Fixed points of set-valued maps in locally complete spaces
- Carlos Bosch^{1}Email author,
- César L García^{1},
- Thomas Gilsdorf^{2},
- Claudia Gómez-Wulschner^{1} and
- Rigoberto Vera^{1}
https://doi.org/10.1186/s13663-017-0607-y
© The Author(s) 2017
Received: 9 March 2017
Accepted: 6 July 2017
Published: 1 August 2017
Abstract
We prove an extension of the Pareto optimization criterion to locally complete locally convex vector spaces to guarantee the existence of fixed points of set-valued maps.
Keywords
fixed points set-valued map Pareto optimizationMSC
47H10 47H04 46N10 46A031 Introduction
Ekeland’s variational principle, a milestone in the theory of nonlinear optimization, focuses on solving an optimization problem via a perturbed optimization problem. Since its appearance many extensions and equivalent formulations have been shown. Some of them, related to our discussion below, are contained in [1–8]. In [9] Azé and Corvellec generalized, in the setting of metric spaces, a result due to Lim ([10]) on the existence of fixed points for weakly inward multivalued contractions, defined on a nonempty closed subset of a Banach space. Their argument is remarkably simple: on the one hand, it avoids the use of transfinite induction (as in Lim’s paper) and, on the other hand, it uses Ekeland’s variational principle as a main tool to guarantee the existence of fixed points. Ekeland’s principle has been shown equivalent to other optimization statements, in particular, and of interest in this paper, is the equivalence to an optimization criterion of Pareto (see [11]) for which the existence of optima (critical points for dynamical systems) has been shown in complete metric spaces (see [12]). Our main result (Theorem 3.2) is a modification of that optimization criterion of Pareto, which shows how the existence of fixed points for set-valued maps can be extended to the setting of locally complete spaces. Among the consequences of Theorem 3.2, we provide a simple argument of Azé and Corvellec’s theorem ([9], Theorem 2.3). Although our results are set in the context of locally complete spaces, let us say that related to Ekeland’s type variational principles, some other forms of completeness have been use in the literature (e.g., quasi-metric spaces with Q-functions, fuzzy metric spaces, or sequentially lower complete spaces). Some of these include equivalences to Ekeland’s variational principle, for instance, to Caristi fixed point theorem or Takahashi minimization theorem, all of them with their natural connection to solution to equilibrium or fixed point theorems for set-valued maps (see, for instance [11, 13–19]).
2 Preliminaries
Throughout this paper \((X, \tau )\) will denote a Hausdorff locally convex space where the topology τ is generated by a saturated family of seminorms \(\{\rho _{j}\}_{j \in J}\). If B is a subset of X which is balanced and convex, we will call B a disk. Let \(X_{B}\) be the linear span of B, endowed with the topology generated by the Minkowski gauge of B, \(\rho _{B}\). When B is bounded \(\rho _{B}\) is a norm, and the norm topology is finer than the topology inherited from X. If \((X_{B}, \rho _{B})\) is a Banach space we say that B is a Banach disk. We say that X is a locally complete space if each closed, bounded disk is a Banach disk. Local complete spaces are also known as \(c^{\infty }\) or convenient spaces. The notion of local completeness has become important in all sorts of applications, for instance, in nonlinear distribution theory (see [1, 20] and the references therein), or as a context where existence and uniqueness for nonlinear integro-differential equations can be shown (see [21]). By lsc we refer to a lower semicontinuous functions, \(f \colon X \to \mathbb{R}\cup \{\infty \}\), which are proper, that is, their effective domain, \(\operatorname{dom}(f) := \{x : f(x) < \infty \}\), is nonempty.
It is important to note that, in general, \(e_{j}(Tx,Tz)\neq e_{j}(Tz,Tx)\). For instance, take \(X = \mathbb{R}^{2}\) equipped with the usual Euclidean norm d and consider the subsets \(A = \{(a,2) : -1 \leq a \leq 1\}\) and \(B = \{(x,y) : x^{2} + y^{2}\leq 1\}\). An easy computation shows that \(e_{d}(A, B) = \sqrt{5} - 1\) while \(e_{d}(B, A) = 3\).
Now we provide sufficient conditions on the functions \(e_{j}\) to get lower semicontinuity and continuity properties for the functions \(\phi _{1j}\) and \(\phi _{2j}\). We start with the following.
Proposition 2.1
If \(M_{j}\in \mathbb{R}^{+}\) is such that \(e_{j}(Tx, Tz) \leq M_{j}\rho _{j}(x - z)\) for every \(x, z \in X\) then \(\phi _{1j}(x)\leq \phi _{1j}(z) + 2M_{j}\rho _{j}(x - z)\). Hence, \(\phi _{1j}\) is \(\rho _{j}\)-continuous. In particular since τ is a uniformity we see that \(\phi _{1j}\) is τ-uniformly continuous.
Proof
For the function \(\phi _{2j}\), the result corresponding to Proposition 2.1 is as follows.
Proposition 2.2
If \(M_{j} \in \mathbb{R}^{+}\) is such that \(e_{j}(Tx, Tz)\leq M_{j}\rho _{j}(x - z)\) for all \(x, z\in X\) then \(\phi _{2j}\) is \(\rho _{j}\)-continuous hence τ-continuous.
Proof
Definition 2.3
For \(j\in J\), \(\phi _{j} \colon X \rightarrow [0, \infty )\) will be the function defined by \(\phi _{j}(x) = \phi _{1j}(x) + \phi _{2j}(x)\).
Lemma 2.4
Let \(A\subset X\) be any closed subset, and suppose that \(\phi _{1j}\) and \(\phi _{2j}\) are \(\rho _{j}\)-lsc. Suppose that \(T \colon A \rightarrow 2^{A}\) is such that Tx is \(\rho _{j}\)-sequentially compact and for each \(x\in A\) and each \(n \in \mathbb{N}\) there exists \(y_{n}\in Tx\) with the following properties: \(\phi _{j}(y_{n})\leq \phi _{1j}(x) + \frac{1}{n}\) and \(\rho _{j}(x-y_{n}) < \phi _{2j}(x)+\frac{1}{n}\). Then there exists \(y\in Tx\) such that \(\phi _{j}(y) + \rho _{j}(x - y)\leq \phi _{j}(x)\).
Proof
To close this section we define the sets that we will consider as target values for the dynamical systems in Theorem 3.2. Note that Lemma 2.4 is tailored to provide conditions for these sets to be nonempty.
Definition 2.5
Lemma 2.6
If \(\phi _{j}\) is lsc then the set \(C_{x}^{j}\) is \(\rho _{j}\)-closed.
Proof
Let \((y_{n})\) be a sequence in \(C_{x}^{j}\) such that \(y_{n} \stackrel{\rho _{j}}{\rightarrow } y\in X\).
Corollary 2.7
If \(\phi _{j}\) is lsc and for each \(y \in C_{x}^{j}\) and \(y + Z_{\rho _{j}}\subset C_{x}^{j}\), where \(Z_{\rho _{j}} = \{x\in X : \rho _{j}(x) = 0\}\) is the zero set of \(\rho _{j}\), then \(C_{x}^{j}\) is a \(\rho _{j}\)-closed linear subspace of X.
Definition 2.8
Let \(A\subset X\) be any closed subset. If \(T \colon A \rightarrow 2^{A}\) is a dynamical system and \(\rho _{j}\) is one of the seminorms defining the topology τ, \(x^{\ast }\in A\) is a \(\rho _{j}\)-critical point of T if \(Tx^{\ast }\subset x^{\ast } + Z_{\rho _{j}}\).
Note that when \(\rho _{j}\) is a norm, to be a \(\rho _{j}\)-critical point means that \(Tx^{\ast } = \{x^{\ast }\}\).
3 Modified Pareto case
Recall that the topology τ in the lcs X is generated by a saturated family of seminorms \(\{\rho _{j} : j \in J\}\). Theorem 1 of [11] provides with sufficient conditions for a given dynamical system \(T \colon A \to 2^{A}\), over a locally complete subset A of X, to have critical points. In [11], Theorem 1, one essentially sees that if for every \(x \in A\) and every \(u \in Tx\), \(c_{j}\rho _{j}(x - u) \le \Phi (x) - \Phi (u)\) for every \(j \in J\) and for some function \(\Phi \colon A \to \mathbb{R}\) lsc and bounded below (\(c_{j}\) are positive scalars such that \(\bigcap_{j \in J}\{x \in X : c_{j}\rho _{j}(x) \le 1\}\) is a nonzero Banach disk) then T has a critical point. In our setting, if we suppose that, for a fixed \(j \in J\), A is \(\rho _{j}\)-complete and \(\Phi \colon A \to \mathbb{R}\) is \(\rho _{j}\)-lsc, that is, \(\rho _{j}(x_{n} - x) \rightarrow 0\) implies \(\Phi (x)\leq \liminf_{n} \Phi (x_{n})\) (which in turn implies τ-lsc of Φ) then we have the following proposition.
Proposition 3.1
If for each \(x\in A\) and each \(u \in Tx\) we have \(\rho _{j}(x - u)\leq \Phi (x) - \Phi (u)\), then T has a \(\rho _{j}\)-critical point, \(x_{j}^{\ast }\in A\).
The proof goes along the same lines as that of Theorem 1 in [11] and we omit it.
Theorem 3.2
Main Result
- 1.
The seminorm \(\rho _{j}\), from the family of seminorms defining the topology τ, is such that the subset A is \(\rho _{j}\)-sequentially complete.
- 2.
The function \(\phi _{j}\) (see definition 2.3) satisfies the condition \(\phi _{j}(x)\leq \liminf \phi _{j}(x_{n})\) whenever \(\rho _{j}(x - x_{n}) \to 0\).
- 3.
\(C_{x}^{j}\neq \emptyset \) for all \(x\in A\).
Proof
Corollary 3.3
If \(x_{j}^{*}\in T_{j}x_{j}^{*}\) then \(T_{j}x_{j}^{*} = x_{j}^{*} + Z_{\rho _{j}}\).
A weak version (in the sense of weak topology) of Theorem 3.2 can be obtained as follows. Consider \(X^{\prime } = (X,\tau )^{\prime }\), the topological dual of X, and the saturated family of seminorms \(\{\rho _{f} : f\in X^{\prime }\}\), where \(\rho _{f}(x) = \vert f(x) \vert \) and \(Z_{f} = \operatorname{ker}(f)\). Assume as before that \(A \subset X\) is \(\rho _{f}\)-complete for some fixed, but arbitrary, \(f \in X^{\prime }\), and let \(\Phi \colon A \to \mathbb{R}\) be a lsc function. Then we have the following.
Proposition 3.4
We also have the following restatement of Theorem 3.2, where the continuous linear functions \(f\in X^{\prime }\) take the place of the indices \(j\in J\).
Theorem 3.5
- 1.
The function \(\phi _{f} = \vert f \vert \) satifies \(\phi _{f}(x)\leq \liminf \phi _{f}(x_{n})=\liminf_{n} \vert f(x_{n}) \vert \) (lower semicontinuity).
- 2.
\(C_{x}^{f}\neq \emptyset\) for all \(x\in A\).
Proof
Just notice that if we consider the weak topology in place of τ on X, then we have the same hypotheses as in Theorem 3.2 above. □
4 Applications to fixed point theory
Theorem 4.1
Equivalently, for all \(k\in \mathbb{N}\), \(\rho _{f}(x^{\ast } - h^{k}(x^{\ast })) = 0\). Moreover, if \(x^{\ast \ast }\in A\) also satisfies (5) then \(f(x^{\ast }) = f(x^{\ast \ast })\).
Also note that if \(A = X\) then as a consequence of inequality (4) we see that \(\hat{h} \colon X/Z_{f} \rightarrow X/Z_{f}\), such that \(\hat{h}([x])=[h(x)]\) is a function. Furthermore, if \(A = X\) and the identities in (5) hold, then the function ĥ has a unique fixed point.
Proof of Theorem 4.1.
As a consequence we see that the function \(e_{f}\) satisfies \(e_{f}(Tx, Tz) \le M\vert f(x - z) \vert \) for all \(x, z \in A\).
Note that for the given function h and the seminorm \(\rho _{f}\) we can use Proposition 2.1 to see that the function \(\phi _{1f}(x) = \operatorname{diam}_{f} Tx\) is Lipschitz and thus \(\rho _{f}\)-uniformly continuous. Also, since \(x \in Tx\), \(\phi _{2f}(x) = 0\) and \(\phi _{f}(x) = \phi _{1f}(x)\). Therefore \(\phi _{f}\) is \(\rho _{f}\) uniformly continuous and satisfies condition \((2)\) in Theorem 3.2. Observe that since \(x \in Tx\) we have trivially from Lemma 2.4 that \(\phi _{f}(x) + \rho _{f}(x - x) = \phi _{f}(x)\). Also \(x\in C_{x}^{f} = \{y\in Tx \mid \phi _{f}(y) + \rho _{f}(x - y)\leq \phi _{f}(x)\}\), that is, \(C_{x}^{f}\neq \emptyset \), which is condition \((3)\) in Theorem 3.2.
By Lemma 2.6, and since \(\rho _{f}\) is continuous, we see that \(C_{x}^{f}\) is \(\rho _{f}\)-closed, thus \(\rho _{f}\)-sequentially complete since X is \(\rho _{f}\)-sequentially complete. By Theorem 3.2 we conclude that, for the map \(T_{f}\colon A \rightarrow 2^{A}\), defined via \(T_{f}(x) = C_{x}^{f}\), there exists \(x^{*}\in A\) such that \(T_{f}(x^{*})\subset x^{*} + Z_{f}\).
Now, since \(x^{*}\in T_{f}(x^{*})\), by Corollary 3.3 we see that \(T_{f}(x^{*}) = x^{*} + Z_{f}\). Hence, for each \(y\in Tx^{*}\) there exists \(z\in Z_{f}\) (\(f(z)=0\)) such that \(y = x^{*} + z\). From which we obtain \(\operatorname{diam}_{f}Tx^{*} = 0\). That is, \(f(x^{*}) = f(h^{k}(x^{*}))\) for all \(k\in \mathbb{N}\). Equivalently \(\rho _{f}(x^{*} - h^{k}(x^{*})) = 0\) for all \(k\in \mathbb{N}\).
There is an interesting connection with the condition of metrical inwardness and the corresponding fixed point theorem in Caristi [23], p.247. Metrical inwardness hold in our setting if we take a fixed function \(f\in X^{\prime }\) such that for each \(x\in A\) there exists \(u\in A\) such that \(f(x - u)\) and \(f(u - hx)\) are both positive numbers (or both negative) where h is a function satisfying the conditions of Theorem 4.1.
All we did in the previous paragraphs can be repeated if we change the weak topology for a generic Hausdorff topology for a lcs \((X, \tau )\). Indeed, if we use a fixed seminorm \(\rho _{j}\) among those that generate the topology τ, then we have the following result corresponding to Proposition 4.1.
Proposition 4.2
A consequence of (6) for \(A = X\) is that the relation \(\hat{h}\colon X/Z_{\rho _{j}} \rightarrow X/Z_{\rho _{j}}\) defined by \(\hat{h}([x]) = [h(x)]\) is a function. A consequence of (7) for \(A = X\), is that the function ĥ has a unique fixed point.
Proof of Proposition 4.2
To prove Proposition 4.2 let us define the function \(T \colon X \rightarrow 2^{X}\) via \(Tx = \{x, h(x), h^{2}(x),\ldots \}\).
The trick now is to follow the argument in Theorem 4.1 but replacing the seminorm \(\rho _{f}\) by the seminorm \(\rho _{j}\) in order to get \(x^{\ast }\in X\) such that, by Theorems 2.4, 3.2, and Corollary 3.3, \(\overline{Tx^{\ast }}^{\rho _{j}} = x^{\ast } + Z_{\rho _{j}}\). Finally, since \(\hat{0}\in Z_{\rho _{j}}\) we obtain \(x^{\ast }\in Tx^{\ast }\). Uniqueness is proved in the same way as in Theorem 4.1. □
Since \(\{h^{n(k - j)}(x^{\ast })\vert n\in \mathbb{N}\}\) is bounded, since it is finite with at most k points, and \(0 < M < 1\), the sequence \(\{M^{nj} \vert f(h^{n(k - j)}(x^{\ast }) - x^{\ast \ast })\vert \}\) converges to 0 as \(n \rightarrow \infty \); in other words, \(\vert f(x^{\ast } - x^{\ast \ast })\vert = 0\) hence \(\vert f(h((x^{\ast })) - h(x^{\ast \ast }))\vert = 0\). We conclude that \(\rho _{f}(x^{\ast } - x^{\ast \ast }) = 0 = \rho _{f}(h(x^{\ast }) - h(x^{\ast \ast }))\). In general, \(\vert f(h^{nj}(x^{\ast }) - h^{nj}(x^{\ast \ast })) \vert = 0 = \vert f(h^{nk}(x^{\ast }) - h^{nk}(x^{\ast \ast })) \vert \).
If we define \(T^{\prime \prime }\colon D_{r}^{f}(x) \rightarrow 2^{X}\) to be \(T^{\prime \prime }x=T_{f}^{\prime }x\), \(T^{\prime \prime } = T_{f}^{\prime }|_{D_{r}^{f}(x)}\), then \(T_{f}^{\prime }\) satisfies the hypotheses of Theorem 3.5. Thus, there exists \(z^{\ast }\in D_{r}^{f}(x)\subset A_{f}\) such that \(z^{\ast }\in T^{\prime \prime }z^{\ast }\), in other words, \(z^{\ast }\in A_{f}\cap F_{T}^{f} = \emptyset \) and the set \(A_{f}\) must be empty.
5 Conclusions
The paper can be considered as an extension of the Pareto optimization criterion to locally complete locally convex vector spaces [11] with some applications to fixed points. Local completeness is a very weak completeness property. This type of spaces is becoming the convenient setting for several applications. Here in this setting we get a fixed point theorem and as an application we obtain the results of Azé and Corvellec’s [9].
Declarations
Acknowledgements
The authors were partially supported by the Asociación Mexicana de Cultura, A.C. Research by T. Gilsdorf was done while on the faculty at the ITAM. The authors gratefully acknowledge all the remarks and suggestions made by the anonymous referees.
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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