Abstract
In this paper, we consider a kind of second-order delay differential system. By taking some transforms, the property of delay is reflected in the boundary condition. The wonder is that the corrseponding first-order system is exactly the so-called P-boundary value problem of Hamiltonian system which has been studied deeply by many mathematicians, including the authors of this paper. Firstly, we define the relative Morse index \(\mu _Q(A,B)\) for the delay system and give the relationship with the P-index \(i_{P}(\gamma _{R})\) of Hamiltonian system. Secondly, by this index, topology degree and saddle point reduction, the existence of periodic solutions is established for this kind of delay differential system.
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1 Introduction and Main Results
Delay models usually appear in some biological modeling. They have been used to describe several aspects of infectious disease dynamics: primary infection [8], drug therapy [34] and immune response [9], to name a few. Delays have also appeared in the study of chemostat models [48], circadian rhythms [39], epidemiology [10], the respiratory system [42], tumor growth [43] and neural networks [4]. Delay effects even appeared in the population dynamics of many species [40, 41].
In 1974, Kaplan and Yorke [21] considered the periodic solutions of the following kinds of delay differential equations
and
with odd function f. They turned their problems into the problems of periodic solution of autonomous Hamiltonian system, it was proved that there existed an energy surface of the Hamiltonian function containing at least one periodic solution. Since then many papers (see [15, 16, 22, 23, 25] and the references therein) used Kaplan and Yorke’s original idea to search for periodic solutions of more general differential delay equations of the following form
The existence of periodic solutions of above delay differential equation has been investigated by Nussbaum in [35] using different techniques. Recently, many results on delay differential systems were obtained, readers may refer to the references [7, 18,19,20, 24, 33, 36, 46, 47] and the references therein. Specially, delay differential equation also has been used to study the COVID-19, the time-delay process is introduced to describe the latent period and treatment cycle, see [45], where the delay differential system derived from the COVID-19 model is a first order equation coupling with some second order delay differential-integral equations. The authors of this paper also have some results on the existence of periodic solutions of above delay differential equation, see [27, 28, 30, 44].
In 1994, Bainov and Domoshnitsky [3] considered the stability of the following second-order delay differential systems
and Agarwal et al. [1] further improved their result. For other results on second-order delay differential equations, readers may refer to [11, 14, 26], and the references therein.
In this paper, we will consider the following delay differential system
with \(x\in C^2(\mathbb {R}, \mathbb {R}^n)\), \(v\in C^1(\mathbb {R}^n,\mathbb {R})\). By taking some transformations, we will transfer the delay differential system (DDS) into the following second order Hamiltonian system
where \(z:\mathbb {R}\rightarrow \mathbb {R}^{N}\), \(A, Q\in \mathcal {L}({\mathbb {R}}^N)\) with \(\mathcal {L}({\mathbb {R}}^N)\) the set of real square matrixes on \({\mathbb {R}}^N\). More generally, in the system (HS) we assume \(A,\, Q\) satisfy the following conditions which including (DDS) as a special case (see Section 3.2 below for details)
and there exists \(k\in {\mathbb {N}}^+\) such that
with \(I_N\) the identity map on \({\mathbb {R}}^N\). The function V satisfies the following condition
(\(V_0\)) \(V\in C^1(\mathbb {R}^{N},\mathbb {R})\) and
The system (HS) is different from the classical second order Hamiltonian system since the matrix A is neither positive or negative definite and the corresponding variational problem is strongly indefinite.
If \(V\in C^2(\mathbb {R}^{N},\mathbb {R})\), for any solution \({\hat{z}}(t)\) of (HS), linearized system at \({\hat{z}}(t)\) is
with \(B(t)=V''({\hat{z}}(t))\), and we have \(B(1)Q=QB(0)\). So, we define the space as
In the next section, we will define the index pair \((\mu _Q(A,B), \upsilon _Q(A,B))\). With this index, we have the following results.
Theorem 1.1
Assume V satisfies (\(V_0\)) and the following condition.
(\(V_1\)) \(V':{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}^{N}\) is Lipschitz continuous
with its Lipschitz constant \(l_V>0\).
(\(V^\pm _2\)) There exists \(M_1,\;M_2,\;K>0\), \(B(t)\in \mathcal L_Q(S_1,{\mathcal {L}}({\mathbb {R}}^{N}))\) with \(B(t)\equiv B\), such that
with
and
Then (HS) has at least one solution.
Theorem 1.2
Assume V satisfying conditions (\(V_0\)), (\(V_1\)) and the following condition
(\(V_3\)) There exists \(B\in C(\mathbb {R}^{N},\mathcal {L}_s({\mathbb {R}}^{N}))\) such that
with
(\(V_4\)) There exist \(B_1,\;B_2\in {\mathcal {L}}_Q(S_1,{\mathcal {L}}({\mathbb {R}}^{N}))\) satisfying
and
Then (HS) has at least one solution.
Since \(Q^k=I_N\), the solutions obtained in Theorem 1.1-1.2 in fact are k-periodic with Q-symmetric. As application of the above two theorems, we will treat the delay differential system (DDS) and obtain two results which are stated in Theorems 3.2–3.3.
2 Variational Setting
Let \(S^1:=\mathbb {R}/(k\mathbb {Z})\) and \({\textbf{E}}\) the closed subspace of \(W^{1,2}(S^1,\mathbb {R}^{N})\) defined by
with the norm
and the corresponding inner product \((\cdot ,\cdot )_{\textbf{E}}\). Define the functional \(\varphi \) on \(\textbf{E}\) by
The critical points of \(\varphi \) are the solutions of (HS).
Define the Hilbert space
Define the unbounded self-adjoint operator \( {\hat{A}}:{\textbf{L}}\rightarrow {\textbf{L}}\) by
with \(D({\hat{A}})\subset {\textbf{L}}\) the domain of \({\hat{A}}\) and we have \({\textbf{E}}=D(|{\hat{A}}|^{1/2})\). Without confusion, we still denote it by A for simplicity. We have A is unbounded from below and above, so the functional \(\varphi \) is strongly indefinite in this sense. Now, we will use the method of saddle point reduction to overcome this difficulty and get the definition of relative Morse index. On the other hand, the system (HS) can be translated to the first order Hamiltonian systems which has the P-index defined in [12, 13, 29, 31, 32], we will give the relation between these two indices.
2.1 Relative Morse Index
For any \(B\in \mathcal {L}_Q(S_1,{\mathcal {L}}({\mathbb {R}}^{N}))\), consider the linearized system (LHS). We know that B defines a self-adjoint operator on \({\textbf{L}}\) by
Without confusion, we still denote it by B, that is to say \({\mathcal {L}}_Q(S_1,{\mathcal {L}}({\mathbb {R}}^{N}))\subset {\mathcal {L}}_s({\textbf{L}})\) the set of bounded self-adjoint operators on \({\textbf{L}}\). So (LHS) can also be rewritten as the following linear operator equation
with \(A={\hat{A}}\) defined in (2.2) and \(B\in {\mathcal {L}}_Q(S_1,{\mathcal {L}}({\mathbb {R}}^{N}))\subset {\mathcal {L}}_s({\textbf{L}})\).
Now, we will give the definition of the relative Morse index and display the relationship with spectral flow. Generally, for any bounded self-adjoint Fredholm operator F on \({\textbf{E}}\), there is a unique F-invariant orthogonal splitting
where \({\textbf{E}}^0(F)\) is the null space of F, F is positive definite on \({\textbf{E}}^+(F)\) and negative definite on \({\textbf{E}}^-(F)\). We denote by \(P_F\) the orthogonal projection from \({\textbf{E}}\) to \({\textbf{E}}^-(F)\). For any compact self-adjoint operator T on \({\textbf{E}}\), \(P_F-P_{F-T}\) is compact (see Lemma 2.7 of [49]). Then by Fredholm operator theory, \(P_F|_{{\textbf{E}}^-{(F-T)}}:{\textbf{E}}^-{(F-T)}\rightarrow {\textbf{E}}^-{(F)}\) is a Fredholm operator. Here and in the sequel, we denote by \(\mathrm ind (\cdot )\) the Fredholm index of a Fredholm operator.
Definition 2.1
For any bounded self-adjoint Fredholm operator F and a compact self-adjoint operator T on \({\textbf{E}}\), the relative Fredholm index pair \((\mu _F(T), \upsilon _F(T))\) is defined by
and
On the other hand, let \(\{F_\theta |\theta \in [0,1]\}\) be a continuous path of self-adjoint Fredholm operators on the Hilbert space \({\textbf{E}}\). The following proposition displays the relationship between spectral flow and the relative Fredholm index defined above. It is well known that the concept of spectral flow \(Sf(F_\theta )\) was first introduced by Atiyah, Patodi and Singer in [2], and then extensively studied in [5, 17, 37, 38, 49].
Proposition 2.2
(See [6, Proposition 3].) Suppose that, for each \(\theta \in [0,1]\), \(F_\theta -F_0\) is a compact operator on \({\textbf{E}}\), then
Thus, from Definition 2.1,
where \(F_\theta =F-\theta T\). Moreover, if \(\sigma (T)\subset [0,\infty )\) and \(0\notin \sigma _P(T)\) the set of point spectrum of T, from the definition of spectral flow, we have
Up to now, we have defined the relative Fredholm index pair \((\mu _F(T), \upsilon _F(T))\) in general abstract setting and displayed the relationship with the spectral flow. Now, we can define our relative Morse index pair \((\mu _Q(A,B), \upsilon _Q(A,B))\) for our problem. The operator A defined a bounded self-adjoint Fredholm operator \(\tilde{A}\) on \({\textbf{E}}\) by
On the other hand, since the embedding map \(i:{\textbf{E}}\hookrightarrow \textbf{L}\) is compact, the dual operator \(i^*: \textbf{L}\rightarrow {\textbf{E}}\) is compact and for any \(B\in \mathcal {L}_s({\textbf{L}})\) , \(i^*B\) is a compact self-adjoint operator on \( {\textbf{E}}\). By Definition 2.1, we have the relative Fredholm index pair \((\mu _{{\tilde{A}}}(i^*B), \upsilon _{\tilde{A}}(i^*B))\), so we have the following definition.
Definition 2.3
Let the operator A defined in (2.2), for any \(B\in {\mathcal {L}}_Q(S_1,{\mathcal {L}}({\mathbb {R}}^{N}))\), the index pair \((\mu _Q(A,B), \upsilon _Q(A,B))\) is defined by
2.2 Relationship with P-Index
In this part, we will transfer (HS) into a first-order Hamiltonian system with P-boundary condition which is called P-boundary problem, and we will get that the index pair \((\mu _Q(A,B),\upsilon _Q(A,B))\) coincides with the P-index pair \((i_P(\gamma ),\nu _P(\gamma ))\) defined in [12, 13, 29, 31, 32].
Firstly, let us recall the P-boundary problem. Recall that the symplectic group is defined as
where \(J=\left( \begin{matrix} 0&{}-I_N\\ I_N&{}0 \end{matrix} \right) \), \(I_N\) is the identity matrix on \({\mathbb {R}}^{N}\), and \({\mathcal {L}}({\mathbb {R}}^{2N})\) is the space of \(2N\times 2N\) real matrices. The P-boundary problem is the following Hamiltonian system
where \(P\in Sp(2N)\) and \(H\in C^1({\mathbb {R}}\times {\mathbb {R}}^{2N},{\mathbb {R}})\) satisfying
Clearly, if \(H\in C^1({\mathbb {R}}\times {\mathbb {R}}^{2N},{\mathbb {R}})\), we have \(P^TH''(t+1,Px)P=H''(t,x)\). Let \(x:[0,1]\rightarrow {\mathbb {R}}^{2N}\) be a solution of (2.10), linearzing the Hamiltonian system \(\dot{x}(t)=JH'(t,x(t))\) at x we get a Hamiltonian system
with \(R(t)=H''(t,x(t))\) satisfying
The fundamental solution of (2.11) is a symplectic path \(\gamma _{R}\in C^1([0,+\infty ),Sp(2N))\) with \(\gamma _{R}(0)=I\). For any such symplectic path \(\gamma \), there is a so called Maslov P-index pair \((i_P(\gamma ),\nu _P(\gamma ))\in {\mathbb {Z}}\times \{0,1,\cdots , 2N\}\). The Maslov P-index theory for a symplectic path was first studied in [12, 29] independently for any symplectic matrix P with different treatment. The Maslov P-index theory was generalized in [31] to the Maslov \((P,\omega )\)-index theory for any \(P\in Sp(2N)\) and all \(\omega \in U=\{z\in {\mathbb {C}}||z|=1\}\). When the symplectic matrix P is orthogonal, the \((P,\omega )\)-index theory and its iteration theory were studied in [13] and it has been generalized in [32] . When \(\omega = 1\), the Maslov \((P,\omega )\)-index theory coincides with the Maslov P-index theory.
Secondly, let us consider system (HS). Denote
then system (HS) can be transformed to the following first-order Hamiltonian system
with
From (1.1), we have \(H(Px)=H(x)\) and \(P^TJP=J\), so \(P\in SP(2N)\). That is to say by (2.13), we can transform system (HS) into the so called P-boundary problem (2.10). Similarly, by (2.13), for any \(B\in {\mathcal {L}}_Q(S_1,{\mathcal {L}}({\mathbb {R}}^{N}))\), we can transform system (LHS) into the linear system
with \(R:=\left( \begin{matrix} B(t) &{}0\\ 0 &{}-A^{-1} \end{matrix} \right) \) and satisfying \( R(t+1)=(P^{-1})^TR(t)P^{-1}. \) Denote the corresponding fundamental solution by \(\gamma _{R}\), so we have the P-pair \((i_{P}(\gamma _{R}),\nu _{P}(\gamma _{R}))\).
Lastly, from the property of the P-index pair, Proposition 2.2 and Definition 2.3, we have
where the constant \(k_0\in \mathbb {Z}\).
2.3 Saddle Point Reduction
Consider system (HS) with V satisfying conditions (\(V_0\)) and (\(V_1\)). We will consider the method of saddle point reduction without assuming the nonlinear term \(V\in C^2({\mathbb {R}}^{N},\mathbb {R})\), then we will give some abstract critical point theorems. Let \(E_A(z)\) the spectrum measure of A, since A has compact resolvent, we can choose \(l>l_V\), such that
Consider the following projection maps on \({\textbf{L}}\)
with I the identity map on \({\textbf{L}}\). Then we have the following decomposition
where \({\textbf{L}}^*_A:=P^*_A{\textbf{L}}\)(\(*=\bot , 0\)) and \({\textbf{L}}^0_A\) is finite dimensional subspace of \({\textbf{L}}\). Denote \(A^*\) the restriction of A on \({\textbf{L}}^*\)(\(*=\bot , 0\)), thus we have \((A^\bot )^{-1}\) are bounded self-adjoint linear operators on \({\textbf{L}}^\bot \) respectively and satisfying
Define the functional \(\Phi \) on \({\textbf{L}}\) by
Then from conditions (\(V_0\)) and (\(V_1\)), we have \(\Phi \in C^1({\textbf{L}},{\mathbb {R}})\) and
System (HS) can be rewritten as the following abstract self-adjoint operator equation on \({\textbf{L}}\)
which is equivalent to the following operator equations
and
where \(z^*=P^*_Az\)(\(*=\bot , 0\)), for simplicity, we rewrite \(x:=z^0\). From (2.18) and(2.20), we have \((A^\bot )^{-1}P^\bot _A \Phi '\) is contraction map on \({\textbf{L}}^+\oplus {\textbf{L}}^-\) for any \(x\in {\textbf{L}}^0\). So there is a map \(z^\bot (x):{\textbf{L}}^0\rightarrow {\textbf{L}}^\bot \) satisfying
and we have the following properties.
Proposition 2.4
-
(1)
The map \(z^\bot (x):{\textbf{L}}^0\rightarrow {\textbf{L}}^\bot \) is continuous, in fact we have
$$\begin{aligned} \Vert z^\bot (x+h)-z^\bot (x)\Vert _{\textbf{L}}\le \frac{l_V}{l-l_V}\Vert h\Vert _{{\textbf{L}}},\;\;\forall x,h\in {\textbf{L}}^0. \end{aligned}$$(2.24) -
(2)
\(\Vert z^\bot (x)\Vert _{\textbf{L}}\displaystyle \le \frac{l_V}{l-l_V}\Vert x\Vert _{{\textbf{L}}}+\frac{1}{l-l_V}\Vert \Phi '(0)\Vert _{{\textbf{L}}}\).
Proof
-
(1)
For any \(x,\;h\in {\textbf{L}}^0\), we have
$$\begin{aligned}&\Vert z^\bot (x+h)-z^\bot (x)\Vert _{{\textbf{L}}}\\ {}&\quad =\Vert (A^\bot )^{-1}P^\bot _A \Phi '(z^\bot (x+h)+x+h)-(A^\bot )^{-1}P^\bot _A \Phi '(z^\bot (x)+x)\Vert _{{\textbf{L}}}\\&\quad \le \frac{1}{l}\Vert \Phi '(z^\bot (x+h)+x+h)-\Phi '(z^\bot (x)+x)\Vert _{{\textbf{L}}}\\&\quad \le \frac{l_V}{l}\Vert z^\bot (x+h)-z^\bot (x)+h\Vert _{{\textbf{L}}}\\&\quad \le \frac{l_V}{l}\Vert z^\bot (x+h)-z^\bot (x)\Vert _{{\textbf{L}}}+\frac{l_V}{l}\Vert h\Vert _{{\textbf{L}}}. \end{aligned}$$So we have \(\Vert z^\bot (x+h)-z^\bot (x)\Vert _{\textbf{L}}\le \frac{l_V}{l-l_V}\Vert h\Vert _{{\textbf{L}}}\) and the map \(z^\bot (x):{{\textbf{L}}}^0\rightarrow {{\textbf{L}}}^\bot \) is continuous.
-
(2)
Similarly,
$$\begin{aligned} \Vert z^\bot (x)\Vert _{{\textbf{L}}}&=\Vert (A^\bot )^{-1}P^\bot _A \Phi '(z^\bot (x)+x)\Vert _{{\textbf{L}}}\\&\le \frac{1}{l}\Vert \Phi '(z^\bot (x)+x)\Vert _{{\textbf{L}}}\\&\le \frac{1}{l}\Vert \Phi '(z^\bot (x)+x)-\Phi '(0)\Vert _{{\textbf{L}}}+\frac{1}{l}\Vert \Phi '(0)\Vert _{{\textbf{L}}}\\&\le \frac{l_V}{l}(\Vert z^\bot (x)\Vert _{{\textbf{L}}}+\Vert x\Vert _{{\textbf{L}}})+\frac{1}{l}\Vert \Phi '(0)\Vert _{{\textbf{L}}}. \end{aligned}$$So we have \(\Vert z^\bot (x)\Vert _{\textbf{L}}\le \frac{l_V}{l-l_V}\Vert x\Vert _{{\textbf{L}}}+\frac{1}{l-l_V}\Vert \Phi '(0)\Vert _{{\textbf{L}}}\).
\(\square \)
Remark 2.5
It’s easy to see \({\textbf{E}}=D(|A|^{\frac{1}{2}})\), with the equivalent norm
From (2.23), we have \(z^\bot (x)\in D(A)\subset {\textbf{E}}\), and
-
(1)
The map \(z^\bot (x):{\textbf{L}}^0\rightarrow {\textbf{E}}\) is continuous, and
$$\begin{aligned} \Vert (z^\bot )(x+h)-(z^\bot )(x)\Vert _{\textbf{E}}\le \frac{l_V \cdot l^\frac{1}{2}}{l-l_V}\Vert h\Vert _{{\textbf{L}}},\;\;\forall x,h\in {\textbf{L}}^0. \end{aligned}$$(2.25) -
(2)
\(\Vert (z^\bot )(x)\Vert _{\textbf{E}}\displaystyle \le \frac{l^\frac{1}{2}}{l-l_V}(l_V\cdot \Vert x\Vert _{{\textbf{L}}}+\Vert \Phi '(0)\Vert _{{\textbf{L}}})\).
Proof
The proof is similar to Proposition 2.4, we only prove (1).
where the last inequality depends on the fact that \(\Vert z^\bot \Vert _{\textbf{E}}\ge l^\frac{1}{2}\Vert z^\bot \Vert _{\textbf{L}}\), so we have (2.25).
Now, define the map \(z:{\textbf{L}}^0\rightarrow {\textbf{L}}\) by
Define the functional \(a:{\textbf{L}}^0\rightarrow \mathbb {R}\) by
With standard discussion, the critical points of a correspond to the solutions of (OE), and we have
Lemma 2.6
Assume V satisfies (\(V_0\)), (\(V_1\)), then we have \(a\in C^1({\textbf{L}}^0,\mathbb {R})\),
Further more, if \(V\in C^2({\mathbb {R}}^{N},{\mathbb {R}})\), we have \(a\in C^2({\textbf{L}}^0,{\mathbb {R}})\) and
Proof
For any \(x,h\in {\textbf{L}}^0\), write
for simplicity, that is to say
and from (2.24), we have
where \(C=\displaystyle \frac{l}{l-l_V}\). Let \(h\rightarrow 0\) in \({\textbf{L}}^0\), and for any \(x\in {\textbf{L}}^0\), we have
From (2.29) we have
Since \(z^\pm (x)\) is the solution of (2.23) and from the definition of \(\eta (x,h)\), we have
so we have
and we have proved (2.27). If \(\Phi \in C^2({\textbf{L}},{\mathbb {R}})\), from (2.23) and by implicit function theorem, we have \(z^\pm \in C^1({\textbf{L}}^0,{\textbf{L}}^\pm )\). From (2.23) and (2.27), we have
and
that is to say \(a\in C^2({\textbf{L}}^0,{\mathbb {R}})\). \(\square \)
3 The Proofs of Main Results with Applications
3.1 The Proofs of Main Results
Proof of Theorem 1.1
Now, we consider the case of (\(V^-_2\)). Since A has compact resolvent, 0 is at most an isolate point spectrum of \(A-B\) with finite dimensional eigenspace, that is to say there exists \(\varepsilon _0>0\) small enough, such that \((-\varepsilon _0,0)\cap \sigma (A-B)=\emptyset \). For any \(\varepsilon \in (0,\varepsilon _0)\) and \(\lambda \in [0,1]\), consider the following two-parameters equation
with I the identity map on \({\textbf{L}}.\) If \(\varepsilon =0\) and \(\lambda =1\), it is (HS). We divide the following proof into four steps.
Step 1 There exists a constant C independent of \(\varepsilon \) and \(\lambda \), such that if \(z_{\varepsilon ,\lambda }\) is a solution of (\(HS_{\varepsilon ,\lambda }\)),
Since \((-\varepsilon _0,0)\cap \sigma (A-B)=\emptyset \), we have \((\varepsilon -\varepsilon _0,\varepsilon )\cap \sigma (\varepsilon \cdot I+A-B)=\emptyset \). Consider the orthogonal splitting
where \(\varepsilon \cdot I+A-B\) is negative definite on \({\textbf{L}}^-_{\varepsilon \cdot I+A-B}\), and positive define on \({\textbf{L}}^+_{\varepsilon \cdot I+A-B}\). Thus, if \(z\in {\textbf{L}}\), we have the splitting
with \(z^-\in {\textbf{L}}^-_{\varepsilon \cdot I+A-B}\) and \(z^+\in {\textbf{L}}^+_{\varepsilon \cdot I+A-B}\). If \(z_{\varepsilon ,\lambda }\) is a solution of (\(HS_{\varepsilon ,\lambda }\)) with its splitting \(z_{\varepsilon ,\lambda }=z^-_{\varepsilon ,\lambda }+z^+_{\varepsilon ,\lambda }\) defined above, then we have
Since \((\varepsilon -\varepsilon _0,\varepsilon )\cap \sigma (\varepsilon \cdot I+A-B)=\emptyset \), we have
Since r is bounded, for \((\varepsilon ,\lambda )\in (0,\frac{\varepsilon _0}{2})\times [0,1]\), we have
Therefor, we have
Step 2 For any \((\varepsilon ,\lambda )\in (0,\frac{\varepsilon _0}{2})\times [0,1]\), (\(HS_{\varepsilon ,\lambda }\)) has at least one solution. Here, we use the topology degree theory. Since \(0\notin \sigma (\varepsilon \cdot I+A-B)\), (\(HS_{\varepsilon ,\lambda }\)) can be rewritten as
Denote by \(f(\varepsilon ,\lambda ,z):=\lambda (\varepsilon \cdot I+A-B)^{-1}G(z)\) for simplicity. From the compactness of \((\varepsilon \cdot I+A-B)^{-1}\) and condition (\(V^-_2\)), Leray Schauder degree theory can be used to the map
From the result received in Step 1, we have
where \(R_\varepsilon >\frac{C}{\varepsilon }\) is a constant only depends on \(\varepsilon \), and \(B(0,R_\varepsilon ):=\{z\in {\textbf{L}} |\Vert z\Vert _{\textbf{L}}<R_\varepsilon \}\).
Step 3 For \(\lambda =1\), \(\varepsilon \in (0,\varepsilon _0/2)\), denote by \(z_\varepsilon \) one of the solutions of (\(HS_{\varepsilon ,1}\)). We have \(\Vert z_\varepsilon \Vert _{\textbf{H}}\le C\). In this step, C denotes various constants independent of \(\varepsilon \).
From the boundedness received in Step 1, we have
Now, consider the orthogonal splitting
where \(A-B\) is zero on \({\textbf{L}}^0_{A-B}\), and \({\textbf{L}}^\bot _{A-B}\) is the orthonormal complement space of \({\textbf{L}}^0_{A-B}\). Let \(z_\varepsilon =z^0_\varepsilon +z^\bot _\varepsilon \) with \(z^0_\varepsilon \in {\textbf{L}}^0_{A-B}\) and \(z^\bot _\varepsilon \in {\textbf{L}}^\bot _{A-B}\). Since 0 is an isolated point in \(\sigma (A-B)\), from (3.1), we have
Additionally, since \(G(z_\varepsilon )\), \(z^\bot _\varepsilon \) and \(\varepsilon z_\varepsilon \) are bounded in \({\textbf{L}}\), we have
On the other hand, from (3.14) in (\(H^-_2\)), we have
where \(\Omega (K):=\left\{ t\in S^1||z_\varepsilon (t)|>K\right\} \). From (3.3) and (3.4), we have
Moreover, since \({\textbf{L}}^0_{A-B}\) is a finite dimensional space, all norms are equivalent, from (3.2) and (3.5), we have prove the boundedness of \(\Vert z_\varepsilon \Vert _{\textbf{L}}\).
Step 4 Passing to a sequence of \(\varepsilon _n\rightarrow 0\), there exists \(z\in {\textbf{L}}\) such that
Here, we will use the method of saddle point reduction. Recall the constant \(l>l_V\) satisfying condition (2.15), the projections \(P^0_{A}\), \(P^\bot _{A}\) defined in (2.16), the decomposition \( {\textbf{L}}={\textbf{L}}^0_{A}\oplus {\textbf{L}}^\bot _{A} \) defined in (2.17), and we have
with \(\delta =l-l_V\). Let \(\varepsilon ^\prime :=\min \{\varepsilon _0,\delta \}\), for \(\varepsilon \in (0,\frac{\varepsilon ^\prime }{2})\), denote by \(A_\varepsilon :=\varepsilon \cdot I+A\). Then \(A_\varepsilon \) has the same invariant subspace with A, so we can also denote by \(A^*_\varepsilon :=A_\varepsilon |_{{\textbf{L}}^*}\) (\(*=0,\bot \)), and we have
Since \(z_\varepsilon \) satisfies (\(HS_{\varepsilon ,1}\)), so we have
with \(\Phi \) defined in (2.19), and
Since \({\textbf{L}}^0\) is a finite dimensional space and \(\Vert z_\varepsilon \Vert _{\textbf{L}}\le C\), there exists a sequence \(\varepsilon _n\rightarrow 0\) and \(z^0\in {\textbf{L}}^0\), such that
For simplicity, we rewrite \(z^*_n:=z^*_{\varepsilon _n}\)(\(*=\bot ,0\)), \(A_n:=\varepsilon _n+A\) and \(A^\bot _n:=A^\bot _{\varepsilon _n} \). So, we have
Since \((A_n^\bot )^{-1}-(A_m^\bot )^{-1}=(\varepsilon _m-\varepsilon _n)(A_n^\bot )^{-1}(A_m^\bot )^{-1}\) and \(\{z_n\}\) is bounded in \({\textbf{L}}\), we have
So we have
therefore, there exists \(z^\bot \in {\textbf{L}}^\bot \), such that \(\displaystyle \lim _{n\rightarrow \infty }\Vert z^\bot _n- z^\bot \Vert _{\textbf{L}}=0\). Thus, we have
with \(z=z^\bot +z^0\). Last, let \(n\rightarrow \infty \) in (\(HS_{\varepsilon _n,1}\)), we have z is a solution of (HS). \(\square \)
In the proof of Theorem 1.2, we need the following Lemma.
Lemma 3.1
Let \(B_1,B_2\in {\mathcal {L}}_Q(S_1,{\mathcal {L}}({\mathbb {R}}^{N}))\).
-
(1)
If \(B_1< B_2\), then we have
$$\begin{aligned} \mu _Q(A,B_2)-\mu _Q(A,B_1)=\displaystyle \sum _{\lambda \in [0,1)}\upsilon _Q(A,B_1+\lambda (B_2-B_1)). \end{aligned}$$(3.8) -
(2)
If \(B_1\le B_2,\;\mu _Q(A,B_1)=\mu _Q(A,B_2),\; \textrm{and}\; \upsilon _Q(A,B_2)=0\), then there exists \(\varepsilon >0\), such that for all \(B\in {\mathcal {L}}_Q(S_1,{\mathcal {L}}({\mathbb {R}}^{N}))\) with
$$\begin{aligned} B_1\le B \le B_2, \end{aligned}$$we have
$$\begin{aligned} \sigma (A-B)\cap (-\varepsilon ,\varepsilon )=\emptyset . \end{aligned}$$
Proof
- (1)
-
(2)
Since \(\upsilon _Q(A,B_2)=0\), there is \(\varepsilon >0\), such that
$$\begin{aligned} \upsilon _Q(A,B_{2}+\lambda \varepsilon )=0,\;\;\forall \lambda \in [0,1]. \end{aligned}$$From (3.8) we have \(\mu _Q(A,B_2+\varepsilon \cdot I)=\mu _Q(A,B_2)\), and from \(\mu _Q(A,B_1)=\mu _Q(A,B_2)\), we have \(\upsilon _Q(A,B_1)=0\). So we can choose \(\varepsilon >0\) small enough, such that
$$\begin{aligned} \upsilon _Q(A,B_{1}-\varepsilon \cdot I)=\upsilon _Q(A,B_{2}+\varepsilon \cdot I)=0. \end{aligned}$$Since
$$\begin{aligned}B_{1}-\varepsilon \cdot I\le B-\varepsilon I<B+\varepsilon I\le B_{2}+\varepsilon \cdot I, \end{aligned}$$it follows that \(\mu _Q(A,B-\varepsilon I)=\mu _Q(A,B+\varepsilon I)\). Note that by (2.7)
$$\begin{aligned} \sum _{-\varepsilon < t \le \varepsilon } \upsilon _Q(A,B-t \cdot I)=\mu _Q(A,B+\varepsilon )-\mu _Q(A,B-\varepsilon )=0. \end{aligned}$$We have \(0\notin \sigma (A-B-\eta ),\;\forall \eta \in (-\varepsilon ,\varepsilon )\), thus the proof is complete. \(\square \)
Proof of Theorem 1.2
Consider the following one-parameter equation
with \(\lambda \in [0,1]\). Denote by
Since V satisfies condition (\(V_1\)) and \(B_1\in {\mathcal {L}}_Q(S_1,{\mathcal {L}}({\mathbb {R}}^{N}))\), we have \(\Phi '_\lambda :{\textbf{L}}\rightarrow {\textbf{L}}\) is Lipschitz continuous, and there exists \(l^\prime >0\) independed of \(\lambda \) such that \(l^\prime \notin \sigma (A)\) and
Now, replace \(l_V\) by \(l^\prime \) in (2.16), we have the projections \(P^*_{A,l^\prime }\)(\(*=\bot ,0\)) and the splitting
with \({\textbf{L}}^*_{A,l^\prime }=P^*_{A,l^\prime }{\textbf{L}}\)(\(*=\bot ,0\)). Thus \(A^\bot \) has bounded inverse on \({\textbf{L}}^\bot _{A,l^\prime }\) with
for some \(c>0\). Without confusion, we still use \(z^\bot \) and \(z^0\) to represent the splitting
with \(z^*\in {\textbf{L}}^*_{A,l^\prime }\)(\(*=\bot ,0\)). Now, we divide the remainder of the proof into three steps. The number \(C>0\) denotes various constants independent of \(\lambda \).
Step 1 If z is a solution of (\(HS_\lambda \)), then we have \(\Vert z^\bot (z^0)\Vert _{\textbf{L}}\le C\Vert z^0\Vert _{{\textbf{L}}}+C\) Since \(A z=\Phi _\lambda ^\prime (z)\), we have
So we have \(\Vert z^\bot (z^0)\Vert _{\textbf{L}}\le \frac{l^\prime }{c}\Vert z^0\Vert _{{\textbf{L}}}+\frac{1}{c}\Vert \Phi ^\prime _\lambda (0)\Vert _{{\textbf{L}}}\). Thus, we have prove this step.
Step 2 We claim that the set of all the solutions (\(z,\lambda \)) of (\(HS_\lambda \)) are a priori bounded.
If not, there exists a sequence \(\{(z_n,\lambda _n)\}\) with \(\lambda _n\in [0,1]\) solving the problem (\(HS_\lambda \)) with \(\Vert z_n\Vert _{{\textbf{L}}}\rightarrow \infty \). Without lose of generality, assume \(\lambda _n\rightarrow \lambda _0\in [0,1]\). From step 1, we have \(\Vert z^0_n\Vert _{\textbf{L}}\rightarrow \infty \). Denote by
and \(\bar{B}_n:=(1-\lambda _n)B_1+\lambda _nB(z_n)\), from condition (V\(_3\)) we have \( Ay_n=\bar{B}_ny_n+\frac{o(\Vert z_n\Vert _{\textbf{L}})}{\Vert z_n\Vert _{\textbf{L}}}, \) that is
Decompose \(y_n=y^{\bot }_n+y^0_n\) with \(y^*_n=z^*_n/\Vert z_n\Vert _{\textbf{L}},\;*=\bot ,0\), we have
That is to say
for n large enough. Since \(B_1(t)\le B(z)\le B_2(t)\), we have \(B_1\le \bar{B}_n\le B_2\). Duo to condition (V\(_4\)) and Lemma 3.1, we write \({\textbf{L}}={\textbf{L}}^+_{A-{\bar{B}}_n}\bigoplus {\textbf{L}}^-_{A-{\bar{B}}_n}\) with \(A-{\bar{B}}_n\) is positive and negative define on \({\textbf{L}}^+_{A-{\bar{B}}_n}\) and \({\textbf{L}}^-_{A-{\bar{B}}_n}\) respectively. Re-decompose \(y_n=\bar{y}^+_n+\bar{y}^-_n\) respect to \({\textbf{L}}^+_{A-{\bar{B}}_n}\) and \({\textbf{L}}^-_{A-{\bar{B}}_n}\). From (\(V_4\)) and (3.9), we have
Since \(\Vert z_n\Vert _{\textbf{L}}\rightarrow \infty \) and \(\Vert y_n\Vert _{\textbf{L}}=1\), we have \(\Vert y^0_n\Vert _{\textbf{L}}\rightarrow 0\) which contradicts to (3.10), so we have \(\{z_n\}\) is bounded.
Step 3 By Leray–Schauder degree, there is a solution of (HS).
Since the solutions of (\(HS_\lambda \)) are bounded, there is a number \(R>0\) large eoungh, such that all of the solutions \(z_\lambda \) of (\(HS_\lambda \)) are in the ball \(B(0,R):=\{z\in {\textbf{L}}| \Vert z\Vert _{\textbf{L}}<R\}\). So we have the Larey-Schauder degree
is well defined and independent of \(\lambda \in [0,1]\), so
That is to say (HS) has at least one solution.
3.2 Applications
Consider the following delay differential system
with \(x\in C^2(\mathbb {R}, \mathbb {R}^n)\), \(v\in C^1(\mathbb {R}^n,\mathbb {R})\).
Now, we will transform the system (DDS) into the system (HS). Set \(\tau =1\) for simplicity. Let
with \(x_k(t)=x(t-(k-1))\) and \(k=1,2,\cdots , m\). If x(t) is a solution of (DDS), z(t) is a solution of the following system
where \(A_{nm}=\left( \begin{matrix} 0 &{}I_n&{} \cdots &{} I_n\\ I_n &{}0&{}\cdots &{} I_n\\ \cdots &{}\cdots &{}\cdots &{}\cdots \\ I_n&{}I_n&{}\cdots &{}0 \end{matrix} \right) _{nm}\), and \(Q_{nm}=\left( \begin{matrix} 0 &{}0&{} \cdots &{} 0&{}I_n\\ I_n &{}0&{}\cdots &{}0&{} 0\\ 0&{} I_n&{}\cdots &{}0&{}0\\ \cdots &{}\cdots &{}\cdots &{}\cdots &{}\cdots \\ 0&{}0&{}\cdots &{}I_n&{}0 \end{matrix} \right) _{nm}\), with \(I_n\) the identity map on \(\mathbb {R}^n\), \(z:\mathbb {R}\rightarrow \mathbb {R}^{nm}\). The function \(V:\textbf{R}^{nm}\rightarrow \textbf{R}\),
On the other hand, if z(t) is a solution of (HS\(_2\)), \(x_1(t)\) is a solution of (DDS).
Let \(A:=-A^{-1}_{nm}\) and \(Q:=Q_{nm}\), it is easy to see that A, Q and V defined here satisfy the conditions in (1.1), (1.2) and (1.3), so (HS\(_2\)) is a specific case of (HS). Corresponding to Theorem 1.1 and 1.2, we have the following results.
Theorem 3.2
Assume \(v\in C^1(\textbf{R}^n,\textbf{R})\) satisfies the following conditions.
(\(v_1\)) \(v':{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^{n}\) is Lipschitz continuous
with its Lipschitz constant \(l_v>0\).
(\(v^\pm _2\)) There exists \(M_1,\;M_2,\;K>0\), \(b(t)\in \mathcal L(S_1,{\mathcal {L}}({\mathbb {R}}^{n}))\) with \(b(t)\equiv b\), such that
with
and
Then (DDS) has at least one solution.
Theorem 3.3
Assume \(v\in C^1(\textbf{R}^n,\textbf{R})\) satisfies conditions (\(v_1\)) and the following condition
(\(v_3\)) There exists \(b\in C(\mathbb {R}^{n},\mathcal {L}_s({\mathbb {R}}^{n}))\) such that
with
(\(v_4\)) There exist \(b_1,\;b_2\in {\mathcal {L}}([0,1],{\mathcal {L}}({\mathbb {R}}^{n}))\) satisfying
and \(B_1,\;B_2\in {\mathcal {L}}_Q(S^1,{\mathcal {L}}({\textbf{R}}^{nm}))\) satisfying
with
Then (DDS) has at least one solution.
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The first author is partially supported by NNSF of China(11790271, 12171108), Guangdong Basic and Applied basic Research Foundation(2020A1515011019), Innovation and Development Project of Guangzhou University. The second author is partially supported by PSF of China (188576).
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Liu, C., Wang, Q. The Existence of Periodic Solutions for Second-Order Delay Differential Systems. J Dyn Diff Equat 35, 1993–2011 (2023). https://doi.org/10.1007/s10884-022-10226-2
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DOI: https://doi.org/10.1007/s10884-022-10226-2