The Existence of Periodic Solutions of Delay Differential Equations by <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06569958pt" id="M1" height="14.3174pt" version="1.1" viewBox="-0.0657574 -14.2517 18.7661 14.3174" width="18.7661pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-70" d="M593 650H142L136 622C219 616 224 611 209 531L130 120C114 39 108 34 23 28L17 0H501C518 32 557 130 568 158L540 169C519 132 495 93 469 70C439 43 409 35 332 35C272 35 241 37 225 50C211 61 211 89 222 149L253 320H350C441 320 443 312 441 239H471L510 434H481C455 366 448 358 360 358H260L303 572C311 613 312 615 348 615H431C509 615 522 603 532 581C544 555 546 531 545 491L574 493C583 555 592 631 593 650Z"/></g><g transform="matrix(.012,0,0,-0.012,10.571,-7.578)"><path id="g54-36" d="M556 236V289H337V504H275V289H56V236H275V-4H337V236H556Z"/></g></svg>-</span>Conley Index Theory

Xiao, Huafeng

doi:https://doi.org/10.1155/2022/3396716

Journal of Function Spaces

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Research Article | Open Access

Volume 2022 | Article ID 3396716 | https://doi.org/10.1155/2022/3396716

The Existence of Periodic Solutions of Delay Differential Equations by -Conley Index Theory

Huafeng Xiao^1,2

Academic Editor: Paola Rubbioni

Received22 Dec 2021

Accepted06 Mar 2022

Published20 Apr 2022

Abstract

In this paper, the -Conley index theory has been used to study the existence of periodic solutions of nonautonomous delay differential equations (in short, DDEs). The variational structure for DDEs is built, and the existence of periodic solutions of DDEs is transferred to that of critical points of the associated function. When DDEs are -nonresonant, some sufficient conditions are obtained to guarantee the existence of periodic solutions. When the system is -resonant at infinity, by making use a second disturbing of the original functional, some sufficient conditions are obtained to guarantee the existence of periodic solutions to DDEs.

1. Introduction

In the last several decades, there has been increasing interest in the dynamical properties of delay differential equations due to their important applications in many fields such as biological, physical, and social sciences. Many rich mathematical investigations and interesting results on DDEs have been available in the literature. In particular, the existence of periodic solutions is of great interest. As far as the author’s knowledge, it was Jones who first studied the existence of periodic solutions of DDEs. Since then, some methods had been developed to search for periodic solutions of DDEs, such as fixed point theorems (cf. [1–4]), cone mapping method (cf. [5]), coincidence degree theory (cf. [6, 7]), Poincaré-Bendixson theorem (cf. [8–10]), and Hopf bifurcation theorem (cf. [11, 12]).

In 1973, Rabinowitz built the variational approach. It had been developed quickly. The variational approach had been used in two different ways to study the existence of periodic solutions of DDEs. Firstly, it can be used indirectly. More specially, in 1974, Kaplan and Yorke studied the following equation:

They (cf. [13]) translated the existence of periodic solutions of (1) to that of an associated plane ordinary differential equations. Li and his cooperators developed Kaplan and Yorke’s technique and applied to the following equation:

They showed that the associated equation of (2) is a Hamiltonian system (resp., general Hamiltonian system) (cf. [14–16]). By noting the symmetric property, Fei (cf. [17, 18]) made use of variational methods to study the existence of periodic solutions of the associated Hamiltonian system and obtain the existence of multiple periodic solutions of (2). In 2019, Li et al. (cf. [19]) proved the multiplicity of periodic solutions to the following equation:

For more results in this direction, we refer to [20] and the reference therein.

In 2005, Guo and Yu firstly established a variational structure for DDEs below:

They converted the existence of periodic solutions of (4) to that of critical points of the associated variational functional. By using of pseudo-index theory, they (cf. [21]) proved multiple periodic solutions for (4). Later, Guo and Yu and Zheng and Guo studied the following systems:

They (cf. [22, 23]) proved the existence of multiple periodic solutions of (5). In 2009, C.J. Guo and Z.M. Guo studied second order system as follows:

They (cf. [24]) obtained some sufficient conditions to guarantee the existence of multiple periodic solutions of (6). For more results on this direction, we refer to [25] and the reference therein.

As for nonautonomous DDEs, Yu and Xiao studied the following system:

By making use of the Maslov-type index theory, they (cf. [26]) proved the existence of multiple periodic solutions. For more reference on this direction, we refer to [27].

It is well-known that there are many index theories in variational approach. However, only a few had been used to study the existence and multiplicity of periodic solutions to DDEs. A nature question is that whether those index theories can be used to study the existence of periodic solutions to DDEs. It is well known that -Conley index, which is a relative Morse index, was introduced by Abbondandolo [28, 29]. It has been used to study periodic solutions of Hamiltonian systems by proving the Morse-Conley relations. Motivated by [21, 28, 29], we use -Conley index to study the existence of periodic solutions of the following equations: where . Assume that

() is odd with respect to variable and -periodic with respect to variable , i.e., for any , uniformly for and uniformly for

() there exists a twice continuously differentiable function , such that the gradient of is , i.e.,

() there exists a -periodic continuous loop of symmetric matrices and a function such that where as , uniformly with respect to

() is bounded

Theorem 1. Assume that satisfies -. Assume that (8) is -nonresonant at infinity (see Definition 31). If all the -periodic solutions of (8) are -nonresonant, then there is an odd number of them and the following relation holds: where the sum is taken over all the -periodic solutions and is a Laurent polynomial with positive coefficients.

Assume the following:

() as , uniformly with respect to

() is bounded

Theorem 2. Assume that ()-(), (), and () hold. Assume that is a -nonresonant periodic solution of the system (8). If then (8) has at least another -periodic solution.

The paper is organized as follows: in Section 2, we present the space which the variational functional is built; in Section 3, we summarize some basic knowledge on the -Conley index; in Section 4, we will study system which is -nonresonant system at infinity; in Section 5, we will study system which is -resonant system at infinity.

2. Preparation

Denote by . A similar discussion as reference [21], we can build the Hilbert space . If , it has Fourier expansion where . The space equips with the following norm and inner product: where .

Proposition 3 (see [30]). For every , is compactly embedded into the Banach space .
Let . If for every , then is called a weak derivative of , denoted by .

Now, the variational function defined on is

Defined an operator by extending the bilinear forms

It is easily to check that is a linear bounded operator on .

Define

Then, can be rewritten as

Now we state a useful lemma, which can be proved as Proposition B.37 in the book of Rabinowitz [30] and Appendix A3 in the book of Hofer and Zehnder [31].

Lemma 4. Assume that ()-() are satisfied; then, and

Moreover, the map is completely continuous.

Define an operator

Clearly, is a bounded linear operator and satisfies .

Next, we set

Then, is a closed space of . It is easily to check that is a linear bounded self-adjoint operator on . If , it has the following Fourier expansion:

Lemma 5. Assume that ()-() are satisfied. Then, is twice continuously differentiable. The existence of -periodic solutions for (1) is equivalent to the existence of critical points of functional restricted to . Moreover, if is a critical point of , the linear operator is compact.

Proof. Based on the definition of and Lemma 4, is twice continuously differentiable on .

By a standard argument as in [32] (cf. Proposition 4.1.3) and [21] (cf. Lemma 4), we can prove the second and third assertions.

Remark 6. Since () and () imply that of (), Lemma 5 holds when satisfies ()-(), (), and ().

Now, we restrict our discussion on the space . Then, can be computed explicitly in terms of the Fourier expansion of :

Define the following family of subspaces of .

Then, on and on for all .

Set

Then, . Notice that the null space of is . Then, is a self-adjoint invertible Fredholm operator.

3. -Conley Index Theory

3.1. The -Dimension and the -Morse Index

Let be a real Hilbert space. We denote by and the Hilbert norm and scalar product on , respectively. Assume that has an orthogonal splitting where both and are infinite dimensional.

Definition 7. Two closed subspaces and of are called commensurable if the quotient projections and are compact.

Let denote the orthogonal projection onto and denote the orthogonal complement of . Two closed subspaces and of are commensurable if and only if both restricted to and restricted to are compact. Subsequently, both and are finite.

Definition 8. Let be a closed subspace of commensurable with . The -dimension of is defined as Let be a twice differentiable functional on and be its second order differential at . Denote by the associated self-adjoint operator:

Definition 9. A critical point of is called nondegenerate if is invertible. A critical point of is called weakly nondegenerate if is a Fredholm operator.

If is a weakly nondegenerate critical point of , splits into three closed subspaces. where , , and are positive, null, and negative eigenspaces of , respectively. Obviously, is finite dimensional. And there exists such that

Definition 10. Let be a weakly nondegenerate critical point of . Assume moreover that the negative eigenspace of is commensurable with . The -Morse index is the finite relative integer The large -Morse index is the finite relative integer

Proposition 11 (see [29]). Let be a Fredholm operator whose negative eigenspace is commensurable with . Then, where the maximum is taken over the family of all closed subspaces of of which are commensurable with and such that is strictly negative on .

Lemma 12 (see [29]). Denote by the negative eigenspace of the self-adjoint operator . Let be the set of invertible self-adjoint operators whose negative eigenspace is commensurable with . Then, is relatively closed in the set of invertible operators. The function is continuous.

Lemma 13 (see [32]). Assume that is a self-adjoint Fredholm operator, and that is a self-adjoint compact operator. Then, the negative eigenspaces of and of are commensurable.

3.2. The -Cohomology

Let be the weakest topology on such that all the bounded linear functions and the quotient projection are continuous. Equivalently, is the product topology between the weak topology of and the strong topology of . If is commensurable with , then the topologies and coincide. Thus, the topology depends only on the commensurability class of .

Definition 14. A subset of is -locally compact if is -locally compact, for every finite dimensional subspace of .

Obviously, every bounded -closed subset of is -locally compact.

Definition 15. An -pair is a topological pair of subset of such that and are -closed and -locally compact.

Definition 16. An -continuous map is an -morphism if (1)It has the formwhere maps bounded sets into -precompact sets and is a linear automorphism of such that . (2) is bounded for every bounded

Definition 17. An -continuous map is an -homotopy if (1)It has the formwhere maps bounded sets into -precompact sets and is a linear automorphism of such that . (2) is bounded for every bounded

Theorem 18 (see [29]). There exists a generalised cohomology , with coefficients in , which acts on the category of -pairs and -morphisms. More precisely, (1)Controvariant Functoriality. If is the identity map, is the identity homomorphism on . If and are -morphisms, then .(2)Homotopy Invariance. If two -morphisms are -homotopic, then .(3)Strong Excision. If and are -closed -locally compact subsets of and is the inclusion map, then is an isomorphism.(4)Naturality of the Coboundary. Given three -closed and -locally compact sets such that , there exists a family of coboundary homomorphismsIf is a -morphism such that , then (5)Long Exact Sequence. Given three -closed and -locally compact sets such that , denote by and the inclusion maps. Then, the following sequence of homomorphisms is exact(6)Dimension Property. Let be a closed subspace of , commensurable with . Let be a closed ball in and be its relative boundary in . Then,

Proposition 19 (see [29]). Assume that are -closed -locally compact subsets of . (1)If there exists a -homotopysuch that , , and for every , then (2)If there exists a -homotopysuch that , , and for every , then

3.3. The -Conley Theory

If is a twice continuously differentiable real valued function on . Let be the critical set of , i.e., the set . Denote by the set .

Recall that the Palais-Smale sequence (denoted by (PS) sequence) is a sequence of elements of such that is bounded and converges to zero.

We assume that

(A1) Each sublevel is -closed and -locally compact

(A2) Every bounded (PS) sequence has a converging subsequence

(A3) is globally Lipschitz

(A4) The flow defined by is an -homotopy.

Since is globally Lipschitz, the flow exists globally. If and , we will use both notations . Besides, (A3) implies that is bounded on bounded sets. Recall that an isolated critical set is an isolated subset of .

Definition 20. Let be a compact isolated critical set. An -index pair for is a topological pair in such that (1) is a bounded -pair(2) and . Here is the interior part of with respect to the topology of , and is the interior part of with respect to the topology induced on by the strong topology of (3) is strongly positively invariant with respect to ; if and for , then (4) is an exit set for : if and is not contained in , then there exists such that Moreover, an -index pair for is called strict if .

Definition 21. An elementary critical set is a compact isolated critical set such that is constant on .

Lemma 22 (see [29]). Let be an elementary critical set and let be a neighborhood of which is -closed. Then, there exists a strict -index pair for such that .

Lemma 23 (see [29]). Let be a compact isolated critical set. Let be an -index pair for and a strict -index pair for such that . Then,

Lemma 24 (see [29]). Let be an elementary critical set. If and are -index pairs for , then We recall that the -Poincaré polynomial of an -pair is defined as In general, is a formal Laurent series, whose coefficients are nonnegative integers or .

Definition 25. Let be an elementary critical set of . The -critical group of is the vector space where is an -index pair for . The -Morse polynomial of is the -Poincaré polynomial of :

Definition 26. Assume that . An elementary critical set is called -nondegenerate critical manifold if it is a finite dimensional compact -manifold embedded in and for every : (1) is a weak nondegenerate critical point, and the kernel of coincides with the tangent space of in (2)The negative eigenspace of is commensurable with

Proposition 11 easily implies that if the -nondegenerate critical manifold is connected, the -Morse index is constant on . In this case, we set

Proposition 27 (see [29]). If is a connected -nondegenerate critical manifold of a functional of class , then where is the standard Poincaré polynomial of .

Now we state a result which follows immediately from Morse-Conley relations.

Theorem 28 (see [29]). Let be an isolated critical set of , and let be an -index pair for . Assume that , where are pairwise disjoint elementary critical sets. Then, there exists a Laurent series with positive or infinite coefficients such that If consists of -nondegenerate critical manifolds, the above relation is an equality between true Laurent polynomials with finite coefficients.

In the case of a Morse function, we have the following immediate corollary.

Corollary 29 (see [32]). Assume that are nondegenerate critical points of with finite -Morse index. If is an -index pair for the critical set , there exists a Laurent polynomial with positive coefficients such that Now we state a useful lemma.

Lemma 30 (see [29]). Let be an orthogonal splitting such that is commensurable with (and therefore, is commensurable with ). Let and be the orthogonal projections onto and , respectively. Assume that there exists such that for every , the following inequalities hold: Then, the critical set of is compact, and is an -index pair for .

4. Systems with Nonresonant at Infinity

In this section, we consider system (8) satisfying -. If is a -periodic solutions of (8), we can linearise system (8) near obtaining the -periodic system where is the Hessian of with respect to variable .

Definition 31. A -periodic solution of system (8) is called -resonant if the linearised system (58) has nontrivial -periodic solutions. Otherwise, it is called -nonresonant.

The corresponding functional is quadratic where is the self-adjoint operator such that

Lemma 5 implies that is a compact operator. Let , , and be the positive, kernel, and negative eigenspaces of .

The linearisation of (8) at infinity is

Definition 32. The linear system (61) is said -resonant if it has nontrivial -periodic solutions. Otherwise, it is said -nonresonant. The asymptotically linear system (8) is said -resonant at infinity if system (61) is -resonant, -nonresonant at infinity in the opposite case.

Let be the self-adjoint operator on defined by extending bilinear form and then, is a compact operator. Define

Therefore,

Let be the positive, kernel, and negative eigenspaces of , respectively. Let be the orthogonal projections onto , respectively.

Now, we want to check that (A1)-(A4) hold under assumptions (F1)-(F4), so that it is possible to apply the -Conley theory to .

Lemma 33 (see [32]). Let be a finite codimensional subspace of . If are positive constants, the functional has -locally compact sublevels.

Lemma 34. Assume that satisfies (F2) and (F3). Then, the sublevels of are -closed and also -locally compact. So (A1) holds.

Proof. Notice that the variational functional is Its quadratic part is lower semicontinuous and convex on and thus weakly lower semicontinuous on . Since it is strongly continuous on , it is -lower semicontinuous. By (F3), has quadratic growth. It follows that is continuous on , if is large enough. By Proposition 3, is weakly continuous on , and therefore, it is also -continuous. We conclude that is -lower semicontinuous.
By (F3), .Since embeds compactly into , for every , we can find a finite codimensional subspace of such that Then, Since is finite codimensional in , Lemma 33 implies that the right side of (68) has -locally compact sublevels. Then, also has -locally compact sublevels.

Lemma 35. Assume that satisfies (F2) and (F3). Then, all the bounded (PS) sequences are precompact. So (A2) holds.

Proof. Assume that is a bounded (PS) sequence. The boundedness of yields that there exists a subsequence which converges weakly to some .
According to Lemma 4, is completely continuous. Then, there exists a subsequence such that the sequence converges strongly. Since , then converges strongly. Together with the fact that is invertible implies that must converge. Thus, (A2) is satisfied.

Lemma 36. Assume that satisfies and . Then, is globally Lipschitz, and (A3) holds.

Proof. Since , we need to only check that the second part in the right side of inequality is globally Lipschitz. By a directly computation, we have where is a positive constant. The last inequality is guaranteed by and Proposition 3. This finishes the proof of this lemma.

Lemma 37. Assume that satisfies (F2) and (F3); then, the flow defined by (48) is an -homotopy, and (A4) holds.

Proof. By induction, we can define a sequence of flows It is a standard fact in the theory of ordinary differential equations in Banach spaces that is locally Lipschitz, and converges uniformly on the bounded subsets of to the solution of the Cauchy problem Since maps bounded sets into bounded sets, so does . Therefore, also maps bounded sets into bounded sets.
Since is -continuous, also is -continuous. Thus, both and are -continuous. Moreover, and converge uniformly on bounded sets to and , respectively. Therefore, and are -continuous, for every . Then, is -continuous.
Since solves the following nonhomogeneous equation it can be represented as is a continuous path of invertible operators which preserve the splitting . Set If is bounded and , is bounded, as we have shown before. Therefore, is -continuous, and we conclude that is -continuous.
Finally, since is a diffeomorphism whose inverse is , must be bounded for every bounded .

Proposition 38. Assume that satisfies ()-(). If the system (8) is -nonresonant at infinity, the critical set is compact and is an -index pair for , provided is large enough.

Proof. Since the system is -nonresonant at infinity, the operator is invertible and . Let be such that on and on .
Lemma 35 implies that all the bounded (PS) sequences are precompact: in order to prove that is compact, it is enough to show that it is bounded.
Now we check conditions (55) and (56) of Lemma 30.
Let . Then, By (F3), there exists such that Let be such that Then, Therefore, which is positive when is large enough. A similar discussion can prove (56). Then, we can use Lemma 30.

Proof of Theorem 39. Notice that () and () imply that has quadratic growth. Therefore, Lemmas 34–37 show that we can apply the -Conley theory to the functional .
By Proposition 38, if is large enough, the pair is an -index pair for the critical set of , which is compact. Then, Theorem 18 is Then, both (48) and Definition 10 imply Theorem 1 follows from the above two equalities.

5. Systems with Resonant at Infinity

In this section, we want to study the existence of periodic solutions of asymptotically linear systems which is resonant at infinity.

Since we consider resonant system in this section, the variational functional does not satisfy (PS) condition. So we first make a perturbation. We prove that the perturbable functional satisfies (PS) condition. Since critical points of first perturbable functional may not to be nondegenerate, we make a second perturbation and critical points of the secondly perturbable functional are nondegenerate. Finally, we prove the critical points of the secondly perturbable functional are the critical points of the original functional.

Let be a nondecreasing function such that for and for . For , we define two new functionals on , and , of class of as

The gradients of these functionals are

Critical points of such that are also critical points of , and thus -periodic solutions of system (8). Let us check that the perturbed functionals satisfy the (PS) condition.

Lemma 40. For every , every sequence such that contains a convergent subsequence.

Proof. Let . Since the proofs of two cases are the same, we only prove this lemma for . Multiplying both sides of (86) by and integrating over , we have By () and Proposition 3, there exist two positive constants and such that According to the assumption, there are small enough such that which implies that the sequence is bounded. Arguing similarly, we can prove that the sequence is bounded.
Next, we prove the sequence is bounded. Since Suppose that as . For large , we have . It follows that Subsequently, we have which contradicts with the fact that is unbounded. Thus, the sequence is bounded and so is . Up to subsequence, it converges weakly to . Standard arguments show that this convergence is strong.

Now we can check that critical points of have priori boundedness in and .

Lemma 41. There exists , independent of , such that, for every critical point of , we have .

Proof. Let be a critical point of . Set . Multiplying both sides of (86) by and integrating over , we have So and thus, must be bounded. Similarly, we can prove the boundedness of .

Lemma 42. Assume that ()-(), (), and () hold. For every , there exists , independent of , such that the following property holds: for every critical point of , if , , and , we have where and are the kernel and negative eigenspace of .

Proof. We firstly prove the second inequality of this lemma. If , we have We claim that . By () and (), there exist , large enough and small enough such that Set . By Proposition 3, there exists such that , where denotes the -norm and . Computing directly, we obtain By assumptions . Let be such that for every with ,, there holds Then, The subspace is finite dimensional. Subsequently, there exists such that Since and are equivalent on , there exists such that Denote by . If , then . It follows that Substituting (103) into (98), we get The arbitrary of induces that . Therefore, if is a critical point of , then for all , we have Subsequently, is strictly positive on . Therefore, .
Repeating the above argument, we conclude that is strictly negative on . Thus, .

Lemma 43. The maps and are proper Fredholm maps of index .

Proof. Notice that can be written as The first term is an invertible linear operator. The last two terms are compact operators. Assuming that converges, it is easy to prove that has a converging subsequence. Thus, are proper operator. Moreover, the differentials of the last two terms are compact self-adjoint operators. Therefore, is a Fredholm operator of index 0, for every .

Consider the constant , fixed in Lemma 42. We assume, by contradiction, that the functionals have no critical points such that , apart from .

According to Lemma 43, the critical set of is compact. Since has no critical points such that , there exists such that there are no critical points such that

As we can see, the perturbable functionals satisfy (PS) condition. However, the critical points of maybe not to be a nondegenerate set. Thus, we need a second perturbation to grantee the critical points of perturbable functional are nondegenerate.

Let be a nondecreasing function such that for and for . Set for all . is a smooth functional on . always belongs to , and there exists a constant such that for all . For , we define two new functional on as

The gradients of these functional are

Lemma 44. If is small enough, for every critical point of , there holds

Proof. From (86) and (109), we get Therefore, and it is enough to assume . By a similar argument, we estimate the bound for .

Lemma 45. If is small enough, there are no critical points of such that

Proof. By Lemma 44, we can assume that is small enough that for every critical point of .
Since has no critical point with , by Lemma 43, there exists such that Therefore, if is a critical point of with , we have This implies that which is contradiction, provided is small enough.

Now we can work with second perturbation functionals. Firstly, we check some proposition of those functionals.

Lemma 46. Every bounded (PS) sequence for admits a converging subsequence; thus, (A2) holds.

Proof. Let be a bounded (PS) sequence for . Up to its subsequence, it converges weakly. Since is weakly continuous and is completely continuous, by (109), converges strongly. By Lemma 43, converges strongly.

Lemma 47. The functionals satisfy the assumption (A1), (A3), and (A4).

Proof. Since Lemma 34 implies that is -lower semicontinuous. The last two terms are weakly continuous and thus also -continuous. So the sublevels of are -closed. Arguing similarly as in the proof of Lemma 34, the sublevels of are -locally compact and (A1) holds.
Computing directly, we obtain According to Lemma 36, is globally Lipschitz. Since and are both functions, the second and third items are globally Lipschitz. We need only the last item is globally Lipschitz. But, which yields that is globally Lipschitz when is fixed. Thus, (A3) holds.
Since can be seen as the sum of the quadratic form and of a function which is continuous from the weak to the strong topology of , then (A4) follows from Proposition 15.1 in [28].

Now we are in a position to use -Conley index theory. We claim that the critical points of are compact and prove neighborhoods of zero are their -Conley index pairs.

Lemma 48. The critical set of is compact, and is an -index pair for , with respect to the functional , provided is large enough and is small enough. The same conclusion also works where the -index pair is replaced by

Proof. We just prove the first claim, since the second one can be proved with analogous arguments.
Let be the flow of the vector field . We use Lemma 30 to prove this lemma. In order to use Lemma 30, we should check conditions (55) and (56).
Let . Then, Then, (55) holds when .
Now let . Then, Since and , we have that In the first case, noticing that is small enough, we have which is negative for . In the second case, we can assume that . So . We have which is negative if Then, (56) holds. Applying Lemma 30, we prove this lemma.

Proof of Theorem 49. We claim that we can choose and , such that and are so small that these of Lemmas 44, 45, and 48 hold and such that all critical points of are nondegenerate.
Since we assume the system is -nonresonant periodic solution, the critical point is nondegenerate. Lemma 45 implies all the other critical points such that If is in a neighborhood of such a critical point , then Therefore, is a continuously differentiable Fredholm map of index 0. By an infinite dimensional Sard-Smale theorem, the set of its critical values has first category. Therefore, we can choose and , such that and are so small that the theses of Lemmas 44, 45, and 48 hold and such that and are regular values for and , respectively.
If is a critical point of different from , then and the linear map is invertible. The same result holds for .
Lemmas 46 and 47 imply that we can use -Conley index theory and get the following Morse relations: where the sum is taken over the finite set of all critical points of such that . If , there is existence of a critical point such that Therefore, But by Lemmas 42 and 44, every critical point has -Morse index which is a contradiction.
If , arguing similarly as the previews case can result a contradiction which finishes our proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

This project is supported by the National Natural Science Foundation of China (No. 11871171).

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