Periodic Solutions and Nontrivial Periodic Solutions for a Class of Rayleigh-Type Equation with Two Deviating Arguments

Abstract

The Rayleigh equation with two deviating arguments 𝑥^′′(𝑡)+𝑓(𝑥^′(𝑡))+𝑔₁(𝑡,𝑥(𝑡−𝜏₁(𝑡)))+𝑔₂(𝑡,𝑥(𝑡−𝜏₂(𝑡)))=𝑒(𝑡) is studied. By using Leray-Schauder index theorem and Leray-Schauder fixed point theorem, we obtain some new results on the existence of periodic solutions, especially for the existence of nontrivial periodic solutions to this equation. The results are illustrated with two examples, which cannot be handled using the existing results.

1. Introduction

Consider the Rayleigh equation with two deviating arguments in the form of 𝑥^′′(𝑡)+𝑓𝑥^′(𝑡)+𝑔₁𝑡,𝑥𝑡−𝜏₁(𝑡)+𝑔₂𝑡,𝑥𝑡−𝜏₂(𝑡)=𝑒(𝑡), (1) where 𝑓∈𝐶(𝑅,𝑅),𝑔_𝑖∈𝐶(𝑅×𝑅,𝑅),𝑖=1,2,𝑒,𝜏_𝑖()𝑔∈𝐶𝑅,𝑅,𝑖=1,2,_𝑖(𝑡+𝑇,𝑥)=𝑔_𝑖(𝑡,𝑥),𝜏_𝑖(𝑡+𝑇)=𝜏_𝑖(𝑡)𝑒()(𝑡).,𝑖=1,2,𝑡+𝑇=𝑒 (2)

The dynamic behaviors of Rayleigh equation have been widely investigated due to their applications in many fields such as physics, mechanics, and the engineering technique fields. For example, an excess voltage of ferroresonance, known as some kind of nonlinear resonance having long duration that arises from the magnetic saturation of inductance in an oscillating circuit of a power system, and a boosted excess voltage can give rise to some problems in relay protection. To probe this mechanism, a mathematical model was proposed in [1–3], which is a special case of the Rayleigh equation with two delays. This implies that (1) can represent analog voltage transmission. In a mechanical problem, 𝑓 usually represents a damping or friction term, 𝑔_𝑖 represents the restoring force, 𝑒 is an externally applied force, and 𝜏_𝑖 is the time lag of the restoring force (see [4]). Some other examples in practical problems concerning physics and engineering technique fields can be found in [5–7].

At the same time, the periodic solutions for Rayleigh equations with two deviating arguments have been studied by authors [8–10] under the assumption of 𝑓(0)=0or𝑓(𝑡,0)=0. (3)

It is not difficult to see that if 𝑔₁(𝑡,0)+𝑔₂(𝑡,0)≢𝑒(𝑡), then the periodic solution obtained in [8–10] must be nontrivial. But if 𝑔₁(𝑡,0)+𝑔₂(𝑡,0)≡𝑒(𝑡), then the periodic solution obtained in [8–10] may be trivial under the assumption of 𝑓(0)=0 or 𝑓(𝑡,0)=0. And if the periodic solution is unique, then it must be trivial. Thus, it is worth discussing the existence of the nontrivial periodic solutions of Rayleigh equations with two deviating arguments in this case.

The main purpose of this paper is to establish sufficient conditions for the existence of periodic solution, especially for the existence of nontrivial periodic solutions of (1) by using the Leray-Schauder index theory. We remark that our methods are different from those used in [8–10] to some degree. In particular, two examples are also given to illustrate the effectiveness of our results.

For ease of exposition, throughout this paper, we assume that 𝑇>0.

On the other hand, the following assumptions are used in this paper.(𝐻₁)𝑓∈𝐶(𝑅,𝑅), 𝑔_𝑖∈𝐶(𝑅²,𝑅), 𝜏_𝑖∈𝐶(𝑅,𝑅), 𝑖=1,2, 𝑒∈𝐶(𝑅,𝑅), 𝑔_𝑖(𝑡+𝑇,𝑥)=𝑔_𝑖(𝑡,𝑥), 𝜏_𝑖(𝑡+𝑇)=𝜏_𝑖(𝑡), 𝑖=1,2, and 𝑒(𝑡+𝑇)=𝑒(𝑡).(𝐻₂)𝑓(0)=0, and there exists 𝛾>0 such that 𝑥𝑓(𝑥)≥𝛾|𝑥|², for all 𝑥∈𝑅 (or 𝑥𝑓(𝑥)≤−𝛾|𝑥|², for all 𝑥∈𝑅).(𝐻₃)𝑔_𝑖 is differentiable with respect to 𝑡, and there exist 𝑎_𝑖≥0, 𝑏_𝑖>0, 𝑖=1,2, such that 𝑔^′_𝑖𝑡(𝑡,𝑥)≤𝑎_𝑖+𝑏_𝑖|𝑥|,∀(𝑡,𝑥)∈𝑅²,𝑖=1,2. (4)(𝐻₄) There exist 𝑙_𝑖>0 such that |𝑔_𝑖(𝑡,𝑥₁)−𝑔_𝑖(𝑡,𝑥₂)|≤𝑙_𝑖|𝑥₁−𝑥₂|, for all 𝑡∈𝑅, 𝑥₁,𝑥₂∈𝑅, 𝑖=1,2.(𝐻₅) There exists 𝑑>0, such that for |𝑥|>𝑑, we have 𝑔₁(𝑡,𝑥)+𝑔₂(𝑡,𝑥)−𝑒(𝑡)sign𝑥>0,∀𝑡∈𝑅. (5)(𝐻₆) There exist integers 𝑛_𝑖 such that 𝛿_𝑖≜max_{𝑡∈[0,𝑇]}|𝜏_𝑖(𝑡)−𝑛_𝑖𝑇|≤𝑇, 𝑖=1,2.

The main results of the present paper are as follows.

Theorem 1. If (𝐻₁)–(𝐻₆) hold, and (𝑏₁+𝑏₂)𝛾⁻¹𝑇²+2√2(𝑙₁𝛿₁+𝑙₂𝛿₂)𝜋²𝛾⁻¹<2𝜋², then (1) has at least one periodic solution.

The proof of Theorem 1 will be given in Section 3.

Remark 2. If 𝑔₁(𝑡,0)+𝑔₂(𝑡,0)≢𝑒(𝑡), then the periodic solution obtained by Theorem 1 must be nontrivial. If 𝑔₁(𝑡,0)+𝑔₂(𝑡,0)≡𝑒(𝑡), we could not conclude whether or not the periodic solution is nontrivial.
So we give the following conditions:(𝐻₇)𝑔^′_𝑖𝑥(𝑡,𝑥), 𝑖=1,2 exist and are continuous, 𝑔^′_1𝑥(𝑡,0)+𝑔^′_2𝑥(𝑡,0)≡const.<0.(𝐻₈)lim_𝑥→0(𝑓(𝑥)/𝑥)=𝑟≥𝛾.

Theorem 3. If (𝐻₁)–(𝐻₈) hold, and 𝑔₁(𝑡,0)+𝑔₂(𝑡,0)≡𝑒(𝑡), (𝑏₁+𝑏₂)𝛾⁻¹𝑇²+2√2(𝑙₁𝛿₁+𝑙₂𝛿₂)𝜋²𝛾⁻¹<2𝜋², then (1) has at least one nontrivial periodic solution.

The proof of Theorem 3 will be given in Section 4.

2. Preliminaries

In this section, to establish the periodic solutions of (1), we provide some background definitions and some well-known results, which are crucial in our arguments.

Let 𝑋 be a real Banach space, and let 𝐴∶𝑋→𝑋 be a completely continuous operator.

Definition 4 (see [11, 12]). If 𝑥₀ is an isolated fixed point of 𝐴, then the fixed point index at 𝑥₀ of 𝐴 is defined by 𝑖𝐴,𝑥₀=deg(𝐼−𝐴,Ω,𝜃), (6) where Ω is a neighborhood of 𝑥₀, which satisfies that 𝑥₀ is the unique fixed point in Ω of 𝐴.

Definition 5 (see [11, 12]). If there exits 𝑥₀∈𝑋 with 𝑥₀≠𝜃 such that 𝐴𝑥₀=𝜆𝑥₀, 𝜆∈𝑅, then 𝜆 is called an eigenvalue of operator 𝐴 and 𝑥₀ is called the eigenfunction of operator 𝐴 corresponding to 𝜆.

Definition 6. Let 𝑢∶𝑅→𝑅 be continuous. 𝑢(𝑡) is said to be periodic on 𝑅 if 𝑢(𝑡+𝑇)=𝑢(𝑡),∀𝑡∈𝑅. (7)

Lemma 7 (see [11, 12], index theorem of Leray-Schauder). Suppose that 𝑥₀ is a fixed point of 𝐴, 𝐴 is Fréchet differentiable at 𝑥₀, and 1 is not the eigenvalue of 𝐴^′(𝑥₀). Then 𝑥₀ is an isolated fixed point of 𝐴, and 𝑖𝐴,𝑥₀=𝑖𝐴^′𝑥₀,0=(−1)^𝛽, (8) where 𝛽 is equal to the sum of the algebraic multiplicities of all of the eigenvalues 𝜆_𝑖>1 of 𝐴^′(𝑥₀).

Lemma 8 (see [11, 12], fixed point theorem of Leray-Schauder). Let 𝑋 be a real Banach space and 𝐴∶𝑋→𝑋 be a completely continuous operator. If {𝑥∶𝑥∈𝑋,𝑥=𝜆𝐴𝑥,0<𝜆<1} (9) is bounded, then 𝐴 has a fixed point 𝑥^∗∈Ω, where {‖𝑥‖},{}.Ω=𝑥∶𝑥∈𝑋,≤𝑙𝑙=sup𝑥∶𝑥∈𝑋,𝑥=𝜆𝐴𝑥,0<𝜆<1 (10)

Lemma 9. Suppose 𝑥(𝑡)∈𝐶¹[0,𝑇] and 𝑥(0)=𝑥(𝑇)=0. Then ∫^𝑇₀|𝑥(𝑡)|²𝑑𝑡≤(𝑇²/𝜋²)∫^𝑇₀|𝑥^′(𝑡)|²𝑑𝑡.

Lemma 10 (see [13]). Let 0≤𝛼≤𝑇 be constant, 𝑠∈𝐶(𝑅,𝑅) be periodic with period 𝑇, and max_{𝑡∈[0,𝑇]}|𝑠(𝑡)|≤𝛼. Then for any 𝑢∈𝐶¹(𝑅,𝑅) which is periodic with period 𝑇, we have ^𝑇₀|𝑢(𝑡)−𝑢(𝑡−𝑠(𝑡))|²𝑑𝑡≤2𝛼^2𝑇₀𝑢^′(𝑡)²𝑑𝑡. (11)

3. Proof of Theorem 1

In this section, we will use Lemma 8 to prove Theorem 1.

Proof. Let 𝑋={𝑥∶𝑥∈𝐶(𝑅,𝑅),𝑥(𝑡+𝑇)=𝑥(𝑡)},𝑌=𝑥∶𝑥∈𝐶¹(𝑅,𝑅),𝑥(𝑡+𝑇)=𝑥(𝑡). (12)
Then 𝑋 and 𝑌 are real Banach spaces endowed with the norm ‖𝑥‖_∞=max_{[]𝑡∈0,𝑇}|𝑥(𝑡,‖𝑥‖=‖𝑥‖)|_∞+𝑥^′_∞, (13) respectively.
Choosing 𝑚>0 with 𝑚≠(2𝑘𝜋/𝑇)²(𝑘=1,2,…), then 𝑥^′′(𝑡)+𝑚𝑥(𝑡)=0 (14) has only trivial solution in 𝑌.
In fact, it is easy to see that the general solution of 𝑥^′′(𝑡)+𝑚𝑥(𝑡)=0 is 𝑥(𝑡)=𝑐₁sin√𝑚𝑡+𝑐₂cos√𝑚𝑡. (15)
By the periodic boundary conditions we obtain that 𝑥=0 is its unique solution in 𝑌. Then for ℎ∈𝑋, −𝑥^′′(𝑡)−𝑚𝑥(𝑡)=ℎ(𝑡) (16) has unique solution 𝑥∈𝑌. Writing 𝑥=𝐾ℎ, then 𝐾∶𝑋→𝑌 is a completely continuous operator.
Define an operator 𝐺∶𝑌→𝑋 by (𝐺𝑥)(𝑡)=𝑓𝑥^′(𝑡)+𝑔₁𝑡,𝑥𝑡−𝜏₁(𝑡)+𝑔₂𝑡,𝑥𝑡−𝜏₂(𝑡)−𝑚𝑥(𝑡)−𝑒(𝑡),𝑥∈𝑌. (17)
Then 𝐺∶𝑌→𝑋 is continuous and bounded. Let 𝐴=𝐾𝐺∶𝑌→𝑌. Then 𝐴 is also a completely continuous operator. By Lemma 8, if {𝑥∶𝑥∈𝑌,𝑥=𝜆𝐴𝑥,0<𝜆<1} (18) is bounded in 𝑌, then 𝐴 has a fixed point in 𝑌. Thus, (1) has periodic solution.
Now suppose that 𝑥∈𝑌, 0<𝜆<1 satisfying 𝑥=𝜆𝐴𝑥. Then 𝑥(𝑡) is a solution of 𝑥^′′(𝑡)+𝜆𝑓𝑥^′(𝑡)+𝜆𝑔₁𝑡,𝑥𝑡−𝜏₁(𝑡)+𝜆𝑔₂𝑡,𝑥𝑡−𝜏₂(𝑡)+()(𝑡)(𝑡).1−𝜆𝑚𝑥=𝜆𝑒 (19)
Let 𝑡, 𝑡 be the maximum point and minimum point of 𝑥(𝑡) on [0,𝑇], respectively. Then 𝑥^′𝑡=0,𝑥^′′𝑡≤0,𝑥^′𝑡=0,𝑥^′′𝑡≥0. (20)
Noticing 𝑓(0)=0, we have 𝜆𝑔₁𝑡,𝑥𝑡−𝜏₁𝑡+𝜆𝑔₂𝑡,𝑥𝑡−𝜏₂𝑡+(1−𝜆)𝑚𝑥𝑡−𝜆𝑒𝑡≥0,𝜆𝑔₁𝑡,𝑥𝑡−𝜏₁𝑡+𝜆𝑔₂𝑡,𝑥𝑡−𝜏₂𝑡+(1−𝜆)𝑚𝑥𝑡−𝜆𝑒𝑡≤0, (21) and hence, there exists 𝜉∈[𝑡,𝑡] (or [𝑡,𝑡]) such that 𝜆𝑔₁𝜉,𝑥𝜉−𝜏₁(𝜉)+𝜆𝑔₂𝜉,𝑥𝜉−𝜏₂(𝜉)−𝜆𝑒(𝜉)+(1−𝜆)𝑚𝑥(𝜉)=0, (22) which implies that 𝜆𝑔₁𝜉,𝑥𝜉−𝜏₁(𝜉)+𝑔₂𝜉,𝑥𝜉−𝜏₂(𝜉)(𝜉(𝜉)+()𝑚𝑥(𝜉−𝑒)×sign𝑥1−𝜆)=0. (23)
From (𝐻₅) we know that |𝑥(𝜉)|≤𝑑 (see [10]). So we have |𝑥(𝑡)|=𝑥(𝜉)+^𝑡_𝜉𝑥^′(𝑠)𝑑𝑠≤𝑑+√𝑇𝑥^′_𝐿2, (24) and then, ‖𝑥‖_∞≤𝑑+√𝑇𝑥^′_𝐿2, (25) where ‖⋅‖_𝐿² is the norm of 𝐿²[0,𝑇].
Multiplying (19) with 𝑥^′(𝑡) and integrating from 0 to 𝑇, then we have 𝜆^𝑇₀𝑓𝑥^′(𝑡)𝑥^′(𝑡)𝑑𝑡=−𝜆^𝑇₀𝑔₁𝑡,𝑥𝑡−𝜏₁(𝑡)𝑥^′(𝑡)𝑑𝑡+𝜆^𝑇₀𝑔₂𝑡,𝑥𝑡−𝜏₂(𝑡)𝑥^′(𝑡)𝑑𝑡+𝜆^𝑇₀𝑒(𝑡)𝑥^′(𝑡)𝑑𝑡. (26)
By (𝐻₂) we know that ^𝑇₀𝑓𝑥^′(𝑡)𝑥^′(𝑡)𝑑𝑡≥𝛾^𝑇₀𝑥^′(𝑡)²𝑑𝑡. (27)
By Hölder's inequality, from (26) and (27) we have 𝛾^𝑇₀𝑥^′2(𝑡)≤𝑑𝑡_𝑇0𝑔₁𝑡,𝑥𝑡−𝜏₁(𝑡)𝑥^′(𝑡)+𝑑𝑡_𝑇0𝑔₂𝑡,𝑥𝑡−𝜏₂(𝑡)𝑥^′(𝑡)𝑑𝑡+‖𝑒‖_𝐿²𝑥^′_𝐿2≤^𝑇₀𝑔₁𝑡,𝑥𝑡−𝜏₁(𝑡)−𝑔₁(𝑡,𝑥(𝑡))𝑥^′(𝑡+)𝑑𝑡_𝑇0𝑔₁(𝑡,𝑥(𝑡))𝑥^′(𝑡)𝑑𝑡+^𝑇₀𝑔₂𝑡,𝑥𝑡−𝜏₂(𝑡)−𝑔₂(𝑡,𝑥(𝑡))𝑥^′(𝑡+)𝑑𝑡_𝑇0𝑔₂(𝑡,𝑥(𝑡))𝑥^′(𝑡)𝑑𝑡+‖𝑒‖_𝐿²𝑥^′_𝐿2. (28)
Since ∫₀^𝑥(𝑡)𝑔_𝑖(𝑡,𝑣)𝑑𝑣, 𝑖=1,2 are 𝑇-periodic, differentiable, and 𝑑𝑑𝑡₀^𝑥(𝑡)𝑔_𝑖(𝑡,𝑣)𝑑𝑣=𝑔_𝑖(𝑡,𝑥(𝑡))𝑥^′(𝑡)+₀^𝑥(𝑡)𝑔^′_𝑖𝑡(𝑡,𝑣)𝑑𝑣,𝑖=1,2, (29) we have ^𝑇₀𝑔_𝑖(𝑡,𝑥(𝑡))𝑥^′(𝑡)𝑑𝑡=−^𝑇₀𝑑𝑡₀^𝑥(𝑡)𝑔^′_𝑖𝑡(𝑡,𝑣)𝑑𝑣,𝑖=1,2. (30)
Combining (28) and (30) with (𝐻₃) and (𝐻₄) we obtain 𝛾^𝑇₀𝑥^′2(𝑡)𝑑𝑡≤𝑙₁^𝑇₀𝑥(𝑡)−𝑥𝑡−𝜏₁(𝑡)𝑥^′(𝑡)𝑑𝑡+𝑙₂^𝑇₀𝑥(𝑡)−𝑥𝑡−𝜏₂(𝑡)𝑥^′(𝑡+)𝑑𝑡_𝑇0𝑑𝑡₀^|𝑥(𝑡)|𝑎₁+𝑏₁|𝑣|+𝑑𝑣_𝑇0𝑑𝑡₀^|𝑥(𝑡)|𝑎₂+𝑏₂|𝑣|𝑑𝑣+‖𝑒‖_𝐿²𝑥^′_𝐿2≤𝑙₁𝑥^′_𝐿2^𝑇₀𝑥(𝑡)−𝑥𝑡−𝜏₁(𝑡)−𝑛₁𝑇²𝑑𝑡^1/2+𝑙₂𝑥^′_𝐿2^𝑇₀𝑥(𝑡)−𝑥𝑡−𝜏₂(𝑡)−𝑛₂𝑇²𝑑𝑡^1/2+𝑎₁+𝑎₂^𝑇₀|𝑥(𝑡𝑏)|𝑑𝑡+¹+𝑏₂2^𝑇₀|𝑥(𝑡)|²+‖𝑒‖𝑑𝑡_𝐿2𝑥^′_𝐿2. (31)
By Lemma 9 we have ^𝑇₀|𝑥(𝑡)|²𝑑𝑡=_𝜉^𝜉+𝑇|𝑥(𝑡)|²𝑑𝑡=_𝜉^𝜉+𝑇𝑥(𝑡)−𝑥(𝜉)²(𝜉)𝑑𝑡+2𝑥_𝜉^𝜉+𝑇𝑥(𝑡)𝑑𝑡−𝑥(𝜉)²𝑇≤𝑇²𝜋²_𝜉^𝜉+𝑇𝑥^′(𝑡)²𝑑𝑡+2𝑑^𝑇₀|𝑥(𝑡−𝑥(𝜉)|𝑑𝑡)²𝑇≤𝑇²𝜋^2𝑇₀𝑥^′(𝑡)²𝑑𝑡+2𝑑^𝑇₀|𝑥(𝑡)|𝑑𝑡. (32)
By (32) and Hölder's inequality we have ^𝑇₀|𝑥(𝑡≤√)|𝑑𝑡𝑇^𝑇₀|𝑥(𝑡)|²𝑑𝑡^1/2≤√𝑇𝑇²𝜋^2𝑇₀𝑥^′(𝑡)²𝑑𝑡+2𝑑^𝑇₀|𝑥(𝑡)|𝑑𝑡^1/2≤𝑇^3/2𝜋^𝑇₀𝑥^′(𝑡)²𝑑𝑡^1/2+√2𝑑𝑇^𝑇₀|𝑥(𝑡)|𝑑𝑡^1/2≤𝑇^3/2𝜋^𝑇₀𝑥^′(𝑡)²𝑑𝑡^1/2+14^𝑇₀|𝑥(𝑡)|𝑑𝑡+2𝑑𝑇. (33)
From (33) we have ^𝑇₀|𝑥(𝑡)|𝑑𝑡≤4𝑇^3/23𝜋^𝑇₀𝑥^′(𝑡)²𝑑𝑡^1/2+8𝑑𝑇3. (34)
By (𝐻₆) and Lemma 10 we have ^𝑇₀𝑥(𝑡)−𝑥𝑡−𝜏_𝑖(𝑡)−𝑛_𝑖𝑇²𝑑𝑡^1/2≤√2𝛿_𝑖^𝑇₀𝑥^′(𝑡)²𝑑𝑡^1/2,𝑖=1,2. (35)
Thus, it follows from (31), (32), (34), and (35) that 𝛾𝑥^′2_𝐿2≤√2𝑙₁𝛿₁+√2𝑙₂𝛿₂+𝑏₁+𝑏₂𝑇²2𝜋²𝑥^′2_𝐿2+4𝑇^3/2𝑎3𝜋¹+𝑎₂+𝑏₁+𝑏₂𝑑+‖𝑒‖_𝐿²𝑥^′_𝐿2+8𝑑𝑇3𝑎₁+𝑎₂+𝑏₁+𝑏₂𝑑. (36)
Combining this with (𝑏₁+𝑏₂)𝛾⁻¹𝑇²+2√2(𝑙₁𝛿₁+𝑙₂𝛿₂)𝜋²𝛾⁻¹<2𝜋², we know that there exists 𝑐₁ such that ‖𝑥^′‖_𝐿²≤𝑐₁. Then ‖𝑥‖_∞≤𝑑+√𝑇𝑐₁≜𝑀₁. (37)
Multiplying (19) with 𝑥^′′(𝑡) and integrating from 0 to 𝑇, then we have 𝑥^′′2_𝐿2≤−𝜆^𝑇₀𝑔₁𝑡,𝑥𝑡−𝜏₁(𝑡)𝑥^′′(𝑡)𝑑𝑡−𝜆^𝑇₀𝑔₂𝑡,𝑥𝑡−𝜏₂(𝑡)𝑥^′′(𝑡)−()𝑚𝑑𝑡1−𝜆^𝑇₀𝑥(𝑡)𝑥^′′(𝑡)𝑑𝑡+𝜆^𝑇₀𝑒(𝑡)𝑥^′′(𝑡)≤𝑔𝑑𝑡_1𝑀1+𝑔_2𝑀1√𝑇𝑥^′′_𝐿2+𝑚𝑀₁√𝑇𝑥^′′_𝐿2+‖𝑒‖_𝐿²𝑥^′′_𝐿2, (38) where 𝑔_1𝑀1=max_{[],‖𝑥‖𝑡∈0,𝑇∞≤𝑀1}𝑔₁((𝑡,𝑔𝑡,𝑥))_2𝑀1=max_{[],‖𝑥‖𝑡∈0,𝑇∞≤𝑀1}𝑔₂(𝑡,𝑥(𝑡)). (39)
Thus, 𝑥^′′_𝐿2≤𝑔_1𝑀1+𝑔_2𝑀1√𝑇+𝑚𝑀₁√𝑇+‖𝑒‖_𝐿²≜𝑀₂. (40)
Selecting 𝜂∈[0,𝑇] such that 𝑥^′(𝜂)=0, then we have 𝑥^′(𝑡)≤^𝑇₀𝑥^′′(𝑡)𝑑𝑡≤√𝑇𝑀₂. (41)
Thus, from (35) and (41) we know that ‖𝑥‖≤𝑀₁+√𝑇𝑀₂≜𝑀. It follows that {𝑥∶𝑥∈𝑌,𝑥=𝜆𝐴𝑥,0<𝜆<1} (42) is bounded. Therefore, by Lemma 8, we obtain that 𝐴 has a fixed point 𝑥^∗∈Ω, where Ω={𝑥∶𝑥∈𝑌,‖𝑥‖≤𝑀}, and hence, it follows that (1) has a periodic solution.

4. Proof of Theorem 3

In this section, we will use Lemma 7 to prove Theorem 3.

Proof. Let 𝑋,𝑌,𝐴 be defined and same as in Section 3. We prove that 𝐴 is Fréchet differentiable at 0 and𝐴^′(0)𝑥(𝑡)=𝐾𝑟𝑥^′(𝑡)+𝑔^′_1𝑥(𝑡,0)+𝑔^′_2𝑥(𝑡,0)𝑥(𝑡)−𝑚𝑥(𝑡). (43)
In fact, by 𝑔₁(𝑡,0)+𝑔₂(𝑡,0)≡𝑒(𝑡), we have 𝐴0=0. Let (𝐵𝑥)(𝑡)=𝐾𝑟𝑥^′(𝑡)+𝑔^′_1𝑥(𝑡,0)+𝑔^′_2𝑥(𝑡,0)𝑥(𝑡)−𝑚𝑥(𝑡). (44)
We have ‖‖=𝐾𝑓𝑥𝐴𝑥−𝐵𝑥_′(𝑡)+𝑔₁𝑡,𝑥𝑡−𝜏₁(𝑡)+𝑔₂𝑡,𝑥𝑡−𝜏₂(𝑡)−𝑚𝑥(𝑡)−𝑒(𝑡)−𝐾𝑟𝑥^′(𝑡)+𝑔^′_1𝑥(𝑡,0)+𝑔^′_2𝑥()𝑥(𝑡)(𝑡≤‖𝐾‖𝑓𝑥𝑡,0−𝑚𝑥)_′(𝑡)𝑥^′(𝑡)−𝑟𝑥^′(𝑡)+𝑔₁𝑡,𝑥𝑡−𝜏₁(𝑡)+𝑔₂𝑡,𝑥𝑡−𝜏₂(𝑡)(𝑡)−𝑔−𝑒_′1𝑥(𝑡,0)+𝑔^′_2𝑥()𝑥(𝑡)≤‖𝐾‖𝑓𝑥𝑡,0_′(𝑡)𝑥^′(𝑡)−𝑟+𝑔^′_1𝑥𝑡,𝜃₁𝑥𝑡−𝜏₁(𝑡)−𝑔^′_1𝑥(𝑡,0)+𝑔^′_2𝑥𝑡,𝜃₂𝑥𝑡−𝜏₂(𝑡)−𝑔^′_2𝑥(𝑡,0)‖𝑥‖, (45) where 𝜃_𝑖=𝜃_𝑖(𝑡)∈(0,1), 𝑖=1,2. Thus, lim_{‖𝑥‖→0}‖𝐴𝑥−𝐴0−𝐵𝑥‖‖𝑥‖=0, (46) which implies that 𝐴^′(0) exists and 𝐴^′(0)=𝐵.
Now, we prove that 1 is not the eigenvalue of 𝐴^′(0).
In fact, if there exists 𝑥∈𝑌 satisfying 𝐴^′(0)𝑥=𝑥, then 𝑥 is a solution of −𝑥^′′(𝑡)−𝑚𝑥(𝑡)=𝑟𝑥^′(𝑡)+𝑔^′_1𝑥(𝑡,0)+𝑔^′_2𝑥()𝑥(𝑡)(𝑡).𝑡,0−𝑚𝑥 (47)
Integrating (47) from 0 to 𝑇, we obtain ^𝑇₀𝑔^′_1𝑥(𝑡,0)+𝑔^′_2𝑥(𝑡,0)𝑥(𝑡)𝑑𝑡=0. (48)
By (𝐻₇), there exists 𝑡₀∈[0,𝑇] such that 𝑥(𝑡₀)=0.
Multiplying (47) with 𝑥^′(𝑡) and integrating it from 0 to 𝑇, we have ^𝑇₀𝑟𝑥^′2𝑑𝑡=−^𝑇₀𝑔^′_1𝑥(𝑡,0)+𝑔^′_2𝑥(𝑡,0)𝑥(𝑡)𝑥^′(𝑡)𝑑𝑡. (49)
By (𝐻₇) we have ^𝑇₀𝑟𝑥^′2𝑑𝑡=0, (50) which implies that 𝑥^′(𝑡)≡0, and hence 𝑥(𝑡)≡0. It follows that 1 is not the eigenvalue of 𝐴^′(0). By Lemma 7 we have 𝑖(𝐴,0)=𝑖𝐴^′(0),0=(−1)^𝛽. (51)
Next we prove that 𝛽=1. We can prove that−𝑥^′′(𝑡)−𝑚𝑥(𝑡)=𝜆𝑟𝑥^′(𝑡)+𝑔^′_1𝑥(𝑡,0)+𝑔^′_2𝑥(𝑡,0)𝑥(𝑡)−𝑚𝑥(𝑡) (52) has only one eigenvalue in (0, 1) and the algebraic multiplicity of it is 1.
In fact, it is easy to see that 𝜆∈(0,1) is the eigenvalue of (52) if and only if 𝑚+𝜆(𝑔^′_1𝑥(𝑡,0)+𝑔^′_2𝑥(𝑡,0)−𝑚)=0, that is, 𝜆=𝜆₀=𝑚𝑚−𝑔^′_1𝑥(𝑡,0)−𝑔^′_2𝑥()∈().𝑡,00,1 (53)
In this case, (52) degenerates to 𝑥^′′(𝑡)+𝑚𝑚−𝑔^′_1𝑥(𝑡,0)−𝑔^′_2𝑥(𝑡,0)𝑟𝑥^′(𝑡)=0. (54)
It is not difficult to see that all the solutions of (54) are 𝑥(𝑡)≡const. Therefore the geometric multiplicity of 𝜆₀=𝑚/(𝑚−𝑔^′_1𝑥(𝑡,0)−𝑔^′_2𝑥(𝑡,0)) is 1.
If (𝐼−𝜆₀𝐴^′(0))²𝑥(𝑡)=0, then (𝐼−𝜆₀𝐴^′(0))𝑥(𝑡)=𝑐=const, and hence, 𝑥(𝑡) satisfies −𝑥^′′(𝑡)−𝑚𝑥(𝑡)=𝜆₀𝑟𝑥^′(𝑡)+𝑔^′_1𝑥(𝑡,0)+𝑔^′_2𝑥(𝑡,0)𝑥(𝑡)−𝑚𝑥(𝑡)+𝑚𝑐, (55) which implies that −𝑥^′′(𝑡)=𝜆₀𝑟𝑥^′(𝑡)+𝑚𝑐. (56)
Integrating (56) from 0 to 𝑇, we obtain 𝑐=0, which implies that 𝑥(𝑡)≡const. It follows that the algebraic multiplicity of 𝜆₀=𝑚/(𝑚−𝑔^′_1𝑥(𝑡,0)−𝑔^′_2𝑥(𝑡,0)) is also 1, and therefore, 𝛽=1. So 𝑖(𝐴,0)=−1. Letting 𝛿>0 small enough, then we have deg𝐼−𝐴,𝐵_𝛿,0=−1. (57)
On the other hand, by the proof of Theorem 1, we know that there exists 𝑀>0 such that if 𝑥∈𝑌, 0<𝜆<1 satisfying 𝑥=𝜆𝐴𝑥, then ‖𝑥‖≤𝑀. Suppose 𝐴𝑥≠𝑥 for 𝑥∈𝑌,‖𝑥‖=𝑀. By the homotopy invariance of topological degree, we have deg𝐼−𝐴,𝐵_𝑀,0=1, (58) where 𝐵_𝑀={𝑥∈𝑌∶‖𝑥‖<𝑀}. It follows from (57) and (58) that deg𝐼−𝐴,𝐵_𝑀⧵𝐵_𝛿,0=2, (59) which implies that 𝐴 has a fixed point 𝑥 in 𝐵_𝑀⧵𝐵_𝛿 and 𝑥 is a nontrivial periodic solution of (1).

5. Two Examples

In this section, we give two examples to demonstrate the results obtained in previous sections.

Example 1. Consider the forced Rayleigh-type equation with period 2𝜋: 𝑥^′′(𝑡)+𝑓𝑥^′(𝑡)+𝑔₁𝑡,𝑥𝑡−𝜏₁(𝑡)+𝑔₂𝑡,𝑥𝑡−𝜏₂(𝑡)=𝑒(𝑡), (60) where 𝑓(𝑥)=𝑒^𝑥−1,𝑥≥0,1−𝑒^−𝑥𝑔,𝑥≤0,₁()=1𝑡,𝑥9sin²()𝑔𝑡𝑥𝑡−𝜃cos𝑡+cos𝑡,₂()=1𝑡,𝑥9cos²()𝑒(𝑡)𝑡𝑥𝑡−𝜃sin𝑡+sin𝑡,=sin𝑡,𝜏₁(𝑡)𝜏=𝜃cos𝑡,₂(𝑡)=𝜃sin𝑡,𝜃∈(0,1). (61)(62)

Conclusion. Equation (60) has at least one periodic solution with period 2𝜋.

Proof. By (61) and (62), it is not difficult to see that the condition (𝐻₁) holds, 𝑇=2𝜋, and 𝑓(0𝑔)=0,_′1𝑡(=1𝑡,𝑥)9≤1𝑥sin2𝑡−sin𝑡9|𝑥|+1,∀(𝑡,𝑥)∈𝑅²,𝑔^′_2𝑡(=−1𝑡,𝑥)9≤1𝑥sin2𝑡−cos𝑡9|𝑥|+1,∀(𝑡,𝑥)∈𝑅². (63)
On the other hand, let 𝑑=9, 𝛾=1, 𝛿_𝑖=𝜃, 𝑙_𝑖=1/9, 𝑏_𝑖=1/9, and 𝑖=1,2. If 𝜃∈(0,√2/2), then 𝑥𝑓(𝑥)≥|𝑥|²𝑏,∀𝑥∈𝑅,₁+𝑏₂𝛾⁻¹𝑇²+2√2𝑙₁𝛿₁+𝑙₂𝛿₂𝜋²𝛾⁻¹<2𝜋². (64)
Hence, (𝐻₁)–(𝐻₆) are satisfied. Thus, by Theorem 1, (60) has at least one periodic solution with period 2𝜋.

Example 2. If we replace 𝑔₁(𝑡,𝑥), 𝑔₂(𝑡,𝑥), and 𝑒(𝑡) in Example 1 by𝑔₁𝑡,𝑥𝑡−𝜏₁(𝑡=1)9sin²𝑡cos𝑥𝑡−𝜏₁(𝑡+1)2+1𝑥10𝑡−𝜏₁(𝑡)1sin2,∀(𝑡,𝑥)∈𝑅²,𝑔₂𝑡,𝑥𝑡−𝜏₂(𝑡)=19cos²𝑡cos𝑥𝑡−𝜏₂(𝑡)+12+sin𝑡,∀(𝑡,𝑥)∈𝑅², (65)𝑒(𝑡)=sin𝑡+(1/9)cos(1/2), then we can obtain that 𝑔₁(𝑡,0)+𝑔₂(𝑡,0)≡𝑒(𝑡), for all 𝑡∈𝑅, and 𝑔^′_1𝑡()=1𝑡,𝑥91sin2𝑡cos𝑥+2,𝑔^′_2𝑡()1𝑡,𝑥=−91sin2𝑡cos𝑥+2𝑔+cos𝑡,_′1𝑡(≤1𝑡,𝑥)9|𝑥|+1,𝑔18_′2𝑡(≤1𝑡,𝑥)9|𝑥|+37()18,∀𝑡,𝑥∈𝑅²,𝑔^′_1𝑥()1𝑡,𝑥=−9sin²1𝑡sin𝑥+2+1110sin2,𝑔^′_2𝑥()1𝑡,𝑥=−9cos²1𝑡sin𝑥+2𝑔,∀𝑡∈𝑅,_′1𝑥(𝑡,0)+𝑔^′_2𝑥()1𝑡,0=−91sin2+1110sin2<0,∀𝑡∈𝑅. (66)
Let 𝑑=(10/(9sin(1/2)))+((10sin(1/2))/(9cos(1/2))), 𝛾=1, 𝛿_𝑖=𝜃, 𝑖=1,2, 𝑙₁=19/90, 𝑙₂=1/9, 𝑏₁=1/9, and 𝑏₂=1/9. If 𝜃∈(0,1); then 𝑥𝑓(𝑥)≥|𝑥|²𝑏,∀𝑥∈𝑅,₁+𝑏₂𝛾⁻¹𝑇²+2√2𝑙₁𝛿₁+𝑙₂𝛿₂𝜋²𝛾⁻¹<2𝜋². (67)
Hence, (𝐻₁)–(𝐻₈) are satisfied. Thus, by Theorem 3, (60) has at least one nontrivial periodic solution with period 2𝜋.

Acknowledgments

This work is sponsored by the project NSFC (11171032) and Beijing Excellent Training Grant (2010D005007000002).