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New product-type oscillation criteria for first-order linear differential equations with several nonmonotone arguments
Boundary Value Problems volume 2023, Article number: 80 (2023)
Abstract
We use an improved technique to establish new sufficient criteria of product type for the oscillation of the delay differential equation
with \(b_{l},\sigma _{l}\in C([t_{0},\infty ),[0,\infty ))\) such that \(\sigma _{l}(t)\leq t\) and \(\lim_{t \rightarrow \infty} \sigma _{l}(t)=\infty \), \(l=1,2,\ldots,m\). The obtained results are applicable for the nonmonotone delay case. Their strength is supported by a detailed practical example.
1 Introduction
Consider the first-order differential equation with several delays of the form
with \(b_{l},\sigma _{l}\in C([t_{0},\infty ),[0,\infty ))\) such that \(\sigma _{l}(t)\leq t\) and \(\lim_{t \rightarrow \infty} \sigma _{l}(t)=\infty \), \(l=1,2,\ldots,m\).
Let \(t_{-1}\) be a real number defined by \(t_{-1}= \min_{1\leq l\leq m}\{ { \inf_{t\geq t_{0}} {\sigma _{l}(t)}\}}\). A function \(x(t)\) is called a solution of Eq. (1.1) if \(x\in C([t_{-1},\infty ),\mathbb{R})\) is continuously differentiable on \([t_{0},\infty )\) and satisfies Eq. (1.1) for all \(t \geq t_{0}\). If \(x(t)\) has arbitrary large zeros, then it is said to be oscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory; otherwise, it is nonoscillatory.
Oscillation and delay phenomena appear in various models from real-world applications; see, e.g., [30, 31] for models from mathematical biology, where oscillation and/or delay actions may be formulated by means of cross-diffusion terms. In particular, the oscillation of first-order delay differential equations has numerous applications in the analysis of higher-order differential equations with deviating arguments (e.g., we can investigate the oscillation and asymptotic behavior of higher-order differential equations with deviating arguments by relating the oscillation of these equations to that of associated first-order delay differential equations); see, e.g., [16, 22, 28, 32] for more detail. Indeed, the oscillation of first-order delay differential equations has attracted the attention of many mathematicians; see [1–15, 17–21, 23–27, 29, 33–42] and the references therein.
Note that most known criteria require the delays to be nondecreasing, although in many situations, relaxation of the monotonicity of the delay is required for some equations to be more realistic; see [13]. Indeed, the nonmonotonicity of the delay adds difficulties to the problem. As a result, some known criteria for the monotonic case fail to extend to the nonmonotone one; see Braverman and Karpuz [9]. This motivates us to investigate the oscillation of Eq. (1.1) without restricting the monotonic behavior of the delays. Our focus will be only on the lim sup-type conditions in the product form. Next, we give a brief summary of these criteria. First, we introduce the following important notation:
and
where \(r,l=1,2,\ldots,m\), and \(\varphi _{l}(t)\) and \(\varphi (t)\) are nondecreasing continuous functions such that
and
Furthermore, the number \(\lambda (\alpha )\) is defined as the smaller real root of the equation \({\mathrm{e}}^{\alpha z}=z\), and the number \(Q(\alpha )\) is defined by
The first work in our summary of oscillation criteria is due to Infante et al. [24]. They obtained the following two criteria:
and
where \(\zeta _{l,l}>0\), \(l=1,2,\ldots,m\).
Koplatadze [25] established the following three conditions:
where \(\bar{d}=\liminf_{t\rightarrow \infty } \sum_{l=1}^{m} \int _{\sigma _{l}(t)}^{t} (\prod_{r=1}^{m}b_{r}(u) )^{ \frac{1}{m}}\,du\),
where \(0<\bar{d}\leq \frac{1}{\mathrm{e}}\), \(\epsilon \in (0, \lambda (\bar{d}))\), and finally
where \(\Upsilon _{1}(t)=0\) and \(\Upsilon _{i}(t)={\mathrm{e}}^{\sum _{l=1}^{m} \int _{\sigma _{l}(t)}^{t} (\prod _{r=1}^{m} b_{r}(u) )^{\frac{1}{m}}\Upsilon _{i-1}(u) \,du}\), \(i=2,3,\ldots \) .
Attia et al. [4] introduced the condition
where
with \(\eta >0\), \(\epsilon \in (0, \lambda (\eta ))\), and \(r_{1}=1,2,\ldots,m\).
Bereketoglu et al. [7] defined the sequence \(\{\Phi _{\ell}(t)\}_{\ell \ge 0}\) by
and obtained the condition
Moremedi et al. [33] established the criterion
where \(\Lambda _{0}(t)=\sum_{l=1}^{m} b_{l}(t)\) and
Attia and El-Morshedy [5] improved (1.3) and (1.7) with \(i=3\) and obtained the criterion
where
for \(l=1,2,\ldots,m\), \(\eta >0\), and \(\epsilon \in (0, \lambda (\eta ))\).
In the next section, we obtain several new oscillation criteria for Eq. (1.1). Moreover, we give a practical example to show that our results can be used to test the oscillation of a certain equation, whereas the criteria listed above fail.
2 Main results
We state some important results for Eq. (1.1) when it possesses a positive solution \(x(t)\). In this case, \(x(t)\) is eventually nonincreasing and eventually satisfies the inequalities
and
Therefore [42], [24, Lemma 3.1], [19, Lemma 2.1.2], and the nonincreasing nature of \(x(t)\) imply respectively, for \(l=1,2,\dots,m\), that
and
where \(\zeta, \zeta _{l,l}>0\).
If nothing else is stated, all inequalities are assumed to hold eventually.
Lemma 2.1
If \(x(t)\) is an eventually positive solution of Eq. (1.1), then
where \(\lambda _{l}^{*}=\max \{\lambda ^{*}(\zeta _{l,l}),\lambda ^{*}( \zeta )\}\), and
Proof
Dividing Eq. (1.1) by \(x(t)\) and integrating from u to t, \(u \leq t\), we obtain
which is equivalent to
Therefore
Equation (2.6) leads to the following two inequalities, using (2.2) and (2.3), for all sufficiently small \(\epsilon >0\):
and
Now taking the lower limits as \(t\rightarrow \infty \) and then letting \(\epsilon \rightarrow 0\), we get
and
The last two inequalities are equivalent to (2.4). □
For an easy reference, the sequences \(\{\Omega _{r}^{(n)}(t)\}_{n\geq 0}\), \(r=1,2,\dots,m\), are defined as follows:
where \(\epsilon _{r} \in (0,{\mathrm{e}}^{\max \{\sum _{l=1}^{m} \zeta _{r,l} \bar{\lambda _{l}}, \lambda (\zeta )\zeta _{r} \}} )\), and
Lemma 2.2
Assume that \(x(t)\) is an eventually positive solution of Eq. (1.1), \(n \in \mathbb{N}_{0}\), and \(j\in \{1,2,\dots,m\}\). Then the inequalities \(G_{j,j}^{(n)}(t)<1\) and
are satisfied.
Proof
Since \(x(t)\) is an eventually positive solution of Eq. (1.1), for any sufficiently small \(\epsilon _{r}>0\), inequality (2.4) yields
Combining this inequality with the fact that \(\frac{x(\sigma _{r}(t))}{x(t)}\geq 1\), we obtain
Integrating Eq. (1.1) from \(\varphi _{i}(t)\) to t, \(i=1,2,\ldots,m\), we get
On the other hand, proceeding as in the proof of Lemma 2.1, we obtain (2.5), which yields
and
Substituting into (2.9), we get
Therefore
that is,
Hence
Now by (2.8) it follows that
Continuing in this way, we can prove that
Returning to (2.5), we obtain
Therefore (2.9) implies that
However, (2.10) leads to
Consequently, the previous equation leads to the inequality
This proves that \(G_{i,i}^{(n)}(t)<1\) and
Then the arithmetic–geometric mean leads to
Taking the product of both sides, we get
where \(A^{(n)}(t)=\prod_{r=1}^{m} (\frac{1}{1-G_{r,r}^{(n)}(t)} )\). Therefore
Thus
Then
□
Theorem 2.1
Assume that \(i\in \{1,2,\dots,m\}\) and either one of the following conditions is satisfied for some \(n \in \mathbb{N}_{0}\):
-
(i)
there exists a sequence \(\{c_{k}\}_{k\geq 0}\) such that \(\lim_{k\rightarrow \infty } c_{k}=\infty \) and
$$\begin{aligned} G_{i,i}^{(n)}(c_{k})\geq 1\quad \textit{for all $k \in \mathbb{N}_{0}$}, \end{aligned}$$(2.12) -
(ii)
$$\begin{aligned} &\limsup_{t\rightarrow \infty } \Biggl(\prod _{r=1}^{m} \frac{1}{1-G_{r,r}^{(n)}(t)} \Biggl(\prod _{r=1}^{m} Q (\eta _{r} )+ (m-1 )^{m} \prod_{r=1}^{m} \Biggl( \prod_{\underset{r_{1} \neq r}{r_{1}=1}}^{m} G_{r,r_{1}}^{(n)}(t) \Biggr)^{ \frac{1}{m-1}} \Biggr) \Biggr) \\ &\quad >1. \end{aligned}$$(2.13)
Then Eq. (1.1) is oscillatory.
Proof
We assume for contradiction that Eq. (1.1) has a nonoscillatory solution \(x(t)\). Because of the linearity of Eq. (1.1), there is no loss of generality to assume the existence of a sufficiently large \(T\geq t_{0}\) such that \(x(t)>0\) for all \(t\geq T\). Then Lemma 2.2 leads to \(G_{i,i}^{(n)}(t)<1\) for all \(i=1,2,\dots,m\) and \(n \in \mathbb{N}_{0}\). This contradicts (2.12) and hence proves (i). For the proof of (ii), we see from (2.1) and (2.7) that
which is impossible due to (2.13). □
Next, we define the functions \(C^{(n)}_{r}(t)\) and \(D_{r}^{(n)}(t)\) for some \(n\in \mathbb{N}_{0}\) as follows:
and
where \(\varphi _{r}(t)\) are strictly increasing functions for all \(r=1,2,\dots,m\).
Theorem 2.2
Assume that the function \(\varphi _{r}(t)\) is strictly increasing for each \(r=1,2,\dots,m\). Suppose that for some \(n\in \mathbb{N}_{0}\),
-
(i)
there exists a sequence \(\{d_{k}\}_{k\geq 0}\) such that \(\lim_{k\rightarrow \infty } d_{k}=\infty \),
$$\begin{aligned} C_{r}^{(n)}(d_{k})\geq 1\quad \textit{for some $r\in \{1,2,\dots,m\}$ and all $k\in \mathbb{N}_{0}$}, \end{aligned}$$(2.14)or
-
(ii)
$$\begin{aligned} &\limsup_{t\rightarrow \infty } \Biggl(\prod _{r=1}^{m} \biggl(\frac{1}{1-G_{r,r}^{(n)}(t)} \biggr) \Biggl( \prod_{r=1}^{m} D_{r}^{(n)}(t)+ (m-1 )^{m} \prod_{r=1}^{m} \Biggl( \prod_{\underset {r_{1} \neq r}{r_{1}=1}}^{m} G_{r,r_{1}}^{(n)}(t) \Biggr)^{ \frac{1}{m-1}} \Biggr) \Biggr) \\ &\quad >1. \end{aligned}$$(2.15)
Then Eq. (1.1) is oscillatory.
Proof
As in the proof of the previous theorem, we assume that Eq. (1.1) has an eventually positive solution \(x(t)\). Integrating Eq. (1.1) from t to \(\varphi ^{-1}_{r}(t)\), we have
that is,
Again, integrating Eq. (1.1) from \(\sigma _{r}(u)\) to \(t\leq u \leq \varphi ^{-1}_{r}(t)\), we obtain
Substituting into (2.16), we get
Recalling that (2.5) holds and \(x(t)\) is nonincreasing, it follows that
Since \(\frac{x(\sigma _{l_{1}}(u_{2}))}{x(u_{2})}\geq \Omega ^{(n)}_{l_{1}}(u_{2})\) (from (2.11)), we have
Therefore
which leads to \(C_{r}^{(n)}(t)<1\). This contradicts (2.14) and completes the proof of (i).
To prove (ii), we notice from (2.17) that
Substituting into (2.7) and then taking the upper limit of both sides, we get a contradiction with (2.15). The proof of the theorem is complete. □
Corollary 2.1
Let \(q_{k},\mu _{k}>0\) be such that \(\sigma _{k}(t) \leq t- \mu _{k}\), \(b_{k}(t) \geq q_{k}\) on \((a_{j}, a_{j}+3\mu ^{*} )\), \(k\in \{1,2,\dots,m\}\) and \(j \in \mathbb{N}_{0}\), \(\mu ^{*}={\max \mu _{k}}_{1\leq k \leq m}\), and \(\lim_{j\rightarrow \infty } a_{j}= \infty \). If
where \(D_{i,k}=\frac{q_{k}}{B} ({\mathrm{e}}^{\mu _{i} B}-1 )\) and \(B=\sum_{l=1}^{m} \frac{q_{l}}{1-\mu _{l} q_{l}}\), \(i=1,2,\ldots,m\), then Eq. (1.1) is oscillatory.
Proof
Let \(\varphi _{k}(t)=t-\mu _{k}\), \(k=1,2,\dots,m\). Then
This leads to
Also,
Therefore
Now let
Then (2.20) leads to
It follows that (2.13) with \(n=1\) is satisfied, and hence (ii) of Theorem 2.1 guarantees the oscillation of Eq. (1.1). □
Corollary 2.2
Let \(q_{k},\mu _{k}>0\) be such that \(\sigma _{k}(t) \leq t- \mu _{k}\), \(b_{k}(t) \geq q_{k}\) on \((a_{j}, a_{j}+4\mu ^{*} )\), \(k\in \{1,2,\dots,m\}\), and \(j \in \mathbb{N}\), \(\mu ^{*}={\max \mu _{k}}_{1\leq k \leq m}\), and \(\lim_{j\rightarrow \infty } a_{j}= \infty \). If
where B, \(D_{i,k}\) are defined as in Corollary 2.1, and
then Eq. (1.1) is oscillatory.
Proof
Let \(\varphi _{k}(t)=t-\mu _{k}\), \(k=1,2,\dots,m\). Then
In view of (2.19), we have
Thus
that is,
Also,
This inequality and (2.22) lead to
Let
Then (2.20), (2.21), and (2.23) imply that
Therefore condition (2.15) with \(n=1\) is satisfied, so Eq. (1.1) is oscillatory. The proof is complete. □
The following illustrative example highlights the significance of some of our results. All calculations are done using a Maple code.
Example 2.1
Consider the equation
where \(\sigma _{2}(t)=t-1-0.0001 \sin ^{2} ( 20000 \pi t )\), and
where \(0<\delta <0.1\). Also,
and
where \(\beta \geq 0\), and \(\{b_{k}\}_{k\geq 0}\), \(\{c_{k}\}_{k\geq 0}\) are sequences of positive integers such that \(c_{k}>b_{k}+1\), \(b_{k+1}>c_{k}+3.0001+\delta \), and \(\lim_{k\rightarrow \infty } b_{k}=\infty \). Let us assume that \(\varphi _{i}(t)=\theta _{i}(t)\), \(i=1,2\) (see (1.2) for definition). It is not difficult to see that \(0\leq b_{1}(t) \leq \frac{1}{2{\mathrm{e}}}\), \(0\leq b_{2}(t) \leq \beta \),
Since
we conclude that
On the other hand,
and
It follows that \(\bar{d}=\liminf_{t\rightarrow \infty } \sum_{l=1}^{2} \int _{\sigma _{l}(t)}^{t} (\prod_{l_{1}=1}^{2}b_{l_{1}}(u) )^{\frac{1}{2}}\,du=0\) and \(\zeta _{i,l}=\zeta =\eta _{l}=\eta =Q(\zeta _{i,l})=Q(\eta _{l})=0\) for \(l,i=1,2\). Consequently, conditions (1.4), (1.5), (1.6), (1.8), and (1.11) cannot be applied.
Also, since
and
for all \(\beta \in [0, \frac{1.43}{\mathrm{e}} ]\), where \(A_{1}= (\frac{1}{2\mathrm{e}} {\mathrm{e}}^{0.2 (\frac{1}{2\mathrm{e}}+ \beta )}+\beta {\mathrm{e}}^{1.0001 (\frac{1}{2\mathrm{e}}+\beta )} )\), we have
Consequently, condition (1.10) with \(\ell =1\) fails for all \(\beta \in [0, \frac{1.43}{\mathrm{e}} ]\).
Moreover, we have
and
for all \(\beta \in [0, \frac{2.23}{ {\mathrm{e}}}]\). Consequently,
for all \(\beta \in [0, \frac{2.23}{ {\mathrm{e}}}]\). This means that condition (1.9) with \(\ell =1\) and \(\beta \in [0, \frac{2.23}{ {\mathrm{e}}}]\) is not satisfied. Similarly, condition (1.3) is not satisfied for all \(\beta \in [0, \frac{2.294}{{\mathrm{e}}}]\), and condition (1.7) with \(i=4\) is not satisfied for all \(\beta \in [0, \frac{3}{{\mathrm{e}}}]\).
Next, we show that Eq. (2.24) is oscillatory for all \(\beta \in [\frac{1.3735}{{\mathrm{e}}}, \frac{1.384}{{\mathrm{e}}}]\). Indeed,
From this and (2.25) the parameters of Corollary 2.1 can be chosen as follows:
Let
where \(D_{l,k}\), \(l,k=1,2\), are defined as in Corollary 2.1. Then
where \(B=\frac{1}{2{\mathrm{e}}-0.1}+\frac{\beta}{1-\beta}\). Hence condition (2.18) is satisfied, and Corollary 2.1 implies that Eq. (2.24) is oscillatory for all \(\beta \in [\frac{1.3735}{{\mathrm{e}}}, \frac{1.384}{{\mathrm{e}}}]\).
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The authors express their sincere gratitude to the anonymous reviewers for their constructive comments.
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This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444).
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EA made the major analysis and the original draft preparation. HE revised the calculations, made corrections and provide several improvements. All authors read and approved the final manuscript.
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Attia, E.R., El-Morshedy, H.A. New product-type oscillation criteria for first-order linear differential equations with several nonmonotone arguments. Bound Value Probl 2023, 80 (2023). https://doi.org/10.1186/s13661-023-01743-0
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DOI: https://doi.org/10.1186/s13661-023-01743-0