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

### Example 6.1

Consider the mapping \(\varphi_{1}\colon[0,1]\multimap[0,1]\) defined as (see Figure 1):

$$\varphi_{1}(x):= \textstyle\begin{cases} [0,\frac{1}{4}],& \mbox{for } x\in[0,\frac{1}{4}],\\ \frac{1}{4},& \mbox{for } x\in(\frac{1}{4},\frac{1}{2}),\\ [\frac{1}{4},\frac{3}{4}],& \mbox{for } x=\frac{1}{2},\\ \frac{3}{4},& \mbox{for } x\in(\frac{1}{2},\frac{3}{4}),\\ [\frac{3}{4},1],& \mbox{for } x=\frac{3}{4},\\ 1,& \mbox{for } x\in(\frac{3}{4},1]. \end{cases} $$

Since the graph \(\Gamma_{\varphi_{1}}\) of \(\varphi_{1}\) is closed and \([0,1]\) is a compact AR-space, \(\varphi_{1}\) is obviously an upper semicontinuous map with convex, compact values, *i.e.* a special case of a *J*-mapping 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 non-isolated 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 \(\operatorname {Ind}(\varphi_{1}, U_{\frac{3}{4}})=0\), non-essential, 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 single-valued 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

Consider the mapping \(\varphi_{2}\colon[0,1]\multimap[0,1]\) defined as (see Figure 2):

$$\varphi_{2}(x):= \textstyle\begin{cases} [0,\frac{1}{4}],& \mbox{for } x=0,\frac{1}{4},\\ 0,& \mbox{for } x\in(0,\frac{1}{4}),\\ \frac{1}{4},& \mbox{for } x\in(\frac{1}{4},\frac{1}{2}),\\ [\frac{1}{4},\frac{1}{2}],& \mbox{for } x=\frac{1}{2},\\ \frac{1}{2},& \mbox{for } x\in(\frac{1}{2},\frac{3}{4}),\\ [\frac{1}{2},\frac{3}{4}],& \mbox{for } x=\frac{3}{4},\\ \frac{3}{4},& \mbox{for } x\in(\frac{3}{4},1),\\ [\frac{3}{4},1],& \mbox{for } x=1. \end{cases} $$

By the same reasoning as in Example 6.1, \(\varphi_{2}\) is obviously an approximable *J*-mapping. 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.

Now, we would like to discuss a possible application of Theorem 5.2 and Theorem 5.5 to scalar differential equations and inclusions. Hence, consider the scalar differential inclusion

where \(F(t,x)\equiv F(t+\omega,x)\), for some \(\omega>0\), and assume that \(F\colon[0,\omega]\times \mathbb {R}\multimap \mathbb {R}\) is an upper semicontinuous mapping with convex, compact values, *i.e.*
*F* to be a Marchaud mapping. Let, furthermore, (6) be *dissipative in the sense of Levinson*, *i.e.*

$$ \exists D>0\colon\limsup_{t\to\infty} \bigl|x(t)\bigr|< D, \mbox{ for all solutions }x(\cdot)\mbox{ of }(6). $$

(7)

This already implies (*cf.* [15], pp.63-64) that, under \(F(t,x)\equiv F(t+\omega,x)\), (6) is *uniformly dissipative*, *i.e.*

$$ \forall D_{0}>0 \exists\Delta t>0, D>0\colon \bigl(t_{0}\in \mathbb {R},\>|x_{0}|< D_{0},\>t\geq t_{0}+\Delta t\bigr) \Rightarrow\bigl|x(t)\bigr|< D, $$

(8)

for all solutions \(x(\cdot)=x(\cdot;t_{0},x_{0})\), satisfying \(x(t_{0};t_{0},x_{0})=x_{0}\).

Defining the *Poincaré translation operator* along the trajectories of (6), \(T_{\omega}\colon\mathbb{R}\multimap\mathbb{R}\), namely

$$T_{\omega}(x_{0}):=\bigl\{ x(\omega;0,x_{0});\> x( \cdot;0,x_{0}) \mbox{ is a solution of } (6) \mbox{ with } x(0;0,x_{0})=x_{0}\bigr\} , $$

it is well known (*cf.*
*e.g.* [7], Chapter 3.4) that, unlike in higher dimensions, \(T_{\omega}^{n}\in J(\mathbb {R},\mathbb {R})\), where

$$T_{\omega}^{n}=\underbrace{T_{\omega}\circ\cdots\circ T_{\omega}}_{n\mbox{-}\mathrm{times}}= T_{n\omega}, $$

for all \(n\in \mathbb {N}\). Furthermore, each fixed point, say \(\bar{x}_{0}\in T_{\omega}^{n} (\bar{x}_{0})\), determines an *nω*-periodic solution \(x(\cdot)=x(\cdot;0,\bar{x}_{0})\) of (6), 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 [16] that if \(n>1\) is minimal then, for each \(m\in \mathbb {N}\), there exists a fixed point \(\bar{x}_{m}\in T_{\omega}^{m} (\bar{x}_{m})\) of \(T_{\omega}^{m}\), determining a subharmonic *mω*-periodic solution of (6) with a minimal period. For \(n=1\), the existence of a fixed point \(\bar{x}_{1}\in T_{\omega}^{n} (\bar{x}_{1})\), determining a harmonic *ω*-periodic solution of (6), follows already from the generalised Levinson transformation theory (see [17] and the references therein).

In our context, condition (7) and subsequently (8) imply the existence of a sufficiently large \(n_{0}\in \mathbb {N}\) such that, for every \(n\geq n_{0}\), \(T_{\omega}^{n} |_{[-D,D]}\in J([-D,D],[-D,D])\). Thus, a fixed point of \(T_{\omega}^{n} |_{[-D,D]}\colon[-D,D]\multimap[-D,D]\) determines an *nω*-periodic solution for (6), but in order this fixed point to be essential, we need to satisfy the assumptions of Theorem 5.2 or Corollary 5.6. In the latter case, it would mean to suppose still that \(\dim \operatorname {Fix}(T_{\omega}^{n} |_{[-D,D]} )=\dim \operatorname {Fix}(T_{n\omega } |_{[-D,D]} )=0\), which seems to be rather difficult to verify in general, because generically \(\dim \operatorname {Fix}(T_{\omega}^{n} |_{[-D,D]} )=1\) (*cf.*
*e.g.* [18, 19]), for multivalued maps \(T_{\omega}\) in \(\mathbb {R}\), *i.e.* in the lack of uniquely solvable Cauchy (initial value) problems for (6). If additionally

$$ F(t,x)\operatorname {sgn}x < 0,\quad \mbox{for }|x|\geq D, $$

(9)

then the interval \([-D,D]\) is obviously positively invariant under \(T_{\omega}|_{[-D,D]}\), by which only \(\dim \operatorname {Fix}(T_{\omega}|_{[-D,D]} )=0\) should be verified, but the same obstruction remains there again.

In the trivial case of uniqueness, \(T_{\omega}\colon \mathbb {R}\to \mathbb {R}\) must be strictly increasing (otherwise, we get a contradiction) by which no purely subharmonic (*i.e.* those with \(n>1\)) *nω*-periodic solutions can exist. Moreover, this behavior highly increases the chance that

$$\dim \operatorname {Fix}\bigl(T_{\omega}^{n} |_{[-D,D]} \bigr)=\dim \operatorname {Fix}(T_{\omega}|_{[-D,D]} )=0 $$

holds, even without (9).

Let us therefore give the last simple illustrative related example.

### Example 6.3

Consider the scalar differential inclusion

$$ x'+cx\in-F_{0}(x)+\cos t,\quad \mbox{with }c>1, $$

(10)

where

$$F_{0}(x):= \textstyle\begin{cases} \arctan x,& \mbox{for } |x|< 1,\\ \frac{\pi}{4}+[-\frac{\pi}{4},0],& \mbox{for } x=1,\\ -\frac{\pi}{4}+[0,\frac{\pi}{4}],& \mbox{for } x=-1,\\ 0,& \mbox{for } |x|>1. \end{cases} $$

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 \(F(t,x)\equiv F(t+2\pi,x)\), \(F(t,x)\equiv-F(t+\pi,-x)\), 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 single-valued, because arctan*x* is, for \(|x|\leq1\), Lipschitzian with constant \(L=1\) as well as *cx* with constant \(c>1\).

Moreover, one can easily check that the differential equation

$$ x'+cx=-\arctan x+\cos t $$

(11)

has a unique 2*π*-periodic solution, provided only \(c>1\); for more details, see *e.g.* [7], pp.354-355.

Because of an evident one-to-one 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 2*π*-periodic 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 2*π*-periodic solution \(x(\cdot;0, \bar{x}_{0})\) of (10) can be therefore called *discretely essential*.

### Remark 6.4

Since \(F(t,x)\equiv-F(t+\pi,-x)\), 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 2*π*-periodic solution \(x(\cdot;0, \bar{x}_{0})\equiv x(\cdot+2\pi;0, \bar{x}_{0})\) of (10) in Example 6.3 is *π*-antiperiodic, *i.e.*
\(x(\cdot;0, \bar{x}_{0})\equiv-x(\cdot +\pi;0, \bar{x}_{0})\).