Qualitative Behaviors of Functional Differential Equations of Third Order with Multiple Deviating Arguments

Abstract

This paper considers nonautonomous functional differential equations of the third order with multiple constant deviating arguments. Using the Lyapunov-Krasovskii functional approach, we find certain sufficient conditions for the solutions to be stable and bounded. We give an example to illustrate the theoretical analysis made in this work and to show the effectiveness of the method utilized here.

1. Introduction

In this paper, we consider nonautonomous differential equation of the third order with constant multiple deviating arguments, 𝜏, 𝜏_𝑖, (𝑖=1,2,…,𝑛) as follows: 𝑥^′′′(𝑡)+𝑎(𝑡)𝑥^′′(𝑡)+^𝑛_𝑖=1𝑏_𝑖(𝑡)𝑔_𝑖𝑥^′𝑡−𝜏_𝑖+𝑔𝑥^′(𝑡(𝑥((𝑡))+ℎ𝑡−𝜏))=𝑝𝑡,𝑥,𝑥𝑡−𝜏₁,…,𝑥𝑡−𝜏_𝑛,𝑥^′,…,𝑥^′𝑡−𝜏_𝑛…,𝑥^′′(𝑡),…,𝑥^′′𝑡−𝜏_𝑛. (1.1) Writing (1.1) as a system of first order equations, we have 𝑥^′=𝑦,𝑦^′𝑧=𝑧,_′=−𝑎(𝑡)𝑧−^𝑛_𝑖=0𝑏_𝑖(𝑡)𝑔_𝑖𝑦−ℎ(𝑥)−𝑔𝑦+^𝑛_𝑖=1𝑏_𝑖(𝑡)^𝑡_{𝑡−𝜏𝑖}𝑔^′_𝑖𝑦(𝑠)𝑧(𝑠)𝑑𝑠+^𝑡_𝑡−𝜏ℎ^′(𝑥(𝑠))𝑦(𝑠)𝑑𝑠+𝑝𝑡,𝑥,…,𝑥𝑡−𝜏_𝑛,𝑦,…,𝑦𝑡−𝜏_𝑛,𝑧,…,𝑧𝑡−𝜏_𝑛, (1.2) where 𝜏 and 𝜏_𝑖,   (𝑖=1,2,…,𝑛), are positive constants, that is, fixed delays; the functions 𝑎,   𝑏_𝑖, 𝑔,   𝑔_𝑖,ℎ, and 𝑝 are continuous for their all respective arguments and the primes in (1.1) denote differentiation with respect to 𝑡,  𝑡∈ℜ⁺=[0,∞). It is also assumed that the derivatives 𝑎^′(𝑡)≡(𝑑/𝑑𝑡)𝑎(𝑡),   𝑏^′_𝑖(𝑡)≡(𝑑/𝑑𝑡)𝑏_𝑖(𝑡),   𝑔^′_𝑖(𝑦)≡(𝑑/𝑑𝑦)𝑔_𝑖(𝑦), and ℎ^′(𝑥)≡(𝑑/𝑑𝑥)ℎ(𝑥) exist and are continuous; throughout the paper 𝑥(𝑡),𝑦(𝑡),   and𝑧(𝑡) are abbreviated as 𝑥,𝑦, and 𝑧, respectively. Finally, the existence and uniqueness of solutions of (1.1) are assumed and all solutions considered are supposed to be real valued.

To the best of our knowledge from the literature, in the last five decades, there has been much attention paid to the discussion of stability and boundedness of solutions of nonlinear differential equations of the third order without a deviating argument. For a comprehensive treatment of the subject, we refer the readers to the book of Reissig et al. [1] as a survey and the papers of Ademola et al. [2], Afuwape et al. [3], Ezeilo [4–13], Ezeilo and Tejumola [14, 15], Mehri and Shadman [16], Ogundare [17], Ogundare and Okecha [18], Omeike [19, 20], Ponzo [21, 22], Swick [23–25], Tejumola [26, 27], Tunç [28–34], Tunç and Ateş [35], Tunç and Ayhan [36], and the references cited in these papers for some works on the topic.

Besides, first, in 1973, Sinha [37] studied the stability of solutions of a third order nonlinear differential equation with a deviating argument. Later, some authors dealt with the stability and boundedness of solutions for various third order nonlinear differential equations with a deviating argument. For some related works, one can refer to the papers of Afuwape and Omeike [38], Omeike [39], Sadek [40, 41], Tejumola and Tchegnani [42], Tunç [43–59], Yao and Meng [60], Zhu [61], and the references thereof.

It should be noted that throughout all the above mentioned papers, Lyapunov’s functions or the Lyapunov-Krasovskii functionals have been used as a basic tool to prove the results established there. It is also worth mentioning that the most effective method to study the stability and boundedness of solutions of nonlinear differential equations of higher orders without or with a deviating argument in the literature is still the Lyapunov’s direct method, despite its use for a past long period by now.

The motivation for this paper comes from the above mentioned papers and the books. Our results improve and include the results existing in the literature. This work makes also a contribution to the existing studies made in the literature.

2. Main Results

Let 𝑝(⋅)=0.

Our first result is given by the following theorem.

Theorem 2.1. In addition to the basic assumptions imposed to the functions 𝑎(𝑡), 𝑏(𝑡), 𝑔, 𝑔_𝑖, and ℎ appearing in (1.1), we assume that there exist positive constants 𝑎, 𝛼, 𝛽, 𝛿, 𝑏, 𝑏_𝑖, 𝐵_𝑖, 𝑐, 𝑐₁, and 𝐿_𝑖 such that the following conditions hold:(i)𝑔_𝑖(0)=𝑔(0)=ℎ(0)=0,𝑎(𝑡)≥2𝛼+𝑎, 𝛽_𝑖≤𝑏_𝑖(𝑡)≤𝐵_𝑖, 0<𝑐₁≤ℎ^′(𝑥)𝑔≤𝑐,𝛼𝑏−𝑐>𝛿,^𝑖𝑦𝑦≥𝑏_𝑖,𝑔𝑦𝑦,𝑔≥𝑏,𝑦≠0_′𝑖𝑦≤𝐿_𝑖, (2.1)(ii)[∑^𝑛_𝑖=1{𝛼𝑏_𝑖𝑏_𝑖(𝑡)}−𝑐]𝑦²≥2⁻¹𝛼𝑎^′(𝑡)𝑦²+∑^𝑛_𝑖=1𝑏^′_𝑖(𝑡)∫^𝑦₀𝑔_𝑖(𝜂)𝑑𝜂.
If 𝜏₀<min𝛼𝑏𝑐+2𝛼𝑐+𝛼∑^𝑛_𝑖=1𝐵_𝑖𝐿_𝑖,2𝛼𝑐+∑^𝑛_𝑖=1(2+𝛼)𝐵_𝑖𝐿_𝑖, (2.2) then the zero solution of (1.1) is stable.

Proof. Define the Lyapunov-Krasovskii functional 𝑉(⋅)=𝑉(𝑡,𝑥_𝑡,𝑦_𝑡,𝑧_𝑡) as 2𝑉(⋅)=𝑧²+2𝛼𝑦𝑧+2^𝑛_𝑖=0𝑏_𝑖(𝑡)^𝑦₀𝑔_𝑖𝜂𝑑𝜂+2𝛼^𝑥₀ℎ(𝜉)𝑑𝜉+2^𝑦₀𝑔𝜂(𝑡)𝑦𝑑𝜂+𝛼𝑎²+2ℎ(𝑥)𝑦+𝜆⁰_−𝜏^𝑡_𝑡+𝑠𝑦²(𝜃)𝑑𝜃𝑑𝑠+^𝑛_𝑖=1𝜆_𝑖⁰_−𝜏𝑖^𝑡_𝑡+𝑠𝑧²(𝜃)𝑑𝜃𝑑𝑠, (2.3) where 𝜆 and 𝜆_𝑖 are some positive constants to be chosen later.
Using the assumptions 𝑔_𝑖(𝑦)/𝑦≥𝑏_𝑖, 𝑔(𝑦)/𝑦≥𝑏, (𝑦≠0),ℎ(0)=0, and 0<𝑐₁≤ℎ^′(𝑥)≤𝑐, we have 2𝑏_𝑖(𝑡)^𝑦₀𝑔_𝑖𝜂𝑑𝜂=2𝑏_𝑖(𝑡)^𝑦₀𝑔_𝑖𝜂𝜂𝜂𝑑𝜂≥𝛽_𝑖𝑏_𝑖𝑦²,2^𝑦₀𝑔𝜂𝑑𝜂=2^𝑦₀𝑔𝜂𝜂𝜂𝑑𝜂≥𝑏𝑦²,ℎ²(𝑥)=2^𝑥₀ℎ(𝜉)ℎ^′(𝜉)𝑑𝜉≤2𝑐^𝑥₀ℎ(𝜉)𝑑𝜉, (2.4) so that 2𝑉(⋅)≥𝑧+𝛼𝑦²+𝑏𝑦+𝑏⁻¹ℎ(𝑥)²+2𝛼^𝑥₀ℎ(𝜉)1𝑑𝜉−𝑏ℎ²(𝑥)+^𝑛_𝑖=1𝛽_𝑖𝑏_𝑖𝑦²+𝛼(𝛼+𝑎)𝑦²+𝜆⁰_−𝜏^𝑡_𝑡+𝑠𝑦²(𝜃)𝑑𝜃𝑑𝑠+^𝑛_𝑖=1𝜆_𝑖⁰_−𝜏𝑖^𝑡_𝑡+𝑠𝑧²(𝜃)≥𝑑𝜃𝑑𝑠𝑧+𝛼𝑦²+𝑏𝑦+𝑏⁻¹ℎ(𝑥)²+2𝛼^𝑥₀ℎ(𝜉)𝑑𝜉−2𝑐𝑏^𝑥₀ℎ(𝜉)+𝑑𝜉_𝑛𝑖=1𝛽_𝑖𝑏_𝑖𝑦²+𝛼(𝛼+𝑎)𝑦²+𝜆⁰_−𝜏^𝑡_𝑡+𝑠𝑦²(𝜃)𝑑𝜃𝑑𝑠+^𝑛_𝑖=1𝜆_𝑖⁰_−𝜏𝑖^𝑡_𝑡+𝑠𝑧²(𝜃)𝑑𝜃𝑑𝑠. (2.5) On the other hand, it is obvious that 2𝛼^𝑥₀ℎ(𝜉)𝑑𝜉−2𝑐𝑏^𝑥₀ℎ(𝜉)𝑑𝜉=2𝑏⁻¹(𝛼𝑏−𝑐)^𝑥₀ℎ(𝜉)𝑑𝜉≥𝑐₁𝑏⁻¹(𝛼𝑏−𝑐)𝑥², (2.6) so that 2𝑉(⋅)≥𝑧+𝛼𝑦²+𝑏𝑦+𝑏⁻¹ℎ(𝑥)²+𝑐₁𝑏⁻¹(𝛼𝑏−𝑐)𝑥²+𝛼(𝛼+𝑎)𝑦²+^𝑛_𝑖=1𝛽_𝑖𝑏_𝑖𝑦²+𝜆⁰_−𝜏^𝑡_𝑡+𝑠𝑦²(𝜃)𝑑𝜃𝑑𝑠+^𝑛_𝑖=1𝜆_𝑖⁰_−𝜏𝑖^𝑡_𝑡+𝑠𝑧²(𝜃)𝑑𝜃𝑑𝑠. (2.7) Hence, we can obtain some positive constants 𝐷_𝑗, (𝑗=1,2,3), such that 𝑉(⋅)≥𝐷₁𝑥²+𝐷₂𝑦²+𝐷₃𝑧²≥𝐷₄𝑥²+𝑦²+𝑧², (2.8) where 𝐷₄=min{𝐷₁,𝐷₂,𝐷₃}, since the integrals 𝜆∫⁰_−𝜏∫^𝑡_𝑡+𝑠𝑦²(𝜃)𝑑𝜃𝑑𝑠 and ∑^𝑛_𝑖=1𝜆_𝑖∫⁰_−𝜏𝑖∫^𝑡_𝑡+𝑠𝑧²(𝜃)𝑑𝜃𝑑𝑠 are nonnegative.
Let (𝑥,𝑦,𝑧) be a solution of (1.2). Differentiating the Lyapunov-Krasovskii functional 𝑉(⋅) along this solution, we find 𝑑𝑉(⋅)𝛼𝑑𝑡=−^𝑛_𝑖=1𝑏_𝑖(𝑡)𝑔_𝑖𝑦𝑦⁻¹+𝛼𝑔𝑦𝑦⁻¹−ℎ^′(𝑥)−2⁻¹𝛼𝑎^′(𝑡)𝑦²+^𝑛_𝑖=1𝑏^′_𝑖(𝑡)^𝑦₀𝑔_𝑖𝜂𝑑𝜂−(𝑎(𝑡)−𝛼)𝑧²+𝑧^𝑛_𝑖=1𝑏_𝑖(𝑡)^𝑡_{𝑡−𝜏𝑖}𝑔^′_𝑖𝑦(𝑠)𝑧(𝑠)𝑑𝑠+𝑧^𝑡_𝑡−𝜏ℎ^′(𝑥(𝑠))𝑦(𝑠)𝑑𝑠+𝛼𝑦^𝑛_𝑖=1𝑏_𝑖(𝑡)^𝑡_{𝑡−𝜏𝑖}𝑔^′_𝑖𝑦(𝑠)𝑧(𝑠)𝑑𝑠+𝛼𝑦^𝑡_𝑡−𝜏ℎ^′(𝑥(𝑠))𝑦(𝑠)𝑑𝑠+𝜆𝜏𝑦²−𝜆^𝑡_𝑡−𝜏𝑦²(𝑠)𝑑𝑠+^𝑛_𝑖=1𝜆_𝑖𝜏_𝑖𝑧²−^𝑛_𝑖=1𝜆_𝑖^𝑡_{𝑡−𝜏𝑖}𝑧²(𝑠)𝑑𝑠. (2.9)
Using the assumptions of Theorem 2.1 and the estimate 2|𝑚𝑛|≤𝑚²+𝑛², we get 𝛼^𝑛_𝑖=1𝑏_𝑖(𝑡)𝑔_𝑖𝑦𝑦⁻¹+𝛼𝑔𝑦𝑦⁻¹−ℎ^′(𝑥)−2⁻¹𝛼𝑎^′(𝑡)𝑦²−^𝑛_𝑖=1𝑏^′_𝑖(𝑡)^𝑦₀𝑔_𝑖𝜂≥𝛼𝑑𝜂_𝑛𝑖=1𝑏_𝑖𝑏_𝑖(𝑡)+𝛼𝑏−𝑐−2⁻¹𝛼𝑎^′(𝑡)𝑦²−^𝑛_𝑖=1𝑏^′_𝑖(𝑡)^𝑦₀𝑔_𝑖𝜂=𝑑𝜂_𝑛𝑖=1𝛼𝑏_𝑖𝑏_𝑖(𝑡)−𝑐𝑦²−2⁻¹𝛼𝑎^′(𝑡)𝑦²−^𝑛_𝑖=1𝑏^′_𝑖(𝑡)^𝑦₀𝑔_𝑖𝜂𝑑𝜂+(𝛼𝑏)𝑦²≥(𝛼𝑏)𝑦²,[𝑎(𝑡)−𝛼]𝑧²≥(𝛼+𝑎)𝑧²,𝑧^𝑛_𝑖=1𝑏_𝑖(𝑡)^𝑡_{𝑡−𝜏𝑖}𝑔^′_𝑖𝑦(𝑠)𝑧(𝑠)1𝑑𝑠≤2^𝑛_𝑖=1𝐵_𝑖𝐿_𝑖𝜏_𝑖𝑧²+12^𝑛_𝑖=1𝐵_𝑖𝐿_𝑖^𝑡_{𝑡−𝜏𝑖}𝑧²(𝑠)𝑑𝑠,𝛼𝑦^𝑛_𝑖=1𝑏_𝑖(𝑡)^𝑡_{𝑡−𝜏𝑖}𝑔^′_𝑖𝑦(𝑠)𝑧(𝑠)1𝑑𝑠≤2^𝑛_𝑖=1𝛼𝐵_𝑖𝐿_𝑖𝜏_𝑖𝑦²+12^𝑛_𝑖=1𝛼𝐵_𝑖𝐿_𝑖^𝑡_{𝑡−𝜏𝑖}𝑧²(𝑠)𝑑𝑠,𝑧^𝑡_𝑡−𝜏ℎ^′(𝑥(𝑠𝑦(𝑠)𝑐))𝑑𝑠≤2𝜏𝑧²+𝑐2^𝑡_𝑡−𝜏𝑦²(𝑠)𝑑𝑠,𝛼𝑦^𝑡_𝑡−𝜏ℎ^′(𝑥(𝑠))𝑦(𝑠)𝑑𝑠≤𝛼𝑐2𝜏𝑦²+𝛼𝑐2^𝑡_𝑡−𝜏𝑦²(𝑠)𝑑𝑠, (2.10) so that 𝑑𝑉(⋅)1𝑑𝑡≤−2𝛼𝑏𝑦²−𝑎𝑧²−12𝛼𝑏−𝛼^𝑛_𝑖=1𝐵_𝑖𝐿_𝑖𝜏_𝑖+(2𝜆+𝛼𝑐)𝜏𝑦²−122𝛼−^𝑛_𝑖=1𝐵_𝑖𝐿_𝑖𝜏_𝑖+2^𝑛_𝑖=1𝜆_𝑖𝜏_𝑖+𝑐𝜏𝑧²+2⁻¹(1+𝛼)𝑐−𝜆^𝑡_𝑡−𝜏𝑦²(𝑠)+2𝑑𝑠₋₁(1+𝛼)^𝑛_𝑖=1𝐵_𝑖𝐿_𝑖−^𝑛_𝑖=1𝜆_𝑖^𝑡_{𝑡−𝜏𝑖}𝑧²(𝑠)𝑑𝑠. (2.11) Let 𝜆=(1/2)(1+𝛼)𝑐 and ∑^𝑛_𝑖=1𝜆_𝑖=(1/2)(1+𝛼)∑^𝑛_𝑖=1(𝐵_𝑖𝐿_𝑖). Hence, 𝑑𝑉(⋅)1𝑑𝑡≤−2𝛼𝑏𝑦²−𝑎𝑧²−12𝛼𝑏−𝛼^𝑛_𝑖=1𝐵_𝑖𝐿_𝑖𝜏_𝑖+(2𝜆+𝛼𝑐)𝜏𝑦²−122𝛼−^𝑛_𝑖=1𝐵_𝑖𝐿_𝑖𝜏_𝑖+2^𝑛_𝑖=1𝜆_𝑖𝜏_𝑖+𝑐𝜏𝑧². (2.12) Let 𝜏₀=max{𝜏,𝜏₁,𝜏₂,…,𝜏_𝑛}. Then 𝑑𝑉(⋅)1𝑑𝑡≤−2𝛼𝑏𝑦²−𝑎𝑧²−12𝛼𝑏−𝛼^𝑛_𝑖=1𝐵_𝑖𝐿_𝑖+(2𝜆+𝛼𝑐)𝜏₀𝑦²−122𝛼−^𝑛_𝑖=1𝐵_𝑖𝐿_𝑖+2^𝑛_𝑖=1𝜆_𝑖+𝑐𝜏₀𝑧². (2.13) Thus, if 𝜏₀<min𝛼𝑏𝑐+2𝛼𝑐+𝛼∑^𝑛_𝑖=1𝐵_𝑖𝐿_𝑖,2𝛼𝑐+∑^𝑛_𝑖=1(2+𝛼)𝐵_𝑖𝐿_𝑖, (2.14) then 𝑑𝑉(⋅)1𝑑𝑡≤−2𝛼𝑏𝑦²−𝑎𝑧². (2.15) This completes the proof of Theorem 2.1 (Burton [62], Hale [63], and Krasovskiĭ [64]).

Let 𝑝(⋅)=0.

Our second main result is given by the following theorem.

Theorem 2.2. In addition to all the assumptions of Theorem 2.1, we assume that the condition 𝑝(⋅)≤𝑞(𝑡) (2.16) holds, where |𝑞|∈𝐿¹(0,∞). If 𝜏₀<min𝛼𝑏𝑐+2𝛼𝑐+𝛼∑^𝑛_𝑖=1𝐵_𝑖𝐿_𝑖,2𝛼𝑐+∑^𝑛_𝑖=1(2+𝛼)𝐵_𝑖𝐿_𝑖, (2.17) then, there exists a finite positive constant 𝐾 such that the solution 𝑥(𝑡) of (1.1) defined by the initial function 𝑥(𝑡)=𝜙(𝑡),𝑥^′(𝑡)=𝜙^′(𝑡),𝑥^′′(𝑡)=𝜙^′′(𝑡) (2.18) satisfies |𝑥(𝑡)|≤𝐾,𝑥^′(𝑡)≤𝐾,𝑥^′′(𝑡)≤𝐾 (2.19) for all 𝑡≥𝑡₀, where 𝜙∈𝐶²([𝑡₀−𝑟,𝑡₀],ℜ).

Proof. Under the assumptions of Theorem 2.2, the time derivative of the Lyapunov-Krasovskii functional 𝑉(⋅) satisfies 𝑑𝑉(⋅)1𝑑𝑡≤−2𝛼𝑏𝑦²−𝑎𝑧²+𝛼𝑦+𝑧𝑝(⋅). (2.20) Using the estimates |𝑚|<1+𝑚² and 𝐷₄(𝑥²+𝑦²+𝑧²)≤𝑉(⋅),it follows that 𝑑𝑉(⋅)≤𝛼𝑦+|𝑧|𝑝(⋅𝑑𝑡)≤𝐷₅2+𝑦²+𝑧²𝑞(𝑡)≤2𝐷₅𝑞(𝑡)+𝐷₅𝐷₄⁻¹𝑉(⋅)𝑞(𝑡), (2.21) where 𝐷₅=max{1,𝛼}.
Integrating the above estimate from 0 to 𝑡, using the assumption |𝑞|∈𝐿¹(0,∞) and the Gronwall-Bellman inequality (see Gronwall [65] and Mitrinović [66]), we can conclude that all solutions of (1.1) are bounded.

Example 2.3. Consider the nonlinear differential equation of the third order with two deviating arguments as follows: 𝑥^′′′(𝑡)+111+1+𝑡²𝑥^′′(𝑡)+21+𝑒^−𝑡𝑥^′𝑡−𝜏₁+22+𝑒^−𝑡𝑥^′𝑡−𝜏₂+4𝑥^′(𝑡)()()=4+𝑥𝑡−𝜏+arc𝑡𝑔𝑥𝑡−𝜏1+𝑡²+𝑥²(𝑡)+𝑥²𝑡−𝜏₁+𝑥^′2𝑡−𝜏₂. (2.22) Writing (2.22) as a system of first order equations, we obtain 𝑥^′=𝑦,𝑦^′𝑧=𝑧,_′1=−11+1+𝑡²𝑧−25+2𝑒^−𝑡𝑦−𝑥−arc𝑡𝑔𝑥+21+𝑒^−𝑡𝑡_𝑡−𝜏1𝑧(𝑠)𝑑𝑠+22+𝑒^−𝑡𝑡_𝑡−𝜏2𝑧(𝑠)𝑑𝑠+^𝑡_𝑡−𝜏𝑦(𝑠)+𝑑𝑠_{𝑡𝑡−𝜏}𝑦(𝑠)1+𝑥²(𝑠)4𝑑𝑠+1+𝑡²+⋯+𝑦²𝑡−𝜏₂. (2.23) It follows that (2.22) is special case of (1.1), and when we compare (2.22) with (1.1) we obtain the following estimates: 𝑎(𝑡)1=11+1+𝑡²𝑏≥11=2×5+1,𝛼=5,𝑎=1,₁(𝑡)1=1+𝑒^𝑡1,1≤1+𝑒^𝑡≤2,𝛽₁=1,𝐵₁𝑔=2,₁𝑦=2𝑦,𝑔₁(0)𝑔=0,¹𝑦𝑦=2=𝑏₁,,𝑔𝑦≠0_′1𝑦=2=𝐿₁,^𝑦₀𝑔₁𝜂𝑑𝜂=^𝑦₀2𝜂𝑑𝜂=𝑦²,𝑏₂(𝑡)1=2+𝑒^𝑡1,2≤2+𝑒^𝑡≤3,𝛽₂=2,𝐵₂𝑔=3,₂𝑦=2𝑦,𝑔₂(0)𝑔=0,²𝑦𝑦=2=𝑏₂,,𝑔𝑦≠0_′2𝑦=2=𝐿₂,^𝑦₀𝑔₂𝜂𝑑𝜂=^𝑦₀2𝜂𝑑𝜂=𝑦²,𝑔𝑦(0)𝑔𝑦=4𝑦,𝑔=0,𝑦,ℎ(𝑥)(0)=4=𝑏,𝑦≠0=𝑥+arc𝑡𝑔𝑥,ℎ=0,ℎ^′(𝑥)1=1+1+𝑥²,0<2⁻¹<ℎ^′(𝑥)≤2,𝑐₁=2⁻¹𝑎,𝑐=2,_′(𝑡)=−2𝑡1+𝑡²²,𝑏^′₁(𝑡)1=−𝑒^𝑡,𝑏^′₂(𝑡)1=−𝑒^𝑡,(),𝑝()=4𝑡≥0𝑡,𝑥,…,𝑧1+𝑡²+⋯+𝑦²𝑡−𝜏₂≤41+𝑡²=𝑞(𝑡). (2.24) In view of the above discussion, it follows that 𝛼𝑏−𝑐=4>0,_2𝑖=1𝛼𝑏_𝑖𝑏_𝑖(𝑡)−𝑐𝑦²≥2⁻¹𝛼𝑎^′(𝑡)𝑦²+²_𝑖=1𝑏^′_𝑖(𝑡)^𝑦₀𝑔_𝑖𝜂=𝑑𝜂𝛼𝑏₁𝑏₁(𝑡)+𝛼𝑏₂𝑏₂(𝑡)−𝑐𝑦²=51+𝑒^−𝑡+102+𝑒^−𝑡−4𝑦²=21+15𝑒^−𝑡𝑦²,(),1𝑡≥02𝛼𝑎^′(𝑡)𝑦²+𝑏^′₁(𝑡)^𝑦₀𝑔₁𝜂𝑑𝜂+𝑏^′₁(𝑡)^𝑦₀𝑔₁𝜂𝑑𝜂=−5𝑡1+𝑡²²𝑦²−2𝑒^−𝑡𝑦²,(𝑡≥0),21+15𝑒^−𝑡𝑦²≥−5𝑡1+𝑡²²−2𝑒^−𝑡𝑦²,_∞0𝑞(𝑠)𝑑𝑠=^∞₀41+𝑠²𝑑𝑠=2𝜋<∞, (2.25) that is, |𝑞|∈𝐿¹(0,∞), and 𝜏₀<min𝛼𝑏𝑐+2𝛼𝑐+𝛼∑²_𝑖=1𝐵_𝑖𝐿_𝑖,2𝛼𝑐+∑²_𝑖=1(2+𝛼)𝐵_𝑖𝐿_𝑖5=min,518=536.36 (2.26) Thus, all the assumptions of Theorems 2.1 and 2.2 hold. This shows that the zero solution of (2.22) is stable and all solutions of the same equation are bounded, when 𝑝(⋅)=0 and ≠0, respectively.