# The nonlinear Kneser problem for singular in phase variables second-order differential equations

- Nino Partsvania
^{1, 2}Email author and - Bedřich Půža
^{3}

**2014**:147

https://doi.org/10.1186/s13661-014-0147-x

© Partsvania and Půža; licensee Springer. 2014

**Received: **11 February 2014

**Accepted: **29 May 2014

**Published: **23 September 2014

## Abstract

For the singular in phase variables differential equation

sufficient conditions are found for the existence of a solution satisfying the conditions

where $\phi :C([0,a];{R}_{+})\to {R}_{+}$ is a continuous nondecreasing functional, $c>0$, and $a>0$.

**MSC:** 34B16, 34B40.

## Keywords

## 1 Statement of the problem and formulation of the main results

*the Kneser solution of Eq.*(1.1) if it is twice continuously differentiable in the interval $]0,+\mathrm{\infty}[$, and in this interval it satisfies the inequalities

and the differential equation (1.1).

where $\phi :C([0,a];{R}_{+})\to {R}_{+}$ is a continuous, nondecreasing functional, $a>0$, and $c>0$.

- 30years later since Kneser’s paper was published, in their study of the problem on the distribution of electrons in a heavy atom, Fermi [2] and Thomas [3] had to investigate the problem analogous to the Kneser one for the concrete second-order differential equation${u}^{\u2033}={t}^{-\frac{1}{2}}{u}^{\frac{3}{2}}.$(1.5)

*vice versa*, a Kneser solution of that equation, satisfying the initial condition

is a solution of problem (1.5), (1.6). Therefore, problem (1.5), (1.6) is equivalent to the Kneser problem for Eq. (1.5) with the initial condition (1.7).

After the papers by Fermi and Thomas were published, many mathematicians have been interested in the Kneser-type problems, and such problems have been investigated in detail for a wide class of differential equations and systems.

Most of the results on the solvability and unique solvability of the Kneser problem for second-order nonlinear differential equations, obtained until the beginning of 50s of the last century, are reflected in the monograph by Sansone [4]. From further investigations, first of all the paper by Hartman and Wintner [5], where the Kneser problem for Eq. (1.1) was studied in the case when $f:{R}_{+}\times {R}^{2}\to {R}_{+}$ is a continuous function, should be noted. Kiguradze [6], [7] studied the same problem in the case when the function $f:\phantom{\rule{0.2em}{0ex}}]0,+\mathrm{\infty}[\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{R}^{2}\to R$ has a nonintegrable singularity in the first argument at the point $t=0$.

The Kneser problem for singular in a time variable higher-order nonlinear differential equations first was studied by Kiguradze in [8], where the optimal conditions are established for the solvability of the above-mentioned problem (see, [9], Sect. 13] as well). Analogous results were obtained by Kiguradze and Rachůnková [10] for the Kneser problem with a nonlinear initial condition.

Sufficient conditions for the solvability of the Kneser-type problems for nonlinear differential systems were obtained by Chanturia [11], Coffman [12], Hartman and Wintner [13], Kiguradze and Rachůnková [14], and Rachůnková [15], [16].

As for the general differential equation (1.1) with singularities in phase variables, for it the Kneser problem has been practically unstudied so far. The aim of the present paper is to fill this gap.

*i.e.*, there exists a sequence of positive numbers ${({t}_{k})}_{k=1}^{+\mathrm{\infty}}$ such that

A functional $\phi :C([0,a];{R}_{+})\to {R}_{+}$ is said to be *nondecreasing* if for any $u\in C([0,a];{R}_{+})$ and ${u}_{0}\in C([0,a];{R}_{+})$ the inequality $\phi (u+{u}_{0})\ge \phi (u)$ holds.

For any $x\in {R}_{+}$, we put $\phi (x)=\phi (u)$, where $u(t)\equiv x$.

A Kneser solution $u$ of Eq. (1.1) is called *vanishing at infinity* if ${lim}_{t\to +\mathrm{\infty}}u(t)=0$, and it is called *remote from zero* if ${lim}_{t\to +\mathrm{\infty}}u(t)>0$.

### Theorem 1.1

*If Eq*. (1.1)

*has a Kneser solution*$u$,

*then*

*and*

*where*

### Corollary 1.1

*If condition*(1.11)

*holds and*

*then problem* (1.1), (1.2) *has no Kneser solution*.

### Theorem 1.2

*If*

*then for any positive number*$\delta $*Eq*. (1.1) *has at least one Kneser solution satisfying equality* (1.14).

### Theorem 1.3

*If along with*(1.11)

*the condition*

*is satisfied*, *then Eq*. (1.1) *has at least one vanishing at infinity Kneser solution*.

According to Corollary 1.1, for small $c$ problem (1.1), (1.2) has no Kneser solution. Thus we can expect the solvability of that problem only for large $c$.

### Theorem 1.4

*Let the function*${g}_{1}$

*satisfy condition*(1.16),

*and*

*If*,

*moreover*,

*then problem* (1.1), (1.2) *has at least one Kneser solution*.

### Remark 1.1

the question on the existence of a Kneser solution of problem (1.1), (1.2) remains open.

*i.e.*, the case where inequality (1.8) has the form

From Theorems 1.1, 1.2, and 1.4 we immediately have the following corollary.

### Corollary 1.2

*Let the function*${g}_{1}$

*satisfy identity*(1.22),

*and let the functional*$\phi $

*satisfy condition*(1.20).

*Then the following assertions are equivalent*:

- (i)
*the function*${g}_{0}$*satisfies conditions*(1.11); - (ii)
*Eq*. (1.1)*has at least one remote from zero Kneser solution*; - (iii)
*for any*$\delta >0$,*problem*(1.1), (1.14)*has at least one Kneser solution*; - (iv)
*for any sufficiently large*$c>0$,*problem*(1.1), (1.2)*has at least one Kneser solution*.

The following statement is also valid.

### Corollary 1.3

*Let the function*${g}_{1}$

*satisfy identity*(1.22),

*and the functional*$\phi $

*satisfy condition*(1.20).

*Let*,

*moreover*,

*there exist numbers*$\alpha $

*and*$\beta $

*such that*

*Then the following assertions are equivalent*:

- (i)
$\alpha <2+\mu $, $\beta >2+\mu $;

- (ii)
*Eq*. (1.1)*has at least one remote from zero Kneser solution*; - (iii)
*Eq*. (1.1)*has at least one vanishing at infinity Kneser solution*; - (iv)
*for any*$\delta >0$,*problem*(1.1), (1.14)*has at least one Kneser solution*; - (v)
*for any sufficiently large*$c>0$,*problem*(1.1), (1.2)*has at least one Kneser solution*.

### Remark 1.2

### Remark 1.3

## 2 Auxiliary propositions

### 2.1 Lemmas on *a priori* estimates

and ${g}_{i}:\phantom{\rule{0.2em}{0ex}}]0,+\mathrm{\infty}[\phantom{\rule{0.2em}{0ex}}\to {R}_{+}$ ($i=0,1$) are continuous functions, not equal identically to zero in an arbitrary neighborhood of $+\mathrm{\infty}$, *i.e.*, there exists a sequence of positive numbers ${({t}_{k})}_{k=1}^{+\mathrm{\infty}}$ such that condition (1.9) is satisfied.

A continuous function $u:{R}_{+}\to \phantom{\rule{0.2em}{0ex}}]0,+\mathrm{\infty}[$ is said to be *the Kneser solution of the differential inequality* (2.1) (*of the differential inequality* (2.2)) if it is twice continuously differentiable in the interval $]0,+\mathrm{\infty}[$ and in this interval along with the inequality ${u}^{\prime}(t)<0$ satisfies the differential inequality (2.1) (the differential inequality (2.2)).

#### Lemma 2.1

*If the differential inequality* (2.1) *has a Kneser solution*$u$, *then the function*${g}_{0}$*satisfies condition* (1.11), *and*$u$*admits estimate* (1.12), *where*${v}_{0}$*is the function given by equality* (1.13), *and*$\nu $, $\delta $*are numbers given by equalities* (1.10) *and* (1.14).

#### Proof

Therefore condition (1.11) is satisfied and the function $u$ admits estimate (1.12). □

be fulfilled. A continuous function $u:[0,b]\to \phantom{\rule{0.2em}{0ex}}]0,+\mathrm{\infty}[$ is said to be *a Kneser solution of the differential inequality* (2.1) (*of the differential inequality* (2.2)) *in the interval*$[0,b]$ if it is continuously differentiable in the interval $]0,b]$ and in this interval along with the inequality ${u}^{\prime}(t)<0$ satisfies the differential inequality (2.1) (the differential inequality (2.2)).

The following lemma can be proved analogously to Lemma 2.1.

#### Lemma 2.2

*Let inequality*(2.4)

*be fulfilled and the differential inequality*(2.1)

*in the interval*$[0,b]$

*have a Kneser solution*$u$.

*Then*

*and that solution admits the estimate*

*where*$\delta =u(b)$

*and*

#### Lemma 2.3

*Let along with*(2.4)

*the condition*

*be fulfilled*,

*and let the differential inequality*(2.2)

*in the interval*$[0,b]$

*have a Kneser solution*$u$.

*Then this solution along with*(2.6)

*admits the estimates*

*where*$\delta =u(b)$, $\epsilon =|{u}^{\prime}(b)|$,

*and*

*for*$0\le t\le b$.

#### Proof

Integration of this inequality from $t$ to $b$ results in estimate (2.9).

If along with (2.9) we take into account inequalities (2.12) and (2.13), then the validity of estimate (2.10) becomes evident. □

### 2.2 Lemma on the solvability of a nonlinear Kneser problem on a finite interval

Let $b>a$. Consider the problem on the existence of a Kneser solution of Eq. (1.1) in the interval $[0,b]$ satisfying condition (1.2).

#### Lemma 2.4

*Let condition*(2.8)

*be fulfilled and there exist numbers*$\epsilon >0$, ${\delta}_{0}>0$,

*and*${\delta}^{\ast}>{\delta}_{0}$

*such that*

*and*

*Then problem*(1.1), (1.2)

*has a Kneser solution*$u$

*in the interval*$[0,b]$

*such that*

#### Proof

where ${k}_{0}$ is a sufficiently large natural number.

hold in the interval $]0,b]$. From these inequalities it follows that the sequence ${({u}_{k}(\cdot ;{\delta}_{k}))}_{k={k}_{0}}^{+\mathrm{\infty}}$ is uniformly bounded and equicontinuous on $[0,b]$, and the sequence ${({u}_{k}^{\prime}(\cdot ;{\delta}_{k}))}_{k={k}_{0}}^{+\mathrm{\infty}}$ is uniformly bounded and equicontinuous on every closed interval contained in $]0,b]$.

According to the Arzelà-Ascoli lemma, without loss of generality it can be assumed that ${({u}_{k}(\cdot ;{\delta}_{k}))}_{k={k}_{0}}^{+\mathrm{\infty}}$ is uniformly converging on the interval $[0,b]$, and ${({u}_{k}^{\prime}(\cdot ;{\delta}_{k}))}_{k={k}_{0}}^{+\mathrm{\infty}}$ is uniformly converging on every closed interval contained in $]0,b]$. Then, by virtue of notation (2.20), (2.21) and estimates (2.28), (2.29), from equality (2.26) it follows that the sequence ${({u}_{k}^{\u2033}(\cdot ;{\delta}_{k}))}_{k={k}_{0}}^{+\mathrm{\infty}}$ is also uniformly converging on every closed interval contained in $]0,b]$.

and conditions (2.17) are satisfied.

If now we pass to the limit in equalities (2.26) and (2.27) as $k\to +\mathrm{\infty}$ and take into account estimates (2.28) and (2.29), then it becomes evident that $u$ is a Kneser solution of problem (1.1), (1.2) in the interval $[0,b]$. □

### 2.3 Lemma on the solvability of the Cauchy problem on a finite interval

#### Lemma 2.5

*Let*$b>0$

*and condition*(2.8)

*be fulfilled*.

*Then*,

*for arbitrarily fixed*$\delta >0$

*and*$\epsilon >0$,

*problem*(1.1), (2.19)

*has at least one Kneser solution in the interval*$[0,b]$

*and every such solution admits the estimates*

*where*${w}_{0}$*and*${w}_{1}$*are functions given by equalities* (2.7) *and* (2.14).

#### Proof

But this contradicts equality (2.33). The contradiction obtained proves that ${b}_{0}=0$.

then it becomes obvious that $u$ is a Kneser solution of problem (1.1), (2.19) in the interval $[0,b]$.

Therefore estimates (2.30) and (2.31) are valid. □

### 2.4 Lemma on the existence of a remote from zero Kneser solution of Eq. (1.1)

#### Lemma 2.6

*Let*$\delta $

*be a positive number*,

*and let*${({b}_{k})}_{k=1}^{+\mathrm{\infty}}$, ${({\delta}_{k})}_{k=1}^{+\mathrm{\infty}}$,

*and*${({\epsilon}_{k})}_{k=1}^{+\mathrm{\infty}}$

*be sequences of positive numbers such that*

*and for any natural*$k$,

*the differential equation*(1.1)

*has a Kneser solution*${u}_{k}$

*in the interval*$[0,{b}_{k}]$

*satisfying the conditions*

*Let*, *moreover*, *condition* (1.16) *be satisfied*. *Then the sequence*${({u}_{k})}_{k=1}^{+\mathrm{\infty}}$*contains a uniformly converging on every finite interval of*${R}_{+}$*subsequence*, *whose limit is a Kneser solution of problem* (1.1), (1.14).

#### Proof

it follows that the sequence ${({u}_{{k}_{m}}^{\u2033})}_{m=1}^{+\mathrm{\infty}}$ is also uniformly converging on every finite closed interval contained in $]0,+\mathrm{\infty}[$.

Thus from (2.36), (2.39), (2.40), and (2.45) it follows that $u$ is a Kneser solution of problem (1.1), (1.14). □

## 3 Proof of the main results

### Proof of Theorem 1.1

Let $u$ be a Kneser solution of Eq. (1.1). Then, by condition (1.8), this function is a Kneser solution of the differential inequality (2.1) as well, where $\tau (t)\equiv t$. Due to Lemma 2.1, the function ${g}_{0}$ satisfies condition (1.11), and $u$ admits estimate (1.12). □

### Proof of Corollary 1.1

But this inequality contradicts inequality (1.15). The contradiction obtained proves the validity of the corollary. □

### Proof of Theorem 1.2

Let ${\delta}_{k}=\delta $ ($k=1,2,\dots $), and let ${({b}_{k})}_{k=1}^{+\mathrm{\infty}}$ and ${({\epsilon}_{k})}_{k=1}^{+\mathrm{\infty}}$ be sequences of positive numbers satisfying conditions (2.36) and (2.37). By Lemma 2.5 and condition (1.16), for any natural $k$ the differential equation (1.1) has a Kneser solution in the interval $[0,{b}_{k}]$ satisfying conditions (2.38). Hence, by virtue of Lemma 2.6, it follows the existence of a Kneser solution of problem (1.1), (1.14). □

### Proof of Theorem 1.3

where $r=1+{v}_{1}(0;0)$, while ${v}_{1}$ and ${h}_{0}$ are functions given by equalities (1.18) and (2.44).

from estimates (3.1)-(3.3) it follows the existence of a subsequence ${({u}_{{k}_{m}})}_{m=1}^{+\mathrm{\infty}}$ of the sequence ${({u}_{k})}_{k=1}^{+\mathrm{\infty}}$ such that ${({u}_{{k}_{m}})}_{m=1}^{+\mathrm{\infty}}$ is uniformly converging on every finite closed interval contained in ${R}_{+}$, and ${({u}_{{k}_{m}}^{\prime})}_{m=1}^{+\mathrm{\infty}}$ and ${({u}_{{k}_{m}}^{\u2033})}_{m=1}^{+\mathrm{\infty}}$ are uniformly converging on every finite closed interval contained in $]0,+\mathrm{\infty}[$.

Therefore $u$ is a vanishing at infinity Kneser solution of Eq. (1.1). □

### Proof of Theorem 1.4

holds.

where ${\delta}_{k}=u({b}_{k})$.

is a Kneser solution of problem (1.1), (1.14). On the other hand, if in the equality $\phi ({u}_{{k}_{m}})=c$ we pass to the limit as $m\to +\mathrm{\infty}$, then it becomes clear that $u$ satisfies condition (1.2) as well. Thus $u$ is a Kneser solution of problem (1.1), (1.2). □

To convince ourselves that Corollary 1.3 is valid, it suffices to note that if conditions (1.22)-(1.24) are fulfilled, then each of conditions (1.11), (1.16), (1.17) is satisfied iff $\alpha <2+\mu $ and $\beta >2+\mu $.

## Declarations

### Acknowledgements

For the first author this work is supported by the Shota Rustaveli National Science Foundation (Project # FR/317/5-101/12), and for the second author this work is supported by the Internal Grant Agency at Brno University of Technology (Project # FP-S-13-2148).

## Authors’ Affiliations

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