A Filippov-Type Lemma for Stieltjes Differential Inclusions with Supremum

Abstract

The aim of this paper is to generalize the classical Filippov lemma to the framework of Cauchy differential set-valued problems involving the supremum of the unknown function on a past interval and its Stieltjes derivative with respect to a left-continuous non-decreasing function. To highlight the wide spectrum of the result (coming from the fact that this setting covers the theories of ordinary differential or difference problems, impulsive problems or problems on time scales), as a consequence, a Filippov-type lemma for dynamic inclusions on time scales with supremum is obtained.

1. Introduction

In the wide class of differential equations, a distinct category consists of those equations where the state at present time depends on the past evolution of the system (the so-called functional differential equations); in this class, one can identify the situation where the maximum of the state on a past interval of time is involved, namely the differential equations with maxima. Such problems have been studied for a long time, starting with an example in the automatic control of some technical systems in Popov [1], and their study includes various qualitative properties, such as oscillatory properties, stability, periodicity, or approximation for differential, difference, or integral equations (see Bainov, Hristova [2], Cernea [3] or Otrocol, Rus [4] for a comprehensive discussion of the topic).

Additionally, the theory of differential equations where the usual derivative is replaced by the Stieltjes derivative with respect to a left-continuous non-decreasing function g (Pouso, Rodriguez [5]) significantly grew in the recent few years (e.g., Frigon, Pouso [6], Monteiro, Satco [7], Marraffa, Satco [8], Pouso, Márquez Albés [9], Satco, Smyrlis [10,11]) due to its interesting real-world applications (e.g., Fernandez et al. [12], Frigon, Pouso [6], Pouso, Márquez Albés [9]).

This growth is also motivated by the fact that Stieltjes differential equations are deeply related to measure differential problems (Cao, Sun [13], Di Piazza et al. [14], Federson et al. [15], Monteiro, Satco [16], Pouso, Rodriguez [5], Satco [17]); thus, their study allows one to cover usual differential problems (for an absolutely continuous map g), discrete problems (when g is a sum of step functions), impulsive equations (for g being a sum of an absolutely continuous map with step maps), dynamic equations on time scales (Federson et al. [15]), and generalized differential equations (see Slavík [18], Kurzweil [19], Schwabik [20]).

Additionally, the classical Filippov lemma (Aubin, Cellina [21]) allows one to find a solution of a differential inclusion which satisfies a specific estimation comparing to a given “quasi-solution”. It was generalized in many directions (to the case of impulsive functional inclusions in Djebali et al. [22], to general topologic and measure spaces in Borisenko [23]—see also Tolstonogov [24]—to measure differential problems in Fryszkowski, Sadowski [25], to Stieltjes differential inclusions in Satco, Smyrlis [11] and to inclusions with maxima in Cernea [3], to give only a few examples).

In this work, having in mind the major implications of Filippov-type lemmas in obtaining qualitative features of differential problems (for instance, to relaxation properties, e.g., Marraffa, Satco [26], Cernea [3]), a Filippov-type result is obtained for differential inclusions with supremum driven by the Stieltjes derivative:

$$\left\{\begin{array}{c}{x}_g^{\prime }\left(t\right)\in F\left(t,x\left(t\right),\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}x\left(s\right)\right),{\mu }_g-a.e.t\in [0,1),\hfill \\ \\ x\left(0\right)=x^0;\hfill \end{array}\right.$$

(1)

here, $g:[0,1]\to \mathbb{R}$ is non-decreasing and left-continuous, and $x^0\in \mathbb{R}$ and $\left[\alpha \right(t),\beta (t\left)\right]\subset [0,t]$ for every $t\in [0,1]$.

If g is continuous, then the supremum is attained, and one can speak about differential problems with maxima. On the other hand, as we shall see by an example, the supremum is not necessarily a maximum when g is discontinuous.

As a corollary, the first existence result for Lipschitz differential inclusions with supremum is deduced in this wide setting of differential problems driven by Stieltjes derivative.

Notably, Stieltjes differential inclusions with a constant (resp. variable) delay can be thus investigated by taking $\alpha \left(t\right)=\beta \left(t\right)=t-h$ with h a non-negative constant (respectively, $\alpha \left(t\right)=\beta \left(t\right)\le t$).

Moreover, the set-valued framework is taken into consideration (more general than the single-valued setting of differential equations), where differential problems with maxima or supremum were studied, as far as the authors know, only in the case where g is the identity map (e.g., Cernea [3]) or when the fractional derivative is considered (Cernea [27]).

Dynamic problems on time scales can be seen as Stieltjes differential problems (Frigon, Pouso [6]); therefore, an application to dynamic inclusions on time scales with supremum is also included. It is important to note that the obtained Filippov-type lemma is new (comparing to the only such result, given in Rafaqat et al. [28]) even in the case where the supremum of the map is not present in the multifunction on the right-hand side.

As a final remark concerning the generality of the outcome, it has to be mentioned that several known results are covered by taking F a single-valued map (for usual differential problems with maxima Bainov, Hristova [2] or impulsive equations supremum Hristova, Georgieva [29]), or by particularizing $\alpha$ and $\beta$ (for instance, ([3], Theorem 1) can be inferred when $g\left(t\right)=t\in [0,1]$, $\alpha \left(t\right)=0$ and $\beta \left(t\right)=t$).

2. Notations and Auxiliary Results

Let $g:[0,1]\to \mathbb{R}$ be a left-continuous non-decreasing function. The measurability with respect to (w.r.t.) the $\sigma$-algebra defined by g will be called g-measurability, and the Stieltjes measure generated by $g$ will be denoted by ${\mu }_g$ (see ([30], Example 6.14) or ([31], paragraph 1.5)).

The Lebesgue–Stieltjes (LS-) integrability w.r.t. g ([31]; also see [32]) represents the abstract Lebesgue integrability w.r.t. the Stieltjes measure ${\mu }_g$. Let $L_g^1\left([0,1]\right)$ be the space of real functions LS-integrable w.r.t. g, endowed with the usual topology given by the norm

$${\parallel f\parallel }_g^1={\int }_{[0,1)}\left|f\left(s\right)\right|d{\mu }_g\left(s\right),\mathrm{whenever}f\in L_g^1\left([0,1]\right).$$

The g-topology on $[0,1]$ is the topology with the basis of the class of all sets

$$V_{g,\delta }\left(t\right)=\{t^{\prime }\in [0,1];\mid g\left(t^{\prime }\right)-g\left(t\right)\mid <\delta \},\delta >0,t\in [0,1].$$

Definition 1. 

A function $f:[0,1]\to \mathbb{R}$ is said to be g-continuous ([6]) if it is continuous w.r.t. the g-topology, i.e., if for any $t\in [0,1]$ and any $\epsilon >0$ there exists ${\delta }_{t,\epsilon }>0$ such that

$$\mid g\left(t\right)-g\left(t^{\prime }\right)\mid <{\delta }_{t,\epsilon }\Rightarrow |f\left(t\right)-f\left(t^{\prime }\right)|<\epsilon .$$

In [6], it was proved that a g-continuous function might not have the right limit $f(t+)$ at the discontinuity points of g and it might not be even bounded. For this reason, the space ${\mathcal{BC}}_g\left([0,1]\right)$ of real bounded, g-continuous functions was considered ([6]). It was proved that, endowed with the norm

$${\parallel f\parallel }_C=\underset{t\in [0,1]}{sup}\left|f\left(t\right)\right|$$

it becomes a Banach space.

Unlike for maps continuous in the usual topology, the supremum might not be necessarily a maximum for g-continuous functions, as it can be seen in the following example.

Example 1. 

Let $g:[0,1]\to \mathbb{R}$ be given by

$$g\left(t\right)=\left\{\begin{array}{c}t,\mathrm{if}t\in \left[0,{\displaystyle \frac{1}{2}}\right],\hfill \\ \\ t+1,\mathrm{if}t\in \left({\displaystyle \frac{1}{2}},1\right]\hfill \end{array}\right.$$

and $f:[0,1]\to \mathbb{R}$ by

$$f\left(t\right)=\left\{\begin{array}{c}t,\mathrm{if}t\in \left[0,{\displaystyle \frac{1}{2}}\right],\hfill \\ \\ -t-1,\mathrm{if}t\in \left({\displaystyle \frac{1}{2}},1\right].\hfill \end{array}\right.$$

It can be seen that f is bounded and for every $\epsilon >0$ there exists ${\delta }_{\epsilon }=min\left(\epsilon ,{\displaystyle \frac{1}{2}}\right)$, such that whenever $t^{\prime },t^{″}\in [0,1]$ satisfy $|g\left(t^{\prime }\right)-g\left(t^{″}\right)|<{\delta }_{\epsilon }$, one has $|f\left(t^{\prime }\right)-f\left(t^{″}\right)|<\epsilon$. This implies that f is g-continuous. Yet, obviously, f does not attain its maximum on the interval $\left[{\displaystyle \frac{1}{2}},1\right]$.

A notion of differentiability with respect to a left-continuous non-decreasing function introduced in [5] by an idea in [33] will be used in what follows.

Definition 2. 

The Stieltjes (or g-) derivative of a function $f:[0,1]\to \mathbb{R}$ with respect to g at a point $t\in [0,1]$ is

$$\begin{array}{ccc}\hfill f_g^{\prime }\left(t\right)& =& \underset{t^{\prime }\to t}{lim}{\displaystyle \frac{f\left(t^{\prime }\right)-f\left(t\right)}{g\left(t^{\prime }\right)-g\left(t\right)}}if g is continuous at t,\hfill \\ \hfill f_g^{\prime }\left(t\right)& =& \underset{t^{\prime }\to t+}{lim}{\displaystyle \frac{f\left(t^{\prime }\right)-f\left(t\right)}{g\left(t^{\prime }\right)-g\left(t\right)}}otherwise,\hfill \end{array}$$

if the limits exist. The formula is meaningless on the set

$$\{t\in (0,1):g \mathrm{is} \mathrm{constant} \mathrm{on} (t-\eta ,t+\eta ) \mathrm{for} \mathrm{some} \eta >0\}$$

which is, by [5], ${\mu }_g$-null (it is worthwhile to mention that in the same way as in ([12], Definition 3.7), the definition of the g-derivative can be extended to cover also the points of this set).

The particular cases of the theory of Stieltjes differential equations are obtained by taking particular functions g. For instance, when $g\left(t\right)=t$, one gets ordinary differential equations. In the case where g is a sum of step functions, one finds difference equations. For the case where g is a sum of the identical function with step functions, impulsive problems are obtained problems. So, the advantage of using Stieltjes derivatives is that they allow one to include a wide class of problems in one study.

This notion of derivatives has intensively been used in solving various problems where abrupt modifications (corresponding to discontinuity points of g) and stationary intervals of time (corresponding to intervals where g is constant) occur, such as [6,34] or [9].

The properties to be imposed on the solutions of the considered problem are described below.

Definition 3. 

A function $f:[0,1]\to \mathbb{R}$ is called g-absolutely continuous (g-AC; see [5]) if for every $\epsilon >0$ there is ${\delta }_{\epsilon }>0$ such that for any collection ${\left\{(t_j^{\prime },t_j^{″})\right\}}_{i=1}^m$ of non-overlapping subintervals of $[0,1]$

$$\sum _{j=1}^m(g\left(t_j^{″}\right)-g\left(t_j^{\prime }\right))<{\delta }_{\epsilon }\Rightarrow \sum _{j=1}^m|f\left(t_j^{″}\right)-f\left(t_j^{\prime }\right)|<\epsilon .$$

Obviously, the g-absolute continuity involves g-continuity.

The fundamental theorem of calculus for LS-integrals offers a characterization of g-AC maps in terms similar to usual absolutely continuous maps.

Theorem 1. 

([5], Theorem 5.4) A map $f:[0,1]\to \mathbb{R}$ is g-absolutely continuous if and only if it is g-differentiable ${\mu }_g$-a.e., $f_g^{\prime }$ is LS-integrable w.r.t. g and

$$f\left(t^{\prime }\right)=f\left(t^{″}\right)+{\int }_{[t^{″},t^{\prime })}{f}_g^{\prime }\left(s\right)d{\mu }_g\left(s\right),\mathrm{for}\mathrm{every}0\le t^{″}<t^{\prime }\le 1.$$

For elementary notions of set-valued analysis, the reader is referred to [21,35] or [36]. ${\mathcal{P}}_k\left(\mathbb{R}\right)$ will denote the family of nonempty compact subsets of $\mathbb{R}$. If $A,C\in {\mathcal{P}}_k\left(\mathbb{R}\right)$, the Hausdorff–Pompeiu metric is defined by

$$D(A,C)=max\left(e\right(A,C),e(C,A\left)\right),$$

where the (Pompeiu) excess of the set A over the set C is

$$e(A,C)=\underset{a\in A}{sup}\underset{c\in C}{inf}|a-c|.$$

3. Main Results

Let $g:[0,1]\to \mathbb{R}$ be non-decreasing and left-continuous; thus, $x^0\in \mathbb{R}$ and the g-measurable functions $\alpha ,\beta :[0,1]\to [0,1]$ satisfy $\left[\alpha \right(t),\beta (t\left)\right]\subset [0,t]$ for every $t\in [0,1]$.

This section aims to provide a Filippov-type result for the problem $\left(1\right)$.

In the same way it was carried out in ([4], Lemma 3.1) for continuous maps, the following estimation can be proved.

Lemma 1. 

For every $x,y\in {\mathcal{BC}}_g\left([0,1]\right)$,

$$\left|\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}x\left(s\right)-\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}y\left(s\right)\right|\le \underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}|x\left(s\right)-y\left(s\right)|,\forall t\in [0,1].$$

Proof. 

The assertion easily follows from the inequality

$$\begin{array}{c}\hfill x\left(s\right)=x\left(s\right)-y\left(s\right)+y\left(s\right)\le \left|x\right(s)-y(s\left)\right|+y\left(s\right)\end{array}$$

by passing to the following supremum:

$$\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}x\left(s\right)\le \underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}|x\left(s\right)-y\left(s\right)|+\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}y\left(s\right).$$

□

The main result of this paper is presented in what follows.

Theorem 2. 

Let $F:[0,1]\times \mathbb{R}\times \mathbb{R}\to {\mathcal{P}}_k\left(\mathbb{R}\right)$ have the following properties:

• For every $x,y\in \mathbb{R}$, $t\mapsto F(t,x,y)$ is g-measurable;
• There exist $L_1,L_2\in L_g^1\left([0,1]\right)$ such that for ${\mu }_g$-a.e. $t\in [0,1]$ and for all $x_1,x_2,y_1,y_2\in \mathbb{R}$,

  $$D(F(t,x_1,y_1),F(t,x_2,y_2))\le L_1\left(t\right)\left|x_1-x_2\right|+L_2\left(t\right)\left|y_1-y_2\right|.$$

Then, given a g-absolutely continuous function $x:[0,1]\to \mathbb{R}$, if $\gamma \in L_g^1\left([0,1]\right)$ satisfies

$$d\left(x_g^{\prime }\left(t\right),F\left(t,x\left(t\right),\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}x\left(s\right)\right)\right)\le \gamma \left(t\right),{\mu }_g-a.e.,$$

there exists a solution $\tilde{x}$ of $\left(1\right)$ such that

$$\left|\tilde{x}\left(t\right)-x\left(t\right)\right|\le \left(\left|x^0-x\left(0\right)\right|+{\int }_{[0,t)}\gamma \left(s\right)d{\mu }_g\left(s\right)\right)e^{{\int }_{[0,t)}{L}_1\left(s\right)+L_2\left(s\right)d{\mu }_g\left(s\right)}.$$

Proof. 

The well-known Filippov method of proof will be adapted to the present setting, as in [25]. A Cauchy sequence ${\left(f_n\right)}_n\subset L_g^1\left([0,1]\right)$ will be defined along with a corresponding sequence ${\left(x_n\right)}_n$ of g-absolutely continuous functions, which will bring us to a solution of $\left(1\right)$ with the required property.

Thus, let $x_0:[0,1]\to \mathbb{R}$ be given by $x_0\left(t\right)=x\left(t\right)$ and

$$f_0:[0,1]\to \mathbb{R},f_0\left(t\right)=\left\{\begin{array}{c}{x}_g^{\prime }\left(t\right),\mathrm{if}\mathrm{it}\mathrm{exists},\hfill \\ \\ 0,\mathrm{otherwise}\hfill \end{array}\right.$$

(it is known that $x_g^{\prime }$ exists ${\mu }_g$-a.e. by Theorem 1 and that it belongs to $L_g^1\left([0,1]\right)$).

By hypothesis,

$$d\left(f_0\left(t\right),F\left(t,x_0\left(t\right),\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}{x}_0\left(s\right)\right)\right)\le \gamma \left(t\right),{\mu }_g-a.e.;$$

thus, by the classical Kuratowski–Ryll–Nardzewski theorem, one can choose a g-measurable selection $f_1$ of the g-measurable ([36], Chapter III) closed-valued multifunction

$$F\left(·,x_0(·),\underset{s\in \left[\alpha \right(·),\beta (·\left)\right]}{sup}{x}_0\left(s\right)\right)\cap \left(f_0(·)+\gamma (·)[-1,1]\right),$$

i.e.,

$$|f_1\left(t\right)-f_0\left(t\right)|\le \gamma \left(t\right),{\mu }_g-a.e.$$

In particular, this implies that $f_1\in L_g^1([0,1]$.

Now, denote by $x_1:[0,1]\to \mathbb{R}$

$$x_1\left(t\right)=x^0+{\int }_{[0,t)}{f}_1\left(s\right)d{\mu }_g\left(s\right),t\in [0,1];$$

it can be seen that

$$\begin{array}{ccc}\hfill |x_1\left(t\right)-x_0\left(t\right)|& \le & \left|x\left(0\right)-x^0\right|+\left|{\int }_{[0,t)}{f}_1\left(s\right)-f_0\left(s\right)d{\mu }_g\left(s\right)\right|\hfill \\ & \le & \left|x\left(0\right)-x^0\right|+{\int }_{[0,t)}\gamma \left(s\right)d{\mu }_g\left(s\right),\forall t\in [0,1]\hfill \end{array}$$

and, using Lemma 1,

$$\begin{array}{ccc}& & d\left(f_1\left(t\right),F\left(t,x_1\left(t\right),\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}{x}_1\left(s\right)\right)\right)\hfill \\ & & \le D\left(F\left(t,x_0\left(t\right),\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}{x}_0\left(s\right)\right),F\left(t,x_1\left(t\right),\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}{x}_1\left(s\right)\right)\right)\hfill \\ & & \le L_1\left(t\right)|x_1\left(t\right)-x_0\left(t\right)|+L_2\left(t\right)\left|\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}{x}_0\left(s\right)-\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}{x}_1\left(s\right)\right|\hfill \\ & & \le L_1\left(t\right)|x_1\left(t\right)-x_0\left(t\right)|+L_2\left(t\right)\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}|x_0\left(s\right)-x_1\left(s\right)|\hfill \\ & & \le (L_1\left(t\right)+L_2\left(t\right))\left(|x\left(0\right)-x^0|+{\int }_{[0,t)}\gamma \left(s\right)d{\mu }_g\left(s\right)\right).\hfill \end{array}$$

Next, the right-hand side of the inequality is g-measurable (in fact, it belongs to $L_g^1([0,1]$), so if

$${\gamma }_1\left(t\right)=(L_1\left(t\right)+L_2\left(t\right))\left(|x\left(0\right)-x^0|+{\int }_{[0,t)}\gamma \left(s\right)d{\mu }_g\left(s\right)\right),$$

again by Kuratowski-Ryll-Nardzewski theorem, one can find a g-measurable selection $f_2$ of the nonempty closed-valued g-measurable multifunction

$$\begin{array}{c}\hfill t\mapsto F\left(t,x_1\left(t\right),\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}{x}_1\left(s\right)\right)\cap \left(f_1\left(t\right)+{\gamma }_1\left(t\right)[-1,1]\right).\end{array}$$

This implies that

$$|f_1\left(t\right)-f_2\left(t\right)|\le {\gamma }_1\left(t\right),{\mu }_g-a.e.$$

(so, $f_2\in L_g^1([0,1]$) and we define $x_2:[0,1]\to \mathbb{R}$ by $x_2\left(t\right)=x^0+{\int }_{[0,t)}{f}_2\left(s\right)d{\mu }_g\left(s\right)$, for each $t\in [0,1]$. It follows that

$$|x_2\left(t\right)-x_1\left(t\right)|\le {\int }_{[0,t)}{\gamma }_1\left(s\right)d{\mu }_g\left(s\right).$$

One proceeds further in the same way to obtain a selection $f_n\in L_g^1\left([0,1]\right)$ of

$$\begin{array}{c}\hfill F\left(·,x_{n-1}(·),\underset{s\in \left[\alpha \right(·),\beta (·\left)\right]}{sup}{x}_{n-1}\left(s\right)\right)\cap \left(f_{n-1}(·)+{\gamma }_{n-1}(·)[-1,1]\right)\end{array}$$

for every $n\ge 1$ and to define $x_n:[0,1]\to \mathbb{R}$ by

$$x_n\left(t\right)=x^0+{\int }_{[0,t)}{f}_n\left(s\right)d{\mu }_g\left(s\right).$$

(2)

One can see as before that

$$\begin{array}{ccc}\hfill |f_n\left(t\right)-f_{n-1}\left(t\right)|& \le & L_1\left(t\right)|x_{n-1}\left(t\right)-x_{n-2}\left(t\right)|\hfill \\ & +& L_2\left(t\right)\left|\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}{x}_{n-1}\left(s\right)-\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}{x}_{n-2}\left(s\right)\right|\hfill \\ & \le & (L_1\left(t\right)+L_2\left(t\right)){\int }_{[0,t)}{\gamma }_{n-2}\left(s\right)d{\mu }_g\left(s\right),{\mu }_g-a.e.\hfill \end{array}$$

If one denotes by

$${\gamma }_{n-1}\left(t\right)=(L_1\left(t\right)+L_2\left(t\right)){\int }_{[0,t)}{\gamma }_{n-2}\left(s\right)d{\mu }_g\left(s\right),t\in [0,1],$$

then for all $t\in [0,1]$,

$$|x_n\left(t\right)-x_{n-1}\left(t\right)|\le {\int }_{[0,t)}{\gamma }_{n-1}\left(s\right)d{\mu }_g\left(s\right).$$

Similarly to [25], it can be proved by mathematical induction that for every $n\ge 1$,

$${\int }_{[0,t)}{\gamma }_n\left(s\right)d{\mu }_g\left(s\right)\le C_n\left(t\right)·\left(\left|x\left(0\right)-x^0\right|+{\int }_{[0,t)}\gamma \left(s\right)d{\mu }_g\left(s\right)\right),t\in [0,1]$$

where

$$C_n\left(t\right)={\int }_{[0,t)}{\int }_{[0,t_n)}...{\int }_{[0,t_2)}(L_1\left(t_n\right)+L_2\left(t_n\right))...(L_1\left(t_1\right)+L_2\left(t_1\right))d{\mu }_g\left(t_1\right)...d{\mu }_g\left(t_n\right).$$

By applying ([25], Proposition 4.5), one obtains

$$C_n\left(t\right)\le {\displaystyle \frac{1}{n!}}{\left({\int }_{[0,t)}{L}_1\left(s\right)+L_2\left(s\right)d{\mu }_g\left(s\right)\right)}^n;$$

hence, it follows that ${\left(f_n\right)}_n\subset L_g^1\left([0,1]\right)$ is Cauchy, so it converges towards a function $f\in L_g^1\left([0,1]\right)$.

The sequence ${\left(x_n\right)}_n\subset {\mathcal{BC}}_g\left([0,1]\right)$ is Cauchy as well, and so it tends to some $\tilde{x}\in {\mathcal{BC}}_g\left([0,1]\right)$. By passing to the limit in $\left(2\right)$, one gets, by dominated convergence theorem,

$$x\left(t\right)=x^0+{\int }_{[0,t)}f\left(s\right)d{\mu }_g\left(s\right),\mathrm{for}\mathrm{every}t\in [0,1].$$

Using the hypotheses on F, for ${\mu }_g$-a.e. $t\in [0,1]$ and every $\epsilon >0$, there exists $n_{\epsilon ,t}\in \mathbb{N}$ such that

$$F\left(t,x_n\left(t\right),\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}{x}_n\left(s\right)\right)\subset F\left(t,\tilde{x}\left(t\right),\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}\tilde{x}\left(s\right)\right)+\epsilon ·[-1,1],\forall n>n_{\epsilon ,t};$$

it follows that

$$f\left(t\right)\in F\left(t,\tilde{x}\left(t\right),\underset{s\in \left[\alpha \right(t),\beta (t\left)\right]}{sup}\tilde{x}\left(s\right)\right),{\mu }_g-a.e.$$

Finally, as for each $n\ge 1$,

$$\begin{array}{ccc}& & |x_n\left(t\right)-x\left(t\right)|\hfill \\ & & \le \sum _{i=1}^n|x_i\left(t\right)-x_{i-1}\left(t\right)|\hfill \\ & & \le \sum _{i=1}^n{\int }_{[0,t)}{\gamma }_{i-1}\left(s\right)d{\mu }_g\left(s\right)\hfill \\ & & \le \left(|x\left(0\right)-x^0|+{\int }_{[0,t)}\gamma \left(s\right)d{\mu }_g\left(s\right)\right)+\sum _{i=2}^n{C}_{i-1}\left(t\right)·\left(|x\left(0\right)-x^0|+{\int }_{[0,t)}\gamma \left(s\right)d{\mu }_g\left(s\right)\right)\hfill \\ & & \le \left(|x\left(0\right)-x^0|+{\int }_{[0,t)}\gamma \left(s\right)d{\mu }_g\left(s\right)\right)\left(1+\sum _{i=2}^n{\displaystyle \frac{1}{(i-1)!}}{\left({\int }_{[0,t)}{L}_1\left(s\right)+L_2\left(s\right)d{\mu }_g\left(s\right)\right)}^{i-1}\right)\hfill \\ & & \le e^{{\int }_{[0,t)}{L}_1\left(s\right)+L_2\left(s\right)d{\mu }_g\left(s\right)}·\left(|x\left(0\right)-x^0|+{\int }_{[0,t)}\gamma \left(s\right)d{\mu }_g\left(s\right)\right),\hfill \end{array}$$

the proof is over. □

In particular, the following corollary can be deduced (one thus generalizes ([3], Theorem 1) which is available in the case where $g\left(t\right)=t$ on $[0,1]$ and $\alpha \left(t\right)=0$, $\beta \left(t\right)=t$).

Corollary 1. 

Suppose that $F:[0,1]\times \mathbb{R}\times \mathbb{R}\to {\mathcal{P}}_k\left(\mathbb{R}\right)$ is jointly measurable and Lipschitz in the last two arguments.

If

$$\parallel L_1{\parallel }_g^1+{\parallel L_2\parallel }_g^1<1,$$

then given a g-absolutely continuous function $x:[0,1]\to \mathbb{R}$, if $\gamma \in L_g^1\left([0,1]\right)$ satisfies

$$d\left(x_g^{\prime }\left(t\right),F\left(t,x\left(t\right),\underset{s\in [0,t]}{sup}x\left(s\right)\right)\right)\le \gamma \left(t\right),{\mu }_g-a.e.,$$

there exists a solution $\tilde{x}$ of $\left(1\right)$ such that

$$\parallel \tilde{x}{-x\parallel }_C\le {\displaystyle \frac{|x^0{-x\left(0\right)|+\parallel \gamma \parallel }_g^1}{1-\parallel L_1{\parallel }_g^1-{\parallel L_2\parallel }_g^1}}.$$

Proof. 

The hypotheses of Theorem 2 are obviously satisfied, so all one has to do is remark that for any $\xi \in (0,1)$,

$$e^{\xi }\le {\displaystyle \frac{1}{1-\xi }}.$$

□

It is useful to mention that the main theorem involves the following existence result for nonconvex Stieltjes differential inclusions with supremum.

Corollary 2. 

Let $F:[0,1]\times \mathbb{R}\times \mathbb{R}\to {\mathcal{P}}_k\left(\mathbb{R}\right)$ satisfy the assumptions of Theorem 2.

Then, the differential problem $\left(1\right)$ has g-absolutely continuous solutions on $[0,1]$.

Remark 1. 

Using Theorem 2, a relaxation result could be obtained for Stieltjes differential inclusions with supremum following the idea in [26]. Note that the method applied in [3] to this purpose is not applicable here since the key result connecting the adherence of the integral of a multifunction and the adherence of the integral of its closed convex hull ([37], Corollary 4.3) works only for nonatomic measures, while ${\mu }_g$ is not nonatomic (more precisely, it has atoms whenever g has discontinuity points).

4. Application to Dynamic Inclusions on Time Scales with Supremum

It will be shown in what follows that Theorem 2 involves a Filippov-type result for dynamic inclusions on time scales with supremum.

Let $\mathbb{T}\subset [0,T]$ be a bounded time scale (i.e., a nonempty and closed subset of $\mathbb{R}$, with the usual topology). Suppose that $0=min\mathbb{T}$, denote by $T=max\mathbb{T}$ and suppose

$$\mathbb{T}\setminus \dot{\mathbb{T}}\mathrm{is}\mathrm{of}\mathrm{null}\mathrm{measure},$$

meaning that the time scale is typical ([6]).

For the convenience of the reader, the basic notions of time-scale analysis will be briefly recalled (see [38]).

The forward jump operator $\sigma :\mathbb{T}\to \mathbb{T}$ is given by $\sigma \left(t\right)=inf\{s\in \mathbb{T}:s>t\}$ (by convention, $\sigma \left(T\right)=T$).

A point $t\in \mathbb{T}$ is called right-dense, resp. right-scattered, if $\sigma \left(t\right)=t$, resp. $\sigma \left(t\right)>t$. It was proved (in [39], Lemma 3.1) that only, at most, countably many points (${\tau }_i,i\in \mathbb{N}$) are right-scattered. The Lebesgue measure ${\mu }_{\Delta }$ on $\mathbb{T}$ was studied in [39], as well as the Lebesgue $\Delta$-integral (see also [38,40]).

(Ref. [39], Theorem 5.2) asserts that if $f:\mathbb{T}\to \mathbb{R}$, for any $a<b\in [0,T]$,

$${\int }_{{[a,b)}_{\mathbb{T}}}f\left(s\right)\Delta s={\int }_{[a,b)\cap \mathbb{T}}f\left(s\right)ds+\sum _{i:{\tau }_i\in [a,b)}f\left({\tau }_i\right)(\sigma \left({\tau }_i\right)-{\tau }_i)$$

(the time-scale interval ${[a,b)}_{\mathbb{T}}=\{t\in \mathbb{T}:a\le t<b\}$).

The idea of this section is to transform the problem on time scales into a Stieltjes differential problem; to achieve this aim, as in [41], consider the left-continuous and non-decreasing map $g:[0,T]\to \mathbb{R}$ given by

$$g\left(t\right)=inf\{s\in \mathbb{T}:s\ge t\}$$

which has the properties

$$g\left(t\right)=t\mathrm{and}\sigma \left(t\right)=g(t+)\mathrm{whenever}t\in \mathbb{T}.$$

For any function $x:\mathbb{T}\to \mathbb{R}$, its Slavik extension ([41]) is

$$x^g:[0,T]\to \mathbb{R},x^g\left(t\right)=x\left(g\left(t\right)\right).$$

It is known (e.g., [8]) that

$${\int }_{{[a,b)}_{\mathbb{T}}}f\left(s\right)\Delta s={\int }_{[a,b)}{f}^g\left(s\right)d{\mu }_g\left(s\right).$$

Definition 4. 

([38]) The function $f:\mathbb{T}\to \mathbb{R}$ is called Δ-differentiable at $t\in \mathbb{T}$ if there is $f^{\Delta }\left(t\right)\in \mathbb{R}$ (the Δ-derivative of f at t), such that for any $\epsilon >0$, there is a neighborhood of t on which

$$\left|f\left(\sigma \left(t\right)\right)-f\left(s\right)-f^{\Delta }\left(t\right)[\sigma \left(t\right)-s]\right|\le \epsilon |\sigma \left(t\right)-s|.$$

The focus in this section is on a dynamic inclusion on time scales with supremum:

$$\left\{\begin{array}{c}{x}^{\Delta }\left(t\right)\in \tilde{F}\left(t,x\left(t\right),\underset{s\in {[\alpha \left(t\right),\beta \left(t\right)]}_{\mathbb{T}}}{sup}x\left(s\right)\right),t\in \mathbb{T},\hfill \\ \\ x\left(0\right)=x^0,\hfill \end{array}\right.$$

(3)

where $\tilde{F}:\mathbb{T}\times \mathbb{R}\times \mathbb{R}\to {\mathcal{P}}_k\left(\mathbb{R}\right)$ and $\alpha ,\beta :\mathbb{T}\to \mathbb{R}$ are $\Delta$-measurable and satisfy the condition $0\le \alpha \left(t\right)\le \beta \left(t\right)\le t$ for each $t\in \mathbb{T}$.

Definition 5. 

A map $x:\mathbb{T}\to \mathbb{R}$ is said to be a solution of this problem if it is Δ-absolutely continuous and, moreover, there exists a Lebesgue Δ-integrable selection f of $\tilde{F}(·,x(·),{sup}_{s\in {[\alpha (·),\beta (·)]}_{\mathbb{T}}}x\left(s\right))$ with $x^{\Delta }\left(t\right)=f\left(t\right)$, ${\mu }_{\Delta }$-a.e. and $x\left(0\right)=x^0$.

Let us recall ([42]) the following notion.

Definition 6. 

One says that $x:\mathbb{T}\to \mathbb{R}$ is Δ-absolutely continuous if for every $\epsilon >0$ one can find ${\delta }_{\epsilon }>0$ such that

$$\sum _{j=1}^m|x\left(t_j^{″}\right)-x\left(t_j^{\prime }\right)|<\epsilon$$

for any family ${\left\{(t_j^{\prime },t_j^{″})\right\}}_{i=1}^m$ of non-overlapping intervals with $t_j^{\prime },t_j^{″}\in \mathbb{T}$ and ${\sum }_{j=1}^m(t_j^{″}-t_j^{\prime })<{\delta }_{\epsilon }$.

The key result for the connection between the dynamic problem on time scales and a Stieltjes differential problem is the following.

Lemma 2. 

A function $x:\mathbb{T}\to \mathbb{R}$ is a solution of problem $\left(3\right)$ if and only if $x^g$ is a solution of

$$\left\{\begin{array}{c}{y}_g^{\prime }\left(t\right)\in \tilde{F}\left(g\left(t\right),y\left(t\right),\underset{s\in \left[\alpha \right(g\left(t\right)),\beta (g\left(t\right)\left)\right]}{sup}y\left(s\right)\right),{\mu }_g-\mathrm{a}.\mathrm{e}.t\in [0,T]\hfill \\ x\left(0\right)=x^0.\hfill \end{array}\right.$$

(4)

Proof. 

For every solution x of $\left(3\right)$, there exists a selection f of $\tilde{F}\left(·,x(·),{sup}_{s\in {[\alpha (·),\beta (·)]}_{\mathbb{T}}}x\left(s\right)\right)$ such that

$$x^{\Delta }\left(t\right)=f\left(t\right),{\mu }_{\Delta }-a.e.t\in \mathbb{T}.$$

In the same way as in [6], one can prove that at any point $t\in \mathbb{T}$ where the $\Delta$-derivative exists,

$$x^{\Delta }\left(t\right)={\left(x^g\right)}_g^{\prime }\left(t\right),$$

hence, ${\left(x^g\right)}_g^{\prime }\left(t\right)=f\left(t\right)$ for every $t\in \mathbb{T}$. Additionally, taking into account the definition of the Lebesgue $\Delta$-measure ([39], Section 2) and the fact that the g-derivative of $x^g$ is not defined on $[0,T]\setminus \mathbb{T}$ since g is constant (but it is well-known, from [5], that ${\mu }_g([0,T]\setminus \mathbb{T})=0$), one gets

$${\left(x^g\right)}_g^{\prime }\left(t\right)=f\left(t\right),{\mu }_g-\mathrm{a}.\mathrm{e}.t\in [0,T].$$

Moreover, since $\mathbb{T}$ is compact, it follows that for every $t\in \mathbb{T}$, we have

$$\underset{s\in {[\alpha \left(t\right),\beta \left(t\right)]}_{\mathbb{T}}}{sup}x\left(s\right)=\underset{s\in \left[\alpha \right(g\left(t\right)),\beta (g\left(t\right)\left)\right]}{sup}x\left(g\left(s\right)\right)$$

and as in $\mathbb{T}$, the equality $g\left(t\right)=t$ holds, it follows that

$$f\left(t\right)\in \tilde{F}\left(t,x\left(t\right),\underset{s\in {[\alpha \left(t\right),\beta \left(t\right)]}_{\mathbb{T}}}{sup}x\left(s\right)\right)=\tilde{F}\left(g\left(t\right),x\left(g\left(t\right)\right),\underset{s\in \left[\alpha \right(g\left(t\right)),\beta (g\left(t\right)\left)\right]}{sup}x\left(g\left(s\right)\right)\right).$$

With the remark that $x^g\left(0\right)=x\left(0\right)$, it can be stated that $x^g$ is indeed a solution of $\left(4\right)$.

Conversely, if y is a solution of $\left(4\right)$, then the map $x:\mathbb{T}\to \mathbb{R}$ given by $x\left(t\right)=y\left(t\right)$ for every $t\in \mathbb{T}$ is a solution of the dynamic problem $\left(3\right)$ since

$$x^{\Delta }\left(t\right)={\left(x^g\right)}_g^{\prime }\left(t\right)=y_g^{\prime }\left(t\right),\mathrm{for}t\in \mathbb{T}.$$

□

Using this auxiliary result, it is now possible to infer a Filippov-type lemma for dynamic inclusions on time scales with supremum from Theorem 2 (in an obvious manner).

Theorem 3. 

Let $\alpha ,\beta :\mathbb{T}\to \mathbb{R}$ be Δ-measurable and satisfy the condition $0\le \alpha \left(t\right)\le \beta \left(t\right)\le t$ for each $t\in \mathbb{T}$. Let $\tilde{F}:\mathbb{T}\times \mathbb{R}\times \mathbb{R}\to {\mathcal{P}}_k\left(\mathbb{R}\right)$ verify the following hypotheses:

• For every $x,y\in \mathbb{R}$, $t\mapsto \tilde{F}(t,x,y)$ is Δ-measurable;
• There exist $L_1,L_2$ Lebesgue Δ-integrable on $\mathbb{T}$ such that ${\mu }_{\Delta }$-a.e. on $\mathbb{T}$, for all $x_1,x_2,y_1,y_2\in \mathbb{R}$, as follows:

  $$D(\tilde{F}(t,x_1,y_1),\tilde{F}(t,x_2,y_2))\le L_1\left(t\right)\left|x_1-x_2\right|+L_2\left(t\right)\left|y_1-y_2\right|.$$

Then, given a Δ-absolutely continuous function $x:\mathbb{T}\to \mathbb{R}$, if a Lebesgue Δ-integrable map $\gamma :\mathbb{T}\to \mathbb{R}$ satisfies

$$d\left(x^{\Delta }\left(t\right),F\left(t,x\left(t\right),\underset{s\in {[\alpha \left(t\right),\beta \left(t\right)]}_{\mathbb{T}}}{sup}x\left(s\right)\right)\right)\le \gamma \left(t\right),{\mu }_{\Delta }-a.e.,$$

there exists a solution $\overline{x}$ of $\left($ such that

$$\left|\overline{x}\left(t\right)-x\left(t\right)\right|\le \left(\left|x^0-x\left(0\right)\right|+{\int }_{{[0,t)}_{\mathbb{T}}}\gamma \left(s\right)\Delta s\right)e^{{\int }_{{[0,t)}_{\mathbb{T}}}{L}_1\left(s\right)+L_2\left(s\right)\Delta s}.$$

Remark 2. 

Even in the case where the supremum is not present in the problem $\left(3\right)$, Theorem 3 gives a new Filippov-type result for dynamic inclusions on time scales (different from ([28], Theorem 4.2), where the estimation on the right-hand side can be expressed in terms of an exponential function adjusted to the time-scale setting—see ([38], Theorem 2.77)).

5. Conclusions

The classical Filippov lemma is generalized to the wide setting of differential inclusions with a Stieltjes derivative and supremum; the first existence result for Lipschitz differential inclusions with supremum and Stieltjes derivative is then deduced.

An application to dynamic inclusions on time scales with supremum is also included; it is important to note that the obtained Filippov-type lemma is new (in comparison to the only such result, given in [28]) even in the case where the supremum of the map is not present in the multifunction on the right-hand side.

To support the generality of the outcome, it has to be mentioned that the presented study covers several known results by taking a particular function g or a single-valued F map (for usual differential problems with maxima or impulsive equations with supremum) or by particularizing $\alpha$ and $\beta$ (such as in the case of Stieltjes differential inclusions with a constant or variable delay).