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.
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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 limn→∞ 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 limn→∞ 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 limn→∞ 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
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:
where Γ is an arbitrary index set. If Γ is a singleton, then problem (1.3) becomes the following equilibrium problem:
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 supt→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 PF(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 limn→∞γ n,i = 0 and {c n,i } ⊂ [0, ∞) such that limn→∞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 PF(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 supm→∞lim supn→∞||x n - Tmx 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 limn→∞γ n,i = 0 and {c n,i } ⊂ [0, ∞) such that limn→∞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 infn→∞r k,n > 0 for all k ∈ {1, 2, ... M}. Let {x n } and {u n } be sequences generated by the following algorithm:
where , as n → ∞, where ρ n = sup{||x n - v|| : v ∈ Ω} < ∞. Then {x n } converges strongly to PΩx1.
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
It follows from the definition of S i and Lemma 2.1(ii), we get
By virtue of the convexity of ||·||2, one has
Substituting (3.2) and (3.3) into (3.4), we obtain
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 Qn+1implies that Ω ⊂ Qn +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 = PΩx1, then
Since Ω is a nonempty closed convex subset of H, there exists a unique q ∈ Ω such that q = PΩx1.
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 xn+1∈ C n ⋂ Q n ⊂ Q n implies that
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 limn→∞||x n - x1|| exists, this implies
It is easy to get
Since xn+1∈ C n , we have
So, we get limn→∞||y n - xn+1|| = 0. It follows that
Next we will show that
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},
Therefore, by the convexity of ||·||2, (3.5) and the nonexpansivity of , we get
It follows that
From (3.10) and (3.13), we obtain (3.11). Then, we have
Combining (3.8) and (3.14), we have
It follows that
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
From (3.10), (3.14), the conditions 0 < a ≤ α n ≤ 1 and 0 < δ ≤ β n ≤ 1 - λ, we obtain
Next, we prove that
It is obvious that the relations hold: h(n) = h(n - N) + 1, i(n) = i(n - N).
Therefore,
Applying Lemma 2.5 and (3.16), we get (3.19). Using the uniformly continuity of S i , we obtain
this together with (3.17) yields
We also have
for any i = 1, 2, ... N, which gives that
Moreover, for each i ∈ {1, 2, ... N}, we obtain that
Step 5. The following implication holds:
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 = PΩx1.
From (3.6), (3.24) and Lemma 2.3, we conclude that x n → q, where q = PΩx1. □
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 {γn,i} ⊂ [0, ∞) such that limn→∞γn,i= 0 and {cn,i} ⊂ [0, ∞) such that limn→∞c n , i = 0. Let λ = max{λ i : 1 ≤ i ≤ N}, γ n = max{γn,i: 1 ≤ i ≤ N} and c n = max{cn,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 infn→∞r n > 0 for all k ∈ {1, 2, ... M}.
Let {x n } and {u n } be sequences generated by the following algorithm:
where, as n → ∞, where ρ n = sup{||x n - v|| : v ∈ Ω} < ∞. Then {x n } converges strongly to PΩx1.
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 {γn,i} ⊂ [0, ∞) such that limn→∞γn,i= 0 and {c n,i } ⊂ [0, ∞) such that limn→∞c n,i = 0. Let λ= max{λ i : 1 ≤ i ≤ N}, γ n = max{γn,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:
where, as n → ∞, where ρ n = sup{||x n - v|| : v ∈ Ω} < ∞. Then {x n } converges strongly to PΩx1.
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 xn+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.
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Acknowledgements
The authors would like to thank the reviewers for their good suggestions. This research is supported by Fundamental Research Funds for the Central Universities (ZXH2011C002).
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PD carried out the proof of convergence of the theorems and realization of numerical examples. JZ carried out the check of the manuscript. All authors read and approved the final manuscript.
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Duan, P., Zhao, J. Strong convergence theorems for system of equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense. Fixed Point Theory Appl 2011, 13 (2011). https://doi.org/10.1186/1687-1812-2011-13
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DOI: https://doi.org/10.1186/1687-1812-2011-13