Existence and Uniqueness of Periodic Solutions for a Second-Order Nonlinear Differential Equation with Piecewise Constant Argument

Abstract

Based on a continuation theorem of Mawhin, a unique periodic solution is found for a second-order nonlinear differential equation with piecewise constant argument.

1. Introduction

Qualitative behaviors of first-order delay differential equations with piecewise constant arguments are the subject of many investigations (see, e.g., [1–19]), while those of higher-order equations are not.

However, there are reasons for studying higher-order equations with piecewise constant arguments. Indeed, as mentioned in [10], a potential application of these equations is in the stabilization of hybrid control systems with feedback delay, where a hybrid system is one with a continuous plant and with a discrete (sampled) controller. As an example, suppose that a moving particle with time variable mass 𝑟(𝑡) is subjected to a restoring controller −𝜙(𝑥[𝑡]) which acts at sampled time [𝑡]. Then Newton's second law asserts that

(𝑟(𝑡)𝑥^′(𝑡))^′=−𝜙(𝑥[𝑡]). (1.1) Since this equation is “similar” to the harmonic oscillator equation

(𝑟(𝑡)𝑥^′(𝑡))^′+𝜅𝑥(𝑡)=0, (1.2) we expect that the well-known qualitative behavior of the later equation may also be found in the former equation, provided appropriate conditions on 𝑟(𝑡) and 𝜙(𝑥) are imposed.

In this paper we study a slightly more general second-order delay differential equation with piecewise constant argument:

(𝑟(𝑡)𝑥^′(𝑡))^′+𝑓(𝑡,𝑥([𝑡]))=𝑝(𝑡), (1.3) where 𝑓(𝑡,𝑥) is a real continuous function defined on 𝑅² with positive integer period 𝜔 for 𝑡;𝑟(𝑡) and 𝑝(𝑡) are continuous function defined on 𝑅 with period 𝜔,𝑟(𝑡)>0 for 𝑡∈𝑅 and ∫^𝜔₀𝑝(𝑡)𝑑𝑡=0.

By a solution of (1.3) we mean a function 𝑥(𝑡) which is defined on 𝑅 and which satisfies the following conditions: (i) 𝑥^′(𝑡) is continuous on 𝑅, (ii) 𝑟(𝑡)𝑥^′(𝑡) is differentiable at each point 𝑡∈𝑅, with the possible exception of the points [𝑡]∈𝑅 where one-sided derivatives exist, and (iii) substitution of 𝑥(𝑡) into (1.3) leads to an identity on each interval [𝑛,𝑛+1)⊂𝑅 with integral endpoints.

In this note, existence and uniqueness criteria for periodic solutions of (1.3) will be established. For this purpose, we will make use of a continuation theorem of Mawhin. Let 𝑋 and 𝑌 be two Banach spaces and 𝐿∶Dom𝐿⊂𝑋→𝑌 is a linear mapping and 𝑁∶𝑋→𝑌 a continuous mapping. The mapping 𝐿 will be called a Fredholm mapping of index zero if dimKer𝐿=codimIm𝐿<+∞, and Im𝐿 is closed in 𝑌. If 𝐿 is a Fredholm mapping of index zero, there exist continuous projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑌→𝑌 such that Im𝑃=Ker𝐿 and Im𝐿=Ker𝑄=Im(𝐼−𝑄). It follows that 𝐿_{|Dom𝐿∩Ker𝑃}∶(𝐼−𝑃)𝑋→Im𝐿 has an inverse which will be denoted by 𝐾_𝑃. If Ω is an open and bounded subset of 𝑋, the mapping 𝑁 will be called 𝐿-compact on Ω if 𝑄𝑁(Ω) is bounded and 𝐾_𝑃(𝐼−𝑄)𝑁∶Ω→𝑋 is compact. Since Im𝑄 is isomorphic to Ker𝐿, there exists an isomorphism 𝐽∶Im𝑄→Ker𝐿.

Theorem 1 A (Mawhin's continuation theorem [18]). Let 𝐿 be a Fredholm mapping of index zero, and let 𝑁 be 𝐿-compact on Ω. Suppose that(i)for each 𝜆∈(0,1), 𝑥∈𝜕Ω,𝐿𝑥≠𝜆𝑁𝑥;(ii)for each 𝑥∈𝜕Ω∩Ker𝐿,𝑄𝑁𝑥≠0 and deg(𝐽𝑄𝑁,Ω∩Ker,0)≠0.Then the equation 𝐿𝑥=𝑁𝑥 has at least one solution in Ω∩dom𝐿.

2. Existence and Uniqueness Criteria

Our main results of this paper are as follows.

Theorem 2.1. Suppose that there exist constants 𝐷>0 and 𝛿⩾0 such that(i)𝑓(𝑡,𝑥)sgn𝑥>0 for 𝑡∈𝑅 and |𝑥|>𝐷,(ii)lim_{𝑥→−∞}max_{0≤𝑡≤𝜔}(𝑓(𝑡,𝑥)/𝑥)≤𝛿 (or lim_𝑥→+∞max_{0≤𝑡≤𝜔}(𝑓(𝑡,𝑥)/𝑥)≤𝛿).If 𝜔²𝛿(max_{0≤𝑡≤𝜔}(1/𝑟(𝑡)))<1, then (1.3) has an 𝜔-periodic solution. Furthermore, the 𝜔-periodic solution is unique if in addition one has the following.(iii)𝑓(𝑡,𝑥) is strictly monotonous in 𝑥 and there exists nonnegative constant 𝑏<(4/𝜔²)min_{0≤𝑡≤𝜔}𝑟(𝑡) such that|𝑓(𝑡,𝑥₁)−𝑓(𝑡,𝑥₂)|≤𝑏|𝑥₁−𝑥₂|,(𝑡,𝑥₁),(𝑡,𝑥₂)∈𝑅². (2.1)

Theorem 2.2. Suppose that there exist constants 𝐷>0 and 𝛿⩾0 such that(i^′)𝑓(𝑡,𝑥)sgn𝑥<0 for 𝑡∈𝑅 and |𝑥|>𝐷,(ii^′)lim_{𝑥→−∞}max_{0≤𝑡≤𝜔}(𝑓(𝑡,𝑥)/𝑥)⩾−𝛿 (or lim_𝑥→+∞max_{0≤𝑡≤𝜔}(𝑓(𝑡,𝑥)/𝑥)⩾−𝛿).If 𝜔²𝛿(max_{0≤𝑡≤𝜔}(1/𝑟(𝑡)))<1, then (1.3) has an 𝜔-periodic solution. Furthermore, the 𝜔-periodic solution is unique if in addition one has the following.(iii)𝑓(𝑡,𝑥) is strictly monotonous in 𝑥 and there exists nonnegative constant 𝑏<(4/𝜔²)min_{0≤𝑡≤𝜔}𝑟(𝑡) such that (2.1) holds.

We only give the proof of Theorem 2.1, as Theorem 2.2 can be proved similarly.

First we make the simple observation that 𝑥(𝑡) is an 𝜔-periodic solution of the following equation:

𝑟(𝑡)𝑥^′(𝑡)=𝑟(0)𝑥^′(0)−^𝑡₀(𝑓(𝑠,𝑥([𝑠]))−𝑝(𝑠))𝑑𝑠, (2.2) if, and only if, 𝑥(𝑡) is an 𝜔-periodic solution of (1.3). Next, let 𝑋_𝜔 be the Banach space of all real 𝜔-periodic continuously differentiable functions of the form 𝑥=𝑥(𝑡) which is defined on 𝑅 and endowed with the usual linear structure as well as the norm ‖𝑥‖₁=∑¹_𝑖=0max_{0≤𝑖≤𝜔}|𝑥^(𝑖)(𝑡)|. Let 𝑌_𝜔 be the Banach space of all real continuous functions of the form 𝑦=𝛼𝑡+ℎ(𝑡) such that 𝑦(0)=0, where 𝛼∈𝑅 and ℎ(𝑡)∈𝑋_𝜔, and endowed with the usual linear structure as well as the norm ‖𝑦‖₂=|𝛼|+‖ℎ‖₁. Let the zero element of 𝑋_𝜔 and 𝑌_𝜔 be denoted by 𝜃₁ and 𝜃₂ respectively.

Define the mappings 𝐿∶𝑋_𝜔→𝑌_𝜔 and 𝑁∶𝑋_𝜔→𝑌_𝜔, respectively, by

𝐿𝑥(𝑡)=𝑟(𝑡)𝑥^′(𝑡)−𝑟(0)𝑥^′(0),𝑁𝑥(𝑡)=−^𝑡₀(𝑓(𝑠,𝑥([𝑠]))−𝑝(𝑠))𝑑𝑠. (2.3)(2.4) Let

ℎ(𝑡)=−^𝑡₀𝑡(𝑓(𝑠,𝑥([𝑠]))−𝑝(𝑠))𝑑𝑠+𝜔^𝜔₀𝑓(𝑠,𝑥([𝑠]))𝑑𝑠. (2.5) Since ℎ∈𝑋_𝜔 and ℎ(0)=0, 𝑁 is a well-defined operator from 𝑋_𝜔 to 𝑌_𝜔. Let us define 𝑃∶𝑋_𝜔→𝑋_𝜔 and 𝑄∶𝑌_𝜔→𝑌_𝜔, respectively, by

𝑃𝑥(𝑡)=𝑥(0),𝑛∈𝑍 (2.6) for 𝑥=𝑥(𝑡)∈𝑋_𝜔 and

𝑄𝑦(𝑡)=𝛼𝑡 (2.7) for 𝑦(𝑡)=𝛼𝑡+ℎ(𝑡)∈𝑌_𝜔.

Lemma 2.3. Let the mapping 𝐿 be defined by (2.3). Then Ker𝐿=𝑅. (2.8)

Proof. It suffices to show that if 𝑥(𝑡) is a real 𝜔-periodic continuously differentiable function which satisfies 𝑟(𝑡)𝑥^′(𝑡)=𝑟(0)𝑥^′(0),𝑡∈𝑅, (2.9) then 𝑥(𝑡) is a constant function. To see this, note that for such a function 𝑥=𝑥(𝑡),𝑥^′(𝑡)=𝑟(0)𝑥^′(0)𝑟(𝑡),𝑡∈𝑅. (2.10) Hence by integrating both sides of the above equality from 0 to 𝑡, we see that 𝑥(𝑡)=𝑥(0)+𝑟(0)𝑥^′(0)^𝑡₀𝑑𝑠𝑟(𝑠),𝑡∈𝑅. (2.11) Since 𝑟(𝑡) is positive, continuous, and periodic, ^∞₀𝑑𝑠𝑟(𝑠)=∞. (2.12) Since 𝑥(𝑡) is bounded, we may infer from (2.11) that 𝑥^′(0)=0. But then (2.9) implies 𝑥^′(𝑡)=0 for 𝑡∈𝑅. The proof is complete.

Lemma 2.4. Let the mapping 𝐿 be defined by (2.3). Then Im𝐿={𝑦∈𝑋_𝜔∣𝑦(0)=0}⊂𝑌_𝜔. (2.13)

Proof. It suffices to show that for each 𝑦=𝑦(𝑡)∈𝑋_𝜔 that satisfies 𝑦(0)=0, there is a 𝑥=𝑥(𝑡)∈𝑋_𝜔 such that 𝑦(𝑡)=𝑟(𝑡)𝑥^′(𝑡)−𝑟(0)𝑥^′(0),𝑡≥0. (2.14) But this is relatively easy, since we may let 1𝛼=∫^𝜔₀,(𝑑𝑠/𝑟(𝑠))𝑥(𝑡)=^𝑡₀𝑦(𝑠)𝑟(𝑠)𝑑𝑠−𝛼^𝜔₀𝑦(𝑠)𝑟(𝑠)𝑑𝑠^𝑡₀𝑑𝑠𝑟(𝑠),𝑡≥0. (2.15)(2.16) Then it may easily be checked that (2.14) holds. The proof is complete.

Lemma 2.5. The mapping 𝐿 defined by (2.3) is a Fredholm mapping of index zero.

Indeed, from Lemmas 2.3 and 2.4 and the definition of 𝑌_𝜔,dimKer𝐿=codimIm𝐿=1<+∞. From (2.13), we see that Im𝐿 is closed in 𝑌_𝜔. Hence 𝐿 is a Fredholm mapping of index zero.

Lemma 2.6. Let the mapping 𝐿,𝑃, and 𝑄 be defined by (2.3), (2.6), and (2.7), respectively. Then Im𝑃=Ker𝐿 and Im𝐿=Ker𝑄.

Indeed, from Lemmas 2.3 and 2.4 and defining conditions (2.6) and (2.7), it is easy to see that Im𝑃=Ker𝐿 and Im𝐿=Ker𝑄.

Lemma 2.7. Let 𝐿 and 𝑁 be defined by (2.3) and (2.4), respectively. Suppose that Ω is an open and bounded subset of 𝑋_𝜔. Then 𝑁 is 𝐿-compact on Ω.

Proof. It is easy to see that for any 𝑥∈Ω,𝑡𝑄𝑁𝑥(𝑡)=−𝜔^𝜔₀𝑓(𝑠,𝑥([𝑠]))𝑑𝑠, (2.17) so that ‖𝑄𝑁𝑥‖₂1=|𝜔^𝜔₀𝑓(𝑠,𝑥([𝑠]))𝑑𝑠|,(𝐼−𝑄)𝑁𝑥(𝑡)=−^𝑡₀𝑡(𝑓(𝑠,𝑥([𝑠]))−𝑝(𝑠))𝑑𝑠+𝜔^𝜔₀𝑓(𝑠,𝑥([𝑠]))𝑑𝑠,𝑡≥0. (2.18)(2.19) These lead us to 𝐾_𝑃(𝐼−𝑄)𝑁𝑥(𝑡)=−^𝑡₀1𝑟(𝑣)𝑑𝑣^𝑣₀(𝑓(𝑠,𝑥([𝑠]))−𝑝(𝑠))𝑑𝑠+𝛼(^𝜔₀𝑑𝑣𝑟(𝑣)^𝑣₀(𝑓(𝑠,𝑥([𝑠]))−𝑝(𝑠))𝑑𝑠)^𝑡₀1+1𝑟(𝑣)𝑑𝑣𝜔^𝑡₀𝑣𝑟(𝑣)𝑑𝑣^𝜔₀−𝛼𝑓(𝑠,𝑥([𝑠]))𝑑𝑠𝜔(^𝜔₀𝑣𝑑𝑣𝑟(𝑣)^𝜔₀𝑓(𝑠,𝑥([𝑠]))𝑑𝑠)^𝑡₀1𝑟(𝑣)𝑑𝑣, (2.20) where 𝛼 is defined by (2.15). By (2.18), we see that 𝑄𝑁(Ω) is bounded. Noting that (2.7) holds and 𝑁 is a completely continuous mapping, by means of the Arzela-Ascoli theorem we know that 𝐾_𝑃(𝐼−𝑄)𝑁(Ω) is relatively compact. Thus 𝑁 is 𝐿-compact on Ω. The proof is complete.

Lemma 2.8. Suppose that   𝑔(𝑡) is a real, bounded and continuous function on [𝑎,𝑏) and lim_𝑥→𝑏⁻𝑔(𝑡) exists. Then there is a point 𝜉∈(𝑎,𝑏) such that ^𝑏_𝑎𝑔(𝑠)𝑑𝑠=𝑔(𝜉)(𝑏−𝑎). (2.21)

The above result is only a slight extension of the integral mean value theorem and is easily proved.

Lemma 2.9. Suppose that condition (i) in Theorem 2.1 holds. Suppose further that 𝑥(𝑡)∈𝑋_𝜔 satisfies ^𝜔₀𝑓(𝑠,𝑥([𝑠]))𝑑𝑠=0. (2.22) Then there is 𝑡₁∈[0,𝜔] such that |𝑥(𝑡₁)|≤𝐷.

Proof. From (2.22) and Lemma 2.8, we have  𝜉_𝑖∈(𝑖−1,𝑖) for 𝑖=1,…,𝜔 such that ^𝜔_𝑖=1𝑓(𝜉_𝑖,𝑥(𝑖−1))=^𝜔_𝑖=1^𝑖_𝑖−1𝑓(𝑠,𝑥([𝑠]))𝑑𝑠^𝜔₀𝑓(𝑠,𝑥([𝑠]))𝑑𝑠=0. (2.23) In case 𝜔=1, from the condition (i) in Theorem 2.1 and (2.23), we know that |𝑥(0)|≤𝐷. Suppose 𝜔≥2. Our assertion is true if one of 𝑥(0),𝑥(1),…,𝑥(𝜔−1) has absolute value less than or equal to 𝐷. Otherwise, there should be 𝑥(𝜂₁) and 𝑥(𝜂₂) among 𝑥(0),𝑥(1),… and 𝑥(𝜔−1) such that 𝑥(𝜂₁)>𝐷 and 𝑥(𝜂₂)<−𝐷. Since 𝑥(𝑡) is continuous, in view of the intermediate value theorem, there is 𝑥(𝜂₃) such that −𝐷≤𝑥(𝜂₃)≤𝐷, (here 𝜂₁>𝜂₃>𝜂₂ or 𝜂₂>𝜂₃>𝜂₁). Since 𝑥(𝑡) is periodic, there is 𝑡₁∈[0,𝜔] such that |𝑥(𝑡₁)|=|𝑥(𝜂₃)|≤𝐷. The proof is complete.

Now, we consider that following equation:

𝑟(𝑡)𝑥^′(𝑡)−𝑟(0)𝑥^′(0)=−𝜆^𝑡₀(𝑓(𝑠,𝑥([𝑠]))−𝑝(𝑠))𝑑𝑠, (2.24) where 𝜆∈(0,1).

Lemma 2.10. Suppose that conditions (i) and (ii) of Theorem 2.1 hold. If 𝜔²𝛿(max_{0≤𝑡≤𝜔}(1/𝑟(𝑡)))<1, then there are positive constants 𝐷₀ and 𝐷₁ such that for any 𝜔-periodic solution 𝑥(𝑡) of (2.24), |𝑥^(𝑖)(𝑡)|≤𝐷_𝑖,𝑡∈[0,𝜔];𝑖=0,1. (2.25)

Proof. Let 𝑥(𝑡) be a 𝜔-periodic solution of (2.24). By (2.24) and our assumption that ∫^𝜔₀𝑝(𝑠)𝑑𝑠=0, we have ^𝜔₀𝑓(𝑠,𝑥([𝑠]))𝑑𝑠=0. (2.26) By Lemma 2.9, there is 𝑡₁∈[0,𝜔] such that |𝑥(𝑡₁)|≤𝐷. (2.27) Since 𝑥(𝑡) and 𝑥^′(𝑡) are with period 𝜔, thus for any 𝑡∈[𝑡₁,𝑡₁+𝜔], we have 𝑥(𝑡)=𝑥(𝑡₁)+^𝑡_𝑡1𝑥^′(𝑠)𝑑𝑠,𝑥(𝑡)=𝑥(𝑡₁+𝜔)+^𝑡_𝑡1+𝜔𝑥^′(𝑠)𝑑𝑠=𝑥(𝑡₁)+^𝑡_𝑡1+𝜔𝑥^′(𝑠)𝑑𝑠. (2.28) From (2.28), we see that for any 𝑡∈[𝑡₁,𝑡₁+𝜔],|𝑥(𝑡)|≤|𝑥(𝑡₁1)|+2^𝑡1_𝑡^+𝜔₁|𝑥^′(𝑠)|𝑑𝑠=|𝑥(𝑡₁1)|+2^𝜔₀|𝑥^′(𝑠)|𝑑𝑠. (2.29) It is easy to see from (2.27) and (2.29) that for any 𝑡∈[0,𝜔]|𝑥(𝑡)|≤|𝑥(𝑡₁1)|+2^𝜔₀|𝑥^′1(𝑠)|𝑑𝑠≤𝐷+2^𝜔₀|𝑥^′(𝑠)|𝑑𝑠. (2.30) In view of the condition 𝜔²𝛿(max_{0≤𝑡≤𝜔}(1/𝑟(𝑡)))<1, we know that there is a positive number 𝜀 such that 𝜂₁∶=𝜔²(𝛿+𝜀)(max_{0≤𝑡≤𝜔}1𝑟(𝑡))<1. (2.31) From condition (ii), we see that there is a 𝜌>𝐷 such that for 𝑡∈𝑅 and 𝑥<−𝜌,𝑓(𝑡,𝑥)𝑥<𝛿+𝜀. (2.32) Let 𝐸₁𝐸={𝑡∣𝑡∈[0,𝜔],𝑥([𝑡])<−𝜌},₂𝐸={𝑡∣𝑡∈[0,𝜔],|𝑥([𝑡])|≤𝜌},₃=[0,𝜔]⧵(𝐸₁∪𝐸₂𝑀),₀=max_{0≤𝑡≤𝜔,|𝑥|≤𝜌}|𝑓(𝑡,𝑥)|. (2.33)(2.34)(2.35)(2.36) By (2.32) and (2.33), we have _𝐸1|𝑓(𝑠,𝑥([𝑠]))|𝑑𝑠≤(𝛿+𝜀)_𝐸1|𝑥([𝑠])|𝑑𝑠≤(𝛿+𝜀)𝜔max_{0≤𝑡≤𝜔}|𝑥(𝑡)|. (2.37) From (2.34) and (2.36), we have _𝐸2|𝑓(𝑠,𝑥([𝑠]))|𝑑𝑠≤𝜔𝑀₀. (2.38) In view of condition (i), (2.26), (2.37), and (2.38), we get _𝐸3|𝑓(𝑠,𝑥([𝑠]))|𝑑𝑠=_𝐸3𝑓(𝑠,𝑥([𝑠]))𝑑𝑠=−_𝐸1𝑓(𝑠,𝑥([𝑠]))𝑑𝑠−_𝐸2𝑓(𝑠,𝑥([𝑠]))𝑑𝑠≤_𝐸1|𝑓(𝑠,𝑥([𝑠]))|𝑑𝑠+_𝐸2|𝑓(𝑠,𝑥([𝑠]))|𝑑𝑠≤(𝛿+𝜀)𝜔max_{0≤𝑡≤𝜔}|𝑥(𝑡)|+𝜔𝑀₀. (2.39) It follows from (2.37), (2.38), and (2.39) that ^𝜔₀|𝑓(𝑠,𝑥([𝑠]))|𝑑𝑠=_𝐸1|𝑓(𝑠,𝑥([𝑠]))|𝑑𝑠+_𝐸2|𝑓(𝑠,𝑥([𝑠]))|𝑑𝑠+_𝐸3|𝑓(𝑠,𝑥([𝑠]))|𝑑𝑠≤2(𝛿+𝜀)𝜔max_{0≤𝑡≤𝜔}|𝑥(𝑡)|+2𝜔𝑀₀. (2.40) Since 𝑥(0)=𝑥(𝜔), thus there is a 𝑡₁∈(0,𝜔) such that 𝑥^′(𝑡₁)=0. In view of (2.24) and the fact that 𝑥^′(𝑡₁)=0, we conclude that for any 𝑡∈[𝑡₁,𝑡₁+𝜔],|𝑟(𝑡)𝑥^′(𝑡)|=|𝑟(𝑡₁)𝑥^′(𝑡₁)−𝜆^𝑡_𝑡1(𝑓(𝑠,𝑥([𝑠]))−𝑝(𝑠))𝑑𝑠|=|−𝜆^𝑡_𝑡1(𝑓(𝑠,𝑥([𝑠]))−𝑝(𝑠))𝑑𝑠|≤|^𝑡_𝑡1(𝑓(𝑠,𝑥([𝑠]))−𝑝(𝑠))𝑑𝑠|≤^𝑡1_𝑡^+𝜔₁|𝑓(𝑠,𝑥([𝑠]))|𝑑𝑠+^𝑡1_𝑡^+𝜔₁≤|𝑝(𝑠)|𝑑𝑠_𝜔0|𝑓(𝑠,𝑥([𝑠]))|𝑑𝑠+^𝜔₀|𝑝(𝑠)|𝑑𝑠. (2.41) From (2.40) and (2.41), we see that max_{0≤𝑡≤𝜔}|𝑥^′(𝑡)|≤(max_{0≤𝑡≤𝜔}1𝑟(𝑡)){2(𝛿+𝜀)𝜔max_{0≤𝑡≤𝜔}|𝑥(𝑡)|+2𝜔𝑀₀+max_{0≤𝑡≤𝜔}|𝑝(𝑡)|}. (2.42) It follows from (2.30), (2.31), and (2.42) that max_{0≤𝑡≤𝜔}1|𝑥(𝑡)|≤𝐷+2^𝜔₀|𝑥^′(𝑠)|𝑑𝑠≤𝜔²(max_{0≤𝑡≤𝜔}1𝑟(𝑡))(𝛿+𝜀)max_{0≤𝑡≤𝜔}|𝑥(𝑡)|+𝑀₁=𝜂₁max_{0≤𝑡≤𝜔}|𝑥(𝑡)|+𝑀₁, (2.43) where 𝑀₁=𝐷+(max_{0≤𝑡≤𝜔}1𝑟(𝑡))(2𝜔𝑀₀+max_{0≤𝑡≤𝜔}|𝑝(𝑡)|). (2.44) Let 𝐷₀=𝑀₁/(1−𝜂₁), then from (2.43) we have max_{0≤𝑡≤𝜔}|𝑥(𝑡)|≤𝐷₀. (2.45) From (2.42) and (2.45), for any 𝑡∈[0,𝜔], we have max_{0≤𝑡≤𝜔}|𝑥^′(𝑡)|≤𝐷₁, (2.46) where 𝐷₁=(max_{0≤𝑡≤𝜔}1𝑟(𝑡)){2(𝛿+𝜀)𝜔𝐷₀+2𝜔𝑀₀+max_{0≤𝑡≤𝜔}|𝑝(𝑡)|}. (2.47) The proof is complete.

Lemma 2.11. Suppose that condition (iii) of Theorem 2.1 is satisfied. Then (1.3) has at most one 𝜔-periodic solution.

Proof. Suppose that 𝑥₁(𝑡) and 𝑥₂(𝑡) are two 𝜔-periodic solutions of (1.3). Set 𝑧(𝑡)=𝑥₁(𝑡)−𝑥₂(𝑡). Then we have (𝑟(𝑡)𝑧^′(𝑡))^′+𝑓(𝑡,𝑥₁([𝑡]))−𝑓(𝑡,𝑥₂([𝑡]))=0. (2.48)Case 2 (i). For all 𝑡∈[0,𝜔],𝑧(𝑡)≠0. Without loss of generality, we assume that 𝑧(𝑡)>0, that is, 𝑥₁(𝑡)>𝑥₂(𝑡) for 𝑡∈[0,𝜔]. Integrating (2.48) from 0 to 𝜔, we have ^𝜔₀[𝑓(𝑡,𝑥₁(𝑥₁([𝑡])))−𝑓(𝑡,𝑥₂([𝑡]))]𝑑𝑡=0. (2.49) Combining condition (iii) and 𝑥₁(𝑡)>𝑥₂(𝑡), either 𝑓(𝑡,𝑥₁([𝑡]))−𝑓(𝑡,𝑥₂([𝑡]))>0,𝑡∈[0,𝜔] (2.50) or 𝑓(𝑡,𝑥₁([𝑡]))−𝑓(𝑡,𝑥₂([𝑡]))<0,𝑡∈[0,𝜔] (2.51) holds. This is contrary to (2.49).Case 2 (ii). There exist 𝜉∈[0,𝜔] such that 𝑧(𝜉)=0. As in the proof of (2.30) in Lemma 2.10, we have max_{0≤𝑡≤𝜔}1|𝑧(𝑡)|≤|𝑧(𝜉)|+2^𝜔₀|𝑧^′1(𝑠)|𝑑𝑠=2^𝜔₀|𝑧^′(𝑠)|𝑑𝑠. (2.52)On the other hand, since 𝑧(0)=𝑧(𝜔), thus there is a 𝑡₁∈(0,𝜔) such that 𝑧^′(𝑡₁)=0. In view of (2.48), we conclude that for any 𝑡∈[𝑡₁,𝑡₁+𝜔],𝑟(𝑡)𝑧^′(𝑡)=𝑟(𝑡₁)𝑧^′(𝑡₁)−^𝑡_𝑡1(𝑓(𝑠,(𝑥₁([𝑠])))−𝑓(𝑠,𝑥₂([𝑠])))𝑑𝑠,𝑟(𝑡)𝑧^′(𝑡)=𝑟(𝑡₁+𝜔)𝑧^′(𝑡₁+𝜔)−^𝑡_𝑡1+𝜔(𝑓(𝑠,(𝑥₁([𝑠])))−𝑓(𝑠,𝑥₂([𝑠])))𝑑𝑠=𝑟(𝑡₁)𝑧^′(𝑡₁)−^𝑡_𝑡1+𝜔(𝑓(𝑠,(𝑥₁([𝑠])))−𝑓(𝑠,𝑥₂([𝑠])))𝑑𝑠. (2.53) By (2.53) and the fact that 𝑧^′(𝑡₁)=0, we have for any 𝑡∈[𝑡₁,𝑡₁+𝜔],𝑟(𝑡)𝑧^′(𝑡)=𝑟(𝑡₁)𝑧^′(𝑡₁1)−2^𝑡_𝑡1(𝑓(𝑠,(𝑥₁([𝑠])))−𝑓(𝑠,𝑥₂+1([𝑠])))𝑑𝑠2^𝑡1_𝑡^+𝜔(𝑓(𝑠,(𝑥₁([𝑠])))−𝑓(𝑠,𝑥₂1([𝑠])))𝑑𝑠.=−2^𝑡_𝑡1(𝑓(𝑠,(𝑥₁([𝑠])))−𝑓(𝑠,𝑥₂+1([𝑠])))𝑑𝑠2^𝑡1_𝑡^+𝜔(𝑓(𝑠,(𝑥₁([𝑠])))−𝑓(𝑠,𝑥₂([𝑠])))𝑑𝑠. (2.54) It follows that for any 𝑡∈[𝑡₁,𝑡₁+𝜔],|𝑟(𝑡)𝑧^′1(𝑡)|≤2^𝑡1_𝑡^+𝜔₁|𝑓(𝑠,(𝑥₁([𝑠])))−𝑓(𝑠,𝑥₂≤1([𝑠]))|𝑑𝑠2^𝜔₀|𝑓(𝑠,(𝑥₁([𝑠])))−𝑓(𝑠,𝑥₂≤1([𝑠]))|𝑑𝑠2𝑏𝜔max_{0≤𝑡≤𝜔}|𝑧(𝑡)|. (2.55) We know that for any 𝑡∈[0,𝜔],|𝑟(𝑡)𝑧^′1(𝑡)|≤2𝑏𝜔max_{0≤𝑡≤𝜔}|𝑧(𝑡)|. (2.56) From (2.56), we have max_{0≤𝑡≤𝜔}|𝑧^′(𝑡)|≤𝑏𝜔2(max_{0≤𝑡≤𝜔}1𝑟(𝑡))max_{0≤𝑡≤𝜔}|𝑧(𝑡)|. (2.57) By (2.52), we get max_{0≤𝑡≤𝜔}𝜔|𝑧(𝑡)|≤2max_{0≤𝑡≤𝜔}|𝑧^′(𝑡)|. (2.58) It is easy to see from (2.57) and (2.58) that max_{0≤𝑡≤𝜔}|𝑧(𝑡)|≤𝑏𝜔²4(max_0≤t≤𝜔1𝑟(𝑡))max_{0≤𝑡≤𝜔}|𝑧(𝑡)|. (2.59) By condition (iii) of Theorem 2.1, we see that (𝑏𝜔²/4)(max_{0≤𝑡≤𝜔}(1/𝑟(𝑡)))<1. Thus (2.58) leads us to max_{0≤𝑡≤𝜔}|𝑧(𝑡)|=0, which is contrary to 𝑥₁≠𝑥₂. So (1.3) has at most one 𝜔-periodic solution. The proof is complete.

We now turn to the proof of Theorem 2.1. Suppose 𝜔²𝛿(max_{0≤𝑡≤𝜔}(1/𝑟(𝑡)))<1. Let 𝐿,𝑁,𝑃, and 𝑄 be defined by (2.3), (2.4), (2.6), and (2.7), respectively. By Lemma 2.10, there are positive constants 𝐷₀ and 𝐷₁ such that for any 𝜔-periodic solution 𝑥(𝑡) of (2.24) such that (2.25) holds. Set

Ω={𝑥∈𝑋_𝜔∣‖𝑥‖₁<𝐷}, (2.60) where 𝐷 is a fixed number which satisfies 𝐷>𝐷+𝐷₀+𝐷₁. It is easy to see that Ω is an open and bounded subset of 𝑋_𝜔. Furthermore, in view of Lemmas 2.5 and 2.7, 𝐿 is a Fredholm mapping of index zero and 𝑁 is 𝐿-compact on Ω. Noting that 𝐷>𝐷₀+𝐷₁, by Lemma 2.10, for each 𝜆∈(0,1) and 𝑥∈𝜕Ω,𝐿𝑥≠𝜆𝑁𝑥. Next note that a function 𝑥∈𝜕Ω∩Ker𝐿 must be constant: 𝑥(𝑡)≡𝐷 or 𝑥(𝑡)≡−𝐷. Hence by (i) and (2.17), 𝑥(𝑡)≡−𝐷. Hence by conditions (i), (iii) and (2.17),

𝑡𝑄𝑁𝑥(𝑡)=−𝜔^𝜔₀𝑡𝑓(𝑠,𝑥([𝑠]))𝑑𝑠=−𝜔^𝜔₀𝑓(𝑠,𝑥)𝑑𝑠, (2.61) so 𝑄𝑁𝑥≠𝜃₂. The isomorphism 𝐽∶Im𝑄→Ker𝐿 is defined by 𝐽(𝑡𝛼)=𝛼 for 𝛼∈𝑅 and 𝑡∈𝑅. Then

1𝐽𝑄𝑁𝑥=−𝜔^𝜔₀1𝑓(𝑠,𝑥)𝑑𝑠𝜔≠0. (2.62) In particular, we see that if 𝑥=𝐷, then

1𝐽𝑄𝑁𝑥=−𝜔^𝜔₀𝑓(𝑠,𝐷)𝑑𝑠<0, (2.63) and if 𝑥=−𝐷, then 1𝐽𝑄𝑁𝑥=−𝜔^𝜔₀𝑓(𝑠,−𝐷)𝑑𝑠>0. (2.64) Consider the mapping

𝐻(𝑥,𝜇)=𝜇𝑥+(1−𝜇)𝐽𝑄𝑁𝑥,0≤𝜇≤1. (2.65) From (2.63) and (2.65), for each 𝜇∈[0,1] and 𝑥=𝐷, we have

𝐻(𝑥,𝜇)=𝜇𝐷+(1−𝜇)−1𝜔^𝜔₀𝑓(𝑠,𝐷)𝑑𝑠<0. (2.66) Similarly, from (2.64) and (2.65), for each 𝜇∈[0,1] and 𝑥=−𝐷, we have

𝐻(𝑥,𝜇)=𝜇𝐷+(1−𝜇)−1𝜔^𝜔₀𝑓(𝑠,−𝐷)𝑑𝑠<0. (2.67) By (2.66) and (2.67), 𝐻(𝑥,𝜇) is a homotopy. This shows that

deg(𝐽𝑄𝑁𝑥,Ω∩Ker𝐿,𝜃₁)=deg(−𝑥,Ω∩Ker𝐿,𝜃₁)≠0. (2.68) By Theorem A, we see that equation 𝐿𝑥=𝑁𝑥 has at least one solution in Ω∩Dom𝐿. In other words, (1.3) has an 𝜔-periodic solution 𝑥(𝑡). Furthermore, if (iii) is satisfied, from Lemma 2.11, we know that (1.3) has an 𝜔-periodic solution only. The proof is complete.

3. Example

Consider the equation

(𝑥^′(𝑡)exp(−2−cos2𝜋𝑡5))^′+(3−sin2𝜋𝑡5)arctan𝑥([𝑡])=cos2𝜋𝑡5, (3.1) and we can show that it has a nontrivial 5-periodic solution. Indeed, take

𝑟(𝑡)=exp(2−cos2𝜋𝑡5),𝑝(𝑡)=cos2𝜋𝑡5,1𝑓(𝑡,𝑥)=100(3−sin2𝜋𝑡5)arctan𝑥. (3.2) We see that min_0≤𝑡≤5𝑟(𝑡)=𝑒. Let 𝐷>0 and 𝛿=𝑏=1/25. Then condition (i) of Theorem 2.1 is satisfied:

lim_{𝑥→−∞}max_{0≤𝑡≤𝜔}𝑓(𝑡,𝑥)𝑥=1.25 (3.3) Let 𝐷>0 and 𝛿=𝑏=1/25. Then conditions (i), (ii) and (iii), of Theorem 2.1 are satisfied. Note further that 5²𝛿(max_{0≤𝑡≤𝜔}(1/𝑟(𝑡)))=𝑒⁻¹<1. Therefore (3.1) has exactly one 5-periodic solution. Furthermore, it is easy to see that any solution of (3.1) must be nontrivial. We have thus shown the existence of a unique nontrivial 5-periodic solution of (3.1).

Acknowledgment

The first author is supported by the Natural Science Foundation of Guangdong Province of China under Grant no. 9151008002000012.