A general composite iterative method for generalized mixed equilibrium problems, variational inequality problems and optimization problems

Let H be a real Hilbert space with inner product 〈·, ·〉 and induced norm || · ||. Let C be a nonempty closed convex subset of H and S : C → C be a self-mapping on C. Let us denote by F(S) the set of fixed points of S and by P _C the metric projection of H onto C.

Let B : C → H be a nonlinear mapping and φ : C → ℝ be a function, and Θ be a bifunction of C × C into ℝ, where ℝ is the set of real numbers.

which was recently introduced by Peng and Yao [1]. The set of solutions of the problem (1.1) is denoted by GMEP(Θ, φ, B). Here, some special cases of the problem (1.1) are stated as follows:

which was studied by Ceng and Yao [2] (see also [3]). The set of solutions of the problem (1.2) is denoted by MEP (Θ,φ).

The set of solutions of the problem (1.3) is denoted by EP(Θ).

The set of solutions of the problem (1.4) is denoted by V I(C, B).

The problem (1.1) is very general in the sense that it includes, as special cases, fixed point problems, optimization problems, variational inequality problems, minmax problems, Nash equilibrium problems in noncooperative games, and others; see [2, 4–6].

Recently, in order to study the problem (1.3) coupled with the fixed point problem, many authors have introduced some iterative schemes for finding a common element of the set of the solutions of the problem (1.3) and the set of fixed points of a countable family of nonexpansive mappings; see [7–16] and the references therein.

In 2008, Su et al. [17] gave an iterative scheme for the problem (1.3), the problem (1.4) for an inverse-strongly monotone mapping, and fixed point problems of non-expansive mappings. In 2009, Yao et al. [18] considered an iterative scheme for the problem (1.2), the problem (1.4) for a Lipschitz and relaxed-cocoercive mapping and fixed point problems of nonexpansive mappings, and in 2008, Peng and Yao [1] studied an iterative scheme for the problem (1.1), the problem (1.4) for a monotone, and Lipschitz continuous mapping and fixed point problems of nonexpansive mappings.

where T is a nonexpansive mapping, f is a contraction with constant k ∈ (0, 1), {α _n }, {β _n }⊂ [0, 1], and {r _n } ⊂ (0, ∞). He showed that the sequences {x _n } and {u _n } generated by (1.5) converge strongly to a point in F(T ) ∩ EP (Θ) under suitable conditions.

where $\Omega ={\bigcap }_{n=1}^{\infty }{C}_n,C_1,C_2,\cdots$ are infinitely many closed convex subsets of H such that ${\bigcap }_{n=1}^{\infty }{C}_n\ne \varnothing ,$u ∈ H, μ ≥ 0 is a real number, A is a strongly positive bounded linear operator on H (i.e., there is a constant $\bar{\gamma }>0$ such that $⟨Ax,x⟩\ge \bar{\gamma }{∥x∥}^2$, ∀x ∈ H) and h is a potential function for γ f (i.e., h'(x) = γ f(x) for all x ∈ H). For this kind of optimization problems, see, for example, Bauschke and Borwein [19], Combettes [20], Deutsch and Yamada [21], Jung [22], and Xu [23] when $\Omega ={\bigcap }_{i=1}^N{C}_i$; and h(x) = 〈x, b〉 for a given point b in H.

In 2010, using the method of Yao et al. [3], Jaiboon and Kumam [24] also introduced a general iterative method for finding a common element of the set of solutions of the problem (1.2), the set of fixed points of a sequence {T _n } of nonexpansive mappings, and the set of solutions of the problem (1.4) for a α-inverse-strongly monotone mapping. We point out that in the main results of [3, 24], the condition of the sequentially continuity from the weak topology to the strong topology for the derivative K' of the function K : C → ℝ is very strong. Even if $K\left(x\right)=\frac{{∥x∥}^2}{2}$, then K' (x) = x is not sequentially continuous from the weak topology to the strong topology.

In this article, inspired and motivated by above mentioned results, we introduce a new iterative method for finding a common element of the set of solutions of a generalized mixed equilibrium problem (1.1), the set of fixed points of a countable family of nonexpansive mappings, and the set of solutions of the variational inequality problem (1.4) for an inverse-strongly monotone mapping in a Hilbert space. We show that under suitable conditions, the sequence generated by the proposed iterative scheme converges strongly to a common element of the above three sets, which is a solution of a certain optimization problem. The results of this article can be viewed as an improvement and complement of the recent results in this direction.

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. In the following, we write x _n ⇀ x to indicate that the sequence {x _n } converges weakly to x. x _n → x implies that {x _n } converges strongly to x.

First, we know that a mapping f : C → C is a contraction on C if there exists a constant k ∈ (0, 1) such that ||f(x) - f(y)|| ≤ k||x - y||, x, y ∈ C. A mapping T : C → C is called nonexpansive if ||Tx - Ty|| ≤ ||x - y||, x, y ∈ C.

holds for every y ∈ H with y ≠ x.

Hence, if λ ≤ 2α, then I - λF is a nonexpansive mapping of C into H. The following result for the existence of solutions of the variational inequality problem for inverse-strongly monotone mappings was given in Takahashi and Toyoda [25].

Proposition Let C be a bounded closed convex subset of a real Hilbert space, and F be an α-inverse-strongly monotone mapping of C into H. Then, V I(C, F) is nonempty.

Then, Q is maximal monotone and 0 ∈ Qv if and only if v ∈ V I(C, F ); see [26, 27].

For solving the equilibrium problem for a bifunction Θ : C × C → ℝ, let us assume that Θ and φ satisfy the following conditions:

(A1) Θ(x, x) = 0 for all x ∈ C;

(A2) Θ is monotone, that is, Θ(x, y) + Θ (y, x) ≤ 0 for all x, y ∈ C;

(A4) for each x ∈ C, y α Θ (x, y) is convex and lower semicontinuous;

(A5) For each y ∈ C, x α Θ (x, y) is weakly upper semicontinuous;

(B2) C is a bounded set;

The following lemmas were given in [1, 4].

We also need the following lemmas for the proof of our main results.

Then, lim_n→∞s _n = 0.

Then, for each y ∈ C, {T _n y} converges strongly to some point of C. Moreover, let T be a mapping of C into itself defined by Ty = lim_n→∞T _n y for all y ∈ C. Then, lim_n→∞sup{||Tz - T _n z|| : z ∈ C} = 0.

The following lemma can be found in [3](see also Lemma 2.1 in [22]).

where h is a potential function for γ f.

In this section, we introduce a new composite iterative scheme for finding a common point of the set of solutions of the problem (1.1), the set of fixed points of a countable family of nonexpansive mappings, and the set of solutions of the problem (1.4) for an inverse-strongly monotone mapping.

where {α _n }, {β _n } ⊂ [0, 1], λ _n ∈ [a, b] ⊂ (0, 2α) and r _n ∈ [c, d] ⊂ (0, 2β). Let {α _n }, {λ _n } and {β _n } satisfy the following conditions:

(C1) α _n → 0 (n → ∞);${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$;

(C2) β _n ⊂ [0, a) for all n ≥ 0 and for some a ∈ (0, 1);

(C3) ${\sum }_{n=1}^{\infty }|{\alpha }_{n+1}-{\alpha }_n|<\infty$, ${\sum }_{n=1}^{\infty }|{\beta }_{n+1}-{\beta }_n|<\infty$, ${\sum }_{n=1}^{\infty }|{\lambda }_{n+1}-{\lambda }_n|<\infty$, ${\sum }_{n=1}^{\infty }|r_{n+1}-r_n|<\infty$.

where h is a potential function for γ f.

Let us divide the proof into several steps. From now on, we put z _n = P _C (u _n -λ _n Fu _n ) and w _n = P _C (y _n - λ _n Fy _n ).