# D-Stability of the Initial Value Problem for Symmetric Nonlinear Functional Differential Equations

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

**Definition**

**1**

## 3. Notation

- $\mathbb{R}$ is the space of real numbers with norm $|\xb7|$;
- ${I}_{\psi}=[{t}_{0},\psi \left({t}_{0}\right)]$;
- L is the Banach space of Lebesgue integrable functions $p:[{t}_{0},\psi \left({t}_{0}\right)]\to \mathbb{R}$, with norm ${\parallel p\parallel}_{L}={\int}_{{t}_{0}}^{\psi \left({t}_{0}\right)}\left|p\left(s\right)\right|ds$;
- D is the Banach space of absolutely continuous functions $x:[{t}_{0},\psi \left({t}_{0}\right)]\to \mathbb{R}$, with norm ${\parallel x\parallel}_{D}={\int}_{{t}_{0}}^{\psi \left({t}_{0}\right)}\parallel {x}^{\prime}\left(\xi \right)\parallel d\xi +\parallel x\left({t}_{0}\right)\parallel $;
- ${g}_{i},{\nu}_{i},{p}_{i},{\mu}_{i},{\tau}_{i}$, $i=1,\dots ,m$, q are Lebesgue measurable functions, ${g}_{i},{p}_{i}$, $i=1,\dots ,m$, are essentially bounded functions on $\mathbb{R}$;
- The function ${\mu}_{i}\circ \psi $ defined by ${\mu}_{i}\left(\psi \left(t\right)\right)$ is called the composite function (or superposition) of ${\mu}_{i}$, $\psi $ and ${\mu}_{i}\circ \psi ={\psi}^{j}\circ {\mu}_{i},i,j=1,2,\dots ,m,$ means that ${\mu}_{i}\left(\psi \left(t\right)\right)=\psi (\underset{j}{\underbrace{\psi (\dots}}\psi \left({\mu}_{i}\left(t\right)\right)\underset{j}{\underbrace{\dots \left)\right)}}$.

## 4. Symmetric Properties

**Lemma**

**1**

**.**If there exist such integers ${j}_{i},{r}_{i},{k}_{i}$, $i=1,2,\dots ,m$, $m\in \mathbb{N}$, that deviations of the argument ${\mu}_{i},{\nu}_{i}$ and ${\tau}_{i}$, $i=1,2,\dots ,m$ have the following properties:

- The increasing function $\psi $ generates an increasing numerical sequence$$\dots <{\psi}^{-2}\left({t}_{0}\right)<{\psi}^{-1}\left({t}_{0}\right)<{t}_{0}<\psi \left({t}_{0}\right)<{\psi}^{2}\left({t}_{0}\right)<\dots ;$$
- Every point from Sequence (9) from (1) divides $\mathbb{R}$ on a specified quantity of intervals$$[{\psi}^{j}\left({t}_{0}\right),{\psi}^{j+1}\left({t}_{0}\right)],\phantom{\rule{2.em}{0ex}}j\in \mathbb{Z};$$
- Assume that the number j is a number of the interval (10).

**Definition**

**2**

**Lemma**

**2**

**Lemma**

**3**

**.**Assume that function $y:{I}_{\psi}\to \mathbb{R}$ is a solution of the equation

## 5. Definitions of the $\mathit{D}$-Stability

**Definition**

**3**

**.**The linear Equation

**-stable**if Problem (18), (8) have a unique solution $\mathit{y}\in \mathit{D}$ for arbitrary $\mathit{q}\in \mathit{L}$, $\mathbf{\alpha}\in \mathbb{R}$ and this solution continuously depends on $\{\mathit{q},\mathbf{\alpha}\}$; this mean that for arbitrary $\mathbf{\epsilon}>\mathbf{0}$, there exists $\mathbf{\delta}>\mathbf{0}$ such that $\parallel {\mathit{y}}_{\mathbf{1}}-\mathit{y}\parallel <\mathbf{\epsilon}$ if $\parallel {\mathit{q}}_{\mathbf{1}}{-\mathit{q}\parallel}_{\mathit{L}}<\phantom{\rule{3.33333pt}{0ex}}\mathbf{\delta}$, $|{\mathbf{\alpha}}_{\mathbf{1}}-\mathbf{\alpha}|<\mathbf{\delta}$, where ${\mathit{y}}_{\mathbf{1}}$ is the solution of the Problem (18), (8) with $\mathit{q}={\mathit{q}}_{\mathbf{1}}$, $\mathbf{\alpha}={\mathbf{\alpha}}_{\mathbf{1}}$.

**Definition**

**4**

**.**The nonlinear equation $\mathcal{N}y=0$ has a $\mathit{D}$

**-stable**property in the vicinity of the trivial solution $\mathit{y}\equiv \mathbf{0}$ if there exist such ${\mathbf{\delta}}_{\mathbf{0}}>\mathbf{0}$ that for every pair $\{\mathit{q},\mathbf{\alpha}\}\in \mathit{L}\times \mathbb{R}$, satisfying conditions ${\parallel \mathit{q}\parallel}_{\mathit{L}}<{\mathbf{\delta}}_{\mathbf{0}}$, $\left|\mathbf{\alpha}\right|<{\mathbf{\delta}}_{\mathbf{0}}$, the problem (19) has a unique solution $\mathit{x}\in \mathit{D}$ and for arbitrary $\mathbf{\epsilon}>\mathbf{0}$ there exists such $\mathbf{\delta}=\mathbf{\delta}(\mathit{x},\mathbf{\epsilon})>\mathbf{0}$ that $\parallel {\mathit{y}}_{\mathbf{1}}{-\mathit{y}\parallel}_{\mathit{D}}<\mathbf{\epsilon}$ if $\parallel {\mathit{q}}_{\mathbf{1}}{-\mathit{q}\parallel}_{\mathit{L}}<\mathbf{\delta}$, $|{\mathbf{\alpha}}_{\mathbf{1}}-\mathbf{\alpha}|<\mathbf{\delta}$, where ${\mathit{y}}_{\mathbf{1}}$ is the solution of the problem (19) with $\mathit{q}={\mathit{q}}_{\mathbf{1}}$, $\mathbf{\alpha}={\mathbf{\alpha}}_{\mathbf{1}}$ and $\parallel {\mathit{q}}_{\mathbf{1}}{\parallel}_{\mathit{L}}<{\mathbf{\delta}}_{\mathbf{0}}$, $|{\mathbf{\alpha}}_{\mathbf{1}}|<{\mathbf{\delta}}_{\mathbf{0}}$.

## 6. A Unique and Symmetric Solution for the Initial Value Problem for Linear Scalar Functional Differential Equations

**Remark**

**1.**

#### 6.1. Unique Solvability

**Lemma**

**4.**

**Lemma**

**5.**

**Definition**

**5.**

**Lemma**

**6.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2**

**.**Suppose that the operator $l:C[{I}_{\psi},\mathbb{R}]\to L[{I}_{\psi},\mathbb{R}]$ in the linear functional differential Equation (31) is positive. Assume also that one can specify some integers r and m, $m\ge r\ge 1$, a real number $\rho \in (1,+\infty )$, some constants ${\left\{{\beta}_{k}\right\}}_{k=1}^{m}\subset [0,+\infty )$ and ${\left\{{b}_{i}\right\}}_{i=1}^{r}\subset [0,+\infty )$, and certain absolutely continuous vector-functions ${u}_{0},{u}_{1},\dots ,{u}_{r-1}$ constructed by the recurrence relation

**Remark**

**2.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4**

**.**Suppose that in the linear functional differential equation (31) with initial condition (23), operator $l:C[{I}_{\psi},\mathbb{R}]\to L[{I}_{\psi},\mathbb{R}]$ is positive and, moreover, there exists an absolutely continuous function ${u}_{0}:{I}_{\psi}\to \mathbb{R}$ with property (45), a natural number m, non-negative integers k and $r\ge 1$, and real numbers ρ, $\rho >1$, $c\in (0,1)$ such that, for almost every t from the interval ${I}_{\psi}$, the following inequality is satisfied:

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

#### 6.2. Symmetric Solution

**Theorem**

**5.**

## 7. $\mathit{D}$-Stability of Nonlinear Functional Differential Equations

**Theorem**

**6.**

**Proof.**

**Corollary**

**4.**

**Theorem**

**7.**

## 8. Application

#### 8.1. Unique Solvability of the Initial Value Problem for Linear Scalar Functional Differential Equations

**Theorem**

**8.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

#### 8.2. Symmetric Property of the Unique Solution of the Initial Value Problem for Linear Scalar Functional Differential Equations

**Theorem**

**9.**

**Proof.**

#### 8.3. D-Stability of Nonlinear Functional Differential Equations

**Remark**

**5.**

**Corollary**

**5.**

**Theorem**

**10.**

**Proof.**

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Function $p\left(t\right)$ on the interval $[{\psi}^{-1}\left(0\right),0]\cup [0,\psi \left(0\right)]\cup [\psi \left(0\right),\psi \left(\psi \left(0\right)\right)].$

**Figure 2.**Function $a\left(t\right)$ on the interval $[{\psi}^{-1}\left(0\right),0]\cup [0,\psi \left(0\right)]\cup [\psi \left(0\right),\psi \left(\psi \left(0\right)\right)].$

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**MDPI and ACS Style**

Dilna, N.; Fečkan, M.; Solovyov, M.
*D*-Stability of the Initial Value Problem for Symmetric Nonlinear Functional Differential Equations. *Symmetry* **2020**, *12*, 1761.
https://doi.org/10.3390/sym12111761

**AMA Style**

Dilna N, Fečkan M, Solovyov M.
*D*-Stability of the Initial Value Problem for Symmetric Nonlinear Functional Differential Equations. *Symmetry*. 2020; 12(11):1761.
https://doi.org/10.3390/sym12111761

**Chicago/Turabian Style**

Dilna, Natalia, Michal Fečkan, and Mykola Solovyov.
2020. "*D*-Stability of the Initial Value Problem for Symmetric Nonlinear Functional Differential Equations" *Symmetry* 12, no. 11: 1761.
https://doi.org/10.3390/sym12111761