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Existence and uniqueness criterion of a periodic solution for a third-order neutral differential equation with multiple delay
Boundary Value Problems volume 2023, Article number: 25 (2023)
Abstract
In this paper, we study the existence and uniqueness of a periodic solution for a third-order neutral delay differential equation (NDDE) by applying Mawhin’s continuation theorem of coincidence degree and analysis techniques. An illustrative example is given as an application to support our results. To confirm the accuracy of our results, we also present a plot of the behavior of the periodic solution.
1 Introduction
Neutral delay differential equations (NDDEs) are a family of differential equations depending on the past as well as the present state that involve derivatives with delays as well as the function itself. The study of the neutral functional differential equations is essentially based on the questions of the action and estimates of the spectral radii of the operators in the spaces of discontinuous functions, for example, in the spaces of summable or essentially bounded functions.
NDDEs have many interesting applications in various branches of science such as, physics, electrical control and engineering, physical chemistry, and mathematical biology, etc., see [4].
The existence and uniqueness of periodic solutions for NDDE are of great interest in mathematics and its applications to the modeling of various practical problems, see [11, 13, 15]. There have been many papers written on the various aspects of the theory of periodic function differential equations (FDE) and periodic NDDE, see for example [1–3, 5–7, 9, 10, 12, 14, 16–21, 23, 24].
In 2014, Xin and Zhao [24] established sufficient conditions for the existence of a periodic solution to the following neutral equation with variable delay
In 2018, Mahmoud and Farghaly [19] studied the sufficient conditions for the existence of a periodic solution for a kind of third-order generalized NDDE with variable parameter
where \(|c(t)|\neq 1, c, \delta \in C^{2}(\mathbb{R},\mathbb{R})\) and c, δ are ω-periodic functions for some \(\omega > 0, \tau , e \in C[0,\omega ]\) and \(\int ^{\omega}_{0}e(t)\,dt=0\); f, g, and h are continuous functions.
In 2022, Taie and Alwaleedy [22] investigated the existence and uniqueness of a periodic solution for the third-order neutral functional differential equation
where, \(|d(t)| \neq 1\), \(d, \delta \in C^{3}(\mathbb{R}, \mathbb{R})\) are ω-periodic functions for some \(\omega > 0\), \(\dot{\delta}(t) < 1\) for all \(t\in [0, \omega ]; a, b, c_{i}, e (i = 1, 2, \ldots, n) \) are continuous periodic functions defined on \(\mathbb{R}\) with period \(\omega > 0\), such that a, b, \(c_{i}\) have the same sign and \(\int ^{\omega}_{0}{e(t)\,dt}=0\); f, g are continuous functions defined on \(\mathbb{R}^{2}\) and periodic in the first argument.
The aim of this paper is to investigate sufficient conditions ensuring the existence and uniqueness of a periodic solution for the following third-order NDDE
where, \(\gamma _{i}, e:\mathbb{R}\rightarrow \mathbb{R}\) are T-periodic, \(|\alpha | \neq 1\), \(\gamma \in C^{2}(\mathbb{R},\mathbb{R})\), γ are T-periodic functions for some \(T> 0, \gamma , e \in C[0,T]\), and \(\int ^{T}_{0}e(t)\,dt=0\); f, g, and h are continuous functions defined on \(\mathbb{R}^{2}\) and periodic in t with \(f(u(t))=f(u(T))\), \(g(t,u(t))=g(t+T,u(t+T))\), \(h(x(t))=h(x(t+T))\), and \(g(t,0)=0\).
2 Preparation
Let \(C_{T}=\{x \in C(\mathbb{R},\mathbb{R}): x(t+T)=x(t), t\in \mathbb{R} \}\) with the norm \(\|x\|_{\infty}= \max_{t\in [0,T]} |x(t)|\), then \((C_{T}, \|\cdot \|_{\infty})\) is a Banach space. Here, the neutral operator \(\mathcal{A}\) is a natural generalization of the familiar operator \(\mathcal{A}_{1}=x(t)-cx(t-\delta )\), \(\mathcal{A}_{2}=x(t)-c(t)x(t- \delta )\). However, \(\mathcal{A}\) possesses a more complicated nonlinearity than \(A_{1}\), \(A_{2}\). Then, for example the neutral operator \(\mathcal{A}_{1}\) is homogeneous in the following estimate \(\frac{d}{dt}(A_{1} x)(t)=(A_{1}{\dot{x}})(t)\), but the neutral operator \(\mathcal{A}\) is inhomogeneous in general. Hence, many of the new results for differential equations with the neutral operator \(\mathcal{A}\), will not be a direct extension of known theorems for NDDEs.
Moreover, define an operator \(\mathcal{A} :C_{T}\rightarrow C_{T} \) as
where, \(|\alpha |\neq 1\), \(\gamma \in C^{2}(\mathbb{R},\mathbb{R})\) is T-periodic for some \(T>0\).
Lemma 2.1
([24])
If \(|\alpha |\neq 1\), then the operator \(\mathcal{A}\) has a continuous inverse \(\mathcal{A}^{-1}\) on \(C_{T}\), satisfying
-
(1)
\((A^{-1}f ) (t)= \scriptsize\Biggl\{\begin{array}{l@{\quad }l} f(t)+\sum^{\infty}_{j=1}{\alpha ^{j} } f(s-\sum^{j-1}_{i=1}{\gamma (D_{i}) )}, & \textit{for } \vert \alpha \vert < 1, \forall f\in C_{T}, \\ -\frac{f(t+\gamma (t))}{\alpha}-\sum^{\infty}_{j=1} \frac{1}{\alpha ^{j+1}}f (s+\gamma (t)+\sum^{j-1}_{i=1} \gamma (D_{i}) ), &\textit{for } \vert \alpha \vert >1, \forall f\in C_{T}; \end{array} \)
-
(2)
\(\vert (A^{-1}f ) (t) \vert \leq \frac{\|f\|}{|1-|\alpha ||}\), \(\forall f\in C_{T}\);
-
(3)
\(\int ^{T}_{0} \vert (A^{-1}f ) (t) \vert \,dt\leq \frac{1}{|1-|\alpha ||}\int _{0}^{T}{|f(t)|\,dt}\), \(\forall f\in C_{T}\);
where \(D_{1}=t\), \(D_{j+1}=t-\sum_{i=1}^{j} \gamma (D_{i})\), \(j=1,2,\ldots \) .
Let X and Y be real Banach spaces and \(L:D(L)\subset X\rightarrow Y\) be a Fredholm operator with index zero, here \(D(L)\) denotes the domain of L. This means that \(ImL\) is closed in Y and \(\dim \operatorname{Ker} L=\dim (Y/\operatorname{Im} L)<+\infty \). Consider supplementary subspaces \(X_{1}\), \(Y_{1}\), of X, Y, respectively, such that \(X=\operatorname{Ker} L \oplus X_{1}\), \(Y=\operatorname{Im} L\oplus Y_{1}\), and let \(P_{1}:X\rightarrow \operatorname{Ker} L\) and \(Q_{1}:Y\rightarrow Y_{1}\) denote the natural projections. Clearly, \(\operatorname{Ker} L\cap (D(L)\cap X_{1})=\{0\}\), thus the restriction \(L_{P_{1}}:=L|_{D(L)\cap X_{1}}\) is invertible. Let \(L_{P_{1}}^{-1}\) denote the inverse of \(L_{P_{1}}\).
Let Ω be an open bounded subset of X with \(D(L)\cap \Omega \neq \emptyset \). A map \(N:\overline{\Omega}\rightarrow Y\) is said to be L-compact in Ω̅ if \(Q_{1}N(\overline{\Omega})\) is bounded and the operator \(L_{P_{1}}^{-1}(I-Q_{1})N:\overline{\Omega}\rightarrow X\) is compact.
Lemma 2.2
(Gaines and Mawhin [8])
Suppose that X and Y are two Banach spaces, and \(L:D(L)\subset X\rightarrow Y\) is a Fredholm operator with index zero. Furthermore, \(\Omega \subset X\) is an open bounded set and \(N:\overline{\Omega}\rightarrow Y\) is L-compact on Ω̅. Assume that the following conditions hold:
-
(1)
\(Lx\neq \lambda Nx\), for all \(x\in \partial \Omega \cap D(L)\), \(\lambda \in (0,1)\);
-
(2)
\(Nx\notin \operatorname{Im} L\), for all \(x\in \partial \Omega \cap \operatorname{Ker} L\);
-
(3)
\(\deg \{JQ_{1}N,\Omega \cap \operatorname{Ker} L,0\}\neq 0\), where \(J:\operatorname{Im} Q_{1}\rightarrow \operatorname{Ker} L\) is an isomorphism.
Then, the equation \(Lx=Nx\) has a solution in \(\overline{\Omega}\cap D(L)\).
3 Existence result
In this section, we will study the existence of a periodic solution for (1.1).
Now, we rewrite (1.1) in the following form:
Here, if \(x(t)=(x_{1}(t),x_{2}(t),x_{3}(t))^{\top}\) is a T-periodic solution to (3.1), then \(x_{1}(t)\) must be a T-periodic solution to (1.1). Thus, the problem of finding a T-periodic solution for (1.1) reduces to finding one for (3.1).
Recall that \(C_{T}=\{\phi \in C(\mathbb{R},\mathbb{R}): \phi (t+T)\equiv \phi (t) \}\) with the norm \(\|\phi \|=\max_{{t\in [0,T]}}|\phi (t)| \). Define \(X=Y=C_{T}\times C_{T}= \{x=(x_{1}(\cdot ),x_{2}(\cdot ),x_{3}(\cdot )) \in C(\mathbb{R},\mathbb{R}^{3}) : x(t) = x(t+T), t \in \mathbb{R}\}\) with the norm \(\|x\|=\max \{\|x_{1}\|,\|x_{2}\|,\|x_{3}\|\}\). Clearly, X and Y are Banach spaces. Moreover, define
by
Also, we can define \(N:X\rightarrow Y\) by
Then, (3.1) can be converted to the abstract equation \(Lx=Nx\). From the definition of L, we obtain
Therefore, we find that L is a Fredholm operator with index zero. Let \(P_{1}:X\rightarrow \operatorname{Ker} L\) and \(Q_{1}:Y\rightarrow \operatorname{Im} Q_{1}\subset \mathbb{R}^{3}\) be defined by
then \(\operatorname{Im} P_{1}=\operatorname{Ker} L\: \text{and} \: \operatorname{Ker} Q_{1}=\operatorname{Im} L\). Set \(L_{P_{1}}=L|_{(D(L)\cap \operatorname{Ker} P_{1})}\) and \(L_{P_{1}}^{-1}: \operatorname{Im} L\rightarrow (D(L)\cap \operatorname{Ker} P_{1})\) denotes the inverse of \(L_{P_{1}}\), it follows that
where
From (3.2), we obtain
Thus, from (3.3) and (3.4), it is clear that \(Q_{1}N\) and \(L_{P_{1}}^{-1}(I-Q_{1})N\) are continuous, and \(Q_{1}N(\overline{\Omega})\) is bounded, and then \(L_{P_{1}}^{-1}(I-Q_{1})N(\overline{\Omega})\) is compact for any open bounded \(\Omega \subset X\), which means N is L-compact on Ω̄.
Now, we will present the following hypotheses that will be used repeatedly during our work:
-
(H1)
There exists a positive constant \(k_{1}\) such that \(|f(u)|\leq k_{1}\), for \(u\in \mathbb{R}\);
-
(H2)
There exist positive constants \(k_{2}\), \(h_{1}\) such that \(|g(t,u)|\leq k_{2}\), \(|h(x)|\leq h_{i}\), for \((t,u)\in \mathbb{R}\times \mathbb{R} \) and \((t,x)\in \mathbb{R}\times \mathbb{R}\);
-
(H3)
There exists a positive constant D such that \(|h(x)|>\frac{bk_{2}}{c_{i}}\) and \(x[f(u)+g(t,v)+h(x)]\neq 0\), for \(t,u,v,x\in \mathbb{R}\) and \(|x|>D\);
-
(H4)
There exist positive constants \(b_{o}\), \(c_{0}\) such that \(|h(x_{1})-h(x_{2})|\leq b_{o}|x_{1}-x_{2}|\), \(|g(t,u_{1})-g(t,u_{2})|\leq c_{o}|u_{1}-u_{2}|\) for all \(t, x_{1}, x_{2}, u_{1}, u_{2} \in \mathbb{R}\).
The following theorem is our main result on the existence of a periodic solution for (1.1).
Theorem 3.1
Suppose that assumptions (H1)–(H4) hold. Assume that the following assumption is satisfied:
If \(|\alpha |<1\) and
-
(i)
\(1-|\alpha |-|\alpha | \gamma _{1}(\gamma _{1}-2)-M_{4}>0\), where
$$\begin{aligned}& M_{4}= \frac{1}{2} (\sqrt{M_{3}}+\alpha \gamma _{2}T ),\\& \begin{aligned} M_{3}={}& \Biggl( bk_{2}+b_{0} c\sum_{i=1}^{n} \biggl\Vert \frac{1}{(1-\dot{\gamma}_{i})} \biggr\Vert _{\infty} D \\ & {} +nc\max \bigl\{ \bigl\vert h(t, 0) \bigr\vert :0\leq t \leq T\bigr\} + \Vert e \Vert _{\infty} \Biggr)M_{1} T, \end{aligned}\\& M_{1}=1+\alpha (1+\gamma _{1}), \\& \gamma _{1}=\max_{t\in [0,T]}{ \vert \dot{\gamma} \vert }, \qquad \gamma _{2}= \max_{t\in [0,T]}{ \vert \ddot{\gamma} \vert }; \qquad c=\max_{t\in [0,T]}{ \vert c_{i} \vert }, \end{aligned}$$
then equation (1.1) has at least one T-periodic solution.
Proof
We know that (3.1) has a T-periodic solution, if and only if, the following operator equation
has a T-periodic solution. From (3.2), we see that N is L-compact in Ω̄, where Ω is an open bounded subset of \(X_{T}\). For \(\lambda \in (0,1]\), define \(\Omega _{1}=\{x \in C_{T} : Lx=\lambda Nx\}\). Then, \(x = (x_{1},x_{2}, x_{3})^{\top}\in \Omega _{1}\) satisfies:
Substituting of \(x_{3}(t)=\frac{1}{\lambda ^{2}}\frac{d^{2}}{dt^{2}}(\mathcal{A}x_{1})(t)\) into the third equation of (3.6), we obtain
By integrating both sides of (3.7) over \([0,T]\), we find
which implies that there is at least one point \(t_{1}\), such that
By using (H2), we have
In view of (H3) we see that \(|x_{1}(t_{1}-\gamma (t_{1}))|\leq D\). Since \(x_{1}(t)\) is periodic with period T, \(t_{1}-\gamma (t_{1})=nT+\eta \), \(\eta \in [0,T]\) and n is an integer, then \(|x_{1}(\eta )|\leq D\).
Thus, for \(t\in [\eta ,\eta +T]\), we obtain
and
Combining the above two inequalities, we obtain
Since \(x_{1}(0)=x_{1}(T)\), there is a constant \(\zeta \in [0,T]\) such that \({\dot{x}}_{1}(\zeta )=0\). Thus, we have
and
Combining the inequalities (3.10) and (3.11), we have
Now, by differentiating (2.1) with respect to t, we obtain
Since \(\gamma _{1}=\max_{t\in [0,T]}{|\dot{\gamma}(t)|}\) and from (3.9), we find
Then,
where
Also, we find
Then, we obtain
Therefore, from the definition of the operator \(\mathcal{A}\), we find
Then, we can write the above equation as
Now, by multiplying both sides of (3.7) by \(\frac{d}{dt}((Ax_{1})(t))\) and integrating it from 0 to T, we obtain
Therefore, we obtain
Then, from the assumption (H4) we obtain
Now, by using (3.14), we can see that
By the assumptions (H1) and (H2), we conclude
Thus, by (3.9), we obtain
For positive constants \(M_{2}\) and \(M_{3}\), the above inequality becomes
where
By applying Lemma 2.1, we obtain
Substituting from (3.15) and by using the conditions of Theorem 3.1, we find
From (3.9) and by using the Schwarz inequality, we conclude
It follows that
Applying the inequality \((m+n)^{r}\leq m^{r}+n^{r}\) for all \(m,n>0\), \(0< r<1\), implies from (3.16) that
Using (3.12), we find
where
Then, we conclude
Since \(|1 -|\alpha || -\alpha \gamma _{1}(\gamma _{1} -2)-M_{4}>0\), we can conclude that there exists a positive constant \(D_{1}\), such that
It follows from (3.12) that
Thus, from (3.9) we obtain
where
Using the first equation of system (3.6), we have
which mean that there exists a constant \(t_{1}\in [0,T]\), such that \(x_{2}(t_{1})=0\), then from (3.16) we find
Therefore, we obtain
where
From the second equation of system (3.6), we have
then, there is a constant \(t_{2}\in [0,T]\), such that \(x_{3}(t_{2})=0\), hence
By the third equation of system (3.6), we have
Using (H1), (H2), and (H4), we obtain
To prove condition \((1)\) of Lemma 2.2, we assume that for any \(\lambda \in (0,1)\) and any \(x=x(t)\) in the domain of L, which also belongs to ∂Ω, we must have \(Lx\neq \lambda Nx\). For otherwise in view of (3.6), we obtain
Let \(D_{5}=\max \{D_{2}, D_{3}, D_{4}\}+1\), \(\Omega =\{x=(x_{1}, x_{2}, x_{3})^{\top}: \|x\|< D_{5}\}\), then we see that x belongs to the interior of Ω, which is contrary to the assumption that \(x\in \partial \Omega \). Therefore, condition \((1)\) of Lemma 2.2 is satisfied. Now, for all \(x\in \partial \Omega \cap \operatorname{Ker} L\)
If \(Q_{1}Nx=0\), then \(x_{2}(t)=0\), \(x_{3}(t)=0\), \(x_{1}=D_{5}\) or \(-D_{5}\). However, if \(x_{1}(t)=D_{5}\), then by \(H_{3}\) we obtain
from which there exists a point \(t_{2}\) such that \(h(t_{2},D_{5})=0\). From assumption (H3), we have \(D_{5}\leq D\), which yields a contradiction. Similarly if \(x_{1}=-\mathcal{M}_{4}\). Therefore, we have \(Q_{1}Nx\neq 0\), hence for all \(x\in \partial \Omega \cap \operatorname{Ker} L\), \(x\notin \operatorname{Im} L\), so condition \((2)\) of Lemma 2.2 is satisfied.
Define the isomorphism \(J:\operatorname{Im} Q_{1}\rightarrow \operatorname{Ker} L\) as follows:
Let \(H(\mu ,x)=\mu x+(1-\mu )JQ_{1}Nx\), \((\mu ,x)\in [0,1]\times \Omega \), then for all \((\mu ,x)\in (0,1)\times (\partial \Omega \cap \operatorname{Ker} L)\),
Since \(\int ^{T}_{0}e(t)\,dt=0\), we can obtain
for all \((\mu ,x)\in (0,1)\times (\partial \Omega \cap \operatorname{Ker} L)\).
Using (H3), it is obvious that \(x^{\top}H(\mu ,x)\neq 0\), for all \((\mu ,x)\in (0,1)\times (\partial \Omega \cap \operatorname{Ker} L)\). Hence,
Hence, condition \((3)\) of Lemma 2.2 is satisfied. By applying Lemma 2.2, we conclude that equation \(Lx=Nx\) has a solution \(x=(x_{1},x_{2},x_{3})^{\top}\) on \(\bar{\Omega}\cap D(L)\), thus (1.1) has a T-periodic solution \(x(t)\). □
4 Uniqueness result
Suppose that
then we have the following uniqueness result.
Theorem 4.1
Suppose that all conditions of Theorem 3.1 hold and \(h(x)\) is a monotone strictly decreasing function in x and \(|\alpha |<1\) and assume that
-
(H5)
There exists a positive constant \(k_{3} \) such that \(f(u(t))=k_{3}\), for all \(u\in \mathbb{R}\);
-
(H6)
There exists a positive constant L such that \(|g(t,u)-g(t,v)|\leq L|u-v|\); for all \(u,v\in \mathbb{R}\).
such that
Then, equation (1.1) has at most one T-periodic solution.
Proof
Assume that \(r_{1}(t) \) and \(r_{2}(t) \) are two T-periodic solutions of (1.1), then we have \(z(t) =r_{1}(t)-r_{2}(t)\). Thus, (1.1) takes the form
Since \(f(u)=k_{3}\), we obtain
By integrating (4.1) from 0 to T and using the condition H6, we obtain
Using the integral mean-value theorem, it follows that there exists a constant \(s_{1}\in [0,T]\) such that
Let \(\bar{\gamma}=s_{1}-\gamma _{i}(s_{1})=nT+\zeta \), where \(\zeta \in [0,T] \) and n is an integer. Hence, from equation (4.2) together with condition \((H6)\) implies that there exists a constant \(\zeta \in [0,T] \) such that
We can write
Again, we have
Hence, we have
By using the Schwartz inequality, we find
Therefore, we obtain
From the definition of the operator, we have
Multiplying (4.1) by \(\dddot{z}(t)\) and integrating it over \([0,T]\), we find
By using condition \(H_{4}\), we obtain
Hence, we have
From the definition of the operator \(\mathcal{A}\), we have
Now, by applying the Schwartz inequality, we obtain
Then, we obtain
By substituting from (4.6) into (4.5), we obtain
Substituting from (4.7) into (4.4) and using the Schwarz inequality, we find
Since \(z(0)=z(T)\), there exists a constant \(\xi \in [0,T]\), such that \(\dot{z}(\xi )=0\) and
Also, for \(t\in [0, T]\), we have
By combining (4.9) and (4.10), we obtain
Therefore, by using the Schwartz inequality, we have
hence, we obtain
therefore, we obtain
Since \(\dot{z}(t)\) is a periodic function for \(t\in [0,T]\) by using the above similar technique we obtain
which, together with the Schwartz inequality, implies
then, we obtain
By substituting (4.15) into (4.13), we obtain
By using (4.13), (4.15), (4.16), and (4.3), (4.8) becomes
From Lemma 2.1, we have
Substituting (4.18) into (4.17), we conclude
Hence, we conclude
Since
we find
Since \(\mathcal{A}z(t)\), \(\frac{d}{dt}((\mathcal{A}z)(t))\), \(\frac{d^{2}}{dt^{2}}((\mathcal{A}z)(t))\), and \(\frac{d^{3}}{dt^{3}}((\mathcal{A}z)(t))\) are T-periodic and continuous functions, we have
Now, applying Lemma 2.1 in [12], we obtain
Hence, we conclude \(r_{1}(t)\equiv r_{2}(t)\) for all \(t\in \mathbb{R}\). □
Hence, (1.1) has a unique T-periodic solution.
5 Example
Consider the following third-order NDDE:
Comparing (5.1) to (1.1), we find \(f(u)=\cos ^{2}{4t}\), \(a=\frac{1}{6}\), \(\alpha =\frac{1}{130}\), \(g(t,u)=\sin 4t\cos u\), \(b=\frac{1}{120}\), \(h(t,x)=\frac{4}{\pi}x(t-\frac{1}{150}\sin{4t})\), \(h(t,0)=0\), \(b_{o}=\frac{4}{\pi}\), \(c=\frac{1}{10}\), \(\gamma (t)=\frac{1}{150}\sin 14t\), \(\dot{\gamma}(t)=\frac{4}{150}\cos{4t}\), \(e(t)=\cos 4t\), and let \(T=\frac{\pi}{4}\).
Also, we have
and
Therefore, by taking \(n=c=k_{2}=1\), we obtain
Hence, we find
To verify how to obtain (3.18) from (3.17), we calculate the following
Then, (3.17) becomes
Therefore, we obtain
which can be considered as a quadratic inequality, its roots are
which implies that
The rest of the proof is clear. Hence, by Theorem 3.1, (5.1) has at least one \(\frac{\pi}{8}\)-periodic solution.
Now, by taking \(k_{3}=1\) and \(c_{0}=1\), we have
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D. A. M. Bakhit wrote the manuscript R. O . A. Taie reviewed the manuscript
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Taie, R.O.A., Bakhit, D.A.M. Existence and uniqueness criterion of a periodic solution for a third-order neutral differential equation with multiple delay. Bound Value Probl 2023, 25 (2023). https://doi.org/10.1186/s13661-023-01711-8
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DOI: https://doi.org/10.1186/s13661-023-01711-8
MSC
- 34C25
Keywords
- Existence and uniqueness
- Neutral delay differential equation
- Mawhin’s continuation theorem