On a finite family of variational inclusions
Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 RESEARCH Open Access On a finite family of variational inclusions with the constraints of generalized mixed equilibrium and fixed point problems Lu-Chuan Ceng1,2 , Chi-Ming Chen3 and Chin-Tzong Pang4,5* * Correspondence: [email protected] 4 Department of Information Management, Yuan Ze University, Chung-Li, 32003, Taiwan 5 Innovation Center for Big Data and Digital Convergence, Yuan Ze University, Chung-Li, 32003, Taiwan Full list of author information is available at the end of the article Abstract In this paper, we introduce two iterative algorithms for finding common solutions of a finite family of variational inclusions for maximal monotone and inverse-strongly monotone mappings with the constraints of two problems: a generalized mixed equilibrium problem and a common fixed point problem of an infinite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under suitable conditions. MSC: Primary 49J30; 47H09; secondary 47J20; 49M05 Keywords: generalized mixed equilibrium; variational inclusion; nonexpansive mapping; asymptotically strict pseudocontractive mapping in the intermediate sense; maximal monotone mapping; inverse-strongly monotone mapping 1 Introduction Let H be a real Hilbert space with inner product ·, · and norm · , C be a nonempty closed convex subset of H and PC be the metric projection of H onto C. Let S : C → H be a nonlinear mapping on C. We denote by Fix(S) the set of fixed points of S and by R the set of all real numbers. A mapping V is called strongly positive on H if there exists a constant γ¯ ∈ (, ] such that Vx, x ≥ γ¯ x , ∀x ∈ H. A mapping S : C → H is called L-Lipschitz-continuous if there exists a constant L > such that Sx – Sy ≤ Lx – y, ∀x, y ∈ C. In particular, if L = then S is called a nonexpansive mapping; if L ∈ (, ) then A is called a contraction. Let ϕ : C → R be a real-valued function, A : H → H be a nonlinear mapping and Θ : C × C → R be a bifunction. We consider the generalized mixed equilibrium problem (GMEP) ©2014 Ceng et al.; 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. Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 2 of 38 [] of finding x ∈ C such that Θ(x, y) + ϕ(y) – ϕ(x) + Ax, y – x ≥ , ∀y ∈ C. (.) We denote the set of solutions of GMEP (.) by GMEP(Θ, ϕ, A). The GMEP (.) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others. The GMEP is further considered and studied in, e.g., [–]. Throughout this paper, it is assumed as in [] that Θ : C × C → R is a bifunction satisfying conditions (H)-(H) and ϕ : C → R is a lower semicontinuous and convex function with restriction (H), where (H) Θ(x, x) = for all x ∈ C; (H) Θ is monotone, i.e., Θ(x, y) + Θ(y, x) ≤ for any x, y ∈ C; (H) Θ is upper-hemicontinuous, i.e., for each x, y, z ∈ C, lim sup Θ tz + ( – t)x, y ≤ Θ(x, y); t→+ (H) Θ(x, ·) is convex and lower semicontinuous for each x ∈ C; (H) for each x ∈ H and r > , there exist a bounded subset Dx ⊂ C and yx ∈ C such that for any z ∈ C \ Dx , Θ(z, yx ) + ϕ(yx ) – ϕ(z) + yx – z, z – x < . r Let Θ , Θ : C × C → R be two bifunctions, and B , B : C → H be two nonlinear mappings. Consider the system of generalized equilibrium problems (SGEP): find (x∗ , y∗ ) ∈ C × C such that Θ (x∗ , x) + B y∗ , x – x∗ + ∗ ∗ ∗ Θ (y , y) + B x , y – y + x∗ μ y∗ μ – y∗ , x – x∗ ≥ , ∗ ∗ – x , y – y ≥ , ∀x ∈ C, ∀y ∈ C, (.) where μ and μ are two positive constants. ∞ Let {Tn }∞ n= be an infinite family of nonexpansive self-mappings on C and {λn }n= be a sequence of nonnegative numbers in [, ]. For any n ≥ , define a self-mapping Wn on H as follows: ⎧ Un,n+ = I, ⎪ ⎪ ⎪ ⎪ ⎪ Un,n = λn Tn Un,n+ + ( – λn )I, ⎪ ⎪ ⎪ ⎪ ⎪ U n,n– = λn– Tn– Un,n + ( – λn– )I, ⎪ ⎪ ⎪ ⎪ . ⎪ . ⎪ ⎪ ⎨. Un,k = λk Tk Un,k+ + ( – λk )I, ⎪ ⎪ ⎪Un,k– = λk– Tk– Un,k + ( – λk– )I, ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ Un, = λ T Un, + ( – λ )I, ⎪ ⎪ ⎩ Wn = Un, = λ T Un, + ( – λ )I. (.) Such a mapping Wn is called the W -mapping generated by Tn , Tn– , . . . , T and λn , λn– , . . . , λ . Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 3 of 38 Let f : H → H be a contraction and V be a strongly positive bounded linear operator on H. Assume that ϕ : H → R is a lower semicontinuous and convex functional, that Θ, Θ , Θ : H × H → R satisfy conditions (H)-(H), and that A, B , B : H → H are inverse-strongly monotone. Very recently, motivated by Yao et al. [], Cai and Bu [] introduced the following hybrid extragradient-like iterative algorithm: ⎧ (Θ,ϕ) ⎪ ⎨zn = Srn (xn – rn Axn ), yn = TμΘ (I – μ B )TμΘ (I – μ B )zn , ⎪ ⎩ xn+ = αn (u + γ f (xn )) + βn xn + (( – βn )I – αn (I + μV ))Wn yn , (.) ∀n ≥ , for finding a common solution of GMEP (.), SGEP (.), and the fixed point problem of an infinite family of nonexpansive mappings {Ti }∞ i= on H, where {rn } ⊂ (, ∞), {αn }, {βn } ⊂ (, ), and x , u ∈ H are given. The authors proved the strong convergence of the se quence generated by the hybrid iterative algorithm (.) to a point x∗ ∈ ( ∞ i= Fix(Ti )) ∩ GMEP(Θ, ϕ, A) ∩ SGEP(G) under some suitable conditions, where SGEP(G) is the fixed point set of the mapping G := TμΘ (I – μ B )TμΘ (I – μ B ). This point x∗ also solves the following optimization problem: x∈( μ Vx, x + x – u – h(x), n= Fix(Tn ))∩GMEP(Θ,ϕ,A)∩SGEP(G) ∞ min (OP) where h : H → R is the potential function of γ f . Let B be a single-valued mapping of C into H and R be a set-valued mapping with D(R) = C. Consider the following variational inclusion: find a point x ∈ C such that ∈ Bx + Rx. (.) We denote by I(B, R) the solution set of the variational inclusion (.). In particular, if B = R = , then I(B, R) = C. If B = , then problem (.) becomes the inclusion problem introduced by Rockafellar []. It is known that problem (.) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory, etc. Let a set-valued mapping R : D(R) ⊂ H → H be maximal monotone. We define the resolvent operator JR,λ : H → D(R) associated with R and λ as follows: JR,λ = (I + λR)– , ∀x ∈ H, where λ is a positive number. In , Huang [] studied problem (.) in the case where R is maximal monotone and B is strongly monotone and Lipschitz-continuous with D(R) = C = H. Subsequently, Zeng et al. [] further studied problem (.) in the case which is more general than Huang’s []. Moreover, the authors [] obtained the same strong convergence conclusion as in Huang’s result []. In addition, the authors also gave the geometric convergence rate estimate for approximate solutions. Also, various types of iterative algorithms for solving variational inclusions have been further studied and developed; for more details, refer to [, –] and the references therein. Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 4 of 38 In , for the case where C = H, Yao et al. [] introduced and analyzed an iterative algorithms for finding a common element of the set of solutions of the GMEP (.), the set of solutions of the variational inclusion (.) for maximal monotone and inverse-strongly monotone mappings and the set of fixed points of a countable family of nonexpansive mappings on H. Recently, Kim and Xu [] introduced the concept of asymptotically κ-strict pseudocontractive mappings in a Hilbert space. Definition . Let C be a nonempty subset of a Hilbert space H. A mapping S : C → C is said to be an asymptotically κ-strict pseudocontractive mapping with sequence {γn } if there exist a constant κ ∈ [, ) and a sequence {γn } in [, ∞) with limn→∞ γn = such that n S x – Sn y ≤ ( + γn )x – y + κ x – Sn x – y – Sn y , ∀n ≥ , ∀x, y ∈ C. Subsequently, Sahu et al. [] considered the concept of asymptotically κ-strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian. Definition . Let C be a nonempty subset of a Hilbert space H. A mapping S : C → C is said to be an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {γn } if there exist a constant κ ∈ [, ) and a sequence {γn } in [, ∞) with limn→∞ γn = such that lim sup sup Sn x – Sn y – ( + γn )x – y – κ x – Sn x – y – Sn y ≤ . (.) n→∞ x,y∈C Put cn := max{, supx,y∈C (Sn x – Sn y – ( + γn )x – y – κx – Sn x – (y – Sn y) )}. Then cn ≥ (∀n ≥ ), cn → (n → ∞), and (.) reduce to the relation n S x – Sn y ≤ ( + γn )x – y + κ x – Sn x – y – Sn y + cn , ∀n ≥ , ∀x, y ∈ C. (.) Whenever cn = for all n ≥ in (.), then S is an asymptotically κ-strict pseudocontractive mapping with sequence {γn }. The authors [] derived the weak and strong convergence of the modified Mann iteration processes for an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {γn }. More precisely, they first established one weak convergence theorem for the following iterative scheme: x = x ∈ C chosen arbitrarily, xn+ = ( – αn )xn + αn Sn xn , ∀n ≥ , ∞ where < δ ≤ αn ≤ – κ – δ, ∞ n= αn cn < ∞, and n= γn < ∞; and then obtained another strong convergence theorem for the following iterative scheme: ⎧ ⎪ x = x ∈ C chosen arbitrary, ⎪ ⎪ ⎪ n ⎪ ⎪ ⎨yn = ( – αn )xn + αn S xn , Cn = {z ∈ C : yn – z ≤ xn – z + θn }, ⎪ ⎪ ⎪ Qn = {z ∈ C : xn – z, x – xn ≥ }, ⎪ ⎪ ⎪ ⎩x = P ∀n ≥ , n+ Cn ∩Qn x, where < δ ≤ αn ≤ – κ, θn = cn + γn Δn , and Δn = sup{xn – z : z ∈ Fix(S)} < ∞. Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 5 of 38 Inspired by the above facts, we in this paper introduce two iterative algorithms for finding common solutions of a finite family of variational inclusions for maximal monotone and inverse-strongly monotone mappings with the constraints of two problems: a generalized mixed equilibrium problem and a common fixed point problem of an infinite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under suitable conditions. The results presented in this paper are the supplement, extension, improvement, and generalization of the previously known results in this area. 2 Preliminaries Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by ·, · and · , respectively. Let C be a nonempty closed convex subset of H. We write xn x to indicate that the sequence {xn } converges weakly to x and xn → x to indicate that the sequence {xn } converges strongly to x. Moreover, we use ωw (xn ) to denote the weak ω-limit set of the sequence {xn }, i.e., ωw (xn ) := x ∈ H : xni x for some subsequence {xni } of {xn } . Definition . A mapping A : C → H is called (i) monotone if Ax – Ay, x – y ≥ , ∀x, y ∈ C; (ii) η-strongly monotone if there exists a constant η > such that Ax – Ay, x – y ≥ ηx – y , ∀x, y ∈ C; (iii) ζ -inverse-strongly monotone if there exists a constant ζ > such that Ax – Ay, x – y ≥ ζ Ax – Ay , ∀x, y ∈ C. It is easy to see that the projection PC is -inverse-strongly monotone (in short, -ism). Inverse-strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields. Definition . A differentiable function K : H → R is called: (i) convex, if K(y) – K(x) ≥ K (x), y – x , ∀x, y ∈ H, where K (x) is the Frechet derivative of K at x; (ii) strongly convex, if there exists a constant σ > such that σ K(y) – K(x) – K (x), y – x ≥ x – y , ∀x, y ∈ H. It is easy to see that if K : H → R is a differentiable strongly convex function with constant σ > then K : H → H is strongly monotone with constant σ > . Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 6 of 38 The metric (or nearest point) projection from H onto C is the mapping PC : H → C which assigns to each point x ∈ H the unique point PC x ∈ C satisfying the property x – PC x = inf x – y =: d(x, C). y∈C Some important properties of projections are gathered in the following proposition. Proposition . For given x ∈ H and z ∈ C: (i) z = PC x ⇔ x – z, y – z ≤ , ∀y ∈ C; (ii) z = PC x ⇔ x – z ≤ x – y – y – z , ∀y ∈ C; (iii) PC x – PC y, x – y ≥ PC x – PC y , ∀y ∈ H. (This implies that PC is nonexpansive and monotone.) By using the technique of [], we can readily obtain the following elementary result. Proposition . (see [, Lemma and Proposition ]) Let C be a nonempty closed convex subset of a real Hilbert space H and let ϕ : C → R be a lower semicontinuous and convex function. Let Θ : C × C → R be a bifunction satisfying the conditions (H)-(H). Assume that (i) K : H → R is strongly convex with constant σ > and the function x → y – x, K (x) is weakly upper semicontinuous for each y ∈ H; (ii) for each x ∈ H and r > , there exist a bounded subset Dx ⊂ C and yx ∈ C such that for any z ∈ C \ Dx , Θ(z, yx ) + ϕ(yx ) – ϕ(z) + K (z) – K (x), yx – z < . r Then the following hold: (a) for each x ∈ H, Sr(Θ,ϕ) (x) = ∅; (Θ,ϕ) is single-valued; (b) Sr (Θ,ϕ) is nonexpansive if K is Lipschitz-continuous with constant ν > and (c) Sr K (x ) – K (x ), u – u ≥ K (u ) – K (u ), u – u , ∀(x , x ) ∈ H × H, (Θ,ϕ) where ui = Sr (xi ) for i = , ; (d) for all s, t > and x ∈ H (Θ,ϕ) (Θ,ϕ) (Θ,ϕ) K Ss x – K St x , Ss(Θ,ϕ) x – St x ≤ s – t (Θ,ϕ) (Θ,ϕ) K Ss x – K (x), Ss(Θ,ϕ) x – St x ; s (e) Fix(Sr(Θ,ϕ) ) = MEP(Θ, ϕ); (f ) MEP(Θ, ϕ) is closed and convex. In particular, whenever Θ : C × C → R is a bifunction satisfying the conditions (H)-(H) and K(x) = x , ∀x ∈ H, then, for any x, y ∈ H, (Θ,ϕ) S x – Sr(Θ,ϕ) y ≤ Sr(Θ,ϕ) x – Sr(Θ,ϕ) y, x – y r Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 (Θ,ϕ) (Sr Page 7 of 38 is firmly nonexpansive) and (Θ,ϕ) |s – t| (Θ,ϕ) S(Θ,ϕ) x – x , S x – St x ≤ s s s ∀s, t > , x ∈ H. In this case, Sr(Θ,ϕ) is rewritten as Tr(Θ,ϕ) . If, in addition, ϕ ≡ , then Tr(Θ,ϕ) is rewritten as TrΘ (see [, Lemma .] for more details). We need some facts and tools in a real Hilbert space H which are listed as lemmas below. Lemma . Let X be a real inner product space. Then we have the following inequality: x + y ≤ x + y, x + y, ∀x, y ∈ X. Lemma . Let H be a real Hilbert space. Then the following hold: (a) x – y = x – y – x – y, y for all x, y ∈ H; (b) λx + μy = λx + μy – λμx – y for all x, y ∈ H and λ, μ ∈ [, ] with λ + μ = ; (c) If {xn } is a sequence in H such that xn x, it follows that lim sup xn – y = lim sup xn – x + x – y , n→∞ n→∞ ∀y ∈ H. Lemma . ([, Lemma .]) Let H be a real Hilbert space. Given a nonempty closed convex subset of H and points x, y, z ∈ H and given also a real number a ∈ R, the set v ∈ C : y – v ≤ x – v + z, v + a is convex (and closed). Lemma . ([, Lemma .]) Let C be a nonempty subset of a Hilbert space H and S : C → C be an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {γn }. Then n S x – Sn y ≤ κx – y + –κ + ( – κ)γn x – y + ( – κ)cn for all x, y ∈ C and n ≥ . Lemma . ([, Lemma .]) Let C be a nonempty subset of a Hilbert space H and S : C → C be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {γn }. Let {xn } be a sequence in C such that xn – xn+ → and xn – Sn xn → as n → ∞. Then xn – Sxn → as n → ∞. Lemma . (Demiclosedness principle [, Proposition .]) Let C be a nonempty closed convex subset of a Hilbert space H and S : C → C be a continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {γn }. Then I – S is demiclosed at zero in the sense that if {xn } is a sequence in C such that xn x ∈ C and lim supm→∞ lim supn→∞ xn – Sm xn = , then (I – S)x = . Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 8 of 38 Lemma . ([, Proposition .]) Let C be a nonempty closed convex subset of a Hilbert space H and S : C → C be a continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {γn } such that Fix(S) = ∅. Then Fix(S) is closed and convex. Remark . Lemmas . and . give some basic properties of an asymptotically κstrict pseudocontractive mapping in the intermediate sense with sequence {γn }. Moreover, Lemma . extends the demiclosedness principles studied for certain classes of nonlinear mappings; see [] for more details. ∞ ∞ Lemma . ([, p.]) Let {an }∞ n= , {bn }n= , and {δn }n= be sequences of nonnegative real numbers satisfying the inequality an+ ≤ ( + δn )an + bn , ∀n ≥ . ∞ ∞ If ∞ n= δn < ∞ and n= bn < ∞, then limn→∞ an exists. If, in addition, {an }n= has a subsequence which converges to zero, then limn→∞ an = . Recall that a Banach space X is said to satisfy the Opial condition [] if, for any given sequence {xn } ⊂ X which converges weakly to an element x ∈ X, we have the inequality lim sup xn – x < lim sup xn – y, n→∞ n→∞ ∀y ∈ X, y = x. It is well known in [] that every Hilbert space H satisfies the Opial condition. Lemma . (see [, Proposition .]) Let C be a nonempty closed convex subset of a real Hilbert space H and let {xn } be a sequence in H. Suppose that xn+ – p ≤ ( + λn )xn – p + δn , ∀p ∈ C, n ≥ , where {λn } and {δn } are sequences of nonnegative real numbers such that ∞ n= δn < ∞. Then {PC xn } converges strongly in C. ∞ n= λn < ∞ and Lemma . (see []) Let C be a closed convex subset of a real Hilbert space H. Let {xn } be a sequence in H and u ∈ H. Let q = PC u. If {xn } is such that ωw (xn ) ⊂ C and satisfies the condition xn – u ≤ u – q, for all n, then xn → q as n → ∞. Lemma . (see [, Lemma .]) Let C be a nonempty closed convex subset of a real Hilbert space H. Let {Tn }∞ n= be a sequence of nonexpansive self-mappings on C such that ∞ n= Fix(Tn ) = ∅ and let {λn } be a sequence in (, b] for some b ∈ (, ). Then, for every x ∈ C and k ≥ the limit limn→∞ Un,k x exists. Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Remark . (see [, Remark .]) It can be known from Lemma . that if D is a nonempty bounded subset of C, then for > there exists n ≥ k such that for all n > n sup Un,k x – Uk x ≤ . x∈D Remark . (see [, Remark .]) Utilizing Lemma ., we define a mapping W : C → C as follows: Wx = lim Wn x = lim Un, x, n→∞ n→∞ ∀x ∈ C. Such a W is called the W -mapping generated by T , T , . . . and λ , λ , . . . . Since Wn is nonexpansive, W : C → C is also nonexpansive. Indeed, observe that for each x, y ∈ C Wx – Wy = lim Wn x – Wn y ≤ x – y. n→∞ If {xn } is a bounded sequence in C, then we put D = {xn : n ≥ }. Hence, it is clear from Remark . that for an arbitrary > there exists N ≥ such that for all n > N Wn xn – Wxn = Un, xn – U xn ≤ sup Un, x – U x ≤ . x∈D This implies that lim Wn xn – Wxn = . n→∞ Lemma . (see [, Lemma .]) Let C be a nonempty closed convex subset of a real Hilbert space H. Let {Tn }∞ n= be a sequence of nonexpansive self-mappings on C such that ∞ Fix(T ) = ∅, and let {λn } be a sequence in (, b] for some b ∈ (, ). Then Fix(W ) = n n= ∞ n= Fix(Tn ). Lemma . (see [, Theorem . (Demiclosedness Principle)]) Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be nonexpansive. Then I – T is demiclosed on C. That is, whenever {xn } is a sequence in C weakly converging to some x ∈ C and the sequence {(I – T)xn } strongly converges to some y, it follows that (I – T)x = y. Here I is the identity operator of H. Recall that a set-valued mapping R : D(R) ⊂ H → H is called monotone if, for all x, y ∈ D(R), f ∈ R(x), and g ∈ R(y) imply f – g, x – y ≥ . A set-valued mapping R is called maximal monotone if R is monotone and (I + λR)D(R) = H for each λ > , where I is the identity mapping of H. We denote by G(R) the graph of R. It is known that a monotone mapping R is maximal if and only if, for (x, f ) ∈ H × H, f – g, x – y ≥ for every (y, g) ∈ G(R), we have f ∈ R(x). We illustrate the concept of maximal monotone mapping with the following example. Page 9 of 38 Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Let A : C → H be a monotone, k-Lipschitz-continuous mapping and let NC v be the normal cone to C at v ∈ C, i.e., NC v = w ∈ H : v – u, w ≥ , ∀u ∈ C . Define Tv = Av + NC v if v ∈ C, ∅ if v ∈/ C. Then T is maximal monotone and ∈ Tv if and only if Av, y – v ≥ for all y ∈ C (see []). Assume that R : D(R) ⊂ H → H is a maximal monotone mapping. Let λ > . In terms of Huang [] (see also []), we have the following property for the resolvent operator JR,λ : H → D(R). Lemma . JR,λ is single-valued and firmly nonexpansive, i.e., JR,λ x – JR,λ y, x – y ≥ JR,λ x – JR,λ y , ∀x, y ∈ H. Consequently, JR,λ is nonexpansive and monotone. Lemma . (see []) Let R be a maximal monotone mapping with D(R) = C. Then for any given λ > , u ∈ C is a solution of problem (.) if and only if u ∈ C satisfies u = JR,λ (u – λBu). Lemma . (see []) Let R be a maximal monotone mapping with D(R) = C and let B : C → H be a strongly monotone, continuous, and single-valued mapping. Then for each z ∈ H, the equation z ∈ (B + λR)x has a unique solution xλ for λ > . Lemma . (see []) Let R be a maximal monotone mapping with D(R) = C and B : C → H be a monotone, continuous and single-valued mapping. Then (I + λ(R + B))C = H for each λ > . In this case, R + B is maximal monotone. Lemma . (see []) 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 the minimization problem g x∗ = inf g(x), x∈C then g (x), x – x∗ ≥ , ∀x ∈ C. In particular, if x∗ solves (OP), then u + γ f – (I + μV ) x∗ , x – x∗ ≤ . Page 10 of 38 Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 11 of 38 3 Strong convergence theorems In this section, we introduce and analyze an iterative algorithm for finding common solutions of a finite family of variational inclusions for maximal monotone and inversestrongly monotone mappings with the constraints of two problems: a generalized mixed equilibrium problem and a common fixed point problem of an infinite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. Under appropriate conditions imposed on the parameter sequences we will prove strong convergence of the proposed algorithm. Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ be a bifunction from C × C to R satisfying (H)-(H) and ϕ : C → R be a lower semicontinuous and convex functional. Let Ri : C → H be a maximal monotone mapping and let A : H → H and Bi : C → H be ζ -inverse-strongly monotone and ηi inverse-strongly monotone, respectively, where i ∈ {, , . . . , N}. Let S : C → C be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some ≤ κ < with sequence {γn } ⊂ [, ∞) such that limn→∞ γn = and {cn } ⊂ [, ∞) such that limn→∞ cn = . Let {Tn }∞ n= be a sequence of nonexpansive self-mappings on C and {λn } be a sequence in (, b] for some b ∈ (, ). Let V be a γ¯ -strongly positive bounded linear operator and f : H → H be an l-Lipschitzian mapping with γ l < ( + μ)γ¯ . Assume that N Ω := ( ∞ n= Fix(Tn )) ∩ GMEP(Θ, ϕ, A) ∩ ( i= I(Bi , Ri )) ∩ Fix(S) is nonempty and bounded. Let Wn be the W -mapping defined by (.) and {αn }, {βn } and {δn } be three sequences in (, ) such that limn→∞ αn = and κ ≤ δn ≤ d < . Assume that: (i) K : H → R is strongly convex with constant σ > and its derivative K is Lipschitz-continuous with constant ν > such that the function x → y – x, K (x) is weakly upper semicontinuous for each y ∈ H; (ii) for each x ∈ H, there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any y ∈/ Dx , Θ(y, zx ) + ϕ(zx ) – ϕ(y) + K (y) – K (x), zx – y < ; r (iii) < lim infn→∞ βn ≤ lim supn→∞ βn < ; (iv) {λi,n } ⊂ [ai , bi ] ⊂ (, ηi ), ∀i ∈ {, , . . . , N}, and {rn } ⊂ [, ζ ] satisfies < lim inf rn ≤ lim sup rn < ζ . n→∞ n→∞ Pick any x ∈ H and set C = C, x = PC x . Let {xn } be a sequence generated by the following algorithm: ⎧ un = Sr(Θ,ϕ) (I – rn A)xn , ⎪ n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪zn = JRN ,λN,n (I – λN,n BN )JRN– ,λN–,n (I – λN–,n BN– ) · · · JR ,λ,n (I – λ,n B )un , ⎪ ⎨k = δ z + ( – δ )Sn z , n n n n n ⎪yn = αn (u + γ f (xn )) + βn kn + (( – βn )I – αn (I + μV ))Wn zn , ⎪ ⎪ ⎪ ⎪ ⎪ Cn+ = {z ∈ Cn : yn – z ≤ xn – z + θn }, ⎪ ⎪ ⎩ xn+ = PCn+ x , ∀n ≥ , (.) where θn = (αn + γn )Δn + cn , Δn = sup{xn – p + u + (γ f – I – μV )p : p ∈ Ω} < ∞, and = –sup αn < ∞. If Sr(Θ,ϕ) is firmly nonexpansive, then the following statements hold: n≥ Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 12 of 38 (I) {xn } converges strongly to PΩ x ; (II) {xn } converges strongly to PΩ x , which solves the optimization problem min x∈Ω μ Vx, x + x – u – h(x), (OP) provided γn + cn + xn – yn = o(αn ) additionally, where h : H → R is the potential function of γ f . Proof Since limn→∞ αn = and < lim infn→∞ βn ≤ lim supn→∞ βn < , we may assume, without loss of generality, that αn ≤ ( – βn )( + μV )– . Since V is a γ¯ -strongly positive bounded linear operator on H, we know that V = sup Vu, u : u ∈ H, u = . Observe that ( – βn )I – αn (I + μV ) u, u = – βn – αn – αn μVu, u ≥ – βn – αn – αn μV ≥ , that is, ( – βn )I – αn (I + μV ) is positive. It follows that ( – βn )I – αn (I + μV ) = sup ( – βn )I – αn (I + μV ) u, u : u ∈ H, u = = sup – βn – αn – αn μVu, u : u ∈ H, u = ≤ – βn – αn – αn μγ¯ . Put Λin = JRi ,λi,n (I – λi,n Bi )JRi– ,λi–,n (I – λi–,n Bi– ) · · · JR ,λ,n (I – λ,n B ) for all i ∈ {, , . . . , N} and n ≥ , and Λn = I, where I is the identity mapping on H. Then we have that zn = ΛN n un . We divide the rest of the proof into several steps. Step . We show that {xn } is well defined. It is obvious that Cn is closed and convex. As the defining inequality in Cn is equivalent to the inequality (xn – zn ), z ≤ xn – zn + θn , by Lemma . we know that Cn is convex and closed for every n ≥ . First of all, we show that Ω ⊂ Cn for all n ≥ . Suppose that Ω ⊂ Cn for some n ≥ . Take p ∈ Ω arbitrarily. Since p = Sr(Θ,ϕ) (p – rn Ap), A is ζ -inverse strongly monotone and n ≤ rn ≤ ζ , we have (I – rn A)xn – Sr(Θ,ϕ) (I – rn A)p un – p = Sr(Θ,ϕ) n n ≤ (I – rn A)xn – (I – rn A)p Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 13 of 38 = (xn – p) – rn (Axn – Ap) = xn – p – rn xn – p, Axn – Ap + rn Axn – Ap ≤ xn – p – rn ζ Axn – Ap + rn Axn – Ap = xn – p + rn (rn – ζ )Axn – Ap ≤ xn – p . (.) Since p = JRi ,λi,n (I – λi,n Bi )p, Λin p = p, and Bi is ηi -inverse-strongly monotone, where ηi ∈ (, ηi ), i ∈ {, , . . . , N}, by Lemma . we deduce that N– zn – p = JRN ,λN,n (I – λN,n BN )ΛN– n un – JRN ,λN,n (I – λN,n BN )Λn p N– ≤ (I – λN,n BN )ΛN– n un – (I – λN,n BN )Λn p = ΛN– un – ΛN– p – λN,n BN ΛN– un – BN ΛN– p n n n n N– N– ≤ ΛN– + λN,n (λN,n – ηN ) BN ΛN– n un – Λn p n un – BN Λn p N– ≤ ΛN– n un – Λn p .. . ≤ Λn un – Λn p = un – p . (.) Combining (.) and (.), we have zn – p ≤ xn – p. (.) By Lemma .(b), we deduce from (.) and (.) that kn – p = δn (zn – p) + ( – δn ) Sn zn – p = δn zn – p + ( – δn ) Sn zn – p – δn ( – δn ) zn – Sn zn ≤ δn zn – p + ( – δn ) ( + γn )zn – p + κ zn – Sn zn + cn – δn ( – δn ) zn – Sn zn = + γn ( – δn ) zn – p + ( – δn )(κ – δn ) zn – Sn zn + ( – δn )cn ≤ ( + γn )zn – p + ( – δn )(κ – δn ) zn – Sn zn + cn ≤ ( + γn )zn – p + cn . (.) Set V¯ = I + μV . Then, for γ l ≤ ( + μ)γ¯ , by Lemma . we obtain from (.), (.), and (.) yn – p = αn u + γ f (xn ) + βn kn + ( – βn )I – αn V¯ Wn zn – p Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 = αn u + γ f (xn ) – V¯ p + βn (kn – p) + ( – βn )I – αn V¯ (Wn zn – p) = αn u + γ f (p) – V¯ p + αn γ f (xn ) – f (p) + βn (kn – p) + ( – βn )I – αn V¯ (Wn zn – p) ≤ αn γ f (xn ) – f (p) + βn (kn – p) + ( – βn )I – αn V¯ (Wn zn – p) + αn u + γ f (p) – V¯ p , yn – p ≤ αn γ f (xn ) – f (p) + βn kn – p + ( – βn )I – αn V¯ (Wn zn – p) + αn u + γ f (p) – V¯ p , yn – p ≤ αn γ lxn – p + βn kn – p + ( – βn – αn – αn μγ¯ )Wn zn – p + αn u + γ f (p) – V¯ p yn – p ≤ αn ( + μ)γ¯ xn – p + βn kn – p + – βn – αn ( + μ)γ¯ zn – p + αn u + γ f (p) – V¯ p + yn – p ≤ αn ( + μ)γ¯ xn – p + βn kn – p + – βn – αn ( + μ)γ¯ zn – p + αn u + γ f (p) – V¯ p + yn – p ≤ αn ( + μ)γ¯ xn – p + βn ( + γn )zn – p + cn + – βn – αn ( + μ)γ¯ zn – p + αn u + γ f (p) – V¯ p + yn – p ≤ αn ( + μ)γ¯ xn – p + βn ( + γn )zn – p + cn + – βn – αn ( + μ)γ¯ ( + γn )zn – p + cn + αn u + γ f (p) – V¯ p + yn – p = αn ( + μ)γ¯ xn – p + – αn ( + μ)γ¯ ( + γn )zn – p + cn + αn u + γ f (p) – V¯ p + yn – p ≤ αn ( + μ)γ¯ ( + γn )xn – p + cn + – αn ( + μ)γ¯ ( + γn )xn – p + cn + αn u + γ f (p) – V¯ p + yn – p = ( + γn )xn – p + cn + αn u + γ f (p) – V¯ p + yn – p , which hence yields + γn αn u + γ f (p) – V¯ p + cn xn – p + – αn – αn – αn α n + γn αn = + cn u + γ f (p) – V¯ p + xn – p + – αn – αn – αn α n + γn α n + γn u + γ f (p) – V¯ p + cn ≤ + xn – p + – αn – αn – αn αn + γn = xn – p + cn xn – p + u + γ f (p) – V¯ p + – αn – αn yn – p ≤ Page 14 of 38 Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 15 of 38 ≤ xn – p + (αn + γn ) xn – p + u + γ f (p) – V¯ p + cn ≤ xn – p + (αn + γn )Δn + cn = xn – p + θn , (.) where θn = (αn + γn )Δn + cn , Δn = sup{xn – p + u + γ f (p) – V¯ p : p ∈ Ω} < ∞, and = –sup αn < ∞ (due to {αn } ⊂ (, ) and limn→∞ αn = ). Hence p ∈ Cn+ . This implies n≥ that Ω ⊂ Cn for all n ≥ . Therefore, {xn } is well defined. Step . We prove that xn – kn → as n → ∞. Indeed, let v = PΩ x . From xn = PCn x and v ∈ Ω ⊂ Cn , we obtain xn – x ≤ v – x . (.) This implies that {xn } is bounded and hence {un }, {zn }, {kn }, and {yn } are also bounded. Since xn+ ∈ Cn+ ⊂ Cn and xn = PCn x , we have xn – x ≤ xn+ – x , ∀n ≥ . Therefore limn→∞ xn – x exists. From xn = PCn x , xn+ ∈ Cn+ ⊂ Cn , by Proposition .(ii) we obtain xn+ – xn ≤ x – xn+ – x – xn , which implies lim xn+ – xn = . n→∞ (.) It follows from xn+ ∈ Cn+ that yn – xn+ ≤ xn – xn+ + θn and hence xn – yn ≤ xn – xn+ + xn+ – yn ≤ xn – xn+ + xn – xn+ + θn = xn – xn+ + θn . From (.) and limn→∞ θn = , we have lim xn – yn = . n→∞ Also, utilizing Lemmas . and .(b) we obtain from (.), (.), and (.) yn – p = αn u + γ f (xn ) – V¯ Wn zn + βn (kn – p) + ( – βn )(Wn zn – p) ≤ βn (kn – p) + ( – βn )(Wn zn – p) + αn u + γ f (xn ) – V¯ Wn zn , yn – p = βn kn – p + ( – βn )Wn zn – p – βn ( – βn )kn – Wn zn + αn u + γ f (xn ) – V¯ Wn zn yn – p (.) Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 16 of 38 ≤ βn kn – p + ( – βn )zn – p – βn ( – βn )kn – Wn zn + αn u + γ f (xn ) – V¯ Wn zn yn – p ≤ βn ( + γn )zn – p + cn + ( – βn )zn – p – βn ( – βn )kn – Wn zn + αn u + γ f (xn ) – V¯ Wn zn yn – p ≤ βn ( + γn )zn – p + cn + ( – βn ) ( + γn )zn – p + cn – βn ( – βn )kn – Wn zn + αn u + γ f (xn ) – V¯ Wn zn yn – p = ( + γn )zn – p + cn – βn ( – βn )kn – Wn zn + αn u + γ f (xn ) – V¯ Wn zn yn – p ≤ ( + γn )xn – p + cn – βn ( – βn )kn – Wn zn + αn u + γ f (xn ) – V¯ Wn zn yn – p, which leads to βn ( – βn )kn – Wn zn ≤ xn – p – yn – p + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn yn – p ≤ xn – yn xn – p + yn – p + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn yn – p. Since limn→∞ αn = , limn→∞ γn = , and limn→∞ cn = , it follows from (.) and condition (iii) that lim kn – Wn zn = . n→∞ (.) Note that yn – kn = αn u + γ f (xn ) – V¯ Wn zn + ( – βn )(Wn zn – kn ), which yields xn – kn ≤ xn – yn + yn – kn ≤ xn – yn + αn u + γ f (xn ) – V¯ Wn zn + ( – βn )(Wn zn – kn ) ≤ xn – yn + αn u + γ f (xn ) – V¯ Wn zn + ( – βn )Wn zn – kn ≤ xn – yn + αn u + γ f (xn ) – V¯ Wn zn + Wn zn – kn . So, from (.), (.), and limn→∞ αn = , we get lim xn – kn = . n→∞ (.) Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 17 of 38 Step . We prove that xn – un → , un – zn → , zn – Wzn → , and zn – Sn zn → as n → ∞. Indeed, taking into consideration that < lim infn→∞ rn ≤ lim supn→∞ rn < ζ , we may assume, without loss of generality, that {rn } ⊂ [c, d] ⊂ (, ζ ). From (.) and (.) it follows that kn – p ≤ + γn ( – δn ) zn – p + ( – δn )(k – δn ) zn – Sn zn + ( – δn )cn ≤ zn – p + γn zn – p + cn ≤ zn – p + γn xn – p + cn . (.) Next we prove that lim xn – un = . (.) n→∞ For p ∈ Ω, we find that (I – rn A)xn – Sr(Θ,ϕ) (I – rn A)p un – p = Sr(Θ,ϕ) n n ≤ (I – rn A)xn – (I – rn A)p = xn – p – rn (Axn – Ap) ≤ xn – p + rn (rn – ζ )Axn – Ap . (.) By (.), (.), and (.), we obtain kn – p ≤ zn – p + γn xn – p + cn ≤ un – p + γn xn – p + cn ≤ xn – p + rn (rn – ζ )Axn – Ap + γn xn – p + cn , which implies that c(ζ – d)Axn – Ap ≤ rn (ζ – rn )Axn – Ap ≤ xn – p – kn – p + γn xn – p + cn ≤ xn – kn xn – p + kn – p + γn xn – p + cn . From limn→∞ γn = , limn→∞ cn = , and (.), we have lim Axn – Ap = . n→∞ and Lemma .(a), we have By the firm nonexpansivity of Sr(,ϕ) n un – p = Sr(Θ,ϕ) (I – rn A)xn – Sr(Θ,ϕ) (I – rn A)p n n (.) Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 18 of 38 ≤ (I – rn A)xn – (I – rn A)p, un – p = (I – rn A)xn – (I – rn A)p + un – p – (I – rn A)xn – (I – rn A)p – (un – p) xn – p + un – p – xn – un – rn (Axn – Ap) = xn – p + un – p – xn – un + rn Axn – Ap, xn – un – rn Axn – Ap , ≤ which implies that un – p ≤ xn – p – xn – un + rn Axn – Apxn – un . (.) Combining (.) and (.), we have kn – p ≤ zn – p + γn xn – p + cn ≤ un – p + γn xn – p + cn ≤ xn – p – xn – un + rn Axn – Apxn – un + γn xn – p + cn , which implies xn – un ≤ xn – p – kn – p + rn Axn – Apxn – un + γn xn – p + cn ≤ xn – kn xn – p + kn – p + rn Axn – Apxn – un + γn xn – p + cn . From limn→∞ γn = , limn→∞ cn = , (.), and (.), we know that (.) holds. Next we show that limn→∞ Bi Λin un –Bi p = , i = , , . . . , N . It follows from Lemma . that i Λ un – p = JR ,λ (I – λi,n Bi )Λi– un – JR ,λ (I – λi,n Bi )p n n i i,n i i,n ≤ (I – λi,n Bi )Λi– n un – (I – λi,n Bi )p + λi,n (λi,n – ηi ) Bi Λi– un – Bi p ≤ Λi– n un – p n ≤ un – p + λi,n (λi,n – ηi ) Bi Λi– un – Bi p n ≤ xn – p + λi,n (λi,n – ηi ) Bi Λi– n un – Bi p . Combining (.) and (.), we have kn – p ≤ zn – p + γn xn – p + cn ≤ Λin un – p + γn xn – p + cn (.) Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 19 of 38 ≤ xn – p + λi,n (λi,n – ηi ) Bi Λi– n un – Bi p + γn xn – p + cn , together with {λi,n } ⊂ [ai , bi ] ⊂ (, ηi ), i ∈ {, , . . . , N}, implies ≤ λi,n (ηi – λi,n ) Bi Λi– un – Bi p ai (ηi – bi ) Bi Λi– n un – Bi p n ≤ xn – p – kn – p + γn xn – p + cn ≤ xn – kn xn – p + kn – p + γn xn – p + cn . From limn→∞ γn = , limn→∞ cn = , and (.), we obtain lim Bi Λi– n un – Bi p = , n→∞ i = , , . . . , N. (.) By Lemma . and Lemma .(a), we obtain i Λ un – p = JR ,λ (I – λi,n Bi )Λi– un – JR ,λ (I – λi,n Bi )p n n i i,n i i,n i ≤ (I – λi,n Bi )Λi– n un – (I – λi,n Bi )p, Λn un – p = (I – λi,n Bi )Λi– un – (I – λi,n Bi )p + Λi un – p n n – (I – λi,n Bi )Λi– un – (I – λi,n Bi )p – Λi un – p n n i + Λ un – p ≤ Λi– n un – p n – Λi– un – Λi un – λi,n Bi Λi– un – Bi p n n n ≤ un – p + Λin un – p – Λi– un – Λi un – λi,n Bi Λi– un – Bi p n n n ≤ xn – p + Λin un – p – Λi– un – Λi un – λi,n Bi Λi– un – Bi p , n n n which implies i Λ un – p n i i– ≤ xn – p – Λi– n un – Λn un – λi,n Bi Λn un – Bi p i – λ Bi Λi– un – Bi p = xn – p – Λi– n un – Λn un i,n n i i– + λi,n Λi– n un – Λn un , Bi Λn un – Bi p i ≤ xn – p – Λi– n un – Λn un i Bi Λi– un – Bi p . + λi,n Λi– n un – Λn un n (.) Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 20 of 38 Combining (.) and (.) we get kn – p ≤ zn – p + γn xn – p + cn ≤ Λin un – p + γn xn – p + cn i ≤ xn – p – Λi– n un – Λn un i– + λi,n Λn un – Λin un Bi Λi– n un – Bi p + γn xn – p + cn , which implies i– Λ un – Λi un n n i Bi Λi– un – Bi p ≤ xn – p – kn – p + λi,n Λi– n un – Λn un n + γn xn – p + cn i Bi Λi– un – Bi p ≤ xn – kn xn – p + kn – p + λi,n Λi– n un – Λn un n + γn xn – p + cn . From (.), (.), limn→∞ γn = , and limn→∞ cn = , we have i lim Λi– n un – Λn un = , n→∞ i = , , . . . , N. (.) From (.) we get un – zn = Λn un – ΛN n un N ≤ Λn un – Λn un + Λn un – Λn un + · · · + ΛN– n un – Λn un → as n → ∞. (.) By (.) and (.), we have xn – zn ≤ xn – un + un – zn → as n → ∞. (.) From (.) and (.), we have zn+ – zn ≤ zn+ – xn+ + xn+ – xn + xn – zn → as n → ∞. (.) By (.), (.), and (.), we get kn – zn ≤ kn – xn + xn – un + un – zn → as n → ∞. (.) Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 21 of 38 We observe that kn – zn = ( – δn ) Sn zn – zn . From δn ≤ d < and (.), we have lim Sn zn – zn = . n→∞ (.) We note that n S zn – Sn+ zn ≤ Sn zn – zn + zn – zn+ + zn+ – Sn+ zn+ + Sn+ zn+ – Sn+ zn . From (.), (.), and Lemma ., we obtain lim Sn zn – Sn+ zn = . n→∞ (.) On the other hand, we note that zn – Szn ≤ zn – Sn zn + Sn zn – Sn+ zn + Sn+ zn – Szn . From (.), (.), and the uniform continuity of S, we have lim zn – Szn = . n→∞ (.) In addition, note that zn – Wzn ≤ zn – kn + kn – Wn zn + Wn zn – Wzn . So, from (.), (.), and Remark . it follows that lim zn – Wzn = . n→∞ (.) Step . we prove that xn → v = PΩ x as n → ∞. Indeed, since {xn } is bounded, there exists a subsequence {xni } which converges weakly to some w. From (.) and (.)-(.), we see that uni w, Λm ni uni w, and zni w, where m ∈ {, , . . . , N}. Since S is uniformly continuous, by (.) we get limn→∞ zn – Sm zn = for any m ≥ . Hence from Lemma ., we obtain w ∈ Fix(S). In the meantime, utilizing Lemma ., we deduce from (.) and zni w that w ∈ Fix(W ) = ∞ n= Fix(Tn ) N (due to Lemma .). Next, we prove that w ∈ m= I(Bm , Rm ). As a matter of fact, since Bm is ηm -inverse-strongly monotone, Bm is a monotone and Lipschitz-continuous mapping. It follows from Lemma . that Rm + Bm is maximal monotone. Let (v, g) ∈ G(Rm + Bm ), i.e., m– g – Bm v ∈ Rm v. Again, since Λm n un = JRm ,λm,n (I – λm,n Bm )Λn un , n ≥ , m ∈ {, , . . . , N}, we have m– m Λm– n un – λm,n Bm Λn un ∈ (I + λm,n Rm )Λn un , Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 22 of 38 that is, m– m– m Λ un – Λm n un – λm,n Bm Λn un ∈ Rm Λn un . λm,n n In terms of the monotonicity of Rm , we get v – Λm n un , g m– m m– – Bm v – Λ un – Λn un – λm,n Bm Λn un ≥ λm,n n and hence v – Λm n un , g m– m m– ≥ v – Λm u , B v + u – Λ u – λ B Λ u Λ n m n n m,n m n n n n λm,n n m– m m m m– m Λ un – Λn un = v – Λn un , Bm v – Bm Λn un + Bm Λn un – Bm Λn un + λm,n n m– m m– m m u , B Λ u – B u u , u – Λ u + v – Λ Λ ≥ v – Λm m n n n n n m n n n n n n . λm,n n In particular, m m m– v – Λm ni uni , g ≥ v – Λni uni , Bm Λni uni – Bm Λni uni m– m m + v – Λni uni , Λ uni – Λni uni . λm,ni ni m– m m– Since Λm n un – Λn un → (due to (.)) and Bm Λn un – Bm Λn un → (due to the m Lipschitz-continuity of Bm ), we conclude from Λni uni w and {λi,n } ⊂ [ai , bi ] ⊂ (, ηi ), i ∈ {, , . . . , N} that lim v – Λm ni uni , g = v – w, g ≥ . i→∞ It follows from the maximal monotonicity of Bm + Rm that ∈ (Rm + Bm )w, i.e., w ∈ I(Bm , Rm ). Therefore, w ∈ N m= I(Bm , Rm ). (Θ,ϕ) Next, we show that w ∈ GMEP(Θ, ϕ, A). In fact, from zn = Srn (I – rn A)xn , we know that Θ(un , y) + ϕ(y) – ϕ(un ) + Axn , y – un + K (un ) – K (xn ), y – un ≥ , rn ∀y ∈ C. From (H) it follows that ϕ(y) – ϕ(un ) + Axn , y – un + K (un ) – K (xn ), y – un ≥ Θ(y, un ), rn ∀y ∈ C. Replacing n by ni , we have ϕ(y) – ϕ(uni ) + Axni , y – uni + ∀y ∈ C. K (uni ) – K (xni ) , y – uni ≥ Θ(y, uni ), rni (.) Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 23 of 38 Put ut = ty + ( – t)w for all t ∈ (, ] and y ∈ C. Then, from (.), we have ut – uni , Aut ≥ ut – uni , Aut – ϕ(ut ) + ϕ(uni ) – ut – uni , Axni K (uni ) – K (xni ) , ut – uni + Θ(ut , uni ) – rni ≥ ut – uni , Aut – Auni + ut – uni , Auni – Axni – ϕ(ut ) + ϕ(uni ) K (uni ) – K (xni ) , ut – uni + Θ(ut , uni ). – rni Since uni – xni → as i → ∞, we deduce from the Lipschitz-continuity of A and K that Auni – Axni → and K (uni ) – K (xni ) → as i → ∞. Further, from the monotonicity of A, we have ut – uni , Aut – Auni ≥ . So, from (H), we have the weakly lower semicontinuity of ϕ, K (uni )–K (xni ) rni → and uni w, then we have ut – w, Aut ≥ –ϕ(ut ) + ϕ(w) + Θ(ut , w), as i → ∞. (.) From (H), (H), and (.) we also have = Θ(ut , ut ) + ϕ(ut ) – ϕ(ut ) ≤ tΘ(ut , y) + ( – t)Θ(ut , w) + tϕ(y) + ( – t)ϕ(w) – ϕ(ut ) = t Θ(ut , y) + ϕ(y) – ϕ(ut ) + ( – t) Θ(ut , w) + ϕ(w) – ϕ(w) – ϕ(ut ) ≤ t Θ(ut , y) + ϕ(y) – ϕ(ut ) + ( – t)ut – w, Aut = t Θ(ut , y) + ϕ(y) – ϕ(ut ) + ( – t)ty – w, Aut , and hence ≤ Θ(ut , y) + ϕ(y) – ϕ(ut ) + ( – t)y – w, Aut . Letting t → , we have, for each y ∈ C, ≤ Θ(w, y) + ϕ(y) – ϕ(w) + Aw, y – w. This implies that w ∈ GMEP(Θ, ϕ, A). Therefore, w∈ ∞ n= Fix(Tn ) ∩ GMEP(Θ, ϕ, A) ∩ N I(Bi , Ri ) ∩ Fix(S) := Ω. i= This shows that ωw (xn ) ⊂ Ω. From (.) and Lemma . we infer that xn → v = PΩ x as n → ∞. Finally, assume additionally that γn + cn + xn – yn = o(αn ). Note that V is a γ¯ -strongly positive bounded linear operator and f : H → H is an l-Lipschitzian mapping with γ l < ( + μ)γ¯ . It is clear that V¯ x – u + γ f (x) – V¯ y – u + γ f (y) , x – y ≥ ( + μ)γ¯ – γ l x – y , ∀x, y ∈ H. Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 24 of 38 Hence we deduce that V¯ x – (u + γ f (x)) is (( + μ)γ¯ – γ l)-strongly monotone. In the meantime, it is easy to see that V¯ x – (u + γ f (x)) is (V¯ + γ l)-Lipschitzian with constant V¯ + γ l > . Thus, there exists a unique solution p in Ω to the VIP V¯ p – u + γ f (p) , u – p ≥ , ∀u ∈ Ω. Equivalently, p ∈ Ω solves (OP) (due to Lemma .). Consequently, we deduce from (.) and xn → v = PΩ x (n → ∞) that lim sup u + γ f (p) – V¯ p, yn – p n→∞ = lim sup u + γ f (p) – V¯ p, xn – p + u + γ f (p) – V¯ p, yn – xn n→∞ = lim sup u + γ f (p) – V¯ p, xn – p n→∞ = u + γ f (p) – V¯ p, v – p ≤ . Furthermore, by Lemma . we conclude from (.), (.), and (.) that yn – p = αn u + γ f (p) – V¯ p + αn γ f (xn ) – f (p) + βn (kn – p) + ( – βn )I – αn V¯ (Wn zn – p) ≤ αn γ f (xn ) – f (p) + βn (kn – p) + ( – βn )I – αn V¯ (Wn zn – p) + αn u + γ f (p) – V¯ p , yn – p ≤ αn γ f (xn ) – f (p) + βn kn – p + ( – βn )I – αn V¯ (Wn zn – p) + αn u + γ f (p) – V¯ p , yn – p ≤ αn γ lxn – p + βn kn – p + ( – βn – αn – αn μγ¯ )Wn zn – p + αn u + γ f (p) – V¯ p , yn – p ≤ αn γ lxn – p + βn kn – p + – βn – αn ( + μ)γ¯ zn – p + αn u + γ f (p) – V¯ p , yn – p γl xn – p + βn kn – p = αn ( + μ)γ¯ · ( + μ)γ¯ + – βn – αn ( + μ)γ¯ zn – p + αn u + γ f (p) – V¯ p , yn – p (γ l) xn – p + βn kn – p ( + μ) γ¯ + – βn – αn ( + μ)γ¯ zn – p + αn u + γ f (p) – V¯ p , yn – p ≤ αn ( + μ)γ¯ · ≤ αn ( + μ)γ¯ · (γ l) xn – p + βn ( + γn )zn – p + cn ( + μ) γ¯ (.) Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 25 of 38 + – βn – αn ( + μ)γ¯ zn – p + αn u + γ f (p) – V¯ p , yn – p (γ l) xn – p + βn ( + γn )zn – p + cn ( + μ)γ¯ + – βn – αn ( + μ)γ¯ ( + γn )zn – p + cn + αn u + γ f (p) – V¯ p , yn – p ≤ αn (γ l) xn – p + – αn ( + μ)γ¯ ( + γn )zn – p + cn ( + μ)γ¯ + αn u + γ f (p) – V¯ p , yn – p = αn (γ l) ( + γn )xn – p + cn ( + μ)γ¯ + – αn ( + μ)γ¯ ( + γn )xn – p + cn + αn u + γ f (p) – V¯ p , yn – p ( + μ) γ¯ – (γ l) ( + γn )xn – p + cn = – αn ( + μ)γ¯ + αn u + γ f (p) – V¯ p , yn – p ( + μ) γ¯ – (γ l) xn – p + γn xn – p + cn ≤ – αn ( + μ)γ¯ + αn u + γ f (p) – V¯ p , yn – p , ≤ αn which hence yields ( + μ) γ¯ – (γ l) xn – p ( + μ)γ¯ xn – p – yn – p γn xn – p + cn + + u + γ f (p) – V¯ p , yn – p αn αn γn + cn xn – yn xn – p + yn – p + xn – p + ≤ αn αn + u + γ f (p) – V¯ p , yn – p . ≤ Since γn + cn = o(αn ), xn – yn = o(αn ), and xn → v = PΩ x , we infer from (.) and ≤ γ l < ( + μ)γ¯ that as n → ∞ ( + μ) γ¯ – (γ l) v – p ≤ . ( + μ)γ¯ That is, p = v = PΩ x . This completes the proof. Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C × C to R satisfying (H)-(H) and ϕ : C → R be a lower semicontinuous and convex functional. Let Ri : C → H be a maximal monotone mapping and let A : H → H and Bi : C → H be ζ -inverse strongly monotone and ηi -inversestrongly monotone, respectively, for i = , . Let S : C → C be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some ≤ κ < with sequence {γn } ⊂ [, ∞) such that limn→∞ γn = and {cn } ⊂ [, ∞) such Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 26 of 38 that limn→∞ cn = . Let {Tn }∞ n= be a sequence of nonexpansive self-mappings on C and {λn } be a sequence in (, b] for some b ∈ (, ). Let V be a γ¯ -strongly positive bounded linear operator and f : H → H be an l-Lipschitzian mapping with γ l < ( + μ)γ¯ . Assume that Ω := ( ∞ n= Fix(Tn )) ∩ GMEP(Θ, ϕ, A) ∩ I(B , R ) ∩ I(B , R ) ∩ Fix(S) is nonempty and bounded. Let Wn be the W -mapping defined by (.) and {αn }, {βn }, and {δn } be three sequences in (, ) such that limn→∞ αn = and κ ≤ δn ≤ d < . Assume that: (i) K : H → R is strongly convex with constant σ > and its derivative K is Lipschitz-continuous with constant ν > such that the function x → y – x, K (x) is weakly upper semicontinuous for each y ∈ H; (ii) for each x ∈ H, there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any y ∈/ Dx , Θ(y, zx ) + ϕ(zx ) – ϕ(y) + K (y) – K (x), zx – y < ; r (iii) < lim infn→∞ βn ≤ lim supn→∞ βn < ; (iv) {λi,n } ⊂ [ai , bi ] ⊂ (, ηi ) for i = , , and {rn } ⊂ [, ζ ] satisfies < lim inf rn ≤ lim sup rn < ζ . n→∞ n→∞ Pick any x ∈ H and set C = C, x = PC x . Let {xn } be a sequence generated by the following algorithm: ⎧ u = Sr(Θ,ϕ) (I – rn A)xn , ⎪ n ⎪ ⎪ n ⎪ ⎪ zn = JR ,λ,n (I – λ,n B )JR ,λ,n (I – λ,n B )un , ⎪ ⎪ ⎪ ⎨k = δ z + ( – δ )Sn z , n n n n n ⎪ yn = αn (u + γ f (xn )) + βn kn + (( – βn )I – αn (I + μV ))Wn zn , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪Cn+ = {z ∈ Cn : yn – z ≤ xn – z + θn }, ⎪ ⎩ xn+ = PCn+ x , ∀n ≥ , (.) where θn = (αn + γn )Δn + cn , Δn = sup{xn – p + u + (γ f – I – μV )p : p ∈ Ω} < ∞, (Θ,ϕ) and = –sup αn < ∞. If Sr is firmly nonexpansive, then the following statements hold: n≥ (I) {xn } converges strongly to PΩ x ; (II) {xn } converges strongly to PΩ x , which solves the optimization problem min x∈Ω μ Vx, x + x – u – h(x), (OP) provided γn + cn + xn – yn = o(αn ) additionally, where h : H → R is the potential function of γ f . Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C × C to R satisfying (H)-(H) and ϕ : C → R be a lower semicontinuous and convex functional. Let R : C → H be a maximal monotone mapping and let A : H → H and B : C → H be ζ -inverse strongly monotone and ξ -inverse-strongly monotone, respectively. Let S : C → C be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some ≤ κ < with sequence {γn } ⊂ [, ∞) such that Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 27 of 38 limn→∞ γn = and {cn } ⊂ [, ∞) such that limn→∞ cn = . Let {Tn }∞ n= be a sequence of nonexpansive self-mappings on C and {λn } be a sequence in (, b] for some b ∈ (, ). Let V be a γ¯ -strongly positive bounded linear operator and f : H → H be an l-Lipschitzian mapping with γ l < ( + μ)γ¯ . Assume that Ω := ( ∞ n= Fix(Tn )) ∩ GMEP(Θ, ϕ, A) ∩ I(B, R) ∩ Fix(S) is nonempty and bounded. Let Wn be the W -mapping defined by (.) and {αn }, {βn }, and {δn } be three sequences in (, ) such that limn→∞ αn = and κ ≤ δn ≤ d < . Assume that: (i) K : H → R is strongly convex with constant σ > and its derivative K is Lipschitz-continuous with constant ν > such that the function x → y – x, K (x) is weakly upper semicontinuous for each y ∈ H; (ii) for each x ∈ H, there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any y ∈/ Dx , Θ(y, zx ) + ϕ(zx ) – ϕ(y) + K (y) – K (x), zx – y < ; r (iii) < lim infn→∞ βn ≤ lim supn→∞ βn < ; (iv) {ρn } ⊂ [a, b] ⊂ (, ξ ), and {rn } ⊂ [, ζ ] satisfies < lim inf rn ≤ lim sup rn < ζ . n→∞ n→∞ Pick any x ∈ H and set C = C, x = PC x . Let {xn } be a sequence generated by the following algorithm: ⎧ ⎪ un = Sr(Θ,ϕ) (I – rn A)xn , ⎪ n ⎪ ⎪ n ⎪ ⎪ k = δ J ⎨ n n R,ρn (I – ρn B)un + ( – δn )S JR,ρn (I – ρn B)un , yn = αn (u + γ f (xn )) + βn kn + (( – βn )I – αn (I + μV ))Wn JR,ρn (I – ρn B)un , ⎪ ⎪ ⎪ C ⎪ n+ = {z ∈ Cn : yn – z ≤ xn – z + θn }, ⎪ ⎪ ⎩x = P ∀n ≥ , n+ Cn+ x , (.) where θn = (αn + γn )Δn + cn , Δn = sup{xn – p + u + (γ f – I – μV )p : p ∈ Ω} < ∞, (Θ,ϕ) is firmly nonexpansive, then the following statements hold: and = –sup αn < ∞. If Sr n≥ (I) {xn } converges strongly to PΩ x ; (II) {xn } converges strongly to PΩ x , which solves the optimization problem min x∈Ω μ Vx, x + x – u – h(x), (OP) provided γn + cn + xn – yn = o(αn ) additionally, where h : H → R is the potential function of γ f . 4 Weak convergence theorems In this section, we introduce and analyze another iterative algorithm for finding common solutions of a finite family of variational inclusions for maximal monotone and inversestrongly monotone mappings with the constraints of two problems: a generalized mixed equilibrium problem and a common fixed point problem of an infinite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. Under mild conditions imposed on the parameter sequences we will prove weak convergence of the proposed algorithm. Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 28 of 38 Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ be a bifunction from C × C to R satisfying (H)-(H) and ϕ : C → R be a lower semicontinuous and convex functional. Let Ri : C → H be a maximal monotone mapping and let A : H → H and Bi : C → H be ζ -inverse-strongly monotone and ηi -inverse-strongly monotone, respectively, where i ∈ {, , . . . , N}. Let S : C → C be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some ≤ κ < with sequences {γn } ⊂ [, ∞) and {cn } ⊂ [, ∞). Let {Tn }∞ n= be a sequence of nonexpansive self-mappings on C and {λn } be a sequence in (, b] for some b ∈ (, ). Let V be a γ¯ -strongly positive bounded linear operator and f : H → H be an l-Lipschitzian mapping with γ l < ( + μ)γ¯ . Assume that Ω := ( ∞ n= Fix(Tn )) ∩ N GMEP(Θ, ϕ, A) ∩ ( i= I(Bi , Ri )) ∩ Fix(S) is nonempty. Let Wn be the W -mapping defined by (.) and {αn }, {βn } and {δn } be three sequences in (, ) such that < κ + ε ≤ δn ≤ d < . Assume that: (i) K : H → R is strongly convex with constant σ > and its derivative K is Lipschitz-continuous with constant ν > such that the function x → y – x, K (x) is weakly upper semicontinuous for each y ∈ H; (ii) for each x ∈ H, there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any y ∈/ Dx , Θ(y, zx ) + ϕ(zx ) – ϕ(y) + K (y) – K (x), zx – y < ; r ∞ (iii) n= (αn + γn + cn ) < ∞ and < lim infn→∞ βn ≤ lim supn→∞ βn < ; (iv) {λi,n } ⊂ [ai , bi ] ⊂ (, ηi ), ∀i ∈ {, , . . . , N}, and {rn } ⊂ [, ζ ] satisfies < lim inf rn ≤ lim sup rn < ζ . n→∞ n→∞ Pick any x ∈ H and let {xn } be a sequence generated by the following algorithm: ⎧ un = Sr(Θ,ϕ) (I – rn A)xn , ⎪ n ⎪ ⎪ ⎨ zn = JRN ,λN,n (I – λN,n BN )JRN– ,λN–,n (I – λN–,n BN– ) · · · JR ,λ,n (I – λ,n B )un , ⎪kn = δn zn + ( – δn )Sn zn , ⎪ ⎪ ⎩ xn+ = αn (u + γ f (xn )) + βn kn + (( – βn )I – αn (I + μV ))Wn zn , ∀n ≥ . (Θ,ϕ) Then {xn } converges weakly to w = limn→∞ PΩ xn provided Sr (.) is firmly nonexpansive. Proof First, let us show that limn→∞ xn – p exists for any p ∈ Ω. Put Λin = JRi ,λi,n (I – λi,n Bi )JRi– ,λi–,n (I – λi–,n Bi– ) · · · JR ,λ,n (I – λ,n B ) for all i ∈ {, , . . . , N}, n ≥ , and Λn = I, where I is the identity mapping on H. Then we see that zn = ΛN n un . Take p ∈ Ω arbitrarily. Similarly to the proof of Theorem ., we obtain un – p ≤ xn – p, (.) zn – p ≤ un – p, (.) un – p ≤ xn – p + rn (rn – ζ )Axn – Ap , (.) Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 29 of 38 un – p ≤ xn – p – xn – un + rn Axn – Apxn – un , i Λ un – p ≤ xn – p + λi,n (λi,n – ηi ) Bi Λi– un – Bi p , n n (.) i ∈ {, , . . . , N}, i Λ un – p ≤ xn – p – Λi– un – Λi un n n n i Bi Λi– un – Bi p , + λi,n Λi– n un – Λn un n (.) i ∈ {, , . . . , N}. (.) By Lemma .(b) we get kn – p = δn (zn – p) + ( – δn ) Sn zn – p = δn zn – p + ( – δn ) Sn zn – p – δn ( – δn ) zn – Sn zn ≤ δn zn – p + ( – δn ) ( + γn )zn – p + κ zn – Sn zn + cn – δn ( – δn ) zn – Sn zn = + γn ( – δn ) zn – p + ( – δn )(κ – δn ) zn – Sn zn + ( – δn )cn ≤ ( + γn )zn – p + ( – δn )(κ – δn ) zn – Sn zn + cn ≤ ( + γn )zn – p + cn . Repeating the same arguments as in the proof of Theorem . we have ( – βn )I – αn (I + μV ) ≤ – βn – αn – αn μγ¯ . Then by Lemma . we deduce from (.), (.), (.), and ≤ γ l ≤ ( + μ)γ¯ that xn+ – p = αn u + γ f (p) – V¯ p + αn γ f (xn ) – f (p) + βn (kn – p) + ( – βn )I – αn V¯ (Wn zn – p) ≤ αn γ f (xn ) – f (p) + βn (kn – p) + ( – βn )I – αn V¯ (Wn zn – p) + αn u + γ f (p) – V¯ p , xn+ – p ≤ αn γ f (xn ) – f (p) + βn kn – p + ( – βn )I – αn V¯ (Wn zn – p) + αn u + γ f (p) – V¯ p xn+ – p ≤ αn γ lxn – p + βn kn – p + ( – βn – αn – αn μγ¯ )Wn zn – p + αn u + γ f (p) – V¯ p xn+ – p ≤ αn ( + μ)γ¯ xn – p + βn kn – p + – βn – αn ( + μ)γ¯ zn – p (.) Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 30 of 38 + αn u + γ f (p) – V¯ p xn+ – p ≤ αn ( + μ)γ¯ xn – p + βn kn – p + – βn – αn ( + μ)γ¯ zn – p + αn u + γ f (p) – V¯ p xn+ – p ≤ αn ( + μ)γ¯ xn – p + βn ( + γn )zn – p + cn + – βn – αn ( + μ)γ¯ zn – p + αn u + γ f (p) – V¯ p xn+ – p ≤ αn ( + μ)γ¯ xn – p + βn ( + γn )xn – p + cn + – βn – αn ( + μ)γ¯ xn – p + αn u + γ f (p) – V¯ p xn+ – p ≤ αn ( + μ)γ¯ ( + γn )xn – p + cn + βn ( + γn )xn – p + cn + – βn – αn ( + μ)γ¯ ( + γn )xn – p + cn + αn u + γ f (p) – V¯ p xn+ – p = ( + γn )xn – p + cn + αn u + γ f (p) – V¯ p xn+ – p ≤ ( + γn )xn – p + cn + αn u + γ f (p) – V¯ p + xn+ – p , which hence yields xn+ – p + γn αn u + γ f (p) – V¯ p + cn xn – p + – αn – αn – αn αn + γn αn xn – p + = + cn u + γ f (p) – V¯ p + – αn – αn – αn ≤ + (αn + γn ) xn – p + αn u + γ f (p) – V¯ p + cn , ≤ (.) where = –sup αn < ∞ (due to {αn } ⊂ (, ) and limn→∞ αn = ). By Lemma ., we see n≥ from ∞ n= (αn + γn + cn ) < ∞ that limn→∞ xn – p exists. Thus {xn } is bounded and so are the sequences {un }, {zn }, and {kn }. Also, utilizing Lemmas . and .(b) we obtain from (.), (.), and (.) xn+ – p = αn u + γ f (xn ) – V¯ Wn zn + βn (kn – p) + ( – βn )(Wn zn – p) ≤ βn (kn – p) + ( – βn )(Wn zn – p) + αn u + γ f (xn ) – V¯ Wn zn , xn+ – p = βn kn – p + ( – βn )Wn zn – p – βn ( – βn )kn – Wn zn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ βn kn – p + ( – βn )zn – p – βn ( – βn )kn – Wn zn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ βn ( + γn )zn – p + cn + ( – βn )zn – p – βn ( – βn )kn – Wn zn Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 31 of 38 + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ βn ( + γn )zn – p + cn + ( – βn ) ( + γn )zn – p + cn – βn ( – βn )kn – Wn zn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ ( + γn )zn – p + cn – βn ( – βn )kn – Wn zn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ ( + γn )xn – p + cn – βn ( – βn )kn – Wn zn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p, (.) which leads to βn ( – βn )kn – Wn zn ≤ xn – p – xn+ – p + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p. Since limn→∞ αn = , limn→∞ γn = , and limn→∞ cn = , it follows from the existence of limn→∞ xn – p and condition (iii) that lim kn – Wn zn = . n→∞ (.) Note that xn+ – kn = αn u + γ f (xn ) – V¯ Wn zn + ( – βn )(Wn zn – kn ), which yields xn+ – kn ≤ αn u + γ f (xn ) – V¯ Wn zn + ( – βn )Wn zn – kn ≤ αn u + γ f (xn ) – V¯ Wn zn + Wn zn – kn . So, from (.) and limn→∞ αn = , we get lim xn+ – kn = . n→∞ In the meantime, we conclude from (.), (.), (.), and (.) that xn+ – p ≤ βn kn – p + ( – βn )zn – p – βn ( – βn )kn – Wn zn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ βn kn – p + ( – βn )zn – p + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ βn ( + γn )zn – p + ( – δn )(κ – δn ) zn – Sn zn + cn (.) Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 32 of 38 + ( – βn )zn – p + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ βn ( + γn )zn – p + ( – δn )(κ – δn ) zn – Sn zn + cn + ( – βn ) ( + γn )zn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ ( + γn )zn – p + βn ( – δn )(κ – δn ) zn – Sn zn + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ ( + γn )xn – p + βn ( – δn )(κ – δn ) zn – Sn zn + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p, which, together with < κ + ≤ δn ≤ d < , implies that ( – d)βn zn – Sn zn ≤ βn ( – δn )(δn – κ) zn – Sn zn ≤ xn – p – xn+ – p + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p. Consequently, from limn→∞ αn = , limn→∞ γn = , limn→∞ cn = , condition (iii), and the existence of limn→∞ xn – p, we get lim zn – Sn zn = . n→∞ (.) Since kn – zn = ( – δn )(Sn zn – zn ), from (.) we have lim kn – zn = . n→∞ (.) Combining (.), (.), and (.), we have xn+ – p ≤ βn ( + γn )zn – p + cn + ( – βn )zn – p – βn ( – βn )kn – Wn zn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ βn ( + γn )zn – p + cn + ( – βn ) ( + γn )zn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p = ( + γn )zn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ un – p + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ xn – p + rn (rn – ζ )Axn – Ap + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p, which implies rn (ζ – rn )Axn – Ap ≤ xn – p – xn+ – p + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p. Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 33 of 38 From condition (iv), limn→∞ αn = , limn→∞ γn = , limn→∞ cn = , and the existence of limn→∞ xn – p, we get lim Axn – Ap = . (.) n→∞ Combining (.), (.), and (.), we have xn+ – p ≤ zn – p + γn zn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ un – p + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ xn – p – xn – un + rn Axn – Apxn – un + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p, which implies xn – un ≤ xn – p – xn+ – p + rn Axn – Apxn – un + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p. From (.), limn→∞ αn = , limn→∞ γn = , limn→∞ cn = , and the existence of limn→∞ xn – p, we obtain lim xn – un = . (.) n→∞ Combining (.), (.), and (.), we have xn+ – p ≤ zn – p + γn zn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ Λin un – p + γn zn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p + γn xn – p + cn ≤ xn – p + λi,n (λi,n – ηi ) Bi Λi– n un – Bi p + αn u + γ f (xn ) – V¯ Wn zn xn+ – p, which implies λi,n (λi,n – ηi ) Bi Λi– n un – Bi p ≤ xn – p – xn+ – p + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p. From {λi,n } ⊂ [ai , bi ] ⊂ (, ηi ), i ∈ {, , . . . , N}, limn→∞ αn = , limn→∞ γn = , limn→∞ cn = , and the existence of limn→∞ xn – p, we obtain lim Bi Λi– n un – Bi p = , n→∞ i ∈ {, , . . . , N}. (.) Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 34 of 38 Combining (.), (.), and (.), we get xn+ – p ≤ zn – p + γn zn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p ≤ Λin un – p + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p i + λi,n Λi– un – Λi un Bi Λi– un – Bi p ≤ xn – p – Λi– n un – Λn un n n n + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p, which implies i– Λ un – Λi un n n i Bi Λi– un – Bi p ≤ xn – p – xn+ – p + λi,n Λi– n un – Λn un n + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p. From (.), limn→∞ αn = , limn→∞ γn = , limn→∞ cn = , and the existence of limn→∞ xn – p, we obtain i lim Λi– n un – Λn un = , n→∞ i ∈ {, , . . . , N}. (.) By (.), we have un – zn = Λn un – ΛN n un N ≤ Λn un – Λn un + Λn un – Λn un + · · · + ΛN– n un – Λn un → as n → ∞. (.) From (.) and (.), we have xn – zn ≤ xn – un + un – zn → as n → ∞. (.) By (.) and (.), we obtain kn – xn ≤ kn – zn + zn – xn → as n → ∞, (.) which, together with (.) and (.), implies that xn+ – xn ≤ xn+ – kn + kn – xn → as n → ∞. On the other hand, we observe that zn+ – zn ≤ zn+ – xn+ + xn+ – xn + xn – zn . (.) Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 35 of 38 By (.) and (.), we have lim zn+ – zn = . n→∞ (.) We note that zn – Szn ≤ zn – zn+ + zn+ – Sn+ zn+ + Sn+ zn+ – Sn+ zn + Sn+ zn – Szn . From (.), (.), Lemma ., and the uniform continuity of S, we obtain lim zn – Szn = . n→∞ (.) In addition, note that zn – Wzn ≤ zn – kn + kn – Wn zn + Wn zn – Wzn So, from (.), (.), and Remark . it follows that lim zn – Wzn = . n→∞ (.) Since {xn } is bounded, there exists a subsequence {xni } of {xn } which converges weakly to w. From (.) and (.), we have zni w and kni w. From (.) and the uniform continuity of S, we have limn→∞ zn – Sm zn = for any m ≥ . So, from Lemma ., we have w ∈ Fix(S). In the meantime, by (.) and Lemma ., we get w ∈ Fix(W ) = ∞ n= Fix(Tn ) (due to Lemma .). Utilizing similar arguments to those in the proof of Theorem ., we can derive w ∈ GMEP(Θ, ϕ, A) ∩ ( N i= I(Bi , Ri )). Consequently, w ∈ Ω. This shows that ωw (xn ) ⊂ Ω. Next let us show that ωw (xn ) is a single-point set. As a matter of fact, let {xnj } be another subsequence of {xn } such that xnj w . Then we get w ∈ Ω. If w = w , from the Opial condition, we have lim xn – w = lim xni – w n→∞ i→∞ < lim xni – w i→∞ = lim xn – w n→∞ = lim xnj – w j→∞ < lim xnj – w j→∞ = lim xn – w. n→∞ This attains a contradiction. So we have w = w . Put vn = PΩ xn . Since w ∈ Ω, we have xn – vn , vn – w ≥ . By Lemma ., we see that {vn } converges strongly to some w ∈ Ω. Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 36 of 38 Since {xn } converges weakly to w, we have w – w , w – w ≥ . Therefore we obtain w = w = limn→∞ PΩ xn . This completes the proof. Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C × C to R satisfying (H)-(H) and ϕ : C → R be a lower semicontinuous and convex functional. Let Ri : C → H be a maximal monotone mapping and let A : H → H and Bi : C → H be ζ -inverse-strongly monotone and ηi -inverse-strongly monotone, respectively, for i = , . Let S : C → C be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some ≤ κ < with sequences {γn } ⊂ [, ∞) and {cn } ⊂ [, ∞). Let {Tn }∞ n= be a sequence of nonexpansive selfmappings on C and {λn } be a sequence in (, b] for some b ∈ (, ). Let V be a γ¯ -strongly positive bounded linear operator and f : H → H be an l-Lipschitzian mapping with γ l < ( + μ)γ¯ . Assume that Ω := ( ∞ n= Fix(Tn )) ∩ GMEP(Θ, ϕ, A) ∩ I(B , R ) ∩ I(B , R ) ∩ Fix(S) is nonempty. Let Wn be the W -mapping defined by (.) and {αn }, {βn }, and {δn } be three sequences in (, ) such that < κ + ε ≤ δn ≤ d < . Assume that: (i) K : H → R is strongly convex with constant σ > and its derivative K is Lipschitz-continuous with constant ν > such that the function x → y – x, K (x) is weakly upper semicontinuous for each y ∈ H; (ii) for each x ∈ H, there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any y ∈/ Dx , Θ(y, zx ) + ϕ(zx ) – ϕ(y) + K (y) – K (x), zx – y < ; r ∞ (iii) n= (αn + γn + cn ) < ∞ and < lim infn→∞ βn ≤ lim supn→∞ βn < ; (iv) {λi,n } ⊂ [ai , bi ] ⊂ (, ηi ) for i = , , and {rn } ⊂ [, ζ ] satisfies < lim inf rn ≤ lim sup rn < ζ . n→∞ n→∞ Pick any x ∈ H and let {xn } be a sequence generated by the following algorithm: ⎧ un = Sr(Θ,ϕ) (I – rn A)xn , ⎪ n ⎪ ⎪ ⎨ zn = JR ,λ,n (I – λ,n B )JR ,λ,n (I – λ,n B )un , ⎪kn = δn zn + ( – δn )Sn zn , ⎪ ⎪ ⎩ xn+ = αn (u + γ f (xn )) + βn kn + (( – βn )I – αn (I + μV ))Wn zn , (.) ∀n ≥ . Then {xn } converges weakly to w = limn→∞ PΩ xn provided Sr(Θ,ϕ) is firmly nonexpansive. Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C × C to R satisfying (H)-(H) and ϕ : C → R be a lower semicontinuous and convex functional. Let R : C → H be a maximal monotone mapping and let A : H → H and B : C → H be ζ -inverse-strongly monotone and ξ -inverse-strongly monotone, respectively. Let S : C → C be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some ≤ κ < with sequences {γn } ⊂ [, ∞) and Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 Page 37 of 38 {cn } ⊂ [, ∞). Let {Tn }∞ n= be a sequence of nonexpansive self-mappings on C and {λn } be a sequence in (, b] for some b ∈ (, ). Let V be a γ¯ -strongly positive bounded linear operator and f : H → H be an l-Lipschitzian mapping with γ l < ( + μ)γ¯ . Assume that Ω := ( ∞ n= Fix(Tn )) ∩ GMEP(Θ, ϕ, A) ∩ I(B, R) ∩ Fix(S) is nonempty. Let Wn be the W mapping defined by (.) and {αn }, {βn }, and {δn } be three sequences in (, ) such that < κ + ε ≤ δn ≤ d < . Assume that: (i) K : H → R is strongly convex with constant σ > and its derivative K is Lipschitz-continuous with constant ν > such that the function x → y – x, K (x) is weakly upper semicontinuous for each y ∈ H; (ii) for each x ∈ H, there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any y ∈/ Dx , Θ(y, zx ) + ϕ(zx ) – ϕ(y) + K (y) – K (x), zx – y < ; r ∞ (iii) n= (αn + γn + cn ) < ∞ and < lim infn→∞ βn ≤ lim supn→∞ βn < ; (iv) {ρn } ⊂ [a, b] ⊂ (, ξ ), and {rn } ⊂ [, ζ ] satisfies < lim inf rn ≤ lim sup rn < ζ . n→∞ n→∞ Pick any x ∈ H and let {xn } be a sequence generated by the following algorithm: ⎧ un = Sr(Θ,ϕ) (I – rn A)xn , ⎪ n ⎪ ⎪ ⎨ kn = δn JR,ρn (I – ρn B)un + ( – δn )Sn JR,ρn (I – ρn B)un , ⎪xn+ = αn (u + γ f (xn )) + βn kn + (( – βn )I ⎪ ⎪ ⎩ – αn (I + μV ))Wn JR,ρn (I – ρn B)un , ∀n ≥ . (.) Then {xn } converges weakly to w = limn→∞ PΩ xn provided Sr(Θ,ϕ) is firmly nonexpansive. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Author details 1 Department of Mathematics, Shanghai Normal University, Shanghai, 200234, P.R. China. 2 Scientific Computing Key Laboratory of Shanghai Universities, Shanghai, 200234, P.R. China. 3 Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu, 30033, Taiwan. 4 Department of Information Management, Yuan Ze University, Chung-Li, 32003, Taiwan. 5 Innovation Center for Big Data and Digital Convergence, Yuan Ze University, Chung-Li, 32003, Taiwan. Acknowledgements In this research, first author was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133) and Ph.D. Program Foundation of Ministry of Education of China (20123127110002). The second author and third author were supported partly by the National Science Council of the Republic of China. Received: 14 August 2014 Accepted: 14 October 2014 Published: 21 Nov 2014 References 1. Peng, JW, Yao, JC: A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwan. J. Math. 12, 1401-1432 (2008) Ceng et al. Journal of Inequalities and Applications 2014, 2014:462 http://www.journalofinequalitiesandapplications.com/content/2014/1/462 2. Ceng, LC, Hu, HY, Wong, MM: Strong and weak convergence theorems for generalized mixed equilibrium problem with perturbation and fixed point problem of infinitely many nonexpansive mappings. Taiwan. J. Math. 15(3), 1341-1367 (2011) 3. Yao, Y, Liou, YC, Yao, JC: New relaxed hybrid-extragradient method for fixed point problems, a general system of variational inequality problems and generalized mixed equilibrium problems. Optimization 60(3), 395-412 (2011) 4. Cai, G, Bu, SQ: Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces. J. Comput. Appl. Math. 247, 34-52 (2013) 5. Yao, Y, Cho, YJ, Liou, YC: Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur. J. Oper. Res. 212, 242-250 (2011) 6. Ceng, LC, Ansari, QH, Schaible, S: Hybrid extragradient-like methods for generalized mixed equilibrium problems, system of generalized equilibrium problems and optimization problems. J. Glob. Optim. 53, 69-96 (2012) 7. Ceng, LC, Guu, SM, Yao, JC: Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems. Fixed Point Theory Appl. 2012, Article ID 92 (2012) 8. Ceng, LC, Petrusel, A: Relaxed extragradient-like method for general system of generalized mixed equilibria and fixed point problem. Taiwan. J. Math. 16(2), 445-478 (2012) 9. Rockafellar, RT: Monotone operators and the proximal point algorithms. SIAM J. Control Optim. 14, 877-898 (1976) 10. Huang, NJ: A new completely general class of variational inclusions with noncompact valued mappings. Comput. Math. Appl. 35(10), 9-14 (1998) 11. Zeng, LC, Guu, SM, Yao, JC: Characterization of H-monotone operators with applications to variational inclusions. Comput. Math. Appl. 50(3-4), 329-337 (2005) 12. Ceng, LC, Guu, SM, Yao, JC: Iterative approximation of solutions for a class of completely generalized set-valued quasi-variational inclusions. Comput. Math. Appl. 56, 978-987 (2008) 13. Ceng, LC, Guu, SM, Yao, JC: Iterative algorithm for finding approximate solutions of mixed quasi-variational-like inclusions. Comput. Math. Appl. 56, 942-952 (2008) 14. Ceng, LC, Ansari, QH, Wong, MM, Yao, JC: Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems. Comput. Math. Appl. 13(2), 403-422 (2012) 15. Ahmad, R, Ansari, QH, Irfan, SS: Generalized variational inclusions and generalized resolvent equations in Banach spaces. Comput. Math. Appl. 29, 1825-1835 (2005) 16. Ahmad, R, Ansari, QH: Generalized variational inclusions and H-resolvent equations with H-accretive operators. Taiwan. J. Math. 111, 703-716 (2007) 17. Ahmad, R, Ansari, QH: System of vector quasi-variational inclusions with some applications. Nonlinear Anal. 69, 2812-2824 (2008) 18. Kim, TH, Xu, HK: Convergence of the modified Mann’s iteration method for asymptotically strict pseudocontractions. Nonlinear Anal. 68, 2828-2836 (2008) 19. Sahu, DR, Xu, HK, Yao, JC: Asymptotically strict pseudocontractive mappings in the intermediate sense. Nonlinear Anal. 70, 3502-3511 (2009) 20. Ceng, LC, Yao, JC: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 214, 186-201 (2008) 21. Ceng, LC, Yao, JC: A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem. Nonlinear Anal. 72, 1922-1937 (2010) 22. Osilike, MO, Aniagbosor, SC, Akuchu, BG: Fixed points of asymptotically demicontractive mappings in arbitrary Banach space. Panam. Math. J. 12, 77-88 (2002) 23. Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591-597 (1967) 24. Huang, S: Hybrid extragradient methods for asymptotically strict pseudo-contractions in the intermediate sense and variational inequality problems. Optimization 60, 739-754 (2011) 25. Martinez-Yanes, C, Xu, HK: Strong convergence of the CQ method for fixed point iteration process. Nonlinear Anal. 64, 2400-2411 (2006) 26. O’Hara, JG, Pillay, P, Xu, HK: Iterative approaches to convex feasibility problems in Banach spaces. Nonlinear Anal. 64(9), 2022-2042 (2006) 27. 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, Article ID 64363 (2007) 28. Goebel, K, Kirk, WA: Topics on Metric Fixed-Point Theory. Cambridge University Press, Cambridge (1990) 29. Yao, Y, Noor, MA, Zainab, S, Liou, YC: Mixed equilibrium problems and optimization problems. J. Math. Anal. Appl. 354, 319-329 (2009) 10.1186/1029-242X-2014-462 Cite this article as: Ceng et al.: On a finite family of variational inclusions with the constraints of generalized mixed equilibrium and fixed point problems. Journal of Inequalities and Applications 2014, 2014:462 Page 38 of 38
© Copyright 2024