Periodic Solutions of a Type of Liénard Higher Order Delay Functional Differential Equation with Complex Deviating Argument

Abstract

The author has studied the existence of periodic solutions of a type of higher order delay functional differential equations with neutral type by using the theory of coincidence degree, and some new sufficient conditions for the existence of periodic solutions have been obtained.

1. Introduction and Lemma

With the rapid development of modern science and technology, functional differential equation with time delay has been widely applied in many areas such as bioengineering, systems analysis, and dynamics. Functional differential equation with complex deviating argument is an important type of the above function. Because the property of the solution to this kind of equation is impossibly estimated, so the literature on the functional differential equation with complex argument is relatively rare [1]. In recent years, with the maturity of the theory of nonlinear functional analysis and algebraic topology, we have the powerful tools of the study on the functional differential equation with complex deviating argument, so it is possible to study the above equation. Furthermore, the study on the periodic solutions of functional differential equation is always one of the most important subject that people concerned for its widespread use. Many results of the study of Duffing-typed functional differential equation and Liénard-typed functional differential equation have been obtained, for example, the literatures [2–18]. Hitherto, the literature of the discussion of higher order functional differential equations has not been found a lot [19]. In this paper I have studied and derived some sufficient conditions that guarantee the existence of periodic solutions for a type of higher order functional differential equations with complex deviating argument as the following: ^𝑚_𝑖=1𝑎_𝑖𝑥^(𝑖)(𝑡)+𝑓(𝑥(𝑡))𝑥̇(𝑡)+𝛽(𝑡)𝑔(𝑥(𝑥(𝑡)))=𝑝(𝑡)𝑎_𝑖≠0, (∗) and some new results have been obtained.

In order to establish the existence of 𝑇-periodic solutions of (∗), we make some preparations.

Definition 1.1. Let 𝑋, 𝑌 are Banach spaces, and let Ω be an open and bounded subset in 𝑋, and let 𝐿∶Dom(𝐿)⊆𝑋→𝑌 be linear mapping; the mapping 𝐿 will be called a Fredholm mapping of index zero if dimker𝐿=codimIm𝐿<+∝ and Im𝐿 is closed in 𝑌.

Definition 1.2. Let 𝑃∶𝑋→ker(𝐿), let 𝑄∶𝑌→𝑌/Im(𝐿) be projectors, and let 𝑁∶Ω→𝑌 be nonlinear mapping; the mapping 𝑁 will be called 𝐿-compact on Ω if 𝑄𝑁∶Ω→𝑌/Im(𝐿) and (𝐿|_ker(𝑃))⁻¹(𝐼−𝑄)𝑁∶Ω→𝑋 are compact.

Lemma 1.3 (see [20]). Let 𝑋, 𝑌 be Banach spaces; 𝐿∶𝐷_𝐿⊂𝑋→𝑌 is a Fredholm mapping of index zero 𝑃∶𝑋→𝑋; 𝑄∶𝑋→𝑌 are continuous mapping projectors; Ω is an open bounded set in 𝑋; 𝑁∶Ω×[0,1]→𝑌 is 𝐿-Compact on Ω, furthermore suppose that:(a)𝐿𝑥≠𝜆𝑁(𝑥,𝜆),forall𝑥∈𝐷_𝐿∩𝜕Ω,𝜆∈(0,1); (b)𝑄𝑁(𝑥,0)≠0,forall𝑥∈ker(𝐿)∩𝜕Ω; (c)deg(𝑄𝑁(𝑥,0),ker(𝐿)∩Ω,0)≠0, then the equation 𝐿𝑥=𝑁(𝑥,1) has at least one solution on Ω, where deg is Brouwer degree.

2. Main Results and Proof of Theorems

Theorem 2.1. Suppose that 𝑓, 𝛽, 𝑔, 𝑝 are continuous for their variables, respectively, 𝑝(𝑡+𝑇)=𝑝(𝑡),  𝛽(𝑡+𝑇)=𝛽(𝑡)>0, ∫^𝑇₀𝑝(𝑡)𝑑𝑡=0, and furthermore suppose that(a)∃𝐴>0,forall𝑥∈ℝ, when |𝑥|>𝐴, such that 𝑥𝑔(𝑥)>0;(b)∃𝑀>0,forall𝑥∈ℝ, such that |𝑔(𝑥)|≤𝑀;(c)𝑓₁=sup_𝑥∈ℝ|𝑓(𝑥)|<(𝑎_𝑚−𝑘(𝑇^𝑚−1−𝑇^𝑚−2−⋯−𝑇))/𝑇^𝑚−1, where 𝑘=max{|𝑎_𝑖|}, 𝑖=1,2,…,𝑚−1 and 𝑎_𝑚>𝑘(𝑇^𝑚−1+𝑇^𝑚−2+⋯+𝑇), then (∗) has at least one 𝑇-periodic solution.

Proof of Theorem 2.1. In order to use continuation theorem to obtain 𝑇-periodic solution of (∗), we firstly make some required preparations. Let 𝑋=𝑥∈𝑐^𝑚−1(ℝ,ℝ)∣𝑥(𝑡+𝑇)=𝑥(𝑡),𝑌=𝑦∈𝐶(ℝ,ℝ)∣𝑦(𝑡+𝑇)=𝑦(𝑡), (2.1) and the norm of 𝑋 and 𝑌 is ‖𝑥‖=max_{0≤𝑖≤𝑚−1}{|𝑥^(𝑖)|_∞}, |𝑥^(𝑖)|_∞=max_𝑡∈ℝ{|𝑥^(𝑖)(𝑡)|}, 𝑖=1,2,…,𝑚−1, and ‖𝑦‖=max_𝑡∈ℝ{|𝑦(𝑡)|}, respectively; then the 𝑋 and 𝑌 with this norm are Banach spaces.
Firstly, we study the priori bound of 𝑇-periodic solution of following equation: ^𝑚_𝑖=1𝑎_𝑖𝑥^(𝑖)(𝑡)+𝜆𝑓(𝑥(𝑡))𝑥̇(𝑡)+𝜆𝛽(𝑡)𝑔(𝑥(𝑥(𝑡)))=𝜆²𝑝(𝑡). (2.2)
Suppose that 𝑥=𝑥(𝑡)∈𝑋 is an arbitrary 𝑇-periodic solution of (2.2), put 𝑥(𝑡) into, (2.2) and then integrate both sides of (2.2) on [0,𝑇], so ^𝑇₀𝛽(𝑡)𝑔(𝑥(𝑥(𝑡)))𝑑𝑡=0. (2.3)
For the continuity of 𝛽, 𝑔, 𝑥, there must exist a number 𝑡₀∈[0,𝑇] such that 𝛽𝑡₀𝑔𝑥𝑥𝑡₀=0, (2.4) that is, 𝑔𝑥𝑥𝑡₀=0. (2.5)
For the condition (a) of Theorem 2.1, we have 𝑥𝑥𝑡₀≤𝐴. (2.6)
Let 𝑥𝑡₀=𝑛𝑇−𝑡₁,𝑛∈𝑁,𝑡₁∈[0,𝑇], (2.7) so 𝑥𝑡₁=𝑥𝑥𝑡₀≤𝐴. (2.8)
In view of ∀𝑡∈[0,𝑇],𝑥(𝑡)=𝑥𝑡₁+^𝑡_𝑡1𝑥̇(𝑠)𝑑𝑠, (2.9) we have |𝑥(𝑡)|=𝑥𝑡₁+^𝑡_𝑡1𝑥̇(𝑠)𝑑𝑠≤𝐴+^𝑡_𝑡1|𝑥̇(𝑠)|𝑑𝑠≤𝐴+^𝑇₀|𝑥̇(𝑡)|𝑑𝑡, (2.10) that is, 𝑥⁽⁰⁾_∞=|𝑥|_∞≤𝐴+^𝑇₀|𝑥̇(𝑡)|𝑑𝑡. (2.11)
Noting 𝑥(𝑡)=𝑥(𝑡+𝑇), so there must exist the number 𝜉_𝑖∈[0,𝑇] such that 𝑥^(𝑖)(𝜉_𝑖)=0, where 𝑖=1,2,3,…,𝑚−1.
For all𝑡∈[0,𝑇], 𝑥^(𝑖)(𝑡)=𝑥^(𝑖)𝜉_𝑖+^𝑡_𝜉𝑖𝑥^(𝑖+1)(𝑠)𝑑𝑠=^𝑡_𝜉𝑖𝑥^(𝑖+1)(𝑠)𝑑𝑠, (2.12)
we have 𝑥^(𝑖)(𝑡)=^𝑡_𝜉𝑖𝑥^(𝑖+1)(𝑠)𝑑𝑠≤^𝑇₀𝑥^(𝑖+1)(𝑡)𝑑𝑡≤𝑇^𝑇₀𝑥^(𝑖+2)(𝑡)𝑑𝑡≤𝑇²⋅^𝑇₀𝑥^(𝑖+3)(𝑡)𝑑𝑡≤⋯≤𝑇^{𝑚−(𝑖+1)𝑇}₀𝑥^{(𝑖+𝑚−𝑖)}(𝑡)𝑑𝑡=𝑇^{𝑚−(𝑖+1)𝑇}₀𝑥^(𝑚)(𝑡)𝑑𝑡, (2.13) that is, 𝑥^(𝑖)_∞≤𝑇^{𝑚−(𝑖+1)𝑇}₀𝑥^(𝑚)(𝑡)𝑑𝑡,𝑖=1,2,…,𝑚−1. (2.14)
Combining (2.11), (2.14), we get 𝑥⁽⁰⁾_∞=|𝑥|_∞≤𝐴+𝑇^𝑚−1𝑇₀𝑥^(𝑚)(𝑡)𝑑𝑡. (2.15)
By (2.2), we get ^𝑇₀𝑎_𝑚𝑥^𝑚(𝑡)𝑑𝑡≤^𝑇₀𝜆𝑓(𝑥(𝑡))𝑥̇(𝑡)𝑑𝑡+^𝑇₀𝜆𝛽(𝑡)𝑔(𝑥(𝑥(𝑡)))𝑑𝑡+^𝑇₀𝜆²𝑝(𝑡+)𝑑𝑡_𝑇0𝑎₁𝑥̇(𝑡)𝑑𝑡+^𝑇₀𝑎₂𝑥̈(𝑡)𝑑𝑡+⋯+^𝑇₀𝑎_𝑚−3𝑥^(𝑚−3)(𝑡+)𝑑𝑡_𝑇0𝑎_𝑚−2𝑥^(𝑚−2)(𝑡)𝑑𝑡+⋯+^𝑇₀𝑎_𝑚−1𝑥^(𝑚−1)(𝑡)𝑑𝑡, (2.16) where 𝛽₁=max_𝑡∈ℝ𝛽(𝑡),  𝑝₁=max_𝑡∈ℝ{|𝑝(𝑡)|}, and 𝑘=max{|𝑎_𝑖|},𝑖=1,2,3,…,𝑚−1.
Noting (2.14) and the conditions (b), (c) of Theorem 2.1, we have ^𝑇₀𝑎_𝑚𝑥^(𝑚)(𝑡)𝑑𝑡≤𝑓₁𝑇⋅𝑇^𝑚−2𝑇₀𝑥^(𝑚)(𝑡)𝑑𝑡+𝛽₁𝑇𝑀+𝑝₁𝑇+𝑘𝑇⋅𝑇^{𝑚−(1+1)𝑇}₀𝑥^(𝑚)(𝑡)𝑑𝑡+𝑘𝑇⋅𝑇^{𝑚−(2+1)𝑇}₀𝑥^(𝑚)(𝑡)𝑑𝑡+⋯+𝑘𝑇⋅𝑇^{𝑚−(𝑚−1+1)𝑇}₀𝑥^(𝑚)(𝑡)𝑑𝑡, (2.17)
so 𝑎_𝑚^𝑇₀𝑥^(𝑚)(𝑡)𝑑𝑡≤𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑘𝑇+𝑓₁𝑇^𝑚−1𝑇₀𝑥^(𝑚)(𝑡)𝑑𝑡+𝛽₁𝑇𝑀+𝑝₁𝑇, (2.18) where 𝑎_𝑚>𝑇^𝑚−1∑^𝑚_𝑖=1𝑓_𝑖+𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑘𝑇.
Let 𝛽₁𝑇𝑀+𝑝₁𝑇𝑎₀−𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑘𝑇+𝑓₁𝑇^𝑚−1≜𝐴₁, (2.19) that is, ^𝑇₀𝑥^(𝑚)(𝑡)𝑑𝑡≤𝐴₁. (2.20)
Noting (2.14), (2.15), and (2.20), we have 𝑥⁽⁰⁾_∞=|𝑥|_∞≤𝐴+𝑇^𝑚−1𝐴₁≜𝜔₀,𝑥^(𝑖)_∞≤𝑇^{𝑚−(𝑖+1)}𝐴₁≜𝜔_𝑖,𝑖=1,2,…,𝑚−1. (2.21)
Let 𝜔=max_{0≤𝑖≤𝑚}{𝜔_𝑖+1}, and let Ω={𝑥∣𝑥∈𝑋∶‖𝑥‖<𝜔}; then Ω is an open and bounded set in 𝑋.
Let 𝐿∶𝐷_𝐿⊂𝑋→𝑌∶𝑥→𝐿𝑥=^𝑚_𝑖=1𝑎_𝑖𝑥^(𝑖)(𝑡),𝑁∶𝑋×𝐼→𝑌∶𝑥→𝑁(𝑥,𝜆)=−𝑓(𝑥(𝑡))𝑥̇(𝑡)−𝛽(𝑡)𝑔(𝑥(𝑥(𝑡)))+𝜆𝑝(𝑡); (2.22) then the corresponding equation of 𝐿𝑥=𝜆𝑁(𝑥,𝜆) is (2.2).
Now, we define projection operators as follows; (𝐿)1𝑃∶𝑋→ker∶𝑥→𝑃𝑥=𝑇^𝑇₀𝑥(𝑡)𝑌𝑑𝑡,𝑄∶𝑌→(𝐿)1Im∶𝑦→𝑄𝑦=𝑇^𝑇₀𝑦(𝑡)𝑑𝑡. (2.23)
Obviously, 𝑃, 𝑄 are continuous operators, Im(𝑃)=ℝ=ker(𝐿), ker(𝑄)=Im(𝐿), and it is easy to prove that 𝐿 is a Fredholm mapping of index zero and is 𝐿-Compact on Ω.
From the above discussion and the construction of Ω, we know that for all 𝑥∈𝐷_𝐿∩𝜕Ω, 𝜆∈(0,1), 𝐿𝑥≠𝜆𝑁(𝑥,𝜆), therefore the condition (a) of Lemma 1.3 holds.
For arbitrary 𝑥∈ker(𝐿)∩𝜕Ω, ‖𝑥‖=𝜔, by the definition of 𝑄, 𝑁, we have ()=1𝑄𝑁𝑥,0𝑇^𝑇₀(𝑥(𝑡𝑥̇(𝑡)(𝑡)𝑔(𝑥(𝑥(𝑡1−𝑓))−𝛽)))𝑑𝑡=−𝑇^𝑇₀𝛽(𝑡)𝑔(𝑥(𝑥(𝑡)))𝑑𝑡, (2.24) so ()1𝑥𝑄𝑁𝑥,0=−𝑇𝑥^𝑇₀𝛽(𝑡)𝑔(𝑥(𝑥(𝑡1)))𝑑𝑡=−𝑇𝑥𝑔(𝑥)^𝑇₀𝛽(𝑡)𝑑𝑡≠0, (2.25) therefore the condition (b) of Lemma 1.3 holds.
Making a transformation. 𝐻𝑥,𝜇=−𝜇𝑥+1−𝜇𝑄𝑁(𝑥,0),∀𝑥∈𝜕Ω∩ker(𝐿),𝜇∈[0,1], (2.26) we have 𝑥𝐻𝑥,𝜇=−𝜇𝑥²+𝑥1−𝜇𝑄𝑁(𝑥,0)=−𝜇𝑥²−11−𝜇𝑇𝑔(𝑥)𝑥^𝑇₀𝛽(𝑡)𝑑𝑡<0. (2.27)
So 𝑥𝐻(𝑥,𝜇)≠0, that is, 𝐻(𝑥,𝜇)≠0 is a homotopy, deg(𝑄𝑁(𝑥,0),ker(𝐿)∩Ω,0) = deg(−𝐼,ker(𝐿)∩Ω,0)=deg(−𝐼,ℝ∩Ω,0)≠0, where 𝐼 is an identity mapping, and the condition (c) of Lemma 1.3 holds.
From above all, the requirements of Lemma 1.3 are all satisfied, so (∗) has at least one 𝑇-periodic solution under the condition of Theorem 2.1, so the proof of Theorem 2.1 is completed.

Remark 2.2. In Theorem 2.1, if 𝛽(𝑡)<0 and the condition (a) of Theorem 2.1 is when |𝑥|>𝐴, 𝑥𝑔(𝑥)<0, and the rest are unchangeable, then (∗) has at least one 𝑇-periodic solution.

If the 𝑔(𝑥) is not a bounded function, we have the following theorem.

Theorem 2.3. Suppose that 𝑓, 𝛽, 𝑔, 𝑝 are continuous for their variables, respectively, 𝑝(𝑡+𝑇)=𝑝(𝑡), 𝛽(𝑡+𝑇)=𝛽(𝑡)>0, ∫^𝑇₀𝑝(𝑡)𝑑𝑡=0, and furthermore suppose following:(a)∃𝐴>0,forall𝑥∈ℝ, when |𝑥|>𝐴, such that 𝑥𝑔(𝑥)>0;(b)∃𝑀>0,forall𝑥∈ℝ, such that |𝑔(𝑥)|≤𝑀|𝑥|+𝑐;(c)𝑓₁=sup_𝑥∈ℝ|𝑓(𝑥)|<(𝑎_𝑚−𝑘𝑇^𝑚−1−𝑘𝑇^𝑚−2−⋯−𝑘𝑇−𝛽₁𝑇^𝑚)/𝑇^𝑚−1, where 𝑘=max{|𝑎_𝑖|},𝑖=1,2,…,𝑚−1, and 𝑎_𝑚>𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑘𝑇+𝛽₁𝑇^𝑚, then (∗) has at least one 𝑇-periodic solution.

Proof of Theorem 2.3. Banach spaces 𝑋,  𝑌 and the mappings 𝐿, 𝑃, 𝑄, and 𝑁 are the same to Theorem 2.1, and their property are equal to Theorem 2.1, then the corresponding equation of 𝐿𝑥=𝜆𝑁(𝑥,𝜆) is ^𝑚_𝑖=1𝑎_𝑖𝑥^(𝑖)(𝑡)+𝜆𝑓(𝑥(𝑡))𝑥̇(𝑡)+𝜆𝛽(𝑡)𝑔(𝑥(𝑥(𝑡)))=𝜆²𝑝(𝑡). (2.28)
It is similar to Theorem 2.1, there must exist a number 𝑡₁∈[0,𝑇], such that 𝑥𝑡₁≤𝐴, (2.29) and it is easy to obtain 𝑥^(𝑖)_∞≤𝑇^{𝑚−(𝑖+1)𝑇}₀𝑥^(𝑚)(𝑡𝑥)𝑑𝑡,𝑖=1,2,…,𝑚−1,_(0)∞=|𝑥|_∞≤𝐴+𝑇^𝑚−1𝑇₀𝑥^(𝑚)(𝑡)𝑑𝑡. (2.30)
Noting (2.28), (2.30) and the conditions (b), (c) of Theorem 2.3, we have ^𝑇₀𝑎_𝑚𝑥^(𝑚)𝑑𝑡≤^𝑇₀𝜆𝑓(𝑥(𝑡))𝑥̇(𝑡)𝑑𝑡+^𝑇₀𝜆𝛽(𝑡)𝑔(𝑥(𝑥(𝑡)))𝑑𝑡+^𝑇₀𝜆²𝑝(𝑡)𝑑𝑡+𝑘𝑇⋅𝑇^{𝑚−(1+1)𝑇}₀𝑥^𝑚(𝑡)𝑑𝑡+𝑘𝑇⋅𝑇^{𝑚−(2+1)𝑇}₀𝑥^𝑚(𝑡)𝑑𝑡+⋯+𝑘𝑇⋅𝑇^{𝑚−(𝑚−1+1)𝑇}₀𝑥^𝑚(𝑡)𝑑𝑡≤𝑓₁𝑇⋅𝑇^𝑚−2𝑇₀𝑥^𝑚(𝑡)𝑑𝑡+𝛽₁𝑇[𝑀|𝑥(𝑥(𝑡))|+𝑐]+𝑝₁𝑇+𝑘𝑇⋅𝑇^{𝑚−(1+1)𝑇}₀𝑥^𝑚(𝑡)𝑑𝑡+𝑘𝑇⋅𝑇^{𝑚−(2+1)𝑇}₀𝑥^𝑚(𝑡)𝑑𝑡+⋯+𝑘𝑇⋅𝑇^{𝑚−(𝑚−1+1)𝑇}₀𝑥^𝑚(𝑡)𝑑𝑡. (2.31)
So 𝑎_𝑚^𝑇₀𝑥^𝑚(𝑡)𝑑𝑡≤𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑘𝑇+𝑓₁𝑇^𝑚−1𝑇₀𝑥^𝑚(𝑡)𝑑𝑡+𝛽₁𝑇𝑐+𝛽₁𝑇𝑀|𝑥|_∝+𝑝₁𝑇≤𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑘𝑇+𝑓₁𝑇^𝑚−1𝑇₀𝑥^𝑚(𝑡)𝑑𝑡+𝛽₁𝑇𝑐+𝛽₁𝑇𝑀𝐴+𝑇^𝑚−1𝑇₀𝑥^(𝑚)(𝑡)𝑑𝑡+𝑝₁𝑇≤𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑘𝑇+𝑓₁𝑇^𝑚−1+𝛽₁𝑇^𝑚𝑇₀𝑥^𝑚(𝑡)𝑑𝑡+𝛽₁𝑇𝑐+𝛽₁𝑇𝑀𝐴+𝑝₁𝑇, (2.32) where 𝑘=max{|𝑎_𝑖|},𝑖=1,2,3,…,𝑚−1, and 𝑎_𝑚>𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑘𝑇+𝑓₁𝑇^𝑚−1+𝛽₁𝑇^𝑚.
Let 𝛽₁𝑇𝑐+𝛽₁𝑇𝑀𝐴+𝑝₁𝑇𝑎_𝑚−𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑘𝑇+𝑓₁𝑇^𝑚−1+𝛽₁𝑇^𝑚≜𝐴₁, (2.33) that is, ^𝑇₀𝑥^𝑚(𝑡)𝑑𝑡≤𝐴₁. (2.34)
Noting (2.30) and (2.34), we have 𝑥⁽⁰⁾_∞=|𝑥|_∞≤𝐴+𝑇^𝑚−1𝐴₁≜𝜔₀,𝑥^(𝑖)_∞≤𝑇^{𝑚−(𝑖+1)}𝐴₁≜𝜔_𝑖,𝑖=1,2,…,𝑚−1. (2.35)
Let 𝜔=max_{0≤𝑖≤𝑚}{𝜔_𝑖+1}, and we take Ω={𝑥∣𝑥∈𝑋∶‖𝑥‖<𝜔}; then Ω is an open and bounded set in 𝑋.
Similarly to Theorem 2.1, we prove easily that 𝐿 is a Fredholm mapping of index zero and 𝑁 is 𝐿-compact on Ω and the conditions (a), (b), and (c) of Lemma 1.3 hold.
From above all, the requirements of Lemma 1.3 are all satisfied, so (∗) has at least one 𝑇-periodic solution under the condition of Theorem 2.3, so far the proof of Theorem 2.3 is completed.

Remark 2.4. In Theorem 2.3, if 𝛽(𝑡)<0 and the condition (a) of Theorem 2.3 is when |𝑥|>𝐴, 𝑥𝑔(𝑥)<0, and the rest are unchangeable, then (∗) has at least one 𝑇-periodic solution.

If the ∫^𝑇₀𝑝(𝑡)𝑑𝑡≠0, we have the following theorem.

Theorem 2.5. Suppose that 𝑓, 𝛽, 𝑔, 𝑝 are continuous for their variables, respectively, 𝛽(𝑡+𝑇)=𝛽(𝑡)>0, and meet the condition (𝑎) of Theorem 2.1 and furthermore suppose as follows:(a)lim_{|𝑥|→+∝}|𝑔(𝑥)|=+∝;(b)∃𝑎,𝑏,𝑐>0, such that |𝑔(𝑥)|≤𝑎𝑔(𝑥)+𝑏|𝑥|+𝑐;(c)𝑓₁=sup_𝑥∈ℝ|𝑓(𝑥)|(𝑎_𝑚−𝑘𝑇^𝑚−1−𝑘𝑇^𝑚−2−⋯−𝑘𝑇−𝑓₁𝑇^𝑚−1−𝑏𝛽₁𝑇^𝑚)/𝑇^𝑚−1, where 𝑘=max{|𝑎_𝑖|},𝑖=1,2,…,𝑚−1, and 𝑎_𝑚>𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑘𝑇+𝑓₁𝑇^𝑚−1+𝑏𝛽₁𝑇^𝑚, then (∗) has at least one 𝑇-periodic solution.

Proof of Theorem 2.5. Banach spaces 𝑋,  𝑌 and the mappings 𝐿, 𝑃, 𝑄, and 𝑁 are the same to Theorem 2.1, and their property are equal to Theorem 2.1, then the corresponding equation of 𝐿𝑥=𝜆𝑁(𝑥,𝜆) is ^𝑚_𝑖=1𝑎_𝑖𝑥^(𝑖)(𝑡)+𝜆𝑓(𝑥(𝑡))𝑥̇(𝑡)+𝜆𝛽(𝑡)𝑔(𝑥(𝑥(𝑡)))=𝜆²𝑝(𝑡). (2.36)
Suppose that 𝑥=𝑥(𝑡)∈𝑋 is an arbitrary 𝑇-periodic solution of (2.36), put 𝑥(𝑡) into (2.36), and then integrate both sides of (2.36) on [0,𝑇], so ^𝑇₀𝛽(𝑡)𝑔(𝑥(𝑥(𝑡)))𝑑𝑡=^𝑇₀𝜆𝑝(𝑡)𝑑𝑡. (2.37)
For the continuity of 𝛽, 𝑔, 𝑥, there must exist a number 𝑡₁∈[0,𝑇], such that 𝑔𝑥𝑥𝑡₁=𝜆∫^𝑇₀𝑝(𝑡)𝑑𝑡∫^𝑇₀𝛽(𝑡).𝑑𝑡 (2.38)
Combing the condition (a) of Theorem 2.5, there must exist 𝐴₁>0, such that 𝑥𝑥𝑡₁≤𝐴₁. (2.39)
Similarly to Theorem 2.1, we have 𝑥^(𝑖)_∞≤𝑇^{𝑚−(𝑖+1)𝑇}₀𝑥^(𝑚)(𝑡)𝑑𝑡,𝑖=1,2,…,𝑚−1,𝑥⁽⁰⁾_∞=|𝑥|_∞≤𝐴₁+𝑇^𝑚−1𝑇₀𝑥^(𝑚)(𝑡)𝑑𝑡. (2.40)(2.41)
By (2.36), (2.37), (2.39), and (2.41) and the conditions (b), (c) of Theorem 2.5, we have ^𝑇₀𝑎_𝑚𝑥^(𝑚)𝑑𝑡≤^𝑇₀𝜆𝑓(𝑥(𝑡))𝑥̇(𝑡)𝑑𝑡+^𝑇₀𝜆𝛽(𝑡)𝑔(𝑥(𝑥(𝑡)))𝑑𝑡+^𝑇₀𝜆²𝑝(𝑡)𝑑𝑡+𝑘𝑇⋅𝑇^{𝑚−(1+1)𝑇}₀𝑥^𝑚(𝑡)𝑑𝑡+𝑘𝑇⋅𝑇^{𝑚−(2+1)𝑇}₀𝑥^𝑚(𝑡)𝑑𝑡+⋯+𝑘𝑡⋅𝑇^{𝑚−(𝑚−1+1)𝑇}₀𝑥^𝑚(𝑡)𝑑𝑡≤𝑓₁𝑇|𝑥̇|_∝+𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑎_𝑚−1𝑇^𝑇₀𝑥^𝑚(𝑡+)𝑑𝑡_𝑇0𝑎𝛽(𝑡)𝑔(𝑥(𝑥(𝑡)))𝑑𝑡+^𝑇₀𝑏𝛽(𝑡)[|𝑥(𝑥(𝑡))|+𝑐]𝑑𝑡+𝑝₁𝑇≤𝑓₁𝑇𝑇^𝑚−2𝑇₀𝑥^(𝑚)(𝑡)𝑑𝑡+𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑘𝑇^𝑇₀𝑥^𝑚(𝑡)𝑑𝑡+𝑎𝛽₁𝑇𝑝₁+𝑏𝛽₁𝑇|𝑥|_∝+𝑏𝛽₁𝑇𝑐+𝑝₁𝑇≤𝑓₁𝑇^𝑚−1𝑇₀𝑥^(𝑚)(𝑡+)𝑑𝑡𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑘𝑇^𝑇₀𝑥^𝑚(𝑡)𝑑𝑡+𝑏𝛽₁𝑇𝐴₁+𝑇^𝑚−1𝑇₀𝑥^(𝑚)(𝑡)𝑑𝑡+𝑎𝛽₁𝑇𝑝₁+𝑏𝛽₁𝑇𝑐+𝑝₁𝑇≤𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑘𝑇+𝑓₁𝑇^𝑚−1+𝑏𝛽₁𝑇^𝑚𝑇₀𝑥^𝑚(𝑡)𝑑𝑡+𝑏𝛽₁𝑇𝐴₁+𝑎𝛽₁𝑇𝑝₁+𝑏𝛽₁𝑇𝑐+𝑝₁𝑇. (2.42)
So 𝑎_𝑚−𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑘𝑇+𝑓₁𝑇^𝑚−1+𝑏𝛽₁𝑇^𝑚𝑇₀𝑥^𝑚(𝑡)𝑑𝑡≤𝑏𝛽₁𝑇𝐴₁+𝑎𝛽₁𝑇𝑝₁+𝑏𝛽₁𝑇𝑐+𝑝₁𝑇. (2.43)
Let 𝑏𝛽₁𝑇𝐴₁+𝑎𝛽₁𝑇𝑝₁+𝑏𝛽₁𝑇𝑐+𝑝₁𝑇𝑎_𝑚−𝑘𝑇^𝑚−1+𝑘𝑇^𝑚−2+⋯+𝑘𝑇+𝑓₁𝑇^𝑚−1+𝑏𝛽₁𝑇^𝑚≜𝐴₂, (2.44) that is, ^𝑇₀𝑥^𝑚(𝑡)𝑑𝑡≤𝐴₂. (2.45)
Noting (2.40), (2.41), and (2.45), we have 𝑥⁽⁰⁾_∞=|𝑥|_∞≤𝐴+𝑇^𝑚−1𝐴₂≜𝜔₀,𝑥^(𝑖)_∞≤𝑇^{𝑚−(𝑖+1)}𝐴₂≜𝜔_𝑖,𝑖=1,2,…,𝑚−1. (2.46)
For condition (a), there exist 𝑀₀>0 and 𝐴₀> 0, such that |𝑥|>𝑀₀,|𝑔(𝑥)|>𝐴₀; let 𝜔=max_{0≤𝑖≤𝑚}{𝜔_𝑖+1,𝑀₀}, and we take Ω={𝑥∣𝑥∈𝑋∶‖𝑥‖<𝜔}; then Ω is an open and bounded set in 𝑋.
Similarly to Theorem 2.1, we prove easily that 𝐿 is a Fredholm mapping of index zero and 𝑁 is 𝐿-compact on Ω and the conditions (a), (b), and (c) of Lemma 1.3 hold.
From above all, the requirements of Lemma 1.3 are all satisfied, so (∗) has at least one 𝑇-periodic solution under the condition of Theorem 2.5, so the proof of Theorem 2.5 is completed.

Remark 2.6. In Theorem 2.5, if 𝛽(𝑡)<0 and the condition (a) of Theorem 2.1 is when |𝑥|>𝐴, 𝑥𝑔(𝑥)<0, and the rest are unchangeable, then (∗) has at least one 𝑇-periodic solution.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (11101305).