An Iterative Method for Solving a System of Mixed Equilibrium Problems, System of Quasivariational Inclusions, and Fixed Point Problems of Nonexpansive Semigroups with Application to Optimization Problems

and Applied Analysis 3 Definition 1.1. A one-parameter family mapping S {T t : t ∈ R } from C into itself is said to be a nonexpansive semigroup on C if it satisfies the following conditions: i T 0 x x for all x ∈ C, ii T s t T s ◦ T t for all s, t ∈ R , iii for each x ∈ C the mapping t → T t x is continuous, iv ‖T t x − T t y‖ ≤ ‖x − y‖ for all x, y ∈ C and t ∈ R . Remark 1.2. We denote by F S the set of all common fixed points of S, that is, F S : ⋂ t∈R F T t {x ∈ C : T t x x}. Recall the following definitions of a nonlinear mapping B : C → H, the following are mentioned. Definition 1.3. The nonlinear mapping B : C → H is said to be i monotone if 〈 Bx − By, x − y ≥ 0, ∀x, y ∈ C, 1.9 ii β-strongly monotone if there exists a constant β > 0 such that 〈 Bx − By, x − y ≥ β∥x − y∥2, ∀x, y ∈ C, 1.10 iii L-Lipschitz continuous if there exists a constant L > 0 such that ∥Bx − By∥ ≤ L∥x − y∥, ∀x, y ∈ C, 1.11 iv ν-inverse-strongly monotone if there exists a constant ν > 0 such that 〈 Bx − By, x − y ≥ ν∥Bx − By∥2, ∀x, y ∈ C, 1.12 v relaxed c, d -cocoercive if there exists a constants c, d > 0 such that 〈 Bx − By, x − y ≥ −c ∥Bx − By∥2 d∥dx − y∥2, ∀x, y ∈ C. 1.13 The resolvent operator technique for solving variational inequalities and variational inclusions is interesting and important. The resolvent equation technique is used to develop powerful and efficient numerical techniques for solving various classes of variational inequalities, inclusions, and related optimization problems. Definition 1.4. Let M : H → 2 be a multivalued maximal monotone mapping. The singlevalued mapping J M,ρ : H → H, defined by J M,ρ u ( I ρM )−1 u , ∀u ∈ H, 1.14 4 Abstract and Applied Analysis is called resolvent operator associated with M, where ρ is any positive number and I is the identity mapping. Next, we consider a system of quasivariational inclusions problem is to find x∗, y∗ ∈ H ×H such that 0 ∈ x∗ − y∗ ρ1 ( B1y ∗ M1x∗ ) , 0 ∈ y∗ − x∗ ρ2 ( B2x ∗ M2y∗ ) , 1.15 where Bi : H → H and Mi : H → 2 are nonlinear mappings for each i 1, 2. As special cases of the problem 1.15 , we have the following results. 1 If B1 B2 B and M1 M2 M, then the problem 1.15 is reduces to the following. Find x∗, y∗ ∈ H ×H such that 0 ∈ x∗ − y∗ ρ1 ( By∗ Mx∗ ) , 0 ∈ y∗ − x∗ ρ2 ( Bx∗ My∗ ) . 1.16 2 Further, if x∗ y∗ in problem 1.16 , then the problem 1.16 is reduces to the following. Find x∗ ∈ H such that 0 ∈ Bx∗ Mx∗. 1.17 The problem 1.17 is called variational inclusion problem. We denote by VI H,B,M the set of solutions of the variational inclusion problem 1.17 . Next, we consider two special cases of the problem 1.17 . 1 M ∂φ : H → 2 , where φ : H → R ∪ { ∞} is a proper convex lower semicontinuous function and ∂φ is the subdifferential of φ then the quasivariational inclusion problem 1.17 is equivalent to finding x∗ ∈ H such that 〈Bx∗, x − x∗〉 φ x −φ x∗ ≥ 0, for all x ∈ H, which is said to be themixed quasivariational inequality. 2 If M ∂δC, where C is a nonempty closed convex subset of H, and δC : H → 0,∞ is the indicator function of C, that is, δC x ⎧ ⎨ ⎩ 0, x ∈ C, ∞, x / ∈ C, 1.18 then the quasivariational inclusion problem 1.17 is equivalent to the classical variational inequality problem denoted by VI C,B which is to find x∗ ∈ C such that 〈Bx∗, x − x∗〉 ≥ 0, ∀x ∈ C. 1.19 This problem is called Hartman-Stampacchia variational inequality problem see e.g., 22–24 . Abstract and Applied Analysis 5 It is known that problem 1.17 provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity, variational inequalities, optimal control, mathematical economics, equilibria, and game theory. Also various types of variational inclusions problems have been extended and generalized see 25–40 and the references therein . On the other hand, the following optimization problem has been studied extensively by many authors:and Applied Analysis 5 It is known that problem 1.17 provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity, variational inequalities, optimal control, mathematical economics, equilibria, and game theory. Also various types of variational inclusions problems have been extended and generalized see 25–40 and the references therein . 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 , 1.20 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 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. 41–44 when Ω ⋂∞ n 1 Cn and h x 〈x, b〉, where b is a given point inH. Li et al. 45 introduced two steps of iterative procedures for the approximation of common fixed point of a nonexpansive semigroup S {T t : t ∈ R } on a nonempty closed convex subset C in a Hilbert space. Recently, Liu et al. 46 introduced a hybrid iterative scheme for finding a common element of the set of solutions of system of mixed equilibrium problems, the set of common fixed points for nonexpansive semigroup and the set of solution of quasivariational inclusions with multivalued maximal monotone mappings and inversestrongly monotone mappings. Very recently, Hao 47 introduced a general iterative method for finding a common element of solution set of quasivariational inclusion problems and the set of common fixed points of an infinite family of nonexpansive mappings. In this paper, motivated and inspired by Li et al. 45 , Liu et al. 46 , and Hao 47 , we introduce a general implicit iterative algorithm base on viscosity approximation methods with a φ-strongly pseudocontractive mapping which is more general than a contraction mapping for finding a common element of the set of solutions for a system of mixed equilibrium problems, the set of common fixed point for a nonexpansive semigroup, and the set of solutions of system of variational inclusions 1.15 with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mappings in Hilbert spaces. We prove that the proposed iterative algorithm converges strongly to a common element of the above three sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator. The results obtained in this paper extend and improve several recent results in this area. 2. Preliminaries In the sequel, we use xn ⇀ x and xn → x to denote the weak convergence and strong convergence of the sequence {xn} inH, respectively. This collects some results that will be used in the proofs of our main results. Proposition 2.1 see 21 . i The resolvent operator J M,ρ associated withM is single-valued and nonexpansive for all ρ > 0, that is, ∥J M,ρ x − J M,ρ ( y )∥ ≤ ∥x − y∥, ∀x, y ∈ H, ∀ρ > 0. 2.1 6 Abstract and Applied Analysis ii The resolvent operator J M,ρ is 1-inverse-strongly monotone, that is, ∥ ∥J M,ρ x − J M,ρ ( y )∥2 ≤ x − y, J M,ρ x − J M,ρ ( y )〉 , ∀x, y ∈ H. 2.2 Obviously, this immediately implies that ∥ ∥x − y − J M,ρ x − J M,ρ ( y ))∥2 ≤ ∥x − y∥2 − ∥J M,ρ x − J M,ρ ( y ∥2, ∀x, y ∈ H. 2.3 For solving the equilibrium problem for bifunctionΘ : H ×H → R, let us assume that satisfies the following conditions: H1 Θ x, x 0 for all x ∈ H; H2 Θ is monotone, that is, Θ x, y Θ y, x ≤ 0 for all x, y ∈ H; H3 for each y ∈ H, x → Θ x, y is concave and upper semicontinuous; H4 for each y ∈ H, x → Θ x, y is convex; H5 for each y ∈ H, x → Θ x, y is lower semicontinuous. Definition 2.2. Amap η : H ×H → H is called Lipschitz continuous, if there exists a constant L > 0 such that ∥η ( x, y )∥ ≤ L∥x − y∥, ∀x, y ∈ H. 2.4 A differentiable function K : H → R on a convex set H is called i η—convex 7 if K ( y ) −K x ≥ K′ x , ηy, x, ∀x, y ∈ H, 2.5 where K′ x is the Fréchet differentiable of K at x, i η—strongly convex 7 if there exists a constant ν > 0 such that K ( y ) −K x − K′ x , ηx, y ≥ (ν 2 ∥x − y∥2, ∀x, y ∈ H. 2.6 Let Θ : H × H → R be an equilibrium bifunction satisfying the conditions H1 – H5 . Let r be any given positive number. For a given point x ∈ H, consider the following auxiliary problem for MEP for short, MEP x, y to find y ∈ H such that Θ ( y, z ) φ z − φy 1 r 〈 K′ ( y ) −K′ x , ηz, y ≥ 0, ∀z ∈ H, 2.7 Abstract and Applied Analysis 7 where η : H × H → H is a mapping, and K′ x is the Fréchet derivative of a functional K : H → R at x. Let V Θ,φ r : H → H be the mapping such that for each x ∈ H, V Θ,φ r x is the set of solutions of MEP x, y , that is,and Applied Analysis 7 where η : H × H → H is a mapping, and K′ x is the Fréchet derivative of a functional K : H → R at x. Let V Θ,φ r : H → H be the mapping such that for each x ∈ H, V Θ,φ r x is the set of solutions of MEP x, y , that is, V Θ,φ r x { y ∈ H : Θy, z φ z − φy 1 r 〈 K′ ( y ) −K′ x , ηz, y ≥ 0, ∀z ∈ H } , ∀x ∈ H. 2.8 Then the following conclusion holds. Proposition 2.3 see 7 . Let H be a real Hilbert space, φ : H → R be a a lower semicontinuous and convex functional. Let Θ : H × H → R be an equilibrium bifunction satisfying conditions (H1)–(H5). Assume that i η : H ×H → R is Lipschitz continuous with constant σ > 0 such that a η x, y η y, x 0 for all x, y ∈ H; b η ·, · is affine in the first variable; c for each fixed y ∈ H, x → η y, x is continuous from the weak topology to the weak topology; ii K : H → R is η-strongly convex with constant μ > 0, and its derivative K′ is continuous from the weak topology to the strong topology; iii for each x ∈ H, there exists a bounded subset Dx ⊂ H and zx ∈ H such that for all y / ∈ Dx, Θ ( y, zx ) φ zx − φ ( y ) 1 r 〈 K′ ( y ) −K′ x , ηzx, y )〉 < 0. 2.9 Then the following hold: i V Θ,φ r is single valued; ii F V Θ,φ r MEP Θ, ’ ; iii MEP Θ, φ is closed and convex. Lemma 2.4 see 48 . Let C be a nonempty bounded closed and convex subset of a real Hilbert space H. Let S {T t : t ∈ R } be a nonexpansive semigroup on C, then for all h > 0, lim t→∞ sup x∈C ∥∥∥∥ 1 t ∫ t 0 T s x ds − T h ( 1 t ∫ t 0 T s x ds )∥∥∥∥ 0. 2.10 Lemma 2.5 see 49 . Let X be a uniformly convex Banach space, C be a nonempty closed and convex subset of X, and T : C → X be a nonexpansive mapping. Then I − T is demiclosed at zero. 8 Abstract and Applied Analysis Lemma 2.6 see 50 . Assume that A is a strongly positive linear bounded operator on H with coefficient γ > 0 and 0 < ρ ≤ ‖A‖−1. Then ‖I − ρA‖ ≤ 1 − ργ . Lemma 2.7 see 51 . Let X be a Banach space and f : X → X be a φ-strongly pseudocontractive and continuous mapping. Then f has a unique fixed point in X. Lemma 2.8. In a real Hilbert space H, the following inequality holds: ∥ ∥x y ∥ ∥2 ≤ ‖x‖ 2y, x y, ∀x, y ∈ H. 2.11 The following lemma can be found in 52, 53 see also Lemma 2.2 in 54 . Lemma 2.9. Let C be a nonempty closed and convex subset of a real Hilbert space H and g : C → R∪ { ∞} be 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 , 2.12 then 〈 g ′ x , x − x∗ ≥ 0, x ∈ C. 2.13 In particular, if x∗ solves the optimization problem min x∈Ω μ 2 〈Ax, x〉 1 2 ‖x − u‖ − h x , 2.14 then 〈 u ( γf − I μAx∗, x − x∗ ≤ 0, x ∈ C, 2.15 where h is a potential function for γf . The following lemmas can be found in 55, 56 . For the sake of the completeness, one includes its proof in a Hilbert space version. Without loss of generality, one assumes that c, d ∈ 0, 1 and LB ∈ 1,∞ . Lemma 2.10. Let H be a real Hilbert space, B : H → H be an LB-Lipschitzian and relaxed c, d cocoercive mapping. Then, one has ∥I − ρBx − I − ρBy∥2 ≤ ( 1 2ρcLB − 2ρd ρLB ∥x − y∥2, 2.16 where ρ > 0. In particular, if 0 < ρ ≤ 2 d − cLB /LB, then I − ρB is nonexpansive. Abstract and Applied Analysis 9 Proof. For all x, y ∈ H, we haveand Applied Analysis 9 Proof. For all x, y ∈ H, we have ∥ ∥ I − ρB x − I − ρB y∥2 ∥ x − y − ρBx − ρBy ∥2 ∥x − y∥2 − 2ρBx − By, x − y ρ2∥Bx − By∥2 ≤ ∥x − y∥2 − 2ρ [ −c∥Bx − By∥2 d∥x − y∥2 ] ρ2 ∥ ∥Bx − By∥2 ∥ ∥x − y∥2 − 2ρd∥x − y∥2 2ρc∥Bx − By∥2 ρ2∥Bx − By∥2 ≤ ( 1 2ρcLB − 2ρd ρLB )∥ ∥x − y∥2. 2.17 It is clear that, if 0 < ρ ≤ 2 d − cLB /LB, then I − ρB is nonexpansive. This completes the proof. Lemma 2.11. Let H be a real Hilbert space, Mi : H → 2 be the a maximal monotone mapping and Bi : H → H be an Li-Lipschitzian and relaxed ci, di -cocoercive mapping for all i 1, 2. Let Q : H → H be a mapping defined by Qx : J M1,ρ1 [ J M2,ρ2 ( x − ρ2B2x ) − ρ1B1J M2,ρ2 ( x − ρ2B2x )] , ∀x ∈ H. 2.18 If 0 < ρi ≤ 2 di − ciLi /Li for all i 1, 2, then Q : H → H is nonexpansive. Proof. By Lemma 2.10, we know that I − ρ2B2 and I − ρ1B1 are nonexpansive, for all x, y ∈ H, we have ∥Qx −Qy∥ ∥J M1,ρ1 [ J M2,ρ2 ( x − ρ2B2x ) − ρ1B1J M2,ρ2 ( x − ρ2B2x )] −J M1,ρ1 [ J M2,ρ2 ( y − ρ2B2y ) − ρ1B1J M2,ρ2 ( y − ρ2B2y )]∥ ≤ ∥J M2,ρ2 ( x − ρ2B2x ) − ρ1B1J M2,ρ2 ( x − ρ2B2x )] −J M2,ρ2 ( y − ρ2B2y ) − ρ1B1J M2,ρ2 ( y − ρ2B2y )]∥ ∥J M2,ρ2 ( I − ρ2B2 )( I − ρ1B1 ) x − J M2,ρ2 ( I − ρ2B2 )( I − ρ1B1 ) y ∥ ≤ ∥I − ρ2B2 )( I − ρ1B1 ) x − I − ρ2B2 )( I − ρ1B1 ) y ∥∥ ≤ ∥I − ρ2B2 ) x − I − ρ2B2 ) y ∥∥ ≤ ∥x − y∥, 2.19 which implies that Q is nonexpansive. This completes the proof. Lemma 2.12. For all x∗, y∗ ∈ H × H, where y∗ J M2,ρ2 x∗ − ρ2B2x∗ , x∗, y∗ is a solution of the problem 1.15 if and only if x∗ is a fixed point of the mapping Q : H → H defined as in Lemma 2.11. 10 Abstract and Applied Analysis Proof. Let x∗, y∗ ∈ H ×H be a solution of the problem 1.15 . Then, we have y∗ − ρ1B1y∗ ∈ ( I ρ1M1 ) x∗, x∗ − ρ2B2x∗ ∈ ( I ρ2M2 ) y∗, 2.20