Positive Periodic Solutions for a Class of Strongly Coupled Differential Systems with Singular Nonlinearities

This article studies the existence of positive periodic solutions for a class of strongly coupled diﬀerential systems. By applying the ﬁxed point theory, several existence results are established. Our main ﬁndings generalize and complement those in the literature studies.

During the past few decades, the fixed point theory has been widely adopted to investigate the nonperiodic coupled differential systems, and researchers have mainly concentrated on the existence and multiplicity of positive solutions [1][2][3]. Meanwhile, the periodic equations and systems with singular nonlinearities have been dealt via some classical fixed point theorems, such as Schauder's fixed point theorem and fixed point theorems in cones [4][5][6][7][8][9][10][11][12]. What is worth mentioning is the results obtained in [5,6,11,12], where the authors show, under some circumstances, weak singularities are helpful to seek out periodic solutions for not only singular equations [5] but also singular coupled systems [11]. Especially, in [10], Li and Zhang considered the singular equation where a, c ∈ L 1 [0, T] and f ∈ Car([0, T] × (0, ∞), R). By employing a fixed point theorem in cones, they established several existence theorems under the following basic assumption.
(H) ere exist b≻0, b≻0, and λ > 0 such that and pointed out they have not limited themselves to the weak singularities; see [10], Section 3, for more details. Besides, the case a(t) ≡ 0 (the resonant case) has also been studied in [10], eorem 4.1. For other research works related to the resonant case of (2), one may refer to [13][14][15] and references therein. To our knowledge, however, so far, the existing results on strongly coupled periodic singular systems are relatively few. erefore, motivated by the aforementioned papers, we shall establish the existence of positive periodic solutions of system (1) in the present paper to further improve and complement those in the literature studies. To demonstrate our new results, we choose the following differential systems: where p i , q i , e i ∈ L 1 (R/TZ, R) and α i > 0, i � 1, 2. Here, α i > 0 means we need not restrict ourselves to the weak force conditions in our results. e rest of the paper is arranged as follows. In Section 2, we give some required preliminaries and notations. In Section 3, we shall state and prove the existence results for (1) in the nonresonant case. Finally, in Section 4, an existence theorem will be proved for (1) in the resonant case

Preliminaries
e linear boundary value problem is called nonresonant if its unique solution is the trivial one. When (5) and (6) are nonresonant, the well-known Fredholm's alternative ensures the nonhomogeneous equation admits a unique T-periodic solution, which can be expressed as where K(t, s) is Green's function associated to (5) and (6). For given ξ ∈ L 1 [0, T], we denote by ξ * and ξ * , respectively, the essential infimum and supremum of ξ. ξ≻0 means ξ ≥ 0 for a.e. t ∈ [0, T], and it is positive on a set with positive measures. Moreover, if (7) has a unique periodic solution x l for any l ∈ C(R/TZ) and x l is positive on [0, T] when l≻0, then we say (5) satisfies the antimaximum principle. Recently, Hakl and Torres [16] established an explicit criterion to guarantee the antimaximum principle holds for (5). For the sake of convenience, set Lemma 1 (see [16]). If q ≡ 0 and then (5) satisfies the antimaximum principle, where [q(s)] + � max q(s), 0 , whereafter Chu et al. [17] pointed out if (5) admits the antimaximum principle, then In addition, they obtained the following.

Lemma 2.
If q ≡ 0 and (11) holds, then the distance between two consecutive zeroes of a nontrivial solution of (5) is greater than T.
Obviously, Lemma 2 implies K(t, s) does not vanish. As a consequence of Lemmas 1 and 2, Chu et al. established the following. (10) and (11)

Lemma 3. If q ≡ 0 and
Note that Lemma 3 plays an important role in the application of the classical fixed point theorems. Indeed, by Lemma 3, the positivity of some completely continuous operators could be easily obtained.
Remark 1. Clearly, if p(t) ≡ 0 (without damping terms), then (10) and (11) reduce, respectively, to which are conditions used to guarantee the positivity of Green's function corresponds to (5) and (6); see [18] for more details. roughout the paper, we always suppose p i , q i , e i ∈ L 1 (R/TZ; R) and It is not hard to see (H1) implies the antimaximum principle holds for L i x i � 0, and thus, Hence, be the Banach space equipped with the norm Here, en, it is not difficult to check K is a positive cone in E, and for any x � (x 1 , x 2 ) ∈ E, we get by (15) that For Lemma 4 (see [19]). Let E be a Banach space and K⊆E a cone.

The Nonresonant Systems
is section is devoted to establishing the existence results for system (1) in the nonresonant case. To this end, we define Theorem 1. Assume (H1) and (H2) hold. Let then (1)  Proof en, A: E ⟶ E is completely continuous, and a T-periodic solution of (1) is equivalent to a fixed point of A. We shall divide the proof of the case into three steps as follows.
Step 3: we shall show Recall that R � R 0 . For any x ∈ K ∩ zΩ R , we have σR � σ‖x‖ ≤ x 1 + x 2 ≤ ‖x‖ � R, and then by (27), (H1), and (H2), we can obtain In [11], several existence theorems have been established for (32) via Schauder's fixed point theorem, where f i satisfies only the weak force conditions. However, we do not restrict ourselves here to weak singularities, and eorem 1 is still valid for (32) with strong singularities.

Proof.
Let us take into account the auxiliary systems where k 1 , k 2 ∈ (0, (π/T)) are constants introduced as in (H4). Clearly, a solution of (40) is just a solution of original system (38), and vice versa. erefore, to complete the proof, it is enough to show (40) has a positive T-periodic solution.
Let K i (t, s) be Green's function of Hill's equation en, a solution of (40) is equivalent to a fixed point of completely continuous operator A: E ⟶ E with components (A 1 , A 2 ): Moreover, let m i and M i denote the minimum and maximum of K i (t, s), respectively; then, σ i � (m i /M i ) � cos(k i T/2) ∈ (0, 1), and so, σ: � min σ 1 , σ 2 ∈ (0, 1). For σ i � cos(k i T/2), we denote again by K, introduced in (18), the positive cone in E.
To apply Lemma 4, it remains to verify Using (H4), we can easily get □ Remark 5. Many authors have paid their attention to the optimal control of the nonlinear systems, and a number of excellent results have been established. See, for instance, [20][21][22][23] and the references therein. For the optimal control of nonlinear system (1), we shall deal in the forthcoming paper.

Data Availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.