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Fixed point property and approximation of a class of nonexpansive mappings
Fixed Point Theory and Applications volume 2014, Article number: 81 (2014)
We introduce the concept of ψ-firmly nonexpansive mapping, which includes a firmly nonexpansive mapping as a special case in a uniformly convex Banach space. It is shown that every bounded closed convex subset of a reflexive Banach space has the fixed point property for ψ-firmly nonexpansive mappings, an important subclass of nonexpansive mappings. Furthermore, Picard iteration of this class of mappings weakly converges to a fixed point.
MSC:47H06, 47J05, 47J25, 47H10, 47H17.
Throughout this paper, a Banach space E will be over the real scalar field. We denote its norm by and its dual space by . Let , the set of all fixed points for a mapping T and ℕ denote the set of all positive integer.
Let K be a nonempty bounded closed convex subset of E. We say that K has the fixed point property for nonexpansive mapping if for every nonexpansive mapping (i.e. ), K contains a fixed point of T (i.e. ); E has the fixed point property (FPP for short) if any nonempty bounded closed convex subset of E has the fixed point property for nonexpansive mapping; E has the weak fixed point property (WFPP for short) if any weakly compact convex subset of E has the fixed point property for nonexpansive mapping. For a reflexive Banach space, both properties are obviously the same.
The famous question whether a Banach space has the fixed point property (WFPP) had remained open for a long time [1, 2]. It has been answered in the negative by Sadovski  and Alspach  who constructed the following examples, respectively.
Let and . Define by
The above two examples suggest that to obtain positive results in the problem of the existence of fixed points for nonexpansive mappings, it is necessary to impose some restrictions either on T or on the Banach space E. Naturally, the following questions are asked also.
Problem 1.3 Which Banach spaces satisfy the WFPP?
Problem 1.4 Determine a subclass of nonexpansive mappings such that every Banach space satisfies the FPP for this subclass.
Considerable effort in the development of a fixed point theory for nonexpansive mappings, mainly for Problem 1.3, has been done in the last 40 years. A well-known result of Browder  asserts that if E is uniformly convex, then E has the weak fixed point property. This theorem was also proved independently by Göhde . At the same time, Kirk  established a more general result by showing that if E has normal structure, then E has the weak fixed point property. Normal structure is a geometric property somewhat more general than uniform convexity. In , one can see a detailed study of sufficient conditions for this property as well as their permanence properties. It has also been shown  that a condition weaker than normal structure is sufficient to guarantee the weak fixed point property (WFPP). In 1981, it has been showed by Maurey  that the Hardy space and the reflexive subspace of have the weak fixed point property (WFPP). For other examples of Banach spaces with the weak fixed point property see [6, 7, 14] and [15, 16] for more details.
One of our main aims is to give an affirmative answer to Problem 1.4. In other words, we will study fixed point properties of ψ-firmly nonexpansive mapping, an important subclass of nonexpansive mappings, on weakly compact convex subsets of a Banach space.
On the other hand, using the Picard iterative method, the well-known Banach Contraction Principle is obtained: Let be a complete metric space and is a contraction (i.e. , and some ). Then T has a unique fixed point and for each , Picard iteration strongly converges to .
It is known for some time that even in a Hilbert space setting, Picard iteration of a nonexpansive mapping T need not actually converge to a fixed point. However, for some special nonexpansive mapping (or some nonexpansive mapping who is modified necessarily), the weak convergence of such iteration can be proved. Fox example, in the frame of a uniformly convex Banach space E with a Fréchet differentiable norm, Reich  showed that if where I is an identity operator and T is a nonexpansive self-mapping defined on a nonempty bounded closed convex subset K of E, then for each , Picard iteration weakly converges to a fixed point of T; Bruck [20, 21] proved that for each , the Cesàro means of the nonexpansive self-mapping T weakly converges to a fixed point of T. This fact was first established by Baillon  for the spaces (). Naturally, the following question is put forward.
Problem 1.5 Does there exist a subclass of nonexpansive mappings (not contraction) such that Picard iteration (weakly) converges to a fixed point of such mapping?
Another purpose of ours is to show that Picard iteration of ψ-firmly nonexpansive mapping weakly converges to its fixed point. That is, ψ-firmly nonexpansive mapping is actually an answer to Problem 1.5.
We also show that in a uniformly convex space, ψ-firmly nonexpansive mapping includes a firmly nonexpansive mapping and the resolvent of an accretive operator as a special case.
2 Fixed point property for ψ-firmly nonexpansive mapping
The concept of firmly nonexpansive mapping was introduced by Bruck . A mapping T with domain and range in Banach space E is said to be firmly nonexpansive if for all the function
is non-increasing on , or equivalently,
Obviously any firmly nonexpansive mapping is nonexpansive mapping (). The converse is not true (consider the mapping in E). However, there is an interesting observation. For any and consider the by . Banach Contraction Principle guarantees that has a unique fixed point , i.e.,
Since depends on u and t, we can define a family mappings . It is minor technicality to prove that all mappings are firmly nonexpansive. Moreover, for any , (see [, pp.120-122] for a proof). This shows that the fixed point property for firmly nonexpansive mapping coincides with the fixed point property for nonexpansive mapping.
In a Hilbert space, T is firmly nonexpansive if and only if
(see [, pp.127-128] for a proof). Clearly, the inequality (2.2) is equivalent to the following:
In view of the above one might expect firmly nonexpansive mappings to exhibit better behavior than nonexpansive mappings in general. However, from the point of view of fixed point theory, the restriction is mild. Naturally, Song and Chai  introduced the notion of firmly type nonexpansive mapping.
A mapping T is said to be firmly type nonexpansive if for all , there exists such that
Obviously, the firmly type nonexpansive mapping contains the firmly nonexpansive mapping and the resolvent of monotone operator as a special case in Hilbert space. For a detailed proof and more examples, see [, Examples 1-5].
Now we introduce the concept of ψ-firmly nonexpansive mapping which includes the firmly type nonexpansive mapping as a special case ().
A mapping T is said to be ψ-firmly nonexpansive if for all , if there exists a continuous strictly increasing function with such that
In order to achieving the objects mentioned in Section 1, solving Problem 1.4, we need the following fact.
Lemma 2.1 ([, Propositions 9.3.6])
Let C be a weakly compact subset in Banach space E and let be a weakly lower semi-continuous function. Then the function f attains its minimum on C. That is,
Now we show our main results.
Theorem 2.2 Let K be a weakly compact convex subsets of a Banach space E and be a ψ-firmly nonexpansive mapping. Then T has a fixed point, i.e. .
Proof Since K is bounded and convex, it is well known (even for nonexpansive mappings) that there exists a sequence in K such that
Let a real valued function φ be defined on K by
Then φ is convex and continuous, and hence weakly lower semi-continuous (see [, p.12, Proposition 5]). It follows from Lemma 2.1 that there exists such that
Next, we show that . It is immediate from (2.5) that
Thus, we have
Therefore together with (2.6), we have
and so by the property of ψ. This yields the desired conclusion. □
Obviously, we also have the following.
Theorem 2.3 Let K be a nonempty bounded closed convex subset of a reflexive Banach space E and be a ψ-firmly nonexpansive mapping. Then T has a fixed point, i.e. .
Now we show that the firmly nonexpansive mapping is a subclass of ψ-firmly nonexpansive mapping in uniformly convex Banach space.
Lemma 2.4 (Xu [, Theorem 2])
Let and be two fixed real numbers. Then a Banach space is uniformly convex if and only if there exists a continuous strictly increasing convex function with such that
for all and , where .
Theorem 2.5 Let K be a nonempty bounded closed convex subset of a uniformly convex Banach space E and be a firmly nonexpansive mapping. Then T is a ψ-firmly nonexpansive mapping.
Proof Since T is firmly nonexpansive, by (2.1), we have
By Lemma 2.4 (, ), we obtain
Let for all . The desired result is reached. □
Let be an accretive operator. Let , the resolvent of A. It is well known that is nonexpansive, where is range of and I is an identity operator of E. Furthermore, for and and ,
which is referred to as the Resolvent Identity. Now we show that for each , the resolvent of A is an ψ-firmly nonexpansive mapping also.
Example 2.6 Let E be a uniformly convex Banach space and be an accretive operator. Then for each , is a ψ-firmly nonexpansive mapping defined on ().
Proof It follows from the Resolvent Identity (2.9) that
Then we have
Using the same proof techniques as Theorem 2.5, we also have
where . □
The other three similar mappings were introduced by Aoyama et al. . For a subset C of a smooth Banach space E, a mapping is of
type (P) (or firmly nonexpansive-like) if
type (Q) (or firmly nonexpansive type; see Kohsaka et al. ) if
type (R) (or firmly generalized nonexpansive) if
where J is the normalized duality mapping of E and is generalized dual pairs on .
Remark 2.7 The common point between ψ-firmly nonexpansive mapping and the above three mappings is that they all include a firmly nonexpansive mapping in Hilbert spaces as a special case. However, in a uniformly convex Banach space, each firmly nonexpansive mapping is a ψ-firmly nonexpansive mapping, but it is not one of the above three mappings since a uniformly convex Banach space may not be smooth.
Remark 2.8 In the framework of a smooth, strictly convex and reflexive Banach space, the fixed point properties of the above three mappings were studied by Aoyama et al. , Kohsaka et al.  and many mathematical workers. Only in reflexive Banach space, we can obtain the fixed point property of ψ-firmly nonexpansive mappings.
3 Approximation methods of ψ-firmly nonexpansive mappings
We discuss the weak convergence of Picard iteration for ψ-firmly nonexpansive mapping.
Lemma 3.1 Let K be a nonempty closed convex subset of a Banach space E and be ψ-firmly nonexpansive with . If for any given , is defined by Picard iteration sequence
Then is an asymptotic fixed point sequence of T, i.e.
Proof Take . Then
Therefore, we have
Consequently, is non-increasing and bounded, and hence the limit exists. So, is bounded also. It follows from (3.2) that
and hence, by the property of ψ. The desired result is obtained. □
A Banach space E is said to satisfy Opial’s condition  if, for any sequence in E, implies
In particular, Opial’s condition is independent of uniformly convex (smooth) since the spaces satisfy this condition for while it fails for the () spaces. In fact, spaces satisfying Opial’s condition need not even by isomorphic to uniformly convex spaces .
Theorem 3.2 Let K be a weakly compact convex subset of a Banach space E satisfying Opial’s condition and be firmly type nonexpansive. Then for any given , , defined by Picard iteration (3.1) weakly converges to some fixed point of T.
Proof It follows from Theorem 2.2 that . Then following Lemma 3.1, we see that is bounded, the limit exists for each and
The weak compactness of K means that there exists a subsequence of such that weakly converges to some point of K, say . Then using the proof technique of Theorem 2.2, we have since by Opial’s condition,
Next we show that weakly converges to . Let y is another weak limit point of and . Then we can choose a subsequence that weakly converges to y, and hence . Since exists for each , we have
a contradiction, and hence . □
Similarly, we also have the following.
Theorem 3.3 Let K be a nonempty closed convex subset of a reflexive Banach space E satisfying Opial’s condition and be firmly type nonexpansive with . Then for any given , , defined by Picard iteration (3.1) weakly converges to some fixed point of T.
Remark 3.4 Theorem 3.2 is applicable to () and . However, we do not know whether it works in for and .
Recall a Banach space E is said to have (i) a Gâteaux differentiable norm (we also say that E is smooth), if the limit
exists for each x (), ; (ii) a uniformly Gâteaux differentiable norm, if for each y in E, the limit is uniformly attained for bounded ; (iii) a Fréchet differentiable norm, if for each , , the limit is attained uniformly for bounded .
The value of at is denoted by , and the normalized duality mapping from E into is denoted by J, that is,
It is well known (see Brower [, p.44]) that for a smooth Banch space E, the normalized duality mapping J is single-valued, and, moreover,
A Banach space E is said to be (iv) strictly convex if , implies ; (v) uniformly convex if for all , such that implies whenever .
In 1979, Bruck  explicitly introduced the following concept. Let Γ denote the set of strictly increasing convex functions with . A mapping T is said to be of type Γ if there exists such that for all and
Three facts about such mappings are easy to observe. Mappings of type Γ are nonexpansive; affine nonexpansive mappings are of type Γ; Mappings of type Γ have convex fixed point sets. Bruck [20, 21] showed that each nonexpansive mapping is of type Γ in a uniformly convex Banach space. See also [, Proposition 10.3] for a proof.
Theorem 3.5 Let E be a uniformly convex Banach space with a Fréchet differentiable norm and K be a nonempty closed convex subset of E. If is ψ-firmly nonexpansive with , then for any given , , defined by Picard iteration (3.1) weakly converges to some fixed point of T.
Proof It follows from Lemma 3.1 that is bounded, the limit exists for each and
Then is weakly compact. Similarly to the proof of Theorem 3.2, we only need show that has unique weak limit point. Let p and q are two weak limit points of . Then Browder Demiclosedness Principle  means that . Thus both the limits and exist. Then the remainder of the proof is identical to the proof of Theorem 10.6 in Reference [, pp.114-115] with the help of the mappings of type Γ. Which is a repeat works, we omit it. □
By Theorem 2.5, the following corollary about firmly nonexpansive mappings is obvious.
Corollary 3.6 Let E be a uniformly convex Banach space with a Fréchet differentiable norm and K be a nonempty closed convex subset of E. If is firmly nonexpansive with , then for any given , , defined by Picard iteration (3.1) weakly converges to some fixed point of T.
Remark 3.7 Theorem 3.5 is dependent of Theorem 3.2 or 3.3 since the spaces satisfy Opial’s condition for while it fails for the () spaces. On the other hand, spaces satisfying Opial’s condition need not even by isomorphic to uniformly convex spaces .
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The authors would like to thank the editor and the anonymous referee for useful comments and valuable suggestions on the language and structure of our manuscript. This work is partially supported by the National Natural Science Foundation of P.R. China (Grant Nos. 11171094, 11271112) and partially done when he was visiting The Hong Kong Polytechnic University.
The authors declare that they have no competing interests.
The work presented here was carried out in collaboration between all authors. All authors contributed equally and significantly to writing this manuscript. All authors have contributed to, seen and approved the final manuscript.
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Song, Y., Huang, Y. Fixed point property and approximation of a class of nonexpansive mappings. Fixed Point Theory Appl 2014, 81 (2014). https://doi.org/10.1186/1687-1812-2014-81
- ψ-firmly nonexpansive mappings
- fixed points
- reflexive Banach spaces
- Picard iteration