# On a Class of Functional Differential Equations with Symmetries

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## Abstract

**:**

## 1. Introduction and Problem Formulation

## 2. Extension by Symmetry

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

## 3. Equation on a Bounded Interval

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

## 4. Existence of a Unique Symmetric Solution

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

## 5. Proofs

**Proposition**

**1**

**.**Let there exist positive linear operators ${g}_{i}:C\left({\overline{I}}_{0}\right)\to L\left({\overline{I}}_{0}\right)$, $i=0,1$, such that (38) holds for all $u,v$ from $C\left({\overline{I}}_{0}\right)$. If, in addition, the inclusions

**Proposition**

**2**

**.**Assume that $p:C\left({\overline{I}}_{0}\right)\to L\left({\overline{I}}_{0}\right)$ is a linear operator of the form

**Proposition**

**3.**

**Proposition**

**4.**

**Proposition**

**5.**

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Dilna, N.; Fečkan, M.; Rontó, A.
On a Class of Functional Differential Equations with Symmetries. *Symmetry* **2019**, *11*, 1456.
https://doi.org/10.3390/sym11121456

**AMA Style**

Dilna N, Fečkan M, Rontó A.
On a Class of Functional Differential Equations with Symmetries. *Symmetry*. 2019; 11(12):1456.
https://doi.org/10.3390/sym11121456

**Chicago/Turabian Style**

Dilna, Nataliya, Michal Fečkan, and András Rontó.
2019. "On a Class of Functional Differential Equations with Symmetries" *Symmetry* 11, no. 12: 1456.
https://doi.org/10.3390/sym11121456