A further generalization of the Emden-Fowler equation

TL;DR

This paper introduces a broader class of Emden-Fowler equations, analyzes their solutions, and focuses on understanding the asymptotic behavior of these solutions using methods building on prior work.

Contribution

It presents a new generalization of the Emden-Fowler equation and investigates the asymptotic properties of its solutions, extending previous research.

Findings

Characterization of solution asymptotics

Extension of Emden-Fowler equation class

Methodology based on previous work

Abstract

A generalization of the Emden-Fowler equation is presented and its solutions are investigated. This paper is devoted to asymptotic behavior of its solutions. The procedure is entirely based on a previous paper by the author.

Equations

$$y^{\prime \prime }=\varphi (t)|y{|}^{\lambda }\mathrm{sgn}y,$$

$$y^{\prime \prime }=t^{-1/2}{y}^{3/2}.$$

$$y^{(n)}=F(t,y),$$

$$y^{(n)}=\varphi (t)|y{|}^{\lambda }\mathrm{sgn}y,$$

$$\mathrm{liminf}\frac{|f(t)|}{t^p}=\infty ,$$

$$\mathrm{limsup}\frac{|f(t)|}{t^p}=0,$$

$$\mathrm{liminf}|f(t)|/t^{\Pi (f)-ϵ}=\infty \mathrm{and}\mathrm{liminf}|f(t)|/t^{\Pi (f)+ϵ}=0.$$

$$\mathrm{limsup}|f(t)|/t^{\Xi (f)-ϵ}=\infty \mathrm{and}\mathrm{limsup}|f(t)|/t^{\Xi (f)+ϵ}=0.$$

$$F(t,y)=\varphi (t)|y{|}^{\lambda }\mathrm{sgn}y\mathrm{for}\lambda >1$$

$$\frac{F(\cdot ,f(\cdot ))}{(f(\cdot ){)}^{\nu }}\in {\mathcal{S}}_1\mathrm{for}\mathrm{all}f\in {\mathcal{S}}_1.$$

$$\Pi (F(\cdot ,f(\cdot )))\ge \mu \Pi (f)+c$$

$$\Pi (\varphi (\cdot )|y{|}^{\lambda }\mathrm{sgn}y)=\lambda \Pi (y)+\Pi (\varphi ).$$

$$\displaystyle\Big{|}\frac{y^{(n)}}{y^{\nu}}\Big{|}=\Big{|}\frac{y^{\prime}}{y^{1+v_{1}}}\Big{|}^{\mu_{1}}\Big{|}\frac{y^{\prime\prime}}{{(y^{\prime})}^{1+v_{2}}}\Big{|}^{\mu_{2}}\cdots\Big{|}\frac{y^{(n-1)}}{{(y^{(n-2)})}^{1+v_{n-1}}}\Big{|}^{\mu_{n-1}}\Big{|}\frac{y^{(n)}}{{(y^{(n-1)})}^{1+v_{n}}}\Big{|}.$$

$$\displaystyle\Big{|}\frac{y^{(n)}(t)}{(y(t))^{\nu}}\Big{|}<1,$$

$$\frac{y^{(n)}(t)}{(y(t){)}^{\nu }}=\frac{F(t,y(t))}{(y(t){)}^{\nu }},$$

$$y^{(i)}\in {\mathcal{S}}_2\mathrm{or}{y}^{(i)}\in {\mathcal{S}}_3\mathrm{with}\Pi (y^{(i)})=-\infty .$$

$$\mu \Pi (y)+c\le \Pi (F(\cdot ,y(\cdot )))=\Pi (y^{(n)})\le \Pi (y)-n,$$

$$\Pi (y)\le -\frac{n+c}{\mu -1}<0.$$

$$\Pi (y^{(i)})\le \Pi (y)-i\le -\frac{n+c}{\mu -1}-i<0.$$

Full text

1. Introduction

The Emden–Fowler equation is a second-order differential equation taking the form

[TABLE]

where $\phi$ is a continuous function and $\lambda$ is a positive number. This type of equations plays an important role in many areas of theoretical physics. As a particular case it includes the Thomas–Fermi equation in atomic physics

[TABLE]

Eq.(1) is linear when $\lambda=1$, and it is superlinear or sublinear, when $\lambda>1$ or $\lambda<1$, respectively.

Many authors have mainly investigated its asymptotic solutions $y(t)$ as $t\to\infty$. We refer only to [2] and the references therein for such results, because our present results do not share so much relationship to them.

In this paper we propose a more general equation and study its asymptotic solutions. It is an $n$th–order differential equation of the form

[TABLE]

where $F(t,y)$ is a continuous function of two variables $t$ and $y$. This equation is also a generalization of the equation

[TABLE]

considered in [1]; i.e., $F(t,y)=\phi(t)|y|^{\lambda}\,{\rm sgn}\,y$.

Our aim is to study asymptotic behavior of its solutions $y(t)$ of Eq.(2) as $t\to\infty$ under assumptions imposed on the function $F$. The procedure is based on [1], where asymptotic properties of functions are discussed. As for the results of [1], we briefly recall them together with the basic definitions in the next section. Although the notation here is different from that of [1], the substance is all the same.

2. Asymptotic behavior of functions

We assume that all the functions are continuous and real–valued, whose domains are intervals of the type $[a,\infty)$ depending on functions. According to aymptotic properties of functions, we divide such functions into three categories; $\mathcal{S}_{1}$, $\mathcal{S}_{2}$ and $\mathcal{S}_{3}$. In the following ’$\limsup$’ and ’$\liminf$’ mean $\limsup_{t\to+\infty}$ and $\liminf_{t\to+\infty}$, respectively.

• ($\mathcal{S}_{1}$)

$\mathcal{S}_{1}$ denotes the set of all functions $f$ such that

[TABLE]

for all real numbers $p$. Some typical examples include $e^{\sqrt{t}}$, $e^{t^{2}}$. It is obvious that in case of $f\in\mathcal{S}_{1}$ we have $|f|^{\nu}\in\mathcal{S}_{1}$ for $\nu>0$.

• ($\mathcal{S}_{2}$)

$\mathcal{S}_{2}$ denotes the set of all functions $f$ such that

[TABLE]

for all real numbers $p$. Some typical examples include $e^{-\sqrt{t}}$, $e^{-t^{2}}$.

• ($\mathcal{S}_{3}$)

$\mathcal{S}_{3}$ denotes the set of all functions $f$, not belonging to $\mathcal{S}_{1}$ nor to $\mathcal{S}_{2}$.

With each function $f\in\mathcal{S}_{3}$ we associate two values $\Pi(f)$ and $\Xi(f)$. The value $\Pi(f)$ is characterized by the number such that for all $\epsilon>0$

[TABLE]

If there exists no such a number (i.e., $\liminf|f(t)|/t^{p}=0$ for all $p$), put $\Pi(f)=-\infty$.

On the other hand, the value $\Xi(f)$ is characterized by the number such that for all $\epsilon>0$

[TABLE]

If there exists no such a number (i.e., $\limsup|f(t)|/t^{p}=\infty$ for all $p$), put $\Xi(f)=+\infty$.

It is obvious that $\Xi(f)\geq\Pi(f)$ for every $f\in{\mathcal{S}_{3}}$. As its example, taking $f(t)=t+e^{t}\cos t$, we have $\Pi(f)=1$ and $\Xi(f)=+\infty$. For a polynomial $f$ of degree $d$, we have $\Pi(f)=\Xi(f)=d$.

Note that when $\Pi(f)<0$ or $f\in\mathcal{S}_{2}$ there exists an increasing sequence $\{t_{k}\}$ such that $t_{k}\to\infty$ and $\lim_{k\to\infty}f(t_{k})=0$. Furthermore, the smaller $\Pi(f)$ is, the faster $f(t_{k})$ approaches to zero.

The following asymptotic behavior concerning the derivatives will be useful, when we consider some types of differential equations. We implicitly assume the existence of solutions with initial conditions, for which we discuss their asymptotic behavior.

Theorem 1. (see [1, Th3]) Let $f$ be a continuously differentiable function. Then the following properties hold.

• (i)

If both $f$ and its derivative $f^{\prime}$ belong to $\mathcal{S}_{3}$, then $\Pi(f^{\prime})\leq\Pi(f)-1$.

• (ii)

If $f$ belongs to $\mathcal{S}_{2}$, then either $f^{\prime}\in{\mathcal{S}_{2}}$ or $f^{\prime}\in{\mathcal{S}_{3}}$ with $\Pi(f^{\prime})=-\infty$.

• (iii)

If the derivative $f^{\prime}$ belongs to $\mathcal{S}_{1}$, then $f\in{\mathcal{S}_{1}}$.

In particular note that (i) also implies $\Pi(f^{\prime})=-\infty$ for functions with $\Pi(f)=-\infty$. The proofs are straightforward and can be found in [1].

3. Asymptotic behavior of solutions

We study a particular case of Eq.(2) that corresponds to the superlinear case of Eq.(1), i.e., $\lambda>1$. The following assumptions on the function $F$ reflect such a case. Indeed we can easily see that the function

[TABLE]

fulfills these assumptions for appropriate functions $\phi$ and appropriate numbers $\nu$ ($1<\nu<\lambda$), by virtue of [1, Th4 and Th5]. We suppress the variable $t$ for expressing a function (of $t$) itself such as $F(\cdot,y(\cdot))$, for example.

(Assumptions)

• (i)

For some number $\nu>1$ the function $F$ satisfies

[TABLE]

• (ii)

For all $f\in\mathcal{S}_{3}$, $F(\cdot,f(\cdot))\in\mathcal{S}_{3}$ and

[TABLE]

holds, where $\mu$ and $c$ are two constants such that $\mu>1$ and $c>-n$, $n$ being the order of Eq.(2).

As for the right-hand side of Eq.(3) it is shown in [1] that, when $\Pi(\phi)=\Xi(\phi)$ holds and hence this common value is finite, every $y\in\mathcal{S}_{3}$ satisfies

[TABLE]

Therefore, if we choose $\phi$ so that $\Pi(\phi)>-n$, then (ii) is satisfied. This is the case of the Thomas–Fermi equation; $\Pi(t^{-1/2})=-1/2>-2$.

Under the assumptions we first show that a solution of Eq.(2) cannot belong to $\mathcal{S}_{1}$ and then we seek the possibility of solutions among functions in $\mathcal{S}_{2}$ or $\mathcal{S}_{3}$.

Lemma 2. *Under Assumption (i) any function in $\mathcal{S}_{1}$ cannot be a solution of *Eq.(2).

Proof. The proof is by contradiction. Assume that $y\in\mathcal{S}_{1}$ with domain $[a,\infty)$ is a solution of Eq.(2); $y^{(n)}(t)=F(t,y(t))$. By Assumption (i) we have $F(t,y(t))=(y(t))^{\nu}g(t)$ for some function $g\in\mathcal{S}_{1}$ depending on $y$. Since the product of two functions in $\mathcal{S}_{1}$ also belongs to $\mathcal{S}_{1}$, it follows that $F(\cdot,y(\cdot))$ and $y^{(n)}$ also belong to $\mathcal{S}_{1}$. By applying Theorem 1 (iii) several times we see that $y^{(i)}\in\mathcal{S}_{1}$ for $i=0,1,\ldots,n$. For a $v_{1}>0$ put $f=y^{\prime}/y^{1+v_{1}}=-1/v_{1}(1/y^{v_{1}})^{\prime}$. By integrating both sides from $a$ to $t$ yields $1/(y(t))^{v_{1}}=c-v_{1}\int_{a}^{t}f(\tau)d\tau$, where $c$ is a constant. Since $y\in\mathcal{S}_{1}$, we have $\lim_{t\to\infty}y(t)=\infty$ and $\int_{a}^{\infty}f(\tau)d\tau$ is finite. Therefore, there exists a subset $J_{1}$ of $[a,\infty)$ whose Lebesgue measure is finite and only on which $|y^{\prime}/y^{1+v_{1}}|\geq 1$ holds. Repeating this argument, for any $v_{i}>0$ $(1\leq i\leq n)$, there exists a subset $J_{i}$ of $[a,\infty)$ whose Lebesgue measure is finite and only on which $|y^{(i)}/(y^{(i-1)})^{1+v_{i}}|\geq 1$ holds.

For the number $\nu>1$, making appropriate choices of $v_{1},\ldots,v_{n}>0$ and ${\mu}_{1},\ldots,{\mu}_{n-1}>1$, we see that $y^{(n)}/y^{\nu}$ can be written as

[TABLE]

Using these numbers, we have for all $t\in[a,\infty)\backslash J$,

[TABLE]

where $J=J_{1}\cup J_{2}\cdots\cup J_{n}$ and the Lebesgue measure of $J$ is finite. However, rewriting Eq.(2) as

[TABLE]

we see that this equality contradicts with the fact that the right-hand side of this equality is a function belonging to $\mathcal{S}_{1}$ by Assumption (i). Hence any function in $\mathcal{S}_{1}$ cannot be a solution of Eq.(2). This completes the proof.

In order to deduce more results on asymptotic solutions of Eq.(2), we employ Assumption (ii) on $F$. They generalize the assertions of [1, Th8] in some aspects.

Theorem 3. *Let $y$ be a solution of Eq.(2). Then there exists an integer $i$ $(0\leq i\leq n)$ such that for each integer $j$ $(i\leq j\leq n)$ an increasing sequence $\{t_{k}\}$ can be chosen such that $t_{k}\to\infty$ and $\lim_{k\to\infty}y^{(j)}(t_{k})=0$.

Proof.* Recall that Theorem 1 tells us the following; a continuously differentiable function $y\in\mathcal{S}_{2}$ satisfies either $y^{\prime}\in\mathcal{S}_{2}$ or $y^{\prime}\in\mathcal{S}_{3}$ with $\Pi(y^{\prime})=-\infty$, and moreover a continuously differentiable function $y\in\mathcal{S}_{3}$ satisfies either $y^{\prime}\in\mathcal{S}_{2}$ or $y^{\prime}\in\mathcal{S}_{3}$ with $\Pi(y^{\prime})\leq\Pi(y)-1$. Lemma 2 implies that a solution $y$ of Eq.(2) either belongs to $\mathcal{S}_{2}$ or to $\mathcal{S}_{3}$. First consider the case $y\in\mathcal{S}_{2}$. Then we see that $y^{\prime}\in\mathcal{S}_{2}$ or $y^{\prime}\in\mathcal{S}_{3}$ with $\Pi(y^{\prime})=-\infty$. Repeating this argument we have, for $i=0,1,\ldots,n$, either

[TABLE]

Both mean that there exists an increasing sequence $\{t_{k}\}$ such that $t_{k}\to\infty$ and $\lim_{k\to\infty}y^{(i)}(t_{k})=0$ for each $i$ $(0\leq i\leq n)$.

Next assume $y\in\mathcal{S}_{3}$. Then we have either $y^{\prime}\in\mathcal{S}_{2}$ or $y^{\prime}\in\mathcal{S}_{3}$ with $\Pi(y^{\prime})\leq\Pi(y)-1$. Applying Theorem 1 again, in case of $y^{\prime}\in\mathcal{S}_{2}$ it follows that either $y^{\prime\prime}\in\mathcal{S}_{2}$ or $y^{\prime\prime}\in\mathcal{S}_{3}$ with $\Pi(y^{\prime\prime})=-\infty$. In case of $y^{\prime}\in\mathcal{S}_{3}$ we have either $y^{\prime\prime}\in\mathcal{S}_{2}$ or $y^{\prime\prime}\in\mathcal{S}_{3}$ with $\Pi(y^{\prime\prime})\leq\Pi(y^{\prime})-1\leq\Pi(y)-2$. Continuing this process until $y^{(i)}$, we have the following two cases:

• (1)

either $y^{(i)}\in\mathcal{S}_{2}$ or $y^{(i)}\in\mathcal{S}_{3}$ with $\Pi(y^{(i)})=-\infty$, for some $i=1,\ldots,n$;

• (2)

$y^{(i)}\in\mathcal{S}_{3}$ for all $i=0,1,\ldots,n$ and $\Pi(y^{(n)})\leq\Pi(y)-n$.

When case (1) happens, the required assertion follows as above, particularly for all $j$ $(i\leq j\leq n)$. When case (2) happens, due to Assumption (ii) and $c>-n$, we have

[TABLE]

and

[TABLE]

Moreover, for each $i=1,\ldots,n$, it follows that

[TABLE]

Hence in case (2) we see that, for each $i$ $(0\leq i\leq n)$, there exists an increasing sequence for $y^{(i)}$ that satisfies the assertion of this theorem. This completes the proof.

Let us return to Eq.(3) with appropriate functions $\phi$ such as both $\Pi(\phi)$ and $\Xi(\phi)$ are finite. Suppose its solution $y$ is non-oscillatory. Then we can conclude that eventually $y^{(i)}(t)$ $(0\leq i\leq n-1)$ tends to zero monotonically. Otherwise $y^{(i+1)}$ and hence $y^{(n)}$ change signs infinitely many times. Due to the form of Eq.(3), so does $y$ itself, meaning that it becomes eventually oscillatory. Therefore, any solution of Eq.(3) is eventually either oscillatory or tends to zero monotonically.

References

S. Tanimoto, Asymptotic behavior of functions and solutions of some nonlinear differential equations, Journal of Mathematical Analysis and Applications, 138, 511–521, 1989.

• 2

J. S. W. Wong, On the generalized Emden–Fowler equation, SIAM Review, 17, 339–360, 1975.