Abstract
This paper is concerned with the existence of periodic solutions for asymptotically linear second-order delay differential equations. We will establish an index theory for the linear system directly in the sense that we do not need to change the problem of the original linear system into the problem of an associated Hamiltonian system. By using the critical point theory and the index theory, some new existence results are obtained.
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1 Introduction
In the past few decades, many results on the existence of periodic solutions for delay differential equations were obtained by several different approaches, including various fixed point theorems, Hopf bifurcation theorems, coincidence degree theory, coupled system methods, Poincaré–Bendixson theorem, and so on. One can refer to [9, 12, 26, 27] for detailed discussions.
In [28], Kaplan and Yorke introduced a technique studying the existence of periodic solutions of the first-order delay differential equation
that may reduce the existence problem of periodic solutions of (1) to a problem of finding periodic solutions of an associated plane ordinary differential system. Afterwards, Kaplan and York’s original idea was used by many authors to study some general differential delay equations by transforming them into Hamiltonian systems, for example, [7, 8, 10, 20–22, 29–31, 33]. Especially, some known results involving various index theories in Hamiltonian systems were generalized to study delay differential equations. For example, in [29–31], by using Morse–Ekeland index theory for the associated Hamiltonian system, Li and He firstly studied the periodic solutions of delay differential equations with multiple delays via critical point theory. Some multiplicity results on periodic solutions in delay differential equations were given by Fei [20, 21] using \(S^{1}\)-pseudo index theory. In [32, 42], by relative Fredholm index and spectral flow, Liu and Wang constructed an index theory for the coupled Hamiltonian system and obtained some interesting existence and multiplicity results for delay differential equations.
In [24], Guo and Yu firstly applied the critical point theory to (1) directly in the sense that one does not need to transform the original existence problem of (1) to the existence problem for an associated Hamiltonian system while allowing a delay in the variational functional. By using pseudo-index theory introduced by Benci and Rabinowitz [4], they obtained multiple periodic solutions of (1) with odd nonlinearity that grows asymptotically linear both at the origin and at infinity. By establishing suitable variational frameworks for the second-order delay differential equation and high-dimensional case, Guo and Guo [23], Guo and Yu [25] obtained some sufficient conditions on the existence of periodic solutions.
Motivated by [23, 32, 42], we consider the existence of 2τ-periodic solutions for the following nonautonomous second-order delay system:
where \(\tau >0\) is a given constant.
Throughout this paper, we make use of the following hypotheses:
- \((F_{1})\):
-
\(F\in C^{1}({\mathbb{R}}\times {\mathbb{R}}^{n}, {\mathbb{R}})\), \(F(t, 0)=0\).
Furthermore, we assume there exist continuous 2τ periodic symmetric matrix functions \(A_{0}(t)\) and \(A_{\infty}(t)\) such that
- \((F_{0})\):
-
\(f(t, x)=A_{0}(t)x+o(x)\text{ as }|x|\rightarrow 0\),
- \((F_{\infty})\):
-
\(A_{\infty}>A_{0}\) and \(f(t, x)=A_{\infty}(t)x+o(x)\text{ as }|x|\rightarrow \infty \),
where we denote \(\nabla F=f\), and for two continuous \(n\times n\) matrix-valued functions \(B_{1}(t)\), \(B_{2}(t)\), we say that \(B_{1}(t)\leq B_{2}(t)\) if and only if \(\max_{\xi \in {\mathbb{R}}^{n}, |\xi |=1} (B_{1}(t)- B_{2}(t))\xi \xi \leq 0\), and \(B_{1}(t)> B_{2}(t)\) if and only if \(B_{1}(t)\leq B_{2}(t)\) does not hold. Furthermore, we write \(B_{1}\leq B_{2}\) if \(B_{1}(t)\leq B_{2}(t)\) for a.e. \(t\in S^{1}\) and denote by \(B_{1}< B_{2}\) if \(B_{1}\leq B_{2}\) and \(B_{1}(t)< B_{2}(t)\) on a subset of \(S^{1}\) with nonzero measure.
Conditions (\(F_{0}\)) and (\(F_{\infty}\)) are referred to in the literature as asymptotically linear quadratic for the nonlinearity F. This type of conditions was widely used in the problems of periodic solutions of Hamiltonian systems. It is well known that the existence and multiplicity of the periodic solutions are related to the difference between \(A_{0}\) and \(A_{\infty}\), and a quantitative way to measure the difference is the index theory. Ekeland established an index theory for convex linear Hamiltonian systems in [17–19]. In [11, 34, 35, 37, 39], Conley et al. introduced an index theory for symplectic paths. In [38, 44], Long and Zhu defined spectral flows for paths of linear operators, the relative Morse index between two linear operators, and redefined Maslov index for symplectic paths. Liu [32] defined the \((\mathcal{J}, \mathcal{M})\)-index and studied the delay Hamiltonian systems. Dong [15] developed an index theory for abstract operator equations with compact resolvent. We also mention that the authors in [6, 40, 43] established various index theories when essential spectrum emerges. We refer to the excellent books of Abbondandolo [1], Ekeland [18], and Long [36] for a more detailed account of the concept.
In this paper, we establish the existence and multiplicity of periodic solutions for (2) by combining the index theory with a generalized linking theorem developed by Ding and Liu [14]. More precisely, for any 2τ-periodic continuous matrix-valued function \(B(t)\), we consider the following linear delay differential equation:
Note that the spectrum of the operator \(A: Ax=\ddot{x}(t+\tau )\) consists of a sequence of eigenvalues with finite multiplicity, which is unbounded from above and below. This implies that the Morse index of the energy functional corresponding to (3) is infinite. Thus, the main difficulty is to find the finite representation of the indefinite Morse index. By taking advantage of the spectral properties of A, we will modify the linear system (3) so that the Morse index of the modified linear system is finite. Then we will define a relative Morse index. Our approach follows closely that in [16]. Moreover, to combine the relative Morse index and the generalized linking theorem developed by Ding and Liu [14], we need to make a suitable choice of the reduction very carefully and overcome some involved issues.
To state our main results, we first introduce some notations. We denote by \(\nu (B)\) the dimension of the kernel of (3) and call \(\nu (B)\) the nullity of B. For any 2τ-periodic continuous matrix-valued functions \(B_{1}\), \(B_{2}\) with \(B_{2}>B_{1}\), we define
We call \(I(B_{1}, B_{2})\) the relative Morse index between \(B_{1}\) and \(B_{2}\) (see Definition 2). For the definitions of the relative Morse index, we refer to [15, 18, 43].
Our main results are stated as follows.
Theorem 1
Let \((F_{1})\), \((F_{0})\), \((F_{\infty})\) be satisfied. Moreover, we assume that \((F_{0})\) holds with \(\nu (A_{0})=0\) and \((F_{\infty})\) holds with \(\nu (A_{\infty})=0\). Then equation (2) has at least one nonconstant 2τ periodic solution.
Theorem 2
Under the assumption of Theorem 1, if furthermore f is odd, then (2) has at least \(I(A_{0}, A_{\infty})\) pairs of 2τ periodic solutions.
Remark 1
The condition \(\nu (A_{\infty})= 0\) implies that the nonlinearity f is nonresonant at infinity. This assumption is crucial for the proof of the Palais–Smale condition (PS condition for short). We refer to [4, 41] and the references therein for results that allow resonance at infinity under some additional technical conditions such as Landsmann–Lazer condition, Rabinowitz resonant condition, strong resonant conditions, etc.
The paper is organized as follows. In Sect. 2, we formulate the variational setting and develop an index theory to classify the associated linear second-order delay differential equations and define the relative Morse index. After collection in Sect. 2.3 of the abstract critical point theorems, which we needed, we prove our theorems in Sect. 3.
2 Variational structure and classification theory
2.1 Variational setting
In this paper, we establish a variational structure that enables us to reduce the existence of 2τ-periodic solutions of (2) to the existence of critical points of corresponding functional defined on some appropriate function space.
Assume that \(\lambda =\frac{\tau}{\pi}\), \(s=\frac{\pi}{\tau}t\). Then equation (2) is transformed to
and we seek 2π periodic solutions of (4), which, of course, correspond to the 2τ periodic solutions of (2). Define \(Ax=-\ddot{x}(t+\pi )\). Then A is self-adjoint on \(H^{1}(S^{1}, {\mathbb{R}}^{n})\) with domain \(D(A)=H^{2}(S^{1}, {\mathbb{R}}^{n})\) (see Lemma 2.2 of [23]). Let \(\sigma (A)\), \(\sigma _{d}(A)\), \(\sigma _{e}(A)\) denote, respectively, the spectrum, the discrete spectrum, and the essential spectrum of A. It is easy to calculate that
For any 2π-period continuous matrix-valued function \(B_{0}(t)\), we denote \(B_{0}\) as the multiplication operator by \(B_{0}(t)\) in \(L^{2}(S^{1}, {\mathbb{R}}^{n})\). Let \(A_{B_{0}}=A-B_{0}\). By (5), we also have
Throughout this paper, we require \(0\notin \sigma (A_{B_{0}})\), which means Ker\((A-B_{0})=\emptyset \). In fact, if \(0\in \sigma (A_{B_{0}})\), by property (6), there exists \(\epsilon _{0}\) such that, for any \(\epsilon \in (0, \epsilon _{0})\), we have \(0\notin \sigma (A_{B_{0}+\epsilon})\), and then we can replace \(B_{0}\) by \(B_{0}+\epsilon \).
Moreover, (4) can be transformed to
This modification is crucial for the establishment of the linking structure for the problem. More precisely, in Sect. 3, we may assume that \(B_{0}=A_{0}-\epsilon \) for some ϵ small, where \(A_{0}\) is defined as in condition (\(f_{0}\)).
Let \(|A_{B_{0}}|\) be the absolute value of \(A_{B_{0}}\) and \(E_{B_{0}}=D(|A_{B_{0}}|^{\frac{1}{2}})\) be the domain of the self-adjoint operator \(|A_{B_{0}}|^{\frac{1}{2}}\), which is a Hilbert space equipped with the inner product
and the induced norm \(\|z\|_{B_{0}}=(z,z)_{B_{0}}^{\frac{1}{2}}\). By the spectral properties (6), we have \(|A_{B_{0}}|^{-\frac{1}{2}}\) is a compact operator and it follows that
Lemma 1
\(E_{B_{0}}\) embeds continuously into \(H^{1}(S^{1}, {\mathbb{R}}^{n})\), and \(E_{B_{0}}\) embeds compactly into \(L^{p}(S^{1}, {\mathbb{R}}^{n})\) for all \(p\geq 2\).
Let \(\{G_{\lambda}\}_{\lambda \in {\mathbb{R}}}\) denote the spectral family of \(A_{B_{0}}\). We define the following projections:
Here \(\beta >0\) is large enough and \(\pm \beta \notin \sigma (A_{B_{0}}) \). This induces an orthogonal decomposition on \(E_{B_{0}}\):
For any \(x\in E_{B_{0}}\), we have \(x=x^{+} + x^{0} + x^{-}\in E_{\beta,B_{0}}^{+}\oplus E_{\beta,B_{0}}^{0} \oplus E_{\beta,B_{0}}^{-}\) and \(|A_{B_{0}}|x=A_{B_{0}}(x^{+} - x^{0} - x^{-})\).
On \(E_{B_{0}}\) we define the functional
By a standard argument as in [4, 23], the functional \(I_{\beta, B_{0}}\) is continuously differentiable on \(E_{B_{0}}\) and the existence of 2π-periodic solutions \(x(t)\) for (2) is equivalent to the existence of critical points of functional \(I_{\beta, B_{0}}(x)\).
2.2 Index theory
In this subsection, we investigate the linear second-order delay differential equations
where \(B_{0}(t)\), \(B(t)\) are 2π-period continuous matrix-valued functions. Recall that we define \((A_{B_{0}}x)(t)=\ddot{x}(t+\pi )-\lambda ^{2} B_{0}(t)x(t)\) and \((Bx)(t)=B(t)x(t)\). Then (9) can be rewritten as
Let \(x\in E_{B_{0}}\) be a solution of (10). Set \(u=|A_{B_{0}}|^{\frac{1}{2}}x\). Then \(u\in L^{2}(S^{1}, {\mathbb{R}}^{n})\). The projections \(P_{\beta, B_{0}}^{+}\), \(P_{\beta, B_{0}}^{0}\), \(P_{\beta, B_{0}}^{-}\) defined as in Sect. 2.1 also induce a decomposition on \(L^{2}(S^{1}, {\mathbb{R}}^{N})\):
where \(L_{\beta,B_{0}}^{\pm}=P_{\beta,B_{0}}^{\pm }L^{2}(S^{1}, { \mathbb{R}}^{n})\) and \(L_{\beta,B_{0}}^{0}=P_{\beta,B_{0}}^{0} L^{2}(S^{1}, {\mathbb{R}}^{n})\). For any \(u\in L^{2}(S^{1}, {\mathbb{R}}^{n})\), we have \(u=u^{+} + u^{0} + u^{-}\in L_{\beta,B_{0}}^{+}\oplus L_{\beta,B_{0}}^{0} \oplus L_{\beta,B_{0}}^{-}\). Then (10) is equivalent to
Define the associated bilinear form
for any \(u, v\in L^{2}(S^{1}, {\mathbb{R}}^{n})\). From (11), it is easy to get
and
Since \(P_{\beta,B_{0}}^{-}|A_{B_{0}}|^{-\frac{1}{2}}=\int _{-\infty}^{- \beta}|\lambda |^{-\frac{1}{2}}\,d F_{\lambda}\), we have
Thus, by choosing \(\beta >\lambda ^{2}(\|B\|+ \|B_{0}\|) \), we have
It follows that \(P_{\beta,B_{0}}^{-}+P_{\beta,B_{0}}^{-}|A_{B_{0}}|^{-\frac{1}{2}} \lambda ^{2}(B-B_{0})|A_{B_{0}}|^{-\frac{1}{2}}P_{\beta,B_{0}}^{-}\) is invertible. Let \(L_{\beta, B_{0}}^{\ast}=L_{\beta, B_{0}}^{+}\oplus L_{\beta, B_{0}}^{0}\) and define \(L_{\beta, B_{0}, B }:L_{\beta, B_{0}}^{\ast}\rightarrow L_{\beta, B_{0}}^{-}\):
Then it is easy to calculate that, for fixed \(u^{\ast}\in L_{\beta, B_{0}}^{\ast}\), \(L_{\beta, B_{0}, B }u^{\ast}\) is the unique solution for
Moreover, for any \(u^{\ast }\in L_{\beta, B_{0}}^{\ast}\), we define
Then \(\psi _{u^{\ast}}(y)\) is of class \(C^{2}\) on \(L_{\beta, B_{0}}^{-}\), and for any \(w\in L_{\beta, B_{0}}^{-}\),
And it follows from (12) that \(D^{2} \psi _{u^{\ast}}(y)(w, w)<0\). Thus \(\psi _{u^{\ast}}(y)\) has unique maximum at the point \(L_{\beta, B_{0}, B }u^{\ast}\). This yields
Now we define a quadratic form \(\tilde{q}_{\beta, B_{0}, B}\) on \(L^{\ast}_{\beta, B_{0}}\) as
By the definition of \(L_{\beta, B_{0}, B }\), we have
and for any \(u^{\ast}, v^{\ast}\in L^{\ast}_{\beta, B_{0}}\),
This yields
Remark 2
In the spirit, the saddle-point reduction process was developed by Amann and Zehnder [2], Chang [5], and Long [36], although technically our constructions of reduction are different.
Lemma 2
For any 2π-periodic continuous matrix-valued function B, there is a splitting
such that
(1) \(L_{\beta, B_{0}}^{+}(B)\), \(L_{\beta, B_{0}}^{0}(B)\), \(L_{\beta, B_{0}}^{-}(B)\) are \(\tilde{q}_{\beta, B_{0}, B}\)-orthogonal, and \(\tilde{q}_{\beta, B_{0}, B}\) is positive definite, negative definite on \(L_{\beta, B_{0}}^{+}(B)\) and \(L_{\beta, B_{0}}^{-}(B)\), respectively. Moreover, \(\tilde{q}_{\beta, B_{0}, B}(u^{\ast}, u^{\ast})=0\), \(\forall u^{\ast }\in L_{\beta, B_{0}}^{0}(B)\).
(2) \(L_{\beta, B_{0}}^{0}(B)\), \(L_{\beta, B_{0}}^{-}(B)\) are two finite dimensional subspaces.
Proof
Define the self-adjoint operator \(\Lambda _{\beta, B_{0}, B }\) on \(L^{\ast}_{\beta, B_{0}}\) as follows:
Then, by (14), we have
where Id denotes the identity map on \(L_{\beta, B_{0}}^{\ast}\). Recalling that \(|A_{B_{0}}|^{-\frac{1}{2}}\) is a compact operator, we have \(\Lambda _{\beta, B_{0}, B }: L_{\beta, B_{0}}^{\ast}\rightarrow L_{ \beta, B_{0}}^{\ast}\) is self-adjoint and compact. Then there is a basis \(\{ e_{j}\}\in L_{\beta, B_{0}}^{\ast}\) and a sequence \(\mu _{j}\rightarrow 0\) in \({\mathbb{R}}\) such that
Thus, for any \(u^{\ast}\in L_{\beta, B_{0}}^{\ast}\), which can be expressed as \(u^{\ast}=\sum _{j=1}^{\infty }c_{j} e_{j}\), we have
Since \(\mu _{j}\rightarrow 0\), all the coefficients \((1-\mu _{j})\) are positive except a finite number. Thus our lemma follows by
□
Definition 1
For any 2τ-periodic continuous matrix-valued function B, we define
We call \(i_{\beta, B_{0}}(B)\) and \(\nu _{\beta, B_{0}}(B)\) the index and nullity of B, respectively.
Lemma 3
\(\nu _{\beta, B_{0}}(B)=\dim \ker (A-B) \).
Proof
For any \(u^{\ast}\in L_{\beta, B_{0}}^{0}(B)\), we have
This implies
By the definition of \(L_{\beta,B_{0},B}\), we have
Combining (16) and (17), we obtain
Let \(x=|A_{B_{0}}|^{-\frac{1}{2}}(u^{+}- u^{0}- L_{\beta,B_{0},B}u^{\ast})\), where \(u^{+}=P_{\beta, B_{0}}^{+}u^{\ast}\), \(u^{0}=P_{\beta, B_{0}}^{0}u^{\ast}\). Then \(x\in E_{B_{0}}\) and
Hence, \(\dim L_{\beta, B_{0}}^{0}(B)=\dim \ker (A-B)\). □
Remark 3
From Lemma 3, we observe that \(\nu _{\beta, B_{0}}(B)\) is independent of β and \(B_{0}\). Thus, we will write \(\nu (B)\) for convenience.
Recall that for a symmetric bilinear form φ defined on a Hilbert space X, its Morse index is defined as \(m^{-}(\phi )=\max \{ \dim X_{1} | X_{1}\text{ is a subspace of }X \text{ such that } \phi (x,x)<0\text{ for any }x\in X_{1}\backslash \{0\} \}\).
Lemma 4
\(i_{\beta, B_{0}}(B)\) is the Morse index of \(\tilde{q}_{\beta, B_{0},B}\).
Proof
Let \(E_{1}\subset L^{\ast}_{\beta, B_{0}}\) with \(\dim E_{1}=k\) such that
Let \(\{e_{j}\}_{j=1}^{k}\) be linear independent in \(E_{1}\). We have the following decomposition:
We claim that \(e_{1}^{-}, \ldots, e_{k}^{-}\) are linear independent. Arguing indirectly, we assume that there exist not all zero numbers \(\alpha _{j}\in {\mathbb{R}}\) such that
Denote \(e=\sum _{j=1}^{k}\alpha _{j}e_{j}\). On the one hand, we have \(e\in E_{1}\) and
On the other hand, \(e=\sum _{j=1}^{k}\alpha _{j}(e_{j}^{0}+e_{j}^{+})\in L_{\beta, B_{0}}^{0}(B) \oplus L_{\beta, B_{0}}^{+}(B)\), we have
This is a contradiction. Thus, \(e_{1}^{-}, \ldots, e_{k}^{-}\) are linear independent, which implies dim\(L_{\beta, B_{0}}^{-}\geq k\) and \(i_{\beta, B_{0}}(B)\geq m^{-}(\tilde{q}_{\beta, B_{0},B})\).
In addition, by the definition of \(L_{\beta, B_{0}}^{-}\), we have
Thus, \(m^{-}(\tilde{q}_{\beta, B_{0},B})\geq \) dim\(L_{\beta, B_{0}}^{-}=i_{ \beta, B_{0}}(B)\). This completes the proof. □
For any \(B_{1}< B_{2}\), let \(B_{s}=(1-s)B_{1}+sB_{2}\), \(s\in (0,1]\), and let \(i_{\beta, B_{0}}(s)=i_{\beta, B_{0}}(B_{s})\), \(\nu (s)=\nu (B_{s})\).
Lemma 5
For any \(s_{0}\in [0,1)\), there exists \(\delta >0\) such that
Proof
Let \(C_{B_{s}}=(P_{\beta, B_{0}}^{-} + P_{\beta, B_{0}}^{-} | A_{B_{0}}|^{- \frac{1}{2}} \lambda ^{2}(B_{s}-B_{0})| A_{B_{0}}|^{-\frac{1}{2}} P_{ \beta, B_{0}}^{-} )^{-1}\). By direct calculation, we have
and
where \(P_{\beta, B_{0}}^{\ast}=P_{\beta, B_{0}}^{+}+ P_{\beta, B_{0}}^{0}\). Thus, for any \(u^{\ast}\in L_{\beta, B_{0}}^{\ast}(B_{s_{0}})\), there exists \(\delta >0\) such that if \(0< s-s_{0}<\delta \), we have
Note that \(\dim L_{\beta, B_{0}}^{\ast}(B_{s_{0}})<+\infty \), \(L_{\beta, B_{0}}^{\ast}(B_{s_{0}})\) is compact. It is easy to deduce that there exists \(\delta _{0}\) independent of \(u^{\ast}\) such that
for all \(s\in (s_{0}, s_{0}+\delta )\). Thus, by Lemma 4, we have \(i_{\beta, B_{0}}(s_{0})+\nu (s_{0})\leq i_{\beta, B_{0}}(s)\). □
Let \(i_{\beta, B_{0}}(s+0)=\lim_{t\rightarrow s^{+}}i_{\beta, B_{0}}(t)\). We have
Lemma 6
\(i_{\beta, B_{0}}(s+0)=i_{\beta, B_{0}}(s)+\nu (s)\).
Proof
From Lemma 5, we have \(i_{\beta, B_{0}}(s+0)\geq i_{\beta, B_{0}}(s)+\nu (s)\). Thus, it suffices to prove \(i_{\beta, B_{0}}(s+0)\leq i_{\beta, B_{0}}(s)+\nu (s)\). Let \(i_{\beta, B_{0}}(s+0)=k\). Since \(i_{\beta, B_{0}}\) is a finite integer, there exists \(s'>s\) such that
Similar to (15), for each p, there is a basis \(\{ e_{p,j}\}_{j=1}^{k}\subset L_{\beta, B_{0}}^{-}(B_{p})\) such that
Here, \(1-\mu _{p,j}<0\). Since \(\Lambda _{\beta, B_{0}, B_{p} }\) is bounded, we obtain that \(\mu _{p,j}=(\Lambda _{\beta, B_{0}, B_{p} }e_{p,j},e_{p,j})_{2}\) is bounded. Then, for any j, there exists \(\{p_{l}\}_{l}\subseteq (s,s')\) with \(p_{l}\rightarrow s+0\) such that
Recall that \(1-\mu _{p_{l},j}<0\). We have \(\{\frac{1}{\mu _{p_{l},j}}\}\) is bounded. Taking the limit in (18), we have
Moreover, for all \(i, j=1,2,\ldots,k\),
This means, by Definition 1, that \(i_{\beta, B_{0}}(s)+\nu (s)\geq k=i_{\beta, B_{0}}(s+0)\). This completes the proof. □
By Lemma 5 and Lemma 6, we conclude that the index function \(i_{\beta, B_{0}}(s)\) is integer-valued and nondecreasing on \([0, 1)\). Its value at any point s must be equal to the sum of the jumps it incurred in \([0, 1)\). Hence,
Lemma 7
For any \(B_{1}< B_{2}\), we have
By Lemma 7, we observe that the difference between \(i_{\beta, B_{0}}(B_{1})\) and \(i_{\beta, B_{0}}(B_{2})\) is independent of β and \(B_{0}\). We define
Definition 2
For any 2π periodic continuous matrix-valued functions \(B_{1}\) and \(B_{2}\) with \(B_{1}< B_{2}\), we define
and for any \(B_{1}\), \(B_{2}\) we define
where K is a constant and id is the identity map on \(L_{B_{0}}^{\ast}\), satisfying \(K\mathrm{id}>B_{1}\) and \(K\mathrm{id}>B_{2}\). We call \(I(B_{1}, B_{2})\) the relative Morse index between \(B_{1}\) and \(B_{2}\).
Remark 4
If we choose \(k_{1}\), \(k_{2}\) such that \(k_{1}\mathrm{id}\), \(k_{2}\mathrm{id}>B_{1}\) and \(B_{2}\), we have
Thus, the relative Morse index \(I(B_{1}, B_{2})\) depends only on \(B_{1}\), \(B_{2}\) and the operator A and is well defined.
Lemma 8
If \(\nu (B)=0\) for some β large enough, then \(( \tilde{q}_{\beta, B_{0}, B}(u^{\ast},u^{\ast}))^{\frac{1}{2}}\) and \((-\tilde{q}_{\beta,B_{0},B}(u^{\ast},u^{\ast}))^{\frac{1}{2}}\) are equivalent norms on \(L^{+}_{\beta, B_{0}}(B)\) and \(L^{-}_{\beta, B_{0}}(B)\), respectively.
Proof
It is sufficient to prove that, for sufficiently large β, there exists \(\delta >0\) independent of β such that
Arguing indirectly, there exist \(\beta _{k}\rightarrow \infty \), \(\mu _{k}\rightarrow 1\) such that
where \(e_{k}^{\ast}\in L^{\ast}_{\beta _{k}, B_{0}}\) with \(\| e_{k}^{\ast}\|_{2}=1\). This yields
Assume that \(e_{k}^{\ast}=e_{k}^{+}+e_{k}^{0} \in L^{+}_{\beta _{k}, B_{0}}(B) \oplus L^{0}_{\beta _{k}, B_{0}}(B)\). Then, up to a subsequence,
From (19) we have
and
Recall that \(|A_{B_{0}}|^{-\frac{1}{2}}\) is compact and \(\|L_{\beta _{k}, B_{0}, B}\|\rightarrow 0\) as \(\beta _{k}\rightarrow \infty \). Taking the limit as \(k\rightarrow \infty \), we have
and
This yields
This is a contradiction to \(\nu (B)=0\). □
Note that, for any \(x\in E_{B_{0}}=D(|A_{B_{0}}|^{\frac{1}{2}})\), we have \(|A_{B_{0}}|^{\frac{1}{2}}x\in L^{2}(S^{1}, {\mathbb{R}}^{n})\). Thus \(L^{2}(S^{1}, {\mathbb{R}}^{n})\) and \(E_{B_{0}}\) are isomorphic. Define \(E_{\beta,B_{0}}^{\ast}=E_{\beta,B_{0}}^{+} \oplus E_{\beta,B_{0}}^{0}\) and \(\tilde{L}_{\beta, B_{0}, B}:E_{\beta,B_{0}}^{\ast }\rightarrow E_{ \beta,B_{0}}^{-}\)
Definition 3
For any \(x, y\in E_{B_{0}}\) and \(x^{\ast}=P^{\ast}_{\beta, B_{0}}x\), \(y^{\ast}=P^{\ast}_{\beta, B_{0}}y\), we define
-
\(Q_{\beta, B_{0}, B}(x,y)=q_{\beta, B_{0}, B}(|A_{B_{0}}|^{ \frac{1}{2}}x,|A_{\epsilon}|^{\frac{1}{2}}x) = \frac{1}{2}(x^{+}, y^{+})_{B_{0}} -\frac{1}{2}(x^{0}, y^{0})_{B_{0}} - \frac{1}{2}(x^{-}, y^{-})_{B_{0}} - \frac{1}{2}(\lambda ^{2}(B-B_{0}) x,y)_{2} \).
-
For the reduced functional, we define
$$\begin{aligned} \tilde{Q}_{\beta, B_{0}, B}\bigl(x^{\ast}, y^{\ast}\bigr) ={}&\tilde{q}_{\beta, B_{0}, B}\bigl( \vert A_{B_{0}} \vert ^{\frac{1}{2}}x^{\ast}, \vert A_{B_{0}} \vert ^{\frac{1}{2}}x^{ \ast}\bigr) \\ ={}& \frac{1}{2}\bigl(x^{+}, y^{+} \bigr)_{B_{0}} - \frac{1}{2}\bigl(x^{0}, y^{0} \bigr)_{B_{0}} \\ &{}-\frac{1}{2}\bigl(\lambda ^{2}(B-B_{0}) x^{\ast},y^{\ast}\bigr)_{2} - \frac{1}{2}\bigl( \lambda ^{2}(B-B_{0})\tilde{L}_{\beta, B_{0}, B} x^{\ast},y^{ \ast}\bigr)_{2}. \end{aligned}$$(20)
Also, by (13), for any \(x^{\ast}\in E_{\beta, B_{0}}^{\ast}\), we have
Applying Lemma 2, Lemma 7, and Lemma 8 to \(E_{B_{0}}^{\ast}\) and \(\tilde{Q}_{\beta,B}\), we conclude
Lemma 9
(1) The \(E_{B_{0}}^{\ast}\) has the following decomposition:
such that \(\tilde{Q}_{\beta,B}\) is positive definite, zero, and negative definite on \(E^{+}_{\beta, B_{0}}(B)\), \(E^{0}_{\beta, B_{0}}(B)\), and \(E^{-}_{\beta, B_{0}}(B)\), respectively. Furthermore, \(E^{0}_{\beta, B_{0}}(B)\) and \(E^{-}_{\beta, B_{0}}(B)\) are finitely dimensional with
(2) For any \(x^{\ast}\in E_{\beta, B_{0}}^{\ast}\), \(({\bar{Q}}_{\beta,B}(x^{\ast},x^{\ast}))^{\frac{1}{2}}\) and \((-{\bar{Q}}_{\beta,B}(x^{\ast},x^{\ast}))^{\frac{1}{2}}\) are equivalent norms on \(E^{+}_{\beta, B_{0}}(B)\) and \(E^{-}_{\beta, B_{0}}(B)\).
(3) There exists \(\epsilon _{0}>0\) such that, for any \(\epsilon \in (0, \epsilon _{0}]\), we have
Proof
(1) and (2) of Lemma 9 come from Lemma 2 and Lemma 8 directly. By Lemma 7, (3) follows by the fact that the index function and the relative Morse index are all integer-valued. □
2.3 Critical point theorem
To prove Theorem 1 and Theorem 2, we use the following critical point theorems.
Let E be a real Hilbert space with \(E=X\oplus Y\). A sequence \((z_{n})\subset E\) is said to be a \((PS)_{c}\)-sequence if \(\Phi (z_{n})\rightarrow c\) and \(\Phi '(z_{n})\rightarrow 0\). Φ is said to satisfy the \((PS)_{c}\)-condition if any \((PS)_{c}\)-sequence has a convergent subsequence.
Theorem 3
([14, Theorem 2.5])
Let \(e\in Y \backslash \{0\}\) and \(\Omega =\{u=se+v: \|u\|< R, s>0, v\in X \}\). Suppose that
\((I_{1})\) \(\Phi \in C^{1}(E, {\mathbb{R}})\) satisfies the \((PS)_{c}\)-condition for any \(c\in {\mathbb{R}}\);
\((I_{2})\) there is \(r\in (0, R)\) such that \(\rho:=\inf \Phi (X\cap \partial B_{r})>\omega:=\sup \Phi ( \partial \Omega )\), where ∂Ω refers to the boundary of Ω relative to span\(\{e\}\oplus X\), and \(B_{r}=\{u\in E: \|u\|< r \}\).
Then Φ has a critical value \(c\geq \rho \) with
where
Theorem 4
([14, Theorem 2.8])
Assume that ϕ is even and satisfies \((\Phi _{1})\). If
\((I_{3})\) there exists \(r>0\) with \(\inf \Phi (S_{r}Y) >\Phi (0)=0\), where \(S_{r}=\partial B_{r}\);
\((I_{4})\) there exists a finite dimensional subspace \(Y_{0}\subset Y\) and \(R>r\) such that, for \(E_{\ast}=X\oplus Y_{0}\), \(M_{\ast}=\sup \Phi (E_{\ast})<+\infty \) and \(\sigma:=\sup \Phi (E_{\ast}\backslash B_{R})<\rho \),
then Φ possesses at least m distinct pairs of critical points, where \(m=\dim Y_{0}\).
Remark 5
(1) By using the abstract critical point theorems, we do not need to do the saddle point reduction procedure for the variational function \(I_{\beta, B_{0}}\). Correspondingly, the nonlinearity f does not need to be \(C^{1}\). For this kind of critical point theorems, we also refer to [3, 13].
(2) Recall that, in this paper, we only consider the nonresonance case (\(\nu (A_{\infty})=0\)). The PS-condition is sufficient to prove our theorems. Thus we use the PS-condition instead of the Cerami condition in the theorems.
3 Linking structure and proof of the main results
In view of condition \((F_{0})\), we assume \(B_{0}=A_{0}-\epsilon \) for some \(\epsilon >0\) small enough. By (3) of Lemma 9, we have \(\nu (A_{0}-\epsilon )=\nu (A_{0})=0\). Define \(E_{A_{0}-\epsilon}=D(|A_{A_{0}-\epsilon}|^{\frac{1}{2}})\) with the inner product \((\cdot, \cdot )_{A_{0}-\epsilon}\) and the norm \(\|\cdot \|_{A_{0}-\epsilon}\). In what follows, we will write \((\cdot, \cdot )\) and \(\|\cdot \|\) for short. Define the following projections:
Here \(\{F_{\lambda}\}_{\lambda \in {\mathbb{R}}}\) denotes the spectral family of \(A_{A_{0}-\epsilon}\). This deduces the decomposition on \(E_{A_{0}-\epsilon}\):
Substituting \(A_{0}-\epsilon \) for \(B_{0}\) in (8), we define
Applying Lemma 9 to \(E_{A_{0}-\epsilon}^{\ast}=E_{\beta,A_{0}-\epsilon}^{+} \oplus E_{ \beta,A_{0}-\epsilon}^{0}\) for any B, there is a splitting
To apply the abstract Theorems 3 and 4 to \(I_{\beta, A_{0}-\epsilon}\), we choose
Since \(Y_{0}\subseteq E_{A_{0}-\epsilon}^{-}(A_{\infty}+2\epsilon )\), we have \(Y_{0}\) is a finite dimensional subspace with \(\dim Y_{0}\leq i_{\beta, A_{0}-\epsilon}(A_{\infty}+2\epsilon )\). Moreover
Lemma 10
\(\dim Y_{0} \geq I(A_{0}, A_{\infty})\).
Proof
By (20), for any B, we have
Then
It is easy to get \(E^{+}_{\beta, A_{0}-\epsilon}(A_{0}-\epsilon )= E_{\beta, A_{0}- \epsilon}^{+}\), \(E^{-}_{\beta, A_{0}-\epsilon}(A_{0}-\epsilon )= E_{\beta, A_{0}- \epsilon}^{0}\). Recall that \(Y_{0}\subseteq E_{A_{0}-\epsilon}^{\ast}=E^{0}_{\beta, A_{0}- \epsilon}\oplus E^{+}_{\beta, A_{0}-\epsilon}\), we have
and
□
Now we are ready to verify the conditions of Theorems 3 and 4 for the functional \(I_{\beta, A_{0}-\epsilon}\).
Lemma 11
Let \((F_{0})\) and \((F_{\infty})\) be satisfied. Then there exist \(r>0\) and \(\rho >0\) such that
Proof
Assume that \(f_{1}(t, x)=f(t, x)-A_{0}(t)x(t)\). By \((F_{0})\), we have \(f_{1}(t, x)=o(x)\) as \(|x|\rightarrow 0\) and \(f_{1}(t, x)-(A_{\infty}-A_{0})x=o(x)\) as \(|x|\rightarrow \infty \). Then, for any \(\epsilon >0\), there exist δ, \(M_{1}\), \(M_{2}>0\) such that
In conclusion, under \((F_{0})\) and \((F_{\infty})\), given \(p>2\), for \(\epsilon >0\), there is \(C_{\epsilon}>0\) such that
and
where \(F_{1}(t, x)=\int _{S^{1}}f_{1}(t, \theta x)\,d\theta x\). This yields
Thus, for \(x\in E_{A_{0}-\epsilon}^{+}\) with \(\|x\|=r\), we have \(x=x^{+}\) and
Thus, if we assume \(\rho =\frac{1}{4}r^{2}\), then the lemma follows by choosing r small enough. □
Lemma 12
Let (\(F_{0}\)) and (\(F_{\infty}\)) be satisfied and \(\epsilon >0\) be given by Lemma 11. We have
Proof
Arguing indirectly, we assume that for some sequence \(\{x_{j}\}\in X\oplus Y_{0}\) with \(\| x_{j}\|\rightarrow \infty \) there is \(a>0\) such that \(I_{A_{0}-\epsilon}(x_{j})>-a\) for all j. Setting \(w_{j}=x_{j}/ \| x_{j}\|\), we have \(w_{j}\in X\oplus Y_{0}\) and \(\| w_{j}\|=1\). And according to (22), we have the decomposition
Since \(Y_{0} \) is a finite dimensional subspace, we assume that \(w_{j}\rightharpoonup w \in E_{A_{0}-\epsilon}\) and
Assume that \(f_{2}(t,x)=f(t, x)-A_{\infty}(t)x(t)\). Under \((F_{0})\) and \((F_{\infty})\), there is \(M>0\) such that
and
where \(F_{2}(t, x)=\int _{S^{1}}f_{2}(t, \theta x)\,d\theta x\). Thus,
Firstly, by the definition of \(\tilde{Q}_{\beta,A_{0}-\epsilon, A_{\infty}+2\epsilon}\) and taking the limit in (24), we have
Here, \(w^{\ast}_{j}=w_{j}^{+}+w_{j}^{0}\) and \(w^{\ast}=w^{+}+w^{0}\). Besides, by the definition of \(\tilde{Q}_{\beta, A_{0}-\epsilon, A_{\infty}+2\epsilon}\), we have
Since \(A_{\infty }>A_{0}\) and \(w^{0}\neq 0\), we have \(\tilde{Q}_{\beta, A_{0}-\epsilon, A_{\infty}+2\epsilon}(w^{0}, w^{0})<0\). Thus,
and \(w^{\ast}\in E^{-}_{\beta, A_{0}-\epsilon}(A_{\infty}+2\epsilon )\). Moreover, since \(\nu (A_{\infty})=0\), we have \(\nu (A_{\infty}+2\epsilon )=0\) for ϵ small enough. By (2) of Lemma 9, we have \(\sqrt{- \tilde{Q}_{A_{0}-\epsilon, A_{\infty}+2\epsilon}(w^{\ast}, w^{ \ast})}\) is an equivalent norm of \(w^{\ast}\) on \(E^{-}_{A_{0}-\epsilon}(A_{\infty}+2\epsilon )\). Thus, \(w_{j}^{\ast }\rightarrow w^{\ast}=0\).
Secondly, define the following projections:
We have \(P_{1}E_{A_{0}-\epsilon}=E^{-}_{A_{0}-\epsilon}\), \((P_{2}+P_{3})E_{A_{0}-\epsilon}=E^{\ast}_{A_{0}-\epsilon}\), and \(P_{2}E_{A_{0}-\epsilon}\) is a finite dimensional subspace. Recalling that \(w_{j}^{0}\rightarrow 0\) and \(w_{j}^{+}\rightarrow 0\), we obtain
Moreover, by the definition of \(P_{1}\) and \(P_{3}\), we have
Suppose that \(\lambda \|A_{0}-A_{\infty}+\epsilon \|< M_{3}<\beta \). By (24), we obtain
This implies that \(P_{1}w_{j}\rightarrow 0\) in \(E_{A_{0}-\epsilon}\). Hence, \(w_{j}=(P_{1}+P_{2}+P_{3})w_{j}\rightarrow 0\). This is a contradiction to \(\|w_{j}\|=1\). □
Lemma 13
Let \((F_{0})\) and \((F_{\infty})\) be satisfied. Then any \((PS)_{c}\)-sequence of \(I_{\beta, A_{0}-\epsilon}\) is bounded.
Proof
Let \(\{x_{j}\}\subset E\) be such that \(I_{\beta, A_{0}-\epsilon}(x_{j})\rightarrow c \) and \(I_{\beta, A_{0}-\epsilon}'(x_{j})\rightarrow 0\). To prove that \(\{x_{j}\}\) is bounded, we develop a contradiction argument. We assume that, up to a subsequence, \(\| x_{j}\|\rightarrow \infty \) and set \(v_{j}=\frac{ x_{j}}{\| x_{j}\|}\). Then \(\|v_{j}\|=1 \). Without loss of generality, we assume that
Then, up to a subsequence, \(v_{j}(t)\rightarrow v(t)\) a.e. on \(S^{1}\). Since, by (\(F_{\infty}\)), \(|f_{2}(t, x)|=|f(t, x)-A_{\infty}(t) x|=o(|x|)\) as \(|x|\rightarrow \infty \) and \(x_{j}(t)\rightarrow \infty \) if \(v(t)\neq 0\), it is easy to see that
for all \(\varphi \in L^{2}(S^{1}, {\mathbb{R}}^{n})\). From this we deduce that
and
Since \(i(A_{\infty})=0\), we have \(v=0\). Thus, \(v_{j}\rightharpoonup 0\) in \(E_{A_{0}-\epsilon}\) and \(v_{j}\rightarrow 0\) in \(L^{2}(S^{1}, {\mathbb{R}}^{n})\). Note that
This is a contradiction. □
Lemma 14
Any \((PS)_{c}\)-sequence of \(I_{A_{0}-\epsilon}\) has a convergence subsequence.
Proof
By Lemma 13, the \((PS)_{c}\)-sequence is bounded. Assume that \(v_{j}\rightharpoonup v\) and let \(w_{j}=v_{j}-v\). Then we have \(w_{j}^{+}\rightharpoonup 0\) in \(E_{A_{0}-\epsilon}\) and \(w_{j}^{+}\rightarrow 0\) in \(L^{2}(S^{1}, {\mathbb{R}}^{n})\). To establish strong convergence, it suffices to prove that \(\|w_{j}\|\rightarrow 0\). Similar to Lemma 13, the proof follows from the procedure in Lemma 13 to \(\frac{I_{\beta, A_{0}-\epsilon}'(x_{j})}{\|x_{j}\|} (w_{j}^{+}-w_{j}^{0}-w_{j}^{-})\). □
Proof of Theorems 1 and 2
(Existence) With \(X=E^{-}_{A_{0}-\epsilon}\oplus E^{0}_{A_{0}-\epsilon}, Y=E^{+}_{A_{0}- \epsilon}, Y_{0}= E_{A_{0}-\epsilon}^{-}(A_{\infty}+2\epsilon ) \cap E^{+}_{A_{0}-\epsilon}\), the condition \((I_{2})\) of Theorem 3 holds by Lemma 12. Lemma 14 shows that \(I_{\beta, A_{0}-\epsilon}\) satisfies the \((PS)_{c}\)-condition and \((I_{1})\) of Theorem 3 is verified. Therefore, \(I_{\beta, A_{0}-\epsilon}\) has at least one critical point by Theorem 3.
(Multiplicity) Lemma 11 implies that \(I_{\beta, A_{0}-\epsilon}\) satisfies \((I_{3})\) of Theorem 4. Lemma 10 and Lemma 12 say that \(I_{\beta, A_{0}-\epsilon}\) satisfies \((I_{4})\) of Theorem 4 with dim\(Y_{0}\geq I(A_{0}, A_{\infty})\). Moreover, \(I_{\beta, A_{0}-\epsilon}\) is even if f is odd. Therefore, \(I_{\beta, A_{0}-\epsilon}\) has at least \(I(A_{0}, A_{\infty})\) pairs of nontrivial critical points by Theorem 4. □
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Shan, Y. Index theory and multiple solutions for asymptotically linear second-order delay differential equations. Bound Value Probl 2023, 7 (2023). https://doi.org/10.1186/s13661-023-01693-7
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DOI: https://doi.org/10.1186/s13661-023-01693-7