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Viscosity iteration algorithm for a ϱ-strictly pseudononspreading mapping in a Hilbert space
Journal of Inequalities and Applications volume 2013, Article number: 80 (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.
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.
Definition 1.1 A is said to be
strongly monotone if there exists a constant such that
For such a case, A is said to be α-strongly-monotone;
inverse-strongly monotone if there exists a constant such that
For such a case, A is said to be α-inverse-strongly-monotone;
k-Lipschitz continuous if there exists a constant such that
Remark 1.2 Let , where B is a θ-Lipschitz and η-strongly monotone operator on H with and f is a Lipschitz mapping on H with coefficient , . It is a simple matter to see that the operator F is -strongly monotone over H, i.e.,
The classical variational inequality, which is denoted by , is to find such that
A mapping is said to be nonexpansive if
A mapping T is said to be firmly nonexpansive if
T is said to be nonspreading in  if
It is shown in  that (1.2) is equivalent to
These mappings are generalization of a firmly nonexpansive mapping in a Hilbert space. is said to be firmly nonexpansive if
Definition 1.3 is called demicontractive on H if there exists a constant such that
Definition 1.4 
is ϱ-strictly pseudononspreading if there exists such that
for all .
Remark 1.5 It is easy to claim that firmly nonexpansive mapping ⇒ nonspreading mapping ⇒ ϱ-strictly pseudononspreading mapping.
Indeed, from the definition of those mappings, , we obtain
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.
Example 1.6 Let with the norm defined by
, and let C be an orthogonal subspace of X (i.e., , we have ). Then it is obvious that C is a nonempty closed convex subset of X. Now, for any , define a mapping as follows:
To see that T is -strictly pseudononspreading, we break the process of proof into three cases. ,
Case 1: and , observe that
since and .
Case 2: and , we obtain , and .
Case 3: and , we have , and . Thus
From (1), (2) and (3), we obtain that T is -strictly pseudononspreading, i.e.,
We can easily know that , where is defined by the set of fixed points of T.
T is not nonspreading, since for , , we have , and , we obtain
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 :
(A1) T α-demicontractive is equivalent to
(A2) if .
Remark 1.8 Observe that if T is ϱ-strictly pseudononspreading and , then and , we obtain
Remark 1.9 According to Remark 1.7(A1) and the fact that the ϱ-strictly pseudononspreading mapping of T is demicontractive, let . Then we obtain
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 
Let C be a nonempty closed convex subset of a real Hilbert space H and let be a ϱ-strictly pseudononspreading mapping with a nonempty fixed point set . Let and let be a real sequence in such that and . Let , and be sequences in C generated from an arbitrary by
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 
Let f be a contraction on a real Hilbert space H and T be a nonexpansive mapping on H. Starting with an arbitrary initial , define a sequence generated by
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 .
Then the sequence generated by (1.8) converges strongly to the unique solution of the variational inequality
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.
Algorithm 1.14 Let be arbitrary
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 
Assume is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence in ℝ such that
Lemma 2.3 
Let be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence of which satisfies for all . Also, consider the sequence of integers defined by
Then is a nondecreasing sequence verifying , . It holds that , and we have
Lemma 2.4 Let K be a closed convex subset of a real Hilbert space H given and . Then if and only if the following inequality holds:
3 Main results
Let C be a nonempty closed convex subset of a real Hilbert space H and let be a -strictly pseudononspreading mapping with a common nonempty fixed point set . Let f be an L-Lipschitz mapping on H with coefficient . Assume the set is nonempty. Since is closed and convex, the nearest point projection from C onto is well defined. Recall is η-strongly monotone and θ-Lipschitzian on H with , . Let , , consider the following sequence defined by
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.
Proof Let and . Then , we have
From and (3.2), we also have
According to (3.3), (3.1) and Remark 3.1, we obtain
which combined with amounts to
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.
Theorem 3.3 Assume C is a nonempty closed convex subset of a real Hilbert space H and let be a -strictly pseudononspreading mapping with a common nonempty fixed point set . Let f be an L-Lipschitz mapping on H with coefficient and be η-strongly monotone and θ-Lipschitzian on H with , . Let , . Consider the sequences and to be sequences in C generated from an arbitrary by (3.1), where , , , and . Then and converge strongly to the unique element in verifying
which equivalently solves the following variational inequality problem:
Proof According to Lemma 3.2, it is simple to know that , and are bounded. Thus, for and and according to (3.2) and (3.1), we have
Summing (3.9) from to n and dividing by n, we obtain
Since is bounded, then there exists a subsequence of which converges weakly to . Replacing n by in (3.10), we obtain
Since and are bounded, letting in (3.11) yields
Let in (3.12). We obtain that .
Observe that since and are bounded, and , then
We next show that
Indeed, take of such that
where is obtained in (3.7). We may assume that as . From (3.13), we have as , then to arbitrary bounded linear functional g on H, we have
Thus, we obtain as , and . Hence, we have
Moreover, from (3.1), (3.13) and (3.14), we have
As required, finally we show that and .
According to (3.1), (3.4) and (3.16), we obtain
It is easily seen that , and . By Lemma 2.2, we conclude that as , and also converges strongly to the unique element in . In addition, the variational inequality (3.15) can be written as
So, by Lemma 2.4, it is equivalent to the fixed point equation
Remark 3.4 For a nonspreading mapping T, we have in Theorem 3.3 to obtain the following corollary.
Corollary 3.5 Assume C is a nonempty closed convex subset of a real Hilbert space H and let be a nonspreading mapping with a common nonempty fixed point set . Let f be an L-Lipschitz mapping on H with coefficient and be η-strongly monotone and θ-Lipschitzian on H with , . Let , , consider the sequences and to be sequences in C generated from an arbitrary by
where , , and . Then and converge strongly to the unique element in verifying
which equivalently solves the following variational inequality problem:
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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).
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
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Deng, BC., Chen, T. & Li, ZF. Viscosity iteration algorithm for a ϱ-strictly pseudononspreading mapping in a Hilbert space. J Inequal Appl 2013, 80 (2013). https://doi.org/10.1186/1029-242X-2013-80
- nonspreading mapping
- ϱ-strictly pseudononspreading
- fixed point