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Iterative approximation of common element of solution sets of various nonlinear operator problems
Fixed Point Theory and Applications volume 2013, Article number: 295 (2013)
Abstract
In this paper, we prove strong convergence theorem for finding a common element of the set of fixed point of a finite family of nonexpansive mappings and a finite family of -strictly pseudocontractive mappings and the set of a finite family of the set of solution of equilibrium problems by using the new mapping generated by a finite family of nonexpansive mappings and a finite family of -strictly pseudocontractive mappings and a sequences of positive real numbers. Furthermore, by using our main result, we obtain two interesting theorems involving variational inequality problems and variational inclusion problems. In the last section, we give numerical examples to support our main results.
1 Introduction
Let H be a real Hilbert space and C be a nonempty closed convex subset of H. A self mapping is a contraction on C if there exists a constant such that , . Let be a mapping, a point is called a fixed point of T if and only if . In this paper, we use to denote the set of fixed point of T. Recall the following definitions.
Definition 1.1 A mapping is called nonexpansive if and only if for all ,
Definition 1.2 A mapping is called κ-strictly pseudocontractive [1] if and only if there exists a constant such that for all ,
For such case, T is also said to be a κ-strictly pseudo contraction.
Note that the class of κ-strict pseudo-contractions strictly includes the class of nonexpansive mappings, that is T is nonexpansive if and only if T is 0-strict pseudo-contractive.
Let be a bifunction. The equilibrium problem for F is to determine its equilibrium points, i.e., the set
Given , let for all . Then if and only if for all , that is, z is a solution of the variational inequality.
Equilibrium problems, which were introduced in [2] in 1994, have had a great impact and influence in the development of several branches of pure and applied sciences. Numerous problems in physics, minimization problems, Nash equilibria in noncooperative games, optimization and economics reduce to find a solution of (see, for example, [2–4]). Some methods have been proposed to solve the equilibrium problem (see, for example, [5–7]).
In 2007, Takahashi and Takahashi [8] proved the following theorem.
Theorem 1.1 Let C be a nonempty closed convex subset of H. Let F be a bifunction from to ℝ satisfying
-
(A1)
, ;
-
(A2)
F is monotone, i.e., , ;
-
(A3)
,
-
(A4)
, is convex and lower semicontinuous;
and let S be a nonexpansive mapping of C into H such that . Let f be a contraction of H into itself, and let and be sequences generated by and
for all , where and satisfy (C1)-(C3) as follows:
-
(C1)
;
-
(C2)
;
-
(C3)
either or ,
and and .
Then and converge strongly to , where .
In 2010, Kangtunyakarn and Suantai [9] proved the strong convergence theorem by using the S-mapping generated by a finite family of strictly pseudocontractive mappings and a finite family of real number as follows.
Theorem 1.2 Let H be a Hilbert space, let f be an α-contraction on H, and let A be a strongly positive linear bounded self-adjoint operator with coefficient . Assume that . Let be a finite family of -strictly pseudo contraction of H into itself for some and with . Let be the S-mappings generated by and , where , , and for all , , , for all . For a point and , let and be the sequences defined iteratively by
where , and are sequences in . Assume that the following conditions hold:
-
(i)
, and ;
-
(ii)
, for all and , and ;
-
(iii)
for all for some .
Then both and strongly converge to , which solves the following variational inequality
Question Can we prove a strong convergence theorem for finding a common solution of the set of fixed point of a finite family of nonexpansive mappings and a finite family of strictly pseudocontractive mappings and a finite family of the set of solution of equilibrium problems?
Let C be a nonempty closed convex subset of Hilbert space H. Let be a finite family of -strict pseudo-contractions of C into itself, and let be a finite family of nonexpansive mappings of C into itself. For each and , let , where , . We define the mapping as follows:
In Lemma 2.8, under suitable conditions of the real sequences , and for every , we show that and is a nonexpansive mapping.
In this paper, motivated by the ongoing research and Theorems 1.1 and 1.2, we prove strong convergence theorem for finding a common solution of the set of fixed point of a finite family of nonexpansive mappings and a finite family of strictly pseudocontractive mappings and a finite family of the set of solution of equilibrium problems by using the mapping defined by (1.4). Furthermore, in the last section, we prove two interesting theorems involving a finite family of the set of solutions of variational inequality problem and variational inclusion problem. In the last section, we give numerical examples to support our main results.
2 Preliminaries
In this section, we need the following lemmas to prove our main result. Let C be a closed convex subset of a real Hilbert space H, let be the metric projection of H onto C, i.e., for , satisfies the property
The following characterizes the projection .
Lemma 2.1 (See [10])
Given and . Then if and only if the following inequality holds
Lemma 2.2 (See [11])
Let be a sequence of nonnegative real numbers satisfying
where , satisfy the conditions
Then .
Lemma 2.3 (See [12])
Let be a sequence of nonnegative real numbers satisfying
where is a sequence in and is a sequence such that
Then .
Lemma 2.4 (See [13])
Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a self-mapping of C. If S is a κ-strict pseudo-contraction mapping, then S satisfies the Lipschitz condition
For solving the equilibrium problem for a bifunction , let us assume that F satisfies the following conditions:
-
(A1)
, ;
-
(A2)
F is monotone, i.e., , ;
-
(A3)
,
-
(A4)
, is convex and lower semicontinuous.
The following lemma appears implicitly in [2].
Lemma 2.5 (See [2])
Let C be a nonempty closed convex subset of H, and let F be a bifunction of into ℝ satisfying (A1)-(A4). Let and . Then there exists such that
for all .
Lemma 2.6 (See [14])
Assume that satisfies (A1)-(A4). For and , define a mapping as follows:
for all . Then the following hold:
-
(1)
is single-valued;
-
(2)
is firmly nonexpansive, i.e.,
-
(3)
;
-
(4)
is closed and convex.
Lemma 2.7 (See [15])
Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E, and be a nonexpansive mapping. Then is demi-closed at zero.
Definition 2.1 Let C be a nonempty convex subset of real Hilbert space. Let be a finite family of -strict pseudo-contractions of C into itself, and let be a finite family of nonexpansive mappings of C into itself. For each , let , where and . We define the mapping as follows:
This mapping is called the -mapping generated by , and .
Lemma 2.8 Let C be a nonempty closed convex subset of a real Hilbert space. Let be a finite family of -strict pseudo-contractions of C into itself, and let be a finite family of nonexpansive mappings of C into itself with and , and let , , where , , for all and , , for all . Let be the -mapping generated by , and . Then , and is a nonexpansive mapping.
Proof It is easy to see that . Let and . Then we have
By (2.4), we have
which implies that , that is, . It implies that
By (2.3) and (2.5), we have
By (2.6), we have
It implies that
By (2.5) and (2.7), we have . Hence, we have
Since and (2.3), we have
It follows that
which implies that , that is, . Since , we have
By (2.2), we have
It follows that
It implies that
By (2.9) and (2.10), we have . Hence, we have
By continuing in this way, we can show that and for all . Finally, we shall show that . Since
It implies that
which implies that , that is, . It implies that
Then we have . Hence .
Applying (2.4), we have that the mapping is a nonexpansive. □
Lemma 2.9 Let C be a nonempty closed convex subset of a real Hilbert space. Let be a finite family of -strict pseudo-contractions of C into itself, and let be a finite family of nonexpansive mappings of C into itself with , and let , where , and such that as for and . Moreover, for every , let and be the -mapping generated by , and and , and , respectively. Then for every bounded sequence in C.
Proof Let be a bounded sequence in C, and be generated by , and and , and , respectively. For each , we have
and for , by using Lemma 2.4, we obtain
By (2.13) and (2.14), we have
This together with the assumption as (, ), we can conclude that
□
Lemma 2.10 Let C be a nonempty closed convex subset of a real Hilbert space. Let be a finite family of -strict pseudo-contractions of C into itself, and let be a finite family of nonexpansive mappings of C into itself with , and let , where , and such that , for all . For every , let be the -mapping generated by , and . Then for every bounded sequence in C.
Proof Let be a bounded sequence in C. For each and the definition of , we have
For , and using the same method as (2.14) in Lemma 2.9, we have
From (2.16), (2.17), and using the same method as (2.15) in Lemma 2.9, we have
It implies that
□
3 Main result
Theorem 3.1 Let C be a nonempty closed convex subset of Hilbert spaces H, and let f be an α-contraction on H. Let be a bifunction from into ℝ, for every satisfying (A1)-(A4). Let be a finite family of -strict pseudo-contractions of C into itself, and let be a finite family of nonexpansive mappings of C into itself with and , and let , , where , , for all . Let be the -mapping generated by , and . Let and be the sequences generated by and
where is a sequence in . Assume that the following conditions hold:
-
(i)
, ;
-
(ii)
, , for all and ;
-
(iii)
, and , for every ;
-
(iv)
, for every and .
Then the sequence converges strongly to .
Proof Let , we have from Lemma 2.6, we obtain . Since
Again from Lemma 2.6, we have for every . By definition of , we have
Put . By (3.3), we can show by induction that , . This implies that is bounded, and so are , for every and .
Next, we will show that
By nonexpansiveness of , we have
From Lemma 2.10, we have
Since for every . By definition of , we have
similarly,
From (3.7) and (3.8), we obtain
and
By (3.9) and (3.10), we have
It follows that
This implies that
It follows that
It follows that
for every .
Substitute (3.11) into (3.5), we have
By (3.12), (3.6), conditions (iii), (iv) and Lemma 2.3, we have
From (3.11), (3.13) and condition (iv), we have
Let . From for every , we have
It implies that
By definition of and (3.15), we have
It implies that
From conditions (i), (iii) and (3.13), we have
Since
from condition (i), we have
From the definition of , we have
From condition (iii) and (3.17), we have
Since
by (3.13), (3.18) and (3.19), we have
Next, we show that
where . To show this inequality, take a subsequence of such that
Without loss of generality, we may assume that a subsequence of converges weakly to some . From (3.19), we have that converges weakly to q.
Since for all . Without loss of generality, we may assume that
Let be the -mapping generated by , and , where , . By Lemma 2.8, is a nonexpansive mapping, and .
By Lemma 2.9, we have
Since
by (3.20), (3.23), we have
Since converges weakly to q as (3.24) and Lemma 2.7, we have
Next, we show that . To show this, we may assume that
By Lemmas 2.5 and 2.6, for every , we define by
Then we have
From (3.1) and , we have
It implies that
and
By (A2), we have
It implies that
It follows that
Then
It follows that
It implies that
From and (3.17), we have
For every , we have
by (3.17) and (3.26), we have
Since a subsequence of converges weakly to q as , from (3.27) and Lemma 2.7, we have
Then
From Lemma 2.6, we have , . From (3.28), we have
By (3.25) and (3.29), we have
Since as and and (3.22), we have
Finally, we show that converges strongly to . Putting , by nonexpansiveness of , we have
It implies that
This implies that by condition (i), (3.21) and Lemma 2.2, we have that the sequence converges strongly to . By (3.19), we have
This completes the proof. □
4 Applications
In this section, we apply our main result to prove strong convergence theorems involving variational inclusion problems and variational inequality problems. To prove these results, we need definition and lemmas as follows.
A set-valued mapping is called monotone if for all , and imply that . A monotone mapping is maximal if the graph of M is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if for , for every implies that .
Next, we consider the following so-called variational inclusion problem: Find a such that
where , are two nonlinear mappings, and θ is zero vector in H (see, for instance, [16–21]). The set of the solution of (4.1) is denoted by .
Definition 4.1 (See [16])
Let be a multi-valued maximal monotone mapping, then the single-valued mapping defined by
is called the resolvent operator associated with M, where λ is any positive number, and I is an identity mapping.
Lemma 4.1 (See [16])
is a solution of variational inclusion (4.1) if and only if , , i.e.,
Further, if , then is a closed convex subset in H.
Lemma 4.2 (See [6])
The resolvent operator associated with M is single-valued, nonexpansive for all and 1-inverse-strongly monotone.
A mapping A of C into H is called α-inverse strongly monotone, see [22], if there exists a positive real number α such that
for all . The variational inequality problem is to find such that
for all . The set of solutions of the variational inequality is denoted by . We need the following lemma to prove a strong convergence theorem in this section.
Lemma 4.3 (See [23])
Let C be a closed convex subset of Hilbert space H. Let be mappings, and let be defined by with , . Then if and only if .
Theorem 4.4 Let C be a nonempty closed convex subset of Hilbert spaces H, and let f be an α-contraction on H. For every , let be a bifunction from into ℝ satisfying (A1)-(A4), let be an -inverse strongly monotone, and let be a mapping defined by , with . Let be a finite family of -strict pseudo-contractions of C into itself with and , and let , , where , , for all . Let be the -mapping generated by , and . Let and be the sequences generated by and
where is a sequence in . Assume that the following conditions hold:
-
(i)
, ;
-
(ii)
, for all and ;
-
(iii)
, and for every ;
-
(iv)
for every and .
Then the sequence converges strongly to .
Proof First, we show that is a nonexpansive mapping for every . For , we have
Thus, is a nonexpansive mapping, and so is for all . Then we obtain the desired result from Lemma 4.3 and Theorem 3.1. □
Corollary 4.5 Let C be a nonempty closed convex subset of Hilbert spaces H, and let f be an α-contraction on H. For every , let be a bifunction from into ℝ, satisfying (A1)-(A4), let be an -inverse strongly monotone, and let be a mapping defined by , with . Let be a finite family of nonexpansive mappings of C into itself with , and let , , where , , for all . Let be the -mapping generated by , and . Let and be the sequences generated by and
where is a sequence in . Assume that the following conditions hold:
-
(i)
, ;
-
(ii)
, for all and ;
-
(iii)
, and for every ;
-
(iv)
for every and .
Then the sequence converges strongly to .
Proof Since is a finite family of nonexpansive mappings, we have that is a finite family of -strict pseudo-contractive mappings. From Theorem 4.4, we can draw the desired conclusion. □
Theorem 4.6 Let C be a nonempty closed convex subset of Hilbert spaces H, and let f be an α-contraction on H. For every , let be a bifunction from into ℝ satisfying (A1)-(A4). Let be maximal monotone mappings for every , and let be a -inverse strongly monotone mapping for every . Let be a mapping defined by for every with . Let be a finite family of -strict pseudo-contractions of H into itself with and , and let , , where , , for all . Let be the -mapping generated by , and . Let and be the sequences generated by and
where is a sequence in . Assume that the following conditions hold:
-
(i)
, ;
-
(ii)
, for all and ;
-
(iii)
, and , for every ;
-
(iv)
, for every and .
Then the sequence converges strongly to .
Proof By using the same method as (4.4), we have that is a nonexpansive mapping for every . By Lemma 4.2, we have is a nonexpansive mapping for every . Then we obtain the desired result from Theorem 3.1. □
5 Example and numerical results
In the last section, we give numerical examples to support our main results.
Example 5.1 Let ℝ be the set of real numbers. For every , let the mappings , , , defined by
for every .
Suppose that is the -mapping generated by , and , where and for every and . Let the sequences and be generated by (3.1), where , and for every and . Then the sequences and converge strongly to 0.
Solution. For every . It is easy to see that is nonexpansive and is -strictly pseudo contractive mappings with .
Since is the -mapping generated by , and , where and for every and , then we have
for every . From Lemma 2.8, we have . For every and , we can see that . From definition of , we have . Then .
For every and , the mappings , , and , , satisfy conditions in Theorem 3.1. Then from Theorem 3.1, we have the sequences and converge to 0.
Next, we give numerical results to support this example. Let and . For every and , and from Lemma 2.5, there exist such that
Put . Then G is a quadratic function of y with coefficient , , . Next, we compute the discriminant Δ of G as follows:
Since for all . If it has most one solution in ℝ, so . It implies that . Then we have
for all and . From (3.1) and (5.1), we have
for every and . Since , , and (5.2), we can rewrite (3.1) as follows:
for every and .
Put and initial points , in (5.3) we have the following results respectively.
The numerical results for initial points and were shown in Tables 1 (Figure 1(b)) and 2 (Figure 1(a)), respectively. We observe that the sequences and converge to .
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This research was supported by the Research Administration Division of King Mongkut’s Institute of Technology Ladkrabang.
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Kangtunyakarn, A. Iterative approximation of common element of solution sets of various nonlinear operator problems. Fixed Point Theory Appl 2013, 295 (2013). https://doi.org/10.1186/1687-1812-2013-295
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DOI: https://doi.org/10.1186/1687-1812-2013-295