A New General System of Generalized Nonlinear Mixed Composite-Type Equilibria and Fixed Point Problems with an Application to Minimization Problems

and Applied Analysis 3 the equilibrium problems, we also have the problems of finding the fixed points of the nonlinear mappings, which is the subject of current interest in functional analysis. It is natural to construct a unified approach for these problems. In this direction, several authors have introduced some iterative schemes for finding a common element of the set of solutions of the equilibrium problems and the set of fixed points of nonlinear mappings e.g., see 5–18 and the references therein . Recall the following definitions. Definition 1.1. The mapping S : C → C is said to be 1 nonexpansive if ∥ ∥Sx − Sy∥ ≤ ∥x − y∥ , ∀x, y ∈ C, 1.6 2 L-Lipschitzian if there exists a constant L > 0 such that ∥Sx − Sy∥ ≤ L∥x − y∥, ∀x, y ∈ C, 1.7 3 k-strict pseudocontraction 19 if there exists a constant k ∈ 0, 1 such that ∥ ∥Sx − Sy∥2 ≤ ∥x − y∥2 k∥ I − S x − I − S y∥2, ∀x, y ∈ C, 1.8 4 pseudocontractive if ∥Sx − Sy∥2 ≤ ∥x − y∥2 ∥ I − S x − I − S y∥2, ∀x, y ∈ C. 1.9 Clearly, the class of strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. It is easy to see that 1.8 is equivalent to 〈 Sx − Sy, x − y ≤ ∥x − y∥2 − 1 − k 2 ∥ I − S x − I − S y∥2, ∀x, y ∈ C, 1.10 that is, I−S is 1−k /2-inverse-strongly monotone. From 19 , we know that if S is a k-strictly pseudocontractive mapping, then S is Lipschitz continuous with constant 3−k / 1−k , that is, ‖Sx − Sy‖ ≤ 3 − k / 1 − k ‖x − y‖, for all x, y ∈ C. In this paper, we use Fix S {x ∈ C : Sx x} to denote the set of fixed points of S. Definition 1.2. A countable family of mapping {Tn}n 1 : C → C is called a family of k-strict pseudocontraction mappings if there exists a constant k ∈ 0, 1 such that ∥Tnx − Tny ∥2 ≤ ∥x − y∥2 k∥ I − Tn x − I − Tn y ∥2, ∀x, y ∈ C, n ≥ 1. 1.11 On the other hand, let C be a nonempty closed and convex subset of a real Hilbert space H. Let F1, F2, F3 : C × C → R be three bifunctions and let Ψ1,Ψ2,Ψ3,Φ1,Φ2,Φ3 : C → H be six 4 Abstract and Applied Analysis nonlinear mappings and let φ1, φ2, φ3 : C → R be three functions. We consider the following problem of finding x∗, y∗, z∗ ∈ C × C × C such that μ1F1 x∗, x 〈 μ1 Ψ1 Φ1 y∗ x∗ − y∗, x − x∗ 〉 ≥ μ1φ1 x∗ − μ1φ1 x , ∀x ∈ C, μ2F2 ( y∗, y ) 〈 μ2 Ψ2 Φ2 z∗ y∗ − z∗, y − y∗ 〉 ≥ μ2φ2 ( y∗ ) − μ2φ2 ( y ) , ∀y ∈ C, μ3F3 z∗, z 〈 μ3 Ψ3 Φ3 x∗ z∗ − x∗, z − z∗ 〉 ≥ μ3φ3 z∗ − μ3φ3 z , ∀z ∈ C, 1.12 which is called a new general system of generalized nonlinear mixed composite-type equilibria, where μi > 0 for all i 1, 2, 3. Next, we present some special cases of problem 1.12 as follows. 1 If Ψi Ψ, Φi Φ, Fi F, and φi φ for all i 1, 2, 3, then the problem 1.12 reduces to the following new general system of generalized nonlinear mixed compositetype equilibria: find x∗, y∗, z∗ ∈ C × C × C such that μ1F x∗, x 〈 μ1 Ψ Φ y∗ x∗ − y∗, x − x∗ 〉 ≥ μ1φ x∗ − μ1φ x , ∀x ∈ C, μ2F ( y∗, y ) 〈 μ2 Ψ Φ z∗ y∗ − z∗, y − y∗ 〉 ≥ μ2φ ( y∗ ) − μ2φ ( y ) , ∀y ∈ C, μ3F z∗, z 〈 μ3 Ψ Φ x∗ z∗ − x∗, z − z∗ 〉 ≥ μ3φ z∗ − μ3φ z , ∀z ∈ C, 1.13 where μi > 0 for all i 1, 2. 2 If F3 0, Ψ3 Φ3 0, μ3 0, and z∗ x∗, then the problem 1.12 reduces to the following general system of generalized nonlinear mixed composite-type equilibria: find x∗, y∗ ∈ C × C such that μ1F1 x∗, x 〈 μ1 Ψ1 Φ1 y∗ x∗ − y∗, x − x∗ 〉 ≥ μ1φ1 x∗ − μ1φ1 x , ∀x ∈ C, μ2F2 ( y∗, y ) 〈 μ2 Ψ2 Φ2 x∗ y∗ − x∗, y − y∗ 〉 ≥ μ2φ2 ( y∗ ) − μ2φ2 ( y ) , ∀y ∈ C, 1.14 which was introduced and considered by Ceng et al. 20 , where μi > 0 for all i 1, 2. 3 If F3 0, Ψ3 Φ3 0, μ3 0, and φi 0 for all i 1, 2, 3 and z∗ x∗, then the problem 1.12 reduces to the following a general system of generalized equilibria: find x∗, y∗ ∈ C × C such that F1 x∗, x 〈 Ψ1y∗, x − x∗ 〉 1 μ1 〈 x∗ − y∗, x − x∗ ≥ 0, ∀x ∈ C, F2 ( y∗, y ) 〈 Ψ2x∗, y − y∗ 〉 1 μ2 〈 y∗ − x∗, y − y∗ ≥ 0, ∀y ∈ C, 1.15 which was introduced and considered by Ceng and Yao 21 , where μi > 0 for all i 1, 2. Abstract and Applied Analysis 5 4 If Fi F, Ψi Ψ, and Φi Φ, φi φ, for all i 1, 2, 3, then the problem 1.12 reduces to the following generalized mixed equilibrium problem with perturbed mapping: find x∗ ∈ C such thatand Applied Analysis 5 4 If Fi F, Ψi Ψ, and Φi Φ, φi φ, for all i 1, 2, 3, then the problem 1.12 reduces to the following generalized mixed equilibrium problem with perturbed mapping: find x∗ ∈ C such that F ( x∗, y ) φ ( y ) − φ x∗ 〈 Ψ Φ x∗, y − x∗ ≥ 0, ∀y ∈ C, 1.16 which was introduced and considered by Hu and Ceng 22 . 5 If Fi 0, Φi 0, and φi 0 for all i 1, 2, 3, then the problem 1.12 reduces to the following general system of variational inequalities: find x∗, y∗, z∗ ∈ C × C × C such that 〈 μ1Ψ1y∗ x∗ − y∗, x − x∗ 〉 ≥ 0, ∀x ∈ C, 〈 μ2Ψ2z∗ y∗ − z∗, y − y∗ 〉 ≥ 0, ∀y ∈ C, 〈 μ3Ψ3x∗ z∗ − x∗, z − z∗ 〉 ≥ 0, ∀z ∈ C, 1.17 which was introduced and considered by Kumam et al. 23 , where μi > 0 for all i 1, 2, 3. 6 If Fi 0, Φi 0, φi 0 for all i 1, 2, 3, Ψ3 0 and z∗ x∗, then the problem 1.12 reduces to the following general system of variational inequalities: find x∗, y∗ ∈ C×C such that 〈 μ1Ψ1y∗ x∗ − y∗, x − x∗ 〉 ≥ 0, ∀x ∈ C, 〈 μ2Ψ2x∗ y∗ − x∗, y − y∗ 〉 ≥ 0, ∀y ∈ C, 1.18 which was introduced and considered by Ceng et al. 24 , where μi > 0 for all i 1, 2. In 2010, Cho et al. 1 introduced an iterative method for finding a common element of the set of solutions of generalized equilibrium problems 1.2 , the set of solutions for a systems of nonlinear variational inequalities problems 1.18 , and the set of fixed points of nonexpansive mappings in Hilbert spaces. Ceng and Yao 21 introduced and considered a relaxed extragradient-like method for finding a common element of the set of solutions of a system of generalized equilibria, the set of fixed points of a strictly pseudocontractive mapping, and the set of solutions of a equilibrium problem in a real Hilbert space and obtained a strong convergence theorem. The result of Ceng and Yao 21 included, as special cases, the corresponding ones of S. Takahashi and W. Takahashi 10 , Ceng et al. 24 , Peng and Yao 25 , and Yao et al. 26 . Motivated and inspired by the works in the literature, we introduce a new general system of generalized nonlinear mixed composite-type equilibria 1.12 and propose a new iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem, a general system of generalized nonlinear mixed composite-type equilibria, and the set of fixed points of a countable family of strict pseudocontraction mappings. Furthermore, we prove the strong convergence theorem of the purposed iterative scheme in a real Hilbert space. As applications, we apply our results to solve a certain 6 Abstract and Applied Analysis minimization problem related to a strongly positive bounded linear operator. The results presented in this paper extend the recent results of Cho et al. 1 , Ceng and Yao 21 , Ceng et al. 20 , and many authors. 2. Preliminaries A bounded linear operatorA is said to be strongly positive, if there exists a constant γ > 0 such that 〈Ax, x〉 ≥ γ‖x‖, ∀x ∈ H. 2.1 Recall that, a mapping f : C → C is said to be contractive if there exists a constant α ∈ 0, 1 such that ∥f x − fy∥ ≤ α∥x − y∥, ∀x, y ∈ C. 2.2 A mapping A : C → H is called α-inverse-strongly monotone if there exists a constant α > 0 such that 〈 x − y,Ax −Ay ≥ α∥Ax −Ay∥2, ∀x, y ∈ C. 2.3 Let C be a nonempty closed convex subset of a real Hilbert space H. For every point x ∈ H there exists a unique nearest point in C denoted by PCx, such that ‖x − PCx‖ ≤ ∥x − y∥, ∀y ∈ C. 2.4 PC is called the metric projection of H onto C. It is well known that PC is nonexpansive see 27 and for x ∈ H, z PCx ⇐⇒ 〈 x − z, y − z ≤ 0, ∀y ∈ C. 2.5 Let φ : C → R be a real-valued function, G : C → H be a mapping and Θ : H × C × C → R be an equilibrium-like function. Let r be a positive real number. For all x ∈ C, we consider the following problem. Find y ∈ C such that Θ ( Gx, y, z ) φ z − φy 1 r 〈 y − x, z − y ≥ 0, ∀z ∈ C, 2.6 which is known as the auxiliary generalized equilibrium problem. Let T r : H → C be the mapping such that, for all x ∈ H, T r is the solution set of the auxiliary problem 2.6 , that is, T r x { y ∈ C : ΘGx, y, z φ z − φy 1 r 〈 y − z, z − x ≥ 0, ∀z ∈ C } . 2.7 Abstract and Applied Analysis 7 Then, we will assume the Condition Δ 28 as follows: a T r is single-valued; b T r is nonexpansive; c Fix T r GEP C,G,Θ, φ . Notice that the examples of showing the sufficient conditions for the existence of the condition Δ can be found in 6 . Throughout this paper, we assume that a bifunction F : C × C → R and φ : C → R is a lower semicontinuous and convex function satisfy the following conditions: H1 F x, x 0, ∀x ∈ C; H2 F is monotone, that is, F x, y F y, x ≤ 0, ∀x, y ∈ C; H3 for all y ∈ C, x → F x, y is weakly upper semicontinuous; H4 for all x ∈ C, y → F x, y is convex and lower semicontinuous; A1 for all x ∈ H and r > 0, there exist a bounded subset Dx ⊂ C and yx ∈ C such that for all z ∈ C \Dx,and Applied Analysis 7 Then, we will assume the Condition Δ 28 as follows: a T r is single-valued; b T r is nonexpansive; c Fix T r GEP C,G,Θ, φ . Notice that the examples of showing the sufficient conditions for the existence of the condition Δ can be found in 6 . Throughout this paper, we assume that a bifunction F : C × C → R and φ : C → R is a lower semicontinuous and convex function satisfy the following conditions: H1 F x, x 0, ∀x ∈ C; H2 F is monotone, that is, F x, y F y, x ≤ 0, ∀x, y ∈ C; H3 for all y ∈ C, x → F x, y is weakly upper semicontinuous; H4 for all x ∈ C, y → F x, y is convex and lower semicontinuous; A1 for all x ∈ H and r > 0, there exist a bounded subset Dx ⊂ C and yx ∈ C such that for all z ∈ C \Dx, F ( z, yx ) φ ( yx ) − φ z 1 r 〈 yx − z, z − x 〉 < 0; 2.8 A2 C is a bounded set. In order to prove our main results in the next section, we need the following lemmas. Lemma 2.1 see 29 . Let C be a nonempty closed and convex subset of a real Hilbert sapce H. Let F : C × C → R be a bifunction satisfying condition H1 – H4 and let φ : C → R be a lower semicontinuous and convex function. For r > 0 and x ∈ H define a mapping T F,φ r : H → C follows T F,φ r x { y ∈ C : Fy, z φ z − φy 1 r 〈 y − z, z − x ≥ 0, ∀z ∈ C } . 2.9 Assume that either A1 or A2 holds, then the following statements hold i T F,φ r / ∅ for all x ∈ H and T F,φ r is single-valued; ii T F,φ r is firmly nonexpansive, that is, for all x, y ∈ H, ∥∥∥T F,φ r x − T F,φ r y ∥∥ 2 ≤ 〈 T F,φ r x − T F,φ r y, x − y 〉 ; 2.10 iii Fix T F,φ r MEP F, φ ; iv MEP F, φ is closed and convex. Remark 2.2. If φ 0, then T F,φ r is rewritten as T r see 21, Lemma 2.1 for more details . 8 Abstract and Applied Analysis Lemma 2.3 see 30 . Let {xn} and {ln} be bounded sequences in a Banach space X and let {βn} be a sequence in 0, 1 with 0 < lim infn→∞βn ≤ lim supn→∞βn < 1. Suppose xn 1 1−βn ln βnxn for all integers n ≥ 0 and lim supn→∞ ‖ln 1 − ln‖ − ‖xn 1 − xn‖ ≤ 0. Then, limn→∞‖ln − xn‖ 0. Lemma 2.4 see 31 . LetH be a real Hilbert space. Then the following inequalities hold. i ‖λx 1 − λ y‖2 λ‖x‖2 1 − λ ‖y‖2 − λ 1 − λ ‖x − y‖2, ∀x, y ∈ H and λ ∈ 0, 1 . ii ‖x y‖2 ≤ ‖x‖2 2〈y, x y〉, ∀x, y ∈ H. Definition 2.5 see 32 . Let {Tn} be a sequence of mappings from a subset C of a real Hilbert space H into itself. We say that {Tn} satisfies the PT-condition if lim k,l→∞ ρ l 0, 2.11 where ρ l supz∈C{‖Tkz − Tlz‖} < ∞, for all k, l ∈ N. Lemma 2.6 see 32 . Suppose that {Tn} satisfies the PT-condition such that i for each x ∈ C, {Tn} is converse strongly to some point in C ii let the mapping T : C → C defined by Tx limn→∞Tnx for all x ∈ C. Then, limn→∞supω∈C‖Tω − Tnω‖ 0. Lemma 2.7 see 33 . Let C be a closed and convex subset of a strictly convex Banach space X. Let {Tn : n ∈ N} be a sequence of nonexpansive mappings on C. Suppose ⋂∞ n 1 Fix Tn is nonempty. Let {γn} be a sequence of positive numbers with ∑∞ n 1 γn 1. Then a mapping S on C defined by Sx ∑∞ n 1 γnTnx for all x ∈ C is well defined, nonexpansive, and Fix S ⋂∞ n 1 Fix Tn holds. Lemma 2.8 see 19 . Let T : C → H be a k-strict pseudocontraction. Define S : C → H by Sx δx 1−δ Tx for each x ∈ C. Then, as δ ∈ k, 1 , S is nonexpansive such that Fix S Fix T . Lemma 2.9 see 34 . Let C be a closed and convex subset of a real Hilbert spaceH and let S : C → C be a nonexpansive mapping. then, the mapping I − S is demiclosed. That is, if {xn} is a sequence in C such that xn ⇀ z and I − S xn → y, then I − S z y. Lemma 2.10 see 35 . Assume that {an} is a sequence of nonnegative real numbers such that an 1 ≤ 1 − σn an δn, 2.12 where {σn} is a sequence in 0, 1 and {δn} is a sequence in R such that i ∑∞ n 0 σn ∞; ii lim supn→∞ δn/σn ≤ 0 or ∑∞ n 0 |δn| < ∞. then, limn→∞an 0. Abstract and Applied Analysis 9and Applied Analysis 9 Lemma 2.11. Let C be a nonempty closed and convex subset of a real Hilbert spaceH. Let mappings Ψ,Φ : C → H be β̃-inverse-strongly monotone and γ̃-inverse-strongly monotone, respectively. Then, we have ∥ ∥I − μ Ψ Φ x − I − μ Ψ Φ y∥2 ≤ ∥x − y∥2 2μ ( μ − β̃ )∥ ∥Ψx −Ψy∥2 2μ ( μ − γ̃∥Φx −Φy∥2, 2.13 where μ > 0. In particular, if μ ∈ 0,min{β̃, γ̃} , then I − μ Ψ Φ is nonexpansive. Proof. From Lemma 2.4 i , for all x, y ∈ C, we have ∥ ∥I − μ Ψ Φ x − I − μ Ψ Φ y∥2 ∥ ∥x − y − μ Ψ Φ x − Ψ Φ y∥2 ∥∥∥ 1 2 (( x − y − 2μΨx −Ψy 1 2 (( x − y − 2μΦx −Φy ∥∥∥ 2 ≤ 1 2 ∥x − y − 2μΨx −Ψy∥2 1 2 ∥x − y − 2μΦx −Φy∥2 1 2 ∥x − y∥2 − 4μx − y,Ψx −Ψy 4μ2∥Ψx −Ψy∥2 ) 1 2 ∥x − y∥2 − 4μx − y,Φx −Φy 4μ2∥Φx −Φy∥2 ) ≤ 1 2 ∥x − y∥2 4μ ( μ − β̃ ∥Ψx −Ψy∥2 ) 1 2 ∥x − y∥2 4μμ − γ̃∥Φx −Φy∥2 ) ∥x − y∥2 2μ ( μ − β̃ ∥Ψx −Ψy∥2 2μμ − γ̃∥Φx −Φy∥2. 2.14 It is clear that, if 0 < μ ≤ min{β̃, γ̃}, then I − μ Ψ Φ is nonexpansive. This completes the proof. Lemma 2.12. Let C be a nonempty closed and convex subset of a real Hilbert spaceH. Let mappings Ψi,Φi : C → H i 1, 2, 3 be β̃i-inverse-strongly monotone and γ̃i-inverse-strongly monotone, respectively. Let Q : C → C be the mapping defined by Qx T F1,φ1 μ1 [ T F2,φ2 μ2 [ T F3,φ3 μ3 ( x − μ3 Ψ3 Φ3 x ) − μ2 Ψ2 Φ2 T F3,φ3 μ3 ( x − μ3 Ψ3 Φ3 x )] − μ1 Ψ1 Φ1 T F2,φ2 μ2 [ T F3,φ3 μ3 ( x − μ3 Ψ3 Φ3 x ) −μ2 Ψ2 Φ2 T F3,φ3 μ3 ( x − μ3 Ψ3 Φ3 x )]] , ∀x ∈ C. 2.15 If μi ∈ 0,min{β̃i, γ̃i} i 1, 2, 3 , then Q is nonexpansive. 10 Abstract and Applied Analysis Proof. From Lemma 2.11, for all x, y ∈ C, we have ∥ ∥Qx −Qy∥ ∥ ∥ ∥T F1,φ1 μ1 [ T F2,φ2 μ2 [ T F3,φ3 μ3 ( x − μ3 Ψ3 Φ3 x ) − μ2 Ψ2 Φ2 T F3,φ3 μ3 ( x − μ3 Ψ3 Φ3 x )] − μ1 Ψ1 Φ1 T F2,φ2 μ2 [ T F3,φ3 μ3 ( x − μ3 Ψ3 Φ3 x ) −μ2 Ψ2 Φ2 T F3,φ3 μ3 ( x − μ3 Ψ3 Φ3 x )]] − T F1,φ1 μ1 [ T F2,φ2 μ2 [ T F3,φ3 μ3 ( y − μ3 Ψ3 Φ3 y ) − μ2 Ψ2 Φ2 T F3,φ3 μ3 ( y − μ3 Ψ3 Φ3 y )] − μ1 Ψ1 Φ1 T F2,φ2 μ2 [ T F3,φ3 μ3 ( y − μ3 Ψ3 Φ3 y ) −μ2 Ψ2 Φ2 T F3,φ3 μ3 ( y − μ3 Ψ3 Φ3 y )]]∥∥ ∥ ≤ ∥∥∥T F2,φ2 μ2 [ T F3,φ3 μ3 ( x − μ3 Ψ3 Φ3 x ) − μ2 Ψ2 Φ2 T F3,φ3 μ3 ( x − μ3 Ψ3 Φ3 x )] − μ1 Ψ1 Φ1 T F2,φ2 μ2 [ T F3,φ3 μ3 ( x − μ3 Ψ3 Φ3 x ) −μ2 Ψ2 Φ2 T F3,φ3 μ3 ( x − μ3 Ψ3 Φ3 x )] − [ T F2,φ2 μ2 [ T F3,φ3 μ3 ( y − μ3 Ψ3 Φ3 y ) − μ2 Ψ2 Φ2 T F3,φ3 μ3 ( y − μ3 Ψ3 Φ3 y )] − μ1 Ψ1 Φ1 T F2,φ2 μ2 [ T F3,φ3 μ3 ( y − μ3 Ψ3 Φ3 y ) −μ2 Ψ2 Φ2 T F3,φ3 μ3 ( y − μ3 Ψ3 Φ3 y )∥∥ ∥∥T F2,φ2 μ2 ( I − μ2 Ψ2 Φ2 ) T F3,φ3 μ3 ( I − μ3 Ψ3 Φ3 )( I − μ1 Ψ1 Φ1 ) x −T F2,φ2 μ2 ( I − μ2 Ψ2 Φ2 ) T F3,φ3 μ3 ( I − μ3 Ψ3 Φ3 )( I − μ1 Ψ1 Φ1 ) y ∥∥ ≤ ∥∥∥ ( I − μ2 Ψ2 Φ2 ) T F3,φ3 μ3 ( I − μ3 Ψ3 Φ3 )( I − μ1 Ψ1 Φ1 ) x −I − μ2 Ψ2 Φ2 ) T F3,φ3 μ3 ( I − μ3 Ψ3 Φ3 )( I − μ1 Ψ1 Φ1 ) y ∥∥∥ ≤ ∥∥T F3,φ3 μ3 ( I − μ3 Ψ3 Φ3 )( I − μ1 Ψ1 Φ1 ) x −T F3,φ3 μ3 ( I − μ3 Ψ3 Φ3 )( I − μ1 Ψ1 Φ1 ) y ∥∥∥ ≤ ∥I − μ3 Ψ3 Φ3 )( I − μ1 Ψ1 Φ1 ) x − I − μ3 Ψ3 Φ3 )( I − μ1 Ψ1 Φ1 ) y ∥∥ ≤ ∥I − μ1 Ψ1 Φ1 ) x − I − μ1 Ψ1 Φ1 ) y ∥∥ ≤ ∥x − y∥, 2.16 which implies that Q : C → C is nonexpansive. This completes the proof. Abstract and Applied Analysis 11 Lemma 2.13. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let Fi : C × C → R i 1, 2, 3 be a bifunction satisfying conditions H1 – H4 and let Ψi,Φi : C → H i 1, 2, 3 be a nonlinear mapping. Suppose that μi i 1, 2, 3 be a real positive number. Let φi : C → R i 1, 2, 3 be a lower semicontinuous and convex function. Assume that either condition A1 or A2 holds. Then, for x∗, y∗, z∗ ∈ C × C × C is a solution of the problem 1.12 if and only if x∗ ∈ Fix Q , y∗ T F2,φ2 μ2 z∗ − μ2 Ψ2 Φ2 z∗ and z∗ T F3,φ3 μ3 x ∗ − μ3 Ψ3 Φ3 x∗ , where Q is the mapping defined as in Lemma 2.12. Proof. Let x∗, y∗, z∗ ∈ C × C × C be a solution of the problem 1.12 . Then, we have μ1F1 x∗, x 〈 μ1 Ψ1 Φ1 y∗ x∗ − y∗, x − x∗ 〉 ≥ μ1φ1 x∗ − μ1φ1 x , ∀x ∈ C, μ2F2 ( y∗, y ) 〈 μ2 Ψ2 Φ2 z∗ y∗ − z∗, y − y∗ 〉 ≥ μ2φ2 ( y∗ ) − μ2φ2 ( y ) , ∀y ∈ C, μ3F3 z∗, z 〈 μ3 Ψ3 Φ3 x∗ z∗ − x∗, z − z∗ 〉 ≥ μ3φ3 z∗ − μ3φ3 z , ∀z ∈ C, 2.17and Applied Analysis 11 Lemma 2.13. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let Fi : C × C → R i 1, 2, 3 be a bifunction satisfying conditions H1 – H4 and let Ψi,Φi : C → H i 1, 2, 3 be a nonlinear mapping. Suppose that μi i 1, 2, 3 be a real positive number. Let φi : C → R i 1, 2, 3 be a lower semicontinuous and convex function. Assume that either condition A1 or A2 holds. Then, for x∗, y∗, z∗ ∈ C × C × C is a solution of the problem 1.12 if and only if x∗ ∈ Fix Q , y∗ T F2,φ2 μ2 z∗ − μ2 Ψ2 Φ2 z∗ and z∗ T F3,φ3 μ3 x ∗ − μ3 Ψ3 Φ3 x∗ , where Q is the mapping defined as in Lemma 2.12. Proof. Let x∗, y∗, z∗ ∈ C × C × C be a solution of the problem 1.12 . Then, we have μ1F1 x∗, x 〈 μ1 Ψ1 Φ1 y∗ x∗ − y∗, x − x∗ 〉 ≥ μ1φ1 x∗ − μ1φ1 x , ∀x ∈ C, μ2F2 ( y∗, y ) 〈 μ2 Ψ2 Φ2 z∗ y∗ − z∗, y − y∗ 〉 ≥ μ2φ2 ( y∗ ) − μ2φ2 ( y ) , ∀y ∈ C, μ3F3 z∗, z 〈 μ3 Ψ3 Φ3 x∗ z∗ − x∗, z − z∗ 〉 ≥ μ3φ3 z∗ − μ3φ3 z , ∀z ∈ C, 2.17 ⇔ x∗ T F1,φ1 μ1 ( y∗ − μ1 Ψ1 Φ1 y∗ ) , y∗ T F2,φ2 μ2 ( z∗ − μ2 Ψ2 Φ2 z∗ ) , z∗ T F3,φ3 μ3 ( x∗ − μ3 Ψ3 Φ3 x∗ ) , 2.18 ⇔ x∗ T F1,φ1 μ1 [ T F2,φ2 μ2 [ T F3,φ3 μ3 ( x∗ − μ3 Ψ3 Φ3 x∗ ) − μ2 Ψ2 Φ2 T F3,φ3 μ3 ( x∗ − μ3 Ψ3 Φ3 x∗ )] − μ1 Ψ1 Φ1 T F2,φ2 μ2 [ T F3,φ3 μ3 ( x∗ − μ3 Ψ3 Φ3 x∗ ) −μ2 Ψ2 Φ2 T F3,φ3 μ3 ( x∗ − μ3 Ψ3 Φ3 x∗ )]] . 2.19 This completes the proof. Corollary 2.14 see 20 . Let C be a nonempty closed and convex subset of a real Hilbert space H. Let Fi : C × C → R i 1, 2 be a bifunction satisfying conditions H1 – H4 and let Ψi,Φi : C → H i 1, 2 be a nonlinear mapping. Suppose that μi i 1, 2 be a real positive number. Let φi : C → R i 1, 2 be a lower semicontinuous and convex function. Assume that either condition A1 or A2 holds. Then, for x∗, y∗ ∈ C × C is a solution of the problem 1.14 if and only if x∗ ∈ Fix Q , y∗ T F2,φ2 μ2 x∗ − μ2 Ψ2 Φ2 x∗ , where G is the mapping defined by Qx T F1,φ1 μ1 [ T F2,φ2 μ2 ( x − μ2 Ψ2 Φ2 x ) − μ1 Ψ1 Φ1 T F2,φ2 μ2 ( x − μ2 Ψ2 Φ2 x )] , ∀x ∈ C. 2.20 Corollary 2.15 see 21 . Let C be a nonempty closed and convex subset of a real Hilbert space H. Let Fi : C × C → R i 1, 2 be a bifunction satisfying conditions H1 – H4 and let Ψi : C → 12 Abstract and Applied Analysis H i 1, 2 be a nonlinear mapping. Suppose that μi i 1, 2 is a real positive number. Assume that either condition A1 or A2 holds. Then, for x∗, y∗ ∈ C × C is a solution of the problem 1.15 if and only if x∗ ∈ Fix Q , y∗ T2 μ2 x∗ − μ2Ψ2x∗ , where Q is the mapping defined by Qx T1 μ1 [ T2 μ2 ( x − μ2Ψ2x ) − μ1Ψ1T2 μ2 ( x − μ2Ψ2x )] , ∀x ∈ C. 2.21 Corollary 2.16 see 23 . Let C be a nonempty closed and convex subset of a real Hilbert space H. For given x∗, y∗, z∗ ∈ C × C × C is a solution of the problem 1.17 if and only if x∗ ∈ Fix Q , y∗ PC z∗ − μ2Ψ2z∗ and z∗ PC x∗ − μ3Ψ3x∗ , where Q is the mapping defined by Qx PC PC [ PC ( x − μ3Ψ3x ) − μ2Ψ2PC ( x − μ3Ψx )] −μ1Ψ1PC [ PC ( x − μ3Ψ3x ) − μ2Ψ2PC ( x − μ3Ψ3x )]] , ∀x ∈ C. 2.22 Corollary 2.17 see 24 . Let C be a nonempty closed and convex subset of a real Hilbert space H. For given x∗, y∗ ∈ C × C is a solution of the problem 1.18 if and only if x∗ ∈ Fix Q , y∗ PC x∗ − μ2Ψ2x∗ , where Q is the mapping defined by Qx PC [ PC ( x − μ2Ψ2x ) − μ1Ψ1PC ( x − μ2Ψ2x )] , ∀x ∈ C. 2.23 3. Main Results We are now in a position to prove the main result of this paper. Theorem 3.1. Let C be a nonempty closed and convex subset of a real Hilbert space H such that C ± C ⊂ C. Let φ, φi : C → R i 1, 2, 3 be lower semicontinuous and convex functionals, Θ : H × C × C → R be an equilibrium-like function, G : C → H be a mapping, and Fi : C × C → R i 1, 2, 3 be a bifunction satisfying conditions H1 – H4 . Assume that either condition A1 or A2 holds. Let B : C → H be β-inverse-strongly monotone, Ψi,Φi : C → H i 1, 2, 3 be β̃i-inverse-strongly monotone and γ̃i-inverse-strongly monotone, respectively. Let {Tn}n 1 : C → C be a family of k-strict pseudocontraction mappings. Define a mapping Snx : δx 1 − δ Tnx, for all x ∈ C, δ ∈ k, 1 and n ≥ 1. Assume that the condition Δ is satisfied and Ω : ∞n 1 Fix Tn ∩ Fix Q ∩ GEP C,G,Θ, φ / ∅, where Q is defined as in Lemma 2.13. Let μ > 0, γ > 0, and r > 0 be three constants. Let f : C → C be a contraction mapping with a coefficient α ∈ 0, 1 and let A be a strongly positive bounded linear operator on C with a coefficient γ ∈ 0, 1 such that 0 < γ < 1 μ γ /α. For x1 ∈ C, let the sequence {xn} defined by Θ ( Gxn, un, y ) φ ( y ) − φ un 1 r 〈 y − un, un − xn − rBxn 〉 ≥ 0, ∀y ∈ C, zn T F3,φ3 μ3 ( xn − μ3 Ψ3 Φ3 xn ) , yn T F2,φ2 μ2 ( zn − μ2 Ψ2 Φ2 zn ) , vn T F1,φ1 μ1 ( yn − μ1 Ψ1 Φ1 yn ) , xn 1 αnγf xn βnxn (( 1 − βn ) I − αn ( I μA ))[ γ1Snxn γ2un γ3vn ] , ∀n ≥ 1, 3.1 Abstract and Applied Analysis 13 where γ1, γ2, γ3 ∈ 0, 1 such that γ1 γ2 γ3 1, μ1 ∈ 0,min{β̃1, γ̃1} , μ2 ∈ 0,min{β̃2, γ̃2} , μ3 ∈ 0,min{β̃3, γ̃3} , r ∈ 0, 2β and {αn}, {βn} are two sequences in 0, 1 . Suppose that {Tn} satisfies the PT-condition. Let T : C → C be the mapping defined by Tx limn→∞Tnx for all x ∈ C and suppose that Fix T ⋂∞ n 1 Fix Tn . Assume the following conditions are satisfied: C1 limn→∞αn 0 and ∑∞ n 1 αn ∞, C2 0 < lim infn→∞βn ≤ lim supn→∞βn < 1. Then the sequence {xn} defined by 3.1 converges strongly to x̂ ∈ Ω, where x̂ is the unique solution of the variational inequality 〈γf − I μAx̂, v − x̂〉 ≤ 0, ∀v ∈ Ω, 3.2and Applied Analysis 13 where γ1, γ2, γ3 ∈ 0, 1 such that γ1 γ2 γ3 1, μ1 ∈ 0,min{β̃1, γ̃1} , μ2 ∈ 0,min{β̃2, γ̃2} , μ3 ∈ 0,min{β̃3, γ̃3} , r ∈ 0, 2β and {αn}, {βn} are two sequences in 0, 1 . Suppose that {Tn} satisfies the PT-condition. Let T : C → C be the mapping defined by Tx limn→∞Tnx for all x ∈ C and suppose that Fix T ⋂∞ n 1 Fix Tn . Assume the following conditions are satisfied: C1 limn→∞αn 0 and ∑∞ n 1 αn ∞, C2 0 < lim infn→∞βn ≤ lim supn→∞βn < 1. Then the sequence {xn} defined by 3.1 converges strongly to x̂ ∈ Ω, where x̂ is the unique solution of the variational inequality 〈γf − I μAx̂, v − x̂〉 ≤ 0, ∀v ∈ Ω, 3.2 or equivalently, x̂ PΩ γf − μA x̂, where PΩ is a metric projection mapping from C onto Ω, and x̂, ŷ, ẑ is a solution of the problem 1.12 , where ŷ T F2,φ2 μ2 ẑ−μ2 Ψ2 Φ2 ẑ and ẑ T F3,φ3 μ3 x̂− μ3 Ψ3 Φ3 x̂ . Proof. Note that from the conditions C1 and C2 , we may assume, without loss of generality, that αn ≤ 1−βn 1 μ‖A‖ −1 for all n ∈ N. SinceA is a linear bounded self-adjoint operator on C, by 2.2 , we have ‖A‖ sup{|〈Au, u〉| : u ∈ C, ‖u‖ 1}. 3.3 Observe that 〈(( 1 − βn ) I − αn ( I μA )) u, u 〉 1 − βn − αn − αnμ〈Au, u〉 ≥ 1 − βn − αn − αnμ‖A‖ ≥ 0. 3.4 This show that 1 − βn I − αn I μA is positive. It follows that ∥1 − βn ) I − αn ( I μA )∥ sup ∣1 − βn ) I − αn ( I μA )) u, u 〉∣ : u ∈ C, ‖u‖ 1 sup { 1 − βn − αn − αnμ〈Au, u〉 : u ∈ C, ‖u‖ 1 } ≤ 1 − βn − αn ( 1 μγ ) ≤ 1 − βn − αn ( 1 μ ) γ. 3.5 First, we show that {xn} is bounded. Taking x∗ ∈ Ω, it follows from Lemma 2.13 that x∗ T F1,φ1 μ1 [ T F2,φ2 μ2 [ T F3,φ3 μ3 ( x∗ − μ3 Ψ3 Φ3 x∗ ) − μ2 Ψ2 Φ2 T F3,φ3 μ3 ( x∗ − μ3 Ψ3 Φ3 x∗ )] − μ1 Ψ1 Φ1 T F2,φ2 μ2 [ T F3,φ3 μ3 ( x∗ − μ3 Ψ3 Φ3 x∗ ) −μ2 Ψ2 Φ2 T F3,φ3 μ3 ( x∗ − μ3 Ψ3 Φ3 x∗ )]] . 3.6 14 Abstract and Applied Analysis Putting y∗ T F2,φ2 μ2 z ∗ − μ2 Ψ2 Φ2 z∗ and z∗ T F3,φ3 μ3 x∗ − μ3 Ψ3 Φ3 x∗ , we obtain x∗ T F1,φ1 μ1 y ∗ − μ1 Ψ1 Φ1 y∗ . Notice that un T r I − rB xn. Since T r is nonexpansive and B is β-inverse-strongly monotone, we have ‖un − x∗‖2 ∥ ∥ ∥T r I − rB xn − T r I − rB x∗ ∥ ∥ ∥ 2 ≤ ‖ I − rB xn − I − rB x∗‖2 ‖ xn − x∗ − r Bxn − Bx∗ ‖ ‖xn − x∗‖2 − 2r〈xn − x∗, Bxn − Bx∗〉 r‖Bxn − Bx∗‖2 ≤ ‖xn − x∗‖2 r ( r − 2β‖Bxn − Bx∗‖2 ≤ ‖xn − x∗‖2, 3.7