A new hybrid general iterative algorithm for common solutions of generalized mixed equilibrium problems and variational inclusions

Let C be a closed convex subset of a real Hilbert space H with the inner product 〈·, ·〉 and the norm || · ||. Let F be a bifunction of C × C into , where is the set of real numbers, Ψ : C → H be a mapping and $\phi :C\to \mathcal{R}$ be a real-valued function. The generalized mixed equilibrium problem for finding x ∈ C such that

$F\left(x,y\right)+〈\mathrm{\Psi }x,y-x〉+\phi \left(y\right)-\phi \left(x\right)\ge 0,\forall y\in C.$

(1.1)

The set of solutions of (1.2) is denoted by MVI(C, φ, Ψ).

The set of solutions of (1.3) is denoted by MEP(F, φ).

The set of solutions of (1.4) is denoted by EP(F). See, e.g. [3–6] and the references therein.

The set of solutions of (1.5) is denoted by VI(C, Ψ).

The set of solutions of (1.6) is denoted by Argmin(φ).

where A is a linear bounded operator, F(S) is the fixed point set of a nonexpansive mapping S and y is a given point in H[8].

where θ is the zero vector in H. The set of solutions of problem (1.8) is denoted by I(B, M). The variational inclusion has been extensively studied in the literature. See, e.g. [11–15] and the reference therein.

A set-valued mapping M : H → 2 ^H is called monotone if for all x, y ∈ H, f ∈ M(x) and g ∈ M(y) imply 〈x - y, f - g〉 ≥ 0. A monotone mapping M is maximal if its graph G(M) := {(f, x) ∈ H × H : f ∈ M(x)} of M is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if for (x, f) ∈ H × H, 〈x - y, f - g〉 > 0 for all (y, g) ∈ G(M) imply f ∈ M(x).

Then M is a maximal monotone and θ ∈ Mν if and only if v ∈ VI(C, B) [16].

is called the resolvent operator associated with M, where λ is any positive number and I is the identity mapping. In the worth mentioning that the resolvent operator is nonexpansive, 1-inverse-strongly monotone and that a solution of problem (1.8) is a fixed point of the operator J_M,λ(I - λB) for all λ > 0 (see also [17]).

where {α _n } ⊂ (0, 1) satisfies certain conditions, converge strongly to a fixed point of S (say $\bar{x}\in C$) which is the unique solution of the following variational inequality.

where u ∈ C, {α _n } ⊂ (0, 1) and {λ _n } ⊂ [a, b] for some a, b with 0 < a < b < 2β. They proved that under certain appropriate conditions imposed on {α _n } and {λ _n }, the sequence {x _n } converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping (say $\bar{x}\in C$) which solve some variational inequality.

where {α _n } ⊂ (0, 1) and A is a strongly positive linear bounded operator. They proved that if C = H then the sequence {x _n } converges strongly to a fixed point of S (say $\bar{x}\in H$) which is the unique solution of the following variational inequality.

for all n ∈ ℕ, where {α _n } ⊂ [0, 1) and {r _n } ⊂ (0, ∞) satisfy some appropriate conditions. Furthermore, they proved {x _n } and {u _n } converge strongly to the same point z, where z = P_{F(S)∩VI(C,A)∩ EP(F)}f(z).

where {α _n } ⊂ (0, 1), λ ∈ (0, 2α] and μ ∈ (0, 2β]. The sequence {x _n } converges strongly to a common element of the set of fixed points of nonexpansive mapping, the set of solutions of the variational inequality and the generalized equilibrium problem.

In this article, we modify by Marino and Xu [8], Su et al. [20] and Tan and Chang [14], the purpose of this article, we show that under some control conditions the sequence {x _n } converges strongly to a common element of the set of fixed points of nonexpansive mappings, the solution of the generalized mixed equilibrium problems and the solution of the variational inclusions in a real Hilbert space.

The following characterizes the projection P _C . We recall some lemmas which will be needed in the rest of this article.

Lemma 2.1. The function u ∈ C is a solution of the variational inequality (1.5) if and only if u ∈ C satisfies the relation u = P _C (u - λ Ψu) for all λ > 0.

Lemma 2.2. For a given z ∈ H, u ∈ C, u = P _C z ⇔ 〈u - z, v - u〉 ≥ 0, ∀v ∈ C.

Lemma 2.3. [21]Let M : H → 2 ^H be a maximal monotone mapping and let B : H → H be a monotone and Lipshitz continuous mapping. Then the mapping L = M + B : H → 2 ^H is a maximal monotone mapping.

Lemma 2.4. [22]Each Hilbert space H satisfies Opial's condition, that is, for any sequence {x _n } ⊂ H with x _n ⇀ x, the inequality lim inf_n→∞||x _n - x|| < lim inf_n→∞||x _n - y||, hold for each y ∈ H with y ≠ x.

where {γ _n } ⊂ (0, 1) and {δ _n } is a sequence in such that

1. (i)

   ${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_n=\mathrm{\infty }$.

    
2. (ii)

   $\mathsf{\text{lim}}\underset{n\to \mathrm{\infty }}{\mathsf{\text{sup}}}\frac{{\delta }_n}{{\gamma }_n}\le 0$ or ${\sum }_{n=1}^{\mathrm{\infty }}\left|{\delta }_n\right|<\mathrm{\infty }$.

    

Then lim_n→∞a _n = 0.

implies x = Sx.

For solving the mixed equilibrium problem, let us assume that the bifunction $F:C\times C\to \mathcal{R}$ and $\phi :C\to \mathcal{R}$ satisfies the following conditions:

(A1) F(x, x) = 0 for all x ∈ C;

(A2) F is monotone, i.e., F(x, y) + F(y, x) ≤ 0 for any x, y ∈ C;

(A3) for each fixed y ∈ C, x ↦ F(x, y) is weakly upper semicontinuous;

(A4) for each fixed x ∈ C, y ↦ F(x, y) is convex and lower semicontinuous;

(B2) C is a bounded set.

Lemma 2.8. [8]Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient$\bar{\gamma }>0$and 0 < ρ ≤ ||A||^-1, then$∥I-\rho A∥\le 1-\rho \bar{\gamma }$.

In this section, we show a strong convergence theorem which solves the problem of finding a common element of F(S), GMEP(F₁, φ₁, B₁), GMEP(F₂, φ₂, B₂), I(A₁, M₁) and I(A₂, M₂).

Theorem 3.1. Let H be a real Hilbert space, C be a closed convex subset of H. Let F₁, F₂be bifunctions of C × C into satisfying (A1)-(A4) and A₁, A₂, B₁, B₂ : C → H be β₁, β₂, η, ρ-inverse-strongly monotone mappings, ${\phi }_1,{\phi }_2:C\to \mathcal{R}$be convex and lower semicontinuous functions, f : C → C be a contraction with coefficient α (0 < α < 1), M₁, M₂ : H → 2 ^H be maximal monotone mappings and A is a δ-strongly monotone and μ-strictly pseudo-contraction with δ + μ > 1, γ is a positive real number such that$\gamma <\frac{1}{\alpha }\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)$. Assume that either (B1) or (B2) holds. Let S be a nonexpansive mapping of C into itself such that

$\mathrm{\Theta }:=F\left(S\right)\cap \mathsf{\text{GMEP}}\left(F_1,{\phi }_1,B_1\right)\cap \mathsf{\text{GMEP}}\left(F_2,{\phi }_2,B_2\right)\cap I\left(A_1,M_1\right)\cap I\left(A_2,M_2\right)\ne \varnothing .$

where {α _n }, {ξ _n } ⊂ (0, 1), λ₁ ∈ (0, 2β₁) such that 0 < a₁ ≤ λ₁ < b₁ < 2β₁, λ₂ ∈ (0, 2β₂) such that 0 < a₁ ≤ λ₂ ≤ b₂ < 2β₂, r _n ∈ (0, 2η) with 0 < c ≤ d ≤ 1 - η and s _n ∈ (0, 2ρ) with 0 < e ≤ f ≤ 1 - ρ satisfy the following conditions:

(C1): lim_n→∞α _n = 0, ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty },{\sum }_{n=1}^{\mathrm{\infty }}\left|{\alpha }_{n+1}-{\alpha }_n\right|<\mathrm{\infty }$,

(C2): 0 < lim inf_n→∞ξ _n < lim sup_n→∞ξ _n < 1, ${\sum }_{n=1}^{\mathrm{\infty }}\left|{\xi }_{n+1}-{\xi }_n\right|<\mathrm{\infty }$,

(C3): lim inf_n→∞r _n > 0 and lim_n→∞|r_n+1- r _n | = 0,

(C4): lim inf_n→∞s _n > 0 and lim_n→∞|s_n+1- s _n | = 0.

where h is a potential function for γf (i.e., h'(q) = γf(q) for q ∈ H).

It is clear that if 0 < λ₁ < 2β₁, 0 < λ₂ < 2β₂, 0 < r _n < 2η, 0 < s _n ≤ 2ρ then I - λ₁A₁, I - λ₂A₂, I - r _n B₁, I - s _n B₂ are all nonexpansive. We will divide the proof into six steps.

Put z _n = P _C [α _n γf(x _n ) + (I - α _n A)SP _C y _n ] for all n ≥ 0. From (3.1) and by Lemma 2.9

Therefore {x _n } is bounded, so are {y _n },{z _n },{P _C w _n },{SP _C y _n },{f(x _n )} and {ASP _C y _n }.