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Second-order impulsive differential systems of mixed type: oscillation theorems

Abstract

In this paper, we obtain necessary and sufficient conditions for the oscillation of solutions to a second-order neutral differential equation with impulses. Two examples are provided to show the effectiveness and feasibility of the main results. Our main tool is Lebesgue’s dominated convergence theorem.

Introduction

Nowadays impulsive differential equations are attracting a lot of attention. They appear in the study of several real world problems (see, for instance, [1, 2, 15]). In general, it is well known that several natural phenomena are driven by differential equations, but the description of some real world problems subjected to sudden changes in their stated became very interesting from the mathematical point of view because they should be described considering systems of differential equations with impulses. Examples of the aforementioned phenomena are related to mechanical systems, biological systems, population dynamics, pharmacokinetics, theoretical physics, biotechnology processes, chemistry, engineering, and control theory.

We also stress that the modeling of these phenomena is suitably formulated by evolutive partial differential equations; moreover, moment problem approaches appear also as a natural instrument in control theory of neutral type systems; see [16, 20, 34] and [13], respectively.

The literature related to impulsive differential equations is very wide. Here we mention some recent developments in this field.

In [28], Shen and Wang considered impulsive differential equations of the following form:

{ u ( ι ) + r ( ι ) u ( ι ν ) = 0 , ι ϕ k , ι ι 0 , u ( ϕ k + ) u ( ϕ k ) = I k ( u ( ϕ k ) ) , k N ,
(1.1)

where rC(R,R) and I k C(R,R) for kN, and obtained sufficient conditions that ensure the oscillation and asymptotic behavior of the solutions of problem (1.1).

In [12], Graef et al. considered the problem

{ ( u ( ι ) q ( ι ) u ( ι ζ ) ) + r ( ι ) | u ( ι ν ) | λ sgn u ( ι ν ) = 0 , ι ι 0 , u ( ϕ k + ) = b k u ( ϕ k ) , k N
(1.2)

assuming that q(ι)PC([ ι 0 ,), R + ) (that is, \(q(\iota )\) is piecewise continuous in \([\iota _{0},\infty )\)), obtained sufficient conditions for the oscillation of the solutions of problem (1.2).

In [27], Shen and Zou obtained oscillation criteria for first-order impulsive neutral delay differential equations of the form

{ ( u ( ι ) q ( ι ) u ( ι ζ ) ) + r ( ι ) u ( ι ν 1 ) v ( ι ) u ( ι ν 2 ) = 0 , ν 1 ν 2 > 0 , u ( ϕ k + ) = I k ( u ( ϕ k ) ) , k N
(1.3)

obtaining sufficient conditions that ensure the oscillation of the solutions of (1.3) under the assumptions that q(ι)PC([ ι 0 ,), R + ) and \(b_{k}\leq \frac{I_{k}(u)}{u}\leq 1\).

Karpuz et al. in [14] extended the results contained in [27] by taking the nonhomogeneous counterpart of system (1.3) with variable delays.

Oscillation and nonoscillation properties for a class of second-order neutral impulsive differential equations with constant coefficients and constant delays were studied by Tripathy and Santra in [30], where the authors considered the problem

{ ( u ( ι ) q u ( ι ζ ) ) + r u ( ι ν ) = 0 , ι ϕ k , k N , Δ ( u ( ϕ k ) q u ( ϕ k ζ ) ) + r ˜ u ( ϕ k ν ) = 0 , k N .
(1.4)

Other necessary and sufficient conditions for the oscillation of a class of second-order neutral impulsive systems were established in [32], where Tripathy and Santra studied systems of the form

{ ( p ( ι ) ( u ( ι ) + q ( ι ) u ( ι ζ ) ) ) + r ( ι ) g ( u ( ι ν ) ) , ι ϕ k , k N , Δ ( p ( ϕ k ) ( u ( ϕ k ) + q ( ϕ k ) u ( ϕ k ζ ) ) ) + r ( ϕ k ) g ( u ( ϕ k ν ) ) = 0 , k N .
(1.5)

In [32], in particular, the authors are interested in oscillating systems that, after a perturbation by instantaneous change of state, remain oscillating.

In [26], Santra and Tripathy investigated the oscillatory behavior of the solutions for first-order impulsive neutral delay differential equations of the form

{ ( u ( ι ) q ( ι ) u ( ι ζ ) ) + r ( ι ) g ( u ( ι ν ) ) = 0 , ι ϕ k , ι ι 0 , u ( ϕ k + ) = I k ( u ( ϕ k ) ) , k N , u ( ϕ k + τ ) = I k ( u ( ϕ k τ ) ) , k N
(1.6)

for different values of the neutral coefficient q.

We also mention the paper [24] in which Santra and Dix, using Lebesgue’s dominated convergence theorem, obtained necessary and sufficient conditions for the oscillation of the solutions of the following second-order neutral differential equation with impulses:

{ ( p ( ι ) ( w ( ι ) ) γ ) + j = 1 m r j ( ι ) g j ( u ( ν j ( ι ) ) ) = 0 , ι ι 0 , ι ϕ k , k N , Δ ( p ( ϕ k ) ( w ( ϕ k ) ) γ ) + j = 1 m r ˜ j ( ϕ k ) g j ( u ( ν j ( ϕ k ) ) ) = 0 ,
(1.7)

where

$$ w(\iota )=u(\iota )+q(\iota )u\bigl(\zeta (\iota )\bigr),\qquad \Delta u(a)= \lim _{s\to a^{+}}u(s)-\lim_{s\to a^{-}}u(s).$$

In line with the contents of [24], Tripathy and Santra in [31] examined oscillation and nonoscillation properties for the solutions of the following class of forced impulsive nonlinear neutral differential systems:

{ ( p ( ι ) ( u ( ι ) + q ( ι ) u ( ι ζ ) ) ) + r ( ι ) g ( u ( ι ν ) ) = f ( ι ) , ι ϕ k , k N , Δ ( p ( ϕ k ) ( u ( ϕ k ) + q ( ϕ k ) u ( ϕ k ζ ) ) ) + r ˜ ( ϕ k ) g ( u ( ϕ k ν ) ) = f ˜ ( ϕ k ) , k N
(1.8)

for different values of \(q(\iota )\) and obtained sufficient conditions for the existence of positive bounded solutions of system (1.8).

Finally, we mention the recent work [33] in which Tripathy and Santra obtained some characterizations for the oscillation of solutions of the following second-order neutral impulsive differential system:

{ ( p ( ι ) ( w ( ι ) ) γ ) + j = 1 m r j ( ι ) x α j ( ν j ( ι ) ) = 0 , ι ι 0 , t ϕ k , Δ ( p ( ϕ k ) ( w ( ϕ k ) ) γ ) + j = 1 m h j ( ϕ k ) x α j ( ν j ( ϕ k ) ) = 0 , k N ,
(1.9)

where \(w(\iota )=u(\iota )+q(\iota )u(\zeta (\iota ))\) and \(-1< q(\iota )\leq 0\).

For further details on neutral impulsive differential equations and for recent results related to the oscillation theory for ordinary differential equations, we refer the reader to the papers [36, 8, 9, 11, 2123, 25, 29, 35] and to the references therein. In particular, the study of oscillation of half-linear/Emden–Fowler (neutral) differential equations with deviating arguments (delayed or advanced arguments or mixed arguments) has numerous applications in physics and engineering (e.g., half-linear/Emden–Fowler differential equations arise in a variety of real world problems such as in the study of p-Laplace equations, non-Newtonian fluid theory, the turbulent flow of a polytropic gas in a porous medium, and so forth); see, e.g., the papers [7, 10, 1620] for more details.

Motivated by the aforementioned findings, in this paper we prove necessary and sufficient conditions for the oscillation of solutions to a second-order nonlinear impulsive differential system of the form

( p ( ι ) ( w ( ι ) ) α ) +r(ι)g ( u ( ν ( ι ) ) ) =0,ι ι 0 ,ι ϕ k ,kN,
(1.10)
$$\begin{aligned}& \Delta \bigl(p(\phi _{k}) \bigl(w'(\phi _{k}) \bigr)^{\alpha} \bigr) + \tilde{r}(\phi _{k})g \bigl(u\bigl(\nu ( \phi _{k})\bigr) \bigr)=0 , \end{aligned}$$
(1.11)

where

$$ w(\iota )=u(\iota )+q(\iota )u\bigl(\zeta (\iota )\bigr),\qquad \Delta u(a)= \lim _{s\to a^{+}}u(s)-\lim_{s\to a^{-}}u(s), $$

the functions g, r, , p, q, ν, ζ are continuous and satisfy the conditions stated below; the sequence \(\{\phi _{k}\}\) satisfies \(0<\phi _{1}<\phi _{2}<\cdots <\phi _{k}< \to \infty \) as \(k\to \infty \); and α is the quotient of two positive odd integers.

In this paper we use the following assumptions:

  1. (a)

    νC([0,),R), ζ C 2 ([0,),R), \(\nu (\iota )<\iota \), \(\zeta (\iota )<\iota \), \(\lim_{\iota \to \infty}\nu (\iota )=\infty \), \(\lim_{\iota \to \infty}\zeta (\iota )=\infty \).

  2. (b)

    νC([0,),R), ζ C 2 ([0,),R), \(\nu (\iota )>\iota \), \(\zeta (\iota )<\iota \), \(\lim_{\iota \to \infty}\zeta (\iota )=\infty \).

  3. (c)

    p C 1 ([0,),R), r, r ˜ C([0,),R); \(0< p(\iota )\), \(0\leq r(\iota )\), \(0\leq \tilde{r}(\iota )\) for all \(\iota \geq 0\); \(\sum r(\iota )\) is not identically zero in any interval \([b,\infty )\).

  4. (d)

    q C 2 ([0,), R + ) with \(0\leq q(\iota )\leq a<1\);

  5. (e)

    gC(R,R) is nondecreasing and \(g(\iota )\iota >0\) for \(\iota \neq 0\).

  6. (f)

    \(\lim_{\iota \to \infty}P(\iota )=\infty \), where \(P(\iota )=\int _{0}^{\iota }p^{-1/\alpha}(s) \,\mathrm{d}s\).

Preliminary results

For the sake of simplicity, we set

$$\begin{aligned} &R_{1}(\iota )= r(\iota )g \bigl((1-a)w \bigl(\nu (\iota ) \bigr) \bigr); \\ &R_{(1,k)}= \tilde{r}(\phi _{k})g \bigl((1-a)w \bigl(\nu (\phi _{k}) \bigr) \bigr) . \end{aligned}$$

Lemma 2.1

Suppose that (a)–(f) hold for \(\iota \geq \iota _{0} \), and let u be an eventually positive solution of (1.10)(1.11). Then w satisfies

$$\begin{aligned}& 0< w(\iota ),\qquad w'(\iota )>0,\quad \textit{and} \quad \bigl(p(\iota ) \bigl(w'( \iota ) \bigr)^{\alpha} \bigr)'\leq 0 \quad \textit{for } \iota \geq \iota _{1} . \end{aligned}$$
(2.1)

Proof

Let u be an eventually positive solution. Then \(w(\iota )>0\) and there exists \(\iota _{0} \geq 0\) such that \(u(\iota )>0\), \(u(\nu (\iota ))>0\), \(u(\zeta (\iota ))>0\) for all \(\iota \geq \iota _{0} \). Then (1.10)–(1.11) gives that

( p ( ι ) ( w ( ι ) ) α ) = r ( ι ) g ( u ( ν ( ι ) ) ) 0 for  ι ϕ k , Δ ( p ( ϕ k ) ( w ( ϕ k ) ) α ) = r ˜ ( ϕ k ) g ( u ( ν ( ϕ k ) ) ) 0 for  k N ,
(2.2)

which shows that \(p(\iota ) (w'(\iota ) )^{\alpha}\) is nonincreasing for \(\iota \geq \iota _{0} \), including jumps of discontinuity. Next we claim that for \(w>0\), \(p(\iota ) (w'(\iota ) )^{\alpha}\) is positive for \(\iota \geq \iota _{1} >\iota _{0} \). If not, letting \(p(\iota ) (w'(\iota ) )^{\alpha}\leq 0\) for \(\iota \geq \iota _{1} \), we can choose \(c>0\) such that

$$ p(\iota ) \bigl(w'(\iota ) \bigr)^{\alpha}\leq -c, $$

that is,

$$ w'(\iota )\leq (-c)^{1/\alpha}p^{-1/\alpha}(\iota ) . $$

Integrating both sides from \(\iota _{1} \) to ι, we get

$$ w(\iota )-w(\iota _{1})-\sum_{k=1}^{\infty }w'( \phi _{k})\leq (-c)^{1/ \alpha} \bigl(P(\iota )-P(\iota _{1}) \bigr). $$

Taking limit on both sides as \(\iota \to \infty \), we have \(\lim_{\iota \to \infty}w(\iota )\leq -\infty \), which leads to a contradiction to \(w(\iota )>0\). Hence, \(p(\iota ) (w'(\iota ) )^{\alpha}>0\) for \(\iota \geq \iota _{1} \), that is, \(w'(\iota )>0\) for \(\iota \geq \iota _{1} \). This completes the proof. □

Lemma 2.2

Suppose that (a)–(f) hold for \(\iota \geq \iota _{0} \), and let u be an eventually positive solution of (1.10)(1.11). Then w satisfies

$$\begin{aligned} u(\iota )\geq (1-a)w(\iota )\quad \textit{for } \iota \geq \iota _{1}. \end{aligned}$$
(2.3)

Proof

Assume that u is an eventually positive solution of (1.10)–(1.11). Then \(w(\iota )>0\) and there exists \(\iota \geq \iota _{1} >\iota _{0} \) such that

$$\begin{aligned} u(\iota )&= w(\iota )-q(\iota )u\bigl(\zeta (\iota )\bigr) \\ &\geq w(\iota )-q(\iota )w\bigl(\zeta (\iota )\bigr) \\ &\geq w(\iota )-q(\iota )w(\iota ) \\ &= \bigl(1-q(\iota ) \bigr)w(\iota ) \\ &\geq (1-a)w(\iota ) . \end{aligned}$$

Hence w satisfies (2.3) for \(\iota \geq \iota _{1} \). □

Main results

In Theorem 3.1 we use a constant β, the quotient of two odd positive integers with \(\beta >\alpha \), for which

$$ \frac{g(\iota )}{\iota ^{\beta}}\text{ is nondecreasing for }0< \iota . $$
(3.1)

The existence of such a constant can be established by taking \(g(\iota )=|\iota |^{\delta}\operatorname{sgn}(\iota )\) with \(\beta <\delta \).

Theorem 3.1

Let (b)–(f) and (3.1) hold for \(\iota \geq \iota _{0} \). Then every solution of (1.10)(1.11) is oscillatory if and only if

$$ \int _{0}^{\infty }p^{-1/\alpha}(s) \biggl[ \int _{s}^{\infty }r(\psi ) \,\mathrm{d}\psi +\sum _{\phi _{k}\geq s} \tilde{r}(\phi _{k}) \biggr]^{1/ \alpha} \,\mathrm{d}s =\infty . $$
(3.2)

Proof

Let u be an eventually positive solution of (1.10)–(1.11). Then \(w(\iota )>0\) and there exists \(\iota _{0} \geq 0\) such that \(u(\iota )>0\), \(u(\nu (\iota ))>0\), \(u(\zeta (\iota ))>0\) for all \(\iota \geq \iota _{0} \). Thus, Lemmas 2.1 and 2.2 hold for \(\iota \geq \iota _{1} \). By Lemma 2.1, there exists \(\iota _{2}>\iota _{1} \) such that \(w'(\iota )>0\) for all \(\iota \geq \iota _{2}\). Then there exist \(\iota _{3}>\iota _{2}\) and \(c>0\) such that \(w(\iota )\geq c\) for all \(\iota \geq \iota _{3}\). Next, using Lemma 2.2, we get \(u(\iota )\geq (1-a)w(\iota )\) for all \(\iota \geq \iota _{3}\) and (1.10)–(1.11) become

$$ \begin{gathered} \bigl(p(\iota ) \bigl(w'( \iota ) \bigr)^{\alpha} \bigr)' + R_{1}(\iota ) \leq 0 \quad \text{for } \iota \neq \phi _{k}, \\ \Delta \bigl(p(\phi _{k}) \bigl(w'(\phi _{k}) \bigr)^{\alpha} \bigr) + R_{(1,k)} \leq 0\quad \text{for }k=1,2,\dots . \end{gathered} $$
(3.3)

Integrating (3.3) from ι to ∞, we get

$$\begin{aligned} &\bigl[p(s) \bigl(w'(s) \bigr)^{\alpha}\bigr]_{\iota}^{\infty }+ \int _{\iota}^{ \infty }R_{1}(s) \,\mathrm{d}s+ \sum _{\phi _{k} \geq \iota} R_{(1,k)} \leq 0 . \end{aligned}$$

Since \(p(\iota ) (w'(\iota ) )^{\alpha}\) is positive and nondecreasing, \(\lim_{\iota \to \infty} p(\iota ) (w'(\iota ) )^{\alpha}\) exists, and it is finite and positive. Then

$$ \begin{aligned} p(\iota ) \bigl(w'(\iota ) \bigr)^{\alpha }&\geq \int _{\iota}^{\infty }R_{1}(s) \,\mathrm{d}s+ \sum _{\phi _{k} \geq \iota} R_{(1,k)}, \end{aligned} $$

that is,

$$ \begin{aligned} w'(\iota ) &\geq p^{-1/\alpha}(\iota ) \biggl[ \int _{\iota}^{\infty }R_{1}(s) \,\mathrm{d}s + \sum _{\phi _{k} \geq \iota} R_{(1,k)} \biggr]^{1/\alpha} . \end{aligned} $$
(3.4)

Since

$$ \begin{aligned} g\bigl[(1-a)w \bigl(\nu (\iota ) \bigr)\bigr]&= \frac{g[(1-a)w (\nu (\iota ) )]}{(1-a)^{\beta }w^{\beta } (\nu (\iota ) )} (1-a)^{\beta }w^{\beta } \bigl(\nu (\iota ) \bigr) \\ &\geq \frac{g[c(1-a)]}{c^{\beta }(1-a)^{\beta}} (1-a)^{\beta }w^{ \beta } \bigl(\nu (\iota ) \bigr) \\ &= \frac{g[c(1-a)]}{c^{\beta}} w^{\beta } \bigl(\nu (\iota ) \bigr) , \end{aligned} $$
(3.5)

then we use (3.5) in (3.4) to get

$$\begin{aligned} w'(\iota ) &\geq p^{-1/\alpha}(\iota ) \biggl[ \int _{\iota}^{\infty }r(s) \frac{g[c(1-a)]}{c^{\beta}} w^{\beta } \bigl(\nu (s) \bigr) \,\mathrm{d}s \\ &\quad{}+ \sum_{\phi _{k}\geq \iota} \tilde{r} (\phi _{k}) \frac{g[c(1-a)]}{c^{\beta}} w^{\beta } \bigl(\nu (\phi _{k}) \bigr) \biggr]^{1/ \alpha} . \end{aligned}$$

Next, if we set \(K=\frac {g_{0}[c(1-a)]}{c^{\beta}}\), where \(g_{0}[c(1-a)]= \min \{g[c(1-a)]\}\), the above inequality becomes

$$\begin{aligned} &w'(\iota ) \geq K^{1/\alpha} p^{-1/\alpha}(\iota ) \biggl[ \int _{\iota}^{ \infty }r(s)w^{\beta } \bigl(\nu (s) \bigr) \,\mathrm{d}s+ \sum_{\phi _{k} \geq \iota} \tilde{r} (\phi _{k}) w^{\beta } \bigl(\nu (\phi _{k}) \bigr) \biggr]^{1/\alpha} . \end{aligned}$$

Using (b) and the fact that \(w(\iota )\) is nondecreasing, we have

$$\begin{aligned} w'(\iota ) &\geq K^{1/\alpha} p^{-1/\alpha}(\iota ) \biggl[ \int _{\iota}^{ \infty }r(s) \,\mathrm{d}s + \sum _{\phi _{k}\geq \iota} \tilde{r} (\phi _{k}) \biggr]^{1/\alpha} w^{\beta /\alpha}(\iota ), \end{aligned}$$

i.e.,

$$\begin{aligned} \frac{w'(\iota )}{w^{\beta /\alpha}(\iota )} &\geq K^{1/\alpha} p^{-1/ \alpha}(\iota ) \biggl[ \int _{\iota}^{\infty }r(s) \,\mathrm{d}s + \sum _{ \phi _{k}\geq \iota} \tilde{r} (\phi _{k}) \biggr]^{1/\alpha} . \end{aligned}$$

Integrating both sides from \(\iota _{3}\) to ∞, we get

$$\begin{aligned} &K^{1/\alpha} \int _{\iota _{3}}^{\infty }p^{-1/\alpha}(s) \biggl[ \int _{s}^{ \infty }r(\psi ) \,\mathrm{d}\psi + \sum _{\phi _{k}\geq \iota} \tilde{r} ( \phi _{k}) \biggr]^{1/\alpha} \,\mathrm{d}s \leq \int _{\iota _{3}}^{ \infty }\frac{w'(s)}{w^{\beta /\alpha}(s)} \,\mathrm{d}s < \infty \end{aligned}$$

due to \(\beta >\alpha \), which is a contradiction to (3.2) and hence the sufficiency part of the theorem is proved.

Next we prove the necessary part by a contrapositive argument. If (3.2) does not hold, then for every \(\varepsilon >0\) there exists \(\iota \geq \iota _{0} \), for which

$$\begin{aligned} \int _{\iota}^{\infty }p^{-1/\alpha}(s) \biggl[ \int _{s}^{\infty }r( \psi ) \,\mathrm{d}\psi +\sum _{\phi _{k}\geq s} \tilde{r}(\phi _{k}) \biggr]^{1/\alpha} \,\mathrm{d}s< \varepsilon \quad \text{for } \iota \geq Y, \end{aligned}$$

where \(2\varepsilon = [\max \{g(\frac {1}{1-a})\} ]^{-1/\alpha}>0\).

Let us define the set

$$ V= \biggl\{ u\in C\bigl([0,\infty)\bigr): \frac{1}{2}\leq u(\iota )\leq \frac{1}{1-a} \text{ for all } \iota \geq Y \biggr\} $$

and \(\Phi : V\rightarrow V\) as

$$ (\Phi u) (\iota )= \textstyle\begin{cases} 0 &\text{if } \iota \leq Y, \\\frac{1+a}{2(1-a)}-q(\iota )u(\zeta (\iota )) \\ \quad {}+ \int _{\iota}^{\iota} p^{-1/\alpha}(s) [\int _{s}^{\infty }r( \psi )g (u(\nu (\psi )) ) \,\mathrm{d}\psi \\ \quad {}+\sum_{\phi _{k}\geq s} \tilde{r}(\phi _{k})g (u(\nu (\phi _{k})) ) ]^{1/\alpha} \,\mathrm{d}s &\text{if } \iota >Y . \end{cases} $$

Now we prove that \((\Phi u)(\iota )\in V\). For \(u(\iota )\in V\),

$$\begin{aligned} (\Phi u) (\iota )&\leq \frac{1+a}{2(1-a)}+ \int _{T}^{\iota} p^{-1/ \alpha}(s) \biggl[ \int _{s}^{\infty }r(\psi )g \biggl(\frac{1}{1-a} \biggr) \,\mathrm{d}\psi \\ &\quad{}+\sum_{\phi _{k}\geq s} \tilde{r}(\phi _{k})g \biggl(\frac{1}{1-a} \biggr) \biggr]^{1/\alpha} \,\mathrm{d}s \\ &\leq \frac{1+a}{2(1-a)}+ \biggl[\max \biggl\{ g \biggl(\frac{1}{1-a} \biggr)\biggr\} \biggr]^{1/ \alpha} .\varepsilon \\ &=\frac{1+a}{2(1-a)} +\frac{1}{2}= \frac{1}{1-a}, \end{aligned}$$

and further, for \(u(\iota )\in V\),

$$\begin{aligned} (\Phi u) (\iota )\geq \frac{1+a}{2(1-a)} - q(\iota ).\frac{1}{1-a}+0 \geq \frac{1+a}{2(1-a)} - \frac{a}{1-a} =\frac{1}{2}. \end{aligned}$$

Hence Φ maps from V to V.

Now we are going to find a fixed point for Φ in V, which will give an eventually positive solution of (1.10)–(1.11).

First we define a sequence of functions in V by

$$\begin{aligned}& u_{0}(\iota )=0 \quad \text{for } \iota \geq _{0}, \\& u_{1}(\iota )=(\Phi u_{0}) (\iota )= \textstyle\begin{cases} 0 &\text{if } \iota < Y, \\ \frac {1}{2} &\text{if } \iota \geq Y, \end{cases}\displaystyle \\& u_{n+1}(\iota ) = (\Phi u_{n}) (\iota )\quad \text{for }n \geq 1, \iota \geq Y. \end{aligned}$$

Here we see \(u_{1}(\iota )\geq u_{0}(\iota )\) for each fixed ι and \(\frac {1}{2}\leq u_{n-1}(\iota )\leq u_{n}(\iota )\leq \frac {1}{1-a}\) for \(\iota \geq Y\) for all \(n\geq 1\). Thus \({u_{n}}\) converges point-wise to a function u. By Lebesgue’s dominated convergence theorem u is a fixed point of Φ in V, which shows that it has a nonoscillatory solution. This completes the proof of the theorem. □

In Theorem 3.2 we take a constant β, the quotient of two odd positive integers with \(\beta < \alpha \), for which

$$ \frac{g(\iota )}{\iota ^{\beta}}\text{ is nonincreasing for }0< \iota . $$
(3.6)

The existence of such a constant can be established by taking \(g(\iota )=|\iota |^{\delta}\operatorname{sgn}(\iota )\) with \(\beta >\delta \). The assumption upon β can be withdrawn by taking \(|u|^{\beta}\operatorname{sgn}(u)\) instead of \(u^{\beta}\).

Theorem 3.2

Let (a), (c)–(f), and (3.6) hold for \(\iota \geq \iota _{0} \). Then every solution of (1.10)(1.11) is oscillatory if

$$ \begin{aligned} &\frac{1}{(2c)^{\beta}} \Biggl[ \int _{0}^{\infty } r(\psi )g\bigl[c(1-a) P \bigl(\nu ( \psi ) \bigr)\bigr] \,\mathrm{d}\psi \\ &\quad{}+\sum_{k=1}^{\infty }\tilde{r}(\phi _{k})g\bigl[c(1-a)P \bigl(\nu (\phi _{k}) \bigr)\bigr] \Biggr]=\infty \quad \forall c \neq 0 . \end{aligned} $$
(3.7)

Proof

Let \(u(\iota )\) be an eventually positive solution of (1.10)–(1.11). Then, proceeding as in the proof of Theorem 3.1, we have \(\iota _{2}>\iota _{1} >\iota _{0} \) such that inequality (3.4) holds for all \(\iota \geq \iota _{2}\). Using (e), there exists \(\iota _{3}>\iota _{2}\) for which \(P(\iota )-P(\iota _{3})\geq \frac {1}{2} P(\iota )\) for \(\iota \geq \iota _{3}\). Integrating (3.4) from \(\iota _{3}\) to ι, we have

$$ \begin{aligned} w(\iota )-w(\iota _{3})&\geq \int _{\iota _{3}}^{\iota} p^{-1/\alpha}(s) \biggl[ \int _{s}^{\infty }R_{1}(\kappa )\,\mathrm{d} \kappa + \sum_{\phi _{k} \geq s} R_{(1,k)} \biggr]^{1/\alpha} \,\mathrm{d}s \\ &\geq \int _{\iota _{3}}^{\iota} p^{-1/\alpha}(s) \biggl[ \int _{\iota}^{ \infty }R_{1}(\kappa )\,\mathrm{d} \kappa + \sum_{\phi _{k} \geq \iota} R_{(1,k)} \biggr]^{1/\alpha} \,\mathrm{d}s, \end{aligned} $$
(3.8)

that is,

$$\begin{aligned} w(\iota )&\geq \bigl(P(\iota )-P(\iota _{3})\bigr) \biggl[ \int _{\iota}^{\infty }R_{1}( \kappa )\,\mathrm{d} \kappa + \sum_{\phi _{k} \geq \iota} R_{(1,k)} \biggr]^{1/ \alpha} \\ &\geq \frac {1}{2} P(\iota ) \biggl[ \int _{\iota}^{\infty }R_{1}( \kappa )\,\mathrm{d} \kappa + \sum_{\phi _{k} \geq \iota} R_{(1,k)} \biggr]^{1/ \alpha}. \end{aligned}$$
(3.8)

Since \(p(\iota ) (w'(\iota ) )^{\alpha}\) is nonincreasing and positive, then there exist \(c>0\) and \(\iota _{4}>\iota _{3}\) such that \(p(\iota ) (w'(\iota ) )^{\alpha}\leq c^{\alpha}\) for \(\iota \geq \iota _{4}\). Integrating the relation \(w'(\iota ) \leq cp^{-1/\alpha}(\iota )\) from \(\iota _{4}\) to ι, we have

$$\begin{aligned} w(\iota )-w(\iota _{4})&\leq c\bigl(P(\iota )-P(\iota _{4})\bigr), \end{aligned}$$

that is,

$$\begin{aligned} w(\iota )&\leq cP(\iota ) \quad \text{for } \iota \geq \iota _{4} . \end{aligned}$$
(3.9)

Using (3.6) and (3.9), we obtain

$$\begin{aligned} g\bigl[(1-a)w \bigl(\nu (\iota ) \bigr)\bigr]&= \frac{g[(1-a)w (\nu (\iota ) )]}{(1-a)^{\beta }w^{\beta } (\nu (\iota ) )} (1-a)^{\beta }w^{\beta } \bigl(\nu (\iota ) \bigr) \\ &\geq \frac{g[c(1-a)P (\nu (\iota ) )]}{c^{\beta }(1-a)^{\beta }P^{\beta } (\nu (\iota ) )} (1-a)^{\beta }w^{\beta } \bigl(\nu (\iota ) \bigr) \\ &= \frac{g[c(1-a)P (\nu (\iota ) )]}{c^{\beta }P^{\beta } (\nu (\iota ) )} w^{\beta } \bigl(\nu (\iota ) \bigr) \quad \forall \iota \geq \iota _{4} . \end{aligned}$$
(3.10)

Using (3.10) in (3.8), we obtain

$$\begin{aligned} w(\iota )&\geq \frac{1}{2} P(\iota ) \biggl[ \int _{\iota}^{\infty }r( \kappa ) \frac{g[c(1-a)P (\nu (\kappa ) )]}{c^{\beta }P^{\beta } (\nu (\kappa ) )} w^{\beta } \bigl(\nu (\kappa ) \bigr) \,\mathrm{d}\kappa \\ &\quad{}+\sum_{\phi _{k}\geq \iota} \tilde{r}(\phi _{k}) \frac{g[c(1-a)P (\nu (\phi _{k}) )]}{c^{\beta }P^{\beta } (\nu (\phi _{k}) )} w^{\beta } \bigl(\nu (\phi _{k}) \bigr) \biggr]^{1/\alpha} . \end{aligned}$$

Hence,

$$\begin{aligned} w(\iota )\geq \frac{1}{2}P(\iota )U^{1/\alpha}(\iota ) \quad \text{for } \iota \geq \iota _{4}, \end{aligned}$$

where

$$\begin{aligned} U(\iota )&=\frac{1}{c^{\beta}} \biggl[ \int _{\iota}^{\infty }r(\kappa )g\bigl[c(1-a)P \bigl(\nu ( \kappa ) \bigr)\bigr] \frac{w^{\beta } (\nu (\kappa ) )}{P^{\beta } (\nu (\kappa ) )} \,\mathrm{d}\kappa \\ &\quad{}+\sum_{\phi _{k}\geq \iota} \tilde{r}(\phi _{k})g \bigl[c(1-a)P \bigl(\nu ( \phi _{k}) \bigr)\bigr] \frac{w^{\beta } (\nu (\phi _{k}) )}{P^{\beta } (\nu (\phi _{k}) )} \biggr] . \end{aligned}$$

Now,

$$\begin{aligned} U'(\iota )&=-\frac {1}{c^{\beta}} r(\iota )g \bigl[c(1-a)P \bigl(\nu (\iota ) \bigr)\bigr] \frac{w^{\beta } (\nu (\iota ) )}{P^{\beta } (\nu (\iota ) )} \\ &\leq -\frac {1}{(2c)^{\beta}} r(\iota )g\bigl[c(1-a)P \bigl(\nu (\iota ) \bigr) \bigr]U^{\beta /\alpha}\bigl(\nu (\iota ) \bigr)\leq 0 \end{aligned}$$
(3.11)

and

$$\begin{aligned} \quad \Delta U(\phi _{k})= -\frac {1}{(2c)^{\beta}} r(\phi _{k})g\bigl[c(1-a)P \bigl(\nu (\phi _{k}) \bigr) \bigr]U^{\beta /\alpha}\bigl(\nu (\phi _{k}) \bigr)\leq 0, \end{aligned}$$
(3.12)

which shows that \(U(\iota )\) is nonincreasing on \([\iota _{4},\infty )\) and \(\lim_{\iota \to \infty}U(\iota )\) exists. Using (3.11) and (a), we find

$$\begin{aligned} \bigl[U^{1-\beta /\alpha}(\iota ) \bigr]'&= (1-\beta / \alpha )U^{-\beta / \alpha}(\iota )U'(\iota ) \\ &\leq -\frac{1-\beta /\alpha}{(2c)^{\beta}} r(\iota )g\bigl[c(1-a)P \bigl( \nu (\iota ) \bigr) \bigr]U^{\beta /\alpha}\bigl(\nu (\iota ) \bigr)U^{-\beta / \alpha}(\iota ) \\ &\leq -\frac{1-\beta /\alpha}{(2c)^{\beta}} r(\iota )g\bigl[c(1-a)P \bigl( \nu (\iota ) \bigr) \bigr]. \end{aligned}$$
(3.13)

To estimate the discontinuity of \(U^{1-\beta /\alpha}\), we use a Taylor polynomial of order 1 from the function \(h(u)=u^{1-\beta /\alpha}\), with \(0<\beta <\alpha \), about \(u=a\):

$$\begin{aligned} b^{1-\beta /\alpha}-a^{1-\beta /\alpha}\leq (1-\beta /\alpha )a^{- \beta /\alpha}(b-a). \end{aligned}$$

Then

$$\begin{aligned} \Delta U^{1-\beta /\alpha}(\phi _{k})&\leq (1-\beta /\alpha )U^{- \beta /\alpha}(\phi _{k})\Delta U(\phi _{k}) \\ &\leq -\frac{1-\beta /\alpha}{(2c)^{\beta}} r(\phi _{k})g\bigl[c(1-a)P \bigl(\nu (\phi _{k}) \bigr)\bigr]. \end{aligned}$$

Now, integrating (3.13) from \(\iota _{4}\) to ι, we have

$$\begin{aligned} &\bigl[U^{1-\beta /\alpha}(s) \bigr]_{\iota _{4}}^{\iota} -\sum _{\phi _{k} \geq \iota}\Delta \bigl[U^{1-\beta /\alpha}(\phi _{k}) \bigr] \\ &\quad \leq -\frac{1-\beta /\alpha}{(2c)^{\beta}} \int _{\iota _{4}}^{\iota} r(s)g\bigl[c(1-a)P \bigl(\nu (s) \bigr)\bigr]\,\mathrm{d}s, \end{aligned}$$

that is,

$$\begin{aligned} &\frac{1-\beta /\alpha}{(2c)^{\beta}} \Biggl[ \int _{0}^{\infty } r(s)g\bigl[c(1-a) P \bigl(\nu (s) \bigr)\bigr] \,\mathrm{d}s+\sum_{k=1}^{\infty } \tilde{r}(\phi _{k})g\bigl[c(1-a)P \bigl(\nu (\phi _{k}) \bigr)\bigr] \Biggr] \\ &\quad \leq - \bigl[U^{1-\beta /\alpha}(s) \bigr]_{\iota _{4}}^{\iota}< U^{1- \beta /\alpha}( \iota _{4})< \infty , \end{aligned}$$

which contradicts (3.7). This completes the proof. □

Example 3.1

Consider the neutral differential equations

$$\begin{aligned}& \bigl( \bigl( \bigl(u(\iota )+e^{-\iota}u\bigl(\zeta (\iota )\bigr) \bigr)' \bigr)^{1/3} \bigr)'+\iota \bigl(u(\iota -2)\bigr)^{7/3} =0 , \end{aligned}$$
(3.14)
$$\begin{aligned}& \bigl( \bigl( \bigl(u\bigl(3^{k}\bigr)-e^{-3^{k}}x \bigl(\zeta \bigl(3^{k}\bigr)\bigr) \bigr)' \bigr)^{1/3} \bigr)' +(\iota +2) \bigl(u \bigl(3^{k}-2\bigr)\bigr)^{7/3} =0 . \end{aligned}$$
(3.15)

Here \(\alpha = 1/3\), \(p(\iota )=1\), \(0< q(\iota )=e^{-\iota}<1\) \(\nu (\iota )= \iota -2\), \(\phi _{k}=3^{k}\) for kN, \(g(\iota )=\iota ^{7/3}\). For \(\beta =5/3\), we have \(\delta =7/3>\beta =5/3>\alpha =1/3\) and \(g(\iota )/\iota ^{\beta}=\iota ^{2/3}\), which are increasing functions. Now we check (3.2). We have

$$\begin{aligned} & \int _{\iota _{0} }^{\infty} \biggl[ \frac{1}{p(s)} \biggl[ \int _{s}^{ \infty }r(\psi ) \,d\psi +\sum _{\phi _{k}\geq s} \tilde{r}(\phi _{k}) \biggr] \biggr]^{1/\alpha} \,\mathrm{d}s \\ &\quad \geq \int _{\iota _{0} }^{\infty} \biggl[ \frac{1}{p(s)} \biggl[ \int _{s}^{ \infty }r(\psi ) \,d\psi \biggr] \biggr]^{1/\alpha} \,\mathrm{d}s \\ &\quad= \int _{2}^{\infty} \biggl[ \int _{s}^{\infty }\psi \,d\psi \biggr]^{3} \,\mathrm{d}s=\infty . \end{aligned}$$

So, all the conditions of Theorem 3.1 hold. Thus, each solution of (3.14)–(3.15) is oscillatory.

Example 3.2

Consider the neutral differential equations

$$\begin{aligned}& \bigl(e^{-\iota} \bigl( \bigl(u(\iota )+e^{-\iota}u \bigl(\zeta (\iota )\bigr) \bigr)' \bigr)^{11/3} \bigr)' +\frac{1}{\iota +1}\bigl(u(\iota -2)\bigr)^{1/3}=0 , \end{aligned}$$
(3.16)
$$\begin{aligned}& \bigl(e^{-k} \bigl( \bigl(u(k)+e^{-k}u\bigl(\zeta (k)\bigr) \bigr)' \bigr)^{11/3} \bigr)' + \frac{1}{\iota +4}\bigl(u(k-2)\bigr)^{1/3}=0 . \end{aligned}$$
(3.17)

Here \(\alpha = 11/3\), \(p(\iota )=e^{-\iota}\), \(0< q(\iota )=e^{-\iota}<1\), \(\nu (\iota )= \iota -2\), \(\phi _{k}=k\) for kN, \(P(\iota )= \int _{0}^{\iota }e^{3s/11} \,ds=\frac{11}{3}(e^{3\iota /11}-1)\), \(g(\iota )=\iota ^{1/3}\). For \(\beta =7/3\), we have \(\delta =1/3<\beta =7/3<\alpha =11/3\) and \(g(\iota )/\iota ^{\beta}=\iota ^{-2}\), which are decreasing functions. Now we check (3.7). We have

$$\begin{aligned} &\frac{1}{(2c)^{\beta}} \biggl[ \int _{0}^{\infty }r(\psi )gc(1-a) P \bigl(\nu (\psi ) \bigr)\biggr] \,\mathrm{d}\psi +\sum_{k=1}^{\infty } \tilde{r}( \phi _{k})g\bigl[c(1-a)P \bigl(\nu (\phi _{k}) \bigr)\bigr] ] \\ &\quad\geq \frac{1}{(2c)^{7/3}} \int _{0}^{\infty }r(\psi )g\bigl[c(1-a) P \bigl( \nu ( \psi ) \bigr)\bigr] \,\mathrm{d}\psi \\ &\quad= \frac{1}{(2c)^{7/3}} \int _{0}^{\infty}\frac{1}{\psi +1} \biggl[c(1-a) \frac{11}{3} \bigl(e^{3(\psi -2)/11}-1 \bigr) \biggr]^{1/3} \,\mathrm{d}\psi = \infty \quad \forall c>0. \end{aligned}$$

So, all the conditions of Theorem 3.2 hold, and therefore each solution of (3.16)–(3.17) is oscillatory.

Conclusions

In this work, we have undertaken the problem by taking a second-order highly nonlinear neutral impulsive differential system and established necessary and sufficient conditions for the oscillation of (1.10)–(1.11) when the neutral coefficient lies in \([0,1)\). It would be of interest to investigate the oscillation of (1.10)–(1.11) with different neutral coefficients; see, e.g., the papers [1719] for more details. Furthermore, it is also interesting to analyze the oscillation of (1.10)–(1.11) with a nonlinear neutral term; see, e.g., the paper [10] for more details.

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References

  1. Agarwal, R.P., O’Regan, D., Saker, S.H.: Oscillation and Stability of Delay Models in Biology. Springer, New York (2014)

    Book  MATH  Google Scholar 

  2. Bainov, D.D., Simeonov, P.S.: Impulsive Differential Equations: Asymptotic Properties of the Solutions. Series on Advances in Mathematics for Applied Sciences, vol. 28. World Scientific, Singapore (1995)

    Book  MATH  Google Scholar 

  3. Bazighifan, O., Ruggieri, M., Santra, S.S., Scapellato, A.: Qualitative properties of solutions of second-order neutral differential equations. Symmetry 12(9), 1520 (2020). https://doi.org/10.3390/sym12091520

    Article  Google Scholar 

  4. Bazighifan, O., Ruggieri, M., Scapellato, A.: An improved criterion for the oscillation of fourth-order differential equations. Mathematics 8(4), 610 (2020). https://doi.org/10.3390/math8040610

    Article  Google Scholar 

  5. Berezansky, L., Braverman, E.: Oscillation of a linear delay impulsive differential equations. Commun. Appl. Nonlinear Anal. 3, 61–77 (1996)

    MathSciNet  MATH  Google Scholar 

  6. Berezansky, L., Domoshnitsky, A., Koplatadze, R.: Oscillation, Nonoscillation, Stability and Asymptotic Properties for Second and Higher Order Functional Differential Equations. Chapman & Hall/CRC Press, Boca Raton (2020)

    Book  MATH  Google Scholar 

  7. Bohner, M., Hassan, T.S., Li, T.: Fite-Hille-Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments. Indag. Math. 29(2), 548–560 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Diblik, J.: Positive solutions of nonlinear delayed differential equations with impulses. Appl. Math. Lett. 72, 16–22 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Diblik, J., Svoboda, Z., Smarda, Z.: Retract principle for neutral functional differential equation. Nonlinear Anal., Theory Methods Appl. 71(12), 1393–1400 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Džurina, J., Grace, S.R., Jadlovská, I., Li, T.: Oscillation criteria for second-order Emden-Fowler delay differential equations with a sublinear neutral term. Math. Nachr. 293(5), 910–922 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ghosh, T., Santra, S.S., Scapellato, A.: Oscillation results for second-order neutral delay differential equations. AIP Conf. Proc. 2425, 210005 (2022)

    Article  Google Scholar 

  12. Graef, J.R., Shen, J.H., Stavroulakis, I.P.: Oscillation of impulsive neutral delay differential equations. J. Math. Anal. Appl. 268, 310–333 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Infusino, M., Kuhlmann, S.: Infinite dimensional moment problem: open questions and applications. In: Contemp. Math., vol. 697, pp. 187–201. Am. Math. Soc., Providence (2017)

    Google Scholar 

  14. Karpuz, B., Ocalan, O.: Oscillation criteria for a class of first-order forced differential equations under impulse effects. Adv. Dyn. Syst. Appl. 7(2), 205–218 (2012)

    MathSciNet  Google Scholar 

  15. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Oscillation Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

    Book  Google Scholar 

  16. Li, T., Pintus, N., Viglialoro, G.: Properties of solutions to porous medium problems with different sources and boundary conditions. Z. Angew. Math. Phys. 70(3), Art. 86, pp. 1–18 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, T., Rogovchenko, Y.V.: Oscillation of second-order neutral differential equations. Math. Nachr. 288(10), 1150–1162 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, T., Rogovchenko, Y.V.: Oscillation criteria for second-order superlinear Emden-Fowler neutral differential equations. Monatshefte Math. 184(3), 489–500 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, T., Rogovchenko, Y.V.: On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations. Appl. Math. Lett. 105, Art. 106293, pp. 1–7 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, T., Viglialoro, G.: Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime. Differ. Integral Equ. 34(5–6), 315–336 (2021)

    MathSciNet  MATH  Google Scholar 

  21. Luo, Z., Jing, Z.: Periodic boundary value problem for first-order impulsive functional differential equations. Comput. Math. Appl. 55, 2094–2107 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ruggieri, M., Santra, S.S., Scapellato, A.: On nonlinear impulsive differential systems with canonical and non-canonical operators. Appl. Anal. (2021). https://doi.org/10.1080/00036811.2021.1965586

    Article  Google Scholar 

  23. Ruggieri, M., Santra, S.S., Scapellato, A.: Oscillatory behavior of second-order neutral differential equations. Bull. Braz. Math. Soc. (2021). https://doi.org/10.1007/s00574-021-00276-3

    Article  MATH  Google Scholar 

  24. Santra, S.S., Dix, J.G.: Necessary and sufficient conditions for the oscillation of solutions to a second-order neutral differential equation with impulses. Nonlinear Stud. 27(2), 375–387 (2020)

    MathSciNet  MATH  Google Scholar 

  25. Santra, S.S., Scapellato, A.: Some conditions for the oscillation of second-order differential equations with several mixed delays. J. Fixed Point Theory Appl. 24(2), 18 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  26. Santra, S.S., Tripathy, A.K.: On oscillatory first order nonlinear neutral differential equations with nonlinear impulses. J. Appl. Math. Comput. 59, 257–270 (2019). https://doi.org/10.1007/s12190-018-1178-8

    Article  MathSciNet  MATH  Google Scholar 

  27. Shen, J., Zou, Z.: Oscillation criteria for first order impulsive differential equations with positive and negative coefficients. J. Comput. Appl. Math. 217, 28–37 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shen, J.H., Wang, Z.C.: Oscillation and asymptotic behaviour of solutions of delay differential equations with impulses. Ann. Differ. Equ. 10(1), 61–68 (1994)

    MATH  Google Scholar 

  29. Tripathy, A.K.: Oscillation criteria for a class of first order neutral impulsive differential-difference equations. J. Appl. Anal. Comput. 4, 89–101 (2014)

    MathSciNet  MATH  Google Scholar 

  30. Tripathy, A.K., Santra, S.S.: Characterization of a class of second order neutral impulsive systems via pulsatile constant. Differ. Equ. Appl. 9(1), 87–98 (2017)

    MathSciNet  MATH  Google Scholar 

  31. Tripathy, A.K., Santra, S.S.: On the forced impulsive oscillatory nonlinear neutral systems of the second order. Nonlinear Oscil. 23(2), 274–288 (2020)

    Google Scholar 

  32. Tripathy, A.K., Santra, S.S.: Necessary and sufficient conditions for oscillation of a class of second order impulsive systems. Differ. Equ. Dyn. Syst. 30(2), 433–450 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tripathy, A.K., Santra, S.S.: Necessary and sufficient conditions for oscillations to a second-order neutral differential equations with impulses. Kragujev. J. Math. 47(1), 81–93 (2023)

    Google Scholar 

  34. Viglialoro, G., Woolley, T.E.: Solvability of a Keller-Segel system with signal-dependent sensitivity and essentially sublinear production. Appl. Anal. 99(14), 2507–2525 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  35. Yu, J., Yan, J.: Positive solutions and asymptotic behavior of delay differential equations with nonlinear impulses. J. Math. Anal. Appl. 207, 388–396 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The researchers would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project. The authors are very grateful to the anonymous referees for their detailed comments and valuable suggestions.

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Santra, S.S., Scapellato, A. & Moaaz, O. Second-order impulsive differential systems of mixed type: oscillation theorems. Bound Value Probl 2022, 67 (2022). https://doi.org/10.1186/s13661-022-01648-4

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MSC

  • 34C10
  • 34K11

Keywords

  • Oscillation
  • Delay
  • Neutral
  • Lebesgue’s dominated convergence theorem