- Open Access
Viscosity iteration algorithm for a ϱ-strictly pseudononspreading mapping in a Hilbert space
© Deng et al.; licensee Springer 2013
- Received: 12 September 2012
- Accepted: 25 January 2013
- Published: 28 February 2013
In this paper, we discuss the strong convergence of the viscosity approximation method in Hilbert spaces relatively to the computation of fixed points of an operator in ϱ-strictly pseudononspreading. Under suitable conditions, some strong convergence theorems are proved. Our work improves previous results for nonspreading mappings.
- nonspreading mapping
- ϱ-strictly pseudononspreading
- fixed point
Throughout this paper, we always assume that H is a real Hilbert space endowed with an inner product and its induced norm denoted by and , respectively. Let C be a nonempty, closed and convex subset of H and let be a nonlinear mapping.
- (ii)strongly monotone if there exists a constant such that
- (iii)inverse-strongly monotone if there exists a constant such that
- (iv)k-Lipschitz continuous if there exists a constant such that
Definition 1.4 
for all .
Remark 1.5 It is easy to claim that firmly nonexpansive mapping ⇒ nonspreading mapping ⇒ ϱ-strictly pseudononspreading mapping.
Clearly, every nonspreading mapping is ϱ-strictly pseudononspreading. The following example shows that the class of ϱ-strictly pseudononspreading mappings is more general than the class of nonspreading mappings. Let us give an example of a ϱ-strictly pseudononspreading mapping satisfying the condition of Definition 1.4.
To see that T is -strictly pseudononspreading, we break the process of proof into three cases. ,
since and .
Case 2: and , we obtain , and .
We can easily know that , where is defined by the set of fixed points of T.
Since our class of maps contains the class of nonspreading mappings, it also contains the class of firmly nonexpansive mappings.
Remark 1.7 
Let T be an α-demicontractive mapping on H with and for :
(A2) if .
In 2011, Osilike and Isiogugu  introduced the following propositions and proved a strong convergence theorem somewhat related to a Halpern-type iteration algorithm for a ϱ-strictly pseudononspreading mapping in Hilbert spaces.
Proposition 1.10 
Let C be a nonempty closed convex subset of a real Hilbert space H and let be a ϱ-strictly pseudononspreading mapping. If , then it is closed and convex.
Proposition 1.11 
Let C be a nonempty closed convex subset of a real Hilbert space H and let be a ϱ-strictly pseudononspreading mapping. Then is demiclosed at 0.
Theorem 1.12 
Then and converge strongly to , where is a metric projection of H onto .
In 2010, Tian  introduced the following theorem for finding an element of a set of solutions to the fixed point of a nonexpansive mapping in a Hilbert space.
Theorem 1.13 
where B is a θ-Lipschitz and η-strongly monotone operator on H with , and . Assume also that a sequence is a sequence in satisfying the following conditions:
(c1) and ,
(c2) or .
In this paper, we combine Theorem 1.12 and Theorem 1.13 and introduce the following general iterative algorithm for a ϱ-strictly pseudononspreading mapping T.
where is η-strongly monotone and boundedly Lipschitzian, f is an L-Lipschitz mapping on H with coefficient and , .
Under suitable conditions, some strong convergence theorems are proved in the following chapter.
Throughout this paper, we write to indicate that the sequence converges weakly to x. implies that converges strongly to x. The following lemmas are useful for main results.
Definition 2.1 A mapping T is said to be demiclosed if for any sequence which weakly converges to y, and if the sequence strongly converges to z, then .
Lemma 2.2 
Lemma 2.3 
where , , , and is a sequence in satisfying the following conditions:
(c2) or .
Remark 3.1 
Let H be a real Hilbert space. Let B be a θ-Lipschitzian and η-strongly monotone operator on H with , . Let , let and , then for , S is a contraction with a constant .
Before stating our main result, we introduce some lemmas for algorithm (3.1) as follows.
Lemma 3.2 The sequence is generated by (3.1) with being a ϱ-strictly pseudononspreading mapping on H and . Then is bounded.
Putting , we clearly obtain . Hence and are bounded. From (3.3), we have that is also bounded. □
Now we are in a position to claim the main result.
Let in (3.12). We obtain that .
As required, finally we show that and .
Remark 3.4 For a nonspreading mapping T, we have in Theorem 3.3 to obtain the following corollary.
This work is supported in part by the National Natural Science Foundation of China (71272148), the Ph.D. Programs Foundation of Ministry of Education of China (20120032110039) and China Postdoctoral Science Foundation (Grant No. 20100470783).
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