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

Authors: Ruipeng Chen; Guangchen Zhang; Jiayin Liu
Journal: Complexity
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

In this paper, we are concerned with the existence of positive periodic solutions of the strongly coupled differential systems:(1)Lixi=fit,x1,x2+eit,i=1,2,where Lixi=xi′′+pitxi′+qitxi is a linear differential operator with pi,qi∈L1ℝ/Tℤ,ℝ. In addition, we assume ei∈L1ℝ/Tℤ,ℝ and fi∈Carℝ/Tℤ×0,∞×0,∞,ℝ, that is, fi|0,T:0,T×0,∞×0,∞⟶ℝ is a L1-Caratheodory function, and it is singular at x1,x2=0,0.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 [1–3]. 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 [4–12]. 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(2)x″+atx=ft,x+ct,where a,c∈L10,T and f∈Car0,T×0,∞,ℝ. By employing a fixed point theorem in cones, they established several existence theorems under the following basic assumption.(H˜) There exist b≻0,b^≻0, and λ>0 such that(3)0≤b^txλ≤ft,x≤btxλ,∀x>0,a.e.t∈0,T,and pointed out they have not limited themselves to the weak singularities; see [10], Section 3, for more details. Besides, the case at≡0 (the resonant case) has also been studied in [10], Theorem 4.1. For other research works related to the resonant case of (2), one may refer to [13–15] and references therein.To our knowledge, however, so far, the existing results on strongly coupled periodic singular systems are relatively few. Therefore, 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:(4)x1′′+p1tx1′+q1tx1=1x1+x2α1+e1t,x2′′+p2tx2′+q2tx2=1x1+x2α2+e2t,where pi,qi,ei∈L1ℝ/Tℤ,ℝ and αi>0,i=1,2. Here, αi>0 means we need not restrict ourselves to the weak force conditions in our results.The rest of the paper is arranged as follows. In Section2, 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 Lixi=xi′′,i=1,2.


## 2. Preliminaries

The linear boundary value problem(5)x″+ptx′+qtx=0,(6)x0=xT,x′0=x′T,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(7)x″+ptx′+qtx=ltadmits a unique T-periodic solution, which can be expressed as(8)xt=∫0TKt,slsds,where Kt,s is Green’s function associated to (5) and (6).For givenξ∈L10,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 xl for any l∈Cℝ/Tℤ and xl 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(9)σpt=e∫0tpsds,σ1pt=σpT∫0tσpsds+∫tTσpsds.Lemma 1 (see [16]).
Ifq≡0 and(10)∫0Tqsσpsσ1−ps≥0,(11)sup0≤t≤T∫tt+Tσ−psds·∫tt+Tqs+σpsds≤4,then (5) satisfies the antimaximum principle, where qs+=maxqs,0, whereafter Chu et al. [17] pointed out if (5) admits the antimaximum principle, then Kt,s≥0 on 0,T×0,T. In addition, they obtained the following.Lemma 2.
Ifq≡0 and (11) holds, then the distance between two consecutive zeroes of a nontrivial solution of (5) is greater than T.Obviously, Lemma2 implies Kt,s does not vanish. As a consequence of Lemmas 1 and 2, Chu et al. established the following.Lemma 3.
Ifq≡0 and (10) and (11) hold, then Kt,s>0 on 0,T×0,T.Note that Lemma3 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, ifpt≡0 (without damping terms), then (10) and (11) reduce, respectively, to(12)∫0Tqsds>0,qs+1<4T,which are conditions used to guarantee the positivity of Green’s function corresponds to (5) and (6); see [18] for more details.
Throughout the paper, we always supposepi,qi,ei∈L1ℝ/Tℤ;ℝ and fi∈Carℝ/Tℤ×0,∞×0,∞,ℝ. Furthermore,(H1) The linear equationLixi=0 is nonresonant, and corresponding Green’s function Kit,s>0 on 0,T×0,T,i=1,2.(H2) There arebi≻0,b^i≻0, and αi>0 so that, for a.e.t∈0,T,(13)0≤b^itx1+x2αi≤fit,x1,x2≤bitx1+x2αi,∀x1>0,x2>0,i=1,2.
It is not hard to see (H1) implies the antimaximum principle holds forLixi=0, and thus, σi=mi/Mi∈0,1, where(14)mi=min0≤t,s≤TKit,s,Mi=max0≤t,s≤TKit,s,i=1,2.
Hence,(15)σ≔minσ1,σ2∈0,1.
Setℝn=∏i=1nℝ,ℝ+n=∏i=1nℝ+, where ℝ=−∞,+∞ and ℝ+=0,+∞). For x=x1,…,xn∈ℝn, we write x=∑i=1nui. Let(16)E=xt=x1t,x2t∈Cℝ,ℝ2:xit+T=xit,t∈ℝ,i=1,2be the Banach space equipped with the norm(17)x=x1∞+x2∞,x=x1,x2∈E.
Here,xi∞=supt∈0,Txit,i=1,2.
A vector functionx1t,x2t∈E is called a positive T-periodic solution of (1) if it satisfies (1) and xi>0 on 0,T, i=1,2. Let(18)K=x=x1,x2∈E:xit≥σixi∞,t∈0,T,i=1,2.
Then, it is not difficult to checkK is a positive cone in E, and for any x=x1,x2∈E, we get by (15) that(19)x1t+x2t≥σx.
Forr>0, let Ωr=x∈K:x<r; then, ∂Ωr=x∈K:x=r.Lemma 4 (see [19]).
LetE be a Banach space and K⊆E a cone. Suppose Ω1,Ω2⊆E are open bounded subsets satisfying 0∈Ω1,Ω¯1⊆Ω2. If A:K∩Ω¯2\Ω1⟶K is completely continuous and satisfiesthere isψ∈K\0 such that u≠Au+λψ for u∈K∩∂Ω1 and λ>0,Au≤u,∀u∈K∩∂Ω2,orthere isψ∈K\0 such that u≠Au+λψ for u∈K∩∂Ω2 and λ>0,Au≤u,∀u∈K∩∂Ω1,then A admits a fixed point in K∩Ω¯2\Ω1.


## 3. The Nonresonant Systems

This section is devoted to establishing the existence results for system (1) in the nonresonant case. To this end, we define(20)γit=∫0TKit,seisds,i=1,2,Bit=∫0TKit,sbisds,B^it=∫0TKit,sb^isds,i=1,2.Theorem 1.
Assume (H1) and (H2) hold. Let(21)Ri=b^i∗ei∗1/αi,R0=minR1,R2.(i)
Ifei∗<0, b^i∗>0, and(22)∑i=12γi∗≤R0−∑i=12Bi∗σR0αi,then (1) has a positive T-periodic solution.(ii)
Ifei∗≥0i=1,2, then (1) has a positive T-periodic solution.Proof.
(i)
LetA:E⟶E be an operator defined by A=A1,A2 with(23)Aixt=∫0TKit,sfis,x1s,x2sds+γit,i=1,2.Then,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 1: if we chooser≪1, then for ψ=1,1∈K,(24)x≠Ax+λψ,∀x∈K∩∂Ωr,λ>0.Or else, if there arex0=x10,x20∈K∩∂Ωr and λ0>0 so that x0=Ax0+λ0,λ0, then by the definition of K and (15), we get(25)σr=σx0≤x10+x20≤x0=r.This together with (H2) implies(26)xi0t=Aix0t+λ0=∫0TKit,sfis,x10s,x20sds+γit+λ0≥∫0TKit,sb^isx10s+x20sαids+γi∗+λ0≥B^i∗rαi+γi∗+λ0>r,i=1,2,which contradictsx0=r since r≪1.Step 2: forR:=R0>r, A maps K∩Ω¯R\Ωr into K.SinceR=R0, for any x1,x2∈0,∞×0,∞ satisfying x1+x2∈0,R] and a.e.t∈0,T, we can deduce from (H2) that(27)fit,x1,x2+eit≥b^itx1+x2αi+eit≥b^i∗x1+x2αi+ei∗≥0,i=1,2,and subsequently, forx∈K∩Ω¯R∖Ωr,(28)Aixt=∫0TKit,sfis,x1s,x2s+eisds≥mi∫0Tfis,x1s,x2s+eisds=σi∫0TMifis,x1s,x2s+eisds≥σiAix∞,i=1,2,which meansA maps K∩Ω¯R∖Ωr into K.Step 3: we shall show(29)Ax≤x,∀x∈K∩∂ΩR.Recall thatR=R0. For any x∈K∩∂ΩR, we have σR=σx≤x1+x2≤x=R, and then by (27), (H1), and (H2), we can obtain(30)0≤Aixt=∫0TKit,sfis,x1s,x2s+eisds≤∫0TKit,sbisx1s+x2sαids+γi∗≤Bi∗σRαi+γi∗,i=1,2,which yieldsAix∞≤Bi∗/σRαi+γi∗,i=1,2. This together with (22) shows(31)Ax=A1x∞+A2x∞≤R=x.And accordingly, Lemma4 ensures (1) admits a positive T-periodic solution.(ii)
To deal with this case, we only need to show (24)–(29) are still satisfied. Indeed, if we choose r≪1, then we can easily prove by (H2) that (24) holds true. Moreover, since ei∗≥0, R≫1 could be chosen so that (27) and (29) are all satisfied. Consequently, Lemma 4 yields (1) has a positive T-periodic solution.Remark 2.
Obviously, (H2) reduces to (H˜) under some special circumstances; hence, Theorem 1 generalizes [10], Theorem 3.1.Remark 3.
Whenpit≡0, f1t,x1,x2≡f1t,x2, and f2t,x1,x2≡f2t,x1, system (1) becomes(32)x1′′+q1tx1=f1t,x2+e1t,x2′′+q2tx2=f2t,x1+e2t.
In [11], several existence theorems have been established for (32) via Schauder’s fixed point theorem, where fi satisfies only the weak force conditions. However, we do not restrict ourselves here to weak singularities, and Theorem 1 is still valid for (32) with strong singularities.Theorem 2.
Assume (H1) and (H3) There arebi≻0,b^i≻0,di≻0, and αi,ηi>0 such that, for a.e.t∈0,T,(33)0≤b^itx1+x2αi≤fit,x1,x2≤bitx1+x2αi+ditx1+x2ηi,∀x1,x2>0.Then, the following results hold true:(i)
Ifei∗<0, b^i∗>0, and(34)∑i=12γi∗≤R0−∑i=12Bi∗σR0αi−∑i=12Di∗R0ηi,then (1) has a positive T-periodic solution.(ii)
Ifei∗≥0i=1,2, then (1) has a positive T-periodic solution.Here,R0 is defined as in Theorem 1.Proof.
(i)
Chooser≪1 and set R=R0. By (H3) and an argument similar to the proof of Theorem 1 (i), we can get (24) and (27). To apply Lemma 4, we just need to show(35)Ax≤x,∀x∈K∩∂ΩR.In fact, for anyx=x1,x2∈K∩∂ΩR, it follows from (27) and (H3) that(36)0≤Aixt=∫0TKit,sfis,x1s,x2sds+γit≤∫0TKit,sbisx1s+x2sαids+∫0TKit,sdisx1s+x2sηids+γi∗≤Bi∗σRαi+Di∗Rηi+γi∗,i=1,2,and then (34) yields Ax=A1x∞+A2x∞≤R=x. Therefore, Lemma 4 implies (1) has a positive T-periodic solution.(ii)
Using Lemma4 and similar to the proof of Theorem 1 (ii), we can easily get the conclusions.Remark 4.
Jiang et al. [4] studied the singular equation(37)u″+atu=btuα+dtuη+ct,where α,η>0 are constants, bt,dt∈C0,T are nonnegative, and ct∈C0,T. For the positone case, they supposed ct≥0, which means c∗≥0, and for the semipositone case, a strong force condition was required. See [4], Corollaries 3.2 and 4.3. Evidently, Theorem 2 generalizes [10], Theorem 3.2, and the corresponding ones in [4].


## 4. The Resonant Systems

Let us consider the singular systems(38)x1″=f1t,x1,x2+e1t,x2″=f2t,x1,x2+e2t.Theorem 3.
Assume (H2) and (H4) There arek1,k2∈0,π/T so that, for any x1,x2∈ℝ+2\0,0 and a.e.t∈0,T,(39)fit,x1,x2+eit+ki2xi≥0,i=1,2.
Ifb^i∗>0 and ei∗<0, then (38) has a positive T-periodic solution.Proof.
Let us take into account the auxiliary systems(40)x1″+k12x1=f1t,x1,x2+e1t+k12x1,x2″+k22x2=f2t,x1,x2+e2t+k22x2,where k1,k2∈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. Therefore, to complete the proof, it is enough to show (40) has a positive T-periodic solution.
LetKit,s be Green’s function of Hill’s equation xi′′+ki2xi=0,i=1,2. Then, a solution of (40) is equivalent to a fixed point of completely continuous operator A:E⟶E with components A1,A2:(41)Aixt=∫0TKit,sfis,x1s,x2s+ki2xis+eisds,i=1,2.
Moreover, letmi and Mi denote the minimum and maximum of Kit,s, respectively; then, σi=mi/Mi=coskiT/2∈0,1, and so, σ:=minσ1,σ2∈0,1. For σi=coskiT/2, we denote again by K, introduced in (18), the positive cone in E.
ChooseR≫1/σ sufficiently large and define ΩR=x∈E:x<R. Since ei∗<0, we have for x∈K∩∂ΩR,(42)∫0TKit,sbisx1+x2αi+eisds≤∫0TKit,sbisσRαi+eisds<0,and subsequently, by (H4),(43)0≤Aixt=∫0TKit,ski2xis+fis,x1s,x2s+eisds≤∫0TKit,sbisx1+x2αi+eis+ki2xisds<ki2xi∞∫0TKit,sds=ki21ki2xi∞=xi∞,i=1,2,this yields Ax≤x,∀x∈K∩∂ΩR.
By (H2) andb^i∗>0, there exists r<R small enough such that, for any x=x1,x2 satisfying σr≤x1+x2≤r,(44)fit,x1,x2+eit≥b^itx1+x2αi+eit≥b^i∗rαi+ei∗≥0,which implies(45)fit,x1,x2+eit+ki2xi≥ki2xi,σr≤x1+x2≤r.
LetΩr=x∈E:x<r and ψ≡1,1T∈K; then,(46)x≠Ax+λψ,∀x∈K∩∂Ωr,λ>0.
Otherwise, if there arex0=x10,x20∈K∩∂Ωr and λ0>0 such that x0=Ax0+λ0ψ, then from (45), it follows(47)xi0t=Aix0t+λ0=∫0TKit,sfis,x10s,x20s+eis+ki2xi0sds+λ0≥∫0TKit,ski2xi0sds+λ0.
SettingΥ=mint∈0,Txi0t, we obtain from (47) that Υ≥Υ+λ0, which contradicts λ0>0.
To apply Lemma4, it remains to verify(48)AK∩Ω¯R∖Ωr⊆K.
Using (H4), we can easily get(49)Aix∞≤Mi∫0TKit,sfis,x1s,x2s+eis+ki2xisds,i=1,2,and thus,(50)Aixt≥mi∫0Tfis,x1s,x2s+eis+ki2xisds=σiMi∫0Tfis,x1s,x2s+eis+ki2xisds=σiAix∞,i=1,2,which means (48) holds true.
Consequently, Lemma4 ensures (40) has a positive T-periodic solution, and accordingly, system (38) admits a positive T-periodic solution.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, [20–23] and the references therein. For the optimal control of nonlinear system (1), we shall deal in the forthcoming paper.



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