit{EP}) is closed and convex and S(\mathit{EP})=F({T}_{r}).
Lemma 2.2 [24]
Let C, H, F, and {T}_{r}x be as in Lemma 2.1. Then the following holds:
for all s,t>0 and x\in H.
Lemma 2.3 [24]
Let C be a nonempty closed convex subset of a real Hilbert space H. Let a mapping A:C\to H be αinversestrongly monotone and r>0 be a constant. Then we have
for all x,y\in C. In particular, if 0\le r\le 2\alpha, then IrA is nonexpansive.
Lemma 2.4 [33]
Let C be a closed convex subset of a real Hilbert space H and let T:C\to C be a nonexpansive mapping. Then the mapping IT is demiclosed, that is, if \{{x}_{n}\} is a sequence in C such that {x}_{n}\to u weakly and (IT){x}_{n}\to v strongly, then (IT)u=v.
3 Convergence results
In this section, first, we give our main results.
Part I: Assumptions on the setting of C, F, A, and T:
(A1) C is a nonempty closed convex subset of a real Hilbert space H;
(A2) F:C\times C\to R is a bifunction satisfying the conditions (C1)(C4);
(A3) A:C\to H is an αinversestrongly monotone mapping;
(A4) T:C\to C is a nonexpansive mapping.
Part II: Parameter restrict:
λ is a constant satisfying a\le \lambda \le b, where [a,b]\subset (0,2\alpha ).
Part III: Perturbations:
\{{u}_{t}\}\subset H is a net satisfying {lim}_{t\to 0+}{u}_{t}=u\in H.
Algorithm 3.1 For any t\in (0,1\frac{\lambda}{2\alpha}), define a net \{{z}_{t}\}\subset C by the implicit manner:
for all y\in C.
Remark 3.2 We show that the net \{{z}_{t}\} is well defined. Next, we prove that (3.1) can be rewritten as
for all t\in (0,1\frac{\lambda}{2\alpha}).
In fact, for any t\in (0,1\frac{\lambda}{2\alpha}), {u}_{t}\in H, and x\in H, we find z such that, for all y\in C,
From Lemma 2.1, we get immediately
Now, we can define a mapping
for all t\in (0,1\frac{\lambda}{2\alpha}). Again, from Lemma 2.1, we know that {T}_{\lambda} is nonexpansive. Thus, for any x,y\in C, we have
From Lemma 2.3, I\frac{\lambda}{1t}A is nonexpansive for all t\in (0,1\frac{\lambda}{2\alpha}). Note that T is also nonexpansive. By (3.3), we deduce
for all x,y\in C. This indicates that {\vartheta}_{t} is a contraction on C and so it has a unique fixed point, denoted by {z}_{t}, in C. That is, {z}_{t}={T}_{\lambda}(t{u}_{t}+(1t)T{z}_{t}\lambda AT{z}_{t}). Hence \{{z}_{t}\} is well defined.
Theorem 3.3 Suppose that F(T)\cap S(\mathit{MEP})\ne \mathrm{\varnothing}. Then the net \{{z}_{t}\} defined by (3.1) converges strongly as t\to 0+ to {P}_{F(T)\cap S(\mathit{MEP})}(u).
Proof Pick up z\in F(T)\cap S(\mathit{MEP}). It is obvious that z={T}_{\lambda}(z\lambda Az) for all \lambda >0. So, we have
for all t\in (0,1\frac{\lambda}{2\alpha}). Then we have
Using the convexity of \parallel \cdot \parallel and the inversestrong monotonicity of A, we derive
By the assumption, we have \lambda 2(1t)\alpha \le 0 for all t\in (0,1\frac{\lambda}{2\alpha}). Then, from (3.4) and (3.5), it follows that
and so
Since {lim}_{t\to 0+}{u}_{t}=u, there exists a positive constant M>0 such that {sup}_{t}\{\parallel {u}_{t}\parallel \}\le M. Then, from (3.7), we deduce that \{{z}_{t}\} is bounded. Hence \{T{z}_{t}\} and \{AT{z}_{t}\} are also bounded.
From (3.4) and (3.5), we obtain
and so
This implies that
Next, we show \parallel {z}_{t}T{z}_{t}\parallel \to 0. Since {T}_{\lambda} is firmly nonexpansive (see Lemma 2.1), we have
Since I\lambda A/(1t) is nonexpansive, we have
Thus we have
It follows that
and so
Since \parallel AT{z}_{t}Az\parallel \to 0 by (3.8), we deduce
Therefore, we have
From (3.4), it follows that
which implies that
where {M}_{1} is a constant such that
Next, we show that \{{z}_{t}\} is relatively normcompact as t\to 0+. Assume that \{{t}_{n}\}\subset (0,1) is a sequence such that {t}_{n}\to 0+ as n\to \mathrm{\infty}. Put {z}_{n}:={z}_{{t}_{n}}. From (3.10), it follows that
for all z\in F(T)\cap S(\mathit{MEP}). Since \{{z}_{n}\} is bounded, without loss of generality, we may assume that {z}_{n}\rightharpoonup \tilde{x}\in C. From (3.9), we have
We can use Lemma 2.4 to (3.12) to deduce \tilde{x}\in F(T). Further, we show that \tilde{x} is also in S(\mathit{MEP}). Since {z}_{n}={T}_{\lambda}({t}_{n}{u}_{n}+(1{t}_{n})T{z}_{n}\lambda AT{z}_{n}) for any y\in C, we have
From (C2), it follows that
Put {x}_{t}=ty+(1t)\tilde{x} for all t\in (0,1\frac{\lambda}{2\alpha}) and y\in C. Then we have {x}_{t}\in C and so, from (3.13), it follows that
Since \parallel {z}_{n}T{z}_{n}\parallel \to 0, we have \parallel A{z}_{n}AT{z}_{n}\parallel \to 0. Further, since A is monotone, we have \u3008{x}_{t}{z}_{n},A{x}_{t}A{z}_{n}\u3009\ge 0. So, from (C4), it follows that, as n\to \mathrm{\infty},
Also, it follows from (C1), (C4), and (3.14) that
and hence
Letting t\to 0, we have
for all y\in C. This implies \tilde{x}\in \mathit{EP}. Therefore, we can substitute \tilde{x} for z in (3.8) to get
for all \tilde{x}\in F(T)\cap S(\mathit{MEP}). By (3.5), we know that \parallel AT{z}_{n}Az\parallel \to 0 for any z\in F(T)\cap S(\mathit{MEP}). Then we get \parallel AT{z}_{n}A\tilde{x}\parallel \to 0. Consequently, the weak convergence of \{{z}_{n}\} (and \{T{z}_{n}\}) to \tilde{x} actually implies that {z}_{n}\to \tilde{x}. This proves the relative normcompactness of the net \{{z}_{t}\} as t\to 0+.
Now, we return to (3.11) and take the limit as n\to \mathrm{\infty} to get
for all z\in F(T)\cap S(\mathit{MEP}). Equivalently, we have
for all z\in F(T)\cap S(\mathit{MEP}). This clearly implies that
Therefore, \tilde{x} is the unique cluster point of the net \{{z}_{t}\}. Hence the whole net \{{z}_{t}\} converges strongly to \tilde{x}={P}_{F(T)\cap S(\mathit{MEP})}(u). This completes the proof. □
4 Induced algorithms and corollaries

(I)
Taking T=I in (3.1), we get the following.
Algorithm 4.1 For any t\in (0,1\frac{\lambda}{2\alpha}), define a net \{{z}_{t}\}\subset C by the implicit manner:
for all y\in C.
Corollary 4.2 Suppose that S(\mathit{MEP})\ne \mathrm{\varnothing}. Then the net \{{z}_{t}\} defined by (4.1) converges strongly as t\to 0+ to {P}_{S(\mathit{MEP})}(u).

(II)
Taking F=0 in (4.1), we get the following.
Algorithm 4.3 For any t\in (0,1\frac{\lambda}{2\alpha}), define a net \{{z}_{t}\}\subset C by the implicit manner:
for all y\in C.
Corollary 4.4 Suppose that S(\mathit{VI})\ne \mathrm{\varnothing}. Then the net \{{z}_{t}\} defined by (4.2) converges strongly as t\to 0+ to {P}_{S(\mathit{VI})}(u).

(III)
Taking A=0 in (4.1), we get the following.
Algorithm 4.5 For any t\in (0,1), define a net \{{z}_{t}\}\subset C by the implicit manner:
for all y\in C.
Corollary 4.6 Suppose that S(\mathit{EP})\ne \mathrm{\varnothing}. Then the net \{{z}_{t}\} defined by (4.3) converges strongly as t\to 0+ to {P}_{S(\mathit{EP})}(u).
5 Minimumnorm solutions
In many problems, one needs to find a solution with the minimum norm. In an abstract way, we may formulate such problems as finding a point {x}^{\u2020} with the property:
where C is a nonempty closed convex subset of a real Hilbert space H. In other words, {x}^{\u2020} is the (nearest point or metric) projection of the origin onto C, that is,
where {P}_{C} is the metric (or nearest point) projection from H onto C.
A typical example is the leastsquares solution of the constrained linear inverse problem:
where A is a bounded linear operator from H to another real Hilbert space {H}_{1} and b is a given point in {H}_{1}. The leastsquares solution is the leastnorm minimizer of the minimization problem:
Motivated by the above leastsquares solution of the constrained linear inverse problems, we study the general case of finding the minimumnorm solutions for the mixed equilibrium problem (1.1), the equilibrium problem (1.2), the variational inequality (1.3), and the fixed point problem.
Now, we state our algorithms which can be inducted from the above section.

(I)
Taking {u}_{t}=0 for all t in (3.1), we get the following.
Algorithm 5.1 For any t\in (0,1\frac{\lambda}{2\alpha}), define a net \{{z}_{t}\}\subset C by the implicit manner:
for all y\in C.
Corollary 5.2 Suppose that F(T)\cap S(\mathit{MEP})\ne \mathrm{\varnothing}. Then the net \{{z}_{t}\} defined by (5.1) converges strongly as t\to 0+ to {P}_{F(T)\cap S(\mathit{MEP})}(0), which is the minimumnorm element in F(T)\cap S(\mathit{MEP}).

(II)
Taking {u}_{t}=0 for all t in (4.1), we get the following.
Algorithm 5.3 For any t\in (0,1\frac{\lambda}{2\alpha}), define a net \{{z}_{t}\}\subset C by the implicit manner:
for all y\in C.
Corollary 5.4 Suppose that S(\mathit{MEP})\ne \mathrm{\varnothing}. Then the net \{{z}_{t}\} defined by (5.2) converges strongly as t\to 0+ to {P}_{S(\mathit{MEP})}(0), which is the minimumnorm element in S(\mathit{MEP}).

(III)
Taking {u}_{t}=0 for all t in (4.2), we get the following.
Algorithm 5.5 For any t\in (0,1\frac{\lambda}{2\alpha}), define a net \{{z}_{t}\}\subset C by the implicit manner:
for all y\in C.
Corollary 5.6 Suppose that S(\mathit{VI})\ne \mathrm{\varnothing}. Then the net \{{z}_{t}\} defined by (5.3) converges strongly as t\to 0+ to {P}_{S(\mathit{VI})}(0), which is the minimumnorm element in S(\mathit{VI}).

(IV)
Taking A=0 in (5.1), we get the following.
Algorithm 5.7 For any t\in (0,1), define a net \{{z}_{t}\}\subset C by the implicit manner:
for all y\in C.
Corollary 5.8 Suppose that F(T)\cap S(\mathit{EP})\ne \mathrm{\varnothing}. Then the net \{{z}_{t}\} defined by (5.4) converges strongly as t\to 0+ to {P}_{F(T)\cap S(\mathit{EP})}(0), which is the minimumnorm element in F(T)\cap S(\mathit{EP}).

(V)
Taking A=0 in (5.2), we get the following.
Algorithm 5.9 For any t\in (0,1), define a net \{{z}_{t}\}\subset C by the implicit manner:
for all y\in C.
Corollary 5.10 Suppose that S(\mathit{EP})\ne \mathrm{\varnothing}. Then the net \{{z}_{t}\} defined by (5.5) converges strongly as t\to 0+ to {P}_{S(\mathit{EP})}(0), which is the minimumnorm element in S(\mathit{EP}).
References
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.
Ceng LC, Schaible S, Yao JC: Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings. J. Optim. Theory Appl. 2008, 139: 403–418. 10.1007/s109570089361y
Ceng LC, AlHomidan S, Ansari QH, Yao JC: An iterative scheme for equilibrium problems and fixed point problems of strict pseudocontraction mappings. J. Comput. Appl. Math. 2009, 223: 967–974. 10.1016/j.cam.2008.03.032
Chadli O, Schaible S, Yao JC: Regularized equilibrium problems with an application to noncoercive hemivariational inequalities. J. Optim. Theory Appl. 2004, 121: 571–596.
Chadli O, Wong NC, Yao JC: Equilibrium problems with applications to eigenvalue problems. J. Optim. Theory Appl. 2003, 117: 245–266. 10.1023/A:1023627606067
Cho YJ, Argyros IK, Petrot N: Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems. Comput. Math. Appl. 2010, 60: 2292–2301. 10.1016/j.camwa.2010.08.021
Cho YJ, Qin X, Kang JI: Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems. Nonlinear Anal. 2009, 71: 4203–4214. 10.1016/j.na.2009.02.106
Chuang CS, Lin LJ, Takahashi W: Halpern’s type iterations with perturbations in Hilbert spaces: equilibrium solutions and fixed points. J. Glob. Optim. 2013. 10.1007/s1089801299116
Colao V, Acedo GL, Marino G: An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings. Nonlinear Anal. 2009, 71: 2708–2715. 10.1016/j.na.2009.01.115
Colao V, Marino G, Xu HK: An iterative method for finding common solutions of equilibrium and fixed point problems. J. Math. Anal. Appl. 2008, 344: 340–352. 10.1016/j.jmaa.2008.02.041
Combettes PL, Hirstoaga A: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.
Jaiboon C, Kumam P: A general iterative method for addressing mixed equilibrium problems and optimization problems. Nonlinear Anal. 2010, 73: 1180–1202. 10.1016/j.na.2010.04.041
Konnov IV, Schaible S, Yao JC: Combined relaxation method for mixed equilibrium problems. J. Optim. Theory Appl. 2005, 126: 309–322. 10.1007/s1095700547160
Moudafi A, Théra M: Proximal and dynamical approaches to equilibrium problems. Lecture Notes in Economics and Mathematical Systems 477. In IllPosed Variational Problems and Regularization Techniques. Springer, Berlin; 1999:187–201.
Moudafi A: Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 2008, 9: 37–43.
Noor MA: Multivalued general equilibrium problems. J. Math. Anal. Appl. 2003, 283: 140–149. 10.1016/S0022247X(03)002518
Noor MA, Rassias TM: On nonconvex equilibrium problems. J. Math. Anal. Appl. 2005, 312: 289–299. 10.1016/j.jmaa.2005.03.069
Peng JW, Yao JC: A new hybridextragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwan. J. Math. 2008, 12: 1401–1432.
Petrot N, Wattanawitoon K, Kumam P: A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces. Nonlinear Anal. Hybrid Syst. 2010, 4: 631–643. 10.1016/j.nahs.2010.03.008
Plubtieng S, Punpaeng R: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 2007, 336: 455–469. 10.1016/j.jmaa.2007.02.044
Qin X, Cho YJ, Kang SM: Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 2010, 72: 99–112. 10.1016/j.na.2009.06.042
Shehu Y: Strong convergence theorems for nonlinear mappings, variational inequality problems and system of generalized mixed equilibrium problems. Math. Comput. Model. 2011, 54: 2259–2276. 10.1016/j.mcm.2011.05.035
Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 2007, 331: 506–515. 10.1016/j.jmaa.2006.08.036
Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 2008, 69: 1025–1033. 10.1016/j.na.2008.02.042
Tada A, Takahashi W: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. In Nonlinear Analysis and Convex Analysis. Edited by: Takahashi W, Tanaka T. Yokohama Publishers, Yokohama; 2007:607–617.
Yao Y, Cho YJ, Liou YC: Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur. J. Oper. Res. 2011, 212: 242–250. 10.1016/j.ejor.2011.01.042
Yao Y, Liou YC, Yao JC: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Appl. 2007., 2007: Article ID 64363
Yao Y, Liou YC: Composite algorithms for minimization over the solutions of equilibrium problems and fixed point problems. Abstr. Appl. Anal. 2010., 2010: Article ID 763506
Yao Y, Shahzad N: New methods with perturbations for nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 79
Zegeye H, Ofoedu EU, Shahzad N: Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasinonexpansive mappings. Appl. Math. Comput. 2010, 216: 3439–3449. 10.1016/j.amc.2010.02.054
Zegeye H, Shahzad N: A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems. Nonlinear Anal. 2011, 74: 263–272. 10.1016/j.na.2010.08.040
Zhang SS, Lee JHW, Chan CK: Algorithms of common solutions for quasi variational inclusion and fixed point problems. Appl. Math. Mech. 2008, 29: 571–581. 10.1007/s104830080502y
Zhou H: Convergence theorems of fixed points for λ strict pseudocontractions in Hilbert spaces. Nonlinear Anal. 2008, 69: 456–462. 10.1016/j.na.2007.05.032
Acknowledgements
The first author was supported in part by NSFC 11071279 and NSFC 71161001G0105, the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: NRF2013053358) and the third author was supported in part by NSC 1012628E230001MY3 and NSC 1012622E230005CC3.
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Yao, Y., Cho, Y.J., Liou, YC. et al. Constructed nets with perturbations for equilibrium and fixed point problems. J Inequal Appl 2014, 334 (2014). https://doi.org/10.1186/1029242X2014334
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DOI: https://doi.org/10.1186/1029242X2014334
Keywords
 equilibrium problem
 fixed point problem
 minimization problem
 nonexpansive mapping