Note on essential fixed points of approximable multivalued mappings

A new definition of essential fixed points is introduced for a large class of multivalued maps. Two abstract existence theorems are presented for approximable maps on compact ANR-spaces in terms of a nontrivial fixed point index, or a nontrivial Lefschetz number and a zero topological dimension of the fixed point set. The second one is applied to the periodic dissipative Marchaud differential inclusions for obtaining the existence of a discretely essential subharmonic solution. Three simple illustrative examples are supplied.


Introduction
In the present note, we will consider for the first time the notion of essential fixed points to multivalued maps as defined below. More concretely, we will present two abstract theorems about the existence of essential fixed points to a large class of approximable multivalued maps, on compact ANR-spaces, in terms of a nontrivial fixed point index, or a nontrivial Lefschetz number and a zero topological dimension of the fixed point set.
These two theorems can be regarded as a multivalued generalisation of their analogies in our recent paper [] (cf. also [], Section ), where single-valued maps were exclusively examined for the same goal. On the other hand, unlike in [, ], we do not consider here compact multivalued maps, or even multivalued maps with only a certain amount of compactness like compact absorbing contractions, on arbitrary ANR-spaces. This remains as a challenge for our future research.
In our approach, we again follow the seminal ideas of Fort, Jr. and O'Neil in their classical papers [, ] from the early s. Hence, roughly speaking, the fixed point, say x  , of a given multivalued approximable mapping is essential if any continuous single-valued map which is sufficiently 'near' admits a fixed point in the neighborhood of x  . For a precise formulation, see Definition . below, and for some further results in this field, see the references in []. Let us note that this definition significantly differs from all the other definitions for multivalued maps (see e.g. [, ] and the references therein), because it effectively employs the approximability of given multivalued maps on their graphs by single-valued maps. In this way, topological invariants like a fixed point index can easily be calculated just by means of these single-valued approximations. This profit naturally connects our approach with the classical theory developed by Fort, Jr. [] and O'Neil [].
There are a lot of important applications of the essential fixed point theory like those in economy and the theory of games (see again the references in []). In [], we concentrated to essential multivalued fractals considered as fixed points of the induced (single-valued) Hutchinson-Barnsley operators in hyperspaces. Here, our application concerns periodic solutions of periodic dissipative (in the sense of Levinson) differential inclusions. From our theoretical results, we will deduce that if a periodic dissipative system of Marchaud inclusions possesses at most a finite number of subharmonic periodic solutions or, in particular, entirely bounded solutions, then it admits a discretely essential periodic solution.
As already pointed out in [], the essentiality can be regarded as a sort of structural stability which has a lot to do with the shadowing property for chaotic dynamics. Thus, the main profit consists not only in an additional information as regards the localization of fixed points of 'near' single-valued approximations, but in a numerical reliability at all.
In order to demonstrate the power of the obtained results, three simple illustrative examples are supplied.

Preliminaries
Let X = (X, d) be a metric space. Let us recall that X is an absolute neighborhood retract (written X ∈ ANR) if there exist an open set U in a normed space and two single-valued continuous maps r : U → X and s : X → U such that r • s = id X , where id X stands for the identity on X.
If U is an arbitrary convex set, then X is called an absolute retract (written X ∈ AR).

Evidently: AR ⊂ ANR.
A compact space is called an R δ -set if it is an intersection of a decreasing sequence of compact AR-spaces. For compact sets, in particular: convex ⊂ AR ⊂ R δ .
Let X = (X, d X ) and Y = (Y , d Y ) be metric spaces. By multivalued mappings ϕ : X Y , we understand here those with nonempty, closed values, i.e. that ϕ : X →  Y \ {∅} and a closed set ϕ(x) ⊂ Y is assigned to every point x ∈ X. By a fixed point of ϕ : X Y , we

Approximable multivalued mappings
In the entire text, all topological spaces are metric and all single-valued mappings are continuous. Moreover, we shall consider only upper semicontinuous (u.s.c.) multivalued mappings with compact values. Let X, Y be two metric spaces. We shall use the following notation: f : X → Y , for singlevalued mappings, and ϕ : X Y , for multivalued mappings. For a subset A ⊂ X and ε > , we let where d is the metric in X. For a singleton x ∈ X, i.e. for A = {x}, we simply put O ε (x).
are the graphs of f and ϕ, respectively.
In the following, in X × Y , we shall consider the max metric.
The following property is self-evident (cf. e.g. []): ). If f : X → Y is an ε-approximation of ϕ : X Y , then we write f ∈ a(ϕ; ε). Now, we shall define the main, from our point of view, class of multivalued mappings.
Let U be an open subset of X and let ∂U denote its boundary. We let We shall employ the following lemma.
For every n, there exists x n ∈ ∂U ∩ Fix(f n ). Since f n ∈ a(ϕ; ε), for every n, we get (x n ,ỹ n ) ∈ ϕ such that But ∂U is a compact set, so we can assume that lim n→∞ x n = x ∈ ∂U. Consequently, in view of (), we obtain lim n→∞xn = x, and lim n→∞ỹn = x. Using the fact that ϕ is u.s.c., we deduce that (x, x) ∈ ϕ and x ∈ ∂U. It means that x ∈ Fix(ϕ) ∩ ∂U, which is a contradiction. Now, we shall define the appropriate notion of homotopy in A U (X).
To indicate how large the class A(X, Y ) is, we let for every x ∈ X, and Y is an ANR-space .
Proposition . (cf. [], Theorem .) Let X be a compact space and Y , Z be arbitrary spaces. If ϕ  : X Y and ϕ  : As a direct consequence of Proposition . and Proposition ., we can give the following corollary which is quite appropriate for the class of multivalued Poincaré operators considered in the two concluding sections.
Open Problem  Is it true that acyclic mappings (cf. [, ]) defined on compact ANRspaces are approximable? Let us note that by the acyclicity, we mean the one in the sense of the Čech homology theory with rational coefficients.
Let X be a compact space and B be a closed subset of X. Assume that r : X → B is a continuous mapping and ϕ ∈ A(B, B).
Proposition . Under the above assumption, we claim that, for each ρ > , there exists The proof of Proposition . is quite analogous to the one in [], Proposition .. (cf. also [], Proposition ..). From Proposition ., we immediately have the following.

Fixed point index for approximable mappings
Let X be a compact ANR and U be an open subset of X. We shall define the fixed point index: where Z is the set of integers.
We put Let us observe that Definition .(b) and Lemma . guarantee that the definition () does not depend on the choice of f ε ∈ a(ϕ; ε).
Observe that from the properties (i) and (v) in Proposition ., we have the following.
Let B ⊂ X also be a compact ANR-space and ϕ ∈ A U (X) be such that ϕ(X) ⊂ B. Let ϕ  : B B be defined by the formula ϕ  (x) = ϕ(x), for every x ∈ B. Then ϕ  ∈ A U∩B (B) and Let X be a compact ANR-space and r : X → B be a retraction map. According to Corollary ., if ϕ ∈ A(B, B), thenφ ∈ A(X, X).
Observe that Consequently Ind(ϕ, V ) and Ind(φ, U) are well defined. Using the excision property (ii) of the fixed point index, we get ()

Essential fixed points
In this section, we assume that all the spaces, under our consideration, are compact. Hence, let (X, d) be a compact space. For a given point x  ∈ X and ε > , we denote by the open and closed balls in X, respectively.
Definition . Let x  be an isolated fixed point of ϕ ∈ A(X, X). We say that x  ∈ X is an essential fixed point of ϕ if, for every ε > , there exists δ = δ(ε) >  such that if f ∈ a(ϕ; δ), Observe that if ϕ is a single-valued mapping, then the essentiality in the sense of Definition . coincides with the one presented in [], Definition .. Concretely, an isolated fixed point x  of a single-valued mapping f : For a given ϕ ∈ A(X, X), we put Ess(ϕ) := x ∈ Fix(ϕ); x is an essential fixed point of ϕ .
We have the following.
Theorem . Let X be a compact ANR-space and ϕ ∈ A(X, X). Assume further that x  ∈ Fix(ϕ) is an isolated fixed point and U be an open subset of X such that x  ∈ U and Fix(ϕ) ∩ ∂U = ∅. Then Ind(ϕ, U) =  implies that x  ∈ Ess(ϕ).
Proof Letting ε  > , we can assume without any loss of generality that O ε  (x  ) ⊂ U, where x  ∈ Fix(ϕ) is an isolated fixed point such that x  ∈ U. From the excision property of the fixed point index, it then follows that Thus, ind(f ε , O ε  (x  )) is well defined. Now, for δ = ε  , we apply condition (b) from Definition ., by which we obtain ε  >  such that, for every  < ε ≤ ε  , all the maps f ε , g ε ∈ a(ϕ; ε) are ε  -homotopic, i.e. there exists a homotopy h : X × [, ] → X, linking f with g such that h t : X → X, h t (x) = h(x, t) belongs to a(ϕ; ε  ).
We can assume that ε  ≤ ε  . Consequently, for every two mappings f ε , g ε ∈ a(ϕ; ε  ), we Finally, it follows from the definition of the fixed point index for ϕ that for every  < ε ≤ ε  . Hence, for every  < ε ≤ ε  and for all f ε ∈ a(ϕ; ε  ), we can deduce that Fix(f ε ) ∩ O ε  (x  ) = ∅, which completes the proof.
Let X be a compact ANR-space and B be a retract of X. In view of the arguments presented in the foregoing section for ϕ ∈ A(B, B), we denote byφ ∈ A(X, X) the map defined by the formulaφ = i • ϕ • r, where r is a retraction map and i is an inclusion.
The following proposition is obvious.

The reverse implication is an open problem.
Let X be a compact ANR-space. We let where dim(·) stands for the topological (Lebesgue covering) dimension (see e.g. []). Letting still ={A ⊂ Fix(ϕ); A is nonempty and compact, and there exists a neighborhood V of A in X such that ∂V ∩ Fix(ϕ) = ∅ and ind(ϕ, V ) = }, observe that is nonempty, because Fix(t) ∈ . Thus, we can take V = X and, by the normalization property of the fixed point index, we get We can consider in the partial ordering given by the inclusion of subsets of X. Now, we shall verify the assumptions of the well-known Kuratowski-Zorn lemma. To do it, let us assume that {A i } i∈J is the chain in . We put To prove that A  ∈ , assume that W is an open neighborhood of A  in X. We claim that there exists i ∈ J such that A i ⊂ W . Otherwise, if we would have assumed, on the contrary, that it is not so, then there is a family B i = (X \ W ) ∩ A i , i ∈ J, of nonempty, compact sets which has nonempty, compact intersection B  . Therefore, B  ⊂ X \ W , together with B  ⊂ A  , which is a contradiction, and subsequently A  ∈ . Thus, in view of the Kuratowski-Zorn lemma, we get a minimal element A * in .
We furthermore claim that A * is a singleton. Let z ∈ A * . It is sufficient to show that {z} ∈ . Since A * ∈ , we obtain an open neighborhood V * of A * with the following properties: Now, from the additivity property of the fixed point index, it follows that by which we arrive (in view of the above properties Ind(ϕ, V * ) =  and Ind(ϕ, U) = ) at Ind(ϕ, V z ) = . This already implies that {z} ∈ and, according to Theorem ., we can conclude that z is an essential fixed point of ϕ. This completes the proof.

Simple examples
At first, we will give two simple illustrative examples of application of the main theorems.

Example . Consider the mapping ϕ  : [, ]
[, ] defined as (see Figure ): Since the graph ϕ  of ϕ  is closed and [, ] is a compact AR-space, ϕ  is obviously an upper semicontinuous map with convex, compact values, i.e. a special case of a J-mapping which is, according to Proposition ., approximable. Hence, in order to apply Theorem ., let us observe that since the interval [,   ] is a set of non-isolated fixed points such that dim Fix(ϕ  ) =  (by which Theorem . cannot be applied here) and since the fixed point   is, in view of Ind(ϕ  , U   ) = , non-essential, we must concentrate on the fixed points   and . Since Fix(ϕ  ) ∩ ∂U   = ∅ and Ind(ϕ  , U   ) =  as well as Fix(ϕ  ) ∩ ∂U  = ∅ and Ind(ϕ  , U  ) = , both fixed points are, according to Theorem ., essential.
Let us note that although the essentiality of  easily follows from the classical results for single-valued maps due to Fort, Jr.
Since  Now, we would like to discuss a possible application of Theorem . and Theorem . to scalar differential equations and inclusions. Hence, consider the scalar differential inclusion where Defining the Poincaré translation operator along the trajectories of (), T ω : R R, namely it is well known (cf. e.g. [], Chapter .) that, unlike in higher dimensions, T n ω ∈ J(R, R), where for all n ∈ N. Furthermore, each fixed point, sayx  ∈ T n ω (x  ), determines an nω-periodic solution x(·) = x(·; ,x  ) of (), because it can be entirely prolongated in an ω-periodic way.
On the other hand, although nω need not be its minimal period, we have proved in [] that if n >  is minimal then, for each m ∈ N, there exists a fixed pointx m ∈ T m ω (x m ) of T m ω , determining a subharmonic mω-periodic solution of () with a minimal period. For n = , the existence of a fixed pointx  ∈ T n ω (x  ), determining a harmonic ω-periodic solution of (), follows already from the generalised Levinson transformation theory (see [] and the references therein).
In our context, condition () and subsequently () imply the existence of a sufficiently large n  ∈ N such that, for every n ≥ n  , In the trivial case of uniqueness, T ω : R → R must be strictly increasing (otherwise, we get a contradiction) by which no purely subharmonic (i.e. those with n > ) nω-periodic solutions can exist. Moreover, this behavior highly increases the chance that Example . Consider the scalar differential inclusion where for |x| > .
Moreover, one can easily check that the differential equation has a unique π -periodic solution, provided only c > ; for more details, see e.g.

Application and concluding remarks
Theorem . Let x ∈ F(t, x) be an ω-periodic dissipative Marchaud system, i.e. let F : [, ω] × R n R n be an upper semicontinuous map with convex, compact values and F(t, x) ≡ F(t + ω, x) such that () is satisfied for all solutions of (). If inclusion () possesses at most a finite number of subharmonic solutions x(·), i.e. those with x(t) ≡ x(t + kω), k ∈ N, then at least one of them exists to be discretely essential. In other words, then for some k  ∈ N there exists an essential fixed pointx  ∈ O D () ⊂ R n of the associated Poincaré operator T k  ω = T k  ω : O D () O D (), determining this discretely essential subharmonic.

Corollary . If a periodic dissipative system of Marchaud inclusions possesses at most a finite number of entirely bounded solutions, then it admits a discretely essential (subharmonic) periodic solution.
Open Problem  Is, under the same assumptions, the conclusion of Theorem ., resp. Corollary ., true for k  = , i.e. is among the existing (cf. []) harmonic solutions at least one to be discretely essential?