Oscillation of Emden–Fowler-Type Neutral Delay Differential Equations

Abstract

In this work, we consider a type of second-order functional differential equations and establish qualitative properties of their solutions. These new results complement and improve a number of results reported in the literature. Finally, we provide an example that illustrates our results.

1. Introduction

Consider the class of Emden–Fowler-type neutral delay differential equations of the form:

$${\left(f\left(y\right){\left(u^{\prime }\left(y\right)\right)}^c\right)}^{\prime }+h\left(y\right)w^a\left(\varsigma \left(y\right)\right)=0\mathrm{for}y\ge y_0>0,$$

(1)

where $u\left(y\right)=w\left(y\right)+g\left(y\right)w\left(\vartheta \right(y\left)\right)$ and c and a are the ratios of two odd positive integers. Let us consider the following conditions:

(a)

$f,h,\vartheta ,\varsigma \in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ such that $\vartheta \left(y\right)\le y$, $\varsigma \left(y\right)\le y$ for $y\ge y_0$, $\vartheta \left(y\right)\to \infty$ $\varsigma \left(y\right)\to \infty$ as $y\to \infty$;

(b)

${lim}_{y\to \infty }\mathsf{\Lambda }\left(y\right)=\infty$ where $\mathsf{\Lambda }\left(y\right)={\int }_0^y{\left(f\left(\eta \right)\right)}^{-1/c}d\eta$;

(c)

$g\in C({\mathbb{R}}_{+},{\mathbb{R}}_{-})$ with $-1+{(2/5)}^{1/c}\le -a\le g\left(y\right)\le 0$ for $y\in {\mathbb{R}}_{+}$;

(d)

$g\in C({\mathbb{R}}_{+},{\mathbb{R}}_{-})$ with $-1<-a\le g\left(y\right)\le 0$ for $y\in {\mathbb{R}}_{+}$.

Brands [1] showed that for bounded delays, the solutions to:

$$w^{″}\left(y\right)+h\left(y\right)w(y-\vartheta \left(y\right))=0$$

are oscillatory, if and only if the solutions to $w^{″}\left(y\right)+h\left(y\right)w\left(y\right)=0$ are oscillatory. Baculikova and Džurina [2] studied (1) when $c=a=1$, by considering the assumptions $0\le g\left(y\right)<\infty$ and ${lim}_{y\to \infty }\mathsf{\Lambda }\left(y\right)=\infty$. They obtained sufficient conditions for the oscillation of the solutions of (1), using comparison techniques. In another paper, Baculikova and Džurina [3] considered (1) and established sufficient conditions for the oscillation of the solutions of (1) under the assumptions $0\le g\left(y\right)<\infty$ and ${lim}_{y\to \infty }\mathsf{\Lambda }\left(y\right)=\infty$. Bohner et al. [4] established sufficient conditions for the oscillation of the solutions of (1) when $c=a$ and assuming ${lim}_{y\to \infty }\mathsf{\Lambda }\left(y\right)<\infty$ and $0\le g\left(y\right)<1$. Chatzarakis et al. [5] established sufficient conditions for the oscillation and asymptotic behavior of all solutions of second-order half-linear differential equations of the form:

$$\begin{array}{c}\hfill {\left(f\left(y\right){\left(w^{\prime }\left(y\right)\right)}^a\right)}^{\prime }+h\left(y\right)w^a\left(\vartheta \left(y\right)\right)=0.\end{array}$$

(2)

In subsequent work, Chatzarakis et al. [6] established improved oscillation criteria for (2). Džurina [7] established sufficient conditions for the oscillation of the solutions of (1) when $c=a=1$, by considering the assumptions ${lim}_{y\to \infty }\mathsf{\Lambda }\left(y\right)=\infty$ and $0\le g\left(y\right)\le g_0<\infty$. Fisnarova and Marik [8] considered the equation:

$${\left(f\left(y\right)\mathsf{\Phi }\left(u^{\prime }\left(y\right)\right)\right)}^{\prime }+c\left(y\right)\mathsf{\Phi }\left(w\left(\varsigma \left(y\right)\right)\right)=0,$$

where $u\left(y\right)=w\left(y\right)+g\left(y\right)w\left(\vartheta \right(y\left)\right)$ and $\mathsf{\Phi }\left(y\right)={\left|y\right|}^{p-2}y$, $p\ge 2$. Umar et al. [9] studied stochastic intelligent computing with neuro-evolution heuristics for nonlinear SITRsystem of novel COVID-19 dynamics. Guirao et al. [10] considered novel third-order nonlinear Emden–Fowler delay differential equations, designed a model, as well as found the numerical solutions of third-order nonlinear Emden–Fowler delay differential equations. Touchent et al. [11] established the modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives. Sabir et al. [12] considered second-order singular periodic nonlinear boundary value problems and performed the computing based on neuro-swarm intelligence. Grace et al. [13] established sufficient conditions for the oscillation of the solutions of (1) when $c=a$ and by considering the assumptions ${lim}_{y\to \infty }\mathsf{\Lambda }\left(y\right)<\infty$, ${lim}_{y\to \infty }\mathsf{\Lambda }\left(y\right)=\infty$, and $0\le g\left(y\right)<1$. Karpuz and Santra [14] obtained several sufficient conditions for the oscillatory and asymptotic behavior of the solutions of:

$${\left(f\left(y\right){(w\left(y\right)+g\left(y\right)w\left(\vartheta \left(y\right)\right))}^{\prime }\right)}^{\prime }+h\left(y\right)f\left(w\left(\varsigma \right(y\left)\right)\right)=0,$$

by considering the assumptions ${lim}_{y\to \infty }\mathsf{\Lambda }\left(y\right)<\infty$ and ${lim}_{y\to \infty }\mathsf{\Lambda }\left(y\right)=\infty$, for different ranges of g. Li et al. [15] established sufficient conditions for the oscillation of the solutions of (1), under the assumptions ${lim}_{y\to \infty }\mathsf{\Lambda }\left(y\right)<\infty$ and $g\left(y\right)\ge 0$.

For further work on the oscillation of the solutions to this type of equation, we refer the readers to [16,17,18,19,20,21,22,23,24,25,26,27]. Note that the majority of publications consider only sufficient conditions, and merely a few consider necessary and sufficient conditions. Hence, the objective in this work is to establish both necessary and sufficient conditions for the oscillatory and asymptotic behavior of the solutions of (1) without using the comparison and the Riccati techniques.

Many researchers investigated regularity properties and obtained several existence results of solutions to partial differential equations; see [28,29,30] and the references therein. Furthermore, the authors in [31,32] considered the study of the exact and approximate solutions to differential equations.

Neutral differential equations have several applications in the natural sciences and engineering. For example, they often appear in the study of distributed networks containing lossless transmission lines (see, e.g., [33]). In this paper, we restrict our attention to the study of (1), which includes the class of functional differential equations of the neutral type.

By a solution to Equation (1), we mean a function $w\in \mathrm{C}([Y_w,\infty ),\mathbb{R})$, where $Y_w\ge y_0$, such that $f{u}^{\prime }\in {\mathrm{C}}^1([Y_w,\infty ),\mathbb{R})$, satisfying (1) on the interval $[Y_w,\infty )$. A solution w of (1) is said to be proper if w is not identically zero eventually, i.e., $sup\left\{\right|w\left(y\right)|:y\ge Y>0\}>0$ for all $Y\ge Y_w$. We assume that (1) possesses such solutions. A solution of (1) is called oscillatory if it has arbitrarily large zeros on $[Y_w,\infty )$; otherwise, it is said to be non-oscillatory. (1) itself is said to be oscillatory if all of its solutions are oscillatory.

Remark 1.

When the domain is not specified explicitly, all functional inequalities considered in this paper are assumed to hold eventually, i.e., they are satisfied for all y large enough.

2. Main Results

Lemma 1.

Let (a), (b), and one of (c) or (d) hold. Furthermore, let w be an eventually positive solution of (1). Then, u satisfies one of the following two possible cases:

(i) 

$u\left(y\right)<0$, $u^{\prime }\left(y\right)>0$ and ${\left(f\left(y\right){\left(u^{\prime }\left(y\right)\right)}^c\right)}^{\prime }<0$;

(ii) 

$u\left(y\right)>0$, $u^{\prime }\left(y\right)>0$ and ${\left(f\left(y\right){\left(u^{\prime }\left(y\right)\right)}^c\right)}^{\prime }<0$

for $y_2\ge y_1>0$, where $y\ge y_2$ is sufficiently large.

Proof. 

Suppose that there exists a $y_1\ge Y_w$ such that $w\left(y\right)>0$, $w\left(\vartheta \right(y\left)\right)$, and $w\left(\varsigma \right(y\left)\right)>0$ for $y\ge y_1$. From (1) and (a), we have that:

$${\left(f\left(y\right){\left(u^{\prime }\left(y\right)\right)}^c\right)}^{\prime }=-h\left(y\right)w^a\left(\varsigma \left(y\right)\right)<0\mathrm{for}y\ge y_1.$$

(3)

Consequently, $\left(f\left(y\right){\left(u^{\prime }\left(y\right)\right)}^c\right)$ is nonincreasing on $[y_1,\infty )$. Since $f\left(y\right)>0$, thus either $u^{\prime }\left(y\right)<0$ or $u^{\prime }\left(y\right)>0$ for $y\ge y_2$, where $y_2\ge y_1$.

If $u^{\prime }\left(y\right)>0$ for $y\ge y_2$, then we have (i) and (ii). We prove now that $u^{\prime }\left(y\right)<0$ cannot occur.

If $u^{\prime }\left(y\right)<0$ for $y\ge y_2$, then there exists $\kappa >0$ such that $\left(f\left(y\right){\left(u^{\prime }\left(y\right)\right)}^c\right)\le -\kappa$ for $y\ge y_2$. Integrating over $[y_2,y)\subset [y_2,\infty )$ after dividing by f, we have:

$$u\left(y\right)\le u\left(y_2\right)-{\kappa }^{1/c}{\int }_{y_2}^y{\left(f\left(\eta \right)\right)}^{-1/c}d\eta \mathrm{for}y\ge y_2.$$

(4)

By virtue of Condition (b), ${lim}_{y\to \infty }u\left(y\right)=-\infty$. We consider now the following possible cases, separately.

If w is unbounded, then there exists a sequence $\left\{y_k\right\}$ such that ${lim}_{k\to \infty }{y}_k=\infty$ and ${lim}_{k\to \infty }w\left(y_k\right)=\infty$, where $w\left(y_k\right)=max\{w\left(\eta \right);y_0\le \eta \le y_k\}$. Since ${lim}_{y\to \infty }\vartheta \left(y\right)=\infty$, $\vartheta \left(y_k\right)>y_0$ for all sufficiently large k. By $\vartheta \left(y\right)\le y$,

$$w\left(\vartheta \left(y_k\right)\right)=max\{w\left(\eta \right);y_0\le \eta \le \vartheta \left(y_k\right)\}\le max\{w\left(\eta \right);y_0\le \eta \le y_k\}=w\left(y_k\right).$$

Therefore, for all large k,

$$u\left(y_k\right)=w\left(y_k\right)+g\left(y_k\right)w\left(\vartheta \left(y_k\right)\right)\ge (1+g\left(y_k\right))w\left(y_k\right)>0,$$

which contradicts the fact that ${lim}_{y\to \infty }u\left(y\right)=-\infty$.

If w is bounded, then u is also bounded, which contradicts ${lim}_{y\to \infty }u\left(y\right)=-\infty$. Hence, u satisfies one of the cases (i) and (ii). This completes the proof.  □

Lemma 2.

Let (a), (b), and one of (c) or (d) hold. Furthermore, let w be an eventually positive unbounded solution of (1). Then, u satisfies (ii), only.

Proof. 

Suppose that there exists a $y_1\ge y_0$ such that $w\left(y\right)>0$, $w\left(\vartheta \right(y\left)\right)$, and $w\left(\varsigma \left(y\right)\right)>0$ for $y\ge y_1$. Then, Lemma 1 holds, and u satisfies (i) and (ii) for $y_2\ge y_1$, where $y\ge y_2$. By Lemma 1, if w is unbounded, then $u\left(y\right)>0$. Therefore, (i) cannot occur. This completes the proof.  □

Lemma 3.

Let (a), (b), and one of (c) or (d) hold. Furthermore, let w be an eventually positive solution of (1) and u satisfy (i). Then, ${lim}_{y\to \infty }w\left(y\right)=0$.

Proof. 

Suppose that there exists $y_1\ge y_0$ such that $w\left(y\right)>0$, $w\left(\vartheta \right(y\left)\right)$, and $w\left(\varsigma \right(y\left)\right)>0$, for $y\ge y_1$. Then, Lemma 1 holds, and u satisfies (i) and (ii) for $y_2\ge y_1$, where $y\ge y_2$. Let u satisfy (i) for $y\ge y_2$. Therefore,

$$\begin{array}{ccc}\hfill 0\ge \underset{y\to \infty }{lim}u\left(y\right)& =& \underset{y\to \infty }{lim\; sup}u\left(y\right)\ge \underset{y\to \infty }{lim\; sup}\left(w\left(y\right)-ax\left(\vartheta \right(y\left)\right)\right)\hfill \\ & \ge & \underset{y\to \infty }{lim\; sup}w\left(y\right)+\underset{y\to \infty }{lim\; inf}\left(-aw\left(\vartheta \right(y\left)\right)\right)=(1-a)\underset{y\to \infty }{lim\; sup}w\left(y\right)\hfill \end{array}$$

which implies that ${lim\; sup}_{y\to \infty }w\left(y\right)=0$, and hence, ${lim}_{y\to \infty }w\left(y\right)=0$.  □

2.1. The Case $c>a$

In this subsection, we assume that there exists a constant $b>0$ such that $0<a<b<c$.

Lemma 4.

Let Assumptions (a), (b), (c), or (d) be satisfied. Furthermore, assume w is an eventually positive solution of (1) and u satisfies (ii). Then, there exists $y_2\ge y_1$ and $\kappa >0$ such that for $y\ge y_2$, the following inequalities hold:

$$\begin{array}{c}\hfill u\left(y\right)\le {\kappa }^{1/c}\mathsf{\Lambda }\left(y\right)\end{array}$$

(5)

$$\begin{array}{c}\hfill \left(\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y_1\right)\right){\left[{\int }_y^{\infty }h\left(\zeta \right){\left({\kappa }^{1/c}\mathsf{\Lambda }\left(\varsigma \left(\zeta \right)\right)\right)}^{a-b}{u}^b\left(\varsigma \left(\zeta \right)\right)d\zeta \right]}^{1/c}\le u\left(y\right).\end{array}$$

(6)

Proof. 

Suppose that there exists $y_1\ge y_0$ such that $w\left(y\right)>0$, $w\left(\vartheta \right(y\left)\right)$, and $w\left(\varsigma \left(y\right)\right)>0$ for $y\ge y_1$. Then, Lemma 1 holds, and u satisfies (i) and (ii) for $y_2\ge y_1$, where $y\ge y_2$. Let u satisfy (ii) for $y\ge y_2$. By (ii), $\left(f\left(y\right){\left(u^{\prime }\left(y\right)\right)}^c\right)$ is positive and non-increasing. Therefore, there exists $\kappa >0$ and $y_2\ge y_1$ such that $\left(f\left(y\right){\left(u^{\prime }\left(y\right)\right)}^c\right)\le \kappa$. Integrating the inequality $u^{\prime }\left(y\right)\le {(\kappa /f\left(y\right))}^{1/c}$, we have:

$$u\left(y\right)\le u\left(y_2\right)+{\kappa }^{1/c}\left(\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y_2\right)\right).$$

Since ${lim}_{y\to \infty }\mathsf{\Lambda }\left(y\right)=\infty$, the last inequality becomes:

$$\begin{array}{c}\hfill u\left(y\right)\le {\kappa }^{1/c}\mathsf{\Lambda }\left(y\right)\mathrm{for}y\ge y_2,\end{array}$$

which is (5). Note that $\kappa$ depends on w being evaluated at a time $y_2$. Thus, (6) must include all possible $\kappa$’s.

From (5), we have:

$$\begin{array}{c}\hfill u^a\left(\varsigma \left(y\right)\right)=u^{a-b}\left(\varsigma \left(y\right)\right)u^b\left(\varsigma \left(y\right)\right)\ge {\left({\kappa }^{1/c}\mathsf{\Lambda }\left(\varsigma \left(y\right)\right)\right)}^{a-b}{u}^b\left(\varsigma \left(y\right)\right).\end{array}$$

(7)

Using $u\left(y\right)\le w\left(y\right)$ and (7) in (1) and then integrating the final inequality from y to ∞, we have:

$$\begin{array}{c}\hfill \underset{A\to \infty }{lim}{\left[\left(f\left(\eta \right){\left(u^{\prime }\left(\eta \right)\right)}^c\right)\right]}_y^A+{\int }_y^{\infty }h\left(\eta \right){\left({\kappa }^{1/c}\mathsf{\Lambda }\left(\varsigma \left(\eta \right)\right)\right)}^{a-b}{u}^b\left(\varsigma \left(\eta \right)\right)d\eta \le 0.\end{array}$$

Using that $\left(f\left(y\right){\left(u^{\prime }\left(y\right)\right)}^c\right)$ is positive and non-increasing, we have:

$${\int }_y^{\infty }h\left(\eta \right){\left({\kappa }^{1/c}\mathsf{\Lambda }\left(\varsigma \left(\eta \right)\right)\right)}^{a-b}{u}^b\left(\varsigma \left(\eta \right)\right)d\eta \le \left(f\left(y\right){\left(u^{\prime }\left(y\right)\right)}^c\right)\mathrm{for}y\ge y_1.$$

Therefore,

$$\begin{array}{c}\hfill u^{\prime }\left(y\right)\ge {\left[\frac{1}{f\left(y\right)}{\int }_y^{\infty }h\left(\eta \right){\left({\kappa }^{1/c}\mathsf{\Lambda }\left(\varsigma \left(\eta \right)\right)\right)}^{a-b}{u}^b\left(\varsigma \left(\eta \right)\right)d\eta \right]}^{1/c}.\end{array}$$

(8)

Since $u\left(y\right)\ge 0$, integrating (8) from $y_2$ to y, we obtain:

$$\begin{array}{cc}\hfill u\left(y\right)& \ge {\int }_{y_2}^y{\left[\frac{1}{f\left(\eta \right)}{\int }_{\eta }^{\infty }h\left(\zeta \right){\left({\kappa }^{1/c}\mathsf{\Lambda }\left(\varsigma \left(\zeta \right)\right)\right)}^{a-b}{u}^b\left(\varsigma \left(\zeta \right)\right)d\zeta \right]}^{1/c}d\eta \hfill \\ \hfill & \ge \left(\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y_2\right)\right){\left[{\int }_y^{\infty }h\left(\zeta \right){\left({\kappa }^{1/c}\mathsf{\Lambda }\left(\varsigma \left(\zeta \right)\right)\right)}^{a-b}{u}^b\left(\varsigma \left(\zeta \right)\right)d\zeta \right]}^{1/c},\hfill \end{array}$$

which is (6).  □

Theorem 1.

Under Assumptions (a)–(c), every unbounded solution of (1) oscillates if and only if:

(e) 

${\int }_T^{\infty }h\left(\eta \right){\mathsf{\Lambda }}^a\left(\varsigma \left(\eta \right)\right)d\eta =+\infty$ for every $T>0$.

Proof. 

To prove sufficiency by contradiction, assume that w is a non-oscillatory unbounded solution of (1). Then, there exists $y_1\ge y_0$ such that either $w\left(y\right)>0$ or $w\left(y\right)<0$ for $y\ge y_1$. Assume that $w\left(y\right)>0$, $w\left(\vartheta \right(y\left)\right)>0$ and $w\left(\varsigma \right(y\left)\right)>0$ for $y\ge y_1$. Then, Lemmas 1, 2, and 4 hold for $y\ge y_2$. From Lemma 1, u satisfies (i) and (ii) for $y\ge y_2$. Again, from Lemma 2, u satisfies (ii) only for $y\ge y_2$. Therefore, by Lemma 4, we have:

$$u\left(y\right)>\left(\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y_1\right)\right){▵}^{1/c}\left(y\right)\mathrm{for}\mathrm{all}y\ge y_2,$$

where:

$$\begin{array}{c}\hfill ▵\left(y\right)={\int }_y^{\infty }h\left(\zeta \right){\left({\kappa }^{1/c}\mathsf{\Lambda }\left(\varsigma \left(\zeta \right)\right)\right)}^{a-b}{u}^b\left(\varsigma \left(\zeta \right)\right)d\zeta \ge 0.\end{array}$$

Since ${lim}_{y\to \infty }\mathsf{\Lambda }\left(y\right)=\infty$, there exists $y_3\ge y_2$, such that $\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y_1\right)\ge \frac{1}{2}\mathsf{\Lambda }\left(y\right)$ for $y\ge y_3$. Then:

$$u\left(y\right)>\frac{1}{2}\mathsf{\Lambda }\left(y\right){▵}^{1/c}\left(y\right)\mathrm{for}y\ge y_3,$$

and $u^b/{\left({\kappa }^{1/c}\mathsf{\Lambda }\right)}^b\ge {▵}^{b/c}/{\left(2{\kappa }^{1/c}\right)}^b$. Taking the derivative of ▵, we have:

$$\begin{array}{cc}\hfill {▵}^{\prime }\left(y\right)& =-h\left(y\right){\left({\kappa }^{1/c}\mathsf{\Lambda }\left(\varsigma \left(y\right)\right)\right)}^{a-b}{u}^b\left(\varsigma \left(y\right)\right)\hfill \\ \hfill & \le -h\left(y\right){\left({\kappa }^{1/c}\mathsf{\Lambda }\left(\varsigma \left(y\right)\right)\right)}^a{▵}^{b/c}\left(\varsigma \left(y\right)\right){\left(2{\kappa }^{1/c}\right)}^{-b}\le 0.\hfill \end{array}$$

Therefore, $▵\left(y\right)$ is non-increasing so ${▵}^{b/c}\left(\varsigma \left(y\right)\right)/{▵}^{b/c}\left(y\right)\ge 1$, and:

$${\left({▵}^{1-b/c}\left(y\right)\right)}^{\prime }=(1-b/c){▵}^{-b/c}\left(y\right){▵}^{\prime }\left(y\right)\le -(1-b/c)2^{-b}{\kappa }^{(a-b)/c}h\left(y\right){\mathsf{\Lambda }}^a\left(\varsigma \left(y\right)\right).$$

Integrating this inequality from $y_3$ to y, we have:

$${\left[{▵}^{1-b/c}\left(\eta \right)\right]}_{y_3}^y\le -(1-b/c)2^{-b}{\kappa }^{(a-b)/c}{\int }_{y_3}^{y}h\left(\eta \right){\mathsf{\Lambda }}^a\left(\varsigma \left(\eta \right)\right)d\eta .$$

Since $b/c<1$ and $▵\left(y\right)$ is positive and non-increasing, we have:

$${\int }_{y_3}^{y}h\left(\eta \right){\mathsf{\Lambda }}^a\left(\varsigma \left(\eta \right)\right)d\eta \le \frac{2^b{\kappa }^{(b-a)/c}}{(1-b/c)}{▵}^{1-b/c}\left(y_3\right).$$

This contradicts (e) and proves the oscillation of all solutions.

If $w\left(y\right)<0$ for $y\ge y_1$, then we set $x\left(y\right):=-w\left(y\right)$ for $y\ge y_1$ in (1). Therefore,

$$\left(f{\left({\overline{u}}^{\prime }\right)}^c\right)\left(y\right)+h\left(y\right)x^a\left(\varsigma \left(y\right)\right)=0\mathrm{for}y\ge y_1,$$

where $\overline{u}\left(y\right)=x\left(y\right)+g\left(y\right)x\left(\vartheta \left(y\right)\right)$. Then, proceeding as above, we reach the same contradiction. This proves the oscillation of all unbounded solutions of (1).

Next, we show that (e) is a necessary condition. Suppose that (e) does not hold; so, for some $\kappa >0$, the integral in (e) is finite. Therefore, there exists $y\ge y_0$ such that:

$${\int }_Y^{\infty }h\left(\eta \right){\mathsf{\Lambda }}^a\left(\varsigma \left(\eta \right)\right)d\eta \le \frac{{\kappa }^{1-a/c}}{5}.$$

Let us consider the closed subset M of continuous functions:

$$\begin{array}{cc}& M=\{w:w\in C([y_0,\infty ),\mathbb{R}),w\left(y\right)=0\mathrm{for}y\in (y_0,Y]\mathrm{and}\\ & (\frac{\kappa }{5}{)}^{1/c}[\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y\right)]\le w\left(y\right)\le {\kappa }^{1/c}[\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y\right)]\mathrm{for}y\ge y_0\}.\end{array}$$

We define the operator $\mathsf{\Phi }:M\to C([y_0,+\infty ),\mathbb{R})$ by:

$$\left(\mathsf{\Phi }w\right)\left(y\right)=\left\{\begin{array}{c}0,y\in (y_0,Y]\hfill \\ -g\left(y\right)w\left(\vartheta \left(y\right)\right)+{\int }_Y^y{\left[\frac{1}{f\left(\eta \right)}[\frac{\kappa }{5}+{\int }_{\eta }^{\infty }h\left(\zeta \right)w^a\left(\varsigma \left(\zeta \right)\right)d\zeta ]\right]}^{1/c}d\eta ,y\ge Y.\hfill \end{array}\right.$$

For every $w\in M$ and $y\ge Y$, we have:

$$\begin{array}{ccc}\hfill \left(\mathsf{\Phi }w\right)\left(y\right)& \ge & {\int }_Y^y{\left[\frac{1}{f\left(\eta \right)}[\frac{\kappa }{5}+{\int }_{\eta }^{\infty }h\left(\zeta \right)w^a\left(\varsigma \left(\zeta \right)\right)d\zeta ]\right]}^{1/c}d\eta \hfill \\ & \ge & {\int }_Y^y{\left[\frac{1}{f\left(\eta \right)}\frac{\kappa }{5}\right]}^{1/c}d\eta ={\left(\frac{\kappa }{5}\right)}^{1/c}[\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y\right)].\hfill \end{array}$$

For every $w\in M$ and $y\ge Y$, we have $w\left(y\right)\le {\kappa }^{1/c}\mathsf{\Lambda }\left(y\right)$ and $w^a\left(y\right)\le {\kappa }^{a/c}{\mathsf{\Lambda }}^a\left(y\right)$. Then:

$$\begin{array}{ccc}\hfill \left(\mathsf{\Phi }w\right)\left(y\right)& \le & -g\left(y\right)w\left(\vartheta \left(y\right)\right)+{\int }_Y^y{\left[\frac{1}{f\left(\eta \right)}\left(\frac{\kappa }{5}+\frac{\kappa }{5}\right)\right]}^{1/c}d\eta \hfill \\ & \le & a{\kappa }^{1/c}[\mathsf{\Lambda }\left(\vartheta \left(y\right)\right)-\mathsf{\Lambda }\left(y\right)]+{(2\kappa /5)}^{1/c}\left[\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y\right)\right]\hfill \\ & \le & a{\kappa }^{1/c}[\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y\right)]+{(2\kappa /5)}^{1/c}\left[\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y\right)\right]\hfill \\ & =& \left(a+{(2/5)}^{1/c}\right){\kappa }^{1/c}\left[\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y\right)\right]\le {\kappa }^{1/c}\left[\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y\right)\right]\hfill \end{array}$$

which implies that $\left(\mathsf{\Phi }w\right)\left(y\right)\in M$. Let us define now a sequence of continuous function $v_n:[y_0,+\infty )\to \mathbb{R}$ by the recursive formula:

$$v_0\left(y\right)=\left\{\begin{array}{c}0,y\in (y_0,Y]\hfill \\ \frac{\kappa }{5}[\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y\right)],y\ge Y.\hfill \end{array}\right.$$

$$v_n\left(y\right)=\left(\mathsf{\Phi }{v}_{n-1}\right)\left(y\right),n\ge 1$$

It is easy to verify that for $n>1$,

$${\left(\frac{\kappa }{5}\right)}^{1/c}\left[\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y\right)\right]\le u_{n-1}\left(y\right)\le u_n\left(y\right)\le {\kappa }^{1/c}\left[\mathsf{\Lambda }\left(y\right)-\mathsf{\Lambda }\left(y\right)\right].$$

Therefore, the pointwise limit of the sequence exists. Let ${lim}_{y\to \infty }{v}_n\left(y\right)=v\left(y\right)$ for $y\ge y_0$. By Lebesgue’s dominated convergence theorem, $v\in M$ and $\left(\mathsf{\Phi }v\right)\left(y\right)=v\left(y\right)$, where $v\left(y\right)$ is a solution of (1) on $[T,\infty )$ such that $v\left(y\right)>0$. Hence, (e) is necessary. This completes the proof.  □

Example 1. 

Consider the delay differential equation:

$${(e^{-y}{({(w(y)-e^{-y}w(y-1))}^{\prime })}^{3/5})}^{\prime }+y{\left(w(y-2)\right)}^{1/3}=0,y\ge 0.$$

(9)

Here, $c=3/5$, $a=1/3$, $f\left(y\right)=e^{-y}$, $-1<g\left(y\right)=-e^{-y}\le 0$, $\vartheta \left(y\right)=y-1$, $\varsigma \left(y\right)=y-2$, $\mathsf{\Lambda }\left(y\right)={\int }_0^y{e}^{5\eta /3}d\eta =\frac{3}{5}\left(e^{5y/3}-1\right)$. For $b=1/2$ and $p\le q$, we have $0<a<b<c$ and $p^{a-b}=p^{-1/6}\ge q^{a-b}=q^{-1/6}$. To check (e), we have:

$$\begin{array}{c}\hfill {\int }_0^{\infty }h\left(\eta \right){\mathsf{\Lambda }}^a\left(\varsigma \left(\eta \right)\right)d\eta ={\int }_0^{\infty }\eta {\left(\frac{3}{5}\left(e^{5(\eta -2)/3}-1\right)\right)}^{1/3}d\eta =\infty ,\end{array}$$

since the integral approaches $+\infty$ as $\eta \to +\infty$. Therefore, all the assumptions of Theorem 1 hold. Thus, every unbounded solution of (9) oscillates.

Theorem 2.

Under Assumptions (a)–(c), every solution of (1) oscillates or ${lim}_{y\to \infty }w\left(y\right)=0$ if and only if (e) holds.

Proof. 

We show sufficiency by contradiction. Assume that w is an eventually positive solution of (1). Then, there exists $y_1\ge y_0$ such that $w\left(y\right)>0$, $w\left(\vartheta \right(y\left)\right)>0$, and $w\left(\varsigma \right(y\left)\right)>0$ for $y\ge y_1$. Then, Lemmas 1, 3, and 4 hold for $y\ge y_2$. From Lemma 1, u satisfies (i) and (ii) for $y\ge y_2$.

If u satisfies (i), then ${lim}_{y\to \infty }w\left(y\right)=0$ due to Lemma 3.

If u satisfies (ii), then we proceed as in Theorem 1 to get a contradiction to (e). Thus, (e) is a sufficient condition.

The case where w is a negative solution is similar, and we omit it here.

The necessary part is the same as in Theorem 1. Thus, the proof of the Theorem is complete.  □

2.2. The Case $c<a$

In this subsection, we assume that there exists a constant $b>0$ such that $a>b>c>0$.

Lemma 5.

Assume (a), (b), and one of (c) or (d) are satisfied. Let w be an eventually positive solution of (1) and u satisfy (ii). Then, there exists $y_2\ge y_1$ and $\kappa >0$ such that for $y\ge y_2$, the following inequality holds:

$$\begin{array}{c}\hfill u^a\left(\varsigma \left(y\right)\right))\ge {\kappa }^{a-b}{u}^b\left(\varsigma \left(y\right)\right).\end{array}$$

(10)

Proof. 

Suppose that there exists $y_1\ge y_0$ such that $w\left(y\right)>0$, $w\left(\vartheta \right(y\left)\right)$, and $w\left(\varsigma \left(y\right)\right)>0$ for $y\ge y_1$. Then, Lemma 1 holds, and u satisfies (i) and (ii) for $y_2\ge y_1$, where $y\ge y_2$. Let u satisfy (ii) for $y\ge y_2$. By (ii), it follows that $u^{\prime }\left(y\right)>0$, so u is increasing, and $u\left(y\right)\ge u\left(y_2\right)$ for $y\ge y_2$. Thus:

$$u\left(\varsigma \left(y\right)\right)\ge u\left(\varsigma \left(y_2\right)\right):=\kappa >0\mathrm{for}y\ge y_2:=y_1.$$

Therefore,

$$\begin{array}{c}\hfill u^a\left(\varsigma \left(y\right)\right)=u^{a-b}\left(\varsigma \left(y\right)\right)u^b\left(\varsigma \left(y\right)\right)\ge {\kappa }^{a-b}{u}^b\left(\varsigma \left(y\right)\right)\mathrm{for}y\ge y_2,\end{array}$$

which is (10).  □

Theorem 3.

Assume that (a), (b), and (d) hold. Furthermore, assume that ${\varsigma }^{\prime }\left(y\right)>1$ and $r^{\prime }\left(y\right)\ge 0$ for $y\in {\mathbb{R}}_{+}$. Then, every solution of (1) oscillates or ${lim}_{y\to \infty }w\left(y\right)=0$ if and only if:

(f) 

${\int }_T^{\infty }{\left[\frac{1}{f\left(\eta \right)}[{\int }_{\eta }^{\infty }h\left(\zeta \right)d\zeta ]\right]}^{1/c}d\eta =+\infty$ for every $T>0$.

Proof. 

We show sufficiency by contradiction. Assume that w is an eventually positive solution of (1). Then, there exists $y_1\ge y_0$ such that $w\left(y\right)>0$, $w\left(\vartheta \right(y\left)\right)>0$, and $w\left(\varsigma \right(y\left)\right)>0$ for $y\ge y_1$. From Lemma 1, u satisfies (i) and (ii) for $y\ge y_2$.

If u satisfies (i), then ${lim}_{y\to \infty }w\left(y\right)=0$ due to Lemma 3.

If u satisfies (ii), then Lemma 5 holds for $y\ge y_2$. Using $u\left(y\right)\le w\left(y\right)$ and (10) in (1) and integrating the final inequality from y to ∞, we have:

$$\begin{array}{c}\hfill \underset{A\to \infty }{lim}{\left[{\left(f\left(\eta \right){\left(u^{\prime }\left(\eta \right)\right)}^c\right)}^{\prime }\right]}_y^A+{\kappa }^{a-b}{\int }_y^{\infty }h\left(\eta \right)u^b\left(\varsigma \left(\eta \right)\right)d\eta \le 0.\end{array}$$

Using that $\left(f\left(y\right){\left(w^{\prime }\left(y\right)\right)}^c\right)$ is positive and non-increasing and $r^{\prime }\left(y\right)\ge 0$, we have:

$${\kappa }^{a-b}{\int }_y^{\infty }h\left(\eta \right)u^b\left(\varsigma \left(\eta \right)\right)d\eta \le \left(f\left(y\right){\left(u^{\prime }\left(y\right)\right)}^c\right)\le \left(f\left(\varsigma \left(y\right)\right){\left(u^{\prime }\left(\varsigma \left(y\right)\right)\right)}^c\right)\le f\left(y\right){\left(u^{\prime }\left(\varsigma \left(y\right)\right)\right)}^c$$

for all $y\ge y_2$. Therefore,

$$\begin{array}{c}\hfill u^{\prime }\left(\varsigma \left(y\right)\right)\ge {\kappa }^{(a-b)/c}{\left[\frac{1}{f\left(y\right)}{\int }_y^{\infty }h\left(\eta \right)u^b\left(\varsigma \left(\eta \right)\right)d\eta \right]}^{1/c}\ge {\kappa }^{(a-b)/c}{u}^{b/c}\left(\varsigma \left(y\right)\right){\left[\frac{1}{f\left(y\right)}{\int }_y^{\infty }h\left(\eta \right)d\eta \right]}^{1/c}\end{array}$$

implies that:

$${\kappa }^{(a-b)/c}{\left[\frac{1}{f\left(y\right)}{\int }_y^{\infty }h\left(\eta \right)d\eta \right]}^{1/c}\le \frac{u^{\prime }\left(\varsigma \left(y\right)\right)}{u^{b/c}\left(\varsigma \left(y\right)\right)}$$

(11)

Since $b/c>1$ and $u^{\prime }\left(\varsigma \left(y\right)\right)\le u^{\prime }\left(\varsigma \left(y\right)\right){\varsigma }^{\prime }\left(y\right)$, integrating (11) from $y_2$ to ∞, we have:

$${\kappa }^{(a-b)/c}{\int }_{y_2}^{\infty }{\left[\frac{1}{f\left(\eta \right)}{\int }_{\eta }^{\infty }h\left(\zeta \right)d\zeta \right]}^{1/c}d\eta \le \frac{w^{1-b/c}\left(\varsigma \left(y_2\right)\right)}{b/c-1}<\infty .$$

This contradicts (f) and proves the oscillation of all solutions.

The case where w is an eventually negative solution is omitted since it can be dealt with similarly.

Next, we show that (f) is necessary. Assume that (f) does not hold, and let there exist $y\ge y_0$ such that:

$${\int }_Y^y{\left[\frac{1}{f\left(\eta \right)}[{\int }_{\eta }^{\infty }h\left(\zeta \right)d\zeta ]\right]}^{1/c}d\eta \le \frac{1-a}{5},$$

Let us consider the closed subset M of continuous functions:

$$M=\left\{w\in C([y_0,\infty ),\mathbb{R}):w\left(y\right)=\frac{1-a}{5},y\in [y_0,Y];\frac{1-a}{5}\le w\left(y\right)\le 1\mathrm{for}y\ge Y\right\}.$$

Then, we define the operator $\mathsf{\Phi }:M\to C([y_0,\infty ),\mathbb{R})$ by:

$$\left(\mathsf{\Phi }w\right)\left(y\right)=\left\{\begin{array}{c}\frac{1-a}{5},y\in [y_0,Y]\hfill \\ -g\left(y\right)w\left(\vartheta \left(y\right)\right)+\frac{1-a}{5}+{\int }_Y^y{\left[\frac{1}{f\left(\eta \right)}[{\int }_{\eta }^{\infty }h\left(\zeta \right)f\left(w\left(\varsigma \right(\zeta \left)\right)\right)d\zeta ]\right]}^{1/c}d\eta ,y\ge Y.\hfill \end{array}\right.$$

For every $x\in M$ and $y\ge Y$, $\left(\mathsf{\Phi }w\right)\left(y\right)\ge \frac{1-a}{5}$ and:

$$\begin{array}{ccc}\hfill \left(\mathsf{\Phi }w\right)\left(y\right)& \le & a+\frac{1-a}{5}+{\int }_Y^y{\left[\frac{1}{f\left(\eta \right)}[{\int }_{\eta }^{\infty }h\left(\zeta \right)d\zeta ]\right]}^{1/c}d\eta \hfill \\ & \le & a+\frac{1-a}{5}+\frac{1-a}{5}=\frac{3a+2}{5}<1\hfill \end{array}$$

which implies that $\mathsf{\Phi }w\in M$. The rest of the proof follows from Theorem 1. The proof of the Theorem is complete.  □

Example 2. 

Consider the delay differential equation:

$${\left({\left({\left(w\left(y\right)-e^{-y}w\left(\vartheta \left(y\right)\right)\right)}^{\prime }\right)}^{1/5}\right)}^{\prime }+(y+1){\left(w(y-2)\right)}^{\frac{7}{3}}=0,y\ge 0.$$

(12)

Here, $c=1/5$, $a=7/3$, $f\left(y\right)=1$, $\varsigma \left(y\right)=t-2$, $\mathsf{\Lambda }\left(y\right)=y$. For $b=4/3$ and $p\le q$, we have $a>b>c>0$ and $p^{a-b}=p\le q^{a-b}=q$. To check (f), we have:

$${\int }_2^{\infty }{\left[{\int }_{\eta }^{\infty }(\zeta +1)d\zeta \right]}^{5}d\eta =\infty .$$

Thus, all the assumptions of Theorem 3 hold. Hence, every solution of (12) oscillates or ${lim}_{y\to \infty }w\left(y\right)=0$.

3. Conclusions

It is worth noting that the necessary and sufficient conditions we established hold when $-1<g\left(y\right)\le 0$. These conditions do not hold in other ranges of $g\left(y\right)$. Therefore, the conditions we obtained hold in a limited range of $g\left(y\right)$.

At this point, we give one remark and two examples to conclude the paper.

Remark 2. 

The results in this paper also hold for equations of the form:

$${\left(f\left(y\right){\left({\left(w\left(y\right)+g\left(y\right)w\left(\vartheta \right(y\left)\right)\right)}^{\prime }\right)}^c\right)}^{\prime }+\sum _{i=1}^m{h}_i\left(y\right)w^{a_i}\left({\varsigma }_i\left(y\right)\right))=0,$$

where $g,f,h_i,{\varsigma }_i$ $(i=1,2,\dots ,m)$ satisfy Assumptions (a)–(d). Then, Theorems 1–3 can be extended, using an index i such that $h_i,a_i,{\varsigma }_i$ fulfill (e) and (f).

We conclude the paper by presenting two examples, which show how Remark 2 can be applied.

Example 3. 

Consider the delay differential equation:

$${\left(e^{-y}{\left({\left(w\left(y\right)-e^{-y}w\left(\vartheta \left(y\right)\right)\right)}^{\prime }\right)}^{3/5}\right)}^{\prime }+\frac{1}{y+1}{\left(w(y-2)\right)}^{1/3}+\frac{1}{y+2}{\left(w(y-1)\right)}^{1/5}=0,y\ge 0.$$

(13)

Here, $c=3/5$, $a_1=1/3$, $a_2=1/5$, $f\left(y\right)=e^{-y}$, $g\left(y\right)=-e^{-y}$, ${\varsigma }_1\left(y\right)=y-2$, ${\varsigma }_2\left(y\right)=y-1$, $\mathsf{\Lambda }\left(y\right)={\int }_0^y{e}^{5s/3}ds=\frac{3}{5}\left(e^{5y/3}-1\right)$. For $b=1/2$ and $p\le q$, we have $0<max\{a_1,a_2\}<b<c$, and $p^{a_1-b}=p^{-1/6}\ge q^{a_1-b}=q^{-1/6}$ and $p^{a_2-b}=p^{-3/10}\ge q^{a_2-b}=q^{-3/10}$. To check (e), we have:

$$\begin{array}{c}\hfill {\int }_0^{\infty }\sum _{i=1}^m{h}_i\left(\eta \right){\mathsf{\Lambda }}^{a_i}\left({\varsigma }_i\left(\eta \right)\right)d\eta \ge {\int }_0^{\infty }{q}_1\left(\eta \right){\mathsf{\Lambda }}^{a_1}\left({\varsigma }_1\left(\eta \right)\right)d\eta ={\int }_0^{\infty }\frac{1}{\eta +1}{\left(\frac{3}{5}\left(e^{5(\eta -2)/3}-1\right)\right)}^{1/3}d\eta =\infty ,\end{array}$$

since the integrand approaches $+\infty$ as $\eta \to +\infty$. Therefore, all the assumptions of Theorem 1 hold. Hence, every unbounded solution of (13) oscillates.

Example 4.

Consider the delay differential equation:

$${\left({\left({\left(w\left(y\right)-e^{-y}w\left(\vartheta \left(y\right)\right)\right)}^{\prime }\right)}^{5/7}\right)}^{\prime }+y{\left(w(y-2)\right)}^{5/3}+(y+1){\left(w(y-1)\right)}^3=0,y\ge 0.$$

(14)

Here, $c=5/7$, $a_1=5/3$, $a_2=3$, $f\left(y\right)=1$, ${\varsigma }_1\left(y\right)=y-2$, ${\varsigma }_2\left(y\right)=y-1$, $\mathsf{\Lambda }\left(y\right)=y$. For $b=4/3$ and $p\le q$, we have $min\{a_1,a_2\}>b>c>0$ and $p^{a_1-b}=p^{1/3}\le q^{a_1-b}=q^{1/3}$ and $p^{a_2-b}=p^{5/3}\le q^{a_2-b}=q^{5/3}$. Clearly, all the assumptions of Theorem 3 hold. Hence, every solution of (14) oscillates or ${lim}_{y\to \infty }w\left(y\right)=0$.