Strong convergence theorems for equilibrium problems and fixed point problems: A new iterative method, some comments and applications

Abstract

In this paper, we introduce a new approach method to find a common element in the intersection of the set of the solutions of a finite family of equilibrium problems and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Under appropriate conditions, some strong convergence theorems are established. The results obtained in this paper are new, and a few examples illustrating these results are given. Finally, we point out that some 'so-called' mixed equilibrium problems and generalized equilibrium problems in the literature are still usual equilibrium problems.

2010 Mathematics Subject Classification: 47H09; 47H10, 47J25.

1 Introduction and preliminaries

Throughout this paper, we assume that H is a real Hilbert space with zero vector θ, whose inner product and norm are denoted by 〈·, ·〉 and || · ||, respectively. The symbols ℕ and ℝ are used to denote the sets of positive integers and real numbers, respectively. Let K be a nonempty closed convex subset of H and T : K → H be a mapping. In this paper, the set of fixed points of T is denoted by F(T). We use symbols → and ⇀ to denote strong and weak convergence, respectively.

For each point x ∈ H, there exists a unique nearest point in K, denoted by P _K x, such that

The mapping P _K is called the metric projection from H onto K. It is well known that P _K satisfies

for every x, y ∈ H. Moreover, P _K x is characterized by the properties: for x ∈ H, and z ∈ K,

Let f be a bi-function from K × K into ℝ. The classical equilibrium problem is to find x ∈ K such that

(1.1)

Let EP(f) denote the set of all solutions of the problem (1.1). Since several problems in physics, optimization, and economics reduce to find a solution of (1.1) (see, e.g., [1, 2]), some authors had proposed some methods to find the solution of equilibrium problem (1.1); for instance, see [1–4]. We know that a mapping S is said to be nonexpansive mapping if for all x, y ∈ K, ||Sx - Sy|| ≤ ||x - y||. Recently, some authors used iterative method including composite iterative, CQ iterative, viscosity iterative etc. to find a common element in the intersection of EP(f) and F(S); see, e.g., [5–11].

Let I be an index set. For each i ∈ I, let f _i be a bi-function from K × K into ℝ. The system of equilibrium problem is to find x ∈ K such that

(1.2)

We know that is the set of all solutions of the system of equilibrium problem (1.2).

For each i ∈ I, if f _i (x, y) = 〈A _i x, y - x〉, where A _i : K → K is a nonlinear operator, then the problem (1.2) becomes the following system of variational inequality problem:

(1.3)

It is obvious that the problem (1.3) is a special case of the problem (1.2).

The following Lemmas are crucial to our main results.

Lemma 1.1 (Demicloseness principle[12]) Let H be a real Hilbert space and K a closed convex subset of H. S : K → H is a nonexpansive mapping. Then the mapping I - S is demiclosed on K, where I is the identity mapping, i.e., x _n ⇀ x in K and (I - S)x _n → y implies that × ∈ K and (I - S)x = y.

Lemma 1.2[13] Let {x _n }and {y _n } be bounded sequences in a Banach space E and let {β _n } be a sequence in [0,1] with 0 < lim inf_n→∞β _n ≤ lim sup_n→∞β _n < 1. Suppose x_n+1= β _n y _n + (1 - β _n )x _n for all integers n ≥ 0 and lim sup_n→∞(||y_n+1- y _n || - ||x_n+1- x _n ||) ≤ 0, then lim_n→∞||y _n - x _n || = 0.

Lemma 1.3[5] Let H be a real Hilbert space. Then the following hold.

• (a) ||x + y||² ≤ ||y||² + 2〈x, x + y〉 for all x, y ∈ H;

• (b) ||αx + (1 - α)y||² = α||x||² + (1 - α) ||y||² - α(1 - α) ||x - y||² for all x, y ∈ H and α ∈ ℝ;

• (c) ||x - y||² = ||x||² + ||y||² - 2 〈x, y〉 for all x, y ∈ H.

Lemma 1.4. [14] Let {a _n } be a sequence of nonnegative real numbers satisfying the following relation:

If

• (i) λ _n ∈ [0,1], or, equivalently, ;

• (ii) or ,

then .

Lemma 1.5[1] Let K be a nonempty closed convex subset of H and F be a bi-function of K × K into ℝ satisfying the following conditions.

(A1) F(x, x) = 0 for all × ∈ 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,

(A4) for each × ∈ K, y → F(x, y) is convex and lower semi-continuous.

Let r > 0 and × ∈ H. Then, there exists z ∈ K such that

Lemma 1.6[3] Let K be a nonempty closed convex subset of H and let F be a bi-function of K × K into R satisfying (A1) - (A4). For r > 0 and × ∈ H, define a mapping T _r : H → K as follows:

for all × ∈ H. Then the following hold:

• (i) T _r is single-valued;

• (ii) T _r is firmly nonexpansive, that is, for any x, y ∈ H,

  

• (iii) F(T _r ) = EP (F);

• (iv) EP(F) is closed and convex.

2 Main results and their applications

Let I = {1, 2,..., k} be a finite index set, where k ∈ ℕ. For each i ∈ I, let f _i be a bi-functions from K × K into ℝ satisfying the conditions (A1)-(A4). Denote by

For each (i, n) ∈ I × ℕ, applying Lemmas 1.5 and 1.6, is a firmly nonexpansive single-valued mapping such that is closed and convex. For each i ∈ I, let , n ∈ ℕ.

First, let us consider the following example.

Example A Let f _i : [-1, 0]×[-1,0] → ℝ be defined by f _i (x, y) = (1+x²ⁱ)(x - y), i = 1, 2, 3. It is easy to see that for any i ∈ {1, 2, 3}, f _i (x, y) satisfies the conditions (A1)-(A4) and . Let Sx = x³ and , ∀ x ∈ [-1, 0] Then g is a -contraction from K into itself and S : K → K is a nonexpansive mapping with . Let λ ∈ (0, 1), {r _n } ⊂ [1, + ∞) and {α _n } ⊂ (0,1) satisfy the conditions (i) lim_n→∞α _n = 0, and (ii) , or equivalently, ; e.g., let , {α _n } ⊂ (0, 1) and {r _n } ⊂ [1, + ∞) be given by

Define a sequence {x _n } by

(2.1)

Then the sequences {x _n } and , i = 1, 2, 3, defined by (2.1) all strongly converge to 0.

Proof

• (a) By Lemmas 1.5 and 1.6, (2.1) is well defined.

• (b) Let K = [-1, 0]. For each i ∈ {1, 2, 3}, define

  

We claim that for each v ∈ K and any i ∈ {1, 2, 3}, there exists a unique z = 0 ∈ K such that

or, equivalently,

Obviously, z = 0 is a solution of the problem . On the other hand, there does not exist z ∈ [-1, 0) such that z - y ≤ 0 and . So z = 0 is the unique solution of the problem .

• (c) We notice that (2.1) is equivalent with (2.2), where

  

  (2.2)

It is easy to see that {x _n } ⊂ [-1, 0], so, by (b), for all n ∈ ℕ. We need to prove x _n → 0 as n → ∞. Since z _n = 0 for all n ∈ ℕ, we have y _n = (1 -λ)x _n and

(2.3)

for all n ∈ ℕ. For any n ∈ ℕ, from (2.3), we have

(2.4)

Hence {|x _n |} is a strictly deceasing sequence and |x _n | ≥ 0 for all n ∈ ℕ. So exists.

On the other hand, for any n, m ∈ ℕ with n > m, using (2.4), we obtain

which implies . Therefore {x _n } strongly converges to 0. □

In this paper, motivated by the preceding Example A, we introduce a new iterative algorithm for the problem of finding a common element in the set of solutions to the system of equilibrium problem and the set of fixed points of a nonexpansive mapping. The following new strong convergence theorem is established in the framework of a real Hilbert space H.

Theorem 2.1 Let K be a nonempty closed convex subset of a real Hilbert space H and I = {1, 2,..., k} be a finite index set. For each i ∈ I, let f _i be a bi-function from K × K into ℝ satisfying (A1)-(A4). Let S : K → K be a nonexpansive mapping with . Let λ, ρ ∈ (0, 1) and g : K → K is a ρ-contraction. Let {x _n } be a sequence generated in the following manner:

If the above control coefficient sequences {α _n } ⊂ (0, 1) and {r _n } ⊂ (0, +∞) satisfy the following restrictions:

(D1) , and ;

(D2) and.

then the sequences {x _n } and, for all i ∈ I, converge strongly to an element c = P_Ωg(c) ∈ Ω. The following conclusion is immediately drawn from Theorem 2.1.

Corollary 2.1 Let K be a nonempty closed convex subset of a real Hilbert space H. Let f be a bi-function from K × K into ℝ satisfying (A1)-(A4) and S : K → K be a nonexpansive mapping with Ω = EP(f) ∩F(S) ≠ ∅. Let λ, ρ ∈ (0,1) and g : K → K is a ρ-contraction. Let {x _n } be a sequence generated in the following manner:

If the above control coefficient sequences {α _n } ⊂ (0, 1) and {r _n } ⊂ (0, +∞) satisfy all the restrictions in Theorem 2.1, then the sequences {x _n } and {u _n } converge strongly to an element c = P_Ωg(c) ∈ Ω, respectively.

If f _i (x, y) ≡ 0 for all (x, y) ∈ K × K in Theorem 2.1 and all i ∈ I, then, from the algorithm (D _H ), we obtain , ∀ i ∈ I. So we have the following result.

Corollary 2.2 Let K be a nonempty closed convex subset of a real Hilbert space H. Let S : K → K be a nonexpansive mapping with F(S) ≠ ∅. Let λ, ρ ∈ (0, 1) and g : K → K is a ρ-contraction. Let {x _n } be a sequence generated in the following manner:

If the above control coefficient sequences {α _n } ⊂ (0, 1) satisfy, and, then the sequences {x _n } converge strongly to an element c = P_Ωg(c) ∈ F (S).

As some interesting and important applications of Theorem 2.1 for optimization problems and fixed point problems, we have the following.

Application (I) of Theorem 2.1 We will give an iterative algorithm for the following optimization problem with a nonempty common solution set:

where h _i (x), i ∈ {1, 2,..., k}, are convex and lower semi-continuous functions defined on a closed convex subset K of a Hilbert space H (for example, h _i (x) = xⁱ , x ∈ K := [0, 1], i ∈ {1, 2,..., k}).

If we put f _i (x, y) = h _i (y) - h _i (x), i ∈ {1, 2,..., k}, then is the common solution set of the problem (OP), where denote the common solution set of the following equilibrium:

For i ∈ {1, 2,..., k}, it is obvious that the f _i (x, y) satisfies the conditions (A1)-(A4). Let S = I (identity mapping), then from (D _H ), we have the following algorithm

(2.5)

where x₁ ∈ K, λ ∈ (0, 1), g : K → K is a ρ-contraction. From Theorem 2.1, we know that {x _n } and , i ∈{1,2,..., k}, generated by (2.5), strongly converge to an element of if the coefficients {α _n } and {r _n } satisfy the conditions of Theorem 2.1.

Application (II) of Theorem 2.1 Let H, K, I, λ, ρ, g be the same as Theorem 2.1. Let A₁, A₂,..., A _k : K → K be k nonlinear mappings with . For any i ∈ I, put f _i (x, y) = 〈x - A _i x, y - x〉, ∀ x, y ∈ K. Since , we have . Let S = I (identity mapping) in the algorithm (D _H ). Then the sequences {x _n } and , defined by the algorithm (D _H ), converge strongly to a common fixed point of {A₁, A₂,..., A _k }, respectively.

The following result is important in this paper.

Lemma 2.1 Let H be a real Hilbert space. Then for any x₁, x₂,... x _k ∈ H and a₁, a₂,..., a _k ∈ [0,1] with, k ∈ ℕ, we have

(2.6)

Proof It is obvious that (2.6) is true if a _j = 1 for some j, so it suffices to show that (2.6) is true for a _j ≠ 1 for all j. The proof is by mathematic induction on k. Clearly, (2.6) is true for k = 1. Let x₁, x₂ ∈ H and a₁, a₂ ∈ [0,1] with a₁ + a₂ = 1. By Lemma 1.3, we obtain

which means that (2.6) hold for k = 2. Suppose that (2.6) is true for k = l ∈ ℕ. Let x₁, x₂,..., x _l , x_l+1∈ H and a₁, a₂,..., a _l , a_l+1∈ [0, 1) with . Let . Then applying the induction hypothesis we have

Hence, the equality (2.6) is also true for k = l + 1. This completes the induction. □

3 Proof of Theorem 2.1

We will proceed with the following steps.

Step 1: There exists a unique c ∈ Ω ⊂ H such that P_Ωg(c) = c.

Since P_Ωg is a ρ-contraction on H, Banach contraction principle ensures that there exists a unique c ∈ H such that c = P_Ωg(c) ∈ Ω.

Step 2: We prove that the sequences {x _n }, {y _n }, {z _n } and , ∀i ∈ I, are all bounded.

First, we notice that (D _H ) is equivalent with (Z _H ), where

For each i ∈ I, we have

(3.1)

For any n ∈ ℕ, from (Z _H ) we have

and

(3.2)

Since g is a ρ-contraction, it follows from (3.2) that

By induction, we obtain

which shows that {x _n } is bounded. Also, we know that {y _n }, {z _n } and , ∀i ∈ I, are all

bounded.

Step 3: We prove lim_n→∞||x_n+1- x _n || = 0.

For each i ∈ I, since , , from (Z _H ), we have

(3.3)

and

(3.4)

By (3.3) and (3.4) and (A2),

which implies

(3.5)

It follows from (3.5) that

(3.6)

Let . For any n ∈ ℕ, since , by (3.6), we have

(3.7)

Set

(3.8)

where β _n = 1 - (1 - λ)(1 - α _n ), n ∈ ℕ. Then for each n ∈ ℕ,

(3.9)

and

(3.10)

For any n ∈ ℕ, since

by (3.7), it follows that

From this and (D1), (D2), we get

(3.11)

By Lemma 1.2 and (3.11),

(3.12)

Owing to (3.9) and (3.12), we obtain

(3.13)

Step 4: We show .

By (3.6), (3.13) and (D2), we have

From (Z _H ), we get

(3.14)

Since ||x _n - y _n || ≤ ||x _n - x_n+1|| + ||x_n+1- y _n ||, by (3.13) and (3.14),

which implies that

By Lemma 1.6,

which yields that

(3.15)

From (3.15) and Lemma 2.1,

Since

where

We have

(3.16)

Letting n → ∞ in the inequality (3.16), we obtain

(3.17)

Furthermore, it is easy to prove that

For any i ∈ I, since

it implies

(3.18)

Step 5: Prove lim sup_n→∞〈g(c) - q, x _n - c〉 ≤ 0.

Take a subsequence of {x _n } such that

(3.19)

Since is bounded, there exists a subsequence of which is still denoted by such that as ℓ → ∞. Notice that for each i ∈ I, by (3.17), so we also have as ℓ → ∞, ∀ i ∈ I.

We want to show z ∈ Ω. First, we show that z ∈ F(S). In fact, since and as ℓ → ∞, by Lemma 1.1, we have (I - S)z = θ or, equivalently, z ∈ F(S).

For each i ∈ I, since , ∀ y ∈ K, it follows from (A2) that

and hence

Applying (3.17) and (A4),

(3.20)

Let y ∈ K be given. Put y _t = ty + (1 - t)z, t ∈ (0, 1). Then y _t ∈ K and f _i (y _t , z) ≤ 0 for all i ∈ I. By (A1) and (A4), we get

For any i ∈ I, by (A3), we have

(3.21)

Hence, from (3.21), . Therefore, we proved . On the other hand, by (3.19), we obtain

(3.22)

Step 6: Finally, we prove {x _n } and , for all i ∈ I, converge strongly to c = P_Ωg(c) ∈ Ω.

From (Z _H ) and (a) of Lemma 1.3, we have

(3.23)

For any n ∈ ℤ, let

and

From (3.23), we have

It is easy to verify that all conditions of Lemma 1.4 are satisfied. Hence, applying Lemma 1.4, we obtain lim_n→∞a _n = 0 which implies

or equivalence, {x _n } strongly converges to c. By (3.17), we can prove that for any i ∈ I, strongly converges to c. The proof of Theorem 2.1 is completed. □

4 Further remarks

Let K be a nonempty closed convex subset of H and f be a bi-function of K × K into ℝ.

Remark 4.1 Recently, some authors introduced the following mixed equilibrium problem (MEP, for short) (see [15–17] and references therein) and generalized equilibrium problem (GEP, for short) (see [18–20] and references therein):

• (a) Mixed equilibrium problem [15–17]:

  

where φ : C → ℝ is a real-valued function.

• (b) Generalized equilibrium problem [18–20]:

  

where A : C → H is a nonlinear operator.

In [15–17], the authors gave some iterative methods for finding the solution of MEP when the bi-function f(x, y) admits the conditions (A1)-(A4) and the real-valued function φ satisfies the following condition:

(A5) φ : C → ℝ is a proper lower semi-continuous and convex function.

However, in this case, we argue that the problem MEP is still the equilibrium problem (1.1). In fact, if we put f₁(x, y) = f(x, y), f₂(x, y) = φ(y) - φ(x) and F(x, y) = f₁(x, y) + f₂(x, y) for each (x, y) ∈ C × C, then f₁(x, y) satisfies the conditions (A1)-(A4), f₂(x, y) satisfies the condition (A5) and the function φ must satisfy the conditions (A1)-(A4). This shows that for each (x, y) ∈ C × C, F(x, y) satisfies the conditions (A1)-(A4). So, when we study the solution of MEP, we only need to study the solution of the equilibrium (1.1). This also shows that some "so-called" mixed equilibrium problem studied in [15–17] is still the equilibrium problem (1.1).

Remark 4. 2 Let us recall some well-known definitions. A mapping T : C → C is said to be

• (1) v-expansive if ||Tx - Ty|| ≥ v||x - y|| for all x, y ∈ C. In particular, if v = 1, then T is called expansive.

• (2) v-strongly monotone if there exists a constant v > 0 such that

  

Clearly, any v-strongly monotone mapping is v-expansive.

• (3) u-inverse strongly monotone if there exists a constant u > 0 such that

  

• (4) L-Lipschitz continuous if ||Tx - Ty|| ≤ L||x - y|| for all x, y ∈ C. In particular, if L = 1, then T is called nonexpansive.

It is easy to see that a u-inverse strongly monotone operator is -Lipschitz continuous.

For the problem GEP, if the nonlinear operator A : C → H is a u-inverse strongly monotone operator and the bi-function f(x, y) admits the conditions (A1)-(A4), we argue that the problem GEP is still the problem (1.1) and so it is indeed not a generalization. In fact, if A is a u-inverse strongly monotone operator from C into H, then A is a continuous operator. So, we obtain easily that the function (x, y) → <Ax, y - x〉, ∀x, y ∈ C, satisfies the conditions (A1)-(A4). Hence, if we put F(x, y) = f(x, y) + 〈Ax, y - x〉 ≥ 0, then the problem GEP studied in [18–20] is still the problem (1.1).

5 Conclusion

The problem MEP studied in [15–17] and the problem GEP studied in [18–20] are still the problem (1.1) studied in the literature [5–11, 21–24] and others.