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On stationary points of nonexpansive setvalued mappings
 Rafa Espínola^{1}Email author,
 Meraj Hosseini^{2} and
 Kourosh Nourouzi^{2}
https://doi.org/10.1186/s1366301504814
© Espínola et al. 2015
Received: 16 October 2015
Accepted: 4 December 2015
Published: 30 December 2015
Abstract
In this paper we deal with stationary points (also known as endpoints) of nonexpansive setvalued mappings and show that the existence of such points under certain conditions follows as a consequence of the existence of approximate stationary sequences. In particular we provide abstract extensions of wellknown fixed point theorems.
Keywords
MSC
1 Introduction and preliminaries
The first relevant work for existence of fixed points for nonexpansive setvalued mappings was provided by Markin [1] in 1968. Then a large and deep theory was developed by several authors (see, for instance, [2–6] or [7], Chapter 15, and references therein). This theory, however, is very far from as much advanced a theory as the corresponding one for singlevalued nonexpansive mappings. The problem of the existence of stationary points has remained almost unexplored for nonexpansive mappings, it being the case that most results about them require contractive like conditions on the mapping as is the case in [8–11]. There has recently been some activity in this direction though. Several authors have begun the study of generalized setvalued nonexpansive mappings through an approach given by the properties of approximate sequences of fixed points where stationary points have appeared in a natural way. See, for instance, [12, 13] and the notion of a strong approximate fixed point sequence (which we call an approximate stationary point sequence here) in [14]. In the present work we show that some of the very wellknown properties implying the existence of fixed points for nonexpansive singlevalued mappings also imply the existence of stationary points in the setvalued case provided approximate stationary point sequences exist.
Approximate fixed point sequences play a major role in metric fixed point theory for both single and setvalued nonexpansive mappings. It is a very wellknown fact, after Nadler’s principle for setvalued contractions, that such sequences always exist provided \(T\colon C\to F(C)\) is nonexpansive (see, for instance, [7], Chapter 15, [15], Theorem 8.23, or [14]). On the contrary, we see that approximate stationary point sequences do not need exist even if the mapping has a fixed point, its values are compact and X is a Hilbert space.
Example 1
Remark
After the referee comments, the authors have learnt about references [16, 17] which deal with close problems to ours. That is, in these works the authors study questions related to stationary points of nonexpansive multivalued mappings provided the existence of approximate stationary point sequences is guaranteed. Although there is some overlapping among [16, 17] and the present work, their goals and ours are different. In [16] the author mainly focuses on conditions involving uniform convexity and wonders about the structure of the set of stationary points while [17] focuses on taking up some of the questions raised in [16]. In the present work, however, we deal with existence of stationary points on more general conditions than those studied in [16, 17] which, in particular, lead to more general versions of some results provided by these references (compare Theorems 3.1 and 3.4 in [16] to, respectively, Theorems 6 and 7 below). Notice also that Theorem 1 below gives an answer to Question 5.3, which remains out of scope of [17], raised in [16].
2 Main results
For our first result we will deal with two different notions of Tinvariant sets.
Definition 1

A subset D of C is said to be Tinvariant if \(Tx\subseteq D\) for all \(x\in D\).

A subset D of C is said to be weakly Tinvariant if \(Tx\cap D\neq\emptyset\) for all \(x\in D\).
Remark 1
Notice that the two notions coincide for singlevalued mappings.
A nonempty, closed, and convex subset D of C will be said to be a minimal Tinvariant set (minimal weakly Tinvariant set) if it does not contain any proper closed and convex subset which is Tinvariant (weakly Tinvariant).
Our first goal is to study the existence of stationary points for nonexpansive setvalued mappings under the conditions of a normal structure (see [7], Chapter 6).
Definition 2
We will use the next two propositions.
Proposition 1
([18], p. 152)
For every weakly compact convex subset C of a Banach space X, \(Z(C)\) is a nonempty, closed, and convex subset of C.
Proposition 2
([18], p. 153)
The next result can be seen as an abstract extension of Kirk’s fixed point theorem.
Theorem 1
Let X be a Banach space and C a nonempty, weakly compact, and convex subset of X with normal structure. Then a nonexpansive mapping \(T\colon C \to K( C)\) has a stationary point if and only if there is a nonempty, closed, and convex subset F of C which is minimal weakly Tinvariant and minimal Tinvariant.
Proof
It is obvious that if T has a stationary point x then \(F=\{ x\}\) fulfills all the requirements. Conversely, let F be the minimal set given by the statement. If \(\operatorname {diam}(F)=0\) then its element is a stationary point. Therefore we can assume that \(\operatorname {diam}(F)>0\). We will show that the set \(Z(F)\) contradicts the minimality of F.
Our next result, inspired by [19], Lemma 1, is a technical one which explores the properties of minimal sets of stationary point free mappings.
Theorem 2
Proof
Suppose that Σ is the set of all nonempty, weakly compact, and convex Tinvariant subsets D of C. The family \(\Sigma\neq \emptyset\), because \(C \in\Sigma\) and it can be partially ordered by set inclusion. An easy application of Zorn’s lemma shows that the family Σ possesses a minimal element E. The diameter of E must be positive since otherwise the T invariancy of E would imply that T has a stationary point which is a contradiction.
Hence \(D \subsetneqq E \) is nonempty, weakly compact, and convex and Tinvariant, i.e., \(D \in\Sigma\) and \(\operatorname {diam}(D) < \operatorname {diam}(E)\), which is a contradiction because E is a minimal member of Σ. □
We have the following immediate corollaries.
Corollary 1
If the Banach space X is reflexive, then the same holds true for any nonempty, closed, convex, and bounded subset C of X.
Corollary 2
Proof
The next technical wellknown results will be needed.
Theorem 3
([18], p. 131)
Let \((x_{n})\) be a bounded sequence in a Banach space X and C a nonempty, weakly compact, and convex subset of X. Then \(Z_{a}(C,\{x_{n}\})\) is nonempty and convex.
Theorem 4
([18], p. 131)
Definition 3
Given \(T\colon C\to2^{C}\setminus\{\emptyset\}\), with C a nonempty, closed, and convex subset of Banach space X, we will say that T has the approximate stationary point sequence property in C if T has an approximate stationary point sequence in any Tinvariant, nonempty, closed, and convex subset of C.
Remember that, as Example 1 exhibits, the existence of such sequences is not guaranteed in general.
Theorem 6
Let X be a Banach space with characteristic of convexity \(\epsilon_{0} ( X ) \leq1\). Let C be a nonempty, weakly compact, and convex subset of X and \(T\colon C\to2^{C}\setminus\{\emptyset\}\) a nonexpansive mapping. Then T has a stationary point if and only if T has the approximate stationary point sequence property.
Proof
Now, the fact that \(Z_{a}(E,\{x_{n}\})\) is Tinvariant follows in a similar way as was shown for set D in the proof of Theorem 2. Therefore, since it is weakly compact and convex too, we meet a contradiction with the minimality of E. □
Theorem 7
Let X be a Banach space which satisfies the Opial property. Let C be a nonempty, weakly compact, and convex subset of X and \(T\colon C\to2^{C}\setminus\{\emptyset\}\) a nonexpansive mapping. Then T has a stationary point if and only if T has the approximate stationary point sequence property.
Proof
In 1974 Lim [6] gave the following extension of the Markin fixed point theorem for uniformly convex spaces.
Theorem 8
Let X be a uniformly convex Banach space, let C be a closed, bounded, and convex subset of X and \(T : C\rightarrow K(C)\) be a nonexpansive mapping. Then T has a fixed point.
In 1990 Kirk and Massa [4] gave the following partial generalization of Lim’s theorem.
Theorem 9
Let C be a closed, bounded, and convex subset of a Banach space X and \(T : C\rightarrow KC(C)\) a nonexpansive mapping, where \(KC(C)\) stands for the collection of nonempty, compact, and convex subsets of C. If the asymptotic center in C of each bounded sequence of X is nonempty and compact, then T has a fixed point.
Motivated by these two theorems we have the following for stationary points.
Theorem 10
Suppose that X is a Banach space such that for each closed convex bounded subset C of X the asymptotic center in C of each bounded sequence is nonempty and compact. Let \(T\colon C\to2^{C}\setminus\{ \emptyset\}\) be a nonexpansive mapping with C weakly compact. Then T has a stationary point if and only if T has the approximate stationary point sequence property.
Proof
In particular, the following corollary holds.
Corollary 3
Let X be a uniformly convex Banach space. Then the same conclusions as in Theorem 10 hold.
Indeed, any Banach space with the \((\mathrm {DL})\)condition has the weak normal structure (see [21], Theorem 3.3).
Before giving the next result, we need the following proposition.
Proposition 3
([6, 22]) Let C be a nonempty bounded subset of a Banach space X and \(\{x_{n}\} \) a bounded sequence in X. Then \(\{x_{n}\}\) has a subsequence that is regular with respect to C.
Theorem 11
Let X be a Banach space with the \((\mathrm {DL})\)condition. Let \(T\colon C\to 2^{C}\setminus\{\emptyset\}\) be a nonexpansive mapping with C nonempty, weakly compact, and convex. Then T has a stationary point if and only if T has the approximate stationary point sequence property.
Proof
Next we list some sufficient conditions that lead to the \((\mathrm {DL})\) property.
Corollary 4
Proof
See [23], Corollary 1. □
Corollary 5
 1.
\(r_{X}(1) >0\),
 2.
\(\Delta_{0}( X ) < 1\).
Proof
See [23], Corollary 2. □
Corollary 6
Proof
See [24], Corollary 3.2. □
Corollary 7
Proof
See [25], Theorem 4. □
Corollary 8
Proof
See [26], Theorem 3.19. □
 1.
\(R_{1}(x) = \Vert x \Vert \) and for all \(k \geq2\), \(R_{k}(x) \leq \Vert x \Vert \).
 2.
\(\lim_{k\rightarrow\infty} R_{k}( x) =0\).
 3.If \(x_{n} \rightharpoonup0\) (weakly convergent to zero), then for all \(k \geq1\)$$\limsup_{n\rightarrow\infty} R_{k}( x_{n}) = \limsup _{n\rightarrow\infty} \Vert x_{n} \Vert . $$
 4.If \(x_{n} \rightharpoonup0\), then for all \(k \geq1\)$$\limsup_{n\rightarrow\infty} R_{k}( x_{n} + x) = \limsup_{n\rightarrow \infty} R_{k}( x_{n} ) + R_{k}( x ). $$
We restate Lemma 2 of [19] next.
Lemma 1
Theorem 12
Let X be a Banach space endowed with a family of seminorms \(\{R_{k}( \cdot) \}\) satisfying Properties 14 stated above, and \(\!\!\cdot\!\!\) be the equivalent norm given above. Let \(T\colon C\to2^{C}\setminus\{ \emptyset\}\) be a nonexpansive mapping with respect to \(\!\!\cdot\!\!\) and C a nonempty, weakly compact, and convex. Then T has a stationary point if and only if T has the approximate stationary point sequence property.
Proof
We only need to prove that T has a stationary point provided it has the approximate stationary point sequence property. Suppose for contradiction that T is stationary point free. By Theorem 2 there exist a minimal, nonempty, Tinvariant, weakly compact, and convex subset E.
Remark 2
Notice that, since approximate fixed point sequence always exists in the singlevalued case for the theorems we have revisited, our results can be regarded as abstract extensions of corresponding results for singlevalued nonexpansive mappings.
Declarations
Acknowledgements
Rafa Espínola was supported by DGES, Grant MTM201234847C0201 and Junta de Andalucía, Grant FQM127. The authors would also like to thank anonymous referee for pointing [16] out as well as making other suggestions, which have improved the presentation of the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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