# Evolution Inclusions in Banach Spaces under Dissipative Conditions

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## Abstract

**:**

## 1. Introduction and Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- (I)
- We prove that the set of limit solutions of (1) is nonempty and closed in $C(I,X)$ when X is a general Banach space and $F(\xb7,\xb7)$ is almost continuous and satisfies a one-sided Perron condition.
- (II)
- We prove that in the case when A generates a compact semigroup, the closure of the set of integral solutions of (1) is exactly the set of limit solutions, which in general does not coincide with the set of integral solutions of the relaxed system. The same result is proved also when $F(t,\xb7)$ is full Perron, but without any restrictions on the semigroup A.

**Hypothesis 1**

**(H1).**

**Hypothesis 2**

**(H2).**

**Hypothesis 3**

**(H3).**

**Remark**

**1.**

## 2. Main Results

#### 2.1. Existence of Limit Solutions

**Lemma**

**1.**

**Proof.**

**Case 1.**If ${t}_{0}$ is a right dense point of ${I}^{\prime}$. Since ${F|}_{{I}^{\prime}\times X}$ is LSC at $({t}_{0},{x}_{0})$, then there exists $\delta \in (0,1/2)$ such that if $t\in {I}^{\prime}$ with $t-{t}_{0}\le \delta $ and $|y-{x}_{0}|\le \delta $ then ${f}_{0}\in F(t,y)+{\epsilon}^{\prime}\mathbb{B}$. We pick

**Case 2.**If ${t}_{0}$ is not a right dense point of ${I}^{\prime}$, let ${y}_{1}(\xb7)$ be the integral solution of the Cauchy problem

**Lemma**

**2.**

- (i)
- the maximal solution $\tilde{v}(\xb7)$ of the scalar differential equation$$\dot{v}\left(t\right)=w(t,v\left(t\right))+l\left(t\right),\phantom{\rule{0.222222em}{0ex}}v\left({t}_{0}\right)=|{x}_{0}-{y}_{0}|,$$
- (ii)
- for every $0<\delta <\epsilon $ there exists a δ–solution $y(\xb7)$ of (1) on I with ${x}_{0}$ replaced by ${y}_{0}$, satisfying$$|x\left(t\right)-y\left(t\right)|\le \tilde{v}\left(t\right),$$

**Proof.**

**Case 1.**${t}_{0}$ is a right dense point of ${I}_{\epsilon}$ (hence it is a right dense point also for ${I}_{\delta}$).

**Case 2.**${t}_{0}$ is not a right dense point of ${I}_{\epsilon}$ but it is a right dense point of ${I}_{\delta}$.

**Case 3.**${t}_{0}$ is not a right dense point of ${I}_{\delta}$.

**Lemma**

**3.**

**Theorem**

**1.**

- (i)
- the maximal solution $\tilde{v}(\xb7)$ of the scalar differential equation$$\dot{v}\left(t\right)=w(t,v\left(t\right))+l\left(t\right),\phantom{\rule{0.222222em}{0ex}}v\left({t}_{0}\right)=|{x}_{0}-{y}_{0}|,$$
- (ii)
- there exists a limit solution $y(\xb7)$ of (1) on I with $y\left({t}_{0}\right)={y}_{0}$ such that$$|x\left(t\right)-y\left(t\right)|\le \tilde{v}\left(t\right)+\epsilon ,$$

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

#### 2.2. Limit and Integral Solutions

**Example**

**1.**

**Hypothesis 3′**

**(H3′).**

**Theorem**

**3.**

**Proof.**

**Remark**

**3.**

**Definition**

**4.**

**Hypothesis 4**

**(H4).**

**Theorem**

**4.**

**Proof.**

#### 2.3. m–Dissipative Inclusions with Compact Semigroup

**(A)**The semigroup $\left\{S\right(\xb7);\phantom{\rule{4pt}{0ex}}t\ge 0\}$ is compact, i.e., $S\left(t\right)$ is a compact operator for every $t>0$.

**Lemma**

**4.**

**Theorem**

**5.**

**Proposition**

**1.**

**Theorem**

**6.**

**Proof.**

#### 2.4. Example

#### 2.5. Applications to Optimal Control

**Proof.**

## 3. Conclusions

- (a)
- The set of limit solutions is nonempty and always $C(I,X)$ closed when the right hand side F is almost continuous with closed bounded values and one-sided Perron in the state variable. Furthermore, every integral solution is also a limit solution.
- (b)
- (c)
- The existence of limit solutions can be also shown for a large class of evolution inclusions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Donchev, T.; Bilal, S.; Cârjă, O.; Javaid, N.; Lazu, A.I.
Evolution Inclusions in Banach Spaces under Dissipative Conditions. *Mathematics* **2020**, *8*, 750.
https://doi.org/10.3390/math8050750

**AMA Style**

Donchev T, Bilal S, Cârjă O, Javaid N, Lazu AI.
Evolution Inclusions in Banach Spaces under Dissipative Conditions. *Mathematics*. 2020; 8(5):750.
https://doi.org/10.3390/math8050750

**Chicago/Turabian Style**

Donchev, Tzanko, Shamas Bilal, Ovidiu Cârjă, Nasir Javaid, and Alina I. Lazu.
2020. "Evolution Inclusions in Banach Spaces under Dissipative Conditions" *Mathematics* 8, no. 5: 750.
https://doi.org/10.3390/math8050750