An equilibrium problem is investigated based on a hybrid projection iterative algorithm. Strong convergence theorems for solutions of the equilibrium problem are established in a strictly convex and uniformly smooth Banach space which also enjoys the Kadec-Klee property.

1. Introduction

Equilibrium problems which were introduced by Fan [1] and Blum and Oettli [2] have had a great impact and influence on the development of several branches of pure and applied sciences. It has been shown that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. It has been shown [3–8] that equilibrium, problems include variational inequalities, fixed point, the Nash equilibrium, and game theory as special cases. A number of iterative algorithms have recently been studying for fixed point and equilibrium problems, see [9–26] and the references therein. However, there were few results established in the framework of the Banach spaces. In this paper, we suggest and analyze a projection iterative algorithm for finding solutions of equilibrium in a Banach space.

2. Preliminaries

In what follows, we always assume that E is a Banach space with the dual space E^*. Let C be a nonempty, closed, and convex subset of E. We use the symbol J to stand for the normalized duality mapping from E to

defined by

()

where 〈·, ·〉 denotes the generalized duality pairing of elements between E and E^*.

Let U_E = {x ∈ E : ∥x∥ = 1} be the unit sphere of E. E is said to be strictly convex if ∥(x + y)/2∥<1 for all x, y ∈ U_E with x ≠ y. It is said to be uniformly convex if for any ϵ ∈ (0,2] there exists δ > 0 such that for any x, y ∈ U_E,

()

It is known that a uniformly convex Banach space is reflexive and strictly convex; for details see [27] and the references therein.

Recall that a Banach space E is said to have the Kadec-Klee property if a sequence {x_n} of E satisfies that x_n⇀x ∈ C, where ⇀ denotes the weak convergence, and ∥x_n∥→∥x∥, where → denotes the strong convergence, and then x_n → x. It is known that if E is uniformly convex, then E enjoys the Kadec-Klee property; for details see [26] and the references therein.

E is said to be smooth provided lim _t→0(∥x + ty∥−∥x∥)/t exists for all x, y ∈ U_E. It is also said to be uniformly smooth if the limit is attained uniformly for all x, y ∈ U_E.

It is well known that if E^* is strictly convex, then J is single valued; if E^* is reflexive, and smooth, then J is single valued and demicontinuous; for more details see [27, 28] and the references therein.

It is also well known that if D is a nonempty, closed, and convex subset of a Hilbert space H, and P_D : H → D is the metric projection from H onto D, then P_D is nonexpansive. This fact actually characterizes the Hilbert spaces, and consequently, it is not available in more general Banach spaces. In this connection, Alber [29] introduced a generalized projection operator Π_D in the Banach spaces which is an analogue of the metric projection in the Hilbert spaces.

Let E be a smooth Banach space. Consider the functional defined by

()

Notice that, in a Hilbert space H, (2.3) is reduced to ϕ(x, y) = ∥x − y∥² for all x, y ∈ H. The generalized projection Π_C : E → C is a mapping that is assigned to an arbitrary point x ∈ E, the minimum point of the functional ϕ(x, y); that is,

, where

is the solution to the following minimization problem:

()

The existence and uniqueness of the operator Π_C follow from the properties of the functional ϕ(x, y) and the strict monotonicity of the mapping J; see, for example, [27, 28]. In the Hilbert spaces, Π_C = P_C. It is obvious from the definition of the function ϕ that

()

Let T : C → C be a mapping. Recall that a point p in C is said to be an asymptotic fixed point of T if C contains a sequence {x_n} which converges weakly to p such that lim _n→∞∥x_n − Tx_n∥ = 0. The set of asymptotic fixed points of T will be denoted by

. T is said to be relatively nonexpansive if

()

The asymptotic behavior of a relatively nonexpansive mapping was studied in [27, 29, 30].

Let f be a bifunction from C × C to ℝ, where ℝ denotes the set of real numbers. In this paper, we consider the following equilibrium problem. Find p ∈ C such that

()

We use EP (f) to denote the solution set of the equilibrium problem (2.3). That is,

()

Given a mapping Q : C → E^*, let

()

Then p ∈ EP (f) if and only if p is a solution of the following variational inequality. Find p such that

()

To study the equilibrium problem (2.8), we may assume that f satisfies the following conditions:

(A1) f(x, x) = 0, for all x ∈ C;

(A2) f is monotone, that is, f(x, y) + f(y, x) ≤ 0, for all x, y ∈ C;

(A3)

()

(A4) for each x ∈ C, y ↦ f(x, y) is convex and weakly lower semicontinuous.

In this paper, we study the problem of approximating solutions of equilibrium problem (2.8) based on a hybrid projection iterative algorithm in a strictly convex and uniformly smooth Banach space which also enjoys the Kadec-Klee property. To prove our main results, we need the following lemmas.

Lemma 2.1. Let E be a strictly convex and uniformly smooth Banach space and C a nonempty, closed, and convex subset of E. Let f be a bifunction from C × C to ℝ satisfying (A1)–(A4). Let r > 0 and x ∈ E. Then

(a)
(see [2]). There exists z ∈ C such that

()

(b)
(see [31]). Define a mapping by

()

Then the following conclusions hold:

(1)
is single valued;
(2)
is a firmly nonexpansive-type mapping; that is, for all x, y ∈ E,

()

(3)
;
(4)
EP (f) is closed and convex;
(5)
is relatively nonexpansive.

Lemma 2.2 (see [29].)Let E be a reflexive, strictly convex, and smooth Banach space and C a nonempty, closed, and convex subset of E. Let x ∈ E, and x₀ ∈ C. Then x₀ = Π_Cx if and only if

()

Lemma 2.3 (see [29].)Let E be a reflexive, strictly convex, and smooth Banach space and C a nonempty, closed, and convex subset of E, and x ∈ E. Then

()

Lemma 2.4 (see [27].)Let E be a reflexive, strictly convex, and smooth Banach space. Then one has the following

()

3. Main Results

Theorem 3.1. Let E be a strictly convex and uniformly smooth Banach space which also enjoys the Kadec-Klee property and C a nonempty, closed, and convex subset of E. Let f be a bifunction from C × C to ℝ satisfying (A1)–(A4) such that EP (f) ≠ ∅. Let {x_n} be a sequence generated by the following manner:

()

where {r_n} is a real number sequence in [r, ∞), where r is some positive real number. Then the sequence {x_n} converges strongly to

Proof. In view of Lemma 2.1, we see that EP (f) is closed and convex. Next, we show that C_n is closed and convex. It is not hard to see that C_n is closed. Therefore, we only show that C_n is convex. It is obvious that C₁ = C is convex. Suppose that C_h is convex for some h ∈ ℕ. Next, we show that C_h+1 is also convex for the same h. Let a, b ∈ C_h+1 and c = ta + (1 − t)b, where t ∈ (0,1). It follows that

()

where a, b ∈ C_h. From the above two inequalities, we can get that

()

where c ∈ C_h. It follows that C_h+1 is closed and convex. This completes the proof that C_n is closed, and convex.

Next, we show that EP (f) ⊂ C_n. It is obvious that EP (f) ⊂ C = C₁. Suppose that EP (f) ⊂ C_h for some h ∈ ℕ. For any z ∈ EP (f) ⊂ C_h, we see from Lemma 2.1 that

()

On the other hand, we obtain from (2.6) that

()

Combining (3.4) with (3.5), we arrive at

()

which implies that z ∈ C_h+1. This shows that EP (f) ⊂ C_h+1. This completes the proof that EP (f) ⊂ C_n.

Next, we show that {x_n} is a convergent sequence and strongly converges to , where . Since , we see from Lemma 2.2 that

()

It follows from EP (f) ⊂ C_n that

()

By virtue of Lemma 2.3, we obtain that

()

This implies that the sequence {ϕ(x_n, x₀)} is bounded. It follows from (2.5) that the sequence {x_n} is also bounded. Since the space is reflexive, we may assume that

. Since C_n is closed and convex, we see that

. On the other hand, we see from the weakly lower semicontinuity of the norm that

()

which implies that

as n → ∞. Hence,

as n → ∞. In view of the Kadec-Klee property of E, we see that

as n → ∞. Notice that x_n+1 = Π_EP (f)x₀ ∈ C_n+1 ⊂ C_n. It follows that

()

Since

and

, we arrive at ϕ(x_n, x₀) ≤ ϕ(x_n+1, x₀). This shows that {ϕ(x_n, x₀)} is nondecreasing. It follows from the boundedness that lim _n→∞ϕ(x_,x₀) exists. It follows that

()

By virtue of

, we find that

()

It follows that

()

In view of (2.5), we see that

()

Since

, we find that

()

It follows that

()

This implies that {Jy_n} is bounded. Note that both E and E^* are reflexive. We may assume that Jy_n⇀y^* ∈ E^*. In view of the reflexivity of E, we see that there exists an element y ∈ E such that Jy = y^*. It follows that

()

Taking liminf _n→∞ on the both sides of the equality above yields that

()

That is,

, which in turn implies that

. It follows that

. Since E^* enjoys the Kadec-Klee property, we obtain from (3.17) that

. Since J⁻¹ : E^* → E is demicontinuous, we find that

. This implies from (3.16) and the Kadec-Klee property of E that

. This in turn implies that lim _n→∞∥y_n − x_n∥ = 0. Since J is uniformly norm-to-norm continuous on any bounded sets, we find that

()

Next, we show that

. In view of Lemma 2.1, we find from

that

()

It follows from condition (A2) and (3.20) that

()

In view of condition (A4), we obtain from (3.17) that

()

For 0 < t < 1 and u ∈ C, define

. It follows that u_t ∈ C, which yields that

. It follows from conditions (A1) and (A4) that

()

That is,

()

Letting t ↓ 0, we find from condition (A3) that

, for all u ∈ C. This implies that

. This shows that

Finally, we prove that . Letting n → ∞ in (3.8), we see that

()

In view of Lemma 2.2, we can obtain that

. This completes the proof.

In the framework of the Hilbert spaces, we have the following.

Corollary 3.2. Let E be a Hilbert space and C a nonempty, closed, and convex subset of E. Let f be a bifunction from C × C to ℝ satisfying (A1)–(A4) such that EP (f) ≠ ∅. Let {x_n} be a sequence generated by the following manner:

()

where {r_n} is a real number sequence in [r, ∞), where r is some positive real number. Then the sequence {x_n} converges strongly to

References

1 Fan K., O. Shisha, A minimax inequality and applications, Inequalities, III, 1972, Academic Press, New York, NY, USA, 103–113, 0341029, ZBL0302.49019.
Google Scholar
2 Blum E. and Oettli W., From optimization and variational inequalities to equilibrium problems, The Mathematics Student. (1994) 63, no. 1–4, 123–145, 1292380, ZBL0888.49007.
Google Scholar
3 Qin X., Cho S. Y., and Kang S. M., Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications, Journal of Computational and Applied Mathematics. (2009) 233, no. 2, 231–240, https://doi.org/10.1016/j.cam.2009.07.018, 2568521, ZBL1204.65081.
10.1016/j.cam.2009.07.018
Web of Science® Google Scholar
4 Huang N. J., Li J., and Thompson H. B., Implicit vector equilibrium problems with applications, Mathematical and Computer Modelling. (2003) 37, no. 12-13, 1343–1356, https://doi.org/10.1016/S0895%2D7177(03)90045%2D8, 1996042, ZBL1080.90086.
10.1016/S0895-7177(03)90045-8
Google Scholar
5 Qin X., Shang M., and Su Y., Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems, Mathematical and Computer Modelling. (2008) 48, no. 7-8, 1033–1046, https://doi.org/10.1016/j.mcm.2007.12.008, 2458216, ZBL1187.65058.
10.1016/j.mcm.2007.12.008
Google Scholar
6 Fuentes M. N., Existence of equilibria in economies with externalities and non-convexities in an infinite-dimensional commodity space, Journal of Mathematical Economics. (2011) 47, 768–776, https://doi.org/10.1016/j.jmateco.2011.10.007.
10.1016/j.jmateco.2011.10.007
Web of Science® Google Scholar
7 Koh A., [email protected], An evolutionary algorithm based on Nash Dominance for equilibrium problems with equilibrium constraints, Applied Soft Computing Journal. (2012) 12, no. 1, 161–173, https://doi.org/10.1016/j.asoc.2011.08.056.
10.1016/j.asoc.2011.08.056
Web of Science® Google Scholar
8 Qin X., Cho Y. J., and Kang S. M., Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, Journal of Computational and Applied Mathematics. (2009) 225, no. 1, 20–30, https://doi.org/10.1016/j.cam.2008.06.011, 2490167, ZBL1165.65027.
10.1016/j.cam.2008.06.011
Web of Science® Google Scholar
9 Petrot N., Wattanawitoon K., and Kumam P., A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces, Nonlinear Analysis. Hybrid Systems. (2010) 4, no. 4, 631–643, https://doi.org/10.1016/j.nahs.2010.03.008, 2680235.
10.1016/j.nahs.2010.03.008
Web of Science® Google Scholar
10 Saewan S. and Kumam P., Modified hybrid block iterative algorithm for convex feasibility problems and generalized equilibrium problems for uniformly quasi-ϕ-asymptotically nonexpansive mappings, Abstract and Applied Analysis. (2010) 2010, 22, 357120, 2669086, https://doi.org/10.1155/2010/357120.
10.1155/2010/357120
Google Scholar
11 Yang S. and Li W., Iterative solutions of a system of equilibrium problems in Hilbert spaces, Advances in Fixed Point Theory. (2011) 1, no. 1, 15–26.
Google Scholar
12 Qin X., Chang S. S., and Cho Y. J., Iterative methods for generalized equilibrium problems and fixed point problems with applications, Nonlinear Analysis. Real World Applications. (2010) 11, no. 4, 2963–2972, https://doi.org/10.1016/j.nonrwa.2009.10.017, 2661959, ZBL1192.58010.
10.1016/j.nonrwa.2009.10.017
Web of Science® Google Scholar
13 Qin X., Cho S. Y., and Kang S. M., Iterative algorithms for variational inequality and equilibrium problems with applications, Journal of Global Optimization. (2010) 48, no. 3, 423–445, https://doi.org/10.1007/s10898%2D009%2D9498%2D8, 2728746.
10.1007/s10898-009-9498-8
Web of Science® Google Scholar
14 Ye J. and Huang J., Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces, Journal of Mathematical and Computational Science. (2011) 1, 1–18.
Google Scholar
15 Kim J. K., Cho S. Y., and Qin X., Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings, Journal of Inequalities and Applications. (2010) 2010, 18, 312602, https://doi.org/10.1155/2010/312602, 2718500, ZBL1206.47076.
10.1155/2010/312602
Google Scholar
16 Kim J. K., [email protected], Anh P. N., and [email protected], Nam Y. M., [email protected], Strong convergence of an extended extragradient method for equilibrium problems and fixed point problems, Journal of the Korean Mathematical Society. (2012) 49, no. 1, 187–200.
10.4134/JKMS.2012.49.1.187
Web of Science® Google Scholar
17 Qin X., Cho S. Y., and Kang S. M., On hybrid projection methods for asymptotically quasi-ϕ-nonexpansive mappings, Applied Mathematics and Computation. (2010) 215, no. 11, 3874–3883, https://doi.org/10.1016/j.amc.2009.11.031, 2578853, ZBL1225.47105.
10.1016/j.amc.2009.11.031
Web of Science® Google Scholar
18 Kang S. M., Cho S. Y., and Liu Z., Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings, Journal of Inequalities and Applications. (2010) 2010, 16, 827082, 2592859, ZBL1187.47050, https://doi.org/10.1155/2010/827082.
10.1155/2010/827082
Google Scholar
19 Qin X. and Su Y., Strong convergence theorems for relatively nonexpansive mappings in a Banach space, Nonlinear Analysis. (2007) 67, no. 6, 1958–1965, https://doi.org/10.1016/j.na.2006.08.021, 2326043, ZBL1124.47046.
10.1016/j.na.2006.08.021
Web of Science® Google Scholar
20 Chang S. S., Lee H. W. J., and Chan C. K., A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications, Nonlinear Analysis. (2010) 73, no. 7, 2260–2270, https://doi.org/10.1016/j.na.2010.06.006, 2674202.
10.1016/j.na.2010.06.006
Web of Science® Google Scholar
21 Qin X., Cho S. Y., and Kang S. M., Strong convergence of shrinking projection methods for quasi-ϕ-nonexpansive mappings and equilibrium problems, Journal of Computational and Applied Mathematics. (2010) 234, no. 3, 750–760, 2606662, https://doi.org/10.1016/j.cam.2010.01.015, ZBL1189.47068.
10.1016/j.cam.2010.01.015
Web of Science® Google Scholar
22 Lv S., Generalized systems of variational inclusions involving (A,η)-monotone mappings, Advanced in Fixed Point Theory. (2011) 1, no. 1, 1–14.
CAS Google Scholar
23 Kim J. K., Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-ϕ-nonexpansive mappings, Fixed Point Theory and Applications. (2011) 2011, https://doi.org/10.1186/1687%2D1812%2D2011%2D10.
10.1186/1687-1812-2011-10
Google Scholar
24 Kim J. K., [email protected], Cho S. Y., and [email protected], Qin X., [email protected], Some results on generalized equilibrium problems involving strictly pseudocontractive mappings, Acta Mathematica Scientia. (2011) 31, no. 5, 2041–2057, https://doi.org/10.1016/S0252%2D9602(11)60380%2D9.
10.1016/S0252-9602(11)60380-9
Web of Science® Google Scholar
25 Qin X., Cho Y. J., and Kang S. M., Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications, Nonlinear Analysis. (2010) 72, no. 1, 99–112, https://doi.org/10.1016/j.na.2009.06.042, 2574921, ZBL1225.47106.
10.1016/j.na.2009.06.042
Web of Science® Google Scholar
26 Qin X., Cho Y. J., Kang S. M., and Zhou H., Convergence of a modified Halpern-type iteration algorithm for quasi-ϕ-nonexpansive mappings, Applied Mathematics Letters. (2009) 22, no. 7, 1051–1055, https://doi.org/10.1016/j.aml.2009.01.015, 2522998.
10.1016/j.aml.2009.01.015
Web of Science® Google Scholar
27 Cioranescu I., Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, 1990, 62, Kluwer Academic Publishers, Dordrecht, The Netherlands, Mathematics and Its Applications, 1079061.
10.1007/978-94-009-2121-4
Google Scholar
28 Takahashi W., Nonlinear Functional Analysis, 2000, Yokohama Publishers, Yokohama, Japan, 1864294.
Google Scholar
29 Alber Y. I., Metric and generalized projection operators in Banach spaces: properties and applications, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, 1996, 178, Marcel Dekker, New York, NY, USA, 15–50, 1386667, ZBL0883.47083.
Google Scholar
30 Censor Y. and Reich S., Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization. (1996) 37, no. 4, 323–339, https://doi.org/10.1080/02331939608844225, 1402641, ZBL0883.47063.
10.1080/02331939608844225
Google Scholar
31 Takahashi W. and Zembayashi K., Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Analysis. (2009) 70, no. 1, 45–57, https://doi.org/10.1016/j.na.2007.11.031, 2468217, ZBL1170.47049.
10.1016/j.na.2007.11.031
Web of Science® Google Scholar

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Approximation of Solutions of an Equilibrium Problem in a Banach Space

Abstract

1. Introduction

2. Preliminaries

3. Main Results

References

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Approximation of Solutions of an Equilibrium Problem in a Banach Space

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1. Introduction

2. Preliminaries

3. Main Results

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