RESEARC H Open Access A general composite iterative method for generalized mixed equilibrium problems, variational inequality problems and optimization problems Jong Soo Jung Correspondence: jungjs@mail. donga.ac.kr Department of Mathematics, Dong-A University, Busan, 604-714, Korea Abstract In this article, we introduce a new general composite iterative scheme for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a variational inequality problem for an inverse-strongly monotone mapping in Hilbert spaces. It is shown that the sequence generated by the proposed iterative scheme converges strongly to a common element of the above three sets under suitable control conditions, which solves a certain optimization problem. The results of this article substantially improve, develop, and complement the previous well-known results in this area. 2010 Mathematics Subject Classifications: 49J30; 49J40; 47H09; 47H10 ; 47J20; 47J25; 47J05; 49M05. Keywords: generalized mixed equilibrium problem, fixed point, nonexpansive map- ping; inverse-strongly monotone mapping, variational inequality; optimization pro- blem, metric projection, strongly positive bounded linear operator 1 Introduction Let H be a real Hilbert space with inner product 〈·, ·〉 and induced norm || · ||. Let C beanonemptyclosedconvexsubsetofH and S : C ® C be a sel f-ma pping on C.Let us denote by F(S)thesetoffixedpointsofS and by P C the metric projection of H onto C . Let B : C ® H be a nonlinear mapping and  : C ® ℝ be a function, and Θ be a bifunction of C × C into ℝ, where ℝ is the set of real numbers. Then, we consider the following generalized mixed equilibrium problem of finding x Î C such that  ( x, y ) + Bx, y − x + ϕ ( y ) − ϕ ( x ) ≥ 0, ∀y ∈ C , (1:1) which was recently introduced by Peng and Yao [1]. The set of solutions of the pro- blem (1.1) is denoted by GMEP(Θ, , B). Here, some special cases of the problem (1.1) are stated as follows: Jung Journal of Inequalities and Applications 2011, 2011:51 http://www.journalofinequalitiesandapplications.com/content/2011/1/51 © 2011 Jung; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attrib ution License (http://creat ivecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. If B = 0, then the problem (1.1) reduces the following mixed equilibrium problem of finding x Î C such that  ( x, y ) + ϕ ( y ) − ϕ ( x ) ≥ 0, ∀y ∈ C, (1:2) which was studied by Ceng and Yao [2] (see also [3]). The set of solutions of the problem (1.2) is denoted by MEP (Θ ,). If  =0andB = 0, then the problem (1.1) reduces the following equilibrium pro- blem of finding x Î C such that  ( x, y ) ≥ 0, ∀y ∈ C . (1:3) The set of solutions of the problem (1.3) is denoted by EP(Θ). If  = 0 and Θ(x, y) = 0 for all x, y Î C, then the problem (1.1) reduces the following variational inequality problem of finding x Î C such that Bx, y − x≥0, ∀ y ∈ C. (1:4) The set of solutions of the problem (1.4) is denoted by VI(C, B). The problem (1.1) is very general in the sense that it i ncludes, as special cases, fixed point problems, optimization problems, variational inequality problems, minmax pro- blems, Nash equilibrium problems in noncooperative games, and others; see [2,4-6]. Recently, in order to study the problem (1.3) coupled with the fixed point problem, many authors have introduced some iterative schemes for finding a common element of the set of the solutions of the problem (1.3) and the set of fixed points of a counta- ble family of nonexpansive mappings; see [7-16] and the references therein. In 2008, Su et al. [17] gave an iterative scheme for the problem (1.3), the problem (1.4) for an inverse-strongly monotone mapping, and fixed point problems of non- expansive mappings. In 2009, Yao et al. [18] considered an iterative scheme for the problem (1.2), the problem (1.4) for a Lipschitz and relaxed-cocoercive mapping and fixed point problems of nonexpansive mappings, and in 2008, Peng and Yao [1] stu- died an iterative scheme for the problem (1.1), the problem (1.4) for a monotone, and Lipschitz continuous mapping and fixed point problems of nonexpansive mappings. In particular, in 2010, Jung [9] introduced the following new composite iterative scheme for finding a com mon element of the set of solutions of the problem (1.3) and the set of fixed points of a nonexpansive mapping: x 1 Î C and ⎧ ⎪ ⎨ ⎪ ⎩ (u n , y)+ 1 r n y − u n , u n − x n ≥0, ∀y ∈ C , y n = α n f (x n )+(1− α n )Tu n , x n+1 = ( 1 − β n ) y n + β n Ty n , n ≥ 1, (1:5) where T is a nonexpansive mapping, f is a contra ction with constant k Î (0, 1), {a n }, {b n }⊂ [0, 1], and {r n } ⊂ (0, ∞). He showed that the sequences {x n }and{u n } generated by (1.5) converge strongly to a point in F(T ) ∩ EP (Θ) under suitable conditions. On the other hand, the followin g optimizatio n problem has been studied extensively by many authors: min x∈ μ 2 Ax, x + 1 2  x − u  2 − h(x), Jung Journal of Inequalities and Applications 2011, 2011:51 http://www.journalofinequalitiesandapplications.com/content/2011/1/51 Page 2 of 23 where  =  ∞ n=1 C n , C 1 , C 2 , ·· · are infinitely many closed convex subsets of H such that  ∞ n =1 C n = ∅ , u Î H, μ ≥ 0 is a real number, A is a strongly positive bounded linear operator on H (i.e., there is a constant ¯ γ > 0 such that Ax, x≥ ¯ γ  x  2 , ∀x Î H) and h is a potential function for g f (i.e., h’ (x)=g f(x) for all x Î H). For this kind of optimi- zation problems, see, for examp le, Bauschke and Borwein [19], Combettes [20], Deutsch and Yamada [21], Jung [22], and Xu [23] when  =  N i =1 C i ;andh(x)=〈x, b〉 for a given point b in H. In 2009, Yao et al. [3] considered the following iterative scheme for the problem (1.2) and optimization problems: ⎧ ⎨ ⎩ (y n , y)+ϕ(y ) − ϕ(y n )+ 1 r K  (y n ) − K  (x n ), y − y n ≥0, ∀y ∈ H, x n+1 = α n (u + γ f (x n )) + β n x n + ((1 − β n )I − α n (I + μA))W n y n , n ≥ 1 , (1:6) where u Î H;{a n }and{b n }aretwosequencesin(0,1),μ >0,r>0, g >0; K’(x)isthe Fréchet derivative of a functional K : H ® ℝ at x;andW n is the so-called W-mapping related to a sequence {T n } of nonexpansive mappings. They showed that under appro- priate conditions, the sequences {x n } and {y n } generated by (1.6) converge strongly to a solution of the optimization problem: min x∈  ∞ n =1 F(T n ) ∩ MEP(,ϕ) μ 2 Ax, x + 1 2 ||x − u|| 2 − h(x) . In 2010, using the method of Yao et al. [3], Jaiboon and Kumam [24] also introduced a general iterative method for finding a common element of the set of solutions of the problem (1.2), the set of fixed points of a sequence {T n } of nonexpansive mappings, and the set of solutions of the problem (1.4) for a a-inverse-strongly monotone map- ping. We point out that in the main result s of [3,24], the condition of the sequentially continuity from the weak topology to the strong topology for the derivative K’ of the function K : C ® ℝ is very strong. E ven if K(x)=  x  2 2 ,thenK’ (x)=x is not sequen- tially continuous from the weak topology to the strong topology. In this article, inspired and motivated by above mentioned results, we introduce a new iterative method for finding a common element of the set of solutions of a gener- alized mixed equilibrium problem (1.1), the set of fixed points of a countable family of nonexpansive mappings, and the set of solutions of the variational inequality probl em (1.4) for an inverse-strongly monotone mapping in a Hilbert space. We show that under suitable conditions, the sequence generated by the proposed iterative scheme converges strongly to a common element of the above three sets, which is a solution of a certain optimization problem. The results of this article can be viewed as an improvement and complement of the recent results in this direction. 2 Preliminaries and lemmas Let H be a real Hilbert space, and let C beanonemptyclosedconvexsubsetofH.In thefollowing,wewritex n ⇀ x to indicate that the sequence {x n }convergesweaklyto x. x n ® x implies that {x n } converges strongly to x. Jung Journal of Inequalities and Applications 2011, 2011:51 http://www.journalofinequalitiesandapplications.com/content/2011/1/51 Page 3 of 23 First, we know that a mapping f : C ® C is a contraction on C if there exists a co n- stant k Î (0, 1) such that ||f(x)-f(y)|| ≤ k||x - y||, x, y Î C.AmappingT : C ® C is called nonexpansive if ||Tx - Ty|| ≤ ||x - y||, x, y Î C . In a real Hilbert space H, we have   λx +(1− λ)y   2 = λ  x  2 +(1− λ)   y   2 − λ(1 − λ)   x − y   2 for all x, y Î H and l Î ℝ . For every point x Î H, there exists the unique nearest point in C, denoted by P C (x), such that | |x − P C ( x ) || ≤ ||x − y| | for all y Î C. P C is called the metric projection of H onto C. It is well known that P C is nonexpansive and P C satisfies x − y, P C (x) − P C (y)≥   P C (x) − P C (y)   2 (2:1) for every x, y Î H. Moreover, P C (x) is characterized by the properties:   x − y   2 ≥   x − P C (x)   2 +   y − P C (x)   2 and u = P C ( x ) ⇔x − u, u − y ≥0forallx ∈ H, y ∈ C . In the context of the variational inequality problem fo r a nonlinear mapping F,this implies that u ∈ VI ( C, F ) ⇔ u = P C ( u − λFu ) ,foranyλ>0 . (2:2) It is also well known that H satisfies the Opial condition, that is, for any sequence {x n } with x n ⇀ x, the inequality lim inf n →∞  x n − x  < lim inf n →∞   x n − y   holds for every y Î H with y ≠ x. A mapping F of C into H is called a-inve rse-s trongly monotone if there exists a con- stant a >0 such that x − y, Fx − Fy≥α   Fx − Fy   2 , ∀x, y ∈ C . We know that if F = I - T, where T is a nonexpansive mapping of C into itself and I is the identity mapping of H,thenF is 1 2 -inverse-strongly monotone a nd VI(C, F )= F(T ). A mapping F of C into H is called strongly monotone if there exists a positive real number h such that x − y, Fx − Fy≥η   x − y   2 , ∀x, y ∈ C . In such a case, we say F is h-strongly monotone. If F is h-strongly monotone and -Lipschitz continuous,thatis,||Fx - Fy|| ≤ ||x - y|| for all x, y Î C,thenF is η κ 2 -inverse-strongly monotone. If F is an a-inverse-strongly monotone mapping of C into H, then it is obvious that F is 1 α -Lipschi tz continuous. We also have that for all x, y Î C and l >0, Jung Journal of Inequalities and Applications 2011, 2011:51 http://www.journalofinequalitiesandapplications.com/content/2011/1/51 Page 4 of 23   (I − λF)x − (I − λF)y   2 =   (x − y) − λ(Fx − Fy)   2 =   x − y   2 − 2λx − y, Fx − Fy + λ 2   Fx − Fy   2 ≤   x − y   2 + λ(λ − 2α)   Fx − Fy   2 . Hence, if l ≤ 2a,thenI - lF is a nonexpansive mapping of C into H. The following result for the existence of solutions of the variational inequality problem for inverse- strongly monotone mappings was given in Takahashi and Toyoda [25]. Proposition Let C be a bounded closed convex subset of a real Hilbert space, and F be an a-inverse-strongly monotone mapping of C into H. Then, V I(C, F) is nonempty. A set-valued mapping Q : H ® 2 H is called monotone if for all x, y Î H, f Î Qx and g Î Qy imply 〈x - y, f - g 〉 ≥ 0. A monotone mapping Q : H ® 2 H is maximal if the graph G(Q)ofQ is not properly contained in the graph of any other monotone map- ping. It is known that a monotone mapping Q is maximal if and only if for (x, f ) Î H × H, 〈x - y, f - g〉 ≥ 0forevery(y, g) Î G(Q) implies f Î Qx.LetF be an inverse- strongly monotone mapping of C into H,andletN C v be the normal cone to C at v, that is, N C v ={w Î H : 〈v - u, w〉 ≥ 0, for all u Î C}, and define Qv =  Fv + N C v, v ∈ C ∅ v ∈ C . Then, Q is maximal monotone and 0 Î Qv if and only if v Î VI(C, F ); see [26,27]. For solving the equilib rium problem for a bifunction Θ : C × C ® ℝ, let us assume that Θ and  satisfy the following conditions: (A1) Θ(x, x) = 0 for all x Î C; (A2) Θ is monotone, that is, Θ(x, y)+Θ (y, x) ≤ 0 for all x, y Î C; (A3) for each x, y, z Î C, lim t ↓ 0 (tz +(1− t)x, y) ≤ (x, y) ; (A4) for each x Î C, y a Θ (x, y) is convex and lower semicontinuous; (A5) For each y Î C, x a Θ (x, y) is weakly upper semicontinuous; (B1) For each x Î H and r>0, there exists a bounded subset D x ⊆ C and y x Î C such that for any z Î C \D x , (z, y x )+ϕ(y x ) − ϕ(z)+ 1 r y x − z, z − x < 0 ; (B2) C is a bounded set; The following lemmas were given in [1,4]. Lemma 2.1 ([4]) Let C be a nonempty closed convex subset of H, and Θ be a bifunc- tion of C × Cintoℝ satisfying (A1)-(A4). Let r >0 and x Î H. Then, there exists z Î C such that (z, y)+ 1 r y − z, z − x≥0, ∀y ∈ C . Lemma 2.2 ([1]) LetCbeanonemptyclosedconvexsubsetofH.LetΘ be a bifunc- tion form C × Ctoℝ satisfying (A1)-(A5) and  : C ® ℝ be a proper lower semicontin- uous and convex function. For r >0 and x Î H, define a mapping S r : H ® C as follows: Jung Journal of Inequalities and Applications 2011, 2011:51 http://www.journalofinequalitiesandapplications.com/content/2011/1/51 Page 5 of 23 S r (x)={z ∈ C : (z, y)+ϕ(y) − ϕ(z)+ 1 r y − z, z − x≥0, ∀y ∈ C } for all z Î H. Assume that either (B1) or (B2) holds. Then, the following hold: (1) For each x Î H, S r (x) ≠ ∅; (2) S r is single-valued; (3) S r is firmly nonexpansive, that is, for any x, y Î H, ||S r x − S r y || 2 ≤S r x − S r y , x − y  ; (4) F(S r )=MEP (Θ, ); (5) MEP (Θ, ) is closed and convex. We also need the following lemmas for the proof of our main results. Lemma 2.3 ([23]) Let {s n } be a sequence of non-negative real numbers satisfying s n+1 ≤ ( 1 − λ n ) s n + β n , n ≥ 1 , where { l n } and {b n } satisfy the following conditions: (i) {l n } ⊂ [0, 1] and  ∞ n =1 λ n = ∞ or, equivalently,  ∞ n =1 (1 − λ n )= 0 , (ii) lim sup n→∞ β n λ n ≤ 0 or  ∞ n =1 |β n | < ∞ , Then, lim n®∞ s n =0. Lemma 2.4 In a Hilbert space, there holds the inequality | |x + y || 2 ≤||x|| 2 +2 y , x + y , ∀x, y ∈ H . Lemma 2.5 (Aoyamaetal.[28])Let C be a nonemp ty closed convex subset of H and {T n } be a sequence of nonexpansive mappings of C into itself. Suppose that ∞  n =1 sup{||T n+1 z − T n z|| : z ∈ C} < ∞ . Then , for each y Î C,{T n y} converges strongly to some point of C. Moreo ver, let T be a mapping of C into itself defined by Ty = lim n®∞ T n y for all y Î C. Then,lim n®∞ sup {||Tz - T n z|| : z Î C}=0. The following lemma can be found in [3](see also Lemma 2.1 in [22]). Lemma 2.6 Let C be a nonempty closed convex subset of a real Hilbert space H and g : C ® ℝ ∪{∞} be a proper lower semicontinuous differentiable convex function. If x* is a solution to the minimization problem g (x ∗ )=inf x ∈ C g(x) , then g  ( x ) , x − x ∗ ≥0, x ∈ C . In particular, if x* solves the optimization problem min x∈C μ 2 Ax, x + 1 2 ||x − u|| 2 − h(x) , Jung Journal of Inequalities and Applications 2011, 2011:51 http://www.journalofinequalitiesandapplications.com/content/2011/1/51 Page 6 of 23 then u + ( γ f − ( I + μA )) x ∗ , x − x ∗ ≤0, x ∈ C , where h is a potential function for g f. 3 Main results In this sect ion, we introduce a new composite iterati ve scheme for finding a common point of the set of solutions of th e problem (1.1), the set of fixed points of a countable fam ily of nonexpansive mappings, and the set of solut ions of the problem (1.4) for an inverse-strongly monotone mapping. Theorem 3.1 Let C be a nonempty closed convex sub set of a real Hilbert space H such that C ± C ⊂ C. Let Θ be a bifunction from C × Ctoℝ satisfying (A1)-(A5) and  : C ® ℝ be a lower semicontinuous and convex function. Let F, B be two a, b-inverse-strongly monotone mappings of C into H, respectively. Let {T n } be a sequence of nonexpansive mappings of C into itself such that  1 :=  ∞ n =1 F( T n ) ∩ VI(C, F) ∩ GMEP(, ϕ, B) = ∅ . Let μ >0and g >0be real numbers. Let f be a contraction of C into itself with constant k Î (0, 1) and A be a strongly positive bounded linearoperatoronCwithconstant ¯γ ∈ ( 0, 1 ) such that 0 <γ < (1+μ) ¯γ k . Assume that either (B1) or (B2) holds. Let u Î C, and let {x n },{y n }, and {u n } be sequences generated by x 1 Î C and ⎧ ⎪ ⎨ ⎪ ⎩ (u n , y)+Bx n , y − u n  + ϕ(y) − ϕ(u n )+ 1 r n y − u n , u n − x n ≥0, ∀y ∈ C, y n = α n (u + γ f (x n )) + (I − α n (I + μA))T n P C (u n − λ n Fu n ), x n+1 = ( 1 − β n ) y n + β n T n P C ( y n − λ n Fy n ) , n ≥ 1, (IS ) where { a n }, {b n } ⊂ [0, 1], l n Î [a, b] ⊂ (0, 2a) and r n Î [c, d] ⊂ (0, 2b). Let {a n }, {l n } and { b n } satisfy the following conditions: (C1) a n ® 0(n ® ∞);  ∞ n =1 α n = ∞ ; (C2) b n ⊂ [0, a) for all n ≥ 0 and for some a Î (0, 1); (C3)  ∞ n =1 |α n+1 − α n | < ∞ ,  ∞ n =1 |β n+1 − β n | < ∞ ,  ∞ n =1 |λ n+1 − λ n | < ∞ ,  ∞ n =1 |r n+1 − r n | < ∞ . Suppose that  ∞ n =1 sup{||T n+1 z − T n z|| : z ∈ D} < ∞ for any bounded subset D of C. Let T b e a mapping of C into itself defined by Tz = lim n®∞ T n zforallzÎ Candsup- pose that F( T)=  ∞ n =1 F( T n ) . Then {x n } and {u n } converge strongly to q Î Ω 1 , which is a solution of the optimization problem: min x∈ 1 μ 2 Ax, x + 1 2 ||x − u|| 2 − h(x), (OP1 ) where h is a potential function for g f. Proof First, from a n ® 0(n ® ∞) in the condi tion (C1), we assume, without loss of generality, that a n ≤ (1 + μ||A ||) -1 and 2 (( 1+μ ) ¯γ − γ k ) α n < 1 for n ≥ 1. We know that if A is bounded linear self-adjoint operator on H, then | |A|| =su p {|Au, u| : u ∈ H, ||u|| =1} . Jung Journal of Inequalities and Applications 2011, 2011:51 http://www.journalofinequalitiesandapplications.com/content/2011/1/51 Page 7 of 23 Observe that (I − α n (I + μA))u, u =1− α n − α n μAu, u  ≥ 1 − α n − α n μ||A|| ≥ 0, which is to say I - a n (I + μA) is positive. It follows that ||I − α n (I + μA)|| =sup{(I − α n (I + μA))u, u : u ∈ H, ||u|| =1 } =sup{1 − α n − α n μAu, u : u ∈ H, ||u|| =1} ≤ 1 − α n (1 + μ ¯γ ) < 1 − α n ( 1+μ ) ¯γ . Let us divide the proof into several step s. From now on, we put z n = P C (u n -l n Fu n ) and w n = P C (y n - l n Fy n ). Step 1: We show that {x n }isbounded.Tothisend,let p ∈  1 :=  ∞ n =1 F( T n ) ∩ VI(C, F) ∩ GMEP(, ϕ, B ) and {S r n } be a sequence of map- pings defined as in Lemma 2.2. Then p = T n p = S r n (p − r n Bp ) and p = P C (p - l n Fp) from (2.2). From z n = P C (u n - l n Fu n )andthefactthatP C and I - l n F are nonexpan- sive, it follows that | |z n − p|| ≤ || ( I − λ n F ) u n − ( I − λ n F ) p|| ≤ ||u n − p|| . Also, by u n = S r n ( x n − r n Bx n ) ∈ C and the b-inverse-strongly monotonicity of B,we have with r n Î (0, 2b), | |u n − p|| 2 ≤||x n − r n Bx n − (p − r n Bp)|| 2 ≤||x n − p|| 2 − 2r n x n − p, Bx n − Bp + r 2 n ||Bx n − Bp|| 2 ≤||x n − p|| 2 + r n (r n − 2β)β||Bx n − Bp|| 2 ≤||x n − p || 2 , that is, ||u n - p|| ≤ ||x n - p||, and so || z n − p|| ≤ || x n − p||. (3:1) Similarly, we have || w n − p|| ≤ ||y n − p||. (3:2) Now, set ¯ A = ( I + μA ) . Then, from (IS) and (3.1), we obtain ||y n − p|| ≤ (1 − (1 + μ ¯γ )α n )||z n − p|| + α n ||u|| + α n γ ||f (x n ) − f (p)|| + α n ||γ f (p) − ¯ Ap|| ≤ (1 − (1 + μ ¯γ )α n )||z n − p|| + α n ||u|| + α n γ k||x n − p|| + α n ||γ f (p) − ¯ Ap|| ≤ (1 − ((1 + μ) ¯γ − γ k)α n )||x n − p|| + α n (||γ f (p) − ¯ Ap|| + ||u||) =(1− ((1 + μ) ¯γ − γ k)α n )||x n − p|| + ((1 + μ) ¯γ − γ k)α n ||γ f (p) − ¯ Ap|| + ||u|| ( 1+μ ) ¯γ − γ k . (3:3) Jung Journal of Inequalities and Applications 2011, 2011:51 http://www.journalofinequalitiesandapplications.com/content/2011/1/51 Page 8 of 23 From (3.2) and (3.3), it follows that | |x n+1 − p|| ≤ (1 − β n )||y n − p|| + β n ||w n − p|| ≤ (1 − β n )||y n − p|| + β n ||y n − p|| = ||y n − p|| ≤ max  ||x n − p||, ||γ f (p) − ¯ Ap|| + ||u|| (1 + μ) ¯γ − γ k  . (3:4) By induction, it follows from (3.4) that ||x n − p|| ≤ max  ||x 1 − p||, ||γ f (p) − ¯ Ap|| + ||u|| (1 + μ) ¯γ − γ k  , n ≥ 1 . Therefore, {x n } is bounded. Hence {u n }, {y n }, {z n }, {w n }, {f(x n )}, {Fu n }, {Fy n }, and { ¯ AT n z n } are bounded. Moreover, since ||T n z n - p|| ≤ ||x n - p|| and ||T n w n - p|| ≤ ||y n - p||, {T n z n }and{T n w n } are also bounded, and since a n ® 0 in the condition (C1), we have ||y n − T n z n || = α n || ( u + γ f ( x n )) − ¯ AT n z n || → 0 ( as n →∞ ). (3:5) Step 2: We show that lim n®∞ ||x n+1 - x n || = 0. Indeed, since I - l n F and P C are non- expansive, we have | |z n − z n−1 || = ||P C (u n − λ n Fu n ) − P C (u n−1 − λ n−1 Fu n−1 )|| ≤||(I − λ n F) u n − (I − λ n F) u n−1 +(λ n − λ n−1 )Fu n−1 | | ≤ || u n − u n−1 || + | λ n − λ n−1 ||| Fu n−1 || . (3:6) Similarly, we get || w n − w n−1 || ≤ ||y n − y n−1 || + | λ n − λ n−1 ||| F y n−1 ||. (3:7) On the other hand, from u n−1 = S r n −1 (x n−1 − r n Bx n−1 ) and u n = S r n (x n − r n Bx n ) ,itfol- lows that (u n−1 , y)+Bx n−1 , y − u n−1  + ϕ(y) − ϕ(u n−1 )+ 1 r n −1 y − u n−1 , u n−1 − x n−1 ≥0, ∀y ∈ C , (3:8) and (u n , y)+Bx n , y − u n  + ϕ(y) − ϕ(u n )+ 1 r n y − u n , u n − x n ≥0, ∀y ∈ C . (3:9) Substituting y = u n into (3.8) and y = u n -1 into (3.9), we obtain (u n−1 , u n )+Bx n−1 , u n −u n−1 +ϕ(u n )−ϕ(u n−1 )+ 1 r n −1 u n −u n−1 , u n−1 −x n−1 ≥ 0 and (u n , u n−1 )+Bx n , u n−1 − u n  + ϕ(u n−1 ) − ϕ(u n )+ 1 r n u n−1 − u n , u n − x n ≥0 . Jung Journal of Inequalities and Applications 2011, 2011:51 http://www.journalofinequalitiesandapplications.com/content/2011/1/51 Page 9 of 23 From (A2), we have u n − u n−1 , Bx n−1 − Bx n + u n−1 − x n−1 r n −1 − u n − x n r n ≥0 , and then u n − u n−1 , r n−1 (Bx n−1 − Bx n )+u n−1 − x n−1 − r n−1 r n (u n − x n )≥0 . Hence, it follows that u n − u n−1 ,(I−r n−l B)x n − (I − r n−1 B)x n−1 + u n−1 − u n + u n − x n − r n−1 r n (u n − x n )≥0 . (3:10) Without loss of generality, let us assume that there exists a real number c such that r n >c >0foralln ≥ 1. Then, by (3.10) and the fact that (I - r n-1 B) is nonex pansive, we have ||u n − u n−1 || 2 ≤u n − u n−1 ,(I − r n−1 B)x n − (I − r n−1 B)x n−1 +(1− r n−1 r n )(u n − x n ) ≤||u n − u n−1 ||  ||(I − r n−1 B)x n − (I − r n−1 B)x n−1 || +     1 − r n−1 r n     ||u n − x n ||  ≤||u n − u n−1 ||  ||x n − x n−1 || +     1 − r n−1 r n     ||u n − x n ||  , which implies that | |u n − u n−1 || ≤ ||x n − x n−1 || + 1 r n |r n − r n−1 |||u n − x n | | ≤||x n − x n−1 || + M 1 c |r n − r n−1 |, (3:11) where M 1 = sup {||u n - x n || : n ≥ 1}. Substituting (3.11) into (3.6), we have | |z n − z n−1 || ≤ ||x n − x n−1 || + M 1 c |r n − r n−1 | + |λ n − λ n−1 |||Fu n−1 || . (3:12) Simple calculations show that y n − y n−1 =(α n − α n−1 )(u + γ f (x n−1 ) − ¯ AT n−1 z n−1 )+α n γ (f (x n ) − f( x n−1 ) ) + ( I − α n ¯ A )( T n z n − T n z n−1 ) + ( I − α n ¯ A )( T n z n−1 − T n−1 z n−1 ) . Jung Journal of Inequalities and Applications 2011, 2011:51 http://www.journalofinequalitiesandapplications.com/content/2011/1/51 Page 10 of 23 [...]... gradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems Taiwan J Math 12, 1401–1432 (2008) 2 Ceng, L-C, Yao, J-C: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems J Comput Appl Math 214, 186–201 (2008) doi:10.1016/j.cam.2007.02.022 3 Yao, YH, Noor, MA, Zainab, S, Liou, Y-C: Mixed equilibrium problems and optimization. .. convergence of composite iterative methods for equilibrium problems and fixed point problems Appl Math Comput 213, 498–505 (2009) doi:10.1016/j.amc.2009.03.048 10 Plubtieng, S, Punpaeng, R: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces J Math Anal Appl 336, 445–468 (2007) 11 Plubtieng, S, Punpaeng, R: A new iterative method for equilibrium problems and fixed... doi:10.1016/j.na.2008.02.042 doi:10.1186/1029-242X-2011-51 Cite this article as: Jung: A general composite iterative method for generalized mixed equilibrium problems, variational inequality problems and optimization problems Journal of Inequalities and Applications 2011 2011:51 Submit your manuscript to a journal and benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on... Liou, Y-C: On iterative methods for equilibrium problems Nonlinear Anal 70, 497–509 (2009) doi:10.1016/j.na.2007.12.021 17 Su, Y, Shang, M, Qin, X: An iterative method of solution for equilibrium problems and optimization problems Nonlinear Anal 69, 2709–2719 (2008) doi:10.1016/j.na.2007.08.045 Jung Journal of Inequalities and Applications 2011, 2011:51 http://www.journalofinequalitiesandapplications.com/content/2011/1/51... Chang, S-S, Joseph Lee, HW, Chan, CK: A new method for solving equilibrium problem, fixed point problem and variational inequality problem with application to optimization Nonlinear Anal 70, 3307–3319 (2009) doi:10.1016/j na.2008.04.035 8 Colao, V, Marino, G, Xu, HK: An iterative method for finding common solutions of equilibrium problems and fixed point problems J Math Anal Appl 344, 340–352 (2008)... R: An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems Nonlinear Anal 71, 3363–3373 (2009) doi:10.1016/j.na.2009.01.236 19 Bauschke, HH, Borwein, JM: On projection algorithms for solving convex feasibility problems SIAM Rev 38, 367–426 (1997) 20 Combettes, PL: Hilbertian convex feasibility problem: convergence of projection methods... Funct Anal Optim 19, 33–56 (1998) 22 Jung, JS: Iterative algorithms with some control conditions for quadratic optimizations Panamer Math J 16(4), 13–25 (2006) 23 Xu, HK: An iterative approach to quadratic optimization J Optim Theory Appl 116, 659–678 (203) 24 Jaiboon, C, Kumam, P: A general iterative method for addressing mixed equilibrium problems and optimizations Nonlinear Anal 73, 1180–1202 (2010)... point problems in Hilbert spaces J Math Anal Appl 331, 506–515 (2007) doi:10.1016/j.jmaa.2006.08.036 14 Wang, S, Hu, C, Chai, G: Strong convergence of a new composite iterative method for equilibrium problems and fixed point problems Appl Math Comput 215, 3891–3898 (2010) doi:10.1016/j.amc.2009.11.036 15 Yao, Y, Liou, YC, Yao, J-C: Convergence theorem for equilibrium problems and fixed point problems. .. point problems of nonlinear mappings and monotone mappings Appl Math Comput 197, 548–558 (2008) doi:10.1016/j.amc.2007.07.075 12 Tada, A, Takahashi, W: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem J Optim Theory Appl 133, 359–370 (2007) doi:10.1007/s10957-007-9187-z 13 Takahashi, A, Takahashi, W: Viscosity approximation methods for equilibrium problems and. .. (B1), we can also establish the new corresponding results for the above mentioned problems: (B2) C is a bounded set; (B3) For each x Î H and r >0, there exist a bounded subset Dx ⊆ C and yx Î C such that for any z Î C\Dx, ϕ(yx ) − ϕ(z) + 1 yx − z, z − x < 0; r (B4) For each x Î H and r >0, there exist a bounded subset Dx ⊆ C and yx Î C such that for any z Î C\Dx, (z, yx ) + 1 yx − z, z − x < 0 r Acknowledgements . RESEARC H Open Access A general composite iterative method for generalized mixed equilibrium problems, variational inequality problems and optimization problems Jong Soo Jung Correspondence:. (1.1) is very general in the sense that it i ncludes, as special cases, fixed point problems, optimization problems, variational inequality problems, minmax pro- blems, Nash equilibrium problems in. H) and h is a potential function for g f (i.e., h’ (x)=g f(x) for all x Î H). For this kind of optimi- zation problems, see, for examp le, Bauschke and Borwein [19], Combettes [20], Deutsch and