Strong convergence theorems for system of equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense

Abstract

Let be N uniformly continuous asymptotically λ _i -strict pseudocontractions in the intermediate sense defined on a nonempty closed convex subset C of a real Hilbert space H. Consider the problem of finding a common element of the fixed point set of these mappings and the solution set of a system of equilibrium problems by using hybrid method. In this paper, we propose new iterative schemes for solving this problem and prove these schemes converge strongly.

MSC: 47H05; 47H09; 47H10.

1. Introduction

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H.

A nonlinear mapping S : C → C is a self mapping of C. We denote the set of fixed points of S by F(S) (i.e., F(S) = {x ∈ C : Sx = x}). Recall the following concepts.

• (1) S is uniformly Lipschitzian if there exists a constant L > 0 such that

  

• (2) S is nonexpansive if

  

• (3) S is asymptotically nonexpansive if there exists a sequence k _n of positive numbers satisfying the property lim_n→∞ k _n = 1 and

  

• (4) S is asymptotically nonexpansive in the intermediate sense [1] provided S is continuous and the following inequality holds:

  

• (5) S is asymptotically λ-strict pseudocontractive mapping [2] with sequence {γ _n } if there exists a constant λ ∈ [0, 1) and a sequence {γ _n } in [0, ∞) with lim_n→∞ γ _n = 0 such that

  

  for all x, y ∈ C and n ∈ ℕ.

• (6) S is asymptotically λ-strict pseudocontractive mapping in the intermediate sense [3, 4] with sequence {γ _n } if there exists a constant λ ∈ [0, 1) and a sequence {γ _n } in [0, ∞) with lim_n→∞ γ _n = 0 such that

  

  (1.1)

for all x, y ∈ C and n ∈ ℕ.

Throughout this paper, we assume that

Then, c _n ≥ 0 for all n ∈ N, c _n → 0 as n → ∞ and (1.1) reduces to the relation

(1.2)

for all x, y ∈ C and n ∈ ℕ.

When c _n = 0 for all n ∈ N in (1.2), then S is an asymptotically λ-strict pseudocontractive mapping with sequence {γ _n }. We note that S is not necessarily uniformly L-Lipschitzian (see [4]), more examples can also be seen in [3].

Let {F _k } be a countable family of bifunctions from C × C to ℝ, where ℝ is the set of real numbers. Combettes and Hirstoaga [5] considered the following system of equilibrium problems:

(1.3)

where Γ is an arbitrary index set. If Γ is a singleton, then problem (1.3) becomes the following equilibrium problem:

(1.4)

The solution set of (1.4) is denoted by EP(F).

The problem (1.3) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games and others; see, for instance, [6, 7] and the references therein. Some methods have been proposed to solve the equilibrium problem (1.3), related work can also be found in [8–11].

For solving the equilibrium problem, let us assume that the bifunction F 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 x, y, z ∈ C, lim sup_t→0F(tz + (1 - t)x, y) ≤ F(x, y);

(A4) F(x,·) is convex and lower semicontionuous for each x ∈ C.

Recall Mann's iteration algorithm was introduced by Mann [12]. Since then, the construction of fixed points for nonexpansive mappings and asymptotically strict pseudocontractions via Mann' iteration algorithm has been extensively investigated by many authors (see, e.g., [2, 6]).

Mann's iteration algorithm generates a sequence {x _n } by the following manner:

where α _n is a real sequence in (0, 1) which satisfies certain control conditions.

On the other hand, Qin et al. [13] introduced the following algorithm for a finite family of asymptotically λ _i -strict pseudocontractions. Let x₀ ∈ C and be a sequence in (0, 1). The sequence {x _n } by the following way:

It is called the explicit iterative sequence of a finite family of asymptotically λ _i -strict pseudocontractions {S₁, S₂,..., S _N }. Since, for each n ≥ 1, it can be written as n = (h - 1)N + i, where i = i(n) ∈ {1, 2,..., N}, h = h(n) ≥ 1 is a positive integer and h(n) → ∞, as n → ∞. We can rewrite the above table in the following compact form:

Recently, Sahu et al. [4] introduced new iterative schemes for asymptotically strict pseudocontractive mappings in the intermediate sense. To be more precise, they proved the following theorem.

Theorem 1.1. Let C be a nonempty closed convex subset of a real Hilbert space H and T: C → C a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence γ _n such that F(T) is nonempty and bounded. Let α _n be a sequence in [0, 1] such that 0 < δ ≤ α _n ≤ 1 - κ for all n ∈ N. Let {x _n } ⊂ C be sequences generated by the following (CQ) algorithm:

where θ _n = c _n + γ _n Δ _n and Δ _n = sup {||x _n - z||: z ∈ F(T)} < ∞. Then, {x _n } converges strongly to P_F(T)(u).

Very recently, Hu and Cai [3] further considered the asymptotically strict pseudocontractive mappings in the intermediate sense concerning equilibrium problem. They obtained the following result in a real Hilbert space.

Theorem 1.2. Let C be a nonempty closed convex subset of a real Hilbert space H and N ≥ 1 be an integer, ϕ : C → C be a bifunction satisfying (A1)-(A4) and A : C → H be an α-inverse-strongly monotone mapping. Let for each 1 ≤ i ≤ N, T _i : C → C be a uniformly continuous k _i -strictly asymptotically pseudocontractive mapping in the intermediate sense for some 0 ≤ k _i < 1 with sequences {γ _n,i } ⊂ [0, ∞) such that lim_n→∞γ _n,i = 0 and {c _n,i } ⊂ [0, ∞) such that lim_n→∞c _n,i = 0. Let k = max{k _i : 1 ≤ i ≤ N}, γ _n = max{γ _n,i : 1 ≤ i ≤ N} and c _n = max{c _n,i : 1 ≤ i ≤ N}. Assume thatis nonempty and bounded. Let {α _n } and {β _n } be sequences in [0, 1] such that 0 < a ≤ α _n ≤ 1, 0 < δ ≤ β _n ≤ 1 - k for all n ∈ N and 0 < b ≤ r _n ≤ c < 2α. Let {x _n } and {u _n } be sequences generated by the following algorithm:

where, as n → ∞, where ρ _n = sup{||x _n - v||: v ∈ F} < ∞. Then, {x _n } converges strongly to P_F(T)x₀.

Motivated by Hu and Cai [3], Sahu et al. [4], and Duan [8], the main purpose of this paper is to introduce a new iterative process for finding a common element of the fixed point set of a finite family of asymptotically λ _i -strict pseudocontractions and the solution set of the problem (1.3). Using the hybrid method, we obtain strong convergence theorems that extend and improve the corresponding results [3, 4, 13, 14].

We will adopt the following notations:

• 1. ⇀ for the weak convergence and → for the strong convergence.

• 2. denotes the weak ω-limit set of {x _n }.

2. Preliminaries

We need some facts and tools in a real Hilbert space H which are listed below.

Lemma 2.1. Let H be a real Hilbert space. Then, the following identities hold.

(i) ||x - y||² = ||x||² - ||y||² - 2〈x - y, y〉, ∀x, y ∈ H.

(ii) ||tx +(1 - t)y||² = t||x||²+(1 - t)||y||² - t(1 - t)||x - y||², ∀t ∈ [0, 1], ∀x, y ∈ H.

Lemma 2.2. ([10]) Let H be a real Hilbert space. Given a nonempty closed convex subset C ⊂ H and points x, y, z ∈ H and given also a real number a ∈ ℝ, the set

is convex (and closed).

Lemma 2.3. ([15]) Let C be a nonempty, closed and convex subset of H. Let {x _n } be a sequence in H and u ∈ H. Let q = P _C u. Suppose that {x _n } is such that ω _w (x _n ) ⊂ C and satisfies the following condition

Then, x _n → q.

Lemma 2.4. ([4]) Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → C a continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense. Then I - T is demiclosed at zero in the sense that if {x _n } is a sequence in C such that x _n ⇀ x ∈ C and lim sup_m→∞lim sup_n→∞||x _n - T^mx _n || = 0, then (I - T)x = 0.

Lemma 2.5. ([4]) Let C be a nonempty subset of a Hilbert space H and T : C → C an asymptotically κ - strict pseudocontractive mapping in the intermediate sense with sequence {γ _n }. Then

for all x, y ∈ C and n ∈ N.

Lemma 2.6. ([6]) Let C be a nonempty closed convex subset of H, let F be bifunction from C × C to ℝ satisfying (A1)-(A4) and let r > 0 and x ∈ H. Then there exists z ∈ C such that

Lemma 2.7. ([5]) For r > 0, x ∈ H, define a mapping T _r : H → C as follows:

for all x ∈ H. Then, the following statements hold:

• (i) T _r is single-valued;

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

  

• (iii) F(T _r ) = EP(F);

• (iv) EP(F) is closed and convex.

3. Main result

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H and N ≥ 1 be an integer, let F _k , k ∈ {1, 2, ... M}, be a bifunction from C × C to ℝ which satisfies conditions (A1)-(A4). Let, for each 1 ≤ i ≤ N, S _i : C → C be a uniformly continuous asymptotically λ _i -strict pseudocontractive mapping in the intermediate sense for some 0 ≤ λ _i < 1 with sequences {γ _n,i } ⊂ [0, ∞) such that lim_n→∞γ _n,i = 0 and {c _n,i } ⊂ [0, ∞) such that lim_n→∞c _n,i = 0. Let λ = max{λ _i : 1 ≤ i ≤ N}, γ _n = max{γ _n,i : 1 ≤ i ≤ N} and c _n = max{c _n,i : 1 ≤ i ≤ N}. Assume thatis nonempty and bounded. Let {α _n } and {β _n } be sequences in [0, 1] such that 0 < a ≤ α _n ≤ 1, 0 < δ ≤ β _n ≤ 1 - λ for all n ∈ ℕ and {r _k,n } ⊂ (0, ∞) satisfies lim inf_n→∞r _k,n > 0 for all k ∈ {1, 2, ... M}. Let {x _n } and {u _n } be sequences generated by the following algorithm:

(3.1)

where , as n → ∞, where ρ _n = sup{||x _n - v|| : v ∈ Ω} < ∞. Then {x _n } converges strongly to P_Ωx₁.

Proof. Denote for every k ∈ {1, 2,..., M} and for all n ∈ ℕ. Therefore . The proof is divided into six steps.

Step 1. The sequence {x _n } is well defined.

It is obvious that C _n is closed and Q _n is closed and convex for every n ∈ ℕ. From Lemma 2.2, we also get that C _n is convex.

Take p ∈ Ω, since for each k ∈ {1, 2,..., M}, is nonexpansive, and , we have

(3.2)

It follows from the definition of S _i and Lemma 2.1(ii), we get

(3.3)

By virtue of the convexity of ||·||², one has

(3.4)

Substituting (3.2) and (3.3) into (3.4), we obtain

(3.5)

It follows that p ∈ C _n for all n ∈ ℕ. Thus, Ω ⊂ C _n .

Next, we prove that Ω ⊂ Q _n for all n ∈ ℕ by induction. For n = 1, we have Ω ⊂ C = Q₁. Assume that Ω ⊂ Q _n for some n ≥ 1. Since , we obtain

As Ω ⊂ C _n ⋂ Q _n by induction assumption, the inequality holds, in particular, for all z ∈ Ω. This together with the definition of Q_n+1implies that Ω ⊂ Q_{n +1}.

Hence Ω ⊂ Q _n holds for all n ≥ 1. Thus Ω ⊂ C _n ⋂ Q _n and therefore the sequence {x _n } is well defined.

Step 2. Set q = P_Ωx₁, then

(3.6)

Since Ω is a nonempty closed convex subset of H, there exists a unique q ∈ Ω such that q = P_Ωx₁.

From , we have

Since q ∈ Ω ⊂ C _n ⋂ Q _n , we get (3.6).

Therefore, {x _n } is bounded. So are {u _n } and {y _n }.

Step 3. The following limits hold:

From the definition of Q _n , we have , which together with the fact that x_n+1∈ C _n ⋂ Q _n ⊂ Q _n implies that

(3.7)

This shows that the sequence {||x _n - x₁||} is nondecreasing. Since {x _n } is bounded, the limit of {||x _n - x₁||} exists.

It follows from Lemma 2.1(i) and (3.7) that

Noting that lim_n→∞||x _n - x₁|| exists, this implies

(3.8)

It is easy to get

(3.9)

Since x_n+1∈ C _n , we have

So, we get lim_n→∞||y _n - x_n+1|| = 0. It follows that

(3.10)

Next we will show that

(3.11)

Indeed, for p ∈ Ω, it follows from the firmly nonexpansivity of that for each k ∈ {1, 2,..., M}, we have

Thus we get

which implies that for each k ∈ {1, 2,..., M},

(3.12)

Therefore, by the convexity of ||·||², (3.5) and the nonexpansivity of , we get

It follows that

(3.13)

From (3.10) and (3.13), we obtain (3.11). Then, we have

(3.14)

Combining (3.8) and (3.14), we have

(3.15)

It follows that

(3.16)

Step 4. Show that ||u _n - S _i u _n || → 0, ||x _n - S _i x _n || → 0, as n → ∞; ∀i ∈ {1, 2,..., N}.

Since, for any positive integer n ≥ N, it can be written as n = (h(n) - 1) N + i(n), where i(n) ∈ {1, 2,..., N}. Observe that

(3.17)

From (3.10), (3.14), the conditions 0 < a ≤ α _n ≤ 1 and 0 < δ ≤ β _n ≤ 1 - λ, we obtain

(3.18)

Next, we prove that

(3.19)

It is obvious that the relations hold: h(n) = h(n - N) + 1, i(n) = i(n - N).

Therefore,

(3.20)

Applying Lemma 2.5 and (3.16), we get (3.19). Using the uniformly continuity of S _i , we obtain

(3.21)

this together with (3.17) yields

We also have

for any i = 1, 2, ... N, which gives that

(3.22)

Moreover, for each i ∈ {1, 2, ... N}, we obtain that

(3.23)

Step 5. The following implication holds:

(3.24)

We first show that . To this end, we take ω ∈ ω _w (x _n ) and assume that as j → ∞ for some subsequence of x _n .

Note that S _i is uniformly continuous and (3.23), we see that , for all m ∈ ℕ. So by Lemma 2.4, it follows that and hence .

Next we will show that . Indeed, by Lemma 2.6, we have that for each k = 1, 2, ..., M,

From (A2), we get

Hence,

From (3.11), we obtain that as j → ∞ for each k = 1, 2, ..., M (especially, ). Together with (3.11) and (A4) we have, for each k = 1, 2, ..., M, that

For any, 0 < t ≤ 1 and y ∈ C, let y _t = ty + (1 - t)ω. Since y ∈ C and ω ∈ C, we obtain that y _t ∈ C and hence F _k (y _t , ω) ≤ 0. So, we have

Dividing by t, we get, for each k = 1, 2, ..., M, that

Letting t → 0 and from (A3), we get

for all y ∈ C and ω ∈ EP(F _k ) for each k = 1, 2, ..., M, i.e., .

Hence (3.24) holds.

Step 6. Show that x _n → q = P_Ωx₁.

From (3.6), (3.24) and Lemma 2.3, we conclude that x _n → q, where q = P_Ωx₁. □

Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H and N ≥ 1 be an integer, let F be a bifunction from C × C to ℝ which satisfies conditions (A1)-(A4). Let, for each 1 ≤ i ≤ N, S _i : C → C be a uniformly continuous λ _i -strict asymptotically pseudocontractive mapping in the intermediate sense for some 0 ≤ λ _i < 1 with sequences {γ_n,i} ⊂ [0, ∞) such that lim_n→∞γ_n,i= 0 and {c_n,i} ⊂ [0, ∞) such that lim_n→∞c _n , _i = 0. Let λ = max{λ _i : 1 ≤ i ≤ N}, γ _n = max{γ_n,i: 1 ≤ i ≤ N} and c _n = max{c_n,i: 1 ≤ i ≤ N}. Assume thatis nonempty and bounded. Let {α _n } and {β _n } be sequences in [0, 1] such that 0 < a ≤ α _n ≤ 1,0 < δ ≤ β _n ≤ 1 - λ for all n ∈ N and {r _n } ⊂ (0,∞) satisfies lim inf_n→∞r _n > 0 for all k ∈ {1, 2, ... M}.

Let {x _n } and {u _n } be sequences generated by the following algorithm:

(3.25)

where, as n → ∞, where ρ _n = sup{||x _n - v|| : v ∈ Ω} < ∞. Then {x _n } converges strongly to P_Ωx₁.

Proof. Putting M = 1, we can draw the desired conclusion from Theorem 3.1.

□

Remark 3.3. Corollary 3.2 extends the theorem of Tada and Takahashi [14] from a nonexpansive mapping to a finite family of asymptotically λ _i -strict pseudocontractive mappings in the intermediate sense.

Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H and N ≥ 1 be an integer, let, for each 1 ≤ i ≤ N, S _i : C → C be a uniformly continuous λ _i -strict asymptotically pseudocontractive mapping in the intermediate sense for some 0 ≤ λ _i < 1 with sequences {γ_n,i} ⊂ [0, ∞) such that lim_n→∞γ_n,i= 0 and {c _n,i } ⊂ [0, ∞) such that lim_n→∞c _n,i = 0. Let λ= max{λ _i : 1 ≤ i ≤ N}, γ _n = max{γ_n,i: 1 ≤ i ≤ N} and c _n = max{c _n,i : 1 ≤ i ≤ N}. Assume thatis nonempty and bounded. Let {α _n } and {β _n } be sequences in [0, 1] such that 0 < a ≤ α _n ≤ 1, 0 <δ ≤ β _n ≤ 1 - λ for all n ∈ ℕ. Let {x _n } and {u _n } be sequences generated by the following algorithm:

(3.26)

where, as n → ∞, where ρ _n = sup{||x _n - v|| : v ∈ Ω} < ∞. Then {x _n } converges strongly to P_Ωx₁.

Proof. If F _k (x, y) = 0, α _n = 1 in Theorem 3.1, we can draw the conclusion easily. □

Remark 3.5. Corollary 3.4 extends the Theorem 4.1 of [4] and Theorem 2.2 of [13], respectively.

4. Numerical result

In this section, in order to demonstrate the effectiveness, realization and convergence of the algorithm in Theorem 3.1, we consider the following simple example ever appeared in the reference [4]:

Example 4.1. Let x = R and C = [0, 1] For each x ∈ C, we define

where 0 < k < 1.

Set C₁ : = [0, 1/2] and C₂ : = (1/2, 1]. Hence,

and

For x ∈ C₁ and y ∈ C₂, we have

Thus

for all x, y ∈ C, n ∈ ℕ and some K > 0. Therefore, T is an asymptotically k-strict pseudocontractive mapping in the intermediate sense.

In the algorithm (3.1), set . We apply it to find the fixed point of T of Example 4.1.

Under the above assumptions, (3.1) is simplified as follows:

In fact, in one dimensional case, the C _n ⋂ Q _n is an closed interval. If we set [a _n , b _n ] := C _n ⋂ Q _n , then the projection point x_n+1of x₁ ∈ C onto C _n ⋂ Q _n can be expressed as:

Since the conditions of Theorem 3.1 are satisfied in Example 4.1, the conclusion holds, i.e., x _n → 0 ∈ F (T).

Now we turn to realizing (3.1) for approximating a fixed point of T. Take the initial guess x₁ = 1/2, 1/5 and 5/8, respectively. All the numerical results are given in Tables 1, 2 and 3. The corresponding graph appears in Figure 1a,b,c.

Table 1 x₁ = 0.5

Table 2 x₁ = 0.2

Table 3

Figure 1

The iteration comparison chart of different initial values. (a) x₁ = 0.5; (b) x₁ = 0.2; (c) .