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# Uncountably many solutions of first-order neutral nonlinear differential equations

- Božena Dorociaková
^{1}Email author, - Martina Kubjatková
^{2}and - Rudolf Olach
^{2}

**2013**:140

https://doi.org/10.1186/1687-1847-2013-140

© Dorociaková et al.; licensee Springer 2013

**Received:**19 December 2012**Accepted:**29 April 2013**Published:**16 May 2013

## Abstract

The article deals with the existence of uncountably many positive solutions which are bounded below and above by positive functions for the first-order nonlinear neutral differential equations. Some examples are included to illustrate the results presented in this article.

**MSC:**34K40, 34K12.

## Keywords

- neutral differential equation
- nonlinear
- existence
- uncountably many positive solutions
- Banach space

## 1 Introduction

*e.g.*, in [8, 9, 11–13]. For example, Erbe

*et al.*[6] established a few oscillation and nonoscillation criteria for linear neutral delay differential equation

Diblík and co-autors in [1–4] studied the existence of positive and oscillatory solutions of differential equations with delay and nonlinear systems in view of Ważievski’s retract principle and later extended to retarded functional differential equations by Rybakowski. Zhou [12] deduced the existence of nonoscillatory solutions of the second-order nonlinear neutral differential equations and Lin *et al.* [9] discussed the existence of nonoscillatory solutions for a third-order nonlinear neutral delay differential equation, and by utilizing Krasnoselskii’s fixed point theorem and Schauder’s fixed point theorem, they developed some sufficient conditions for the existence of uncountably many nonoscillatory solutions bounded by positive constants. Some interesting results about the existence of nonoscillatory solutions of delay differential equations can also be found in [1, 5].

where $\tau >0$, $\sigma \ge 0$, $a\in C([{t}_{0},\mathrm{\infty}),(0,\mathrm{\infty}))$, $p\in C(R,(0,\mathrm{\infty}))$, $f\in C(R,R)$, *f* is a nondecreasing function for $x>0$ and $f(x)>0$, $x>0$.

By a solution of Eq. (1), we mean a function $x\in C([{t}_{1}-\tau ,\mathrm{\infty}),R)$ for some ${t}_{1}\ge {t}_{0}$ such that $x(t)-a(t)x(t-\tau )$ is continuously differentiable on $[{t}_{1},\mathrm{\infty})$ and such that Eq. (1) is satisfied for $t\ge {t}_{1}$.

As much as we know, in the literature there is no result for the existence of uncountably many solutions which are bounded below and above by positive functions. This problem is discussed and treated in this paper.

The following fixed point theorem will be used to prove the main results in the next section.

**Lemma 1.1** ([6, 12] Krasnoselskii’s fixed point theorem)

*Let*

*X*

*be a Banach space*,

*let*Ω

*be a bounded closed convex subset of*

*X*

*and let*${S}_{1}$, ${S}_{2}$

*be maps of*Ω

*into*

*X*

*such that*${S}_{1}x+{S}_{2}y\in \mathrm{\Omega}$

*for every pair*$x,y\in \mathrm{\Omega}$.

*If*${S}_{1}$

*is contractive and*${S}_{2}$

*is completely continuous*,

*then the equation*

*has a solution in* Ω.

## 2 The existence of positive solutions

In this section we consider the existence of uncountably many positive solutions for Eq. (1) which are bounded by two positive functions. We use the notation $m=max\{\tau ,\sigma \}$.

**Theorem 2.1**

*Suppose that there exist bounded from below and from above by the functions*

*u*

*and*$v\in {C}^{1}([{t}_{0},\mathrm{\infty}),(0,\mathrm{\infty}))$

*constants*$c>0$, ${K}_{2}>{K}_{1}\ge 0$

*and*${t}_{1}\ge {t}_{0}+m$

*such that*

*Then Eq*. (1) *has uncountably many positive solutions which are bounded by the functions* *u*, *v*.

*Proof*Let $C([{t}_{0},\mathrm{\infty}),R)$ be the set of all continuous bounded functions with the norm $\parallel x\parallel ={sup}_{t\ge {t}_{0}}|x(t)|$. Then $C([{t}_{0},\mathrm{\infty}),R)$ is a Banach space. We define a closed, bounded and convex subset Ω of $C([{t}_{0},\mathrm{\infty}),R)$ as follows:

Thus we have proved that ${S}_{1}x+{S}_{2}y\in \mathrm{\Omega}$ for any $x,y\in \mathrm{\Omega}$.

Also, for $t\in [{t}_{0},{t}_{1}]$ the inequality above is valid. We conclude that ${S}_{1}$ is a contraction mapping on Ω.

This means that ${S}_{2}$ is continuous.

*ε*. Then, with regard to condition (8), for $x\in \mathrm{\Omega}$ and any $\epsilon >0$, we take ${t}^{\ast}\ge {t}_{1}$ large enough so that

Then $\{{S}_{2}x:x\in \mathrm{\Omega}\}$ is uniformly bounded and equicontinuous on $[{t}_{0},\mathrm{\infty})$, and hence ${S}_{2}\mathrm{\Omega}$ is a relatively compact subset of $C([{t}_{0},\mathrm{\infty}),R)$. By Lemma 1.1 there is an ${x}_{0}\in \mathrm{\Omega}$ such that ${S}_{1}{x}_{0}+{S}_{2}{x}_{0}={x}_{0}$. We conclude that ${x}_{0}(t)$ is a positive solution of (1).

*K*, ${S}_{1}$, ${S}_{2}$ are replaced by $\overline{K}$, ${\overline{S}}_{1}$, ${\overline{S}}_{2}$, respectively. We assume that $x,y\in \mathrm{\Omega}$, ${S}_{1}x+{S}_{2}x=x$, ${\overline{S}}_{1}y+{\overline{S}}_{2}y=y$, which are the bounded positive solutions of Eq. (1), that is,

According to (9) we get that $x\ne y$. Since the interval $[{K}_{1},{K}_{2}]$ contains uncountably many constants, then Eq. (1) has uncountably many positive solutions which are bounded by the functions $u(t)$, $v(t)$. This completes the proof. □

**Corollary 2.1**

*Suppose that there exist bounded from below and from above by the functions*

*u*

*and*$v\in {C}^{1}([{t}_{0},\mathrm{\infty}),(0,\mathrm{\infty}))$

*constants*$c>0$, ${K}_{2}>{K}_{1}\ge 0$

*and*${t}_{1}\ge {t}_{0}+m$

*such that*(2), (4)

*hold and*

*Then Eq*. (1) *has uncountably many positive solutions which are bounded by the functions* *u*, *v*.

*Proof*We only need to prove that condition (10) implies (3). Let $t\in [{t}_{0},{t}_{1}]$ and set

Thus all the conditions of Theorem 2.1 are satisfied. □

**Example 2.1**Consider the nonlinear neutral differential equation

If the function $a(t)$ satisfies (12), then Eq. (11) has uncountably many positive solutions which are bounded by the functions *u*, *v*.

**Example 2.2**Consider the nonlinear neutral differential equation

If the function $a(t)$ satisfies (14), then Eq. (13) has uncountably many solutions which are bounded by the functions *u*, *v*.

**Example 2.3**Consider the nonlinear neutral differential equation

Eq. (11) has uncountably many solutions which are bounded by the functions *u*, *v*.

## Declarations

### Acknowledgements

The research was supported by the grant 1/0090/09 of the Scientific Grant Agency of the Ministry of Education of the Slovak Republic.

## Authors’ Affiliations

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