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Strong convergence of viscosity approximation methods for the fixed-point of pseudo-contractive and monotone mappings
Fixed Point Theory and Applications volume 2013, Article number: 273 (2013)
Abstract
In this paper, we introduce a viscosity iterative process, which converges strongly to a common element of the set of fixed points of a pseudo-contractive mapping and the set of solutions of a monotone mapping. We also prove that the common element is the unique solution of certain variational inequality. The strong convergence theorems are obtained under some mild conditions. The results presented in this paper extend and unify most of the results that have been proposed for this class of nonlinear mappings.
MSC:47H09, 47H10, 47L25.
1 Introduction
Let C be a closed convex subset of a real Hilbert space H. A mapping is called monotone if and only if
A mapping is called α-inverse strongly monotone if there exists a positive real number such that
Obviously, the class of monotone mappings includes the class of the α-inverse strongly monotone mappings.
A mapping is called pseudo-contractive if , we have
A mapping is called κ-strict pseudo-contractive, if there exists a constant such that
A mapping is called non-expansive if
Clearly, the class of pseudo-contractive mappings includes the class of strict pseudo-contractive mappings and non-expansive mappings. We denote by the set of fixed points of T, that is, .
A mapping is called contractive with a contraction coefficient if there exists a constant such that
For finding an element of the set of fixed points of the non-expansive mappings, Halpern [1] was the first to study the convergence of the scheme in 1967
In 2000, Moudafi [2] introduced the viscosity approximation methods and proved the strong convergence of the following iterative algorithm under some suitable conditions
Viscosity approximation methods are very important, because they are applied to convex optimization, linear programming, monotone inclusions and elliptic differential equations. In a Hilbert space, many authors have studied the fixed points problems of the fixed points for the non-expansive mappings and monotone mappings by the viscosity approximation methods, and obtained a series of good results, see [3–18].
Suppose that A is a monotone mapping from C into H. The classical variational inequality problem is formulated as finding a point such that , . The set of solutions of variational inequality problems is denoted by .
Takahashi [19, 20] introduced the following scheme and studied the weak and strong convergence theorem of the elements of , respectively, under different conditions
where T is a non-expansive mapping, A is an α-inverse strong monotone operator.
Recently, Zegeye and Shahzad [21] introduced the algorithms and obtained the strong convergence theorems in a Hilbert space, respectively,
and
where are asymptotically non-expansive mappings, and , are non-expansive mappings.
Our concern is now the following: Is it possible to construct a new sequence that converges strongly to a common element of the intersection of the set of fixed points of a pseudo-contractive mapping and the solution set of a variational inequality problem for a monotone mapping?
2 Preliminaries
Let C be a nonempty closed and convex subset of a real Hilbert space H, a mapping is called the metric projection, if , there exists a unique point in C, denote by such that
It is well known that is a non-expansive mapping, and have the property as follows:
Lemma 2.1 [6]
Let be a sequence of nonnegative real numbers satisfying the following relation:
where is a sequence in (0,1) and is a real sequence such that
-
(i)
;
-
(ii)
or .
Then .
Lemma 2.2 [21]
Let C be a closed convex subset of a Hilbert space H. Let be a continuous monotone mapping, let be a continuous pseudo-contractive mapping, define mappings and as follows: ,
Then the following hold:
-
(i)
and are single-valued;
-
(ii)
and are firmly non-expansive mappings, i.e., , ;
-
(iii)
, ;
-
(iv)
and are closed convex.
Lemma 2.3 [22]
Let and be bounded sequence in a Banach space, and let be a sequence in , which satisfies the following condition:
Suppose that
and
Then .
Let C be a closed convex subset of a Hilbert space H. Let be a continuous monotone mapping, let be a continuous pseudo-contractive mapping. Then we define the mappings as follows: for ,
3 Main results
Theorem 3.1 Let C be a nonempty closed convex subset of a Hilbert space H. Let be a continuous pseudo-contractive mapping, let be a continuous monotone mapping such that , let be a contraction with a contraction coefficient . The mappings and are defined as (2.3) and (2.4), respectively. Let be a sequence generated by
where , let , , be sequences of nonnegative real numbers in and
-
(i)
, ;
-
(ii)
, ;
-
(iii)
;
-
(iv)
, .
Then the sequence converges strongly to , and also is the unique solution of the variational inequality
Proof First, we prove that is bounded. Take , then we have from Lemmas 2.2 that
For , because and are non-expansive, and f is contractive, we have
Therefore, is bounded. Consequently, we get that , and , are bounded.
Next, we show that .
Let , , by the definition of the mapping , we have that
Putting in (3.5), and letting in (3.6), we have that
Adding (3.7) and (3.8), we have that
Since A is a monotone mapping, which implies that
Therefore, we have that
i.e.,
Without loss of generality, let b be a real number such that , , then we have that
where . Then we have from (3.10) and (3.4) that
On the other hand, let , , we have that
Let in (3.12), and let in (3.13), we have that
Adding (3.14) and (3.15), and because T is pseudo-contractive, we have that
Therefore, we have
Hence we have that
where .
Let , hence we have that
Hence we have from (3.17), (3.16), (3.11) and condition (iii) that
Notice conditions (ii) and (iv), we have that
Hence we have from Lemma 2.3 that
Therefore, we have that
Hence we have from (3.10) and (3.16) that
Since , so we have that
According to Lemma 2.1, we have that
In the same way, we have that
Consequently, we have that
Now, we show that .
Since sequence is bounded, then there exists a sub-sequence of and such that . Next, we show that .
Since , we have that
Let , , , then we have that
Since , and also A is monotone, we have that
Consequently,
If , by the continuity of A, we have , . Thus, .
In addition, since , we have that
Let , , , then we have that
Since , if , by the continuity of T, we have that
Let , we have , thus, . Consequently, we conclude that .
Because , we have from (2.1) that
Next, we show that . From formula (3.3), we have that
Let , . According to Lemma 2.1 and formula (3.23), we have that , i.e., the sequence converges strongly to .
According to formula (3.23), we conclude that is the solution of the variational inequality (3.2). Now, we show that is the unique solution of the variational inequality (3.2).
Suppose that is another solution of the variational inequality (3.2). Because is the solution of the variational inequality (3.2), i.e., , . Because , then we have
On the other hand, to the solution , since , so
Adding (3.24) and (3.25), we have that
i.e.,
Hence
Because , hence we conclude that , the uniqueness of the solution is obtained. □
Theorem 3.2 Let C be a nonempty closed convex subset of a Hilbert space H. Let be a continuous pseudo-contractive mapping, let be a continuous monotone mapping such that , let be a contraction with a contraction coefficient . The mappings and are defined as (2.3) and (2.4), respectively. Let be a sequence generated by
where , , are sequences of nonnegative real numbers in and
-
(i)
, ;
-
(ii)
, ;
-
(iii)
, .
Then the sequence converges strongly to , and also is the unique solution of the variational inequality
Proof Putting in Theorem 3.1, we can obtain the result. □
If in Theorem 3.1 and Theorem 3.2, let be a constant mapping, we have the following theorems.
Theorem 3.3 Let C be a nonempty closed convex subset of a Hilbert space H. Let be a continuous pseudo-contractive mapping, let be a continuous accretive mapping such that . The mappings and are defined as (2.3) and (2.4), respectively. Let be a sequence generated by
where , and let , , be sequences of nonnegative real numbers in and
-
(i)
, ;
-
(ii)
, ;
-
(iii)
;
-
(iv)
, .
Then the sequence converges strongly to , and also is the unique solution of the variational inequality
Theorem 3.4 Let C be a nonempty closed convex subset of a Hilbert space H. Let be a continuous pseudo-contractive mapping, let be a continuous monotone mapping such that . The mappings and are defined as (2.3) and (2.4), respectively. Let be a sequence generated by
where , , are sequences of nonnegative real numbers in and
-
(i)
, ;
-
(ii)
, ;
-
(iii)
, .
Then the sequence converges to , and also is the unique solution of the variational inequality
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Acknowledgements
This work was supported by the Natural Science Foundation Project of Chongqing (CSTC, 2012jjA00039) and the Science and Technology Research Project of Chongqing Municipal Education Commission (KJ130712, KJ130731).
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Tang, Y. Strong convergence of viscosity approximation methods for the fixed-point of pseudo-contractive and monotone mappings. Fixed Point Theory Appl 2013, 273 (2013). https://doi.org/10.1186/1687-1812-2013-273
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DOI: https://doi.org/10.1186/1687-1812-2013-273