The hybrid algorithm for the system of mixed equilibrium problems, the general system of finite variational inequalities and common fixed points for nonexpansive semigroups and strictly pseudo-contractive mappings

Abstract

In this article, we introduce a new iterative algorithm by the shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of general system of finite variational inequalities, the set of solutions of fixed points for nonexpansive semigroups and the set of common fixed points for an infinite family of strictly pseudo-contractive mappings in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above four sets under some mind conditions. Our results improve and extend the corresponding recent results in literature work.

1 Introduction

Throughout this article, we assume that C is a closed convex subset of a real Hilbert space H with inner product and norm are denoted by 〈.,.〉 and ∥.∥, respectively.

Let A, B: C → H be two mappings. We consider the following problem of finding (x*, y*) ∈ C × C such that

$$\left\{\begin{array}{c}\hfill 〈\lambda A{y}^{*}+x^{*}-y^{*},x-x^{*}〉\ge 0,\forall x\in C,\hfill \\ \hfill 〈\mu B{x}^{*}+y^{*}-x^{*},x-y^{*}〉\ge 0,\forall x\in C,\hfill \end{array}\right.$$

(1)

which is called a general system of variational inequalities, where λ ≥ 0 and μ ≥ 0 are two constants. The set of solution of (1) is denoted by SVI(C, A, B). In particular, if A = B, then problem (1) reduces to finding (x*, y*) ∈ C × C such that

$$\left\{\begin{array}{c}\hfill 〈\lambda A{y}^{*}+x^{*}-y^{*},x-x^{*}〉\ge 0,\forall x\in C,\hfill \\ \hfill 〈\mu A{x}^{*}+y^{*}-x^{*},x-y^{*}〉\ge 0,\forall x\in C,\hfill \end{array}\right.$$

(2)

which is defined by Verma [1] (see also Verma [2]), and is called the new general system of variational inequalities. Further, if we set B = 0, then problem (1) reduces to the classical variational inequality VI(C, A) which was originally introduced and studied by Stampacchia [3] in 1964.

By the system of variational inequality problems above, we extend into the general system of finite variational inequalities is to find $\left(x_1^{*},x_2^{*},\dots ,x_M^{*}\right)\in C\times C\times \cdots \times C$ and is defined by

$$\left\{\begin{array}{c}〈{\lambda }_M{A}_M{x}_M^{*}+x_1^{*}-x_M^{*},x-x_1^{*}〉\ge 0,\forall x\in C,\hfill \\ 〈{\lambda }_{M-1}{A}_{M-1}{x}_{M-1}^{*}+x_M^{*}-x_{M-1}^{*},x-x_M^{*}〉\ge 0,\forall x\in C,\hfill \\ ⋮\hfill \\ 〈{\lambda }_2{A}_2{x}_2^{*}+x_3^{*}-x_2^{*},x-x_3^{*}〉\ge 0,\forall x\in C,\hfill \\ 〈{\lambda }_1{A}_1{x}_1^{*}+x_2^{*}-x_1^{*},x-x_2^{*}〉\ge 0,\forall x\in C,\hfill \end{array}\right.$$

(3)

where ${\left\{A_l\right\}}_{l=1}^M:C\to H$ is a family of mappings, λ _l ≥ 0, l ∈ {1,2, ..., M}. The set of solution of (3) is denoted by GSVI(C, A _l ). In particular, if $M=2,A_1=B,A_2=A,{\lambda }_1=\mu ,{\lambda }_2=\lambda ,x_1^{*}=x^{*}$, and $x_2^{*}=y^{*}$, then the problem (3) is reduced to the problem (1).

Recall that a mapping T : C → C is said to be a k-strict pseudo-contraction (see also [4]) if there exists 0 ≤ k < 1 such that

$${∥Tx-Ty∥}^2\le {∥x-y∥}^2+k{∥\left(I-T\right)x-\left(I-T\right)y∥}^2,\forall x,y\in C,$$

where I denotes the identity operator on C (see also [5]). If k = 0, a mapping T : C → C is said to be nonexpansive[6], that is,

$$∥Tx-Ty∥\le ∥x-y∥,\forall x,y\in C.$$

If k = 1, a mapping T : C → C is said to be pseudo-contraction, that is,

$${∥Tx-Ty∥}^2\le {∥x-y∥}^2+{∥\left(I-T\right)x-\left(I-T\right)y∥}^2,\forall x,y\in C.$$

Clearly, the class of k-strict pseudo-contraction falls into the one between classes of nonexpansive mappings and pseudo-contraction mappings. We denote the set of fixed points of T by F(T).

Let ℑ = {F _k }_{k ∈ Γ}be a countable family of bifunctions from C × C to ℝ, where ℝ is the set of real numbers and Γ is an arbitrary index set. Let φ : C → ℝ ∪ {+∞} be a proper extended real-valued function. The system of mixed equilibrium problems is to find x ∈ C such that

$$F_k\left(x,y\right)+\phi \left(y\right)\ge \phi \left(x\right),\forall k\in \text{Γ},\forall k\in C.$$

(4)

The set of solutions of (4) is denoted by SMEP(F _k , φ), that is

$$SMEP\left(F_k,\phi \right)=\left\{x\in C:F_k\left(x,y\right)+\phi \left(y\right)\ge \phi \left(x\right),\forall k\in \text{Γ},\forall y\in C\right\}.$$

(5)

If Γ is a singleton, the problem (4) reduces to find the following mixed equilibrium problem (see also Flores-Bazán [7]). For finding x ∈ C such that

$$F\left(x,y\right)+\phi \left(y\right)\ge \phi \left(x\right),\forall y\in C.$$

(6)

The set of solutions of (6) is denoted by MEP(F, φ). Combettes and Hirstoaga [8] introduced the following system of equilibrium problems. For finding x ∈ C such that,

$$F_k\left(x,y\right)\ge 0,\forall k\in \text{Γ},\forall y\in C.$$

(7)

The set of solutions of (7) is denoted by SEP(ℑ), that is,

$$SEP\left(\Im \right)=\left\{x\in C:F_k\left(x,y\right)\ge 0,\forall k\in \text{Γ},\forall y\in C\right\}.$$

(8)

If Γ is a singleton, the problem (7) becomes the following equilibrium problem. For finding x ∈ C such that

$$F\left(x,y\right)\ge 0,\forall y\in C.$$

(9)

The set of solution of (9) is denoted by EP(F). The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, noncooperative games, economics and the equilibrium problem as special cases (see [9–19]). In the last two decades, many articles have appeared in the literature on the existence of solutions of equilibrium problems; see, for example [13] and references therein. Some solution methods have been proposed to solve the mixed equilibrium problems; see, for example, (see [11–14, 16–24]) and references therein.

A family $\mathcal{S}=\left\{S\left(s\right):0\le s<\infty \right\}$ of mappings of C into itself is called a nonexpansive semigroup on C if it satisfies the following conditions:

• (i) S(0)x = x for all x ∈ C;

• (ii) S(s + t) = S(s) S(t) for all s, t ≥ 0;

• (iii) ∥S(s)x - S(s)y∥ ≤ ∥x - y∥ for all x, y ∈ C and s ≥ 0;

• (iv) for all x ∈ C, s ↦ S(s)x is continuous.

We denote by $F\left(\mathcal{S}\right)$ the set of all common fixed points of $\mathcal{S}=\left\{S\left(s\right):s\ge 0\right\}$, i.e., $F\left(\mathcal{S}\right)={\cap }_{s\ge 0}F\left(S\left(s\right)\right)$. It is well known that $F\left(\mathcal{S}\right)$ is closed and convex (see also [25, 26]).

In 2011, Shehu [21] introduced a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of an infinite family of k-strictly pseudocontractive mappings and the set of common solutions to a system of generalized mixed equilibrium problems and the set of solution of variational inequality problems in Hilbert spaces. Starting with an arbitrary $x_0\in C,C_{1,i}=C,C_1={\cap }_{i=1}^{\infty }{C}_{1,i},x_1=P_{C_1}{x}_0$ define sequences {x _n }, {w _n }, {u _n }, {z _n }, and {y_n,i} as follows:

$$\left\{\begin{array}{c}{z}_n=T_{r_n}^{\left(F_1,{\phi }_1\right)}\left(x_n-r_{n}A{x}_n\right)\hfill \\ y_n=T_{{\lambda }_n}^{\left(F_2,{\phi }_2\right)}\left(z_n-{\lambda }_{n}B{z}_n\right)\hfill \\ w_n=P_C\left(u_n-s_{n}D{u}_n\right)\hfill \\ y_{n,i}={\alpha }_{n,i}{w}_n+\left(1-{\alpha }_{n,i}\right)T_i{w}_n,n\ge 1,\hfill \\ C_{n+1,i}=\left\{z\in C_{n,i}:∥y_{n,i}-z∥\le ∥x_n-z∥\right\},n\ge 1,\hfill \\ C_{n+1}={\cap }_{i=1}^{\infty }{C}_{n+1,i},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_0,n\ge 1,\hfill \end{array}\right.$$

(10)

where T _i be a k _i -strictly pseudocontractive mapping and for some 0 ≤ k _i < 1, A, B are α, β-inverse-strongly monotone mappings of C into H, respectively. He proved that if the sequences {α _n,i }, {r _n }, {s _n }, and {λ _n } of parameters satisfies appropriate conditions, then {x _n } is generated by (10) converges strongly to P_Ωx₀, where P_Ω is metric projection on H in to $\Omega :=MEP\left(F_1,{\phi }_1\right)\cap MEP\left(F_2,{\phi }_2\right)\cap VI\left(C,A\right)\cap \left({\cap }_{i=1}^{\infty }F\left(T_i\right)\right)$. For using the hybrid method, we can see [27–29].

In this article, motivated by the above results, we present a new iterative algorithm for finding a common element of the set of solutions for a system of mixed equilibrium problems, the set of common solutions of general system of finite variational inequality problems, the set of solutions of fixed points for nonexpansive semigroup mappings and the set of common fixed points for an infinite family of strictly pseudo-contractive mappings in a real Hilbert space. Then, we prove strong convergence theorem under some mind conditions. The results presented in this article extend and improve the results of Shehu [21] and many authors.

2 Preliminaries

Let H be a real Hilbert space with norm ∥ ⋅ ∥ and inner product 〈⋅,⋅〉, respectively. Let C be a closed convex subset of H. The sequence {x _n } is a sequence in H, x _n ⇀ x means {x _n } converges weakly to x and x _n → x means {x _n } converges strongly to x. In a real Hilbert space H, we have

$${∥x-y∥}^2={∥x∥}^2-{∥y∥}^2-2〈x-y,y〉$$

(11)

and

$${∥\lambda x+\left(1-\lambda \right)y∥}^2=\lambda {∥x∥}^2+\left(1-\lambda \right){∥y∥}^2-\lambda \left(1-\lambda \right){∥x-y∥}^2,\forall x,y\in H,$$

(12)

and λ ∈ ℝ. For every point x ∈ H, there exists a unique nearest point in C, denoted by P _C x, such that

$$∥x-P_{C}x∥\le ∥x-y∥,\forall y\in C.$$

(13)

P _C is called the metric projection of H onto C. It is well known that P _C is a nonexpansive mapping of H onto C and satisfies

$$〈x-y,P_{C}x-P_{C}y〉\ge {∥P_{C}x-P_{C}y∥}^2,\forall x,y\in H.$$

(14)

Moreover, P _C x is characterized by the following properties: P _C x ∈ C and

$$〈x-P_{C}x,y-P_{C}x〉\le 0,$$

(15)

$${∥x-y∥}^2\ge {∥x-P_{C}x∥}^2+{∥y-P_{C}x∥}^2,\forall x\in H,y\in C.$$

(16)

Recall that a mapping A of C into H is called α-inverse-strongly monotone if there exists a positive real number α such that

$$〈Ax-Ay,x-y〉\ge \alpha {∥Ax-Ay∥}^2,\forall x,y\in C.$$

(17)

It is obvious that any α-inverse-strongly monotone mappings A is $\left(\frac{1}{\alpha }\right)$-Lipschitz monotone and continuous mappings.

In order to prove our main results, we need the following Lemmas.

Lemma 2.1. [30]Let V : C → H be a k-strict pseudo-contraction, then

(1) the fixed point set F(V) of V is closed convex so that the projection P_F(V)is well defined;

(2) define a mapping T : C → H by

$$Tx=tx+\left(1-t\right)Vx,\forall x\in C.$$

(18)

If t ∈ [k, 1), then T is a nonexpansive mapping such that F(V) = F(T).

A family of mappings ${\left\{V_i:C\to H\right\}}_{i=1}^{\infty }$ is called a family of uniformly k-strict pseudo-contractions, if there exists a constant k ∈ [0, 1) such that

$${∥V_{i}x-V_{i}y∥}^2\le {∥x-y∥}^2+k{∥\left(1-V_i\right)x-\left(I-V_i\right)y∥}^2,\forall x,y\in C,\forall i\ge 1.$$

Let ${\left\{V_i:C\to C\right\}}_{i=1}^{\infty }$ be a countable family of uniformly k-strict pseudo-contractions. Let ${\left\{T_i:C\to C\right\}}_{i=1}^{\infty }$ be the sequence of nonexpansive mappings defined by (18), i.e.,

$$T_{i}x=tx+\left(1-t\right)V_{i}x,\forall x\in C,\forall i\ge 1,t\in \left[k,1\right).$$

(19)

Let {T _i } be a sequence of nonexpansive mappings of C into itself defined by (19) and let {μ _i } be a sequence of nonnegative numbers in [0,1]. For each n ≥ 1, define a mapping W _n of C into itself as follows:

$$\begin{array}{c}{U}_{n,n+1}=I,\hfill \\ U_{n,n}={\mu }_n{T}_n{U}_{n,n+1}+\left(1-{\mu }_n\right)I,\hfill \\ U_{n,n-1}={\mu }_{n-1}{T}_{n-1}{U}_{n,n}+\left(1-{\mu }_{n-1}\right)I,\hfill \\ ⋮\hfill \\ U_{n,k}={\mu }_k{T}_k{U}_{n,k+1}+\left(1-{\mu }_k\right)I,\hfill \\ U_{n,k-1}={\mu }_{k-1}{T}_{k-1}{U}_{n,k}+\left(1-{\mu }_{k-1}\right)I,\hfill \\ ⋮\hfill \\ U_{n,2}={\mu }_2{T}_2{U}_{n,3}+\left(1-{\mu }_2\right)I,\hfill \\ W_n=U_{n,1}={\mu }_1{T}_1{U}_{n,2}+\left(1-{\mu }_1\right)I.\hfill \end{array}$$

(20)

Such a mapping W _n is nonexpansive from C to C and it is called the W-mapping generated by T₁, T₂, ..., T _n and μ₁, μ₂, ..., μ _n .

For each n, k ∈ ℕ, let the mapping U_n,kbe defined by (20). Then we can have the following crucial conclusions concerning W _n . You can find them in [31]. Now we only need the following similar version in Hilbert spaces.

Lemma 2.2. [31]Let C be a nonempty closed convex subset of a real Hilbert space H. Let T₁, T₂, ... be nonexpansive mappings of C into itself such that${\cap }_{n=1}^{\infty }F\left(T_n\right)$is nonempty, let μ₁, μ₂, ... be real numbers such that 0 ≤ μ _n ≤ b < 1 for every n ≥ 1. Then,

(1) W _n is nonexpansive and$F\left(W_n\right)={\cap }_{i=1}^{n}F\left(T_i\right),\forall n\ge 1$;

(2) for every x ∈ C and k ∈ ℕ, the limit lim_n→∞U_n,kx exists;

(3) a mapping W : C → C defined by

$$Wx:=\underset{n\to \infty }{\text{lim}}{W}_{n}x=\underset{n\to \infty }{\text{lim}}{U}_{n,1}x,\forall x\in C$$

(21)

is a nonexpansive mapping satisfying$F\left(W\right)={\cap }_{i=1}^{\infty }F\left(T_i\right)$and it is called the W-mapping generated by T₁,T₂, ... and μ₁, μ₂, ....

Lemma 2.3. [32]Let C be a nonempty closed convex subset of a Hilbert space H, {T _i : C → C} be a countable family of nonexpansive mappings with${\cap }_{i=1}^{\infty }F\left(T_i\right)\ne \varnothing ,\left\{{\mu }_i\right\}$be a real sequence such that 0 < μ _i ≤ b < 1, ∀i ≥ 1. If D is any bounded subset of C, then

$$\underset{n\to \infty }{\text{lim}}\underset{x\in D}{\text{sup}}∥Wx-W_{n}x∥=0.$$

Lemma 2.4. [33]Each Hilbert space H satisfies Opial's condition, i.e., for any sequence {x _n } ⊂ H with x _n ⇀ x, the inequality

$$\underset{n\to \infty }{\text{lim inf}}∥x_n-x∥<\underset{n\to \infty }{\text{lim inf}}∥x_n-y∥,$$

hold for each y ∈ H with y ≠ x.

Lemma 2.5. [22]Let C be a nonempty bounded closed convex subset of a Hilbert space H and let$\mathcal{S}=\left\{S\left(s\right):0\le s<\infty \right\}$be a nonexpansive semigroup on C, then for any h ≥ 0,

$$\underset{t\to \infty }{\text{lim}}\underset{x\in C}{\text{sup}}∥\frac{1}{t}{\int }_0^{t}S\left(s\right)xds-S\left(h\right)\left(\frac{1}{t}{\int }_0^{t}S\left(s\right)xds\right)∥=0.$$

Lemma 2.6. [34]Let C be a nonempty bounded closed convex subset of H, {x _n } be a sequence in C and$\mathcal{S}=\left\{S\left(s\right):0\le s<\infty \right\}$be a nonexpansive semigroup on C. If the following conditions are satisfied:

(i) x _n ⇀ z;

(ii) lim sup_s→∞lim sup_n→∞∥S(s)x _n - x _n ∥ = 0, then$z\in F\left(\mathcal{S}\right)$.

Lemma 2.7. Let C be a nonempty closed convex subset of Hilbert space H, A _l : C → H be a β _l -inverse-strongly monotone and λ _l ∈ (0, 2β _l ) where l ∈ {1, 2, ..., M}. If$\mathcal{P}:C\to C$is defined by

$$\mathcal{P}\left(x\right)=P_C\left(I-{\lambda }_M{A}_M\right)P_C\left(I-{\lambda }_{M-1}{A}_{M-1}\right)\dots P_C\left(I-{\lambda }_2{A}_2\right)P_C\left(I-{\lambda }_1{A}_1\right)x,\forall x\in C,$$

then is nonexpansive.

Proof. Taking ${\mathcal{P}}_C^l=P_C\left(I-{\lambda }_l{A}_l\right)P_C\left(I-{\lambda }_{l-1}{A}_{l-1}\right)\dots P_C\left(I-{\lambda }_2{A}_2\right)P_C\left(I-{\lambda }_1{A}_1\right),l\in \left\{1,2,3,\dots ,M\right\}$ and ${\mathcal{P}}_C^0=I$, where I is the identity mapping on H. Then we have $\mathcal{P}={\mathcal{P}}_C^M$.

For any x, y ∈ C, we have

$$\begin{array}{ll}\hfill ∥\mathcal{P}\left(x\right)-\mathcal{P}\left(y\right)∥& =∥{\mathcal{P}}_C^{M}x-{\mathcal{P}}_C^{M}y∥\\ =∥P_C\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}x-P_C\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}y∥\\ \le ∥\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}x-\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}y∥\\ \le ∥{\mathcal{P}}_C^{M-1}x-{\mathcal{P}}_C^{M-1}y∥\\ ⋮\\ \le ∥{\mathcal{P}}_C^{0}x-{\mathcal{P}}_C^{0}y∥\\ =∥x-y∥.\end{array}$$

This show that is nonexpansive on C.

Lemma 2.8. Let C be a nonempty closed and convex subset of a real Hilbert space H, A _l :C → H be nonlinear mappings, where l ∈ {1, 2, ..., M}. For$x_l^{*}\in C,l\in \left\{1,2,\dots ,M\right\}$, then$\left(x_1^{*},x_2^{*},\dots ,x_M^{*}\right)$is a solution of problem (3) if and only if

$$\left\{\begin{array}{c}{x}_1^{*}=P_C\left(I-{\lambda }_M{A}_M\right)x_M^{*}\hfill \\ x_2^{*}=P_C\left(I-{\lambda }_1{A}_1\right)x_1^{*}\hfill \\ x_3^{*}=P_C\left(I-{\lambda }_2{A}_2\right)x_2^{*}\hfill \\ \begin{array}{c}⋮\hfill \\ x_M^{*}=P_C\left(I-{\lambda }_{M-1}{A}_{M-1}\right)x_{M-1}^{*},\end{array}\hfill \end{array}\right.$$

(22)

that is

$$x_1^{*}=P_C\left(I-{\lambda }_M{A}_M\right)P_C\left(I-{\lambda }_{M-1}{A}_{M-1}\right)\dots P_C\left(I-{\lambda }_2{A}_2\right)P_C\left(I-{\lambda }_1{A}_1\right)x_1^{*}.$$

Proof. From the problem (3), we can rewrite as

$$\left\{\begin{array}{c}〈x_1^{*}-\left(x_M^{*}-{\lambda }_M{A}_M{x}_M^{*}\right),x-x_1^{*}〉\ge 0,\forall x\in C,\hfill \\ 〈x_M^{*}-\left(x_{M-1}^{*}-{\lambda }_{M-1}{A}_{M-1}{x}_{M-1}^{*}\right),x-x_M^{*}〉\ge 0,\forall x\in C,\hfill \\ ⋮\hfill \\ 〈x_3^{*}-\left(x_2^{*}-{\lambda }_2{A}_2{x}_2^{*}\right),x-x_3^{*}〉\ge 0,\forall x\in C,\hfill \\ 〈x_2^{*}-\left(x_1^{*}-{\lambda }_1{A}_1{x}_1^{*}\right),x-x_2^{*}〉\ge 0,\forall x\in C.\hfill \end{array}\right.$$

(23)

From (15), we conclude that (23) is equivalent to (22).

Lemma 2.9. (Demi-closedness Principle[6]) Assume that S is a nonexpansive self-mapping of a nonempty closed convex subset C of a real Hilbert space H. If S has a fixed point, the I - S is demi-closed: that is, whenever {x _n } is a sequence in C converging weakly to some x ∈ C (for short, x _n ⇀ x), and the sequence {(I - S)x _n } converges strongly to some y (for short, (I - S)x _n → y), it follows that (I - S)x = y. Here I is the identity operator of H.

For solving the system of mixed equilibrium problems, let us assume that bifunction F _k : C × C → ℝ, k = 1,2, ..., N satisfies the following conditions:

(H1) F _k is monotone, i.e., F _k (x, y) + F _k (y, x) ≤ 0, ∀x, y ∈ C;

(H2) for each fixed y ∈ C, x ↦ F _k (x,y) is convex and upper semicontinuous;

(H3) for each fixed x ∈ C, y ↦ F _k (x, y) is convex.

Lemma 2.10. [35]Let C be a nonempty closed convex subset of a real Hilbert space H and let φ be a lower semicontinuous and convex functional from C to ℝ. Let F be a bifunction from C × C to ℝ satisfying (H1)-(H3). Assume that

(i) η : C × C → H is k Lipschitz continuous with constant k > 0 such that;

(a) η(x, y) + η(y, x) = 0, ∀x, y ∈ C,

(b) η(⋅,⋅) is affine in the first variable,

(c) for each fixed x ∈ C, y ↦ η(x, y) is sequentially continuous from the weak topology to the weak topology,

(ii)$\mathcal{K}:C\to \mathcal{R}$is η-strongly convex with constant σ > 0 and its derivative${\mathcal{K}}^{\prime }$is sequentially continuous from the weak topology to the strong topology;

(iii) for each x ∈ C, there exist a bounded subset D _x ⊂ C and z _x ∈ C such that for any y ∈ C\D _x ,

$$F\left(y,z_x\right)+\phi \left(z_x\right)-\phi \left(y\right)+\frac{1}{r}〈{\mathcal{K}}^{\prime }\left(y\right)-{\mathcal{K}}^{\prime }\left(x\right),\eta \left(z_x,y\right)〉<0.$$

For given r > 0, Let$K_r^F:C\to C$be the mapping defined by:

$$K_r^F\left(x\right)=\left\{y\in C:F\left(y,z\right)+\phi \left(z\right)-\phi \left(y\right)+\frac{1}{r}〈{\mathcal{K}}^{\prime }\left(y\right)-{\mathcal{K}}^{\prime }\left(x\right),\eta \left(z,y\right)〉\ge 0,\forall z\in C\right\}$$

(24)

for all x ∈ C. Then the following hold

(1) $K_r^F$ is single-valued;

(2)$K_r^F$is nonexpansive if${\mathcal{K}}^{\prime }$is Lipschitz continuous with constant ν > 0 such that σ ≥ kν;

(3)$F\left(K_r^F\right)=MEP\left(F,\phi \right)$;

(4) MEP(F, φ) is closed and convex.

3 Main result

In this section, we prove a strong convergence theorem in a real Hilbert space.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert Space H. Let {F _k : C × C → ℝ, k = 1, 2, ..., N} be a finite family of bifunctions satisfying conditions (H1)-(H3). Let A _l be β _l -inverse-strongly monotone mappings of C into H, where l ∈ {1, 2, ..., M}. Let$\mathcal{S}=\left\{S\left(s\right):0\le s<\infty \right\}$be a nonexpansive semigroup on C and let {t _n } be a positive real divergent sequence. Let${\left\{V_i:C\to C\right\}}_{i=1}^{\infty }$be a countable family of uniformly k-strict pseudo-contractions,${\left\{T_i:C\to C\right\}}_{i=1}^{\infty }$be a countable family of nonexpansive mappings defined by T _i x = tx + (1 - t)V _i x, ∀x ∈ C, ∀i ≥ 1, t ∈ [k, 1). For l ∈ {1,2, ..., M}, suppose$\Theta :=F\left(\mathcal{S}\right)\cap \left({\cap }_{i=1}^{\infty }F\left(T_i\right)\right)\cap \left({\cap }_{k=1}^{N}SMEP\left(F_k\right)\right)\cap GSVI\left(C,A_l\right)\ne \varnothing$. Let {x _n } be a sequence generated by$x_0\in C,C_{1,i}=C,C_1={\cap }_{i=1}^{\infty }{C}_{1,i},x_1=P_{C_1}{x}_0$and

$$\left\{\begin{array}{c}{u}_n=K_{r_{N,n}}^{F_N}{K}_{r_{N-1,n}}^{F_{N-1}}\dots K_{r_{2,n}}^{F_2}{K}_{r_{1,n}}^{F_1}{x}_n,\hfill \\ w_n=P_C\left(I-{\lambda }_M{A}_M\right)P_C\left(I-{\lambda }_{M-1}{A}_{M-1}\right)\dots P_C\left(I-{\lambda }_2{A}_2\right)P_C\left(I-{\lambda }_1{A}_1\right)u_n,\hfill \\ y_{n,i}={\alpha }_{n,i}{x}_0+\left(1-{\alpha }_{n,i}\right)\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds,\hfill \\ C_{n+1,i}=\left\{z\in C_{n,i}:{∥y_{n,i}-z∥}^2\le {∥x_n-z∥}^2+{\alpha }_{n,i}\left({∥x_0∥}^2+2〈x_n-x_0,z〉\right)\right\},\hfill \\ C_{n+1}={\cap }_{i=1}^{\infty }{C}_{n+1,i},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_0\hfill \end{array}\right.$$

(25)

for every n ≥ 0, where$K_{r_k}^{F_k}:C\to C$, is the mapping defined by (24), r _k > 0, k = 1, 2, ..., N are constants and${\left\{{\alpha }_{n,i}\right\}}_{n=1}^{\infty }\subset \left(0,1\right)$satisfy the following conditions:

(i) η _k : C × C → H is L _k -Lipschitz continuous with constant k = 1,2, ..., N such that

(1) η _k (x, y) + η _k (y, x) = 0, ∀x, y ∈ C,

(2) x ↦ η _k (x, y) is affine,

(3) for each fixed y ∈ C, y ↦ η _k (x, y) is sequentially continuous from the weak topology to the weak topology;

(ii)${\mathcal{K}}_k:C\to \mathcal{R}$is η _k -strongly convex with constant σ _k > 0 and its derivative${\mathcal{K}}_k^{\prime }$is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant ν _k > 0 such that σ _k > L _k ν _k ;

(iii) For each k ∈ {1, 2, ..., N} and for all x ∈ C, there exist a bounded subset D _x ⊂ C and z _x ∈ C such that for any y ∈ C\D _x ,

$$F_k\left(y,z_x\right)+\phi \left(z_x\right)-\phi \left(y\right)+\frac{1}{r_k}〈{\mathcal{K}}^{\prime }\left(y\right)-{\mathcal{K}}^{\prime }\left(x\right),\eta \left(z_x,y\right)〉<0;$$

(iv) lim_n→∞α _n,i = 0,∀i ≥ 1;

(v) {λ _l } ⊂ (0,2β _l ), ∀l = 1, 2, ..., M;

(vi) lim inf_n→∞r _k,n > 0, ∀k = 1, 2, 3, ..., N.

Then, {x _n } converges strongly to P_Θx₀.

Proof. First, we show that I - λ _l A _l for all l ∈ {1, 2, ..., M} is nonexpansive mappings.

Indeed, for all x, y ∈ C and λ _l ∈ (0, 2β _l ), we observe that

$$\begin{array}{ll}\hfill {∥\left(I-{\lambda }_1{A}_l\right)x-\left(I-{\lambda }_1{A}_l\right)y∥}^2& ={∥x-y-{\lambda }_l\left(A_{l}x-A_{l}y\right)∥}^2\\ ={∥x-y∥}^2-2{\lambda }_l〈x-y,A_{l}x-A_{l}y〉+{\lambda }_l^2{∥A_{l}x-A_{l}y∥}^2\\ \le {∥x-y∥}^2-2{\lambda }_1{\beta }_l{∥A_{l}x-A_{l}y∥}^2+{\lambda }_l^2{∥A_{l}x-A_{l}y∥}^2\\ \le {∥x-y∥}^2+{\lambda }_l\left({\lambda }_l-2{\beta }_l\right){∥A_{l}x-A_{l}y∥}^2\\ \le {∥x-y∥}^2,\end{array}$$

(26)

which implies that the mapping I - λ _l A _l is nonexpansive for all l ∈ {1, 2, ..., M}.

Let p ∈ Θ. Taking

$${\Im }_n^k=K_{r_{k,n}}^{F_k}{K}_{r_{k-1,n}}^{F_{k-1}}\dots K_{r_{2,n}}^{F_2}{K}_{r_{1,n}}^{F_1},k\in \left\{1,2,3,\dots ,N\right\},n\in \mathbb{N}$$

and

$${\mathcal{P}}_C^l=P_C\left(I-{\lambda }_l{A}_l\right)P_C\left(I-{\lambda }_{l-1}{A}_{l-1}\right)\dots P_C\left(I-{\lambda }_2{A}_2\right)P_C\left(I-{\lambda }_1{A}_1\right)\text{for}\text{all}l\in \left\{1,2,3,\dots ,M\right\},$$

${\Im }_n^0={\mathcal{P}}_C^0=I$, where I is the identity mapping on H. From the definition of $K_{r_{k,n}}^{F_k}$ and P _C are nonexpansive then ${\Im }_n^k,k\in \left\{1,2,3,\dots ,N\right\}$ and ${\mathcal{P}}_C^l,l\in \left\{1,2,3,\dots ,M\right\}$ also. We note that $u_n={\Im }_n^N{x}_n$ and $p={\Im }_{r_{k,n}}^{F_k}p$, we have

$$∥u_n-p∥=∥{\Im }_n^N{x}_n-{\Im }_n^{N}p∥\le ∥x_n-p∥.$$

(27)

It follows that

$$∥w_n-p∥=∥{\mathcal{P}}_C^l{u}_n-{\mathcal{P}}_C^{l}p∥\le ∥u_n-p∥\le ∥x_n-p∥,\forall l\in \left\{1,2,3,\dots ,M\right\}.$$

(28)

Next, we will divide the proof into five steps.

Step 1. We show that {x _n } is well defined. Let n = 1, then C_1,i= C is closed and convex for each i ≥ 1. Suppose that C _n,i is closed convex for some n > 1. Then, from definition of C_n+1,i, we know that C_n+1,iis closed convex for the same n ≥ 1. Hence, C _n,i is closed convex for n ≥ 1 and for each i ≥ 1. This implies that C _n is closed convex for n ≥ 1. Furthermore, we show that Θ ⊂ C _n . For n = 1, Θ ⊂ C = C_1,i. For n ≥ 2, let p ∈ Θ. Then,

$$\begin{array}{ll}\hfill {∥y_{n,i}-p∥}^2& ={∥{\alpha }_{n,i}\left(x_0-p\right)+\left(1-{\alpha }_{n,i}\right)\left(\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds-p\right)∥}^2\\ \le {\alpha }_{n,i}{∥x_0-p∥}^2+\left(1-{\alpha }_{n,i}\right){∥\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds-p∥}^2\\ \le {\alpha }_{n,i}{∥x_0-p∥}^2+\left(1-{\alpha }_{n,i}\right){∥w_n-p∥}^2\\ ={∥w_n-p∥}^2+{\alpha }_{n,i}\left({∥x_0-p∥}^2-{∥w_n-p∥}^2\right)\\ \le {∥x_n-p∥}^2+{\alpha }_{n,i}\left({∥x_0∥}^2+2〈x_n-x_0,p〉\right),\end{array}$$

(29)

which shows that p ∈ C _n,i , ∀n ≥ 2, ∀i ≥ 1. Thus, Θ ⊂ C _n,i , ∀n ≥ 1, ∀i ≥ 1. Hence, it follows that $\varnothing \ne \Theta \subset C_n,\forall n\ge 1$. This implies that {x _n } is well-defined.

Step 2. We claim that lim_n→∞∥x_n+1- x _n ∥ = 0 and lim_n→∞∥y_n,i- x _n ∥ = 0, for i ≥ 1. Since $x_n=P_{C_n}{x}_0$ and $x_{n+1}=P_{C_{n+1}}{x}_0\in C_{n+1}\subset C_n,\forall n\ge 1$, we have

$$∥x_n-x_0∥\le ∥x_{n+1}-x_0∥.$$

(30)

Also, as Θ ⊂ C _n by (13), it follows that

$$∥x_n-x_0∥\le ∥z-x_0∥,z\in \Theta ,\forall n\ge 1.$$

(31)

Form (30) and (31), we have that lim _n→∞∥x _n - x₀∥ exists. Hence {x _n } is bounded and so are {y_n,i}, ∀i ≥ 1, {w _n }, {u _n }, {W _n w _n }, and $\left\{\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds\right\}$. For m > n ≥ 1, we have that $x_m=P_{C_m}{x}_0\in C_m\subset C_n$. By (16), we obtain

$${∥x_m-x_n∥}^2\le {∥x_n-x_0∥}^2-{∥x_m-x_0∥}^2.$$

(32)

Letting m, n → ∞ and taking the limit in (32), we have ∥x _m - x _n ∥ → 0, which shows that {x _n } is a Cauchy sequence. In particular,

$$\underset{n\to \infty }{\text{lim}}∥x_{n+1}-x_n∥=0.$$

(33)

Since, {x _n } is a Cauchy sequence, we assume that x _n → z ∈ C. Since $x_{n+1}=P_{C_{n+1}}{x}_0\in C_{n+1}$, then for each i ≥ 1,

$${∥y_{n,i}-x_{n+1}∥}^2\le {∥x_n-x_{n+1}∥}^2+{\alpha }_{n,i}\left({∥x_0∥}^2+2〈x_n-x_0,x_{n+1}〉\right)\to 0,n\to \infty .$$

It follows that

$$∥y_{n,i}-x_n∥\le ∥y_{n,i}-x_{n+1}∥+∥x_{n+1}-x_n∥.$$

Therefore

$$\underset{n\to \infty }{\text{lim}}∥y_{n,i}-x_n∥=0,\forall i\ge 1.$$

(34)

Step 3. We claim that the following statements hold:

• (1) ${\text{lim}}_{n\to \infty }∥{\Im }_n^k{x}_n-{\Im }_n^{k-1}{x}_n∥=0,\forall k=1,2,\dots ,N$;

• (2) lim_n→∞∥u _n - x _n ∥ = 0;

• (3) lim_n→∞∥u _n - w _n ∥ = 0.

Indeed, for p ∈ Θ, note that $K_{r_{k,n}}^{F_k},k=1,2,\dots ,N$ is the firmly nonexpansive, so we have

$$\begin{array}{ll}\hfill {∥{\Im }_n^k{x}_n-{\Im }_n^{k}p∥}^2& ={∥K_{r_{k,n}}^{F_k}{\Im }_n^{k-1}{x}_n-K_{r_{k,n}}^{F_k}p∥}^2\\ \le 〈{\Im }_n^k{x}_n-p,{\Im }_n^{k-1}{x}_n-p〉\\ =\frac{1}{2}\left\{{∥{\Im }_n^k{x}_n-p∥}^2+{∥{\Im }_n^{k-1}{x}_n-p∥}^2-{∥{\Im }_n^k{x}_n-{\Im }_n^{k-1}{x}_n∥}^2\right\}.\end{array}$$

Thus, we get

$${∥{\Im }_n^k{x}_n-{\Im }_n^{k}p∥}^2\le {∥{\Im }_n^{k-1}{x}_n-p∥}^2-{∥{\Im }_n^k{x}_n-{\Im }_n^{k-1}{x}_n∥}^2.$$

It follows that

$$\begin{array}{ll}\hfill {∥u_n-p∥}^2& \le {∥{\Im }_n^k{x}_n-{\Im }_n^{k}p∥}^2\\ \le {∥{\Im }_n^{k-1}{x}_n-p∥}^2-{∥{\Im }_n^k{x}_n-{\Im }_n^{k-1}{x}_n∥}^2\\ \le {∥x_n-p∥}^2-{∥{\Im }_n^k{x}_n-{\Im }_n^{k-1}{x}_n∥}^2.\end{array}$$

(35)

From (29) and (35), we have for i ≥ 1,

$$\begin{array}{ll}\hfill {∥y_{n,i}-p∥}^2& \le {\alpha }_{n,i}{∥x_0-p∥}^2+\left(1-{\alpha }_{n,i}\right){∥w_n-p∥}^2\\ \le {\alpha }_{n,i}{∥x_0-p∥}^2+\left(1-{\alpha }_{n,i}\right){∥u_n-p∥}^2\\ \le {\alpha }_{n,i}{∥x_0-p∥}^2+\left(1-{\alpha }_{n,i}\right)\left[{∥x_n-p∥}^2-{∥{\Im }_n^k{x}_n-{\Im }_n^{k-1}{x}_n∥}^2\right],\end{array}$$

it follows that

$$\begin{array}{ll}\hfill \left(1-{\alpha }_{n,i}\right){∥{\Im }_n^k{x}_n-{\Im }_n^{k-1}{x}_n∥}^2& \le {\alpha }_{n,i}{∥x_0-p∥}^2+{∥x_n-p∥}^2-{∥y_{n,i}-p∥}^2\\ \le {\alpha }_{n,i}{∥x_0-p∥}^2+∥x_n-y_{n,i}∥\left(∥x_n-p∥+∥y_{n,i}-p∥\right).\end{array}$$

By the condition (iv) and (34), we have

$$\underset{n\to \infty }{\text{lim}}∥{\Im }_n^k{x}_n-{\Im }_n^{k-1}{x}_n∥=0.$$

(36)

For p ∈ Θ and again since $K_{r_{k,n}}^{F_k},k=1,2,\dots ,N$ is the firmly nonexpansive, we obtain

$$\begin{array}{ll}\hfill {∥u_n-p∥}^2& ={∥{\Im }_n^k{x}_n-{\Im }_n^{k}p∥}^2\\ \le 〈{\Im }_n^k{x}_n-{\Im }_n^{k}p,x_n-p〉\\ =\frac{1}{2}\left\{{∥{\Im }_n^k{x}_n-{\Im }_n^{k}p∥}^2+{∥x_n-p∥}^2-{∥{\Im }_n^k{x}_n-x_n∥}^2\right\}\\ =\frac{1}{2}\left\{{∥u_n-p∥}^2+{∥x_n-p∥}^2-{∥u_n-x_n∥}^2\right\}\end{array}$$

and hence

$${∥u_n-p∥}^2\le {∥x_n-p∥}^2-{∥u_n-x_n∥}^2.$$

(37)

From (29) and (37), for i ≥ 1, we have

$$\begin{array}{ll}\hfill {∥y_{n,i}-p∥}^2& \le {\alpha }_{n,i}{∥x_0-p∥}^2+\left(1-{\alpha }_{n,i}\right){∥w_n-p∥}^2\\ \le {\alpha }_{n,i}{∥x_0-p∥}^2+\left(1-{\alpha }_{n,i}\right){∥u_n-p∥}^2\\ \le {\alpha }_{n,i}{∥x_0-p∥}^2+\left(1-{\alpha }_{n,i}\right)\left[{∥x_n-p∥}^2-{∥u_n-x_n∥}^2\right],\end{array}$$

it follows that

$$\begin{array}{ll}\hfill \left(1-{\alpha }_{n,i}\right){∥u_n-x_n∥}^2& \le {\alpha }_{n,i}{∥x_0-p∥}^2+{∥x_n-p∥}^2-{∥y_{n,i}-p∥}^2\\ \le {\alpha }_{n,i}{∥x_0-p∥}^2+∥x_n-y_{n,i}∥\left(∥x_n-p∥+∥y_{n,i}-p∥\right).\end{array}$$

By the condition (iv) and (34), we have

$$\underset{n\to \infty }{\text{lim}}∥u_n-x_n∥=0.$$

(38)

From (26), we note that

$$\begin{array}{ll}\hfill {∥w_n-p∥}^2& ={∥{\mathcal{P}}_C^M{u}_n-{\mathcal{P}}_C^{M}p∥}^2\\ ={∥P_C\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}{u}_n-P_C\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}p∥}^2\\ \le {∥\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}{u}_n-\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}p∥}^2\\ \le {∥{\mathcal{P}}_C^{M-1}{u}_n-{\mathcal{P}}_C^{M-1}p∥}^2+{\lambda }_M\left({\lambda }_M-2{\beta }_M\right){∥A_M{\mathcal{P}}_C^{M-1}{u}_n-A_M{\mathcal{P}}_C^{M-1}p∥}^2\\ ⋮\\ \le {∥u_n-p∥}^2+\sum _{l=1}^M{\lambda }_l\left({\lambda }_l-2{\beta }_i\right){∥A_l{\mathcal{P}}_C^{l-1}{u}_n-A_l{\mathcal{P}}_C^{l-1}p∥}^2\\ \le {∥x_n-p∥}^2+\sum _{l=1}^M{\lambda }_l\left({\lambda }_l-2{\beta }_l\right){∥A_l{\mathcal{P}}_C^{l-1}{u}_n-A_l{\mathcal{P}}_C^{l-1}p∥}^2.\end{array}$$

(39)

It follows that, for i ≥ 1

$$\begin{array}{ll}\hfill {∥y_{n,i}-p∥}^2& \le {\alpha }_{n,i}{∥x_0-p∥}^2+\left(1-{\alpha }_{n,i}\right){∥w_n-p∥}^2\\ ={\alpha }_{n,i}{∥x_0-p∥}^2+\left(1-{\alpha }_{n,i}\right)\left[{∥x_n-p∥}^2+\sum _{l=1}^M{\lambda }_l\left({\lambda }_l-2{\beta }_l\right){∥A_l{\mathcal{P}}_C^{l-1}{u}_n-A_l{\mathcal{P}}_C^{l-1}p∥}^2\right]\\ \le {\alpha }_{n,i}{∥x_0-p∥}^2+{∥x_n-p∥}^2+\left(1-{\alpha }_{n,i}\right)\sum _{l=1}^M{\lambda }_l\left({\lambda }_l-2{\beta }_l\right){∥A_l{\mathcal{P}}_C^{l-1}{u}_n-A_l{\mathcal{P}}_C^{l-1}p∥}^2,\end{array}$$

(40)

which implies that

$$\begin{array}{c}\left(1-{\alpha }_{n,i}\right){\sum }_{l=1}^M{{\lambda }_l\left(2{\beta }_l-{\lambda }_l\right)∥A_l{\mathcal{P}}_C^{l-1}{u}_n-A_l{\mathcal{P}}_C^{l-1}p∥}^2\\ ={\alpha }_{n,i}{∥x_0-p∥}^2+{∥x_n-p∥}^2-{∥y_{n,i}-p∥}^2\\ \le {\alpha }_{n,i}{∥x_0-p∥}^2+∥x_n-y_{n,i}∥\left(∥x_n-p∥+∥y_{n,i}-p∥\right).\end{array}$$

(41)

By the conditions (iv), (v) and (34), we obtain

$$\underset{n\to \infty }{\text{lim}}∥A_l{\mathcal{P}}_C^{l-1}{u}_n-A_l{\mathcal{P}}_C^{l-1}p∥=0.$$

(42)

On the other hand, we note that

$$\begin{array}{ll}\hfill {∥{\mathcal{P}}_C^M{u}_n-{\mathcal{P}}_C^{M}p∥}^2& ={∥P_C\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}{u}_n-P_C\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}p∥}^2\\ \le 〈\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}{u}_n-\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}p,{\mathcal{P}}_C^M{u}_n-{\mathcal{P}}_C^{M}p〉\\ =\frac{1}{2}\left\{{∥\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}{u}_n-\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}p∥}^2+{∥{\mathcal{P}}_C^M{u}_n-{\mathcal{P}}_C^{M}p∥}^2\right.\\ \left.-{∥\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}{u}_n-\left(I-{\lambda }_M{A}_M\right){\mathcal{P}}_C^{M-1}p-\left({\mathcal{P}}_C^M{u}_n-{\mathcal{P}}_C^{M}p\right)∥}^2\right\}\\ \le \frac{1}{2}\left\{{∥{\mathcal{P}}_C^{M-1}{u}_n-{\mathcal{P}}_C^{M-1}p∥}^2+{∥{\mathcal{P}}_C^M{u}_n-{\mathcal{P}}_C^{M}p∥}^2\right.\\ \left.-{∥\left({\mathcal{P}}_C^{M-1}{u}_n-{\mathcal{P}}_C^{M-1}p-{\mathcal{P}}_C^M{u}_n+{\mathcal{P}}_C^{M}p\right)-{\lambda }_M\left(A_M{\mathcal{P}}_C^{M-1}{u}_n-A_M{\mathcal{P}}_C^{M-1}p\right)∥}^2\right\}\\ =\frac{1}{2}\left\{{∥{\mathcal{P}}_C^{M-1}{u}_n-{\mathcal{P}}_C^{M-1}p∥}^2+{∥{\mathcal{P}}_C^M{u}_n-{\mathcal{P}}_C^{M}p∥}^2-∥{\mathcal{P}}_C^{M-1}{u}_n-{\mathcal{P}}_C^{M-1}p-{\mathcal{P}}_C^M{u}_n\right.\\ {+{\mathcal{P}}_C^{M}p∥}^2+2{\lambda }_M〈{\mathcal{P}}_C^{M-1}{u}_n-{\mathcal{P}}_C^{M-1}p-{\mathcal{P}}_C^M{u}_n+{\mathcal{P}}_C^{M}p,A_M{\mathcal{P}}_C^{M-1}{u}_n-A_M{\mathcal{P}}_C^{M-1}p〉\\ \left.-{\lambda }_M^2{∥A_M{\mathcal{P}}_C^{M-1}{u}_n-A_M{\mathcal{P}}_C^{M-1}p∥}^2\right\}\\ \le \frac{1}{2}\left\{{∥{\mathcal{P}}_C^{M-1}{u}_n-{\mathcal{P}}_C^{M-1}p∥}^2+{∥{\mathcal{P}}_C^M{u}_n-{\mathcal{P}}_C^{M}p∥}^2-∥{\mathcal{P}}_C^{M-1}{u}_n-{\mathcal{P}}_C^{M-1}p-{\mathcal{P}}_C^M{u}_n\right.\\ \left.{+{\mathcal{P}}_C^{M}p∥}^2+2{\lambda }_M∥{\mathcal{P}}_C^{M-1}{u}_n-{\mathcal{P}}_C^{M-1}p-{\mathcal{P}}_C^M{u}_n+{\mathcal{P}}_C^{M}p∥∥A_M{\mathcal{P}}_C^{M-1}{u}_n-A_M{\mathcal{P}}_C^{M-1}p∥\right\},\end{array}$$

which implies that

$$\begin{array}{ll}\hfill {∥{\mathcal{P}}_C^M{u}_n-{\mathcal{P}}_C^{M}p∥}^2& \le {∥{\mathcal{P}}_C^{M-1}{u}_n-{\mathcal{P}}_C^{M-1}p∥}^2-{∥{\mathcal{P}}_C^{M-1}{u}_n-{\mathcal{P}}_C^{M-1}p-{\mathcal{P}}_C^M{u}_n+{\mathcal{P}}_C^{M}p∥}^2\\ +2{\lambda }_M∥{\mathcal{P}}_C^{M-1}{u}_n-{\mathcal{P}}_C^{M-1}p-{\mathcal{P}}_C^M{u}_n+{\mathcal{P}}_C^{M}p∥∥A_M{\mathcal{P}}_C^{M-1}{u}_n-A_M{\mathcal{P}}_C^{M-1}p∥\\ ⋮\\ \le {∥u_n-p∥}^2-\sum _{l=1}^M{∥{\mathcal{P}}_C^{l-1}{u}_n-{\mathcal{P}}_C^{l-1}p-{\mathcal{P}}_C^l{u}_n+{\mathcal{P}}_C^{l}p∥}^2\\ +\sum _{l=1}^{M}2{\lambda }_l∥{\mathcal{P}}_C^{l-1}{u}_n-{\mathcal{P}}_C^{l-1}p-{\mathcal{P}}_C^l{u}_n+{\mathcal{P}}_C^{l}p∥∥A_l{\mathcal{P}}_C^{l-1}{u}_n-A_l{\mathcal{P}}_C^{l-1}p∥\\ \le {∥x_n-p∥}^2-\sum _{l=1}^M{∥{\mathcal{P}}_C^{l-1}{u}_n-{\mathcal{P}}_C^{l-1}p-{\mathcal{P}}_C^l{u}_n+{\mathcal{P}}_C^{l}p∥}^2\\ +\sum _{l=1}^{M}2{\lambda }_l∥{\mathcal{P}}_C^{l-1}{u}_n-{\mathcal{P}}_C^{l-1}p-{\mathcal{P}}_C^l{u}_n+{\mathcal{P}}_C^{l}p∥∥A_l{\mathcal{P}}_C^{l-1}{u}_n-A_l{\mathcal{P}}_C^{l-1}p∥.\end{array}$$

(43)

From (29) and (43), for i ≥ 1, we note that

$$\begin{array}{ll}\hfill {∥y_{n,i}-p∥}^2& \le {\alpha }_{n,i}{∥x_0-p∥}^2+\left(1-{\alpha }_{n,i}\right){∥w_n-p∥}^2\\ ={\alpha }_{n,i}{∥x_0-p∥}^2+\left(1-{\alpha }_{n,i}\right){∥{\mathcal{P}}_C^M{u}_n-{\mathcal{P}}_C^{M}p∥}^2\\ \le {\alpha }_{n,i}{∥x_0-p∥}^2+\left(1-{\alpha }_{n,i}\right)\left\{{∥x_n-p∥}^2-\sum _{l=1}^M{∥{\mathcal{P}}_C^{l-1}{u}_n-{\mathcal{P}}_C^{l-1}p-{\mathcal{P}}_C^l{u}_n+{\mathcal{P}}_C^{l}p∥}^2\right.\\ \left.+\sum _{l=1}^{M}2{\lambda }_l∥{\mathcal{P}}_C^{l-1}{u}_n-{\mathcal{P}}_C^{l-1}p-{\mathcal{P}}_C^l{u}_n+{\mathcal{P}}_C^{l}p∥∥A_l{\mathcal{P}}_C^{l-1}{u}_n-A_l{\mathcal{P}}_C^{l-1}p∥\right\}\\ \le {\alpha }_{n,i}{∥x_0-p∥}^2+{∥x_n-p∥}^2-\left(1-{\alpha }_{n,i}\right)\sum _{l=1}^M{∥{\mathcal{P}}_C^{l-1}{u}_n-{\mathcal{P}}_C^{l-1}p-{\mathcal{P}}_C^l{u}_n+{\mathcal{P}}_C^{l}p∥}^2\\ +\sum _{l=1}^{M}2{\lambda }_l∥{\mathcal{P}}_C^{l-1}{u}_n-{\mathcal{P}}_C^{l-1}p-{\mathcal{P}}_C^l{u}_n+{\mathcal{P}}_C^{l}p∥∥A_l{\mathcal{P}}_C^{l-1}{u}_n-A_l{\mathcal{P}}_C^{l-1}p∥.\end{array}$$

(44)

This implies that

$$\begin{array}{c}\left(1-{\alpha }_{n,i}\right){\sum }_{l=1}^M{∥{\mathcal{P}}_C^{l-1}{u}_n-{\mathcal{P}}_C^{l-1}p-{\mathcal{P}}_C^l{u}_n+{\mathcal{P}}_C^{l}p∥}^2\\ \le {\alpha }_{n,i}{∥x_0-p∥}^2+{∥x_n-p∥}^2-{∥y_{n,i}-p∥}^2\\ +\sum _{l=1}^{M}2{\lambda }_l∥{\mathcal{P}}_C^{l-1}{u}_n-{\mathcal{P}}_C^{l-1}p-{\mathcal{P}}_C^l{u}_n+{\mathcal{P}}_C^{l}p∥∥A_l{\mathcal{P}}_C^{l-1}{u}_n-A_l{\mathcal{P}}_C^{l-1}p∥\\ \le {\alpha }_{n,i}{∥x_0-p∥}^2+∥x_n-y_{n,i}∥\left(∥x_n-p∥+∥y_{n,i}-p∥\right)\\ +\sum _{l=1}^{M}2{\lambda }_l∥{\mathcal{P}}_C^{l-1}{u}_n-{\mathcal{P}}_C^{l-1}p-{\mathcal{P}}_C^l{u}_n+{\mathcal{P}}_C^{l}p∥∥A_l{\mathcal{P}}_C^{l-1}{u}_n-A_l{\mathcal{P}}_C^{l-1}p∥.\end{array}$$

(45)

Then, by condition (iv), (34) and (42), we obtain that

$$\underset{n\to \infty }{\text{lim}}∥{\mathcal{P}}_C^{l-1}{u}_n-{\mathcal{P}}_C^{l-1}p-{\mathcal{P}}_C^l{u}_n+{\mathcal{P}}_C^{l}p∥=0.$$

(46)

Therefore, we have

$$∥u_n-w_n∥=∥{\mathcal{P}}_C^0{u}_n-{\mathcal{P}}_C^l{u}_n∥\le \sum _{l=1}^M{∥{\mathcal{P}}_C^{l-1}{u}_n-{\mathcal{P}}_C^{l-1}p-{\mathcal{P}}_C^l{u}_n+{\mathcal{P}}_C^{l}p∥}^2\to 0\text{as}n\to \infty .$$

(47)

On the other hand, by condition (iv) implies that

$${∥y_{n,i}-\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds∥}^2={\alpha }_{n,i}∥x_0-\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds∥\to 0,\forall i\ge 1$$

(48)

it follows that

$$∥x_n-\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds∥\le ∥x_n-y_{n,i}∥+∥y_{n,i}+\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds∥\to 0.$$

(49)

Step 4. We show that $z\in \Theta :=F\left(\mathcal{S}\right)\cap \left({\cap }_{i=1}^{\infty }F\left(T_i\right)\right)\cap \left({\cap }_{k=1}^{N}SMEP\left(F_k\right)\right)\cap GSVI\left(C,A_l\right),\forall l\in \left\{1,2,\dots ,M\right\}$. Since $\left\{w_{n_i}\right\}$ is bounded, there exists a subsequence $\left\{w_{n_{i_j}}\right\}$ of $\left\{w_{n_i}\right\}$ which converges weakly to z ∈ C. Without loss of generality, we can assume that $w_{n_i}\rightharpoonup z$.

• (1) First, we prove that $z\in F\left(\mathcal{S}\right)$. From (38), (47), and (49), we get

  $$\underset{n\to \infty }{\text{lim}}∥w_n-\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds∥=0.$$

  (50)

Since {W _n w _n } is a bounded and from Lemma 2.5 for any h ≥ 0, we have

$$\underset{n\to \infty }{\text{lim}}∥\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds-S\left(h\right)\left(\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds\right)∥=0,$$

(51)

and since

$$\begin{array}{ll}\hfill ∥w_n-S\left(h\right)w_n∥& \le ∥w_n-\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds∥+∥\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds-S\left(h\right)\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds∥\\ +∥S\left(h\right)\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds-S\left(h\right)w_n∥\\ \le 2∥w_n-\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds∥+∥\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds-S\left(h\right)\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_n{w}_{n}ds∥\end{array}$$

for all 0 ≤ s < ∞. It follows from (50) and (51), we get

$$\underset{n\to \infty }{\text{lim}}∥w_n-S\left(h\right)w_n∥=0.$$

(52)

Indeed, from Lemma 2.6 and (52), we get $z\in F\left(\mathcal{S}\right)$, i.e., z = S(s)z, ∀s ≥ 0.

• (2) Next, we show that $z\in F\left(W\right)={\cap }_{n=1}^{\infty }F\left(W_n\right)={\cap }_{i=1}^{\infty }F\left(T_i\right)$, where $F\left(W_n\right)={\cap }_{i=1}^{n}F\left(T_i\right),\forall n\ge 1$ and F(W _{n+ 1}) ⊂ F(W _n ). Assume that z ∉ F(W), then there exists a positive integer m such that z ∉ F(T _m ) and so $z\notin {\cap }_{i=1}^{m}F\left(T_i\right)$. Hence for any $n\ge m,z\notin {\cap }_{i=1}^{n}F\left(T_i\right)=F\left(W_n\right)$, i.e., z ≠ W _n z. This together with z = S(s)z, ∀s ≥ 0 shows z = S(s)z ≠ S(s)W _n z, ∀s ≥ 0, therefore we have $z\ne \frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)W_{n}zds,\forall n\ge m$. It follows from the Opial's condition and (50) that

  $$\begin{array}{ll}\hfill \underset{i\to \infty }{\text{lim inf}}∥w_{n_i}-z∥& <\underset{i\to \infty }{\text{lim inf}}∥w_{n_i}-\frac{1}{t_{n_i}}{\int }_0^{t_{n_i}}S\left(s\right)W_{n_i}zds∥\\ \le \underset{i\to \infty }{\text{lim inf}}\left(∥w_{n_i}-\frac{1}{t_{n_i}}{\int }_0^{t_{n_i}}S\left(s\right)W_{n_i}{w}_{n_i}ds∥\right.\\ \left.+∥\frac{1}{t_{n_i}}{\int }_0^{t_{n_i}}S\left(s\right)W_{n_i}{w}_{n_i}ds-\frac{1}{t_{n_i}}{\int }_0^{t_{n_i}}S\left(s\right)W_{n_i}zds∥\right)\\ \le \underset{i\to \infty }{\text{lim inf}}∥w_{n_i}-z∥,\end{array}$$

which is a contradiction. Thus, we get $z\in {\cap }_{i=1}^{\infty }F\left(T_i\right)$.

• (3) We prove that $z\in {\cap }_{k=1}^{N}SMEP\left(F_k,\phi \right)$. Since ${\Im }_n^k={\mathcal{K}}_{r_k}^{F_k},k=1,2,\dots ,N$ and $u_n^k={\Im }_n^k{x}_n$, we have

  $$F_k\left({\Im }_n^k{x}_n,x\right)+\phi \left(x\right)-\phi \left({\Im }_n^k{x}_n\right)+\frac{1}{r_k}〈{\mathcal{K}}^{\prime }\left({\Im }_n^k{x}_n\right)-{\mathcal{K}}^{\prime }\left({\Im }_n^{k-1}{x}_n\right),\eta \left(x,{\Im }_n^k{x}_n\right)〉\ge 0,\forall x\in C.$$

It follows that

$$\frac{1}{r_k}〈{\mathcal{K}}^{\prime }\left({\Im }_{n_i}^k{x}_{n_i}\right)-{\mathcal{K}}^{\prime }\left({\Im }_{n_i}^{k-1}{x}_{n_i}\right),\eta \left(x,{\Im }_{n_i}^k{x}_{n_i}\right)〉\ge -F_k\left({\Im }_{n_i}^k{x}_{n_i},x\right)-\phi \left(x\right)+\phi \left({\Im }_{n_i}^k{x}_{n_i}\right)$$

(53)

for all x ∈ C. From (36) and by conditions (i)(3) and (ii), we get

$$\underset{n_i\to \infty }{\text{lim}}\frac{1}{r_k}〈{\mathcal{K}}^{\prime }\left({\Im }_{n_i}^k{x}_{n_i}\right)-{\mathcal{K}}^{\prime }\left({\Im }_{n_i}^{k-1}{x}_{n_i}\right),\eta \left(x,{\Im }_{n_i}^k{x}_{n_i}\right)〉=0.$$

By the assumption and by the condition (H1), we know that the function φ and the mapping x ↦ (-F _k (x, y)) both are convex and lower semicontinuous, hence they are weakly lower semicontinuous. These together with $\frac{{\mathcal{K}}^{\prime }\left({\Im }_{n_i}^k{x}_{n_i}\right)-{\mathcal{K}}^{\prime }\left({\Im }_{n_i}^{k-1}{x}_{n_i}\right)}{r_k}\to 0$ and ${\Im }_{n_i}^k{x}_{n_i}\rightharpoonup z$, we have $0=\underset{n_i\to \infty }{\text{lim inf}}〈\frac{{\mathcal{K}}^{\prime }\left({\Im }_{n_i}^k{x}_{n_i}\right)-{\mathcal{K}}^{\prime }\left({\Im }_{n_i}^{k-1}{x}_{n_i}\right)}{r_k},\eta \left(x,{\Im }_{n_i}^k{x}_{n_i}\right)〉\ge \underset{n_i\to \infty }{\text{lim inf}}\left\{-F_k\left({\Im }_{n_i}^k{x}_{n_i},x\right)-\phi \left(x\right)+\phi \left({\Im }_{n_i}^k{x}_{n_i}\right)\right\}$.

Then, we obtain

$$F_k\left(z,x\right)+\phi \left(x\right)-\phi \left(z\right)\ge 0,\forall x\in C,\forall k=,1,2,\dots ,N.$$

(54)

Therefore $z\in {\cap }_{k=1}^{N}SMEP\left(F_k,\phi \right)$.

• (4) Last, we show that z ∈ GSVI(C, A _l ), ∀l ∈ {1, 2, ..., M}. By the nonexpansivity of in Lemma 2.7, we have

  $$\begin{array}{ll}\hfill ∥w_n-\mathcal{P}\left(w_n\right)∥& =∥P_C\left(I-{\lambda }_M{A}_M\right)P_C\left(I-{\lambda }_{M-1}{A}_{M-1}\right)\dots P_C\left(I-{\lambda }_2{A}_2\right)P_C\left(I-{\lambda }_1{A}_1\right)u_n-\mathcal{P}\left(w_n\right)∥\\ =∥\mathcal{P}\left(u_n\right)-\mathcal{P}\left(w_n\right)∥\\ \le ∥u_n-w_n∥.\end{array}$$

We note that

$$∥x_n-w_n∥\le ∥x_n-u_n∥+∥u_n-w_n∥.$$

Therefore, we conclude that $\underset{n\to \infty }{\text{lim}}∥w_n-\mathcal{P}\left(w_n\right)∥=0$ and lim_n→∞║x _n - w _n ║ = 0. Since is nonexpansive, we get

$$\begin{array}{ll}\hfill ∥x_n-\mathcal{P}\left(x_n\right)∥& \le ∥x_n-w_n∥+∥w_n-\mathcal{P}\left(w_n\right)∥+∥\mathcal{P}\left(w_n\right)-\mathcal{P}\left(x_n\right)∥\\ \le 2∥x_n-w_n∥+∥w_n-\mathcal{P}\left(w_n\right)∥.\end{array}$$

According to Lemma 2.9, we obtain that z ∈ GSVI(C, A _l ) for all l ∈ {1, 2, ..., M}. Hence by (1)-(4), we have z ∈ Θ.

Step 5. Noting that $x_n=P_{C_n}{x}_0$. By (15), we have

$$〈x_0-x_n,y-x_n〉\le 0,\forall y\in C_n.$$

Since Θ ⊂ C _n and by the continuity of inner product, we obtain from the above inequality that

$$〈x_0-z,y-z〉\le 0,\forall y\in C.$$

By (15) again, we conclude that z = P_Θx₀. This completes the proof.

Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert Space H. Let F : C × C → ℝ be a bifunctions satisfying conditions (H1)-(H3). Let A be a β-inverse-strongly monotone mappings of C into H. Let$\mathcal{S}=\left\{S\left(s\right):0\le s<\infty \right\}$be a nonexpansive semigroup on C and let {t _n } be a positive real divergent sequence. Suppose that$\Theta :=F\left(\mathcal{S}\right)\cap MEP\left(F,\phi \right)\cap VI\left(C,A\right)\ne \mathrm{0̸}$. Let {x _n } be a sequence generated by$x_0\in C,C_{1,i}=C,C_1={\cap }_{i=1}^{\infty }{C}_{1,i},x_1=P_{C_1}{x}_0$and

$$\left\{\begin{array}{c}{u}_n=K_r^F{x}_n,\hfill \\ w_n=P_C\left(I-\lambda A\right)u_n,\hfill \\ y_{n,i}={\alpha }_{n,i}{x}_0+\left(1-{\alpha }_{n,i}\right)\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)w_{n}ds,\hfill \\ C_{n+1,i}=\left\{z\in C_{n,i}:{∥y_{n,i}-z∥}^2\le {∥x_n-z∥}^2+{\alpha }_{n,i}\left({∥x_0∥}^2+2〈x_n-x_0,z〉\right)\right\},\hfill \\ C_{n+1}={\cap }_{i=1}^{\infty }{C}_{n+1,i},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_0\hfill \end{array}\right.$$

(55)

for every n ≥ 0, where$K_r^F:C\to C$is the mapping defined by (24), r > 0 is a constant and${\left\{{\alpha }_{n,i}\right\}}_{n=1}^{\infty }\subset \left(0,1\right)$satisfy the conditions (i)-(vi). Then, {x _n } converges strongly to P_Θx₀.

Proof. Putting W _n = I (Identity mapping), M = 1 and N = 1 in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.

4 Deduced theorems

In this section, we deduce Theorem 3.1 to obtain the following four corollaries.

Corollary 4.1. Let C be a nonempty closed convex subset of a real Hilbert Space H. Let$\mathcal{S}=\left\{S\left(s\right):0\le s<\infty \right\}$be a nonexpansive semigroup on C and let {t _n } be a positive real divergent sequence. Suppose that$F\left(\mathcal{S}\right)\ne \varnothing$. Let {x _n } be a sequence generated by$x_0\in C,C_{1,i}=C,C_1={\cap }_{i=1}^{\infty }{C}_{1,i},x_1=P_{C_1}{x}_0$and

$$\left\{\begin{array}{c}{y}_{n,i}={\alpha }_{n,i}{x}_0+\left(1-{\alpha }_{n,i}\right)\frac{1}{t_n}{\int }_0^{t_n}S\left(s\right)x_{n}ds,\hfill \\ C_{n+1,i}=\left\{z\in C_{n,i}:{∥y_{n,i}-z∥}^2\le {∥x_n-z∥}^2+{\alpha }_{n,i}\left({∥x_0∥}^2+2〈x_n-x_0,z〉\right)\right\},\hfill \\ C_{n+1}={\cap }_{i=1}^{\infty }{C}_{n+1,i},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_0\hfill \end{array}\right.$$

(56)

for every n ≥ 0, where${\left\{{\alpha }_{n,i}\right\}}_{n=1}^{\infty }\subset \left(0,1\right)$satisfy the condition (iv). Then, {x _n } converges strongly to$P_{F\left(\mathcal{S}\right)}{x}_0$.

Proof. Putting $W_n=K_{r_k}^{F_k}=I,\forall k=1,2,\dots ,N$ (Identity mappings) and A _l = 0, ∀l ∈ {1, 2, ..., M} in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.

Corollary 4.2. Let C be a nonempty closed convex subset of a real Hilbert Space H. Let${\left\{V_i:C\to C\right\}}_{i=1}^{\infty }$be a countable family of uniformly k-strict pseudo-contractions,${\left\{T_i:C\to C\right\}}_{i=1}^{\infty }$be a countable family of nonexpansive mappings defined by T _i x = tx+(1 - t)V _i x, ∀x ∈ C, ∀i ≥ 1, t ∈ [k, 1). Suppose that$\Theta :={\cap }_{i=1}^{\infty }F\left(T_i\right)\ne \varnothing$. Let {x _n } be a sequence generated by$x_0\in C,C_{1,i}=C,C_1={\cap }_{i=1}^{\infty }{C}_{1,i},x_1=P_{C_1}{x}_0$and

$$\left\{\begin{array}{c}{y}_{n,i}={\alpha }_{n,i}{x}_0+\left(1-{\alpha }_{n,i}\right)W_n{x}_n,\hfill \\ C_{n+1,i}=\left\{z\in C_{n,i}:{∥y_{n,i}-z∥}^2\le {∥x_n-z∥}^2+{\alpha }_{n,i}\left({∥x_0∥}^2+2〈x_n-x_0,z〉\right)\right\},\hfill \\ C_{n+1}={\cap }_{i=1}^{\infty }{C}_{n+1,i},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_0\hfill \end{array}\right.$$

(57)

for every n ≥ 0, where${\left\{{\alpha }_{n,i}\right\}}_{n=1}^{\infty }\subset \left(0,1\right)$satisfy the condition (iv). Then, {x _n } converges strongly to P_Θx₀.

Proof. Taking $K_{r_k}^{F_k}=I,\forall k=1,2,\dots ,N$ (Identity mapping), s = 0 and A _l = 0, ∀l ∈ {1, 2, ..., M} in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.

Corollary 4.3. Let C be a nonempty closed convex subset of a real Hilbert Space H. Let {F _k : C × C → ℝ, k = 1, 2, ..., N} be a finite family of bifunctions satisfying conditions (H1)-(H3). Suppose that$\Theta :={\cap }_{k=1}^{N}SMEP\left(F_k\right)\ne \varnothing$. Let {x _n } be a sequence generated by$x_0\in C,C_{1,i}=C,C_1={\cap }_{i=1}^{\infty }{C}_{1,i},x_1=P_{C_1}{x}_0$and

$$\left\{\begin{array}{c}{u}_n=K_{r_{N,n}}^{F_N}{K}_{r_{N-1,n}}^{F_{N-1}}\dots K_{r_{2,n}}^{F_2}{K}_{r_{1,n}}^{F_1}{x}_n,\hfill \\ y_{n,i}={\alpha }_{n,i}{x}_0+\left(1-{\alpha }_{n,i}\right)u_n,\hfill \\ C_{n+1,i}=\left\{z\in C_{n,i}:{∥y_{n,i}-z∥}^2\le {∥x_n-z∥}^2+{\alpha }_{n,i}\left({∥x_0∥}^2+2〈x_n-x_0,z〉\right)\right\},\hfill \\ C_{n+1}={\cap }_{i=1}^{\infty }{C}_{n+1,i},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_0\hfill \end{array}\right.$$

(58)

for every n ≥ 0, where$K_r^F:C\to C$is the mapping defined by (24), r > 0 is a constant and${\left\{{\alpha }_{n,i}\right\}}_{n=1}^{\infty }\subset \left(0,1\right)$satisfy the conditions (i)-(iv) and (vi). Then, {x _n } converges strongly to P_Θx₀.

Proof. Putting W _n = I (identity mapping), A _l = 0, ∀l ∈ {1, 2, ..., M} and s = 0 in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.

Corollary 4.4. Let C be a nonempty closed convex subset of a real Hilbert Space H. Let A _l be β _l -inverse-strongly monotone mappings of C into H, where l ∈ {1,2, ..., M}. For l ∈ {1, 2, ..., M}, suppose that$\Theta :=GSVI\left(C,A_l\right)\ne \varnothing$. Let {x _n } be a sequence generated by$x_0\in C,C_{1,i}=C,C_1={\cap }_{i=1}^{\infty }{C}_{1,i},x_1=P_{C_1}{x}_0$and

$$\left\{\begin{array}{c}{w}_n=P_C\left(I-{\lambda }_M{A}_M\right)P_C\left(I-{\lambda }_{M-1}{A}_{M-1}\right)\dots P_C\left(I-{\lambda }_2{A}_2\right)P_C\left(I-{\lambda }_1{A}_1\right)x_n,\hfill \\ y_{n,i}={\alpha }_{n,i}{x}_0+\left(1-{\alpha }_{n,i}\right)w_n,\hfill \\ C_{n+1,i}=\left\{z\in C_{n,i}:{∥y_{n,i}-z∥}^2\le {∥x_n-z∥}^2+{\alpha }_{n,i}\left({∥x_0∥}^2+2〈x_n-x_0,z〉\right)\right\},\hfill \\ C_{n+1}={\cap }_{i=1}^{\infty }{C}_{n+1,i},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_0\hfill \end{array}\right.$$

(59)

for every n ≥ 0, where${\left\{{\alpha }_{n,i}\right\}}_{n=1}^{\infty }\subset \left(0,1\right)$satisfy the conditions (iv) and (v). Then, {x _n } converges strongly to$P_{F\left(\mathcal{S}\right)}{x}_0$.

Proof. Taking $W_n=K_{r_k}^{F_k}=I,\forall k=1,2,\dots ,N$ (identity mapping) and s = 0 in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.