Stability and Uniform Boundedness in Multidelay Functional Differential Equations of Third Order

Abstract

We consider a nonautonomous functional differential equation of the third order with multiple deviating arguments. Using the Lyapunov-Krasovskiì functional approach, we give certain sufficient conditions to guarantee the asymptotic stability and uniform boundedness of the solutions.

1. Introduction

Differential equations of third order are valuable tools in the modeling of many phenomena in various fields of science and engineering (Chlouverakis and Sprott [1], Cronin-Scanlon [2], Eichhorn et al. [3], Friedrichs [4], Linz [5], and Rauch [6]). In reality, the stability and boundedness of solutions of certain nonlinear differential equations of the third order have been received intensive attentions by authors (Ademola et al. [7], Afuwape and Castellanos [8], Chukwu [9], Ezeilo ([10, 11]), Hara [12], Mehri and Shadman [13], Ogundare and Okecha [14], Omeike [15], Reissig et al. [16], Swick [17], Tejumola ([18, 19]), Tunç [20–33], and Yoshizawa [34]).

In 2009, Omeike [15] considered the nonlinear differential equation of the third order with the constant delay 𝑟(>0): 𝑥⃛+𝑎(𝑡)𝑥̈+𝑏(𝑡)𝑔(𝑥̇)+𝑐(𝑡)ℎ(𝑥(𝑡−𝑟))=𝑝(𝑡), (1) and he discussed the stability and boundedness of solutions of this equation.

In this paper, instead of the above equation, we consider the nonautonomous differential equation of the third order with multiple deviating arguments: 𝑥⃛+𝑎(𝑡)𝑥̈+𝑛𝑏(𝑡)𝑔(𝑥̇)+𝑐(𝑡)^𝑘_𝑖=1ℎ_𝑖𝑥𝑡−𝑟_𝑖=𝑝(𝑡), (2) where 𝑟_𝑖 are certain positive constants, and 𝑎(𝑡),𝑏(𝑡),𝑐(𝑡),𝑔(𝑥̇),ℎ(𝑥) and 𝑝(𝑡) are real valued and continuous functions in their respective arguments with 𝑔(0)=ℎ(0)=0,𝑘=𝑛. The existence and uniqueness of the solutions of (2) are also assumed.

The motivation for this paper is a result of the researches mentioned regarding ordinary differential equations of the third order. It follows that the equation discussed in [15] is a special case of (2). Our aim is to improve the results established in [15] from one deviating argument to the multiple deviating arguments for the asymptotic stability and uniform boundedness of solutions. This work contributes to and complements previously known results on the topic in the literature, and it may be useful for researchers working on the qualitative behaviors of solutions. It should be noted that in recent years scores of papers have been published on the qualitative behaviors of solutions (stability of solutions, boundedness of the solutions, existence of the periodic solutions, etc.) of the functional differential equations of the second order with multiple deviating arguments. However, very little attention was given to stability and boundedness of functional differential equations of the third order with multiple deviating arguments ([32]). Therefore, it is worth investigating the qualitative behaviors of solutions in multidelay functional differential equations of the third order. This case is the novelty of the present paper. It should also be noted that the results to be established here are different from those in Tunç [20–33] and the literature.

2. Main Results

Let 𝑝(𝑡)≡0.

Theorem 1. One assumes that there exist positive constants 𝑎,𝑏,𝑐,𝜌_𝑖,𝛼,𝜇_𝑖,𝛿₁,𝛿₂,𝛿₅,𝛿₆, and 𝐿 such that the following conditions hold:(i)ℎ(0)=𝑔(0)=0,ℎ_𝑖(𝑥)/𝑥≥𝜇_𝑖,𝑥≠0,  ℎ^′_𝑖(𝑥)≤𝜌_𝑖,𝑔(𝑦)/𝑦≥𝑏>0,𝑦≠0(𝑖=1,2,…,𝑘), (ii)0<𝛿₁≤𝑐(𝑡)≤𝑏(𝑡),−𝐿≤𝑏^′(𝑡)≤𝑐^′(𝑡)≤0,0<𝑎<𝑎(𝑡).

If 𝑏𝜌_𝑖1>𝛼>𝑎,12𝑎^′(𝑡)≤𝛿₂<𝛿₁𝑛𝑏−𝛼^𝑘_𝑖=1𝜌_𝑖,_𝑘𝑖=1𝑟_𝑖<min2𝛿₅(),𝛿𝐿𝑐𝛼+2⁶,𝐿𝑐𝛼 (3)

then every solution 𝑥≡𝑥(𝑡) of (2) is uniform bounded and satisfies 𝑥(𝑡)⟶0,𝑥̇(𝑡)⟶0,𝑥̈(𝑡)⟶0as𝑡⟶∞. (4)

Remark 2. It should be noted that it follows from (ii) that 𝑏(𝑡) and 𝑐(𝑡) are nonincreasing functions on [0,∞). Therefore, since these functions are continuous on this interval and bounded below by 𝛿₁>0, they are bounded on [0,∞) and the limit of each exists as 𝑡→∞. Since 𝐿 in (ii) is an arbitrary selected bound, we can also assume the following estimates: 0<𝛿₁≤𝑐(𝑡)≤𝑏(𝑡)≤𝐿,lim_𝑡→∞𝑐(𝑡)=𝑐₀,lim_𝑡→∞𝑏(𝑡)=𝑏₀,𝛿₁≤𝑐₀≤𝑏₀≤𝐿. (5)

Proof. We write (2) in the system form as follows: ̇̇̇(𝑡)(𝑡)𝑔𝑦(𝑡)𝑥=𝑦,𝑦=𝑧,𝑧=−𝑎𝑧−𝑛𝑏−𝑐^𝑘_𝑖=1ℎ_𝑖(𝑥)+𝑐(𝑡)^𝑘_𝑖=1^𝑡_{𝑡−𝑟𝑖}ℎ^′_𝑖(𝑥(𝑠))𝑦(𝑠)𝑑𝑠. (6)
Define a Lyapunov-Krasovskiì functional ([35]) 𝑉(⋅)=𝑉(𝑡,𝑥_𝑡,𝑦_𝑡,𝑧_𝑡) by 𝑉(⋅)=𝑐(𝑡)𝐻(𝑥)+𝑛𝛼𝑏(𝑡)𝐺𝑦+𝛼𝑐(𝑡)𝑦^𝑘_𝑖=1ℎ_𝑖(𝑥)+12𝑎(𝑡)𝑦²+12𝛼𝑧²++𝑦𝑧_𝑘𝑖=1𝜆_𝑖⁰_−𝑟𝑖^𝑡_𝑡+𝑠𝑦²(𝜃)𝑑𝜃𝑑𝑠, (7) where 𝐻(𝑥)≡∫^𝑥₀∑^𝑘_𝑖=1ℎ_𝑖(𝑠)𝑑𝑠,𝐺(𝑦)≡∫^𝑦₀𝑔(𝜉)𝑑𝜉, and 𝜆_𝑖 are certain positive constants, which will be determined later in the proof.
This functional can be arranged as follows: 𝑉(⋅)=𝑉₁+12𝑉₂+^𝑘_𝑖=1𝜆_𝑖⁰_−𝑟𝑖^𝑡_𝑡+𝑠𝑦²(𝜃)𝑑𝜃𝑑𝑠, (8) where 𝑉₁(𝑡𝐻(𝑥)𝑏(𝑡)=𝑐)+𝑛𝛼𝑐(𝑡)𝐺𝑦+𝛼𝑦^𝑘_𝑖=1ℎ_𝑖(𝑥,𝑉₂)=𝑎(𝑡)𝑦²+𝛼𝑧²+2𝑦𝑧. (9)
Using the assumptions of Theorem 1, it follows that 𝛼𝑎(𝑡)≥𝛼𝑎>1 since 𝛼>𝑎⁻¹ and 𝛼𝑎(𝑡)−1>0.
Thus, there exist constants 𝛿₁>0 and 𝛿₃>0 such that 𝑉₂=𝑎(𝑡)𝑦²+𝛼𝑧²(𝑡𝑧+2𝑦𝑧=𝑎)𝑦+2𝑎(𝑡)²+1(𝑡)[(𝑡)]𝑧4𝑎4𝛼𝑎−1²≥12𝛿₃𝑦²+12𝛿₃𝑧²,𝑉₁≥𝛿₁𝐻(𝑥)+12𝑛𝛼𝑏𝑦²+𝛼𝑦^𝑘_𝑖=1ℎ_𝑖(𝑥) (10) since 𝑏(𝑡)/𝑐(𝑡)≥1,𝑐(𝑡)≥𝛿₁>0,and𝑔(𝑦)/𝑦≥𝑏>0implythat𝐺(𝑦)≥(1/2)𝑏𝑦². Further, using the assumptions of Theorem 1 and 𝑘=𝑛, it follows that 𝐻(𝑥)+12𝑛𝛼𝑏𝑦²+𝛼𝑦^𝑘_𝑖=1ℎ_𝑖(𝑥)=122𝐻(𝑥)+𝑛𝛼𝑏𝑦²+2𝛼𝑦^𝑘_𝑖=1ℎ_𝑖(𝑥=1)2𝛼𝑏𝑏𝑦+ℎ₁(𝑥)²+𝑏𝑦+ℎ₂(𝑥)²+⋯+𝑏𝑦+ℎ_𝑛(𝑥)²(𝑥)−𝛼+2𝐻𝑏^𝑘_𝑖=1ℎ²_𝑖(𝑥=1)2𝛼𝑏𝑏𝑦+ℎ₁(𝑥)²+𝑏𝑦+ℎ₂(𝑥)²+⋯+𝑏𝑦+ℎ_𝑛(𝑥)²+^𝑘_𝑖=1^𝑥₀1−𝛼ℎ^′_𝑖(𝑠)𝑏ℎ_𝑖(𝑠)≥𝑑𝑠_{𝑘𝑖=1𝑥0}1−𝛼𝜌_𝑖𝑏ℎ_𝑖(𝑠)𝑠≥𝑠𝑑𝑠_{𝑘𝑖=1𝑥0}1−𝛼𝜌_𝑖𝑏𝜇_𝑖≥1𝑠𝑑𝑠2^𝑘_𝑖=11−𝛼𝜌_𝑖𝑏𝜇_𝑖𝑥²=𝛿₄2𝑥² (11) so that 𝑉₁≥𝛿₁𝛿₄2𝑥², (12) where 𝛿₄=^𝑘_𝑖=11−𝛼𝜌_𝑖𝑏𝜇_𝑖. (13)
In view of the previous discussion, we can get 𝑉(⋅)≥𝛿₁𝛿₄2𝑥²+𝛿₃4𝑦²+𝛿₃4𝑧²+^𝑘_𝑖=1𝜆_𝑖⁰_−𝑟𝑖^𝑡_𝑡+𝑠𝑦²(𝜃)𝑑𝜃𝑑𝑠. (14)
Using a basic calculation, the time derivative of 𝑉(⋅) along solutions of (6) results in 𝑑𝑉(⋅)𝑑𝑡=𝑐^′(𝑡)𝐻(𝑥)+𝑐(𝑡)𝐻^′(𝑥)+𝑛𝛼𝑏^′(𝑡)𝐺𝑦+𝑛𝛼𝑏(𝑡)𝐺^′𝑦+𝛼𝑐^′(𝑡)𝑦^𝑘_𝑖=1ℎ_𝑖(𝑥)+𝛼𝑐(𝑡)𝑦^′𝑘_𝑖=1ℎ_𝑖+𝛼𝑐(𝑡)𝑦𝑥^′𝑘_𝑖=1ℎ^′_𝑖(𝑥)+12𝑎^′(𝑡)𝑦²+𝑎(𝑡)𝑦𝑦^′+𝛼𝑧𝑧^′+𝑦^′𝑧+𝑦𝑧^′+^𝑘_𝑖=1𝜆_𝑖⁰_−𝑟𝑖𝑦²(𝑡)−𝑦²(𝑡+𝑠)𝑑𝑠=𝑐^′(𝑡)𝐻(𝑥)+𝑐(𝑡)𝑦^𝑘_𝑖=1ℎ_𝑖(𝑥)+𝑛𝛼𝑏^′(𝑡)𝐺𝑦+𝑛𝛼𝑏(𝑡)𝑧𝑔𝑦+𝛼𝑐^′(𝑡)𝑦^𝑘_𝑖=1ℎ_𝑖(𝑥)+𝛼𝑐(𝑡)𝑧^𝑘_𝑖=1ℎ_𝑖(𝑥)+𝛼𝑐(𝑡)𝑦^2𝑘_𝑖=1ℎ^′_𝑖(𝑥)+12𝑎^′(𝑡)𝑦²(𝑡)×(𝑡)(𝑡)𝑔𝑦(𝑡)+𝑎𝑦𝑧+𝛼𝑧−𝑎𝑧−𝑛𝑏−𝑐^𝑘_𝑖=1ℎ_𝑖(𝑥(𝑡(𝑡)))+𝑐^𝑘_𝑖=1^𝑡_{𝑡−𝑟𝑖}ℎ^′_𝑖(𝑥(𝑠))𝑦(𝑠)𝑑𝑠+𝑧²(𝑡)(𝑡)𝑔𝑦(𝑡)+𝑦−𝑎𝑧−𝑛𝑏−𝑐^𝑘_𝑖=1ℎ_𝑖(𝑥(𝑡(𝑡)))+𝑐^𝑘_𝑖=1^𝑡_{𝑡−𝑟𝑖}ℎ^′_𝑖(𝑥(𝑠𝑦(𝑠)+))𝑑𝑠_𝑘𝑖=1𝜆_𝑖𝑟_𝑖𝑦²−^𝑘_𝑖=1𝜆_𝑖^𝑡_{𝑡−𝑟𝑖}𝑦²(𝑠)𝑑𝑠=𝑐^′(𝑡)𝐻(𝑥)+𝛼𝑐^′(𝑡)𝑦^𝑘_𝑖=1ℎ_𝑖(𝑥)+𝑛𝛼𝑏^′(𝑡)𝐺𝑦+12𝑎^′(𝑡)𝑦²−𝑛𝑦𝑏(𝑡)𝑔𝑦−𝛼𝑐(𝑡)𝑦^2𝑘_𝑖=1ℎ^′_𝑖(𝑥)−^𝑘_𝑖=1𝜆_𝑖𝑟_𝑖𝑦²−[𝛼𝑎(𝑡)−1]𝑧²+𝛼𝑧𝑐(𝑡)^𝑘_𝑖=1^𝑡_{𝑡−𝑟𝑖}ℎ^′_𝑖(𝑥(𝑠𝑦(𝑠)(𝑡)))𝑑𝑠+𝑦𝑐^𝑘_𝑖=1^𝑡_{𝑡−𝑟𝑖}ℎ^′_𝑖(𝑥(𝑠𝑦(𝑠)−))𝑑𝑠_𝑘𝑖=1𝜆_𝑖^𝑡_{𝑡−𝑟𝑖}𝑦²(𝑠)𝑑𝑠. (15)
Using ℎ^′_𝑖(𝑥)≤𝜌_𝑖,𝑐(𝑡)≤𝐿, and the estimate 2|𝑒𝑓|≤𝑒²+𝑓², we have 𝛼𝑧𝑐(𝑡)^𝑘_𝑖=1^𝑡_{𝑡−𝑟𝑖}ℎ^′_𝑖(𝑥(𝑠𝑦(𝑠)(𝑡𝑧|))𝑑𝑠≤𝛼𝑐)|^𝑘_𝑖=1^𝑡_{𝑡−𝑟𝑖}𝑐𝑦(𝑠≤1)𝑑𝑠2𝛼𝑐(𝑡)𝑧^2𝑘_𝑖=1𝑟_𝑖𝜌_𝑖+12𝛼𝑐(𝑡)^𝑘_𝑖=1^𝑡_{𝑡−𝑟𝑖}𝜌_𝑖𝑦²(𝑠)≤1𝑑𝑠2𝐿𝛼𝑐𝑧^2𝑘_𝑖=1𝑟_𝑖+12𝐿𝛼𝑐^𝑘_𝑖=1^𝑡_{𝑡−𝑟𝑖}𝑦²(𝑠)(𝑡)𝑑𝑠,𝑦𝑐^𝑘_𝑖=1^𝑡_{𝑡−𝑟𝑖}ℎ^′_𝑖(𝑥(𝑠𝑦(𝑠)(𝑡𝑦))𝑑𝑠≤𝑐)^𝑘_𝑖=1^𝑡_{𝑡−𝑟𝑖}𝜌_𝑖𝑦(𝑠≤1)𝑑𝑠2𝑐(𝑡)𝑦^2𝑘_𝑖=1𝑟_𝑖𝜌_𝑖+12𝑐(𝑡)^𝑘_𝑖=1^𝑡_{𝑡−𝑟𝑖}𝜌_𝑖𝑦²(𝑠)≤1𝑑𝑠2𝐿𝑐𝑦^2𝑘_𝑖=1𝑟_𝑖+12𝐿𝑐^𝑘_𝑖=1^𝑡_{𝑡−𝑟𝑖}𝑦²(𝑠)𝑑𝑠, (16) where 𝑐=max𝜌_𝑖. (17)
Noting the previous discussion, it follows that 𝑑𝑉(⋅)𝑑𝑡≤𝑐^′(𝑡)𝐻(𝑥)+𝑛𝛼𝑏^′(𝑡)𝐺𝑦+𝛼𝑐^′(𝑡)𝑦^𝑘_𝑖=1ℎ_𝑖(𝑥)+12𝑎^′(𝑡)𝑦²−𝑛𝑦𝑏(𝑡)𝑔𝑦−𝛼𝑐(𝑡)𝑦^2𝑘_𝑖=1ℎ^′_𝑖(𝑥)−^𝑘_𝑖=1𝜆_𝑖𝑟_𝑖𝑦²−122(𝛼𝑎(𝑡)−1)−𝐿𝛼𝑐^𝑘_𝑖=1𝑟_𝑖𝑧²+12𝐿𝑐𝑦^2𝑘_𝑖=1𝑟_𝑖+^𝑘_𝑖=112𝐿𝑐(𝛼+1)−𝜆_𝑖^𝑡_{𝑡−𝑟𝑖}𝑦²(𝑠)𝑑𝑠. (18)
If 𝑦=0, then 𝑛𝑦𝑏(𝑡)𝑔𝑦−𝛼𝑐(𝑡)𝑦^2𝑘_𝑖=1ℎ^′_𝑖(𝑥)−^𝑘_𝑖=1𝜆_𝑖𝑟_𝑖𝑦²=0. (19)
If 𝑦≠0, then it follows that 𝑛𝑦𝑏(𝑡)𝑔𝑦−𝛼𝑐(𝑡)𝑦^2𝑘_𝑖=1ℎ^′_𝑖(𝑥)−^𝑘_𝑖=1𝜆_𝑖𝑟_𝑖𝑦²=(𝑡)𝑔𝑦𝑛𝑏𝑦−𝛼𝑐(𝑡)^𝑘_𝑖=1ℎ^′_𝑖(𝑥)−^𝑘_𝑖=1𝜆_𝑖𝑟_𝑖𝑦²≥𝑛𝑏(𝑡)𝑏−𝛼𝑐(𝑡)^𝑘_𝑖=1𝜌_𝑖−^𝑘_𝑖=1𝜆_𝑖𝑟_𝑖𝑦²(𝑡𝑏(𝑡)=𝑐)𝑐(𝑡)𝑛𝑏−𝛼^𝑘_𝑖=1𝜌_𝑖𝑦²−^𝑘_𝑖=1𝜆_𝑖𝑟_𝑖𝑦²≥𝛿₁𝑛𝑏−𝛼^𝑘_𝑖=1𝜌_𝑖−𝛿₁^𝑘−1_𝑖=1𝜆_𝑖𝑟_𝑖𝑦². (20)
Since 0≤𝛿₁≤𝑐(𝑡)≤𝑏(𝑡),1≤𝑏(𝑡)/𝑐(𝑡), and 𝛼𝑎(𝑡)≥𝛼𝑎>1, then 12𝑎^′(𝑡)𝑦²−𝑛𝑦𝑏(𝑡)𝑔𝑦−𝛼𝑐(𝑡)𝑦^2𝑘_𝑖=1ℎ^′_𝑖(𝑥)−^𝑘_𝑖=1𝜆_𝑖𝑟_𝑖𝑦²≤𝛿₂−𝛿₁𝑛𝑏−𝛼^𝑘_𝑖=1𝜌_𝑖+^𝑘_𝑖=1𝜆_𝑖𝑟_𝑖𝑦²=−𝛿₅−^𝑘_𝑖=1𝜆_𝑖𝑟_𝑖𝑦²,2(𝛼𝑎(𝑡)−1)−𝐿𝛼𝑐^𝑘_𝑖=1𝑟_𝑖𝑧²≥𝛿₆−𝐿𝛼𝑐^𝑘_𝑖=1𝑟_𝑖𝑧², (21) where 𝛿₅=𝛿₁(𝑛𝑏−𝛼∑^𝑘_𝑖=1𝜌_𝑖)−𝛿₂>0 and 𝛿₆≡2(𝛼𝑎−1)>0.
Thus, we get 𝑑𝑉(⋅)𝑑𝑡≤𝑐^′(𝑡)𝐻(𝑥)+𝑛𝛼𝑏^′(𝑡)𝐺𝑦+𝛼𝑐^′(𝑡)𝑦^𝑘_𝑖=1ℎ_𝑖(𝑥)−𝛿₅−^𝑘_𝑖=1𝜆_𝑖𝑟_𝑖𝑦²−12𝛿₆−𝐿𝛼𝑐^𝑘_𝑖=1𝑟_𝑖𝑧²+12𝐿𝑐𝑦^2𝑘_𝑖=1𝑟_𝑖+^𝑘_𝑖=112𝐿𝑐(𝛼+1)−𝜆_𝑖^𝑡_{𝑡−𝑟𝑖}𝑦²(𝑠)𝑑𝑠. (22)
Let 𝜆_𝑖=(1/2)𝐿𝑐(𝛼+1). Hence, 𝑑𝑉(⋅)𝑑𝑡≤𝑐^′(𝑡)𝐻(𝑥)+𝑛𝛼𝑏^′(𝑡)𝐺𝑦+𝛼𝑐^′(𝑡)𝑦^𝑘_𝑖=1ℎ_𝑖(𝑥)−𝛿₅−12𝐿𝑐(𝛼+2)^𝑘_𝑖=1𝑟_𝑖𝑦²−12𝛿₆−𝐿𝛼𝑐^𝑘_𝑖=1𝑟_𝑖𝑧². (23)
If 𝑐^′(𝑡)=0, then 𝑐^′(𝑡)𝐻(𝑥)+𝑛𝛼𝑏^′(𝑡)𝐺(𝑦)+𝛼𝑐^′(𝑡)𝑦∑^𝑘_𝑖=1ℎ_𝑖(𝑥)=𝑛𝛼𝑏^′(𝑡)𝐺(𝑦)≤0 since 𝑏^′(𝑡)≤0 and 𝐺(𝑦)≥0. For those 𝑡's such that 𝑐^′(𝑡)<0, we have 𝑐^′(𝑡)𝐻(𝑥)+𝑛𝛼𝑏^′(𝑡)𝐺𝑦+𝛼𝑐^′(𝑡)𝑦^𝑘_𝑖=1ℎ_𝑖(𝑥)=𝑐^′(𝑡𝐻(𝑥)𝑏)+𝑛𝛼^′(𝑡)𝑐^′(𝑡)𝐺𝑦+𝛼𝑦^𝑘_𝑖=1ℎ_𝑖(𝑥)≤𝑐^′(𝑡)𝐻(𝑥)+𝑛𝛼𝐺𝑦+𝛼𝑦^𝑘_𝑖=1ℎ_𝑖(𝑥)≤𝑐^′(𝑡)𝛿₄𝐻(𝑥)≤0. (24)
Thus, 𝑑𝑉(⋅)𝛿𝑑𝑡≤−₅−12𝐿𝑐(𝛼+2)^𝑘_𝑖=1𝑟_𝑖𝑦²−12𝛿₆−𝐿𝛼𝑐^𝑘_𝑖=1𝑟_𝑖𝑧². (25)
Therefore, if ^𝑘_𝑖=1𝑟_𝑖<min2𝛿₅(),𝛿𝐿𝑐𝛼+2⁶,𝐿𝛼𝑐 (26) then we have 𝑑𝑉(⋅)𝑦𝑑𝑡≤−𝛽²+𝑧²forsome𝛽>0. (27)
The proof for Theorem 1 is complete.

Let 𝑝(𝑡)≠0.

Theorem 3. One assumes that all the assumptions of Theorem 1 and the assumption ^𝑡₀𝑝(𝑠)𝑑𝑠<∞ (28) hold. If 𝑏𝜌_𝑖1>𝛼>𝑎,12𝑎^′(𝑡)≤𝛿₂<𝛿₁𝑛𝑏−𝛼^𝑘_𝑖=1𝜌_𝑖,_𝑘𝑖=1𝑟_𝑖<min2𝛿₅(),𝛿𝐿𝑐𝛼+2⁶,𝐿𝑐𝛼 (29) then all solutions of (2) are bounded.

Proof. Equation (2) is equivalent to the system ̇̇̇(𝑡)(𝑡)𝑔𝑦(𝑡)𝑥=𝑦,𝑦=𝑧,𝑧=−𝑎𝑧−𝑛𝑏−𝑐^𝑘_𝑖=1ℎ_𝑖(𝑥(𝑡(𝑡)))+𝑐^𝑘_𝑖=1^𝑡_{𝑡−𝑟𝑖}ℎ^′_𝑖(𝑥(𝑠))𝑦(𝑠)𝑑𝑠+𝑝(𝑡). (30)
Along any solution (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) of (6), we have 𝑉̇₍₃₎(⋅)=𝑉̇₍₂₎(⋅)+𝑦+𝛼𝑧𝑝(𝑡). (31)
Since 𝑉̇₍₂₎(⋅)≤0, then it follows that 𝑉̇₍₃₎(⋅)≤𝑦+𝛼𝑧𝑝(𝑡)≤𝑦+𝛼|𝑧|𝑝(𝑡)≤𝛿₈𝑦+|𝑧|𝑝(𝑡), (32) where 𝛿₈≡max{1,𝛼}. Noting that |𝑚|<1+𝑚², we get 𝑉̇₍₃₎(⋅)≤𝛿₈2+𝑦²+𝑧²𝑝(𝑡)≤2𝛿₈𝑝(𝑡)+𝛿₈‖𝑋‖²𝑝(𝑡)≤2𝛿₈𝑝(𝑡+𝛿)⁸𝛿₇𝑉(⋅)𝑝(𝑡) (33) recalling that 𝛿₇‖𝑋‖²≤𝑉(⋅).
Let 𝜂=max(2𝛿₈,𝛿₈/𝛿₇), then 𝑉̇₍₃₎(⋅)≤𝜂𝑝(𝑡)+𝜂𝑉(⋅)𝑝(𝑡) (34) or 𝑉̇₍₃₎(⋅)−𝜂𝑉(⋅)𝑝(𝑡)≤𝜂𝑝(𝑡). (35)
Multiplying each side of this estimate by the integrating factor exp(−𝜂∫^𝑡₀|𝑝(𝑠)|𝑑𝑠), we get 𝑉̇₍₃₎(⋅)exp−𝜂^𝑡₀𝑝(𝑠(⋅𝑝(𝑡)𝑑𝑠−𝜂𝑉))exp−𝜂^𝑡₀𝑝(𝑠𝑝(𝑡)𝑑𝑠≤𝜂)exp−𝜂^𝑡₀𝑝(𝑠)𝑑𝑠. (36)
Integrating each side of this estimate from 0 to t, we obtain 𝑉(⋅)exp−𝜂^𝑡₀𝑝(𝑠)𝑑𝑠−𝑉(0)≤1−exp−𝜂^𝑡₀𝑝(𝑠)𝑑𝑠 (37) or 𝑉(⋅)≤𝑉(0)exp𝜂^𝑡₀𝑝(𝑠)𝑑𝑠+exp𝜂^𝑡₀𝑝(𝑠)𝑑𝑠−1, (38) where (0,𝑥(0),𝑦(0),𝑧(0))=0.
Since ∫^𝑡₀|𝑝(𝑠)|𝑑𝑠≤𝐴 for all 𝑡, this implies 𝑉(⋅)≤𝑉(0)𝑒^𝜂𝐴+𝑒^𝜂𝐴−1for𝑡≥0. (39)
Since the right-hand side of the last estimate is a constant and 𝑉(⋅)→∞ when 𝑥²+𝑦²+𝑧²→∞, it follows that there exists a positive constant 𝐷 such that |𝑥(𝑡)|≤𝐷,𝑦(𝑡)≤𝐷,|𝑧(𝑡)|≤𝐷∀𝑡≥0. (40)
From the system (30) this implies that |𝑥(𝑡)|≤𝐷,|𝑥̇(𝑡)|≤𝐷,|𝑥̈(𝑡)|≤𝐷∀𝑡≥0. (41)
The proof for Theorem 3 is complete.