*Article* **Evolution Inclusions in Banach Spaces under Dissipative Conditions**

#### **Tzanko Donchev 1,\*, Shamas Bilal 2, Ovidiu Cârj ˘a 3,4, Nasir Javaid 5 and Alina I. Lazu 6**


Received: 23 March 2020; Accepted: 5 May 2020; Published: 9 May 2020

**Abstract:** We develop a new concept of a solution, called the limit solution, to fully nonlinear differential inclusions in Banach spaces. That enables us to study such kind of inclusions under relatively weak conditions. Namely we prove the existence of this type of solutions and some qualitative properties, replacing the commonly used compact or Lipschitz conditions by a dissipative one, i.e., one-sided Perron condition. Under some natural assumptions we prove that the set of limit solutions is the closure of the set of integral solutions.

**Keywords:** m-dissipative operators; limit solutions; integral solutions; one-sided Perron condition; Banach spaces

#### **1. Introduction and Preliminaries**

Let *X* be a real Banach space with the norm |·|, *A* : *D*(*A*) ⊂ *X* ⇒ *X* an m–dissipative operator generating the semigroup {*S*(*t*) : *D*(*A*) → *<sup>D</sup>*(*A*); *t* ≥ 0} and *F* : *I* × *X* ⇒ *X* a multifunction with nonempty, closed and bounded values, where *I* = [*<sup>t</sup>*0, *<sup>T</sup>*].

In this paper, we study evolution inclusions of the form

$$
\dot{\mathbf{x}}(t) \in A\mathbf{x}(t) + F(t, \mathbf{x}(t)), \\
\mathbf{x}(t\_0) = \mathbf{x}\_0 \in \overline{D(A)}.\tag{1}
$$

Notice that many parabolic systems can be written in the form (1). We refer the reader to [1–3] for the general theory of the system (1) when *F* is single valued. In the case when *X*∗ is uniformly convex, the system (1) is comprehensively studied in [4]. We recall also the monograph [5], where (1) is studied in different settings.

An important problem regarding the system (1) is to find the closure of the set of integral solutions. This problem is not solved in the case of general Banach spaces.

We consider the associated Cauchy problem

$$\dot{\mathbf{x}}(t) \in Ax(t) + f(t), \mathbf{x}(t\_0) = \mathbf{x}\_0 \in \overleftarrow{D(A)},\tag{2}$$

where *f*(·) is a Bochner integrable function. We denote by [·, ·]+ the right directional derivative of the norm, i.e., [*x*, *y*]+ = lim *h*→0<sup>+</sup> *<sup>h</sup>*−<sup>1</sup>(|*x* + *hy*|−|*x*|) (see, e.g., ([6], Section 1.2) for definition and properties). Following [7], we say that a continuous function *x* : [*<sup>t</sup>*0, *T*] → *D*(*A*) is an integral solution of (2) on [*<sup>t</sup>*0, *T*] if *<sup>x</sup>*(*<sup>t</sup>*0) = *x*0 and for every *u* ∈ *<sup>D</sup>*(*A*), *v* ∈ *Au* and *t*0 ≤ *τ* < *t* ≤ *T* the following inequality holds

$$|x(t) - u| \le |x(\tau) - u| + \int\_{\tau}^{t} [x(s) - u, f(s) + v]\_{+} ds.$$

**Definition 1.** *The Bochner integrable function g*(·) *is said to be pseudoderivative of the continuous function y*(·) *(with respect to A) if y*(·) *is an integral solution of* (2) *on* [*<sup>t</sup>*0, *T*] *with f*(·) *replaced by g*(·)*.*

Notice that the pseudoderivative *g*(·) (if it exists) depends on *A* and *y*(·). However, along this paper *A* is fixed and we assume without loss of generality that the pseudoderivative depends only on *y*(·). To stress this dependence on *y*, we will denote the pseudoderivative *g*(·) by *gy*(·).

It is well known that for each *x*0 ∈ *D*(*A*) the Cauchy problem (2) has a unique integral solution on [*<sup>t</sup>*0, *<sup>T</sup>*]. Moreover, if *<sup>x</sup>*(·) and *y*(·) are integral solutions of (2) with *<sup>x</sup>*(*<sup>t</sup>*0) = *x*0 and *y*(*<sup>t</sup>*0) = *y*0 then

$$|x(t) - y(t)| \le |x\_0 - y\_0| + \int\_{t\_0}^t [x(s) - y(s), f\_x(s) - f\_y(s)]\_+ ds,\tag{3}$$

$$|\mathbf{x}(t) - \mathbf{y}(t)| \le |\mathbf{x}0 - \mathbf{y}0| + \int\_{t\_0}^{t} |f\_{\mathbf{x}}(\mathbf{s}) - f\_{\mathbf{y}}(\mathbf{s})| d\mathbf{s},\tag{4}$$

for every *t* ∈ [*<sup>t</sup>*0, *T*] (see, e.g., [7]).

We define now the notion of integral solution for the differential inclusion (1). Moreover, following [8], where the semilinear case was considered, we define the notions of *ε*-solution (called outer *ε*-solution in [8]) and limit solution for (1). In the following, B denotes the closed unit ball in *X*.

**Definition 2.** *The function x* : *I* → *D*(*A*) *is said to be an integral solution of* (1) *on I if it is an integral solution of* (2) *such that its pseudoderivative fx*(·) *satisfies fx*(*t*) ∈ *<sup>F</sup>*(*<sup>t</sup>*, *x*(*t*)) *for a.a. t* ∈ *I.*

Consider the following system

$$\begin{cases}
\dot{\mathbf{x}}(t) \in A\mathbf{x}(t) + F(t, \mathbf{x}(t) + \mathbb{B}) + \mathbb{B}, \\
\mathbf{x}(t\_0) = \mathbf{x}\_0. \end{cases} \tag{5}$$

**Definition 3.** *(i) Let ε* > 0*. The continuous function x* : *I* → *D*(*A*) *is said to be an ε–solution of* (1) *on I if it is a solution of* (5) *and its pseudoderivative fx*(·) *satisfies*

$$\int\_I \text{dist}(f\_{\mathfrak{x}}(t), F(t, \mathfrak{x}(t))) dt \le \varepsilon.$$

*(ii) The function <sup>x</sup>*(·) *is said to be a limit solution of* (1) *on I if x*(*t*) = lim*n*→∞ *xn*(*t*) *uniformly on I for some sequence* (*xn*(·)) *of <sup>ε</sup>n–solutions as εn* ↓ 0+*.*

Recall that the distance between a point *u* ∈ *X* and a subset *C* of *X* is given by dist(*u*; *C*) = inf{*u* − *c*; *c* ∈ *<sup>C</sup>*}.

In the literature, we can find different definitions for *ε*–solutions. Maybe the most popular is when its pseudoderivative satisfies *fx*(*t*) ∈ *<sup>F</sup>*(*<sup>t</sup>*, *x*(*t*) + *ε*<sup>B</sup>) a.e. on *I*. However, our definition given above is more convenient for the study of the qualitative properties of the set of integral solutions of (1) in the case when *X* is an arbitrary Banach space.

For ordinary differential inclusions (*A* = 0), the limit solutions are usually called quasitrajectories (cf., [9] ). We prefer the notion of limit solution because it is the original definition of the integral solution in the case of m–dissipative systems (cf. [6]). For ordinary differential inclusions in R*<sup>n</sup>*, the limit solutions are the integral solutions of the relaxed system. In our case, the relaxed system has the form

$$\dot{\mathbf{x}}(t) \in A\mathbf{x}(t) + \overline{\alpha} \,\mathrm{F}(t, \mathbf{x}(t)), \,\mathbf{x}(t\_0) = \mathbf{x}\_{0\prime} \tag{6}$$

where *co <sup>F</sup>*(*<sup>t</sup>*, *x*(*t*)) stands for the closed convex hull of the set *<sup>F</sup>*(*<sup>t</sup>*, *<sup>x</sup>*(*t*)). In this general setting, the limit solutions are not integral solutions of the relaxed system (6).

It is well known that the set of integral solutions of (6) is not necessarily closed in *<sup>C</sup>*(*<sup>I</sup>*, *X*) even if *X* is finite dimensional. For instance, in [10] the author constructed an example in which a sequence (*xn*(·)) of integral solutions of

$$
\dot{\mathfrak{x}}(t) \in A\mathfrak{x}(t) + f\_n(t), \mathfrak{x}(t\_0) = \mathfrak{x}\_{0,\*}
$$

converges uniformly on [*<sup>t</sup>*0, *T*] to a function *<sup>x</sup>*(·), (*fn*(·)) converges weakly in *<sup>L</sup>*<sup>1</sup>(*<sup>t</sup>*0, *T*; *X*) to *f*(·), but *<sup>x</sup>*(·) is not an integral solution of

$$\mathbf{x}'(t) \in A\mathbf{x}(t) + f(t), \; \mathbf{x}(t\_0) = \mathbf{x}\_0.$$

The main results of this paper are summarized as follows.


The limit solutions in the case when *A* is linear were studied in [8]. It was shown there that the limit solutions of (1) and (6) coincide. It is not the case for the nonlinear problem.

Let us now define a few classes of multifunctions which will be used in the following.

We say that *<sup>F</sup>*(·, ·) is lower semicontinuous (LSC) at (*<sup>t</sup>*0, *<sup>x</sup>*0) ∈ *I* × *X* if for every *f*0 ∈ *<sup>F</sup>*(*<sup>t</sup>*0, *<sup>x</sup>*0), every *xk* → *x*0 and every *tk* → *t*0 there exists *fk* ∈ *<sup>F</sup>*(*tk*, *xk*) such that *fk* → *f*0. This definition is equivalent to the following property of the graph: for every *α* ∈ *<sup>F</sup>*(*<sup>t</sup>*0, *<sup>x</sup>*0) and every *ε* > 0, there exists *δ* > 0 such that *α* ∈ *<sup>F</sup>*(*<sup>t</sup>*, *x*) + *<sup>ε</sup>*B, when |*t* − *t*0| ≤ *δ* and |*x* − *<sup>x</sup>*0| ≤ *δ*.

The multifunction *<sup>F</sup>*(·, ·) is called LSC if it is LSC at every (*t*, *x*) ∈ *I* × *X*.

The multifunction *<sup>F</sup>*(·, ·) is called continuous if it is continuous with respect to the Hausdorff distance. We recall that the Hausdorff distance between the bounded sets *B* and *C* is defined by

$$D\_H(B; \mathbb{C}) = \max\{e(B; \mathbb{C}), e(\mathbb{C}; B)\},$$

where *e*(*<sup>B</sup>*; *C*) is the excess of *B* to *C*, defined by *e*(*<sup>B</sup>*; *C*) = sup*x*∈*<sup>B</sup>* dist(*x*; *<sup>C</sup>*).

The multifunction *<sup>F</sup>*(·, ·) is called almost LSC (continuous) if for every *ε* > 0 there exists a compact set *Iε* ⊂ *I* with Lebesgue measure meas(*<sup>I</sup>* \ *Iε*) ≤ *ε* such that *F* |*<sup>I</sup>ε*×*<sup>X</sup>* is LSC (continuous).

Let *v* : *I* × R+ → R+ be Carathéodory and integrally bounded on the bounded sets. As is well known, the scalar differential equation

$$
\dot{r}(t) = v(t, r(t)), \quad r(t\_0) = r\_0 \ge 0,\tag{7}
$$

has maximal solutions *h*(·), i.e., 0 ≤ *r*(*t*) ≤ *h*(*t*) for every solution *<sup>r</sup>*(·) of (7) on the existence interval of *h*(·) (see, e.g., [6]).

We introduce now the standing hypotheses of this paper.

**Hypothesis 1** (**H1**)**.** *The multifunction <sup>F</sup>*(·, ·) *is almost continuous.* **Hypothesis 2** (**H2**)**.** *There exists γ* > 0 *such that F*(*<sup>t</sup>*, *x*) ≤ *γ*(<sup>1</sup> + |*x*|) *for a.a. t* ∈ *I and every x* ∈ *X. We recall that F*(*<sup>t</sup>*, *x*) = sup *y*∈*<sup>F</sup>*(*<sup>t</sup>*,*<sup>x</sup>*)|*y*|*.*

**Hypothesis 3** (**H3**)**.** *(One-sided Perron condition) There exist a Perron function <sup>w</sup>*(·, ·) *and a null set* N ⊂ *I such that such for every x*, *y* ∈ *X, for every ε* > 0 *and for every f* ∈ *<sup>F</sup>*(*<sup>t</sup>*, *x*) *there exists g* ∈ *<sup>F</sup>*(*<sup>t</sup>*, *y*) *such that*

$$[\mathbf{x} - \mathbf{y}, f - \mathbf{g}]\_+ \le w(t, |\mathbf{x} - \mathbf{y}|) + \varepsilon$$

*on I* \ N *.*

We recall that the Carathéodory function *w* : *I* × R+ → R+ is said to be *Perron function* if it is integrally bounded on bounded sets, *<sup>w</sup>*(*<sup>t</sup>*, 0) ≡ 0, *<sup>w</sup>*(*<sup>t</sup>*, ·) is nondecreasing for every *t* ∈ *I* and the zero function is the only solution of the scalar differential equation *r* (*t*) = *<sup>w</sup>*(*<sup>t</sup>*,*<sup>r</sup>*(*t*)), *<sup>r</sup>*(*<sup>t</sup>*0) = 0, on *I*.

Notice that it is more popular to call such kind of functions Kamke functions. We refer the reader to [11], where Perron and Kamke functions are comprehensively studied. That paper is the main reason to use here the notion of Perron (not Kamke) function. In [12] some examples of the Perron (Kamke) functions different from the Lipshitz one are given (see, e.g., Corollary 1.13 and Corollary 1.15).

**Remark 1.** *Due to Gronwall's lemma, there exists a constant M* > 0 *such that* |*x*(*t*)| ≤ *M for every t* ∈ *I and every solution <sup>x</sup>*(·) *of* (5)*. Let N* = 1 + *γ*(<sup>2</sup> + *<sup>M</sup>*)*. Then F*(*<sup>t</sup>*, *x*(*t*) + B) + B ≤ *N for every solution <sup>x</sup>*(·) *of* (5)*.*

*Clearly, for every solution <sup>x</sup>*(·) *of* (5)*, in particular for every ε–solution <sup>x</sup>*(·) *of* (1)*, with the pseudoderivative fx*(·)*, we have that* dist(*fx*(*t*), *<sup>F</sup>*(*<sup>t</sup>*, *x*(*t*))) ≤ 2*N on I, since* | *fx*(*t*)| ≤ *N and F*(*<sup>t</sup>*, *x*(*t*)) ≤ *N for every t* ∈ *I.*

#### **2. Main Results**

The main results are given in three subsections. In the first one, we prove the existence of limit solutions. In the second subsection, we prove the most interesting results of this paper, namely, that the set of limit solutions of (1) is the closure of the set of integral solutions of (1) when *A* generates a compact semigroup or when *<sup>F</sup>*(*<sup>t</sup>*, ·) is full Perron. An example and some applications are discussed in the last two subsections.

#### *2.1. Existence of Limit Solutions*

In this subsection we prove an existence result of *ε*-solutions of the Cauchy problem (1) on *I* and a variant of the well known lemma of Filippov–Pli´s.

First, recall that ¯*t* is said to be a right dense point of a closed subset I ⊂ *I* if for every *τ* > 0 there exists a point *s* ∈ (¯*t*, ¯*t* + *τ*) I. Clearly, ¯*t* is not a right dense point of I if there exists *τ* > 0 such that (¯*t*, ¯*t* + *τ*) I = ∅.

**Lemma 1.** *Assume that <sup>F</sup>*(·, ·) *is almost LSC and satisfies* (H2)*. Then for every ε* > 0 *there exists at least one ε–solution of* (1) *defined on the whole I.*

**Proof.** Let *ε* > 0. We take *ε* ≤ *ε T* − *t*0 + 2*N* . There exists *I* ⊂ *I* a closed set with Lebesgue measure meas(*<sup>I</sup>* ) ≥ *T* − *t*0 − *ε* such that *<sup>F</sup>*|*I* ×*X* is LSC on *I* × *X*.

We take *f*0 ∈ *<sup>F</sup>*(*<sup>t</sup>*0, *<sup>x</sup>*0) arbitrary but fixed and let *f*1(·) be Bochner integrable with *f*1(*t*) ∈ *<sup>F</sup>*(*<sup>t</sup>*, *<sup>x</sup>*0) on *I*. Two cases are possible.

**Case 1.** If *t*0 is a right dense point of *I* . Since *<sup>F</sup>*|*I* ×*X* is LSC at (*<sup>t</sup>*0, *<sup>x</sup>*0), then there exists *δ* ∈ (0, 1/2) such that if *t* ∈ *I* with *t* − *t*0 ≤ *δ* and |*y* − *<sup>x</sup>*0| ≤ *δ* then *f*0 ∈ *<sup>F</sup>*(*<sup>t</sup>*, *y*) + *ε* B. We pick

$$f\_{\mathcal{Y}}(t) = \begin{cases} f\_{0\prime} \ t \in I^{\prime} \\ f\_{1}(t), \ t \in I \backslash I^{\prime} \end{cases}$$

.

Let *y*1(·) be the integral solution of the Cauchy problem

$$
\dot{y}(t) \in Ay(t) + f\_y(t), \\
y(t\_0) = x\_0.
$$

Since lim *t*↓*t*0 *y*1(*t*) = *x*0, we deduce that there exists *τ* ∈ (*<sup>t</sup>*0, *t*0 + *δ*) such that |*y*1(*t*) − *<sup>x</sup>*0| ≤ *δ* whenever *t* ∈ [*<sup>t</sup>*0, *<sup>τ</sup>*). Thus, *f*0 ∈ *<sup>F</sup>*(*<sup>t</sup>*, *y*1(*t*)) + *ε* B for every *t* ∈ [*<sup>t</sup>*0, *τ*) \* *I* and *f*1(*t*) ∈ *<sup>F</sup>*(*<sup>t</sup>*, *y*1(*t*) + B) for *t* ∈ [*<sup>t</sup>*0, *τ*) (*I* \ *I* ). Therefore, *fy*(*t*) ∈ *<sup>F</sup>*(*<sup>t</sup>*, *y*1(*t*) + B) + B for every *t* ∈ [*<sup>t</sup>*0, *<sup>τ</sup>*), i.e., *y*1(·) is a solution of (5) on [*<sup>t</sup>*0, *<sup>τ</sup>*). 

We let *y*(*t*) = *y*1(*t*) for every *t* ∈ [*<sup>t</sup>*0, *<sup>τ</sup>*). Thus, dist(*fy*(*t*), *<sup>F</sup>*(*<sup>t</sup>*, *y*(*t*))) ≤ *ε* for every *t* ∈ [*<sup>t</sup>*0, *τ*) \**I* and, due to Remark 1, dist(*fy*(*t*), *<sup>F</sup>*(*<sup>t</sup>*, *y*(*t*))) ≤ 2*N* for every *t* ∈ [*<sup>t</sup>*0, *τ*) \*(*I* \ *I* ).

**Case 2.** If *t*0 is not a right dense point of *I* , let *y*1(·) be the integral solution of the Cauchy problem

$$
\dot{y}(t) \in Ay(t) + f\_1(t), \ y(t\_0) = x\_0.
$$

Then there exists *τ* > *t*0 such that [*<sup>t</sup>*0, *τ*) ⊂ *I* \ *I* and |*y*1(*t*) − *<sup>x</sup>*0| < *ε* for *t* ∈ [*<sup>t</sup>*0, *<sup>τ</sup>*). Thus, *y*1(·) is a solution of (5) on [*<sup>t</sup>*0, *<sup>τ</sup>*).

We let *y*(*t*) = *y*1(*t*) and *fy*(*t*) = *f*1(*t*) for every *t* ∈ [*<sup>t</sup>*0, *<sup>τ</sup>*). Moreover, dist(*fy*(*t*), *<sup>F</sup>*(*<sup>t</sup>*, *y*(*t*))) ≤ 2*N* for every *t* ∈ [*<sup>t</sup>*0, *<sup>τ</sup>*).

In both cases we let *yτ* = lim*t*↑*τ y*(*t*). We continue the above construction in a similar way by replacing *t*0by *τ* and *x*0by *y<sup>τ</sup>*.

Let [*<sup>t</sup>*0, ¯*t*) be the maximal interval of the existence of solution *y*(·) of (5), with the properties that dist(*fy*(*t*), *<sup>F</sup>*(*<sup>t</sup>*, *y*(*t*))) ≤ *ε* on [*<sup>t</sup>*0, ¯*t*) *I* and dist(*fy*(*t*), *<sup>F</sup>*(*<sup>t</sup>*, *y*(*t*))) ≤ 2*N* on [*<sup>t</sup>*0, ¯*t*) (*I* \ *I* ), where *fy*(·) is the pseudoderivative of *y*(·). Suppose that ¯*t* < *T*. Due to the growth condition lim *t*↑¯*t y*(*t*) exists. Let *y*¯*t* = lim*t*↑¯*t y*(*t*). Then, using a similar construction as above with ¯*t* instead of *t*0 and *y*¯*t* instead of *x*0, we can extend the solution *y*(·) on some interval [*<sup>t</sup>*0, ¯*t* + *<sup>θ</sup>*), *θ* > 0, such that dist(*fy*(*t*), *<sup>F</sup>*(*<sup>t</sup>*, *y*(*t*))) ≤ *ε* on [*<sup>t</sup>*0, ¯*t* + *θ*) *I* and dist(*fy*(*t*), *<sup>F</sup>*(*<sup>t</sup>*, *y*(*t*))) ≤ 2*N* on [*<sup>t</sup>*0, ¯*t* + *θ*) (*I* \ *I* ), which contradicts the maximality of [*<sup>t</sup>*0, ¯*t*). Hence ¯*t* = *T*.

It is clear that the pseudoderivative *fy*(·) satisfies dist(*fy*(*t*), *<sup>F</sup>*(*<sup>t</sup>*, *y*(*t*))) = *ky*(*t*) with *ky*(*t*) ≤ *ε* for every *t* ∈ *I* and *ky*(*t*) ≤ 2*N* for every *t* ∈ *I* \ *I* . One checks easily that *I ky*(*t*)*dt* ≤ *ε*. Hence, *y*(·) is an *ε*–solution of (1) on *I*.

The next lemma will play a crucial role in the sequel.

**Lemma 2.** *Assume* (H1)–(H3)*. Let ε* > 0 *and let <sup>x</sup>*(·) *be an ε–solution of* (1) *on I. Then, there exist l*(·) *positive and bounded on I with I l*(*t*)*dt* ≤ 2*ε and η* > 0 *such that for every y*0 ∈ *D*(*A*) *with* |*<sup>x</sup>*0 − *y*0| < *η we have that:*

*(i) the maximal solution <sup>v</sup>*˜(·) *of the scalar differential equation*

$$
\dot{v}(t) = w(t, v(t)) + l(t), \\
v(t\_0) = |x\_0 - y\_0|\_{\prime}.
$$

*exists on I and*

*(ii) for every* 0 < *δ* < *ε there exists a δ–solution y*(·) *of* (1) *on I with x*0 *replaced by y*0*, satisfying*

$$|x(t) - y(t)| \le \vartheta(t)\_r$$

*for all t* ∈ *I.*

**Proof.** The assertion (i) follows from ([13], Lemma 2.4) (see also Lemma 3 below).

Let *ε* > 0 be fixed and let *fx*(·) be the pseudoderivative of *<sup>x</sup>*(·). Then, due to Definition 3, *fx*(*t*) ∈ *<sup>F</sup>*(*<sup>t</sup>*, *x*(*t*) + B) + B a.e. on *I* and *kx*(*t*) = dist(*fx*(*t*), *<sup>F</sup>*(*<sup>t</sup>*, *x*(*t*))) satisfies *I kx*(*t*)*dt* ≤ *ε*. Moreover, due to Remark 1, *kx*(*t*) ≤ 2*N* for any *t* ∈ *I*.

We take *ε* ≤ *ε* 5(*T* − *t*0 + *N*). We can assume without loss of generality that there exists a compact set*Iε*⊂*I*,withmeas(*<sup>I</sup>*\*Iε*)<*ε* suchthatthefunctions*fx*|*<sup>I</sup>ε*,*kx*|*<sup>I</sup>ε*and*<sup>w</sup>*|*<sup>I</sup>ε*×<sup>R</sup>arecontinuous.

 Let *δ* < *ε*. We can assume that there exists a compact set *Iδ* ⊂ *I* such that *Iε* ⊂ *Iδ*, meas(*<sup>I</sup>* \ *<sup>I</sup>δ*) < *δ* , where *δ* ≤ min + *δ* 5(*Tt*0+*N*),*<sup>ε</sup>* ,, and *<sup>F</sup>*|*<sup>I</sup>δ*×*<sup>X</sup>* is continuous.

 We take *fx* ∈ *<sup>F</sup>*(*<sup>t</sup>*0, *<sup>x</sup>*0) such that | *fx* − *fx*(*<sup>t</sup>*0)| ≤ *kx*(*<sup>t</sup>*0) + *ε* . Let *η* ∈ (0, 1) and *y*0 ∈ *D*(*A*) with |*<sup>x</sup>*0 − *y*0| < *η*. By (H3), there exists *f*1 ∈ *<sup>F</sup>*(*<sup>t</sup>*0, *y*0) such that

$$w[\mathbf{x}\_0 - y\_0, f\_\mathbf{x} - f\_1]\_+ \le w(t\_{0\prime}|\mathbf{x}\_0 - y\_0|) + \varepsilon'.\tag{8}$$

Hence,

 −

, 
$$\mathbb{E}\left[\mathbf{x}\_{0} - y\_{0\prime}f\_{\mathbf{x}}(t\_{0}) - f\_{1}\right]\_{+} \leq \left[\mathbf{x}\_{0} - y\_{0\prime}f\_{\mathbf{x}} - f\_{1}\right]\_{+} + \left|f\_{\mathbf{x}}(t\_{0}) - f\_{\mathbf{x}}\right| \leq w\left(t\_{0\prime}\left|\mathbf{x}\_{0} - y\_{0}\right|\right) + 2\boldsymbol{\varepsilon}^{\prime} + k\_{\mathbf{x}}(t\_{0}).$$

Let *f*(·) be a Bochner integrable function such that *f*(*t*) ∈ *<sup>F</sup>*(*<sup>t</sup>*, *y*0) for every *t* ∈ *I*. We consider the following cases.

**Case 1.** *t*0 is a right dense point of *Iε* (hence it is a right dense point also for *Iδ*).

We pick

$$f\_{\mathcal{Y}}(t) = \begin{cases} f\_{1\prime} & \text{if } t \in I\_{\delta} \\ f(t)\_{\prime} & \text{if } t \in I \backslash I\_{\delta} \end{cases}$$

Let *<sup>y</sup>*<sup>1</sup>(·) be the integral solution of

$$
\dot{y}(t) \in Ay(t) + f\_y(t), \ y(t\_0) = y\_0. \tag{9}
$$

Then, by the continuity of *<sup>F</sup>*|*<sup>I</sup>δ*×*<sup>X</sup>* and *<sup>y</sup>*<sup>1</sup>(·), there exists *τ* > *t*0 such that *f*1 ∈ *<sup>F</sup>*(*<sup>t</sup>*, *y*<sup>1</sup>(*t*)) + *δ* B for every *t* ∈ [*<sup>t</sup>*0, *τ*) *Iδ*.

Due to the continuity of *<sup>y</sup>*<sup>1</sup>(·), the upper semicontinuity of [·, ·]+ and the continuity of *<sup>w</sup>*(·, ·) at (*<sup>t</sup>*0, |*<sup>x</sup>*0 − *y*0|) and of *kx*(·) at *t*0, the number *τ* > *t*0 can be chosen such that |*y*<sup>1</sup>(*t*) − *y*0| ≤ 12 for every *t* ∈ [*<sup>t</sup>*0, *<sup>τ</sup>*), and moreover,

$$\begin{aligned} &\left[\mathbf{x}(t) - \mathbf{y}^1(t), f\_\mathbf{x}(t) - f\_1\right] + \leq \left[\mathbf{x}\_0 - y\_0, f\_\mathbf{x}(t\_0) - f\_1\right] + \epsilon' \\ &\leq w(t\_0, |\mathbf{x} - \mathbf{y}\_0|) + 3\epsilon' + k\_\mathbf{x}(t) \\ &\leq \left| w(t\_0, |\mathbf{x}\_0 - \mathbf{y}\_0|) - w(t, |\mathbf{x}(t) - \mathbf{y}^1(t)|) \right| + w(t, |\mathbf{x}(t) - \mathbf{y}^1(t)|) + 4\epsilon' + k\_\mathbf{x}(t) \\ &\leq w(t, |\mathbf{x}(t) - \mathbf{y}^1(t)|) + 5\epsilon' + k\_\mathbf{x}(t), \end{aligned}$$

for every *t* ∈ [*<sup>t</sup>*0, *τ*) *I<sup>ε</sup>*.

Clearly, due to our choice of *τ*, we have that *fy*(*t*) ∈ *<sup>F</sup>*(*<sup>t</sup>*, *y*<sup>1</sup>(*t*) + B) + B for any *t* ∈ [*<sup>t</sup>*0, *<sup>τ</sup>*), hence *<sup>y</sup>*<sup>1</sup>(·) is a solution of (5) on [*<sup>t</sup>*0, *<sup>τ</sup>*).

We set *y*(*t*) = *y*<sup>1</sup>(*t*) for any *t* ∈ [*<sup>t</sup>*0, *τ*) and let *ky*(*t*) = dist(*fy*(*t*), *<sup>F</sup>*(*<sup>t</sup>*, *y*(*t*))). Then *ky*(*t*) ≤ *δ* for *t* ∈ [*<sup>t</sup>*0, *τ*) *Iδ* and *ky*(*t*) ≤ 2*N* for *t* ∈ [*<sup>t</sup>*0, *τ*) (*I* \ *<sup>I</sup>δ*).

Hence, for any *t* ∈ [*<sup>t</sup>*0, *τ*) *I<sup>ε</sup>*,

$$w[x(t) - y(t), f\_x(t) - f\_y(t)]\_+ \le w(t, |x(t) - y(t)|) + 5\varepsilon' + k\_x(t).$$

On the other hand, for any *t* ∈ [*<sup>t</sup>*0, *τ*) (*I* \ *Iε*) we have that

$$[\mathbf{x}(t) - y(t), f\_{\mathbf{x}}(t) - f\_{\mathbf{y}}(t)]\_{+} \le |f\_{\mathbf{x}}(t) - f\_{\mathbf{y}}(t)| \le 2N \le 2N + w(t, |\mathbf{x}(t) - \mathbf{y}(t)|).$$

**Case 2.** *t*0 is not a right dense point of *Iε* but it is a right dense point of *Iδ*.

Let *<sup>y</sup>*<sup>1</sup>(·) be the integral solution of (9), where *fy*(·) is chosen as in Case 1. Then there exists *τ* > *t*0 such that |*y*<sup>1</sup>(*t*) − *y*0| ≤ 12 for every *t* ∈ [*<sup>t</sup>*0, *<sup>τ</sup>*), and moreover, [*<sup>t</sup>*0, *τ*) ⊂ *I* \ *Iε*. Moreover, we can choose *τ* such that *f*1 ∈ *<sup>F</sup>*(*<sup>t</sup>*, *y*<sup>1</sup>(*t*)) + *δ* B for every *t* ∈ [*<sup>t</sup>*0, *τ*) *Iδ*.

We set, as in the previous case, *y*(*t*) = *y*<sup>1</sup>(*t*) for any *t* ∈ [*<sup>t</sup>*0, *<sup>τ</sup>*). Hence *ky*(*t*) ≤ *δ* for *t* ∈ [*<sup>t</sup>*0, *τ*) *Iδ* and *ky*(*t*) ≤ 2*N* for *t* ∈ [*<sup>t</sup>*0, *τ*) (*I* \ *<sup>I</sup>δ*).

**Case 3.** *t*0 is not a right dense point of *Iδ*.

> In this case, we let *<sup>y</sup>*<sup>1</sup>(·) to be the integral solution of

$$
\dot{y}(t) \in Ay(t) + f(t), \ y(t\_0) = y\_0.
$$

Then there exists *τ* > *t*0 such that |*y*<sup>1</sup>(*t*) − *y*0| ≤ 12 for every *t* ∈ [*<sup>t</sup>*0, *<sup>τ</sup>*), and moreover, [*<sup>t</sup>*0, *τ*) ⊂ *I* \ *Iδ* ⊂ *I* \ *I<sup>ε</sup>*. We have that *<sup>y</sup>*<sup>1</sup>(·) is a solution of (5) on [*<sup>t</sup>*0, *<sup>τ</sup>*).

We let *y*(*t*) = *y*<sup>1</sup>(*t*) and *fy*(*t*) = *f*(*t*) for every *t* ∈ [*<sup>t</sup>*0, *τ*) and hence *ky*(*t*) ≤ 2*N* on [*<sup>t</sup>*0, *<sup>τ</sup>*). Moreover, in both cases 2 and 3, for any *t* ∈ [*<sup>t</sup>*0, *τ*) we have that

$$
\left[\left[\mathbf{x}(t) - \mathbf{y}(t), f\_{\mathbf{x}}(t) - f\_{\mathbf{y}}(t)\right]\_{+} \leq 2N + w(t, |\mathbf{x}(t) - \mathbf{y}(t)|).
$$

We continue the above construction in a similar way by replacing *t*0 by *τ* and *y*0 by *yτ* = lim*t*↑*τy*(*t*).

Finally, reasoning as in the proof of Lemma 1, we define *y*(·) on *I*, solution of (5). Its pseudoderivative *fy*(·) satisfies dist(*fy*(*t*), *<sup>F</sup>*(*<sup>t</sup>*, *y*(*t*))) = *ky*(*t*) with *ky*(*t*) ≤ *δ* for every *t* ∈ *Iδ* and *ky*(*t*) ≤ 2*N* for every *t* ∈ *I* \ *Iδ*. One checks easily that *I ky*(*t*)*dt* ≤ *δ*. Hence, *y*(·) is a *δ*–solution of (1) on *I*.

Moreover, for any *t* ∈ *I<sup>ε</sup>*, we have that

$$[\mathbf{x}(t) - \mathbf{y}(t), f\_{\mathbf{x}}(t) - f\_{\mathbf{y}}(t)]\_{+} \le w(t, |\mathbf{x}(t) - \mathbf{y}(t)|) + \mathfrak{S}\varepsilon' + k\_{\mathbf{x}}(t)$$

and, for any *t* ∈ *I* \ *I<sup>ε</sup>*,

$$
\frac{1}{2} \left[ \mathbf{x}(t) - \mathbf{y}(t), f\_{\mathbf{x}}(t) - f\_{\mathbf{y}}(t) \right]\_{+} \leq 2N + w(t, |\mathbf{x}(t) - \mathbf{y}(t)|).
$$

Furthermore, using (3), we have that

$$\begin{aligned} |\mathbf{x}(t) - \mathbf{y}(t)| &\le |\mathbf{x}\_0 - \mathbf{y}\_0| + \int\_{t\_0}^t [\mathbf{x}(s) - \mathbf{y}(s), f\_\mathbf{x}(s) - f\_\mathbf{y}(s)]\_+ ds \\\\ &\le |\mathbf{x}\_0 - \mathbf{y}\_0| + \int\_{[t\_0, t] \cap I\_\mathbf{e}} (w(s, |\mathbf{x}(s) - \mathbf{y}(s)|) + 5\mathbf{e}' + k\_\mathbf{x}(s)) ds \\ &+ \int\_{[t\_0, t] \cap (I/I\_\mathbf{e})} (w(s, |\mathbf{x}(s) - \mathbf{y}(s)|) + 2N) ds \end{aligned}$$

$$0 \le |\mathbf{x}\_0 - y\_0| + \int\_{t\_0}^t w(s, |\mathbf{x}(s) - y(s)|) ds + \int\_{[t\_0, t] \cap I\_\varepsilon} (\mathbf{5}\varepsilon' + k\_\mathcal{X}(s)) ds + \int\_{[t\_0, t] \cap (I/I\_\varepsilon)} 2\mathcal{N} ds$$

for any *t* ∈ *I*. Let *l*(*t*) = 5*ε* + *kx*(*t*) for *t* ∈ *Iε* and *l*(*t*) = 2*N* for *t* ∈ *I* \ *I<sup>ε</sup>*. Then, for any *t* ∈ *I*,

$$|\mathbf{x}(t) - \mathbf{y}(t)| \le |\mathbf{x}\_0 - \mathbf{y}\_0| + \int\_{t\_0}^t w(\mathbf{s}, |\mathbf{x}(s) - \mathbf{y}(s)|) ds + \int\_{t\_0}^t l(s) ds.$$

Hence, |*x*(*t*) − *y*(*t*)| ≤ *v*˜(*t*) for every *t* ∈ *I*, where *<sup>v</sup>*˜(·) is the maximal solution of the scalar differential equation

$$
\psi(t) = w(t, v(t)) + l(t), \\
v(t\_0) = |x\_0 - y\_0|.
$$

on *I*. Clearly, *l*(·) is bounded on *I* and

$$\int\_I l(s)ds = \int\_{I\_\ell} (5\varepsilon' + k\_\mathcal{X}(s))ds + \int\_{I\backslash I\_\ell} 2N ds \le 5\varepsilon'(T - t\_0) + \varepsilon + 2N\varepsilon' \le 2\varepsilon.$$

The proof is completed.

The proof of the following result follows the same steps as the proof of ([13], Lemma 2.4) and it is omitted.

**Lemma 3.** *Let λ* ∈ *<sup>L</sup>*<sup>1</sup>(*<sup>I</sup>*; R+) *and let v* : *I* × R+ → R+ *be a Carathéodory function, integrally bounded on the bounded sets, with <sup>v</sup>*(*<sup>t</sup>*, ·) *nondecreasing for every t* ∈ *I. If the maximal solution h*(·) *of* (7) *exists on I, then for every ε* > 0 *there exists δ* > 0 *such that the maximal solution <sup>r</sup>*¯(·) *of*

$$
\dot{r}(t) = v(t, r(t)) + \mu(t), \\
 r(t\_0) = \overline{r}\_0 \in [r\_{0\prime} r\_0 + \delta],
$$

*exists on I and r*¯(*t*) ≤ *h*(*t*) + *ε on I, for every function μ*(·) *such that* 0 ≤ *μ*(*t*) ≤ *λ*(*t*) *for t* ∈ *I and Iμ*(*t*)*dt* ≤ *δ.*

Now, by using the previous lemmas, we will prove the following existence result of a limit solution for the Cauchy problem (1).

**Theorem 1.** *Assume* (H1)–(H3)*. Let ε* > 0 *and let <sup>x</sup>*(·) *be an ε–solution of* (1)*. Then, there exist a positive and bounded function l*(·) *with I l*(*t*)*dt* ≤ 2*ε and η* > 0 *such that for every y*0 ∈ *D*(*A*) *with* |*<sup>x</sup>*0 − *y*0| < *η we have that:*

*(i) the maximal solution <sup>v</sup>*˜(·) *of the scalar differential equation*

$$
\dot{v}(t) = w(t, v(t)) + l(t), \\
v(t\_0) = |\mathbf{x}\_0 - \mathbf{y}\_0|. \tag{10}
$$

*exists on I and*

*(ii) there exists a limit solution y*(·) *of* (1) *on I with y*(*<sup>t</sup>*0) = *y*0 *such that*

$$|x(t) - y(t)| \le \vartheta(t) + \varepsilon\_\prime$$

*for every t* ∈ *I.*

**Proof.** Let *δ* > 0 be given by Lemma 3, corresponding to *ε*/2. Take *ε*1 ≤ min{*ε*/2, *<sup>δ</sup>*/2}. By Lemma 2 there exist *l*1(·) a positive and bounded function with *I l*1(*t*)*dt* ≤ 2*ε* and *η* > 0 such that for any *y*0 ∈ *D*(*A*) with |*<sup>x</sup>*0 − *y*0| < *η* there exists *y*1(·) an *<sup>ε</sup>*1–solution of (1) with *y*1(*<sup>t</sup>*0) = *y*0 satisfying

$$|x(t) - y\_1(t)| \le v\_1(t)\_\prime$$

where *<sup>v</sup>*1(·) is the maximal solution of

$$
\psi(t) = w(t, \upsilon(t)) + l\_1(t), \upsilon(t\_0) = |\mathbf{x}\_0 - \mathbf{y}\_0|, \tag{11}
$$

on *I*.

Let *δ*1 > 0 be given by Lemma 3 corresponding to *ε*1/2. Take *ε*2 ≤ min{*ε*1/2, *<sup>δ</sup>*1/2}. By Lemma 2 there exists an *<sup>ε</sup>*2–solution *y*2(·) of (1) on *I* with *y*2(*<sup>t</sup>*0) = *y*0 such that

$$|y\_2(t) - y\_1(t)| \le v\_2(t)\_{\prime\prime}$$

for every *t* ∈ *I*. Here *<sup>v</sup>*2(·) is the maximal solution of

$$
\psi(t) = w(t, \upsilon(t)) + l\_2(t), \upsilon(t\_0) = 0,
$$

where *l*2(·) is positive and bounded on *I* and *I l*2(*t*)*dt* ≤ 2*ε*1 ≤ *δ*. Then, by Lemma 3, *<sup>v</sup>*2(*t*) ≤ *ε*/2 for any *t* ∈ *I*.

We construct by induction a sequence of *<sup>ε</sup>n*–solutions (*yn*(·)) of (1) on *I*, where *εn* ≤ min{*<sup>ε</sup>n*−1/2, *<sup>δ</sup><sup>n</sup>*−1/2}, for any *n* = 2, 3, . . ., such that

$$|y\_{n+1}(t) - y\_n(t)| \le v\_{n+1}(t)\_{\prime\prime}$$

for every *t* ∈ *I*. Here *vn*+<sup>1</sup>(·) is the maximal solution of

$$
\psi(t) = w(t, \upsilon(t)) + l\_{n+1}(t), \upsilon(t\_0) = 0\_r
$$

where *ln*+<sup>1</sup>(·) is positive and bounded on *I* and satisfies *I ln*+<sup>1</sup>(*t*)*dt* ≤ 2*εn* ≤ *δ<sup>n</sup>*−1. Moreover, *vn*+<sup>1</sup>(*t*) ≤ *<sup>ε</sup>n*−1/2 for every *t* ∈ *I* and every *n* = 2, 3, . . .. Therefore,

$$|y\_{n+1}(t) - y\_n(t)| \le \varepsilon\_{n-1}$$

for every *t* ∈ *I* and every *n* = 2, 3, .... Taking into account that ∞ ∑ *<sup>n</sup>*=1 *εn* ≤ *ε*, we conclude that (*yn*(·)) is a Cauchy sequence in *<sup>C</sup>*(*<sup>I</sup>*; *<sup>X</sup>*). Thus, there exists a continuous function *y* : *I* → *X* such that lim *n*→∞ *yn*(*t*) = *y*(*t*) uniformly on *I*. Furthermore, |*x*(*t*) − *y*(*t*)| ≤ *<sup>v</sup>*1(*t*) + *ε*, where *<sup>v</sup>*1(·) is the maximal solution of (11).

The next theorem is a variant of the well known lemma of Filippov–Pli´s. This lemma has numerous applications in optimal control theory and had been proved on different variants by different authors. In the next theorem, we extend this result to the case when the integral solutions do not necessarily exist. Variants of this lemma have been proved in [14,15] for the case of uniformly convex dual space and in [16] for the case when *A* generates a compact semigroup.

**Theorem 2.** *Assume* (H1)–(H3)*. Let <sup>x</sup>*(·) *be an integral solution of the differential inclusion*

$$\begin{cases} \dot{\mathbf{x}}(t) \in Ax(t) + F(t, \mathbf{x}(t)) + \mathbf{g}(t)\mathbb{B},\\ \mathbf{x}(t\_0) = \mathbf{x}\_0 \in \overline{D(A)}, \end{cases} \tag{12}$$

*on I, where g* ∈ *<sup>L</sup>*<sup>1</sup>(*<sup>I</sup>*; <sup>R</sup>+)*. Then for every ε* > 0 *and every y*0 ∈ *D*(*A*) *for which the maximal solution <sup>v</sup>*(·) *of the scalar differential equation*

$$
\psi(t) = w(t, v(t)) + \mathcal{g}(t), \\
v(t\_0) = |\mathbf{x}\_0 - \mathbf{y}\_0|, \tag{13}
$$

*exists on I, there exists a limit solution <sup>z</sup>*(·) *of* (1) *on I with <sup>z</sup>*(*<sup>t</sup>*0) = *y*0*, satisfying*

$$|\mathbf{x}(t) - \mathbf{z}(t)| \le v(t) + \varepsilon\_r$$

*for all t* ∈ *I.*

**Proof.** Let *fx*(·) be the pseudoderivative of *<sup>x</sup>*(·). Then *fx*(*t*) ∈ *<sup>F</sup>*(*<sup>t</sup>*, *x*(*t*)) + *g*(*t*)<sup>B</sup> for every *t* ∈ *I*. Furthermore, for every *ε* > 0 there exists a compact *Iε* ⊂ *I* with Lebesgue measure meas(*<sup>I</sup>* \ *Iε*) < *ε* such that *fx*|*<sup>I</sup>ε*, *<sup>g</sup>*|*<sup>I</sup>ε*, *<sup>F</sup>*|*<sup>I</sup>ε*×*<sup>X</sup>* and *<sup>w</sup>*|*<sup>I</sup>ε*×R<sup>+</sup> are continuous. We fix *ν* > 0 and define the multifunction

$$G(t,\mu) = \overline{\{v \in F(t,\mu) \colon [\mathbf{x}(t) - \mathbf{u}, f\_{\mathbf{x}}(t) - v]\_{+} < w(t, |\mathbf{x}(t) - \mathbf{u}|) + \mathbf{g}(t) + \nu\}}.$$

It follows from (H3) that *<sup>G</sup>*(·, ·) has nonempty closed values. Moreover, *<sup>G</sup>*(·, ·) is almost LSC (the proof follows, with obvious modifications, the same lines as the proof of ([16], Theorem 2). Due to Lemma 1, for every *μ* > 0 the evolution inclusion

$$\begin{cases} \dot{x}(t) \in A x(t) + G(t, x(t)), \\ x(t\_0) = y\_0 \end{cases}$$

has a *μ*–solution *y*(·) defined on the whole *I*. Then, its pseudoderivative *fy*(·) satisfies *fy*(*t*) ∈ *<sup>G</sup>*(*<sup>t</sup>*, *y*(*t*)) + *hy*(*t*)<sup>B</sup> for any *t* ∈ *I*, where *hy*(*t*) ≤ 2*N* on *I* and *I hy*(*s*)*ds* ≤ *μ*. It follows from the properties of [·, ·]+ that

$$[x(t) - y(t), f\_x(t) - f\_y(t)]\_+ \le w(t, |x(t) - y(t)|) + \mathcal{g}(t) + \nu + h\_{\mathcal{Y}}(t).$$

Thus, |*x*(*t*) − *y*(*t*)| ≤ *<sup>r</sup>*(*t*), where *<sup>r</sup>*(·) is the maximal solution of the inequality *r*˙(*t*) ≤ *<sup>w</sup>*(*<sup>t</sup>*,*<sup>r</sup>*(*t*)) + *g*(*t*) + *ν* + *hy*(*t*) with *<sup>r</sup>*(*<sup>t</sup>*0) = |*<sup>x</sup>*0 − *y*0|.

Due to Lemma 3, *<sup>r</sup>*(·) exists on the whole *I* for sufficiently small *ν* and *μ* and moreover, for every *ε* > 0 there exists *κ* > 0 such that *r*(*t*) ≤ *v*(*t*) + *ε* for *μ*, *ν* < *κ*.

Clearly, *y*(·) is a *μ*–solution also of (1). It follows from Theorem 1 that there exists a limit solution *<sup>z</sup>*(·) of (1) such that |*z*(*t*) − *y*(*t*)| ≤ *ε*. The proof is therefore complete thanks to the triangle inequality.

**Remark 2.** *In fact, Theorem 2 says that the solution set of* (1) *depends continuously on small perturbations of the initial condition and the right-hand side.*

#### *2.2. Limit and Integral Solutions*

We start this subsection by giving a simple example to illustrate the notion of limit solutions.

**Example 1.** *Let A* ≡ 0*. We consider the ordinary differential inclusion:*

$$
\dot{\mathbf{x}}(t) \in \mathbb{B}, \ t \in (0, 1), \ \mathbf{x}(0) = 0. \tag{14}
$$

*Here* B *denotes the unit ball in L*<sup>1</sup>(0, 1; <sup>R</sup>*<sup>n</sup>*)*. Clearly, the limit solutions of* (14) *are all Lipschitz functions (of Lipschitz constant* 1*). However, there exists such kind of functions nowhere differentiable, i.e., which are not integral solutions.*

First, we will prove that the set of limit solutions is the closure of the set of integral solutions of (1) when *<sup>F</sup>*(·, ·) satisfies the following stronger assumption than (H3).

**Hypothesis 3 (H3).** *(Full Perron condition) There exists a Perron function <sup>w</sup>*(·, ·) *such that*

$$D\_H(F(t, \mathbf{x}), F(t, y)) \le w(t, |\mathbf{x} - y|)$$

*for every x*, *y* ∈ *X and every t* ∈ *I*.

**Theorem 3.** *Assume* (H1), (H2) *and* (H3 )*. Then* (1) *has integral solutions. Furthermore, the set of integral solutions of* (1) *is dense in the set of limit solutions of* (1)*.*

**Proof.** Let *ε* > 0 and let *y*(·) be an *ε*-solution (1) with the pseudoderivative *fy*(·). Then *fy*(*t*) ∈ *<sup>F</sup>*(*<sup>t</sup>*, *y*(*t*)) + *hy*(*t*)<sup>B</sup> for any *t* ∈ *I*, where *hy*(*t*) ≤ 2*N* on *I* and *Ihy*(*t*)*dt* ≤ *ε*.

Let 0 < *δ* < *ε*. Since the function *<sup>w</sup>*(·, ·) is Perron, there exists 0 < *μ* < *ε* such that *I <sup>w</sup>*(*<sup>t</sup>*, *μ*)*dt* < *δ*. Furthermore, there exists *t*1 > *t*0 such that |*y*(*t*) − *<sup>x</sup>*0| < *μ* for *t* ∈ [*<sup>t</sup>*0, *<sup>t</sup>*1]. Let *z*(*t*) := *y*(*t*) on [*<sup>t</sup>*0, *<sup>t</sup>*1) and denote *z*1 = *<sup>z</sup>*(*<sup>t</sup>*1). By (H1) and (H3 ), there exists a strongly measurable function *f*1(·) such that *f*1(*t*) ∈ *<sup>F</sup>*(*<sup>t</sup>*, *z*<sup>1</sup>) and

$$|f\_y(t) - f\_1(t)| \le w(t, |y(t) - z^1|) + h\_y(t) + \mu\_1$$

a.e. on [*<sup>t</sup>*1, *<sup>T</sup>*]. Consider the problem

$$\begin{cases} \dot{z}(t) \in Az(t) + f\_1(t) \\ z(t\_1) = z^1 \end{cases} \tag{15}$$

and let *<sup>z</sup>*1(·) be a solution of (15) on [*<sup>t</sup>*1, *<sup>T</sup>*]. There exists *t*2 > *t*1 such that |*<sup>z</sup>*1(*t*) − *z*1| < *μ* for any *t* ∈ [*<sup>t</sup>*1, *<sup>t</sup>*2]. Then, on [*<sup>t</sup>*1, *<sup>t</sup>*2],

$$|f\_{\mathcal{Y}}(t) - f\_1(t)| \le w(t, |y(t) - z\_1(t)|) + |w(t, |y(t) - z\_1(t)| + \mu) - w(t, |y(t) - z\_1(t)|) + h\_{\mathcal{Y}}(t) + \mu.$$

Denote *Mw*(*μ*) := sup |*x*|≤2*<sup>N</sup>* |*w*(*<sup>t</sup>*, |*x*| + *μ*) − *<sup>w</sup>*(*<sup>t</sup>*, |*x*|)| and let *z*(*t*) := *<sup>z</sup>*1(*t*) on [*<sup>t</sup>*1, *<sup>t</sup>*2]. Then, *<sup>z</sup>*(·) is a solution of *z*˙(*t*) ∈ *Az*(*t*) + *<sup>F</sup>*(*<sup>t</sup>*, *z*(*t*)) + *<sup>w</sup>*(*<sup>t</sup>*, *μ*)<sup>B</sup> and

$$|f\_{\mathcal{Y}}(t) - f\_{\mathcal{Z}}(t)| \le w(t, |y(t) - z(t)|) + M\_{\mathcal{W}}(\mu) + h\_{\mathcal{Y}}(t) + \mu$$

on [*<sup>t</sup>*1, *<sup>t</sup>*2].

Using the same method as above, as in the proof of Lemma 1, we can extend *<sup>z</sup>*(·) on the whole interval *I*, such that *z*˙(*t*) ∈ *Az*(*t*) + *<sup>F</sup>*(*<sup>t</sup>*, *z*(*t*)) + *<sup>w</sup>*(*<sup>t</sup>*, *μ*)<sup>B</sup> and

$$|f\_y(t) - f\_z(t)| \le w(t, |y(t) - z(t)|) + M\_w(\mu) + h\_y(t) + \mu$$

for any *t* ∈ *I*. Moreover, *I* dist(*fz*(*t*), *<sup>F</sup>*(*<sup>t</sup>*, *z*(*t*)))*dt* ≤ *I <sup>w</sup>*(*<sup>t</sup>*, *μ*) < *δ* on *I*. Hence, *<sup>z</sup>*(·) is a *δ*-solution of (1). Using (4), we ge<sup>t</sup> that |*y*(*t*) − *z*(*t*)| ≤ *<sup>r</sup>*(*t*), where *<sup>r</sup>*(·) is the maximal solution of

$$
\dot{r}(t) \le w(t, r(t)) + M\_w(\varepsilon) + h\_{\mathcal{Y}}(t) + \varepsilon, \ r(t\_0) = 0.
$$

Now, let *εn* ↓ 0 and let (*xn*(·)) be a sequence of *<sup>ε</sup>n*-solutions of (1), constructed as above, with (*fn*(·)) the corresponding sequence of pseudoderivatives. Then

$$|\mathbf{x}\_n(t) - \mathbf{x}\_{n+1}(t)| \le r\_n(t)$$

and

$$|f\_n(t) - f\_{n+1}(t)| \le w(t, r\_n(t)) + M\_w(\varepsilon\_n) + h\_{n+1}(t) + \varepsilon\_n r\_n$$

where *rn*(·) is the maximal solution of

$$
\dot{r}\_n(t) \le w(t, r\_n(t)) + M\_w(\varepsilon\_n) + h\_{n+1}(t), \\
r\_n(t\_0) = 0.
$$

Due to the definitions of *Mw*(*<sup>ε</sup>n*) and since *<sup>w</sup>*(·, ·) is Perron, one can choose (*<sup>ε</sup>n*) such that ∞ ∑ *<sup>n</sup>*=1 |*xn*(*t*) − *xn*+<sup>1</sup>(*t*)| converges uniformly to 0 and (*fn*(·)) converges *L*1-strongly. Therefore, lim *n*→∞ *xn*(*t*) = *x*(*t*) and lim*n*→∞ *fn*(*t*) = *f*(*t*). Then *f*(*t*) ∈ *<sup>F</sup>*(*<sup>t</sup>*, *x*(*t*)) since *<sup>F</sup>*(·, ·) is almost continuous and *x* ˙(*t*) ∈ *Ax*(*t*) + *f*(*t*) with *<sup>x</sup>*(*<sup>t</sup>*0) = *x*0. Therefore, *<sup>x</sup>*(·) is an integral solution of (1).

To prove the second part of the theorem, let *δ* > 0. Let *<sup>z</sup>*(·) be a limit solution of (1). Therefore, for any *ε* > 0 there exists an *ε*-solution *<sup>z</sup>ε*(·) such that |*z*(*t*) − *<sup>z</sup>ε*(*t*)| < *ε* for *t* ∈ *I*. As in the first part of the proof starting from *<sup>z</sup>ε*(·), we can choose *εn* ↓ 0 with *ε*1 = *ε* such that there exists an integral solution *<sup>x</sup>*(·) of (1) with |*x*(*t*) − *<sup>z</sup>ε*(*t*)| < *δ* on *I*. Hence, |*z*(*t*) − *x*(*t*)| < *ε* + *δ* for any *t* ∈ *I*. The proof is completed.

We refer the reader to ([4], pp. 25–27), where the author gives one example of nonexistence of solutions even when *X* = R*<sup>n</sup>*. In this case, the set of limit solutions is nonempty and closed.

In [4] it is also studied another example where the solution set of

$$
\dot{\mathfrak{x}}(t) \in A\mathfrak{x}(t) + \mathcal{K}, \\
\mathfrak{x}(t\_0) = \mathfrak{x}\_0 \in D(A),
$$

with *K* convex compact, is not closed. In this case, since the multivalued term is constant, due to Theorem 3, the set of integral solutions is nonempty and dense in the set of limit solutions.

**Remark 3.** *Consider the relaxed problem* (6)*. The solution set of this problem is not closed, in general. We are not able to prove that it is contained in the set of limit solutions of* (1)*, even if <sup>F</sup>*(*<sup>t</sup>*, ·) *is Lipshitz continuous. Nevertheless, if the solution set of* (1) *is dense in the solution set of* (6)*, then every relaxed solution is also a limit solution. We refer the reader to [16,17], where this type of relaxation theorems are proved in Banach spaces with some additional properties. In our opinion, the limit solution set is more adequate, because it is compact and, under mild assumptions, it is the closure of the solution set of* (1)*.*

**Definition 4.** *(see, e.g., [18]) The m-dissipative operator A is said to be of complete continuous type if for every a* < *b and every* (*fn*(·)) *in <sup>L</sup>*<sup>1</sup>(*<sup>a</sup>*, *b*; *X*) *and* (*xn*(·)) *in <sup>C</sup>*([*<sup>a</sup>*, *b*], *<sup>X</sup>*)*, with xn*(·) *a solution on* [*a*, *b*] *of x* ˙ *n*(*t*) ∈ *Axn*(*t*) + *fn*(*t*), *n* = 1, 2, ...*,* lim*n*→∞ *fn* = *f weakly in <sup>L</sup>*<sup>1</sup>(*<sup>a</sup>*, *b*; *X*) *and* lim*n*→∞ *xn* = *x uniformly in <sup>C</sup>*([*<sup>a</sup>*, *b*], *<sup>X</sup>*)*, it follows that x is a solution on* [*a*, *b*] *of*

$$
\dot{\mathfrak{x}}(t) \in A\mathfrak{x}(t) + f(t).
$$

We need the following assumption:

**Hypothesis 4** (**H4**)**.** *<sup>F</sup>*(·, ·) *has nonempty convex weakly compact values.*

We give now sufficient conditions that the limit solutions to be integral ones.

**Theorem 4.** *Let A be of complete continuous type. If* (H1)–(H4) *hold, then every limit solution of* (1) *is also an integral solution of* (1)*.*

**Proof.** Let (*xn*(·)) be a sequence of *<sup>ε</sup>n*–solutions of (1) with *εn* ↓ 0 such that lim*n*→∞ *xn*(*t*) = *x*(*t*) uniformly on *I*. Consequently, the set M = - *t*∈*I* - ∞ *<sup>n</sup>*=1 {*xn*(*t*)} is compact. Denote by (*fn*(·)) the corresponding sequence of pseudoderivatives, hence *I*dist(*fn*(*t*), *<sup>F</sup>*(*<sup>t</sup>*, *xn*(*t*)))*dt* ≤ *εn* for any natural *n*. Let ¯*fn*(·) ∈

*<sup>L</sup>*<sup>1</sup>(*<sup>I</sup>*; *X*) be such that ¯ *fn*(*t*) ∈ *<sup>F</sup>*(*<sup>t</sup>*, *xn*(*t*)) and | *fn*(*t*) − ¯ *fn*(*t*)| ≤ 3 2dist(*fn*(*t*), *<sup>F</sup>*(*<sup>t</sup>*, *xn*(*t*))) for a.a. *t* ∈ *I*. Take *yn*(·) the solutions of ¯

$$
\dot{y}\_n(t) \in Ay\_n(t) + f\_n(t), \\
y\_n(t\_0) = x\_0.
$$

Due to (4), |*xn*(*t*) − *yn*(*t*)| ≤ *t t*0 | *fn*(*t*) − ¯ *fn*(*t*)|*dt* ≤ 3 2 *εn*. Consequently, (*yn*(·)) converges uniformly to *<sup>x</sup>*(·).

On the other hand, since *<sup>F</sup>*(·, ·) is almost continuous, for any *ε* > 0 there exists a compact set *Iε* ⊂ *I* with meas(*<sup>I</sup>* \ *Iε*) ≤ *ε* such that *<sup>F</sup>*|*<sup>I</sup>ε*×*<sup>X</sup>* is continuous. Therefore, *F* : *Iε* × *X* ⇒ *Xw* is also continuous.

Here *Xw* is *X* endowed with the weak topology. Due to (H4), the set *Kε* := *co*( - *t*∈*Iε* -∞*<sup>n</sup>*=1 *<sup>F</sup>*(*<sup>t</sup>*, *xn*(*t*))) is ¯¯

weakly compact. We have that *fn*(*t*) ∈ *Kε* on *I<sup>ε</sup>*. Moreover, since ( *fn*(·)) is uniformly integrable, it is relatively weakly compact. Then, passing to subsequences, ¯ *fn*(·) → *f*(·) weakly in *<sup>L</sup>*<sup>1</sup>(*<sup>I</sup>*; *<sup>X</sup>*). Moreover, as *<sup>F</sup>*(·, ·) is almost continuous, *f*(*t*) ∈ *<sup>F</sup>*(*<sup>t</sup>*, *x*(*t*)) a.e. on *I*.

Finally, since *A* is of complete continuous type, we ge<sup>t</sup> that *<sup>x</sup>*(·) is the solution of

$$
\dot{\mathfrak{x}}(t) \in A\mathfrak{x}(t) + f(t), \\
\mathfrak{x}(t\_0) = \mathfrak{x}\_0.
$$

The proof is therefore complete.

*2.3. m–Dissipative Inclusions with Compact Semigroup*

In this section, we will study the differential inclusion (1) under the following additional assumption on *A*.

**(A)** The semigroup {*S*(·); *t* ≥ 0} is compact, i.e., *S*(*t*) is a compact operator for every *t* > 0.

Since *F*(*<sup>t</sup>*, *x*(*T*)) ≤ *N* for every solution *<sup>x</sup>*(·) of (5) the following result is a consequence of ([4], Lemma 3.1).

**Lemma 4.** *Under hypotheses* (H1)–(H3) *and* (A)*, the set of integral solutions of* (1) *is <sup>C</sup>*(*<sup>I</sup>*, *X*) *precompact* (*if nonempty*)*.*

Notice also the following theorem which is proved in [19].

**Theorem 5.** *Let <sup>F</sup>*(·, ·) *be almost LSC with closed bounded values and let X be a separable Banach space. Under hypotheses* (H2) *and* (A)*, the set of integral solutions of* (1) *is nonempty.*

As a corollary, one can prove the following variant of Filippov–Pli´s Lemma (see ([16], Theorem 3) for the separable case).

**Proposition 1.** *Assume* (H1)–(H3) *and* (A)*. Let <sup>x</sup>*(·) *be an integral solution of the Cauchy problem*

$$
\dot{\mathfrak{x}}(t) \in A\mathfrak{x}(t) + f\_{\mathfrak{x}}(t), \\
\mathfrak{x}(t\_0) = \mathfrak{x}\_0 \in D(A),
$$

*on I, where* dist(*fx*(*t*); *<sup>F</sup>*(*<sup>t</sup>*, *x*(*t*))) ≤ *g*(*t*) *for all t* ∈ *I and g* ∈ *<sup>L</sup>*<sup>1</sup>(*<sup>t</sup>*0, *T*; <sup>R</sup>+)*. Then for any ε* > 0 *and any y*0 ∈ *<sup>D</sup>*(*A*)*, there exists a solution y*(·) *of the Cauchy problem* (1) *on I with x*0 *replaced by y*0 *such that*

$$|x(t) - y(t)| \le v(t) + \varepsilon\_r$$

*for all t* ∈ *I, where <sup>v</sup>*(·) *is the maximal solution of the scalar differential equation v*˙(*t*) = *<sup>w</sup>*(*<sup>t</sup>*, *v*(*t*)) + *g*(*t*), *<sup>v</sup>*(*<sup>t</sup>*0) = |*<sup>x</sup>*0 − *y*0|*, on I.*

We are ready to prove the following interesting result. **Theorem 6.** *Under hypotheses* (H1)–(H3) *and* (A)*, the set of integral solutions of* (1) *is dense in the set of limit solutions of* (1)*.*

**Proof.** Let *<sup>x</sup>*(·) be a limit solution of (1) on *I*. Then there exists a sequence (*xn*(·)) of *<sup>ε</sup>n*–solutions of (1) with *εn* ↓ 0 such that lim *n*→∞ |*xn*(*t*) − *x*(*t*)| = 0 uniformly on *I*. Then, for any natural *n*, *xn*(·) is a solution of *<sup>x</sup>*˙*n*(*t*) ∈ *Axn*(*t*) + *fn*(*t*), where dist(*fn*(*t*); *<sup>F</sup>*(*<sup>t</sup>*, *xn*(*t*)) = *gn*(*t*) with 0 < *gn*(*t*) ≤ 2*N* on *I* and *Ign*(*t*)*dt* ≤ *εn*. Due to Proposition 1, to every *n* there exists a solution *yn*(·) of (1) such that

$$\left|\mathbf{x}\_{\mathsf{n}}(t) - y\_{\mathsf{n}}(t)\right| \leq v\_{\mathsf{n}}(t) + \frac{\varepsilon}{2^{n}}.$$

where *vn*(·) is the maximal solution of the scalar differential equation *v*˙(*t*) = *<sup>w</sup>*(*<sup>t</sup>*, *v*(*t*)) + *gn*(*t*), *<sup>v</sup>*(*<sup>t</sup>*0) = 0, on *I*. From Lemma 1, we have that lim *n*→∞ *vn*(*t*) = 0 uniformly on *I*. Consequently, lim*n*→∞ |*xn*(*t*) − *yn*(*t*)| = 0 uniformly on *I*, i.e., *x*(*t*) = lim*n*→∞*yn*(*t*) uniformly on *I*.
