Strong convergence theorems based on the viscosity approximation method for a countable family of nonexpansive mappings
© Bagherboum et al.; licensee Springer. 2014
Received: 2 April 2014
Accepted: 8 December 2014
Published: 18 December 2014
In a real Hilbert space, an iterative scheme is considered to obtain a common fixed point for a countable family of nonexpansive mappings. In addition, strong convergence to the common fixed point of this sequence is investigated. As an application, an equilibrium problem is solved. We also state more applications of this procedure to obtain a common fixed point of W-mappings.
MSC:47H09, 47H10, 47J20.
Let H be a real Hilbert space, C be a nonempty closed convex subset of H, and I be an identity mapping on H. The strong (weak) convergence of to x is written by () as .
holds for every with .
A metric (nearest point) projection from a Hilbert space H to a closed convex subset C of H is defined as follows.
for all .
A mapping S from C into itself is called nonexpansive if , for all . is the set of fixed point of S. Note that is closed and convex if S is nonexpansive. A mapping f from C into C is said to be contraction, if there exists a constant such that , for all .
They proved the generated sequence converges strongly to a common element of the set of common fixed points of a countable family of nonexpansive mappings and the set of solutions of the variational inequality.
to find a fixed point of a nonexpansive mapping.
In this work, motivated and inspired by the above results, an iterative scheme based on the viscosity approximation method is utilized to find a common element of the set of common fixed points of a countable family of nonexpansive mappings. Moreover, a strong convergence theorem with different conditions on the parameters is studied. As an application, an equilibrium problem is solved. In addition, a common fixed point for W-mappings is obtained.
The following lemmas will be useful in the sequel.
Lemma 1.1 ()
Let and be bounded sequences in a Banach space X and be a sequence in with . Suppose for all integers and . Then .
Lemma 1.2 ()
, for all and .
Lemma 1.3 ()
Let C be a nonempty closed subset of a Banach space and be a sequence of nonexpansive mappings from C into itself. Suppose . Then, for each , converges strongly to some point of C. If S is a mapping from C into itself which is defined by , for all , then .
2 Strong convergence theorem
In this section, we use the viscosity approximation method to find a common element of the set of common fixed points of a countable family of nonexpansive mappings.
, , ,
, for any bounded subset B of C.
Let S be a mapping from C into itself defined by for all and . Then converges strongly to an element , where .
for all . This shows that is a contraction from H into C. Since H is complete, there exists a unique element of such that .
is bounded, there exists a subsequence of which converges weakly to z. Without loss of generality, assume that . is a sequence in C and C is closed and convex, so . Now, using the fact that , we obtain . Next we show .
This is a contradiction. Thus, .
It is easy to see that , and . Hence, by Lemma 1.2, we find that strongly converges to , where . This completes the proof of this theorem. □
The following example shows that this theorem is not a special case of [, Theorem 3.1].
with initial value , converges strongly to an element (zero) of and .
In this section, we consider the equilibrium problems and W-mappings.
3.1 Equilibrium problems
Equilibrium theory plays a central role in various applied sciences such as physics, mechanics, chemistry, and biology. In addition, it represents an important area of the mathematical sciences such as optimization, operations research, game theory, and financial mathematics. Equilibrium problems include fixed point problems, optimization problems, variational inequalities, Nash equilibria problems, and complementary problems as special cases.
for all .
- 1.If , then the problem (3.1) is reduced to generalized equilibrium problem, i.e., finding such that
- 2.If , then the problem (3.1) is reduced to the mixed equilibrium problem, that is, to find such that
- 3.If , , then the problem (3.1) is reduced to the equilibrium problem, which is to find such that
If , , then the problem (3.1) is reduced to the variational inequality problem (1.2).
Now let be a real-valued function. To solve the generalized mixed equilibrium problem for a bifunction , let us assume that F, φ, and C satisfy the following conditions:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A3) for each , ;
(A4) for each , is convex and lower semicontinuous;
(B2) C is a bounded set.
In what follows we state some lemmas which are useful to prove our convergence results.
Lemma 3.1 ()
For each , ;
- (3)is firmly nonexpansive, i.e., for any ,
MEP is closed and convex.
Lemma 3.2 ()
Let C be a nonempty closed convex subset of a real Hilbert space H. Assume that is a nonexpansive mapping from C into H and a firmly nonexpansive mapping from H into C such that . Then is a nonexpansive mapping from H into itself and .
, for any bounded subset B of C;
, where is a mapping defined by , for all . Moreover, .
for all . It is clear that each α-inverse strongly monotone mapping is monotone and -Lipschitzian and that each μ-Lipschitzian, relaxed -cocoercive mapping with is monotone. Also, if A is an α-inverse strongly monotone, then is a nonexpansive mapping from C to H, provided that (see ).
Now we have the following theorem.
, , ,
, for any bounded subset E of C.
Then converges strongly to an element , where .
for any bounded subset E of C. In addition, the mapping , defined by for all , satisfies .
Therefore, by Theorem 2.1, converges strongly to an element , where . This completes the proof. □
The concept of W-mappings was introduced in [21, 22]. It is now one of the main tools in studying convergence of iterative methods to approach a common fixed point of nonlinear mapping; more recent progress can be found in  and the references cited therein.
One can find the proof of the following lemma in .
is nonexpansive and for all ;
exists, for all and ;
the mapping defined by , for all is a nonexpansive mapping satisfying ; and it is called W-mapping generated by , and .
, , .
Let W be a mapping from C into itself defined by for all . Then converges strongly to an element , where .
Now, by setting in Theorem 2.1 and using Lemma 3.5, we obtain the result. □
Applying Lemma 3.5 and Theorem 3.4, we obtain the following result.
, , ,
Then converges strongly to an element , where .
The authors express their gratitude to the referees for reading this paper carefully.
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