We introduce a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of infinite family of nonexpansive mappings, the set of common solutions to a system of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in a real Hilbert space. We then prove strong convergence of the scheme to a common element of the three sets. We give some applications of our results. Our results extend important recent results.

1. Introduction

Let K be a nonempty closed and convex subset of a real Hilbert space H. A mapping A : K → H is called monotone if

(1.1)

The variational inequality problem is to find an x^* ∈ K such that

(1.2)

(see, e.g., [1]). We will denote the set of solutions to the variational inequality problem (1.2) by VI(K, A).

A mapping A : K → H is called inverse-strongly monotone (see, e.g., [2, 3]) if there exists a positive real number α such that 〈Ax − Ay, x − y〉≥α∥Ax − Ay∥², for all x, y ∈ K. For such a case, A is called α-inverse-strongly monotone.

A mapping T : K → K is said to be nonexpansive if

(1.3)

for all x, y ∈ K. A point x ∈ K is called a fixed point of T if Tx = x. The set of fixed points of T is the set F(T)∶ = {x ∈ K : Tx = x}.

Let φ : K → ℝ be a real-valued function and A : K → H a nonlinear mapping. Suppose that F : K × K into ℝ is an equilibrium bifunction. That is, F(u, u) = 0, for all u ∈ K. The generalized mixed equilibrium problem is to find x ∈ K (see, e.g., [4–6]) such that

(1.4)

for all y ∈ K. We will denote the set of solutions of this generalized mixed equilibrium problem by GMEP(F, A, φ). Thus

(1.5)

If φ = 0, A = 0, then problem (1.4) reduces to equilibrium problem studied by many authors (see, e.g., [7–14]), which is to find x^* ∈ K such that

(1.6)

for all y ∈ K. The set of solutions of (1.6) is denoted by EP (F).

If φ = 0, then problem (1.4) reduces to generalized equilibrium problem studied by many authors (see, e.g., [15–18]), which is to find x^* ∈ K such that

(1.7)

for all y ∈ K. The set of solutions of (1.7) is denoted by GEP (F, A).

If A = 0, then problem (1.4) reduces to mixed equilibrium problem considered by many authors (see, e.g., [19–21]), which is to find x^* ∈ K such that

(1.8)

for all y ∈ K. The set of solutions of (1.8) is denoted by MEP(F, φ).

The generalized mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases (see, e.g., [22]). Numerous problems in physics, optimization, and economics reduce to find a solution of problem (1.4). Several methods have been proposed to solve the fixed point problems, variational inequality problems, and equilibrium problems in the literature. See, for example, [23–33].

One of the iterative processes (see Halpern [34]) which is often used to approximate a fixed point of a nonexpansive mapping T is defined as follows. Take an initial guess x₀ ∈ K arbitrarily and define {x_n} recursively by

(1.9)

where {α_n} is a sequence in [0, 1]. The iteration process (1.9) has been proved to be strongly convergent both in Hilbert spaces [34–36] and uniformly smooth Banach spaces [37, 38] when the sequence {α_n} satisfies the conditions

(i)
lim _n→∞α_n = 0,
(ii)
,
(iii)
either or lim _n→∞α_n/α_n+1 = 1.

Motivated by (1.9), Martinez-Yanes and Xu [39] introduced the following iterative scheme for a single nonexpansive mapping T in a Hilbert space:

(1.10)

where P_K denotes the metric projection of H onto a closed and convex subset K of H. They proved that if {α_n}⊂(0,1) and lim _n→∞α_n = 0, then the sequence {x_n} converges strongly to P_F(T)x₀.

Furthermore, algorithm (1.10) has been modified by many authors for relatively nonexpansive mappings and quasi-ϕ-nonexpansive mappings in Banach spaces (see, e.g., [40–43]).

Recently, Ceng and Yao [44] introduced a new iterative scheme of approximating a common element of the set of solutions to mixed equilibrium problem and set of common fixed points of finite family of nonexpansive mappings in a real Hilbert space H. In the proof process of their results, they imposed the following condition on a nonempty closed and convex subset K of H:

(E)
A : K → ℝ is η-strongly convex and its derivative A^′ is sequentially continuous from weak topology to the strong topology.

We remark here that this condition (E) has been used by many authors for approximation of solution to mixed equilibrium problem in a real Hilbert space (see, e.g., [45, 46]). However, it is observed that condition (E) does not include the case A(x) = ∥x∥²/2 and η(x, y) = x − y. Furthermore, Peng and Yao [19], Wangkeeree and Wangkeeree [47], and other authors replaced condition (E) with these conditions:

(B1)
for each x ∈ H and r > 0, there exist a bounded subset D_x⊆K and y_x ∈ K such that, for any z ∈ K∖D_x,
(1.11)
or
(B2)
K is a bounded set.

Consequently, conditions (B1) and (B2) have been used by many authors in approximating solution to generalized mixed equilibrium (mixed equilibrium) problems in a real Hilbert space (see, e.g., [19, 47]).

In [48], Takahashi et al. proved the following convergence theorem using hybrid method.

Theorem 1.1 (Takahashi et al. [48]). Let K be a nonempty closed and convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of K into itself such that F(T) ≠ ∅. For C₁ = K, , define sequences and of K as follows:

(1.12)

Assume that

satisfies 0 ≤ α_n < α < 1. Then,

converges strongly to P_F(T)x₀.

Motivated by the results of Takahashi et al. [48], Kumam [49] studied the problem of approximating a common element of set of solutions to an equilibrium problem, set of solutions to variational inequality problem, and set of fixed points of a nonexpansive mapping in a real Hilbert space. In particular, he proved the following theorem.

Theorem 1.2 (Kumam [49]). Let K be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from K × K satisfying (A1)–(A4), and let B be a β-inverse-strongly monotone mapping of K into H. Let T be a nonexpansive mapping of K into H such that F(T)∩EP (F)∩VI(K, B) ≠ ∅. For C₁ = K, , define sequences and of K as follows:

(1.13)

Assume that

, and

satisfy

(1.14)

Then,

converges strongly to P_{F(T)∩EP (F)∩VI(K,B)}x₀.

Quite recently, Chantarangsi et al. [50] proved the following convergence theorem for approximation of fixed point of a nonexpansive mapping which is also a common solution to a system of generalized mixed equilibrium problems and variational inequality problem in a real Hilbert space.

Theorem 1.3 (Chantarangsi et al. [50]). Let K be a nonempty closed and convex subset of a real Hilbert space H. Let θ₁, θ₂ be bifunctions from K × K satisfying (A1)–(A4), Ψ₁ an ξ-inverse-strongly monotone mapping of K into H, Ψ₂ a β-inverse-strongly monotone mapping of K into H with assumption (B1) or (B2), and T : K → K a nonexpansive mapping. Let B be an ω-Lipschitz continuous and relaxed (υ, ν) co-coercive mapping of K into H, f : K → K a contraction mapping with coefficient η ∈ (0,1), and A a strongly positive linear bounded selfadjoint operator with coefficient and . Suppose that F∶ = F(T)∩GMEP(θ₁, φ, Ψ₁)∩GMEP(θ₂, φ, Ψ₂)∩VI(K, B) ≠ ∅. Let , , , and be generated by

(1.15)

where {r_n}⊂[a, b]⊂[0,2ξ], {s_n}⊂[c, d]⊂[0,2β], {γ_n}⊂[h, j]⊂(0,1), {γ_n}, {ε_n}, {β_n} are three sequences in (0, 1) satisfying the following conditions:

(C1)
lim _n→∞ε_n = 0 and ,
(C2)
0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1,
(C3)
0 < liminf _n→∞r_n ≤ limsup _n→∞r_n < 2ξ and lim _n→∞ | r_n+1 − r_n | = 0,
(C4)
0 < liminf _n→∞s_n ≤ limsup _n→∞s_n < 2β and lim _n→∞ | s_n+1 − s_n | = 0,
(C5)
{α_n}⊂[e, g]⊂(0, (2(ν − υω²))/ω²), ν > υω² and lim _n→∞ | α_n+1 − α_n | = 0.

Then, {x_n} converges strongly to z = P_F(γf + (I − A))(z).

Motivated by the ongoing research and the above-mentioned results, we modify algorithm (1.10) and introduce a new iterative scheme for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of common solutions to a system of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in a real Hilbert space. Furthermore, we show that our new iterative scheme converges strongly to a common element of the three sets. In the proof process of our results, we use conditions (B1) and (B2) mentioned above. Our result extends many important recent results. Finally, we give some applications of our results.

2. Preliminaries

Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ∥·∥, and let K be a nonempty closed and convex subset of H. The strong convergence of to x is written x_n → x as n → ∞.

For any point u ∈ H, there exists a unique point P_Ku ∈ K such that

(2.1)

P_K is called the metric projection of H onto K. We know that P_K is a nonexpansive mapping of H onto K. It is also known that P_K satisfies

(2.2)

for all x, y ∈ H. Furthermore, P_Kx is characterized by the properties P_Kx ∈ K and

(2.3)

for all y ∈ K and

(2.4)

In the context of the variational inequality problem, (2.3) implies that

(2.5)

If A is an α-inverse-strongly monotone mapping of K into H, then it is obvious that A is a (1/α)-Lipschitz continuous. We also have that, for all x, y ∈ K and r > 0,

(2.6)

So, if r ≤ 2α, then I − rA is a nonexpansive mapping of K into H.

For solving the generalized mixed equilibrium problem for a bifunction F : K × K → ℝ, let us assume that F, φ, and K satisfy the following conditions:

(A1)
F(x, x) = 0 for all x ∈ K,
(A2)
F is monotone, that is, F(x, y) + F(y, x) ≤ 0 for all x, y, ∈K,
(A3)
for each x, y, z ∈ K, lim _t→0F(tz + (1 − t)x, y) ≤ F(x, y),
(A4)
for each x ∈ K, y ↦ F(x, y) is convex and lower semicontinuous,
(B1)
for each x ∈ H and r > 0 there exist a bounded subset D_x⊆K and y_x ∈ K such that for any z ∈ K∖D_x,

(2.7)

(B2)
K is a bounded set.

Then, we have the following lemma.

Lemma 2.1 (Wangkeeree and Wangkeeree [47]). Assume that F : K × K → ℝ satisfies (A1)–(A4), and let φ : K → ℝ be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r > 0 and x ∈ H, define a mapping as follows:

(2.8)

for all z ∈ H. Then, the following hold:

(1)
for each x ∈ H, ,
(2)
is single-valued,
(3)
is firmly nonexpansive, that is, for any x, y ∈ H,
(2.9)
(4)
,
(5)
GMEP(F) is closed and convex.

We will also use the following lemma in our results.

Lemma 2.2 (Baillon and Haddad [51]). Let E be a Banach space, let f be a continuously Fréchet differentiable convex functional on E, and let ∇f be the gradient of f. If ∇f is (1/α)-Lipschitz continuous, then ∇f is α-inverse-strongly monotone.

3. Main Results

Theorem 3.1. Let K be a nonempty closed and convex subset of a real Hilbert space H. For each m = 1,2, let F_m be a bifunction from K × K satisfying (A1)–(A4), φ_m : K → ℝ ∪ {+∞} a proper lower semicontinuous and convex function with assumption (B1) or (B2), A be an α-inverse-strongly monotone mapping of K into H, and B a β-inverse-strongly monotone mapping of K into H, and, for each i = 1,2, …, let T_i : K → K be a nonexpansive mapping such that . Let D be a γ-inverse-strongly monotone mapping of K into H. Suppose that . Let , , , (i = 1,2, …), and be generated by x₀ ∈ K, C_1,i = K, , ,

(3.1)

Assume that

(i = 1,2, …),

, and

satisfy

(i)
0 < a ≤ r_n ≤ b < 2α,
(ii)
0 < c ≤ λ_n ≤ f < 2β,
(iii)
lim _n→∞α_n,i = 0,
(iv)
0 < h ≤ s_n ≤ j < 2γ.

Then,

converges strongly to P_Fx₀.

Proof. Let x^* ∈ F. Then,

(3.2)

Since both I − r_nA and I − λ_nB are nonexpansive for each n ≥ 1 and

, from (2.6), we have that

(3.3)

Therefore,

(3.4)

Let n = 1, then C_1,i = K is closed convex for each i = 1,2, …. Now assume 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,i is closed convex for the same n > 1. Hence, C_n,i is closed convex for n ≥ 1 and for each i = 1,2, …. This implies that C_n is closed convex for n ≥ 1. Furthermore, we show that F ⊂ C_n. For n = 1, F ⊂ K = C_1,i. For n ≥ 2, let x^* ∈ F. Then,

(3.5)

which shows that x^* ∈ C_n,i, for all n ≥ 2, for all i = 1,2, …. Thus, F ⊂ C_n,i, for all n ≥ 1, for all i = 1,2, …. Hence, it follows that F ⊂ C_n, for all n ≥ 1. Since

, for all n ≥ 1, and x_n+1 ∈ C_n+1 ⊂ C_n, for all n ≥ 1, we have that

(3.6)

Also, as F ⊂ C_n by (2.1), it follows that

(3.7)

From (3.6) and (3.7), we have that lim _n→∞∥x_n − x₀∥ exists. Hence,

is bounded and so are

, and

, i = 1,2, …. For m > n ≥ 1, we have that

. By (2.4), we obtain

(3.8)

Letting m, n → ∞ and taking the limit in (3.8), we have that x_m − x_n → 0, m, n → ∞, which shows that

is Cauchy. In particular, lim _n→∞∥x_n+1 − x_n∥ = 0. Since,

is Cauchy, we assume that x_n → z ∈ K.

Since , then

(3.9)

and it follows that

(3.10)

Thus,

(3.11)

Furthermore,

(3.12)

Since 0 < c ≤ λ_n ≤ f < 2β, we have that

(3.13)

Hence, lim _n→∞∥Bz_n − Bx^*∥ = 0. From (3.1), we have that

(3.14)

On the other hand,

(3.15)

and, hence,

(3.16)

Putting (3.16) into (3.14), we have that

(3.17)

It follows that

(3.18)

Therefore, lim _n→∞∥z_n − u_n∥ = 0. Furthermore,

(3.19)

Since 0 < a ≤ r_n ≤ b < 2α, we have that

(3.20)

Hence, lim _n→∞∥Ax_n − Ax^*∥ = 0. From (3.1), we have that

(3.21)

On the other hand,

(3.22)

and, hence,

(3.23)

Putting (3.23) into (3.21), we have that

(3.24)

It follows that

(3.25)

Therefore, lim _n→∞∥x_n − z_n∥ = 0. But y_n,i = α_n,ix₀ + (1 − α_n,i)T_iw_n implies that

(3.26)

Furthermore, we have that

(3.27)

Furthermore,

(3.28)

Thus,

(3.29)

Since 0 < h ≤ s_n ≤ j < 2γ, condition (iii) and ∥y_n,i − x_n∥→0 as n → ∞, we have that lim _n→∞∥Du_n − Dx^*∥ = 0. Now, using (2.2), we obtain

(3.30)

Thus,

(3.31)

Using this last inequality, we obtain from (3.1)

(3.32)

This implies that

(3.33)

Since lim _n→∞α_n,i = 0, ∥y_n,i − x_n∥→0 as n → ∞, and ∥Du_n − Dx^*∥→0 as n → ∞, we have that lim _n→∞∥w_n − u_n∥ = 0. Also since lim _n→∞∥w_n − x_n∥ = 0 and lim _n→∞∥x_n − z∥ = 0, we have that lim _n→∞∥w_n − z∥ = 0. Now,

(3.34)

Hence, lim _n→∞∥w_n − T_iw_n∥ = 0, i = 1,2, …. By lim _n→∞∥w_n − z∥ = 0 and lim _n→∞∥w_n − T_iw_n∥ = 0, i = 1,2, …, we have that

Since , n ≥ 1, we have, for any y ∈ K, that

(3.35)

Furthermore, replacing n by n_j in the last inequality and using (A2), we obtain

(3.36)

Let z_t∶ = ty + (1 − t)z for all t ∈ (0,1] and y ∈ K. This implies that z_t ∈ K. Then, we have that

(3.37)

Since

, j → ∞, we obtain

, j → ∞. Furthermore, by the monotonicity of A, we obtain

. Then, by (A4), we obtain (noting that

)

(3.38)

Using (A1), (A4), and (3.38), we also obtain

(3.39)

and, hence,

(3.40)

Letting t → 0, we have, for each y ∈ K, that

(3.41)

This implies that z ∈ GMEP(F₁, A, φ₁). By following the same arguments, we can show that z ∈ GMEP(F₂, B, φ₂).

Following the arguments of [3, Theorem 3.1, pages 346-347], we can show that z ∈ VI(K, D). Therefore, .

Noting that , we have by (2.3),

(3.42)

for all y ∈ C_n. Since F ⊂ C_n and by the continuity of inner product, we obtain, from the above inequality,

(3.43)

for all y ∈ F. By (2.3) again, we conclude that z = P_Fx₀. This completes the proof.

Corollary 3.2. Let K be a nonempty closed and convex subset of a real Hilbert space H. Let T : K → K be a nonexpansive mapping such that F(T) ≠ ∅. Let be generated by

(3.44)

Assume that

such that lim _n→∞α_n = 0. Then,

converges strongly to P_F(T)x₀.

Remark 3.3. Corollary 3.2 can be viewed as an improvement of Theorem 3.1 of Martinez-Yanes and Xu [39] because we relax the iterative step Q_n in the algorithm of Theorem 3.1 of [39].

4. Applications

Let C be a nonempty closed and convex cone in H and D an operator of C into H. We define the polar of C in H to be the set

(4.1)

Then, the element u ∈ C is called a solution of the complementarity problem if

(4.2)

The set of solutions of the complementarity problem is denoted by C(C, D). We will assume that D satisfies the following conditions:

(E1)
D is γ-inverse strongly monotone,
(E2)
C(C, D) ≠ ∅.

Also, we replace conditions (B1) and (B2) with

(D1)
for each x ∈ H and r > 0, there exist a bounded subset D_x⊆C and y_x ∈ C such that, for any z ∈ C∖D_x,
(4.3)
(D2)
C is a bounded set.

Theorem 4.1. Let C be a nonempty closed and convex subset of a real Hilbert space H. For each m = 1,2, let F_m be a bifunction from C × C satisfying (A1)–(A4), φ_m : C → ℝ ∪ {+∞} a proper lower semicontinuous and convex function with assumption (B1) or (B2), A an α-inverse-strongly monotone mapping of C into H, and B a β-inverse-strongly monotone mapping of C into H, and, for each i = 1,2, …, let T_i : C → C be a nonexpansive mapping such that . Let D be a γ-inverse-strongly monotone mapping of K into H. Suppose that . Let , , , (i = 1,2, …), and be generated by x₀ ∈ C, C_1,i = C, , ,

(4.4)

Assume that

(i = 1,2, …),

, and

satisfy

(i)
0 < a ≤ r_n ≤ b < 2α,
(ii)
0 < c ≤ λ_n ≤ f < 2β,
(iii)
lim _n→∞α_n,i = 0,
(iv)
0 < h ≤ s_n ≤ j < 2γ.

Then,

converges strongly to P_Fx₀.

Proof. Using Lemma 7.1.1 of [52], we have that VI(C, D) = C(C, D). Hence, by Theorem 3.1 we obtain the desired conclusion.

Next we study the problem of finding a minimizer of a continuously Fréchet differentiable convex functional in a Hilbert space.

Theorem 4.2. For each m = 1,2, let F_m be a bifunction from H × H satisfying (A1)–(A4), φ_m : H → ℝ ∪ {+∞} a proper lower semicontinuous and convex function with assumption (B1) or (B2), A an α-inverse-strongly monotone mapping of H into itself, and B a β-inverse-strongly monotone mapping of H into itself, and, for each i = 1,2, …, let T_i : H → H be a nonexpansive mapping such that . Suppose that f is a functional on H which satisfies the following conditions:

(1)
f is a continuously Fréchet differentiable convex functional on H and ∇f is (1/γ)-Lipschitz continuous,
(2)
(∇f) ⁻¹0 = {z ∈ H : f(z) = min _y∈Hf(y)} ≠ ∅.

Suppose that

. Let

(i = 1,2, …), and

be generated by x₀ ∈ K, C_1,i = K,

(4.5)

Assume that

(i = 1,2, …),

, and

satisfy

(i)
0 < a ≤ r_n ≤ b < 2α,
(ii)
0 < c ≤ λ_n ≤ f < 2β,
(iii)
lim _n→∞α_n,i = 0,
(iv)
0 < h ≤ s_n ≤ j < 2γ.

Then,

converges strongly to P_Fx₀.

Proof. We know from condition (i) and Lemma 2.2 that ∇f is an γ-inverse-strongly monotone operator from H into H. Using Theorem 3.1, we have the desired conclusion.

We now study a kind of multiobjective optimization problem with nonempty set of solutions:

(4.6)

where K is a nonempty closed convex subset of a real Hilbert space H, and h_i : K → ℝ, i = 1,2, is a convex and a lower semicontinuous functional. Let us denote the set of solutions to (4.6) by Ω and assume that Ω ≠ ∅.

We will denote the set of solutions of the following two optimization problems by Ω₁ and Ω₂, respectively:

(4.7)

Clearly, if we find a solution x ∈ Ω₁∩Ω₂, then one must have x ∈ Ω.

Now, for each i = 1,2, let F_i : K × K → ℝ be defined by F_i(x, y)∶ = h_i(y) − h_i(x). Let us now find the following equilibrium problem: find x ∈ K such that

(4.8)

for all y ∈ K. It is obvious that F_i satisfies conditions (A1)–(A4) and EP (F_i) = Ω_i, i = 1,2, where EP (F_i) is the set of solutions to (4.8). By Theorem 3.1, we have the following theorem.

Theorem 4.3. Let K be a nonempty closed and convex subset of a real Hilbert space H. For each i = 1,2, let h_i be a lower semicontinuous and convex function such that Ω₁∩Ω₂ ≠ ∅. Let , , , and be generated by x₀ ∈ K, C₁ = K, ,

(4.9)

Assume that

, and

satisfy

(i)
liminf _n→∞r_n > 0,
(ii)
liminf _n→∞λ_n > 0,
(iii)
lim _n→∞α_n = 0.

Then,

converges strongly to

Remark 4.4. Our results in this paper also hold for infinite family of uniformly continuous quasi-nonexpansive mappings in a real Hilbert space.

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Strong Convergence Theorems for Family of Nonexpansive Mappings and System of Generalized Mixed Equilibrium Problems and Variational Inequality Problems

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Strong Convergence Theorems for Family of Nonexpansive Mappings and System of Generalized Mixed Equilibrium Problems and Variational Inequality Problems

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1. Introduction

2. Preliminaries

3. Main Results

4. Applications

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