A System of Generalized Mixed Equilibrium Problems, Maximal Monotone Operators, and Fixed Point Problems with Application to Optimization Problems

and Applied Analysis 3 The generalized mixed equilibrium problems with perturbation is very general in the sense that it includes fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases see, e.g., 4, 5 . Numerous problems in physics, optimization, and economics reduce to find a solution of problem 1.2 . Several methods have been proposed to solve the fixed point problems, variational inequality problems, and equilibrium problems in the literature see, e.g., 6–34 . Let A be a strongly positive bounded linear operator on H; that is, there exists a constant γ > 0 such that 〈Ax, x〉 ≥ γ‖x‖, ∀x ∈ H. 1.7 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. 1.8 A mapping T : C → C is said to be 1 nonexpansive if ∥Tx − Ty∥ ≤ ∥x − y∥, ∀x, y ∈ C, 1.9 2 firmly nonexpansive if ∥Tx − Ty ∥2 ≤ Tx − Ty, x − y, ∀x, y ∈ C, 1.10 3 k-strictly pseudocontractive if there exists a constant k ∈ 0, 1 such that ∥Tx − Ty∥2 ≤ ∥x − y∥2 k∥ I − T x − I − T y∥2, ∀x, y ∈ C. 1.11 We denote by F T the set of fixed points of T , that is, F T {x ∈ C : x Tx}. Recall the following definitions of a nonlinear mapping B : C → H; the following is mentioned. Definition 1.1. The nonlinear mapping B : C → H is said to be i monotone if 〈 Bx − By, x − y ≥ 0, ∀x, y ∈ C, 1.12 ii β-strongly monotone if there exists a constant β > 0 such that 〈 Bx − By, x − y ≥ β∥x − y∥2, ∀x, y ∈ C, 1.13 4 Abstract and Applied Analysis iii ν-inverse-strongly monotone if there exists a constant ν > 0 such that 〈 Bx − By, x − y ≥ ν∥Bx − By∥2, ∀x, y ∈ C. 1.14 LetW : H → 2 be a set-valuedmapping. The setD W defined byD W {x ∈ H : Wx/ ∅} is said to be the domain of W . The set R W defined by R W x∈H Wx is said to be the range ofW . The set G W defined by G W { x, y ∈ H ×H : x ∈ D W , y ∈ R W } is said to be the graph ofW . Recall that W is said to be monotone if 〈 x − y, f − g > 0, ∀x, f, y, g ∈ G W . 1.15 W is said to be maximal monotone if it is not properly contained in any other monotone operator. Equivalently, W is maximal monotone if R I rW H for all r > 0. For a maximal monotone operator M on H and r > 0, we may define the single-valued resolvent Jr I rW −1 : H → D W . It is known that Jr is firmly nonexpansive W−1 0 F Jr , where F Jr denotes the fixed point set of Jr . We discuss the following algorithms for solving the solutions of variational inequality problems and fixed point problems for a nonexpansive mapping see, e.g., 29, 35–43 . In 2010, Chantarangsi et al. 44 introduced a new viscosity hybrid steepest descent method for solving the generalized mixed equilibrium problems 1.2 , variational inequality problems, and fixed point problems of nonexpansive mappings in a real Hilbert space. More precisely, they proved the following theorem. Theorem CCK [see [44]] Let C be a nonempty closed and convex subset of a real Hilbert space H. Let Θ1, Θ2 be two bifunctions satisfying condition H1 – H5 , let Ψ1, Ψ2 be ξ-inverse-strongly monotone mapping and β-inverse-strongly monotone mapping, respectively, and let T : C → C be a nonexpansive mapping. Let B be an ω-Lipschitz continuous and relaxed v, ν cocoercive mapping, f : C → C a contraction mapping with coefficient α ∈ 0, 1 , and A a strongly positive linear bounded self-adjoint operator with coefficient γ > 0 and 0 < γ < γ/α. Suppose that Ω : F T ∩ GMEP Θ1, φ,Ψ1 ∩ GMEP Θ2, φ,Ψ2 ∩ VI C,B . Let {zn}, {un}, {yn}, and {xn} be generated by un V Θ2,φ2 rn xn − rnΨ2xn , vn V Θ1,φ1 μn ( un − μnΨ1un ) , zn PC vn − αnBTvn , yn nγf xn βnxn (( 1 − βn ) I − nA ) zn, xn 1 γnxn ( 1 − γn ) yn, ∀n ≥ 1, 1.16 where {γn} ⊂ a, b ⊂ 0, 2ξ , {sn} ⊂ c, d ⊂ 0, 2β , {γn} ⊂ h, j ⊂ 0, 1 , {γn}, { n}, and {βn} are three sequences in 0, 1 satisfying the following conditions: Abstract and Applied Analysis 5 C1 limn→∞ n 0 and ∑∞ n 1 n ∞, C2 0 < lim infn→∞βn ≤ lim supn→∞βn < 1, C3 0 < lim infn→∞μn ≤ lim supn→∞μn < 2β and limn→∞|μn 1 − μn| 0, C4 0 < lim infn→∞rn ≤ lim supn→∞rn < 2ξ and limn→∞|rn 1 − rn| 0, C5 {αn} ⊂ e, g ⊂ 0, 2 ν − vω2 /ω2 , ν > vω2 and limn→∞|αn 1 − αn| 0. Then, {xn} converges strongly to x∗ PΩ γf I −A x∗ . Very recently, Yu and Liang 45 proved the following convergence theorem of finding a common element in the fixed point set of a strict pseudocontraction and in the zero set of a nonlinear mapping which is the sum of a maximal monotone operator and inverse strongly monotone mapping in a real Hilbert space.and Applied Analysis 5 C1 limn→∞ n 0 and ∑∞ n 1 n ∞, C2 0 < lim infn→∞βn ≤ lim supn→∞βn < 1, C3 0 < lim infn→∞μn ≤ lim supn→∞μn < 2β and limn→∞|μn 1 − μn| 0, C4 0 < lim infn→∞rn ≤ lim supn→∞rn < 2ξ and limn→∞|rn 1 − rn| 0, C5 {αn} ⊂ e, g ⊂ 0, 2 ν − vω2 /ω2 , ν > vω2 and limn→∞|αn 1 − αn| 0. Then, {xn} converges strongly to x∗ PΩ γf I −A x∗ . Very recently, Yu and Liang 45 proved the following convergence theorem of finding a common element in the fixed point set of a strict pseudocontraction and in the zero set of a nonlinear mapping which is the sum of a maximal monotone operator and inverse strongly monotone mapping in a real Hilbert space. Theorem YL [see [45]] Let H be a real Hilbert space and C a nonempty close and convex subset of H. Let W1 : H → 2 and W2 : H → 2 be two maximal monotone operators such that D W1 ⊂ C and D W2 ⊂ C, respectively. Let S : C → C be a k-strict pseudocontractionmapping,A : C → H an α-inverse-strongly monotone mapping, and B : C → H an β-inverse-strongly monotone mapping. Assume that Ω : F S ∩ A W1 −1 0 ∩ B W2 −1 0 / ∅. Let {xn} be a sequence generated by x1 ∈ C, yn Jtn xn − tnBxn , xn 1 αnu βnxn γn [ δnJsn ( yn − snAyn ) 1 − δn Jsn ( yn − snAyn )] , ∀n ≥ 1, 1.17 where u ∈ C is a fixed element, Jsn I snW1 −1, Jtn I tnW2 −1, {sn} is a sequence in 0, 2α , {tn} is a sequence in 0, 2β , and {αn}, {βn}, {γn}, and {δn} are sequences in 0, 1 satisfying the following conditions: C1 limn→∞αn 0 and ∑∞ n 1αn ∞, C2 0 < lim infn→∞βn ≤ lim supn→∞βn < 1, C3 0 < a ≤ sn ≤ b < 2α and limn→∞ sn 1 − sn 0, C4 0 < c ≤ tn ≤ d < 2β and limn→∞ tn 1 − tn 0, C5 0 < c ≤ k ≤ δn < e < 1 and limn→∞ δn 1 − δn 0. Then, the sequence {xn} converges strongly to x∗ PΩx∗. On the other hand, the following optimization problem has been studied extensively by many authors: min x∈Ω μ 2 〈Ax, x〉 1 2 ‖x − u‖ − h x , 1.18 where Ω ⋂∞ n 1 Cn,C1, C2, . . . are infinitely many closed convex subsets of H such that ⋂∞ n 1 Cn / ∅, u ∈ H, μ ≥ 0 is a real number, A is a strongly positive linear bounded operator 6 Abstract and Applied Analysis on H, and h is a potential function for γf i.e., h′ x γf x for all x ∈ H . This kind of optimization problem has been studied extensively by many authors see, e.g., 5, 46–52 for when Ω ⋂∞ n 1 Cn and h x 〈x, b〉, where b is a given point inH. The following questions naturally arise in connection with above the results. Question 1. Could we weaken the control conditions of Theorems CCK and YL in C3 and C4 ? Question 2. Can Theorem YL be extended to finding a common element of the set of solutions of a system generalized mixed equilibrium problems and the set of common fixed points of infinite family of nonexpansive mappings? The purpose of this paper is to give the affirmative answers to these questions mentioned above. Motivated by the iterative process 1.16 and 1.17 , we introduce a new iterative algorithm 3.2 below, for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse-strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Then, we prove the strong convergence theorem of these iterative process in a real Hilbert space. The results presented in the paper improve and extend the recent ones announced by many others. 2. Preliminaries Definition 2.1 see 53 . Let C be a nonempty convex subset of a real Hilbert spaceH. Let Ti, i 1, 2, . . ., be mappings of C into itself. For each j 1, 2, . . ., let αj α j 1, α j 2, α j 3 ∈ I × I × I, where I 0, 1 and αj1 α j 2 α j 3 1. For every n ∈ N, we define the mapping Sn : C → C as follows: Un,n 1 I, Un,n αn1TnUn,n 1 α n 2Un,n 1 α n 3I, Un,n−1 αn−1 1 Tn−1Un,n α n−1 2 Un,n α n−1 3 I, .. Un,k 1 α 1 1 Tk 1Un,k 2 α k 1 2 Un,k 2 α k 1 3 I, Un,k αk1TkUn,k 1 α k 2Un,k 1 α k 3I, .. Un,2 α1T2Un,3 α 2 2Un,3 α 2 3I, Sn Un,1 α1T1Un,2 α 1 2Un,2 α 1 3I. 2.1 Abstract and Applied Analysis 7and Applied Analysis 7 Such a mapping Sn is nonexpansive from C into itself, and it is called S-mapping generated by Tn, Tn−1, . . . , T1 and αn, αn−1, . . . , α1. Lemma 2.2 see 53 . Let C be a nonempty closed convex subset of a real Hilbert space H. Let {Ti}i 1 be nonexpansive mappings of C into itself with F Ti / ∅ and let αj αj1, α j 2, α j 3 ∈ I × I × I, where I 0, 1 , αj1 α j 2 α j 3 1, α j 1 α j 2 ≤ b < 1, and α j 1, α j 2, α j 3 ∈ 0, 1 for all j 1, 2, . . . . For all n ∈ N, let Sn and S be S-mappings generated by Tn, Tn−1, . . . , T1 and αn, αn−1, . . . , α1 and Tn, Tn−1, . . . and αn, αn−1, . . ., respectively. Then, i Sn is nonexpansive and F Sn ⋂n i 1 F Ti , for all n ≥ 1, ii for all x ∈ C and for all positive integer k, the limn→∞Un,k exists, iii the mapping S : C → C defined by Sx : lim n→∞ Snx lim n→∞ Un,1x, ∀x ∈ C 2.2 is a nonexpansive mapping such that F S ⋂∞ i 1 F Ti , and it is called the S-mapping generated by Tn, Tn−1, . . . and αn, αn−1, . . ., iv if K is any bounded subset of C, then lim n→∞ sup x∈K ‖Snx − Sx‖ 0. 2.3 Lemma 2.3 see 54 . 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 that 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 55 . 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. Lemma 2.5 see 56 . Let C be a nonempty closed convex subset of a real Hilbert spaceH,A : C → H a mapping, and W : H → 2 a maximal monotone mapping. Then, F Jr I − rA A W −1 0 , ∀r > 0. 2.4 Lemma 2.6 see 57 . Let H be a real Hilbert space and let M be a maximal monotone operator on H. For r > 0 and x ∈ H, define the resolvent Jrx. Then, the following holds: s − t s 〈Jsx − Jtx, Jsx − x〉 ≥ ‖Jsx − Jtx‖ 2.5 for all s, t > 0 and x ∈ H. For solving the equilibrium problem for bifunction Θ : C × C → R, let us assume that Θ satisfies the following conditions: 8 Abstract and Applied Analysis H1 Θ x, x 0 for all x ∈ C, H2 Θ is monotone; that is, Θ x, y Θ y, x ≤ 0 for all x, y ∈ C, H3 for each y ∈ C, x → Θ x, y is concave and upper semicontinuous, H4 for each y ∈ C, x → Θ x, y is convex, H5 for each y ∈ C, x → Θ x, y is lower semicontinuous. Definition 2.7. A differentiable function K : C → R on a convex set C is called i convex 2 if K ( y ) −K x ≥ K′ x , y − x, ∀x, y ∈ C, 2.6 where K′ x is the Fréchet differentiable of K at x; ii strongly convex 2 if there exists a constant σ > 0 such that K ( y ) −K x − K′ x , x − y ≥ (σ 2 ∥x − y∥2, ∀x, y ∈ C. 2.7 It is easy to see that if K : C → R is a differentiable strongly convex function with constant σ > 0, then K′ : C → H is strongly monotone with constant σ > 0. Let Θ : C × C → R be an equilibrium bifunction satisfying the conditions H1 – H5 . Let r be any given positive number. For a given point x ∈ C, consider the auxiliary mixed equilibrium problem to finding y ∈ C such that Θ ( y, z ) φ z − φy 1 r 〈 K′ ( y ) −K′ x , z − y ≥ 0, ∀z ∈ C, 2.8 where K′ x is the Fréchet differentiable of K at x. Let V Θ,φ r : C → C be the mapping such that for each x ∈ C, V Θ,φ r x is the set of solutions of MEP x, y , that is, V Θ,φ r x { y ∈ C : Θy, z φ z − φy 1 r 〈 K′ ( y ) −K′ x , z − y ≥ 0, ∀z ∈ C } . 2.9 Then, the following conclusion holds. Lemma 2.8 see 58 . 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 functional. Let Θ : C × C → R be a bifunction satisfying the conditions (H1)–(H5). Assume that i K : C → R is strongly convex with constant σ > 0 and the function x → 〈y − x,K′ x 〉 is weakly upper semicontinuous for each y ∈ C; ii for each x ∈ C, there exist a bounded subset Dx ⊂ C and zx such that for all y / ∈ Dx, Θ ( y, zx ) φ zx − φ ( y ) 1 r 〈 K′ ( y ) −K′ x , zx − y 〉 < 0. 2.10 Abstract and Applied Analysis 9and Applied Analysis 9 Then, the following holds: a V Θ,φ r is single-valued mapping; b V Θ,φ r is nonexpansive if K′ is Lipschitz continuous with constant ν > 0 and 〈K′ x1 −K′ x2 , u1 − u2〉 ≥ 〈K′ u1 −K′ u2 , u1 − u2〉, ∀x1, x2 ∈ C, 2.11 where ui V Θ,φ r xi for i 1, 2; c F V Θ,φ r MEP Θ, φ ; d MEP Θ, φ is closed and convex. In particular, whenever Θ : C × C → R is a bifunction satisfying the conditions H1 – H5 and K x ‖x‖2/2, for all x ∈ C, then V Θ,φ r is firmly nonexpansive; that is, for any x, y ∈ C, ∥∥∥V Θ,φ r x − V Θ,φ r y ∥∥∥ 2 ≤ 〈 V Θ,φ r x − V Θ,φ r y, x − y 〉 . 2.12 In this case, V Θ,φ r is rewritten as T Θ,φ r . If, in addition, φ ≡ 0, then T Θ,φ r is rewritten as TΘ r see 59, Lemma 2.1 for more details . Remark 2.9. We remark that Lemma 2.8 is not a consequence of 2, Lemma 3.1 because the condition of the sequential continuity from the weak topology to the strong topology for the derivative K′ of the function K : C → R does not cover the case K x ‖x‖2/2. Lemma 2.10. Let C,H,Θ, φ, and V Θ,φ r be as in Lemma 2.8. Then, the following holds: 〈 K′ ( V Θ,φ s x ) −K′ ( V Θ,φ t x ) , V Θ,φ s x − V Θ,φ t x 〉 ≤ s − t s 〈 K′ ( V Θ,φ s x ) −K′ x , V Θ,φ s x − V Θ,φ t x 〉 , 2.13 for all s, t > 0 and x ∈ C. Proof. By similar argument as in the proof of Proposition 1 in 58 , for all s, t > 0 and x ∈ C, let u V Θ,φ s x and v V Θ,φ t x; we have Θ ( u, y ) φ ( y ) − φ u 1 s 〈 K′ u −K′ x , y − u ≥ 0, ∀x ∈ C, 2.14 Θ ( v, y ) φ ( y ) − φ v 1 t 〈 K′ v −K′ x , y − v ≥ 0, ∀x ∈ C. 2.15 10 Abstract and Applied Analysis Let y v in 2.14 and y u in 2.15 ; we have Θ u, v φ v − φ u 1 s 〈 K′ u −K′ x , v − u ≥ 0, Θ v, u φ u − φ v 1 t 〈 K′ v −K′ x , u − v ≥ 0. 2.16 Adding up the last two inequalities and from the monotonicity of Θ, we obtain that 1 s 〈K′ u −K′ x , v − u〉 1 t 〈K′ v −K′ x , u − v〉 ≥ 0. 2.17 It follows that 〈 K′ v −K′ x t − K ′ u −K′ x s , u − v 〉 ≥ 0. 2.18 We derive from 2.18 that 0 ≤ 〈 K′ v −K′ x − t s ( K′ u −K′ x , u − v 〉 〈 K′ v −K′ u K′ u −K′ x − t s ( K′ u −K′ x , u − v 〉 〈 K′ v −K′ u ( 1 − t s ) ( K′ u −K′ x , u − v 〉 . 2.19 Hence, we obtain that 〈K′ u −K′ v , u − v〉 ≤ s − t s 〈K′ u −K′ x , u − v〉. 2.20 The following lemma can be found in 60, 61 see also 62, Lemma 2.2 . Lemma 2.11. Let C be a nonempty closed convex subset of a real Hilbert space H and g : C → R∪{ ∞} a proper lower semicontinuous differentiable convex function differentiable convex function. If x∗ is a solution to the minimization problem g x∗ inf x∈C g x , 2.21