Note on essential fixed points of approximable multivalued mappings
 Jan Andres^{1}Email author and
 Lech Górniewicz^{2}
https://doi.org/10.1186/s1366301605686
© Andres and Górniewicz 2016
Received: 25 January 2016
Accepted: 28 June 2016
Published: 15 July 2016
Abstract
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 ANRspaces 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.
Keywords
MSC
1 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 ANRspaces, 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 [1] (cf. also [2], Section 12), where singlevalued maps were exclusively examined for the same goal. On the other hand, unlike in [1, 2], 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 ANRspaces. 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 [3, 4] from the early 1950s. Hence, roughly speaking, the fixed point, say \(x_{0}\), of a given multivalued approximable mapping is essential if any continuous singlevalued map which is sufficiently ‘near’ admits a fixed point in the neighborhood of \(x_{0}\). For a precise formulation, see Definition 5.1 below, and for some further results in this field, see the references in [1]. Let us note that this definition significantly differs from all the other definitions for multivalued maps (see e.g. [5, 6] and the references therein), because it effectively employs the approximability of given multivalued maps on their graphs by singlevalued maps. In this way, topological invariants like a fixed point index can easily be calculated just by means of these singlevalued approximations. This profit naturally connects our approach with the classical theory developed by Fort, Jr. [3] and O’Neil [4].
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 [1]). In [1], we concentrated to essential multivalued fractals considered as fixed points of the induced (singlevalued) HutchinsonBarnsley 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 [1], 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’ singlevalued approximations, but in a numerical reliability at all.
In order to demonstrate the power of the obtained results, three simple illustrative examples are supplied.
2 Preliminaries
Let \(X=(X,d)\) be a metric space. Let us recall that X is an absolute neighborhood retract (written \(X\in ANR\)) if there exist an open set U in a normed space and two singlevalued continuous maps \(r \colon U \to X\) and \(s \colon X \to U\) such that \(r\circ s=\mathrm {id}_{X}\), where \(\mathrm {id}_{X}\) stands for the identity on X.
If U is an arbitrary convex set, then X is called an absolute retract (written \(X\in \operatorname {AR}\)). Evidently: \(\operatorname {AR}\subset \operatorname {ANR}\).
A compact space is called an \(R_{\delta}\) set if it is an intersection of a decreasing sequence of compact ARspaces. For compact sets, in particular: \(\operatorname{convex} \subset \operatorname {AR}\subset R_{\delta}\).
Let \(X=(X,d_{X})\) and \(Y=(Y,d_{Y})\) be metric spaces. By multivalued mappings \(\varphi\colon X\multimap Y\), we understand here those with nonempty, closed values, i.e. that \(\varphi\colon X\to2^{Y}\setminus\{\emptyset\}\) and a closed set \(\varphi (x)\subset Y\) is assigned to every point \(x\in X\). By a fixed point of \(\varphi\colon X \multimap Y\), we mean the point \(x_{0}\in X \cap Y\) such that \(x_{0}\in\varphi(x_{0})\).
A mapping \(\varphi\colon X \multimap Y\) is said to be upper semicontinuous (written u.s.c.) if, for every open \(U\subset Y\), the set \(\varphi^{1}(U):=\{x\in X;\> \varphi(x)\subset U\}\) is open in X or equivalently if, for every closed \(U\subset Y\), the set \(\varphi^{1}_{+}(U):=\{x\in X;\> \varphi(x)\cap U\ne\emptyset\}\) is closed in X.
It is well known that, for every u.s.c. mapping \(\varphi\colon X\multimap Y\), its graph \(\Gamma_{\varphi}:=\{(x,y)\in X\times Y;\> y\in\varphi(x)\}\) is a closed subset of \(X\times Y\). The reverse implication does not hold, in general.
On the other hand, if \(\varphi\colon X\multimap Y\) is such that \(\varphi (X)\subset K\), where \(K\subset Y\) is a compact set and the graph \(\Gamma_{\varphi}\) is closed, then φ is u.s.c.
For u.s.c. maps \(\varphi\colon X\multimap Y\) with compact values, if \(K\subset X\) is compact, then so is \(\varphi(K)\subset Y\).
The composition \(\varphi_{2}\circ\varphi_{1}\colon X\multimap Z\) of two u.s.c. maps \(\varphi_{1}\colon X\multimap Y\) and \(\varphi_{2}\colon Y\multimap Z\) with compact values is again u.s.c. with compact values.
If a u.s.c. mapping φ is singlevalued, i.e. \(\varphi \colon X\to Y\), then it is continuous.
For the proofs and more details as regards ANRspaces and multivalued maps, we recommend [7, 8].
3 Approximable multivalued mappings
In the entire text, all topological spaces are metric and all singlevalued 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: \(f\colon X \to Y\), for singlevalued mappings, and \(\varphi\colon X \multimap Y\), for multivalued mappings.
Definition 3.1
In the following, in \(X\times Y\), we shall consider the max metric.

\(\Gamma_{\varphi}\subset O_{\varepsilon}(\Gamma_{\varphi})\) if and only if, for every \(x\in X\), there is \(f(x)\in O_{\varepsilon} (\varphi(O_{\varepsilon }(x)))\).
If \(f\colon X \to Y\) is an εapproximation of \(\varphi \colon X \multimap Y\), then we write \(f\in a(\varphi;\varepsilon)\).
Now, we shall define the main, from our point of view, class of multivalued mappings.
Definition 3.2
 (a)
for every \(\varepsilon>0\), we have \(a(\varphi ;\varepsilon)\ne\emptyset\),
 (b)
for every \(\delta>0\), there exists \(\varepsilon _{0}>0\) such that, for every ε (\(0<\varepsilon\le\varepsilon_{0}\)), if \(f,g\in a(\varphi;\varepsilon)\), then there exists a homotopy \(h\colon X\times[0,1]\to Y\) linking f and g such that \(h_{t}\in a(\varphi;\delta)\), for every \(t\in[0,1]\), where \(h_{t}(x)=h(x,t)\).
We shall employ the following lemma.
Lemma 3.3
Proof
Let, on the contrary, for every \(\varepsilon>0\), there exist \(f_{\varepsilon}\in a(\varphi;\varepsilon)\) such that \(\operatorname {Fix}(f_{\varepsilon}) \cap\partial U\ne\emptyset\). We put \(\varepsilon_{n}=\frac{1}{n}\), \(n=1,2,\ldots\) , and \(f_{n}\in a(\varphi ;\varepsilon)\).
Now, we shall define the appropriate notion of homotopy in \(A_{U}(X)\).
Definition 3.4
Two maps \(\varphi,\psi\in A_{U}(X)\) are called homotopic (written \(\varphi\sim\psi\)) if there exists \(\chi\in A(X\times[0,1],X)\) such that \(\chi(x,0)=\varphi(x)\) and \(\chi(x,1)=\psi(x)\), for every \(x\in X\), and \(x\notin\chi(x,t)\), for every \(x\in\partial U\) and \(t\in[0,1]\).
We recall the following propositions (see e.g. [9–11]).
Proposition 3.5
(cf. [11], Corollary 3.4)
Proposition 3.6
(cf. [11], Theorem 2.5)
Let X be a compact space and Y, Z be arbitrary spaces. If \(\varphi_{1}\colon X\multimap Y\) and \(\varphi_{2}\colon Y\multimap Z\) are approximable, i.e. if \(\varphi_{1}\in A(X,Y)\) and \(\varphi_{2}\in A(Y,Z)\), then so is \(\varphi_{2}\circ\varphi_{1}\colon X\multimap Z\), i.e. \(\varphi _{2}\circ\varphi_{1}\in A(X,Z)\).
As a direct consequence of Proposition 3.5 and Proposition 3.6, we can give the following corollary which is quite appropriate for the class of multivalued Poincaré operators considered in the two concluding sections.
Corollary 3.7
In particular, if \(\varphi\in J(X,Y)\), where X is a compact ANRspace, then \(f\circ\varphi\in A(X,Z)\), for any singlevalued map \(f\colon Y\to Z\).
Open Problem 1
Is it true that acyclic mappings (cf. [7, 8]) 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 \(r\colon X\to B\) is a continuous mapping and \(\varphi\in A(B,B)\). Then we put \(\tilde{\varphi}=i\circ\varphi\circ r\), where \(i\colon B\to X\) is the inclusion map.
Proposition 3.8
Under the above assumption, we claim that, for each \(\rho>0\), there exists \(\varepsilon_{0}>0\) such that, for any ε (\(0<\varepsilon<\varepsilon_{0}\)), if \(f\in a(\varphi;\varepsilon)\), then \(i\circ f\circ r \in a(\tilde{\varphi};\rho)\).
The proof of Proposition 3.8 is quite analogous to the one in [10], Proposition 3.1.2 (cf. also [8], Proposition 22.3.2). From Proposition 3.8, we immediately have the following.
Corollary 3.9
If \(\varphi\in A(B,B)\), then \(\tilde{\varphi}\in A(X,X)\).
4 Fixed point index for approximable mappings
Let \(\varphi\in A_{U}(X)\). According to Lemma 3.3, we choose \(\varepsilon_{1}>0\). Then we use Definition 3.2(b), for \(\delta =\varepsilon_{1}\), and we get \(\varepsilon_{0}>0\) such that Definition 3.2(b) holds true. Let \(f_{\varepsilon}\in a(\varphi;\varepsilon)\). Thus, in view of Lemma 3.3, we have \(\operatorname {Fix}(f_{\varepsilon}) \cap\partial U=\emptyset\). So the fixed point index \(\operatorname {ind}(f_{\varepsilon},U)\) of \(f_{\varepsilon}\) is well defined (cf. [12]).
Below there are collected the most important properties of the above fixed point index (cf. [8, 10, 13]).
Proposition 4.1
 (i)
(Existence) If \(\operatorname {Ind}(\varphi, U) \neq 0\), then \(\operatorname {Fix}(\varphi) \cap U \neq\emptyset\).
 (ii)
(Excision) If \(\{x\in U;\> x\in\varphi(x)\} \subset V \subset U\), then \(\operatorname {ind}(\varphi, V ) = \operatorname {ind}(\varphi, U)\), where V is an open subset of X.
 (iii)(Additivity) Let \(U_{1}\), \(U_{2}\) be two open subsets of X such that \(U=U_{1} \cup U_{2}\), \(U_{1} \cap U_{2} = \emptyset\) and \(\operatorname {Fix}(\varphi) \cap ( \overline{U}\setminus(U_{1} \cup U_{2}) ) =\emptyset\), then$$\operatorname {Ind}(\varphi, U) = \operatorname {Ind}(\varphi, U_{1}) + \operatorname {Ind}(\varphi, U_{2}). $$
 (iv)
(Homotopy) If \(\varphi, \psi\in A_{U}(X)\) are homotopic, then \(\operatorname {Ind}(\varphi, U) = \operatorname {Ind}(\psi, U)\).
 (v)
(Normalization) If \(\varphi\in A(X,X)\), then \(\lambda(\varphi) = \operatorname {Ind}(\varphi, X)\), where \(\lambda(\varphi)\) stands for the ordinary Lefschetz number for φ.
Observe that from the properties (i) and (v) in Proposition 4.1, we have the following.
Corollary 4.2
If \(\varphi\in A (X,X)\) and \(\lambda(\varphi)\ne0\), then \(\operatorname {Fix}(\varphi)\ne\emptyset\).
Let \(B\subset X\) also be a compact ANRspace and \(\varphi\in A_{U}(X)\) be such that \(\varphi(X)\subset B\). Let \(\varphi_{1}\colon B\multimap B\) be defined by the formula \(\varphi_{1}(x)=\varphi(x)\), for every \(x\in B\). Then \(\varphi_{1}\in A_{U\cap B}(B)\) and \(\operatorname {Ind}(\varphi, U) = \operatorname {Ind}(\varphi_{1}, U\cap B)\).
Let X be a compact ANRspace and \(r\colon X\to B\) be a retraction map. According to Corollary 3.9, if \(\varphi\in A(B,B)\), then \(\tilde{\varphi}\in A(X,X)\).
5 Essential fixed points
Definition 5.1
Let \(x_{0}\) be an isolated fixed point of \(\varphi\in A(X,X)\). We say that \(x_{0}\in X\) is an essential fixed point of φ if, for every \(\varepsilon>0\), there exists \(\delta=\delta(\varepsilon)>0\) such that if \(f\in a(\varphi;\delta)\), then \(\operatorname {Fix}(f)\cap O_{\varepsilon}(x_{0}) \ne\emptyset\).
Observe that if φ is a singlevalued mapping, then the essentiality in the sense of Definition 5.1 coincides with the one presented in [1], Definition 2.1. Concretely, an isolated fixed point \(x_{0}\) of a singlevalued mapping \(f\colon X\to X\) is essential if, for every open εneighborhood \(O_{\varepsilon}(x_{0})\) of \(x_{0}\), there exists \(\delta=\delta(\varepsilon)>0\) such that any map \(g\colon X\to X\) which is ‘δnear’ to f, i.e. \(\sup_{x\in X} d(f(x),g(x))<\delta\), has a fixed point in \(O_{\varepsilon}(x_{0})\).
We have the following.
Theorem 5.2
Let X be a compact ANRspace and \(\varphi\in A(X,X)\). Assume further that \(x_{0}\in \operatorname {Fix}(\varphi)\) is an isolated fixed point and U be an open subset of X such that \(x_{0}\in U\) and \(\operatorname {Fix}(\varphi)\cap\partial U =\emptyset\). Then \(\operatorname {Ind}(\varphi, U)\ne0\) implies that \(x_{0}\in \operatorname {Ess}(\varphi)\).
Proof
Letting \(\varepsilon_{0}>0\), we can assume without any loss of generality that \(O_{\varepsilon_{0}}(x_{0})\subset U\), where \(x_{0}\in \operatorname {Fix}(\varphi)\) is an isolated fixed point such that \(x_{0}\in U\). From the excision property of the fixed point index, it then follows that \(\operatorname {Ind}(\varphi, O_{\varepsilon_{0}}(x_{0}))= \operatorname {Ind}(\varphi, U)\ne0\). Applying Lemma 3.3 we can take \(\varepsilon_{1}>0\) such that if \(f_{\varepsilon}\in a(\varphi;\varepsilon)\), for every \(0<\varepsilon\le\varepsilon_{1}\), then \(\operatorname {Fix}(f_{\varepsilon})\cap\partial O_{\varepsilon_{0}}(x_{0}) =\emptyset\). Thus, \(\operatorname {ind}(f_{\varepsilon}, O_{\varepsilon_{0}}(x_{0}))\) is well defined.
Now, for \(\delta=\varepsilon_{1}\), we apply condition (b) from Definition 3.2, by which we obtain \(\varepsilon_{2}>0\) such that, for every \(0<\varepsilon\le\varepsilon_{2}\), all the maps \(f_{\varepsilon},g_{\varepsilon}\in a(\varphi;\varepsilon)\) are \(\varepsilon_{1}\)homotopic, i.e. there exists a homotopy \(h\colon X\times[0,1]\to X\), linking f with g such that \(h_{t}\colon X\to X\), \(h_{t}(x)=h(x,t)\) belongs to \(a(\varphi;\varepsilon_{1})\).
We can assume that \(\varepsilon_{2}\le\varepsilon_{1}\). Consequently, for every two mappings \(f_{\varepsilon},g_{\varepsilon}\in a(\varphi;\varepsilon_{2})\), we have \(\operatorname {ind}(f_{\varepsilon}, O_{\varepsilon_{1}}(x_{0}))=\operatorname {ind}(g_{\varepsilon}, O_{\varepsilon_{1}}(x_{0}))\), for every \(0<\varepsilon\le\varepsilon_{2}\).
Let X be a compact ANRspace and B be a retract of X. In view of the arguments presented in the foregoing section for \(\varphi\in A(B,B)\), we denote by \(\tilde{\varphi}\in A(X,X)\) the map defined by the formula \(\tilde{\varphi}=i \circ\varphi\circ r\), where r is a retraction map and i is an inclusion.
The following proposition is obvious.
Proposition 5.3
If \(x_{0}\in \operatorname {Ess}(\tilde{\varphi})\), then \(x_{0}\in \operatorname {Ess}(\varphi)\).
The reverse implication is an open problem.
Lemma 5.4
(cf. [14])
Let B be a compact space such that \(\dim B=0\). Then, for every \(x\in B\) and for every \(\varepsilon>0\), there exists an open set \(V\subset O_{\varepsilon}(x)\) such that \(x\in V\) and \(\partial V \cap B =\emptyset\).
Now, we are ready to give the main result of this paper.
Theorem 5.5
Let X be a compact ANRspace and \(\varphi\in A^{0}(X,X)\). If \(\lambda(\varphi)\ne0\), then \(\operatorname {Ess}(\varphi)\ne\emptyset\).
Proof
By the hypothesis \(\lambda(\varphi)\ne0\), the fixed point set \(\operatorname {Fix}(\varphi)\) is nonempty and compact, and \(\dim \operatorname {Fix}(\varphi)=0\). Let \(x_{0}\in \operatorname {Fix}(\varphi)\). In view of Lemma 5.4, there exists an open set \(V\subset O_{\varepsilon}(x_{0})\) such that \(\partial V\cap \operatorname {Fix}(\varphi )=\emptyset\), for every \(\varepsilon>0\).
We can consider in Γ the partial ordering given by the inclusion of subsets of X.
Now, we shall verify the assumptions of the wellknown KuratowskiZorn lemma. To do it, let us assume that \(\{A_{i}\}_{i\in J}\) is the chain in Γ. We put \(A_{0}=\bigcap\{A_{i};i\in J\}\).
To prove that \(A_{0}\in\Gamma\), assume that W is an open neighborhood of \(A_{0}\) in X. We claim that there exists \(i\in J\) such that \(A_{i}\subset W\). Otherwise, if we would have assumed, on the contrary, that it is not so, then there is a family \(B_{i}=(X\setminus W)\cap A_{i}\), \(i\in J\), of nonempty, compact sets which has nonempty, compact intersection \(B_{0}\). Therefore, \(B_{0}\subset X\setminus W\), together with \(B_{0}\subset A_{0}\), which is a contradiction, and subsequently \(A_{0}\in\Gamma\). Thus, in view of the KuratowskiZorn lemma, we get a minimal element \(A_{*}\) in Γ.
We furthermore claim that \(A_{*}\) is a singleton. Let \(z\in A_{*}\). It is sufficient to show that \(\{z\}\in\Gamma\). Since \(A_{*}\in\Gamma\), we obtain an open neighborhood \(V_{*}\) of \(A_{*}\) with the following properties: \(V_{*}\subset V\), \(\partial V_{*} \cap \operatorname {Fix}(\varphi) =\emptyset\) and \(\operatorname {Ind}(\varphi, V_{*})\ne0\).
Let W be an arbitrary open neighborhood of z in X. Applying Lemma 5.4, we can choose an open neighborhood \(V_{z}\) of z in \(V_{*}\cap W\) such that \(\operatorname {Fix}(\varphi)\cap\partial V_{z}=\emptyset\). Since \(A_{*}\) is a minimal element of Γ, the compact set \(A_{*}\setminus V_{z}\) is not in Γ, and so there exists an open set \(U\subset V_{*}\) such that \(( A_{*}\setminus U_{z})\subset U\subset V_{*}\), \(\operatorname {Fix}(\varphi)\cap\partial U =\emptyset\), \(\operatorname {Ind}(\varphi,U)= 0\) and \(\operatorname {Ind}(\varphi, V_{*})=\operatorname {Ind}(\varphi, U\cup V_{z})\).
This already implies that \(\{z\}\in\Gamma\) and, according to Theorem 5.2, we can conclude that z is an essential fixed point of φ. This completes the proof. □
Corollary 5.6
If X is a compact ARspace and \(\varphi\in A^{0}(X,X)\), then \(\operatorname {Ess}(\varphi)\ne\emptyset\).
6 Simple examples
At first, we will give two simple illustrative examples of application of the main theorems.
Example 6.1
Since the graph \(\Gamma_{\varphi_{1}}\) of \(\varphi_{1}\) is closed and \([0,1]\) is a compact ARspace, \(\varphi_{1}\) is obviously an upper semicontinuous map with convex, compact values, i.e. a special case of a Jmapping which is, according to Proposition 3.5, approximable.
Hence, in order to apply Theorem 5.2, let us observe that since the interval \([0,\frac{1}{4}]\) is a set of nonisolated fixed points such that \(\dim \operatorname {Fix}(\varphi_{1})=1\) (by which Theorem 5.5 cannot be applied here) and since the fixed point \(\frac{3}{4}\) is, in view of \(\operatorname {Ind}(\varphi_{1}, U_{\frac{3}{4}})=0\), nonessential, we must concentrate on the fixed points \(\frac{1}{2}\) and 1. Since \(\operatorname {Fix}(\varphi_{1})\cap\partial U_{\frac{1}{2}}=\emptyset\) and \(\operatorname {Ind}(\varphi_{1}, U_{\frac{1}{2}})\ne0\) as well as \(\operatorname {Fix}(\varphi_{1})\cap \partial U_{1}=\emptyset\) and \(\operatorname {Ind}(\varphi_{1}, U_{1})\ne0\), both fixed points are, according to Theorem 5.2, essential.
Let us note that although the essentiality of 1 easily follows from the classical results for singlevalued maps due to Fort, Jr. [3] and O’Neil [4], the appropriate application of Theorem 5.2 concerns the essential fixed point \(\frac{1}{2}\).
Example 6.2
By the same reasoning as in Example 6.1, \(\varphi_{2}\) is obviously an approximable Jmapping. Hence, in order to apply Theorem 5.2, resp. Corollary 5.6, it is enough to realize that \(\lambda(\varphi_{2})=1\) and \(\dim \operatorname {Fix}(\varphi_{2})=0\). Thus, \(\operatorname {Ess}(\varphi_{2})\ne\emptyset\).
Since \(\operatorname {Ind}(\varphi_{2}, U_{\frac{1}{4}})=\operatorname {Ind}(\varphi_{2}, U_{\frac{1}{2}}) =\operatorname {Ind}(\varphi_{2}, U_{\frac{3}{4}})=\operatorname {Ind}(\varphi_{2}, U_{1})= 0\), there is (in view of Theorem 5.2) the only essential fixed point 0. Let us note that, because of a multivalued character of \(\varphi_{2}\), the essentiality of 0 cannot be this time deduced by the local application of classical results due to Fort, Jr. [3] and O’Neil [4]. On the other hand, the same easily follows from the locally applied Theorem 5.2.
Let us therefore give the last simple illustrative related example.
Example 6.3
Since condition (9) can easily be verified for \(F(t,x):=F_{0}(x)cx+\cos t\), on \(x\geq D\), where \(D\in(\frac{1}{c},1)\), and \(F(t,x)\equiv F(t+2\pi,x)\), \(F(t,x)\equivF(t+\pi,x)\), the associated Poincaré translation operator along the trajectories of (10), \(T_{2\pi}\colon \mathbb {R}\multimap \mathbb {R}\), satisfies \(T_{2\pi} _{[1,1]}: [1,1]\multimap(1,1)\). Observe that \(T_{2\pi} _{(1,1)}\) is even singlevalued, because arctanx is, for \(x\leq1\), Lipschitzian with constant \(L=1\) as well as cx with constant \(c>1\).
Because of an evident onetoone correspondence between the fixed points \(\bar{x}_{0}=T_{2\pi} _{(1,1)}(\bar{x}_{0})=T_{2\pi} _{[1,1]}(\bar{x}_{0})\) of \(T_{2\pi} _{[1,1]}\) and 2πperiodic solutions \(x(\cdot;0, \bar{x}_{0})\equiv x(\cdot+2\pi; 0, \bar{x}_{0})\) of (11) as well as (10), the unique fixed point \(\bar{x}_{0}\) must be, in view of \(\dim \operatorname {Fix}(T_{2\pi} _{[1,1]} )=0\), essential by means of Corollary 5.6. The determined 2πperiodic solution \(x(\cdot;0, \bar{x}_{0})\) of (10) can be therefore called discretely essential.
Remark 6.4
Since \(F(t,x)\equivF(t+\pi,x)\), we can still prove by means of the modified operator \(\tilde{T}_{\pi} _{[1,1]}=T_{\pi} _{[1,1]}\colon [1,1]\multimap(1,1)\) that the unique discretely essential 2πperiodic solution \(x(\cdot;0, \bar{x}_{0})\equiv x(\cdot+2\pi;0, \bar{x}_{0})\) of (10) in Example 6.3 is πantiperiodic, i.e. \(x(\cdot;0, \bar{x}_{0})\equivx(\cdot +\pi;0, \bar{x}_{0})\).
7 Application and concluding remarks
Nevertheless, we can finally state as a theorem at least a particular case of it.
Theorem 7.1
Let \(x'\in F(t,x)\) be an ωperiodic dissipative Marchaud system, i.e. let \(F\colon[0,\omega]\times \mathbb {R}^{n}\multimap \mathbb {R}^{n}\) be an upper semicontinuous map with convex, compact values and \(F(t,x)\equiv F(t+\omega,x)\) such that (7) is satisfied for all solutions of (6). If inclusion (6) possesses at most a finite number of subharmonic solutions \(x(\cdot)\), i.e. those with \(x(t)\equiv x(t+k\omega )\), \(k\in \mathbb {N}\), then at least one of them exists to be discretely essential. In other words, then for some \(k_{0}\in \mathbb {N}\) there exists an essential fixed point \(\bar{x}_{0}\in O_{D}(0)\subset \mathbb {R}^{n}\) of the associated Poincaré operator \(T_{k_{0}\omega}=T_{\omega}^{k_{0}}\colon\overline{O_{D}(0)}\multimap O_{D}(0)\), determining this discretely essential subharmonic.
Corollary 7.2
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 2
Is, under the same assumptions, the conclusion of Theorem 7.1, resp. Corollary 7.2, true for \(k_{0}=1\), i.e. is among the existing (cf. [17]) harmonic solutions at least one to be discretely essential?
Remark 7.3
For the same conclusion of Theorem 7.1, resp. Corollary 7.2, the Marchaud maps F can be replaced by more general upper Carathéodory maps with convex, compact values. For their definition and more details, see e.g. [7, 8]
Remark 7.4
Declarations
Acknowledgements
The first author was supported by the grant No. 1406958S ‘Singularities and impulses in boundary value problems for nonlinear ordinary differential equations’ of the Grant Agency of the Czech Republic.
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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