- Open Access
General alternative regularization methods for nonexpansive mappings in Hilbert spaces
© Yang and He; licensee Springer. 2014
- Received: 9 March 2014
- Accepted: 4 September 2014
- Published: 26 September 2014
Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product and the norm . Let be a nonexpansive mapping with a nonempty set of fixed points and let be a Lipschitzian strong pseudo-contraction. We first point out that the sequence generated by the usual viscosity approximation method may not converge to a fixed point of T, even not bounded. Secondly, we prove that if the sequence satisfies the conditions: (i) , (ii) and (iii) or , then the sequence generated by a general alternative regularization method: converges strongly to a fixed point of T, which also solves the variational inequality problem: finding an element such that for all . Furthermore, we prove that if T is replaced with the sequence of average mappings () such that , where and are two positive constants, then the same convergence result holds provided conditions (i) and (ii) are satisfied. Finally, an algorithm for finding a common fixed point of a family of finite nonexpansive mappings is also proposed and its strong convergence is proved. Our results in this paper extend and improve the alternative regularization methods proposed by HK Xu.
MSC:47H09, 47H10, 65K10.
- fixed point
- nonexpansive mapping
- strong pseudo-contraction
- viscosity approximation method
- general alternative regularization method
Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product and the norm and let be a α-contractive mapping, i.e., there exists a constant such that holds for all . Let be a nonexpansive mapping, i.e., for all . Throughout this article, the set of fixed points of T, indicated by , is always assumed to be nonempty.
Since is a closed convex subset of H, so the metric projection is valid.
In this case, we also call h a α-strong pseudo-contraction.
It is very easy to see that a α-contractive mapping is a α-strongly pseudo-contractive and α-Lipschitzian mapping, i.e., the class of contractive mappings is a proper subset of the class of Lipschitzian strong pseudo-contractions. The class of Lipschitzian strong pseudo-contractions will be used repeatedly in the sequel.
where K is a nonempty closed convex subset of H and is a nonlinear operator. It is well known that  if is a Lipschitzian and strongly monotone operator, then the variational inequality problem has a unique solution.
where is a sequence in the interval and u is some given element in C. For Halpern’s iteration process, a classical result in the setting of Hilbert spaces is as follows.
Theorem 1.1 ()
and studied its strong convergence in the setting of Hilbert spaces and Banach spaces, respectively. Indeed, in the setting of Hilbert spaces, one can proved that for algorithm (1.2) the same convergence result as that of Theorem 1.1 holds under conditions (i)-(iii) above.
where is a sequence in the interval . Moudafi proved the following result in Hilbert spaces.
Theorem 1.2 ()
Xu studied the viscosity approximation method in the setting of Banach spaces and obtained the strong convergence theorems .
where is a sequence in the interval . In fact, in the setting of Hilbert spaces, it is not difficult to prove by an argument very similar to the proof of Theorem 1.2 that for algorithm (1.4) the same result as that of Theorem 1.2 holds under conditions (i)-(iii) above.
The main purpose of this paper is to consider a very natural question: if algorithms (1.3) and (1.4) can be extended to more general cases, more precisely, can we replace a contractive mapping f with a Lipschitzian strong pseudo-contraction h so that the same convergence result as that of Theorem 1.2 is still guaranteed under conditions (i)-(iii) as above? The answer to this question is negative for algorithm (1.3) unfortunately but is sure for algorithm (1.4) fortunately. In this sense, it seems reasonable to deem that algorithm (1.4) is better that algorithm (1.3).
holds for all , it is easy to see that for any positive constant κ and any , is a κ-Lipschitzian and α-strongly pseudo-contractive mapping.
Since , holds and thus this implies that the sequence generated by (1.6) is not bounded provided .
The rest of this paper is organized as follows. In order to prove our main results, some useful facts and tools are listed as lemmas in the next section. In Section 3, we prove that if a contractive mapping f in algorithm (1.4) is replaced with a Lipschitzian strong pseudo-contraction h, then the same convergence result as that of Theorem 1.2 is still guaranteed under conditions (i)-(iii) as above. Furthermore, we prove that if T is replaced with the sequence of average mappings () such that , where and are two positive constants, then the same result still holds provided conditions (i) and (ii) are satisfied. In the last section, an algorithm for finding a common fixed point of a family of finite nonexpansive mappings is also proposed and its strong convergence is proved.
⇀ for weak convergence and → for strong convergence.
denotes the weak ω-limit set of .
means that B is the definition of A.
We need some facts and tools in a real Hilbert space H, which are listed as lemmas below.
Lemma 2.3 ()
Let C be a closed convex subset of a real Hilbert space H and let be a nonexpansive mapping such that . If a sequence in C is such that and , then .
Lemma 2.5 ()
implies for any subsequence .
where I denotes the identity operator.
Our first result is as follows.
Proof Noting that is a -Lipschitzian and -strongly monotone mapping, so the variational inequality problem (3.4) has a unique solution, which is denoted by . Now we try to prove that .
This means that is bounded, so is .
we get from conditions (i) and (ii) that and hold. Hence, this together with condition (iii) allows us to assert by using Lemma 2.2. From this together with (3.10) one concludes that and hence we obtain by using Lemma 2.3.
there exist two constants and such that for all ,
for any subsequence .
and hence holds. □
In this section, we turn to considering an algorithm for finding a common fixed point of a family of finite nonexpansive mappings.
where for all and .
Our main result in this section is as follows.
there exist two constants and such that for all and ,
Proof Without loss of the generality, we only give the proof for the case where .
This work was supported by National Natural Science Foundation of China (Grant no. 11201476) and in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing.
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