Periodic Solutions for Duffing Type 𝑝-Laplacian Equation with Multiple Constant Delays

Abstract

Using inequality techniques and coincidence degree theory, new results are provided concerning the existence and uniqueness of periodic solutions for the Duffing type 𝑝-Laplacian equation with multiple constant delays of the form (𝜑_𝑝(𝑥^′(𝑡)))^′+𝐶𝑥^′(𝑡)+𝑔₀(𝑡,𝑥(𝑡))+∑^𝑛_𝑘=1𝑔_𝑘(𝑡,𝑥(𝑡−𝜏_𝑘))=𝑒(𝑡). Moreover, an example is provided to illustrate the effectiveness of the results in this paper.

1. Introduction

Referring to the work of Esmailzadeh and Nakhaie-Jazar [1], Duffing type equation is the simplest case of a vibrating system with nonlinear restoring force generator element. This is equivalent to a mechanical vibrating system with either a hard or soft spring. Thus, this equation and its modifications have been extensively and intensively studied. In particular, the existence of periodic solutions for Duffing type equations with and without delays have been discussed by various researchers (see, e.g., [2–8] and the references given therein). However, to the best of our knowledge, the existence and uniqueness of periodic solutions of Duffing type 𝑝-Laplacian equation whose delays more than two have not been sufficiently researched. Motivated by this, we shall consider the Duffing type 𝑝-Laplacian equations with multiple constant delays of the form𝜑_𝑝𝑥^′(𝑡)^′+𝐶𝑥^′(𝑡)+𝑔₀(𝑡,𝑥(𝑡))+^𝑛_𝑘=1𝑔_𝑘𝑡,𝑥𝑡−𝜏_𝑘=𝑒(𝑡), (1.1) where 𝑝>1 and 𝜑_𝑝∶𝑅→𝑅 is given by 𝜑_𝑝(𝑠)=|𝑠|^𝑝−2𝑠 for 𝑠≠0 and 𝜑_𝑝(0)=0, 𝐶 and 𝜏_𝑘 are constants, 𝑒∶𝑅→𝑅 and 𝑔₀, 𝑔_𝑘∶𝑅×𝑅→𝑅 are continuous functions, 𝜏_𝑘 and 𝑒 are 𝑇-periodic, 𝑔₀ and 𝑔_𝑘 are 𝑇-periodic in the first argument, 𝑇>0 and 𝑘=1,2,…,𝑛. The main purpose of this paper is to establish sufficient conditions for the existence and uniqueness of 𝑇-periodic solutions of (1.1). The results of this paper are new and complement previously known results. Moreover, we give an example to illustrate the results.

2. Preliminary Results

Throughout this paper, we will denote𝐶¹_𝑇∶=𝑥∈𝐶¹(𝑅)∶𝑥is𝑇-periodic,|𝑥|_𝑘=^𝑇₀|𝑥(𝑡)|^𝑘𝑑𝑡^1/𝑘(𝑘>0),|𝑥|_∞=max_{[]𝑡∈0,𝑇}|𝑥(𝑡)|. (2.1)

For the periodic boundary value problem𝜑_𝑝𝑥^′(𝑡)^′=𝑓𝑡,𝑥,𝑥^′,𝑥(0)=𝑥(𝑇),𝑥^′(0)=𝑥^′(𝑇), (2.2) where 𝑓∈𝐶(𝑅³,𝑅) is 𝑇-periodic in the first variable, we have the following lemma.

Lemma 2.1 (see [9]). Let Ω be an open bounded set in 𝐶¹_𝑇, if the following conditions hold. (i)For each 𝜆∈(0,1) the problem 𝜑_𝑝𝑥^′(𝑡)^′=𝜆𝑓𝑡,𝑥,𝑥^′,𝑥(0)=𝑥(𝑇),𝑥^′(0)=𝑥^′(𝑇), (2.3) has no solution on 𝜕Ω.(ii)The equation 𝐹(𝑎)1∶=𝑇^𝑇₀𝑓(𝑡,𝑎,0)𝑑𝑡=0 (2.4) has no solution on 𝜕Ω⋂𝑅.(iii)The Brouwer degree of 𝐹deg𝐹,Ω𝑅,0≠0. (2.5) Then, the periodic boundary value problem (2.2) has at least one 𝑇-periodic solution on Ω.

We can easily obtain the homotopic equation of (1.1) as follows:𝜑_𝑝𝑥^′(𝑡)′+𝜆𝐶𝑥^′(𝑡)+𝜆𝑔₀(𝑡,𝑥(𝑡))+^𝑛_𝑘=1𝑔_𝑘𝑡,𝑥𝑡−𝜏_𝑘=𝜆𝑒(𝑡),𝜆∈(0,1). (2.6)

Lemma 2.2. Assume that the following conditions are satisfied.
(𝐴₁) There exists a constant 𝑑>0 such that(1)∑^𝑛_𝑘=0𝑔_𝑘(𝑡,𝑥_𝑘)−𝑒(𝑡)<0for𝑥_𝑘>𝑑,𝑡∈𝑅,𝑘=0,1,2,…,𝑛, (2)∑^𝑛_𝑘=0𝑔_𝑘(𝑡,𝑥_𝑘)−𝑒(𝑡)>0for𝑥_𝑘<−𝑑,𝑡∈𝑅,𝑘=0,1,2,…,𝑛.Moveover, if 𝑥(𝑡) is a 𝑇-periodic solution of (2.6), then |𝑥|_∞1≤𝑑+2√𝑇𝑥^′₂. (2.7)

Proof. Let 𝑥(𝑡) be a 𝑇-periodic solution of (2.6). Then, integrating (2.6) over [0,𝑇], we have ^𝑇₀𝑔₀(𝑡,𝑥(𝑡))+^𝑛_𝑘=1𝑔_𝑘𝑡,𝑥𝑡−𝜏_𝑘−𝑒(𝑡)𝑑𝑡=0. (2.8) Using the integral mean-value theorem, it follows that there exists 𝑡₁∈[0,𝑇] such that 𝑔₀𝑡₁,𝑥𝑡₁+^𝑛_𝑘=1𝑔_𝑘𝑡₁,𝑥𝑡₁−𝜏_𝑘−𝑒𝑡₁=0. (2.9) We now prove that there exists a constant 𝑡₂∈𝑅 such that 𝑥𝑡₂≤𝑑. (2.10) Indeed, suppose otherwise. Then, |𝑥(𝑡)|>𝑑∀𝑡∈𝑅. (2.11) Let 𝜏₀=0. From (𝐴₁), (2.9), and (2.11), we see that there exist 0≤𝑖,𝑗≤𝑛 such that 𝑥𝑡₁−𝜏_𝑖=max_{0≤𝑘≤𝑛}𝑥𝑡₁−𝜏_𝑘≥min_{0≤𝑘≤𝑛}𝑥𝑡₁−𝜏_𝑘=𝑥𝑡₁−𝜏_𝑗, (2.12) which, together with (2.11), implies −𝑑>𝑥𝑡₁−𝜏_𝑖=max_{0≤𝑘≤𝑛}𝑥𝑡₁−𝜏_𝑘or𝑥𝑡₁−𝜏_𝑗=min_{0≤𝑘≤𝑛}𝑥𝑡₁−𝜏_𝑘>𝑑. (2.13) Without loss of generality, we may assume that 𝑥(𝑡₁−𝜏_𝑗)>𝑑 (the situation is analogous for −𝑑>𝑥(𝑡₁−𝜏_𝑖)). Then, we have 𝑥𝑡₁−𝜏_𝑖≥𝑥𝑡₁−𝜏_𝑘≥𝑥𝑡₁−𝜏_𝑗>𝑑,𝑘=0,1,2,…,𝑛. (2.14) According to (2.14) and (𝐴₁), we obtain 0>^𝑛_𝑘=0𝑔_𝑘𝑡₁,𝑥𝑡₁−𝜏_𝑘−𝑒𝑡₁, (2.15) this contradicts the fact (2.9); thus, (2.10) is true.
Let 𝑡₂=𝑚𝑇+𝑡₀ where 𝑡₀∈[0,𝑇] and 𝑚 is an integer. Then, by the same approach used in the proof of inequality (3.3) of [7], we have |𝑥|_∞=max_{𝑡𝑡∈0,𝑡0+𝑇}|𝑥(𝑡)|≤max_{𝑡𝑡∈0,𝑡0+𝑇}1𝑑+2^𝑡_𝑡0𝑥^′(𝑠)𝑑𝑠+^𝑡0_𝑡−𝑇𝑥^′(𝑠1)𝑑𝑠≤𝑑+2√𝑇𝑥′₂. (2.16) This completes the proof of Lemma 2.2.

Lemma 2.3. Let (𝐴₁) holds. Assume that the following condition is satisfied.
(𝐴₂) There exist nonnegative constants 𝑏₀,𝑏₀,𝑏₁,𝑏₂,…,𝑏_𝑛 such that 𝑏₀𝑥₁−𝑥₂²≤−𝑔₀𝑡,𝑥₁−𝑔₀𝑡,𝑥₂𝑥₁−𝑥₂,𝑏₀>𝑏₁+𝑏₂+⋯+𝑏_𝑛,𝑔_𝑘𝑡,𝑥₁−𝑔_𝑘𝑡,𝑥₂≤𝑏_𝑘𝑥₁−𝑥₂, (2.17) for all 𝑡,𝑥₁,𝑥₂∈𝑅,𝑘=0,1,2,…,𝑛.
Then, (1.1) has at most one 𝑇-periodic solution.

Proof. Suppose that 𝑥₁(𝑡) and 𝑥₂(𝑡) are two 𝑇-periodic solutions of (1.1). Set 𝑍(𝑡)=𝑥₁(𝑡)−𝑥₂(𝑡). Then, we obtain 𝜑_𝑝𝑥^′₁(𝑡)−𝜑_𝑝𝑥^′₂(𝑡)′+𝐶𝑥^′₁(𝑡)−𝑥^′₂(𝑡)+𝑔₀𝑡,𝑥₁(𝑡)−𝑔₀𝑡,𝑥₂(𝑡)+^𝑛_𝑘=1𝑔_𝑘𝑡,𝑥₁𝑡−𝜏_𝑘−𝑔_𝑘𝑡,𝑥₂𝑡−𝜏_𝑘=0. (2.18) Multiplying 𝑍(𝑡) and (2.18) and then integrating it from 0 to T, from (𝐴₂) and Schwarz inequality, we get 𝑏₀|𝑍|²₂=𝑏₀^𝑇₀|𝑍(𝑡)|²𝑑𝑡≤−^𝑇₀𝑥₁(𝑡)−𝑥₂(𝑡)𝑔₀𝑡,𝑥₁(𝑡)−𝑔₀𝑡,𝑥₂(𝑡)𝑑𝑡=−^𝑇₀𝜑_𝑝𝑥^′₁(𝑡)−𝜑_𝑝𝑥^′₂(𝑡)𝑥^′₁(𝑡)−𝑥^′₂(𝑡)+𝑑𝑡_{𝑛𝑘=1𝑇0}𝑔_𝑘𝑡,𝑥₁𝑡−𝜏_𝑘−𝑔_𝑘𝑡,𝑥₂𝑡−𝜏_𝑘𝑍(𝑡)≤𝑑𝑡_𝑛𝑘=1𝑏_𝑘^𝑇₀𝑥₁𝑡−𝜏_𝑘−𝑥₂𝑡−𝜏_𝑘𝑍(𝑡≤|)|𝑑𝑡_𝑛𝑘=1𝑏_𝑘^𝑇₀𝑥₁𝑡−𝜏_𝑘−𝑥₂𝑡−𝜏_𝑘²𝑑𝑡^1/2|𝑍|₂=^𝑛_𝑘=1𝑏_𝑘|𝑍|²₂. (2.19) Since 𝑏₀>𝑏₁+𝑏₂+⋯+𝑏_𝑛, we have 𝑍(𝑡)≡0∀𝑡∈𝑅. (2.20) Thus, 𝑥₁(𝑡)≡𝑥₂(𝑡) for all 𝑡∈𝑅. Therefore, (1.1) has at most one 𝑇-periodic solution. The proof of Lemma 2.3 is now complete.

3. Main Results

Theorem 3.1. Let (𝐴₁) and (𝐴₂) hold. Then, (1.1) has a unique 𝑇-periodic solution in 𝐶¹_𝑇.

Proof. By Lemma 2.3, it is easy to see that (1.1) has at most one 𝑇-periodic solution in 𝐶¹_𝑇. Thus, in order to prove Theorem 3.1, it suffices to show that (1.1) has at least one 𝑇-periodic solution in 𝐶¹_𝑇. To do this, we are going to apply Lemma 2.1. Firstly, we claim that the set of all possible 𝑇-periodic solutions of (2.6) in 𝐶¹_𝑇 is bounded.
Let 𝑥(𝑡)∈𝐶¹_𝑇 be a 𝑇-periodic solution of (2.6). Multiplying 𝑥(𝑡) and (2.6) and then integrating it from 0 to T, we have −^𝑇₀𝜑_𝑝𝑥^′(𝑡)𝑥^′(𝑡)𝑑𝑡+𝜆^𝑇₀𝑥(𝑡)𝑔₀(𝑡,𝑥(𝑡))+^𝑛_𝑘=1𝑔_𝑘𝑡,𝑥𝑡−𝜏_𝑘−𝑒(𝑡)𝑑𝑡=0. (3.1) Since 𝑥(0)=𝑥(𝑇), then there exists 𝑡₀∈[0,𝑇] such that 𝑥′(𝑡₀)=0. And since 𝜑_𝑝(0)=0, we have 𝜑_𝑝𝑥^′(𝑡)=^𝑡_𝑡0𝜑_𝑝𝑥^′(𝑠)^′𝑑𝑠≤𝜆^𝑡0_𝑡^+𝑇₀𝑔₀(𝑡,𝑥(𝑡))+^𝑛_𝑘=1𝑔_𝑘𝑡,𝑥𝑡−𝜏_𝑘−𝑒(𝑡)𝑑𝑡, (3.2) where 𝑡∈[𝑡₀,𝑡₀+𝑇].
In view of (3.1), (𝐴₂), and Schwarz inequality, we get 𝑏₀|𝑥|²₂=𝑏₀^𝑇₀|𝑥(𝑡)|²𝑑𝑡≤−^𝑇₀(𝑥(𝑡)−0)𝑔₀(𝑡,𝑥(𝑡))−𝑔₀()1𝑡,0𝑑𝑡=−𝜆^𝑇₀𝜑_𝑝𝑥^′(𝑡)𝑥^′(𝑡)𝑑𝑡+^𝑛_𝑘=1^𝑇₀𝑔_𝑘𝑡,𝑥𝑡−𝜏_𝑘−𝑔_𝑘(𝑥(𝑡)+𝑡,0)𝑑𝑡_{𝑛𝑘=0𝑇0}𝑔_𝑘(𝑡,0)𝑥(𝑡)𝑑𝑡−^𝑇₀𝑥(𝑡)𝑒(𝑡)≤𝑑𝑡_𝑛𝑘=1𝑏_𝑘^𝑇₀𝑥𝑡−𝜏_𝑘|𝑥(𝑡)|𝑑𝑡+^𝑛_𝑘=0^𝑇₀𝑔_𝑘(𝑡,0)|𝑥(𝑡)|𝑑𝑡+√𝑇|𝑒|_∞|𝑥|₂≤^𝑛_𝑘=1𝑏_𝑘^𝑇₀|𝑥(𝑡−𝜏_𝑘)|²𝑑𝑡^1/2|𝑥|₂+√𝑇^𝑛_𝑘=0𝑔_𝑘(𝑡,0)_∞|𝑥|₂+√𝑇|𝑒|_∞|𝑥|₂=^𝑛_𝑘=1𝑏_𝑘|𝑥|²₂+√𝑇^𝑛_𝑘=0𝑔_𝑘(𝑡,0)_∞|𝑥|₂+√𝑇|𝑒|_∞|𝑥|₂. (3.3) It follows that |𝑥|₂≤√𝑇∑^𝑛_𝑘=0𝑔_𝑘(𝑡,0)_∞+√𝑇|𝑒|_∞𝑏₀−∑^𝑛_𝑘=1𝑏_𝑘∶=𝜃. (3.4)
Again from (𝐴₂) and Schwarz inequality, (3.2) and (3.4) yield 𝑥^′_∞^𝑝−1=max_{𝑡𝑡∈0,𝑡0+𝑇}𝜑_𝑝𝑥^′(𝑡)=max_{𝑡𝑡∈0,𝑡0+𝑇}^𝑡_𝑡0𝜑_𝑝𝑥^′(𝑠)^′≤𝑑𝑠_{𝑡0𝑡+𝑇0}𝑔₀(𝑡,𝑥(𝑡))+^𝑛_𝑘=1𝑔_𝑘𝑡,𝑥𝑡−𝜏_𝑘(𝑡=−𝑒)𝑑𝑡_𝑇0𝑔₀(𝑡,𝑥(𝑡))+^𝑛_𝑘=1𝑔_𝑘𝑡,𝑥𝑡−𝜏_𝑘(𝑡≤−𝑒)𝑑𝑡_𝑇0𝑔₀(𝑡,𝑥(𝑡))−𝑔₀(𝑡,0)𝑑𝑡+^𝑛_𝑘=1^𝑇₀𝑔_𝑘𝑡,𝑥𝑡−𝜏_𝑘−𝑔_𝑘(+𝑡,0)𝑑𝑡_{𝑛𝑘=0𝑇0}𝑔_𝑘(𝑡,0)𝑑𝑡+𝑇|𝑒|_∞≤𝑏₀^𝑇₀|𝑥(𝑡)|𝑑𝑡+^𝑛_𝑘=1^𝑇₀𝑏_𝑘𝑥𝑡−𝜏_𝑘𝑑𝑡+^𝑛_𝑘=0𝑇𝑔_𝑘(𝑡,0)_∞+𝑇|𝑒|_∞≤𝑏₀√𝑇|𝑥|₂+^𝑛_𝑘=1𝑏_𝑘√𝑇|𝑥|₂+^𝑛_𝑘=0𝑇𝑔_𝑘(𝑡,0)_∞+𝑇|𝑒|_∞≤𝑏₀√𝑇𝜃+^𝑛_𝑘=1𝑏_𝑘√𝑇𝜃+^𝑛_𝑘=0𝑇𝑔_𝑘(𝑡,0)_∞+𝑇|𝑒|_∞∶=𝜂, (3.5) which, together with (2.7), implies that there exists a positive constant 𝑀>1+(𝜂)^1/(𝑝−1) such that, for all 𝑡∈𝑅, 𝑥^′_∞<𝑀,|𝑥|_∞1≤𝑑+2√𝑇𝑥^′₂≤𝑑+12𝑇𝑥^′_∞<𝑀. (3.6) Set Ω=𝑥∈𝐶¹_𝑇∶|𝑥|_∞≤𝑀,𝑥^′_∞≤𝑀, (3.7) then we know that (2.6) has no 𝑇-periodic solution on 𝜕Ω as 𝜆∈(0,1) and when 𝑥(𝑡)∈𝜕Ω⋂𝑅,𝑥(𝑡)=𝑀 or 𝑥(𝑡)=−𝑀, from (𝐴₂), we can see that 1𝑇^𝑇₀−𝑔₀(𝑡,𝑀)−^𝑛_𝑘=1𝑔_𝑘()(𝑡1𝑡,𝑀+𝑒)𝑑𝑡>0,𝑇^𝑇₀−𝑔₀(𝑡,−𝑀)−^𝑛_𝑘=1𝑔_𝑘(𝑡,−𝑀)+𝑒(𝑡)𝑑𝑡<0, (3.8) so condition (ii) of Lemma 2.1 is also satisfied. Set 𝐻1𝑥,𝜇=𝜇𝑥−1−𝜇𝑇^𝑇₀𝑔₀(𝑡,𝑥)+^𝑛_𝑘=1𝑔_𝑘(𝑡,𝑥)−𝑒(𝑡)𝑑𝑡, (3.9) and when 𝑥∈𝜕Ω⋂𝑅,𝜇∈[0,1], we have 𝑥𝐻𝑥,𝜇=𝜇𝑥²−𝑥11−𝜇𝑇^𝑇₀𝑔₀(𝑡,𝑥)+^𝑛_𝑘=1𝑔_𝑘(𝑡,𝑥)−𝑒(𝑡)𝑑𝑡>0. (3.10) Thus, 𝐻(𝑥,𝜇) is a homotopic transformation and −1deg𝐹,Ω𝑅,0=deg𝑇^𝑇₀𝑔₀(𝑡,𝑥)+^𝑛_𝑘=1𝑔_𝑘(𝑡,𝑥)−𝑒(𝑡)𝑑𝑡,Ω𝑅,0=deg𝑥,Ω𝑅,0≠0, (3.11) so condition (iii) of Lemma 2.1 is satisfied. In view of the previous Lemma 2.1, (1.1) has at least one solution with period 𝑇. This completes the proof.

4. Example and Remark

Example 4.1. Let 𝑝=4,𝑔₀(𝑡,𝑥)=−10𝑒^20+sin𝑡𝑥,𝑔₁(𝑡,𝑥)=−2𝑒^2+sin𝑡sin𝑥, and 𝑔₂(𝑡,𝑥)=−3𝑒^3+cos𝑡cos𝑥 for all 𝑡,𝑥∈𝑅. Then, the following Liénard type 𝑝-Laplacian equation with two constant delays 𝜑_𝑝𝑥^′(𝑡)′+55𝑥^′(𝑡)+𝑔₀(𝑡,𝑥(𝑡))+𝑔₁(𝑡,𝑥(𝑡−1))+𝑔₂(𝑡,𝑥(𝑡−2))=cos𝑡 (4.1) has a unique 2𝜋-periodic solution.

Proof. From (4.1), it is straight forward to check that all the conditions needed in Theorem 3.1 are satisfied. Therefore, (4.1) has at least one 2𝜋-periodic solution.

Remark 4.2. Obviously, the results in [2–5] obtained on Duffing type 𝑝-Laplacian equation with single delay and without multiple delays cannot be applicable to (4.1). This implies that the results of this paper are essentially new.

Acknowledgments

The authors would like to express their sincere appreciation to the editor and anonymous referee for their valuable comments which have led to an improvement in the presentation of the paper. This work was supported by the construct program of the key discipline in Hunan province (Mechanical Design and Theory), the Scientific Research Fund of Hunan Provincial Natural Science Foundation of PR China (Grant no. 11JJ6006), the Natural Scientific Research Fund of Hunan Provincial Education Department of PR China (Grants no. 11C0916, 11C0915, 11C1186), the Natural Scientific Research Fund of Zhejiang Provincial of P.R. China (Grant no. Y6110436), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of P.R. China (Grant no. Z201122436).