Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 83, pp. 1–11. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu EXISTENCE OF POSITIVE PERIODIC SOLUTIONS FOR A NONLINEAR SYSTEM OF SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS XIAO HAN, YUJING HE, HUI WEI Abstract. This article concerns the positive periodic solutions for a system of second-order nonlinear ordinary differential equations, in which the nonlinear term is sublinear in one equation and superlinear in the other equation. By using the fixed point theorem of cone expansion and compression we obtain the existence of positive periodic solutions. 1. Introduction System of ordinary differential equations appear in fields such as applied math- ematics, mathematical physics, mechanical engineering, etc. To find special solu- tions, for example, radial symmetric solutions of elliptic system, it is natural to consider systems of ordinary differential equations (see [3, 4, 11, 12]). In recent decades, the existence of solutions for ordinary differential equation and related questions have attracted extensive attention (see [1, 2, 5, 7, 8, 13, 14]). Dunninger and Wang [3] considered positive and radial symmetric solutions for a class of ellip- tic systems. The corresponding problem was reduced to a Dirichlet boundary value problem for a system of ordinary differential equations. In their work, the nonlinear terms of the two equations are either both sublinear or superlinear, which means that the corresponding solution operators have the properties of the cone compres- sion or the cone expansion. Thus, the result on existence of positive solutions can be obtained by constructing a single cone in the product space C[0, 1] × C[0, 1] and applying the fixed point theorem of cone compression or expansion. Later, Cheng and Zhong [2] studied the Dirichlet boundary value problem of a system of the second-order ordinary differential equations in which the nonlinear terms have the different growth properties and proved the existence of positive solutions by investigating the properties of the fixed point index of the Cartesian product of two cones in the space C[0, 1]. This article is mainly concerned with the periodic behavior of solutions to or- dinary differential equations. Such a problem has always been the focus in the study of ordinary differential equation. Guerrero-Flores et al. [9] studied the sea- sonal SIQRS models with nonlinear infection terms and proved the existence of periodic solutions by using Leray-Schauder degree theory. Kobilzoda and Naimov 2020 Mathematics Subject Classification. 34C25, 47H10. Key words and phrases. Existence ofsolutions; positive periodic solutions; fixed point theorem. ©2022. This work is licensed under a CC BY 4.0 license. Submitted August 11, 2022. Published December 15, 2022. 1 2 X. HAN, Y. HE, H. WEI EJDE-2022/83 [10] considered a class of systems of nonlinear ordinary differential equations on the plane and obtained the existence of positive periodic solutions by giving suit- able estimations and applying the theory of rotation of vector fields. In [6, 16] the authors studied periodic solutions of a system of (generalized) ordinary differential equations by using the bifurcations method. Here we hope to develop the method of fixed point theorem in cone to investigate the periodic solutions of the system of ordinary differential equations. To do so, the first problem we face is to give the Green’s function for the linear ordinary differential equation and study its prop- erties. Then, we need to construct the suitable cone by analyzing the nonlinear problem. Finally, we develop the fixed point theorem in a cone to establish the existence of periodic solutions. We consider the existence of positive periodic solutions for the system of second- order nonlinear ordinary differential equations −u′′(t) + u(t) = g1(t, u) + h1(u, v), 0 < t < 1, −v′′(t) + v(t) = g2(t, u) + h2(u, v), 0 < t < 1, u(0) = u(1), u′(0) = u′(1), v(0) = v(1), v′(0) = v′(1), (1.1) where gi ∈ C([0, 1] × R+,R+) are 1-periodic in t and hi ∈ C(R+ × R+,R+) for i = 1, 2, R+ = [0,+∞). In this article we assume the folloing hypothesis on gi and hi (i = 1, 2): (H1) lim supu→0+ maxt∈[0,1] g1(t,u) u < 1 < lim infu→+∞mint∈[0,1] g1(t,u) u ; (H2) lim supv→+∞maxt∈[0,1] g2(t,v) v < 1 < lim infv→0+ mint∈[0,1] g2(t,v) v ; (H3) limu→0+ h1(u,v) u = 0 uniformly for v ∈ R+; (H4) limv→+∞ h2(u,v) v = 0 uniformly for u ∈ R+, and for any fixed constant M > 0, limu→+∞ h2(u, v) = 0 uniformly for v ∈ [0,M ]. From (H1) it follows that g1(t, u) is superlinear with respect to u at 0 and +∞. Condition (H2) implies that g2(t, v) is sublinear with respect to v at 0 and +∞. (H3) and (H4) show that h1(u, v) is superlinear with respect to u at 0, and h2(u, v) is sublinear with respect to v at +∞. The main results of this article read as follows. Theorem 1.1. Assume that gi ∈ C([0, 1]×R+,R+) are 1-periodic in t and satisfy (H1) and (H2), and hi ∈ C(R+×R+,R+) satisfy (H3) and (H4) for i = 1, 2. Then (1.1) has at least one positive periodic solution. This article is organized as follows. We first give some preliminaries and con- struct the Green’s function for the corresponding homogeneous linear problem in Section 2. Then the proof of Theorem 1.1 is completed in Section 3 and some examples are presented in Section 4 as the application of our main result. 2. Preliminaries In this section, we first construct a cone which can be viewed as the Cartesian product of two cones in C[0, 1], and then we shall transform the problem of finding the positive periodic solutions of (1.1) into a fixed-point index problem in this cone. As we know, C[0, 1] is a Banach space with the norm ‖u‖ = max t∈[0,1] |u(t)|, ∀u ∈ C[0, 1]. EJDE-2022/83 POSITIVE PERIODIC SOLUTIONS FOR A SYSTEM OF ODES 3 The space of non-negative functions belonging to C[0, 1] is defined by C+[0, 1] = {u ∈ C[0, 1] : u(t) ≥ 0}. To find positive periodic solutions of (1.1), we would like to transform the original problem into a fixed point problem of some integral system; thus we first need to construct the Green’s function G(t, s) of the corresponding linear problem −u′′(t) + u(t) = 0, 0 < t < 1, (2.1) u(0) = u(1), u′(0) = u′(1). (2.2) To do so, we consider the Cauchy problems of equation (2.1) with the initial con- ditions u(0) = 1, u′(0) = 0, and u(0) = 0, u′(0) = 1, respectively. It is obvious that u1(t) = cosh t = et + e−t 2 and u2(t) = sinh t = et − e−t 2 are, respectively, the solutions of the above Cauchy problems. We denote κ = u1(1) + u′2(1)− 2 = e+ e−1 − 2. Then, a not very complicated calculation shows that the Green function is G(t, s) = u2(1) κ u1(t)u1(s)− u′1(1) κ u2(t)u2(s) + r(t, s), with r(t, s) =  u′ 2(1)−1 κ u1(t)u2(s)− u1(1)−1 κ u1(s)u2(t), 0 ≤ s ≤ t ≤ 1, u′ 2(1)−1 κ u1(s)u2(t)− u1(1)−1 κ u1(t)u2(s), 0 ≤ t ≤ s ≤ 1. From the expressions u1(t) and u2(t) and noting that u1(1) = u′2(1) = e+ e−1 2 , u′1(1) = u2(1) = e− e−1 2 , it is easy to see that G(t, s) = { e−1 2κ (et−s−1 + es−t), 0 ≤ s ≤ t ≤ 1, e−1 2κ (es−t−1 + et−s), 0 ≤ t ≤ s ≤ 1. (2.3) Lemma 2.1. The Green function G(t, s) given by (2.3) has the following properties: (i) G(t, s) ≥ 0, ∀t, s ∈ (0, 1); (ii) G(t, s) ≤ G(s, s), ∀t, s ∈ [0, 1]; (iii) G(t, s) ≥ 2 √ e e+1G(s, s), for all t, s ∈ [0, 1]. 4 X. HAN, Y. HE, H. WEI EJDE-2022/83 Proof. Firstly, from the expression of G(t, s) given by (2.3), it is easy to obtain the property (i). Because of the symmetry of G(t, s) with respect to t and s, it is sufficient for us to consider one of the cases, for example, the case of 0 ≤ s ≤ t ≤ 1, in which G(t, s) = e− 1 2κ (et−s−1 + es−t). For this case, it is obvious that G(s, s) = e− 1 2κ (e−1 + 1), and G(s, s) G(t, s) = e−1 + 1 et−s−1 + es−t . Denote x = t− s and define f(x) = e−1 + 1 ex−1 + e−x . Then, in this case (that is, 0 ≤ s ≤ t ≤ 1), we have 0 ≤ x ≤ 1 and f(x) = G(s, s) G(t, s) . A simple calculation yields that max x∈[0,1] f(x) = 1 + e 2e1/2 , min x∈[0,1] f(x) = 1, which shows that 1 ≤ f(x) ≤ 1 + e 2e1/2 holds for 0 ≤ x ≤ 1. Therefore, we have 1 ≤ G(s, s) G(t, s) ≤ 1 + e 2e1/2 , which implies the conclusions (ii) and (iii). The proof is complete. � For each h ∈ C[0, 1], we consider the non-homogeneous problem associated with (2.1) and (2.2) having the form −u′′(t) + u(t) = h(t), 0 < t < 1, (2.4) u(0) = u(1), u′(0) = u′(1). (2.5) From the Green function G(t, s) given by (2.3), the periodic solution of (2.4)–(2.5) can be expressed as u(t) = ∫ 1 0 G(t, s)h(s)ds. (2.6) Thus, the problem of finding positive periodic solutions of (1.1) is transformed into the fixed point problem of the integral system u(t) = ∫ 1 0 G(t, s) ( g1(s, u(s)) + h1(u(s), v(s)) )) ds, v(t) = ∫ 1 0 G(t, s) ( g2(s, u(s)) + h2(u(s), v(s)) )) ds, in C+[0, 1]× C+[0, 1]. EJDE-2022/83 POSITIVE PERIODIC SOLUTIONS FOR A SYSTEM OF ODES 5 To prove the existence of fixed point for the above integral system, we introduce a family of operators as following. For each θ ∈ [0, 1] and u, v ∈ C+[0, 1], we define Av(θ, u)(t) = ∫ 1 0 G(t, s) ( (1− θ)u2(s) + θ ( g1(s, u(s)) + h1(u(s), v(s)) )) ds, (2.7) Bu(θ, v)(t) = ∫ 1 0 G(t, s) ( (1− θ) √ v(s) + θ ( g2(s, u(s)) + h2(u(s), v(s)) )) ds. (2.8) It is easy to check that, for any θ ∈ [0, 1], the operators Av(θ, ·) and Bu(θ, ·) : C+[0, 1]→ C+[0, 1]. Now we define the vector operator Tθ(u, v) = ( Av(θ, u)(t), Bu(θ, v)(t) ) , (2.9) then Tθ(·, ·) : C+[0, 1] × C+[0, 1] → C+[0, 1] × C+[0, 1] for all θ ∈ [0, 1], and the positive periodic solutions of (1.1) correspond to the fixed points of the vector operator T1 in C+[0, 1]× C+[0, 1]. We define the cone K in C+[0, 1] by K = { u ∈ C+[0, 1] : u(0) = u(1), u(t) ≥ 2 √ e e+ 1 ‖u‖, ∀t ∈ [ 1 4 , 3 4 ] } , and, for a constant r > 0, we define Kr = {u ∈ K : ‖u‖ < r}, ∂Kr = {u ∈ K : ‖u‖ = r}. (2.10) Lemma 2.2. For each θ ∈ [0, 1], the operator Tθ : K ×K → K ×K is completely continuous. Proof. We first prove that for any θ ∈ [0, 1], the operator Tθ maps K × K into K × K. In fact, for any u, v ∈ K, by using the properties of the Green function G(t, s) given in Lemma 2.1, and taking into consideration the definition of operator Av(θ, u) given in (2.7), it is easy to see that Av(θ, u)(t) = ∫ 1 0 G(t, s) ( (1− θ)u2(s) + θ ( g1(s, u(s)) + h1(u(s), v(s)) )) ds ≥ 2 √ e e+ 1 ∫ 1 0 G(s, s) ( (1− θ)u2(s) + θ ( g1(s, u(s)) + h1(u(s), v(s)) )) ds ≥ 2 √ e e+ 1 ‖Av(θ, u)‖ holds for all t ∈ [1/4, 3/4]. In a similar way we obtain Bu(θ, v)(t) ≥ 2 √ e e+ 1 ‖Bu(θ, v)‖ for t ∈ [1/4, 3/4]. Thus, we have Tθ(u, v) ∈ K ×K, ∀(u, v) ∈ K ×K. Finally, using the Arzelà-Ascoli theorem, it is not difficult to prove that Tθ is completely continuous. � 6 X. HAN, Y. HE, H. WEI EJDE-2022/83 At the end of this section, we make some remarks and introduce two lemmas. Let X be a Banach space and P ⊂ X be a closed convex cone. Assume that Ω ⊂ X is a bounded open set and the operator A : P ∩ Ω → P is completely continuous. If Au 6= u for all u ∈ P ∩ ∂Ω, then the fixed point index i(A,P ∩ Ω, P ) can be defined as in [15]. Furthermore, if i(A,P ∩ Ω, P ) 6= 0, the operator A possesses a fixed point in P ∩ Ω. Lemma 2.3 ([2, 15]). Assume that A : Pr → Pr is completely continuous, where Pr = {u ∈ P : ‖u‖ < r} with ∂Pr = {u ∈ P : ‖u‖ = r} for some constant r > 0. (i) If ‖Au‖ > ‖u‖, for all u ∈ ∂Pr, then i(A,Pr, P ) = 0; (ii) If ‖Au‖‖ < u‖, for all u ∈ ∂Pr, then i(A,Pr, P ) = 1. Lemma 2.4 (Product rule for fixed point index, see [2]). Assume that Pi ⊂ X are closed convex cone in Banach space X and Ai : Pi → Pi are completely continuous operators for i = 1, 2. If Aiui 6= ui for any ui ∈ ∂Pi, then i(A,Pr1 × Pr2 , P1 × P2) = i(A1, Pr1 , P1) · i(A2, Pr2 , P2), where A(u, v) = (A1(u), A2(v)) for (u, v) ∈ P1 × P2, Pri = {u ∈ Pi : ‖u‖ < ri} and ∂Pri = {u ∈ Pi : ‖u‖ = ri} for some constants ri > 0. 3. Proof of main result In this section, we shall prove the existence of positive periodic solutions of (1.1). To do so, we first give the fixed point index of T0, and then we apply the homotopy invariance to obtain the fixed point index of T1. In this process, the following theorem plays a fundamental role. Theorem 3.1. There exist constants 0 < ri < Ri for i = 1, 2 such that, for any θ ∈ [0, 1], we have Tθ(u, v) 6= (u, v), ∀(u, v) ∈ ∂ ( (KR1 \Kr1)× (KR2 \Kr2) ) , where KRi and Kri are defined by (2.10). Proof. The proof is divided into the following four steps. Step 1. Denote r0 = (∫ 1 0 G(s, s)ds )−1 = 2(e+ e−1 − 2) e− e−1 . (3.1) Then, by the assumptions (H1) and (H3), there exist ε ∈ (0, 1/2) and 0 < r1 < min{r0, 1− ε}, such that g1(t, u) ≤ (1− 2ε)u, ∀t ∈ [0, 1], 0 ≤ u ≤ r1, (3.2) h1(u, v) ≤ εu, ∀v ≥ 0, 0 ≤ u ≤ r1. (3.3) Thus, for any θ ∈ [0, 1], it is not difficult to verify that Tθ(u, v) 6= (u, v), ∀(u, v) ∈ ∂Kr1 ×K. In fact, if there exist θ0 ∈ [0, 1] and (u0, v0) ∈ ∂Kr1 ×K such that Tθ0(u0, v0) = (u0, v0). Then, by (2.7) and (2.9), u0 satisfies −u′′0(t) + u0(t) = (1− θ)u2 0(t) + θ ( g1(t, u0(t)) + h1(u0(t), v0(t)) ) , (3.4) EJDE-2022/83 POSITIVE PERIODIC SOLUTIONS FOR A SYSTEM OF ODES 7 u0(0) = u0(1), u′0(0) = u′0(1). (3.5) Noting that 0 < r1 < 1− ε, by (3.2) and (3.3), we have −u′′0(t) + u0(t) ≤ (1− θ)(1− ε)u0(t) + θ(1− ε)u0(t) = (1− ε)u0(t), for 0 ≤ u ≤ r1. Integrating this inequality on [0, 1] yields∫ 1 0 u0(t)dt ≤ (1− ε) ∫ 1 0 u0(t)dt, which implies 1 ≤ 1− ε because of u0 ∈ C+[0, 1]. This contradicts ε ∈ (0, 1 2 ). Step 2. According to assumption (H2), there exist ε > 0 and 0 < ξ < 1 (1+ε)2 such that g2(t, v) ≥ (1 + ε)v, ∀t ∈ [0, 1], 0 ≤ v ≤ ξ. From 0 ≤ v ≤ ξ and 0 < ξ < 1 (1+ε)2 , we have √ v ≥ (1 + ε)v. Taking r2 ∈ (0,min{r0, ξ}) and noting h2(t) ≥ 0, a similar proof as Step 1 shows that Tθ(u, v) 6= (u, v), ∀(u, v) ∈ K × ∂Kr2 , for any fixed θ ∈ [0, 1]. Step 3. By assumption (H1), there exist ε > 0 and M1 > 0 such that g1(t, u) ≥ (1 + ε)u, ∀t ∈ [0, 1], u ≥M1. Furthermore, since g1 is continuous, there exists a constant C1 > 0 such that g1(t, u) ≥ (1 + ε)u− C1, ∀t ∈ [0, 1], u ≥ 0, (3.6) u2 ≥ (1 + ε)u− (1 + ε)2 ≥ (1 + ε)u− C1, u ≥ 0. (3.7) If there exist θ0 ∈ [0, 1] and (u0, v0) ∈ K ×K such that Tθ0(u0, v0) = (u0, v0), then (3.4) and (3.5) hold. Noting h1(t) ≥ 0, by (3.4), (3.6) and (3.7), we have −u′′0(t)+u0(t) ≥ (1−θ) ( (1+ε)u0(t)−C1 ) +θ ( (1+ε)u0(t)−C1 ) = (1+ε)u0(t)−C1. Integrating this inequality on [0, 1] yields∫ 1 0 u0(t)dt ≥ (1 + ε) ∫ 1 0 u0(t)dt− C1, which shows that C1 ≥ ε ∫ 1 0 u0(t)dt. Furthermore, by the definition of K, we have C1 ≥ ε ∫ 1 0 u0(t)dt ≥ ε ∫ 3/4 1/4 2 √ e e+ 1 ‖u0‖dt = ε √ e e+ 1 ‖u0‖, which shows ‖u0‖ ≤ C1(e+ 1)√ eε =: R1. Taking R1 > max{R0, R1} with R0 = ((2e1/2 e+ 1 )3 ∫ 3/4 1/4 G(s, s)ds )−1 = (e+ 1)3(e+ e−1 − 2) 2e3/2(e− e−1) , (3.8) for any θ ∈ [0, 1], we have Tθ(u, v) 6= (u, v), ∀(u, v) ∈ ∂KR1 ×K. 8 X. HAN, Y. HE, H. WEI EJDE-2022/83 Step 4. By assumptions (H2) and (H4), there exist 0 < ε � 1 and M2 > 0 such that g2(t, v) ≤ (1− 2ε)v and h2(u, v) ≤ εv, ∀t ∈ [0, 1], v ≥M2, u ≥ 0. Also, we have g2(t, v) + h2(u, v) ≤ (1− ε)v + C2, ∀t ∈ [0, 1], v ≥ 0, u ≥ 0, where 0 < C2 = 1 1− ε + max t∈[0,1],0≤v≤M2,u≥0 ( g2(t, v) + h2(u, v) ) < +∞, because of the continuity of g2 and h2 and the assumption (H4). Obviously, we have √ v ≤ (1− ε)v + C2, ∀t ∈ [0, 1]. Let R2 = C2(e + 1)/( √ eε). Taking R2 > max{R0, R2}, a similar proof as in Step 3 shows that Tθ(u, v) 6= (u, v), ∀(u, v) ∈ K × ∂KR2 , for each fixed θ ∈ [0, 1]. Finally, the conclusion of this theorem is derived from the results of steps 1–4, and thus the proof is completed. � Proof of Theorem 1.1. By Lemma 2.1 and the definitions of Av(θ, u) and Au(θ, v) given in (2.7) and (2.8), for any u, v ∈ K, we have Av(0, u) = ∫ 1 0 G(t, s)u2(s)ds ≤ ∫ 1 0 G(s, s)u2(s)ds, Bu(0, v) = ∫ 1 0 G(t, s) √ v(s)ds ≤ ∫ 1 0 G(s, s) √ v(s)ds. Therefore, ‖Av(0, u)‖ ≤ ∫ 1 0 G(s, s)ds‖u‖2, ‖Bu(0, v)‖ ≤ ∫ 1 0 G(s, s)ds‖v‖1/2. Moreover, by Lemma 2.1, we have ‖Av(0, u)‖ ≥ ∫ 1 0 G (1 2 , s ) u2(s)ds ≥ 8e3/2 (e+ 1)3 ∫ 3/4 1/4 G(s, s)ds‖u‖2, and ‖Bu(0, v)‖ ≥ ∫ 1 0 G (1 2 , s )√ v(s)ds ≥ (2e1/2 e+ 1 )3/2 ∫ 3/4 1/4 G(s, s)ds‖v‖1/2. Let R0 = (∫ 1 0 G(s, s)ds )2 = ( e− e−1 2(e+ e−1 − 2) )2 , r̄0 = ((2e1/2 e+ 1 )3/2 ∫ 3/4 1/4 G(s, s)ds )2 = 2e3/2 (e+ 1)3 R0. It is obvious that R0 > r̄0, and thus we have ‖Av(0, u)‖ < ‖u‖, ∀r ∈ (0, r0), u ∈ ∂Kr, EJDE-2022/83 POSITIVE PERIODIC SOLUTIONS FOR A SYSTEM OF ODES 9 ‖Av(0, u)‖ > ‖u‖, ∀R ∈ (R0,+∞), u ∈ ∂KR, ‖Bu(0, v)‖ > ‖v‖, ∀r̄ ∈ (0, r̄0), v ∈ ∂Kr̄, ‖Bu(0, v)‖ < ‖v‖, ∀R ∈ (R0,+∞), u ∈ ∂KR, where r0 and R0 are given by (3.1) and (3.8), respectively. By Lemma 2.3, we have i(Av(0, ·),Kr,K) = 1, ∀r ∈ (0, r0), i(Av(0, ·),KR,K) = 0, ∀R ∈ (R0,+∞), i(Bu(0, ·),Kr̄,K) = 0, ∀r̄ ∈ (0, r̄0), i(Bu(0, ·),KR,K) = 1, ∀R ∈ (R0,+∞). Thus, by Lemma 2.4, we have i ( T0, (KR\Kr)× (KR\Kr̄),K ×K ) = i(Av(0, ·),KR\Kr,K) · i(Bu(0, ·),KR\Kr̄,K) = −1. Therefore, by Theorem 3.1 and applying the homotopy invariance, we have i ( Tθ, (KR1\Kr1)× (KR2\Kr2),K ×K ) = i ( T0, (KR1\Kr1)× (KR2 \Kr2),K ×K ) = −1, for any fixed θ ∈ [0, 1], where Ri and ri (i = 1, 2) are given in Theorem 3.1 and satisfy r1 ∈ (0, r0), R1 > R0, r2 ∈ (0, r̄0) and R2 > R0. Consequently, i ( T1, (KR1 \Kr1)× (KR2\Kr2),K ×K ) = −1, which implies that problem (1.1) has a positive periodic solution in K ×K. The proof is complete. � 4. Applications To demonstrate the usefulness of our main theorem, in this section we consider the the existence of positive periodic solutions for −u′′(t) + u(t) = ξ(t)up+1 + up+1 | sin v| v , 0 < t < 1, −v′′(t) + v(t) = η(t)v1−q + v1−q | sinu| u , 0 < t < 1, u(0) = u(1), u′(0) = u′(1), v(0) = v(1), v′(0) = v′(1), (4.1) where p > 0, 0 < q < 1 are constants, and ξ(t), η(t) ∈ C([0, 1];R+) are positive, 1-periodic, continuous functions. Set g1(t, u) = ξ(t)up+1, g2(t, v) = η(t)v1−q, h1(u, v) = { up+1 | sin v| v , v > 0, up+1, v = 0, h2(u, v) = { v1−q | sinu| u , u > 0, v1−q, u = 0. Then a simple computation shows that all the conditions in Theorem 1.1 are satis- fied. Thus we have the following result. 10 X. HAN, Y. HE, H. WEI EJDE-2022/83 Proposition 4.1. Let p > 0, 0 < q < 1. Then, for any positive, 1-periodic, continuous functions ξ(t), η(t) ∈ C([0, 1];R+), system (4.1) has at least one positive 1-periodic solution. More specifically, we present the following example. Example 4.2. Consider the system of ordinary differential equations −u′′(t) + u(t) = (2 + sin(πt))u 5 2 + u2(t) | sin v| v , 0 < t < 1, −v′′(t) + v(t) = (2 + sin(πt))v 1 3 + v 1 4 (t) | sinu| u , 0 < t < 1, u(0) = u(1), u′(0) = u′(1), v(0) = v(1), v′(0) = v′(1). (4.2) By choosing ξ(t) = η(t) = 2 + sin(πt) in system (4.1), it is reduced to the above specific example. By Proposition 4.1, we can conclude that system (4.2) has at least one positive 1-periodic solution. Acknowledgements. This work was supported by the Natural Science Founda- tion of Jilin Province (No. 20210101142JC), by the Key Scientific Research Projects of Colleges and Universities in Henan Province (No. 22A110016), and by the Na- tional Project Cultivation Fund of Luoyang Normal University (No. 2019-PYJJ- 002). References [1] F.M. Atici, G.Sh. Guseinov; On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions, J. Comput. Appl. Math. 132 (2001) 341–356. [2] X. Cheng, C. 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Appl. 317 (2006) 1–13. [14] M. Naito, H. Usami; On the existence and asymptotic behavior of solutions of half-linear ordinary differential equations, J. Differential Equations 318 (2022) 359–383. EJDE-2022/83 POSITIVE PERIODIC SOLUTIONS FOR A SYSTEM OF ODES 11 [15] C. Zhong, X. Fan, W. Chen; Introduction to Nonlinear Functional Analysis (in Chinese), Lanzhou University Press, Lanzhou, 1998. [16] C. Zhu, B. Long; The periodic solution bifurcated from homoclinic orbit for coupled ordinary differential equations, Math. Methods Appl. Sci. 40 (2017) 2834–2846. Xiao Han School of Mathematics, Jilin University, Changchun 130012, China Email address: hanx@jlu.edu.cn Yujing He School of Mathematics, Jilin University, Changchun 130012, China Tianjin Binhai Foreign Language School, Tianjin 300450, China Email address: 1073669517@qq.com Hui Wei (corresponding author) Department of Mathematics, Luoyang Normal University, Luoyang 471934, China Email address: weihui01@163.com 1. Introduction 2. Preliminaries 3. Proof of main result 4. Applications Acknowledgements References