Wen Fixed Point Theory and Applications 2014, 2014:232 http://www.fixedpointtheoryandapplications.com/content/2014/1/232 RESEARCH Open Access Weak and strong convergence of hybrid subgradient method for pseudomonotone equilibrium problem and multivalued nonexpansive mappings Dao-Jun Wen* * Correspondence: [email protected] College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China Abstract In this paper, we introduce a hybrid subgradient method for ﬁnding a common element of the set of solutions of a class of pseudomonotone equilibrium problems and the set of ﬁxed points of a ﬁnite family of multivalued nonexpansive mappings in Hilbert space. The proposed method involves only one projection rather than two as in the existing extragradient method and the inexact subgradient method for an equilibrium problem. We establish some weak and strong convergence theorems of the sequences generated by our iterative method under some suitable conditions. Moreover, a numerical example is given to illustrate our algorithm and our results. MSC: 47H05; 47H09; 47H10 Keywords: pseudomonotone equilibrium problem; multivalued nonexpansive mapping; hybrid subgradient method; ﬁxed point; weak and strong convergence 1 Introduction Let H be a real Hilbert space with inner product ·, · and norm · , respectively. Let K be a nonempty closed convex subset of H. Let F : K × K → R be a bifunction, where R denotes the set of real numbers. We consider the following equilibrium problem: Find x ∈ K such that F(x, y) ≥ , ∀y ∈ K. (.) The set of solution of equilibrium problem is denoted by EP(F, K). It is well known that some important problems such as convex programs, variational inequalities, ﬁxed point problems, minimax problems, and Nash equilibrium problem in noncooperative games and others can be reduced to ﬁnding a solution of the equilibrium problem (.); see [–] and the references therein. Recall that a mapping T : K → K is said to be nonexpansive if Tx – Ty ≤ x – y, ∀x, y ∈ K. A subset K ⊂ H is called proximal if for each x ∈ H, there exists an element y ∈ K such that dist(x, K) := x – y = inf x – z : z ∈ K . ©2014 Wen; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Wen Fixed Point Theory and Applications 2014, 2014:232 http://www.fixedpointtheoryandapplications.com/content/2014/1/232 Page 2 of 14 We denote by B(K), C(K), and P(K) the collection of all nonempty closed bounded subsets, nonempty compact subsets and nonempty proximal bounded subsets of K , respectively. The Hausdorﬀ metric H on B(H) is deﬁned by H(K , K ) := max sup dist(x, K ), sup dist(y, K ) , x∈K ∀K , K ∈ B(H). y∈K Let T : H → H be a multivalued mapping, of which the set of ﬁxed points is denoted by Fix(T), i.e., Fix(T) := {x ∈ Tx : x ∈ K}. A multivalued mapping T : K → B(K) is said to be nonexpansive if H(Tx, Ty) ≤ x – y, ∀x, y ∈ K. (.) T is said to be quasi-nonexpansive if, for all p ∈ Fix(T), H(Tx, p) ≤ x – p, ∀x ∈ K. (.) Recently, the problem of ﬁnding a common element of the set of solutions of equilibrium problems and the set of ﬁxed points of nonlinear mappings has become an attractive subject, and various methods have been extensively investigated by many authors. It is worth mentioning that almost all the existing algorithms for this problem are based on the proximal point method applied to the equilibrium problem combining with a Mann iteration to ﬁxed point problems of nonexpansive mappings, of which the convergence analysis has been considered if the bifunction F is monotone. This is because the proximal point method is not valid when the underlying operator F is pseudomonotone. Another basic idea for solving equilibrium problems is the projection method. However, Facchinei and Pang [] show that the projection method is not convergent for monotone inequality, which is a special case of monotone equilibrium problems. In order to obtain convergence of the projection method for equilibrium problems, Tran et al. [] introduced an extragradient method for pseudomonotone equilibrium problems, which is computationally expensive because of the two projections deﬁned onto the constrained set. Eﬀorts for deducing the computational costs in computing the projection have been made by using penalty function methods or relaxing the constrained convex set by polyhedral convex ones; see, e.g., [–]. In , Santos and Scheimberg [] further proposed an inexact subgradient algorithm for solving a wide class of equilibrium problems that requires only one projection rather than two as in the extragradient method, and of which computational results show the eﬃciency of this algorithm in ﬁnite dimensional Euclidean spaces. On the other hand, iterative schemes for multivalued nonexpansive mappings are far less developed than those for nonexpansive mappings though they have more powerful applications in solving optimization problems; see, e.g., [–] and the references therein. In , Eslamian [] considered a proximal point method for nonspreading mappings and multivalued nonexpansive mappings and equilibrium problems. To be more precise, they proposed the following iterative method: F(un , z) + rn y – un , un – xn ≥ , ∀y ∈ K, xn+ = αn un + βn fn un + γn zn , n ≥ , (.) Wen Fixed Point Theory and Applications 2014, 2014:232 http://www.fixedpointtheoryandapplications.com/content/2014/1/232 Page 3 of 14 where Tn = Tn(mod N) , zn ∈ Tn un , αn + βn + γn = for all n ≥ and fi , Ti are ﬁnite families of nonspreading mappings and multivalued nonexpansive mappings for i = , , . . . , N , respectively. Moreover, he further proved the weak and strong convergence theorems of the iterative sequences under the condition of monotone deﬁned on a bifunction F. In this paper, inspired and motivated by research going on in this area, we introduce a hybrid subgradient method for the pseudomonotone equilibrium problem and a ﬁnite family of multivalued nonexpansive mappings, which is deﬁned in the following way: ⎧ ⎪ ⎨ wn ∈ ∂n F(xn , ·)xn , un = PK (xn – γn wn ), γn = max{σβnn,wn } , ⎪ ⎩ xn+ = αn xn + ( – αn )zn , n ≥ , (.) where Tn = Tn(mod N) , zn ∈ Tn un , and {αn }, {βn }, {n }, and {σn } are nonnegative real sequences. Our purpose is not only to modify the proximal point iterative schemes (.) for the equilibrium problem to a hybrid subgradient method for a class of pseudomonotone equilibrium problems and a ﬁnite family of multivalued nonexpansive mappings, but also to establish weak and strong convergence theorems involving only one projection rather than two as in the extragradient method [] and the inexact subgradient method [] for the equilibrium problem. Our theorems presented in this paper improve and extend the corresponding results of [, , , ]. 2 Preliminaries Let K be a nonempty closed convex subset of a real Hilbert space H with inner product ·, · and norm · , respectively. For every point x ∈ H, there exists a unique nearest point in K , denoted by PK (x), such that x – PK (x) ≤ x – y, ∀y ∈ K. Then PK is called the metric projection of H onto K . It is well known that PK is nonexpansive and satisﬁes the following properties: x – PK (x), PK (x) – y ≥ , ∀x ∈ H, y ∈ K, x – y ≥ x – PK (x) + y – PK (x) , ∀x ∈ H, y ∈ K. (.) (.) Recall also that a bifunction F : K × K → R is said to be (i) r-strongly monotone if there exists a number r > such that F(x, y) + F(y, x) ≤ –rx – y , ∀x, y ∈ K; (ii) monotone on K if F(x, y) + F(y, x) ≤ , ∀x, y ∈ K; (iii) pseudomonotone on K with respect to x ∈ K if F(x, y) ≥ ⇒ F(y, x) ≤ , ∀y ∈ K. (.) Wen Fixed Point Theory and Applications 2014, 2014:232 http://www.fixedpointtheoryandapplications.com/content/2014/1/232 Page 4 of 14 It is clear that (i) ⇒ (ii) ⇒ (iii), for every x ∈ K . Moreover, F is said to be pseudomonotone on K with respect to A ⊆ K , if it is pseudomonotone on K with respect to every x ∈ A. When A ≡ K , F is called pseudomonotone on K . The following example, taken from [], shows that a bifunction may not be pseudomonotone on K , but yet is pseudomonotone on K with respect to the solution set of the equilibrium problem deﬁned by F and K : F(x, y) := y|x|(y – x) + xy|y – x|, ∀x, y ∈ R, K := [–, ]. Clearly, EP(F, K) = {}. Since F(y, ) = for every y ∈ K , this bifunction is pseudomonotone on K with respect to the solution x∗ = . However, F is not pseudomonotone on K . In fact, both F(–., .) = . > and F(., –.) = . > . To study the equilibrium problem (.), we may assume that is an open convex set containing K and the bifunction F : × → R satisfy the following assumptions: (C) F(x, x) = for each x ∈ K and F(x, ·) is convex and lower semicontinuous on K ; (C) F(·, y) is weakly upper semicontinuous for each y ∈ K on the open set ; (C) F is pseudomonotone on K with respect to EP(F, K) and satisﬁes the strict paramonotonicity property, i.e., F(y, x) = for x ∈ EP(F, K) and y ∈ K implies y ∈ EP(F, K); (C) if {xn } ⊆ K is bounded and n → as n → ∞, then the sequence {wn } with wn ∈ ∂n F(xn , ·)xn is bounded, where ∂ F(x, ·)x stands for the -subdiﬀerential of the convex function F(x, ·) at x. Throughout this paper, weak and strong convergence of a sequence {xn } in H to x are denoted by xn x and xn → x, respectively. In order to prove our main results, we need the following lemmas. Lemma . [] Let H be a real Hilbert space. For all x, y ∈ H, we have the following identity: x – y = x – y – x – y, y. Lemma . [] Let H be a real Hilbert space and α, β, γ ∈ [, ] with α + β + γ = . For all x, y, z ∈ H, we have the following identity: αx + βy + γ z = αx + βy + γ z – αβx – y – αγ x – z – βγ y – z . Lemma . [] Let {an } and {bn } be two sequences of nonnegative real numbers such that an+ ≤ an + bn , where ∞ n= bn n ≥ , < ∞. Then the sequence {an } is convergent. Lemma . [] Let K be a nonempty closed convex subset of a real Hilbert space H. Let T : K → C(K) be a multivalued nonexpansive mapping. If xn q and limn→∞ dist(xn , Txn ) = , then q ∈ Tq. Wen Fixed Point Theory and Applications 2014, 2014:232 http://www.fixedpointtheoryandapplications.com/content/2014/1/232 Page 5 of 14 3 Weak convergence Theorem . Let K be a nonempty closed convex subset of a Hilbert space H and F : K × : K → C(K) be a ﬁnite family of K → R be a bifunction satisfying (C)-(C). Let {Ti }N i= Fix(T multivalued nonexpansive mappings such that = N i ) ∩ EP(F, K) = φ and Ti (q) = i= {q} for i = , , . . . , N and q ∈ . For a given point x ∈ K , < c < σn < σ , {αn }, {βn }, and {n } are nonnegative sequences satisfying the following conditions: (i) αn ∈ [a, b] ⊂ (, ); ∞ ∞ ∞ (ii) n= βn = ∞, n= βn < ∞, and n= βn n < ∞. Then the sequence {xn } generated by (.) converges weakly to x ∈ . Proof First, we show the existence of limn→∞ xn – p for every p ∈ . It follows from (.) and Lemmas . and . that xn+ – p = αn (xn – p) + ( – αn )(zn – p) = αn xn – p + ( – αn )zn – p – αn ( – αn )xn – zn = αn xn – p + ( – αn ) dist(zn , Tn p) – αn ( – αn )xn – zn ≤ αn xn – p + ( – αn )H(Tn un , Tn p) – αn ( – αn )xn – zn ≤ αn xn – p + ( – αn )un – p – αn ( – αn )xn – zn = αn xn – p + ( – αn ) xn – p – un – xn + xn – un , p – un – αn ( – αn )xn – zn ≤ xn – p + ( – αn )xn – un , p – un – αn ( – αn )xn – zn . (.) By un = PK (xn – γn wn ) and (.), we have xn – un , p – un ≤ γn wn , p – un . Using un = PK (xn – γn wn ) and xn ∈ K again, we obtain (note that γn = (.) βn ) max{σn ,wn } xn – un = xn – un , xn – un ≤ γn wn , xn – un ≤ γn wn xn – un ≤ βn xn – un , which implies that xn – un ≤ βn . Substituting (.) into (.) yields xn+ – p ≤ xn – p + ( – αn )γn wn , p – un – αn ( – αn )xn – zn = xn – p + ( – αn )γn wn , p – xn + ( – αn )γn wn , xn – un – αn ( – αn )xn – zn ≤ xn – p + ( – αn )γn wn , p – xn + ( – αn )γn wn xn – un – αn ( – αn )xn – zn (.) Wen Fixed Point Theory and Applications 2014, 2014:232 http://www.fixedpointtheoryandapplications.com/content/2014/1/232 Page 6 of 14 ≤ xn – p + ( – αn )γn wn , p – xn + ( – αn )βn – αn ( – αn )xn – zn . (.) Since wn ∈ ∂n F(xn , ·)xn and F(x, x) = for all x ∈ K , we have wn , p – xn ≤ F(xn , p) – F(xn , xn ) + n ≤ F(xn , p) + n . (.) On the other hand, since p ∈ EP(F, K), i.e., F(p, x) ≥ for all x ∈ K , by the pseudomonotonicity of F with respect to p, we have F(x, p) ≤ for all x ∈ K . Replacing x by xn ∈ K , we get F(xn , p) ≤ . Then from (.) and (.), it follows that xn+ – p ≤ xn – p + ( – αn )γn F(xn , p) + ( – αn )γn n + ( – αn )βn – αn ( – αn )xn – zn ≤ xn – p + ( – αn )γn n + ( – αn )βn – αn ( – αn )xn – zn ≤ xn – p + ( – αn )γn n + ( – αn )βn . (.) Applying Lemma . to (.), by condition (ii), we obtain the existence of limn→∞ xn – p = d. Now, we claim that lim supn→∞ F(xn , p) = for every p ∈ . Indeed, since F is pseudomonotone on K and F(p, xn ) ≥ , we have –F(xn , p) ≥ . From (.), we have ( – αn )γn –F(xn , p) ≤ xn – p – xn+ – p + ( – αn )γn n + ( – αn )βn . (.) Summing up (.) for every n, we obtain ≤ ∞ ( – αn )γn –F(xn , p) n= ≤ x – p + ∞ γn n + ∞ n= βn < +∞. (.) n= By the assumption (C), we can ﬁnd a real number w such that wn ≤ w for every n. Setting L := max{σ , w}, where σ is a real number such that < σn < σ for every n, it follows from (i) that ∞ ≤ ( – b) βn –F(xn , p) L n= ≤ ∞ ( – αn )γn –F(xn , p) < +∞, n= which implies that ∞ n= βn –F(xn , p) < +∞. (.) Wen Fixed Point Theory and Applications 2014, 2014:232 http://www.fixedpointtheoryandapplications.com/content/2014/1/232 Page 7 of 14 Combining with –F(xn , p) ≥ and ∞ n= βn = ∞, we can deduced that lim supn→∞ F(xn , p) = as desired. Next, we show that any weak subsequential limit of the sequence of {xn } is an element of = N i= Fix(Ti ) ∩ EP(F, K). To do this, suppose that {xni } is a subsequence of {xn }. For simplicity of notation, without loss of generality, we may assume that xni x as i → ∞. By convexity, K is weakly closed and hence x ∈ K . Since F(·, p) is weakly upper semicontinuous for p ∈ , we have F(x, p) ≥ lim sup F(xni , p) = lim F(xni , p) i→∞ i→∞ = lim sup F(xn , p) = . (.) n→∞ By the pseudomonotonicity of F with respect to p and F(p, x) ≥ , we obtain F(x, p) ≤ . Thus F(x, p) = . Moreover, by the assumption (C), we can deduce that x is a solution of EP(F, K). On the other hand, it follows from (.) and condition (ii) that lim xn – un = . (.) n→∞ From (.) and conditions (i)-(ii), we have αn ( – αn )xn – zn ≤ xn – p – xn+ – p + ( – αn )γn n + ( – αn )βn , taking the limit as n → ∞ yields lim xn – zn = , (.) n→∞ and thus lim dist(xn , Tn un ) ≤ lim xn – zn = . n→∞ n→∞ (.) Using (.) again, we have lim xn+ – xn = lim ( – αn )xn – zn = . n→∞ n→∞ (.) It follows that lim xn+i – xn = , n→∞ i = , , . . . , N. (.) Note that un+ – un ≤ un+ – xn+ + xn+ – xn + xn – un . Combining (.) and (.), we obtain lim un+ – un = . n→∞ (.) Wen Fixed Point Theory and Applications 2014, 2014:232 http://www.fixedpointtheoryandapplications.com/content/2014/1/232 Page 8 of 14 This also implies that lim un+i – un = , i = , , . . . , N. n→∞ (.) Observe that dist(un , Tn+i un ) ≤ un – xn + xn – xn+i + dist(xn+i , Tn+i un+i ) + H(Tn+i un+i , Tn+i un ) ≤ un – xn + xn – xn+i + dist(xn+i , Tn+i un+i ) + un+i – un . Together with (.), (.), (.), and (.), we have lim dist(un , Tn+i un ) = , i = , , . . . , N, n→∞ (.) which implies that the sequence N dist(un , Tn+i un ) n≥ → as n → ∞. (.) i= For i = , , . . . , N , we note also that dist(un , Ti un ) n≥ = dist(un , Tn+(i–n) un ) n≥ = dist(un , Tn+in un ) n≥ ⊂ N dist(un , Tn+i un ) n≥ , i= where i – n = in (mod N) and in ∈ {, , . . . , N}. Therefore, we have lim dist(un , Ti un ) = , n→∞ i = , , . . . , N. (.) Similarly, for i = , , . . . , N , we obtain dist(xn , Ti xn ) ≤ xn – un + dist(un , Ti un ) + H(Ti un , Ti xn ) ≤ xn – un + dist(un , Ti un ). It follows from (.) and (.) that lim dist(xn , Ti xn ) = , n→∞ i = , , . . . , N. (.) Applying Lemma . to (.), we can deduce that x ∈ Fix(Ti ) for i = , , . . . , N and hence x ∈ . Finally, we prove that {xn } converges weakly to an element of . Indeed to verify that the claim is valid it is suﬃcient to show that ωw (xn ) is a single point set, where ωw (xn ) = {x ∈ H : xni x} for some subsequence {xni } of {xn }. Indeed since {xn } is bounded and H is reﬂexive, ωw (xn ) is nonempty. Taking w , w ∈ ωw (xn ) arbitrarily, let {xnk } and {xnj } be Wen Fixed Point Theory and Applications 2014, 2014:232 http://www.fixedpointtheoryandapplications.com/content/2014/1/232 Page 9 of 14 subsequences of {xn } such that xnk w and xnj w , respectively. Since limn→∞ xn – p exists for all p ∈ and w , w ∈ , we see that limn→∞ xn – w and limn→∞ xn – w exist. Now let w = w , then by Opial’s property, lim xn – w = lim xnk – w n→∞ k→∞ < lim xnk – w = lim xn – w k→∞ n→∞ = lim xnj – w < lim xnj – w j→∞ j→∞ = lim xn – w , n→∞ which is a contradiction. Therefore, w = w . This shows that ωw (xn ) is a single point set, i.e., xn x. This completes the proof. Theorem . Let K be a nonempty closed convex subset of a Hilbert space H and F : K × K → R be a bifunction satisfying (C)-(C). Let T : K → C(K) be a multivalued nonexpansive mapping such that = Fix(T) ∩ EP(F, K) = φ and T(q) = {q} for all q ∈ . For a given point x ∈ K , < c < σn < σ , let {xn } be deﬁned by ⎧ ⎪ ⎨ wn ∈ ∂n F(xn , ·)xn , un = PK (xn – γn wn ), γn = max{σβnn,wn } , ⎪ ⎩ xn+ = αn xn + ( – αn )zn , n ≥ , where zn ∈ Tun , {αn }, {βn }, and {n } are nonnegative sequences satisfying the following conditions: (i) αn ∈ [a, b] ⊂ (, ); ∞ ∞ ∞ (ii) n= βn = ∞, n= βn < ∞, and n= βn n < ∞. Then the sequence {xn } converges weakly to x ∈ . Proof Putting N = , then Ti = T, a single multivalued nonexpansive mapping, and the conclusion follows immediately from Theorem .. This completes the proof. 4 Strong convergence To obtain strong convergence results, we either add the control condition limn→∞ αn = , or we remove the condition T(q) = {q} for all q ∈ and adjust the nonempty compact subset C(K) to a proximal bounded subset P(K) of K as follows. Theorem . Let K be a nonempty closed convex subset of a Hilbert space H and F : K × K → R be a bifunction satisfying (C)-(C). Let {Ti }N : K → C(K) be a ﬁnite family of i= multivalued nonexpansive mappings such that = N Fix(T i ) ∩ EP(F, K) = φ and Ti (q) = i= {q} for i = , , . . . , N and q ∈ . For a given point x ∈ K , < c < σn < σ , let {xn } be deﬁned by ⎧ ⎪ ⎨ wn ∈ ∂n F(xn , ·)xn , un = PK (xn – γn wn ), γn = max{σβnn,wn } , ⎪ ⎩ xn+ = αn xn + ( – αn )zn , n ≥ , (.) Wen Fixed Point Theory and Applications 2014, 2014:232 http://www.fixedpointtheoryandapplications.com/content/2014/1/232 Page 10 of 14 where Tn = Tn(mod N) , zn ∈ Tn un , {αn }, {βn }, and {n } are nonnegative sequences satisfying the following conditions: (i) αn ∈ [a, b] ⊂ (, ) and limn→∞ αn = ; ∞ ∞ ∞ (ii) n= βn = ∞, n= βn < ∞, and n= βn n < ∞. Then the sequence {xn } generated by (.) converges strongly to x∗ ∈ . Proof By a similar argument to the proof of Theorem . and (.), we have zn – P (xn ) ≤ zn – xn – xn – P (xn ) . (.) It follows from (.) that xn+ – P (xn+ ) ≤ αn xn – P (xn ) + ( – αn ) zn – P (xn ) ≤ αn xn – P (xn ) + ( – αn )zn – P (xn ) ≤ (αn – )xn – P (xn ) + ( – αn )zn – xn . (.) Combining (.), limn→∞ αn = , and the boundedness of the sequence {xn – P (xn )}, we obtain lim xn+ – P (xn+ ) = . (.) n→∞ By the assumptions (C) and (C), the set is convex (see the proof of Theorem in []). For all m > n, we have (P (xm ) + P (xn )) ∈ , and therefore P (xm ) – P (xn ) = xm – P (xm ) + xm – P (xn ) – xm – P (xm ) + P (xn ) ≤ xm – P (xm ) + xm – P (xn ) – xm – P (xm ) = xm – P (xn ) – xm – P (xm ) . (.) Using (.) with p = P (xn ), we have xm – P (xn ) ≤ xm– – P (xn ) + ( – αm– )γm– m– + ( – αm– )β m– ≤ xm– – P (xn ) + ηm– + ηm– ≤ ··· m– ≤ xn – P (xn ) + ηj , (.) j=n where ηj = ( – αj )γj j + ( – αj )βj . It follows from (.) and (.) that m– P (xm ) – P (xn ) ≤ xn – P (xn ) + ηj – xm – P (xm ) . j=n (.) Wen Fixed Point Theory and Applications 2014, 2014:232 http://www.fixedpointtheoryandapplications.com/content/2014/1/232 Page 11 of 14 Together with (.) and m– j=n ηj < ∞, this implies that {P (xn )} is a Cauchy sequence. Hence {P (xn )} strongly converges to some point x∗ ∈ . Moreover, we obtain x∗ = lim P (xni ) = P (x) = x, i→∞ (.) which implies that P (xn ) → x∗ = x ∈ . Then, from (.), (.), and (.), we can con clude that xn → x∗ . This completes the proof. Theorem . Let K be a nonempty closed convex subset of a Hilbert space H and F : K × K → R be a bifunction satisfying (C)-(C). Let {Ti }N i= : K → P(K) be a ﬁnite family of multivalued mappings such that PTi is nonexpansive, where PTi := {y ∈ Ti x : dist(x, Ti x) = x – y} and = N i= Fix(Ti ) ∩ EP(F, K) = φ. For a given point x ∈ K , < c < σn < σ , let {xn } be deﬁned by ⎧ ⎪ ⎨ wn ∈ ∂n F(xn , ·)xn , un = PK (xn – γn wn ), γn = max{σβnn,wn } , ⎪ ⎩ xn+ = αn xn + ( – αn )zn , n ≥ , (.) where Tn = Tn(mod N) , zn ∈ PTn un , {αn }, {βn }, and {n } are nonnegative sequences satisfying the following conditions: (i) αn ∈ [a, b] ⊂ (, ); ∞ ∞ ∞ (ii) n= βn = ∞, n= βn < ∞, and n= βn n < ∞. Then the sequence {xn } converges strongly to x∗ ∈ . Proof Taking p ∈ , then PTn (p) = {p}. By substituting PT instead of T and similar argument as (.) in the proof of Theorem . we obtain lim dist xn , Ti (xn ) ≤ lim dist xn , PTi (xn ) = . n→∞ n→∞ (.) By compactness of K , there exists a subsequence {xnk } of {xn } such that limk→∞ xnk = x∗ , for some x∗ ∈ K . Since PTi is nonexpansive for i = , , . . . , N , we have dist x∗ , Ti x∗ ≤ dist x∗ , PTi x∗ ≤ x∗ – xnk + dist xnk , PTi (xnk ) + H PTi (xnk ), PTi x∗ ≤ x∗ – xnk + dist xnk , PTi (xnk ) . (.) It follows from (.) and (.) that lim dist x∗ , Ti x∗ = , k→∞ (.) ∗ which implies that x∗ ∈ N i= Fix(Ti ). Since {xnk } converges strongly to x and limn→∞ xn – ∗ x exists (as in the proof of Theorem .), we ﬁnd that {xn } converges strongly to x∗ . This completes the proof. In addition, we supply an example and numerical results to illustrate our method and the main results of this paper. Wen Fixed Point Theory and Applications 2014, 2014:232 http://www.fixedpointtheoryandapplications.com/content/2014/1/232 Page 12 of 14 Example . Let H = R and K := [, ] with usual metric. Consider the nonsmooth equilibrium problem deﬁned by the bifunction F(x, y) = xy(y – x) + xy|y – x|, ∀x, y ∈ K. Clearly, F is pseudomonotone on K . Note that F(x, ·) is convex for x ∈ K and ∂F(x, ·)x = [x , x ] by taking n = for all n ∈ N. (i) Let Tx := [ x , x ] deﬁned on K := [, ]. Note that T is a multivalued nonexpansive mapping and Fix(T) ∩ EP(F, K) = {}. Setting N = , σn = , αn = , βn = n , and xn – x∗ ≤ – as stop criteria, we obtain the results of algorithm (.) with diﬀerent initial points in Table . (ii) Let Tx := [–x, –x] deﬁned on [, ∞) → R . Note that T is not nonexpansive but PT x = {–x} is nonexpansive for all x ∈ [, ∞). Indeed, for each u ∈ Tx, u = –ax, ≤ a ≤ , choose v = –ay. Then |u – v| = –ax – (–ay) = a|x – y| ≤ |x – y| = H(Tx, Ty). On the other hand, for any x, we have ∈ [, ∞) and T = {}. It follows that Fix(T) ∩ EP(F, K) = {}. Setting N = , σn = , αn = , βn = n , and xn – x∗ ≤ – as stop criteria, we obtain the results of algorithm (.) with diﬀerent initial points to be found in Table . The computations are performed by Matlab Ra running on a PC Desktop Intel(R) Core(TM) i-M, CPU @. GHz, MHz, . GB, GB RAM. Remark . Our hybrid subgradient method improves the extragradient method of Tran et al. [] and the inexact subgradient algorithm of Santos and Scheimberg [] for an equilibrium problem in deducing the computational costs of an iterative process. Table 1 Numerical results for an initial point x0 = 0.2, 0.5, 0.8 Iter. (n) xn(1) xn(2) xn(3) 0 1 2 3 4 5 6 0.2000 0.1459 0.0816 0.0314 0.0027 0.0000 0.0000 0.5000 0.3618 0.2157 0.1031 0.0351 0.0059 0.0000 0.8000 0.4782 0.2391 0.1206 0.0524 0.0093 0.0001 Table 2 Numerical results for an initial point x0 = 0.2, 0.5, 0.8 xn(1) xn(2) xn(3) 0 1 2 3 4 5 6 .. . 0.2000 0.1683 0.1247 0.0925 0.0621 0.0319 0.0042 .. . 0.5000 0.4136 0.3914 0.2518 0.1492 0.0991 0.0427 .. . 0.8000 0.6839 0.5284 0.3855 0.2679 0.1732 0.1043 .. . 11 12 0.0000 0.0000 0.0035 0.0001 0.0086 0.0001 Iter. (n) Wen Fixed Point Theory and Applications 2014, 2014:232 http://www.fixedpointtheoryandapplications.com/content/2014/1/232 Remark . Our results generalize the results of Eslamian [], a proximal point method for an equilibrium problem, to a hybrid subgradient method for a pseudomonotone equilibrium problem. Competing interests The author declares to have no competing interests. Acknowledgements The author is grateful to the anonymous referees for valuable remarks suggestions which helped him very much in improving this manuscript. This work was supported by the National Science Foundation of China (11471059, 11271388), Basic and Advanced Research Project of Chongqing (cstc2014jcyjA00037) and Science and Technology Research Project of Chongqing Municipal Education Commission (KJ1400618). Received: 23 April 2014 Accepted: 30 October 2014 Published: 17 Nov 2014 References 1. Blum, E, Oettli, W: From optimization and variational inequality to equilibrium problems. Math. 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