On Inequalities of Lyapunov for Two-Dimensional Nonlinear Dynamic Systems on Time Scales

Abstract

We establish some new Lyapunov-type inequalities for two-dimensional nonlinear dynamic systems on time scales. As for application, boundedness of the Emden-Fowler-type equation is proved.

1. Introduction

In this paper, we establish some Lyapunov-type inequalities for the following two-dimensional nonlinear dynamic system: 𝑥^Δ1Δ2(𝑠,𝑡)=𝛼₁(𝑠,𝑡)𝑥(𝜎(𝑠),𝜎(𝑡))+𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢(),𝑢𝑠,𝑡_Δ1Δ2(𝑠,𝑡)=−𝛽₂(𝑠,𝑡)|𝑥(𝜎(𝑠),𝜎(𝑡))|^𝛽−2𝑥(𝜎(𝑠),𝜎(𝑡))−𝛼₁(𝑠,𝑡)𝑢(𝑠,𝑡), (1) where𝑠,𝑡∈𝕋=a time scale,Δ₁denotes the delta derivative with respect to𝑠, andΔ₂denotes the delta derivative with respect to𝑡.

Lyapunov-type inequalities have proven to be very useful in the study of qualitative behavior of solutions such as oscillation, disconjugacy, and eigenvalue problems for differential equations and difference equations. Since the appearance of Lyapunov's fundamental paper [1], considerable attention has been given to various extensions and improvements of the Lyapunov-type inequality from different viewpoints [2–8]. Although Lyapunov-type inequalities are well developed for the continuous cases, their time scale versions are still in early stages and are worth due attention.

Recently, He et al. in [2] considered the linear Hamiltonian system 𝑥^Δ(𝑡)(𝑡)𝑥(𝜎(𝑡(𝑡)𝑦(𝑡),𝑦=𝛼))+𝛽_Δ(𝑡)=−𝛾(𝑡)𝑥(𝜎(𝑡))−𝛼(𝑡)𝑦(𝑡) (2) and obtained several useful Lyapunov-type inequalities.

Chen et al. in [3] considered the nonlinear system 𝜕²𝑥(𝑠,𝑡)𝜕𝑠𝜕𝑡=𝛼₁(𝑠,𝑡)𝑥(𝑠,𝑡)+𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢(),𝜕𝑠,𝑡₂𝑢(𝑠,𝑡)𝜕𝑠𝜕𝑡=−𝛽₂(𝑠,𝑡)|𝑥(𝑠,𝑡)|^𝛽−2𝑥(𝑠,𝑡)−𝛼₁(𝑠,𝑡)𝑢(𝑠,𝑡) (3) and obtained some interesting Lyapunov-type inequalities for partial differential equations.

In this paper, under the assumption of existence of a nontrivial solution(𝑥(𝑠,𝑡),𝑢(𝑠,𝑡))to the 2-dimensional nonlinear dynamic system (1), some new and interesting Lyapunov-type inequalities are established.

2. Main Results

Throughout this paper, the following mild and natural conditions are assumed:(i)𝛾>1,𝛽>1are real constants,(ii)𝛼₁(𝑠,𝑡),  𝛽₁(𝑠,𝑡),𝛽₂(𝑠,𝑡)∶[𝑠₀,∞)_𝕋×[𝑡₀,∞)_𝕋→𝑅are rd-continuous functions such that𝛽₁(𝑠,𝑡)>0for(𝑠,𝑡)∈[𝑠₀,∞)_𝕋×[𝑡₀,∞)_𝕋, where[𝑠₀,∞)_𝕋∶=[𝑠₀,∞)∩𝕋, and𝜎is the forward jump operator; that is,𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡}.

Theorem 1. If the nonlinear dynamic system (1) has a real solution(𝑥(𝑠,𝑡),𝑢(𝑠,𝑡))which is not identically zero on[𝑎,𝑏]_𝕋×[𝑐,𝑑]_𝕋satisfying𝑥(𝑎,𝑡)=𝑥(𝑏,𝑡)=𝑥(𝑠,𝑐)=𝑥(𝑠,𝑑)=0and𝑥^Δ1(𝑠,𝜎(𝑡))𝑢^Δ2(𝑠,𝑡)+𝑥^Δ2(𝜎(𝑠),𝑡)𝑢^Δ1(𝑠,𝑡)=0for all(𝑠,𝑡)∈[𝑎,𝑏]_𝕋×[𝑐,𝑑]_𝕋, where𝑎,𝑏,𝑐,𝑑∈𝕋with𝑎<𝑏,  𝑐<𝑑, then 2≤^𝑏_𝑎^𝑑_𝑐𝛼₁(𝑠,𝑡)Δ𝑡Δ𝑠+𝑀^{𝛽/𝛼−1𝑏}_𝑎^𝑑_𝑐𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾⋅^𝑏_𝑎^𝑑_𝑐𝛽⁺₂(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛼, (4) where1/𝛼+1/𝛾=1,𝑀∶=max{|𝑥(𝑠,𝑡)|∶𝜎(𝑎)≤𝑠<𝑏,𝜎(𝑐)≤𝑡<𝑑}, and𝛽⁺₂(𝑠,𝑡)∶=max{𝛽₂(𝑠,𝑡),0}.

Proof. From the conditions𝑥(𝑎,𝑡)=𝑥(𝑏,𝑡)=𝑥(𝑠,𝑐)=𝑥(𝑠,𝑑)=0and𝑥(𝑠,𝑡)is not identically zero on[𝑎,𝑏]_𝕋×[𝑐,𝑑]_𝕋, there exists(𝜏₁,𝜏₂)such that|𝑥(𝜏₁,𝜏₂)|=𝑀>0.
Multiplying the first equation of (1) by𝑢(𝑠,𝑡)and the second one by𝑥(𝜎(𝑠),𝜎(𝑡))and adding up, we get 𝑥^Δ1Δ2(𝑠,𝑡)𝑢(𝑠,𝑡)+𝑢^Δ1Δ2(𝑠,𝑡)𝑥(𝜎(𝑠),𝜎(𝑡))=𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−𝛽₂(𝑠,𝑡)|𝑥(𝜎(𝑠),𝜎(𝑡))|^𝛽, (5) and, hence, (𝑥(𝑠,𝑡)𝑢(𝑠,𝑡))^Δ1Δ2=𝑥^Δ1Δ2(𝑠,𝑡)𝑢(𝑠,𝑡)+𝑥^Δ2(𝜎(𝑠),𝑡)𝑢^Δ1(𝑠,𝑡)+𝑥^Δ1(𝑠,𝜎(𝑡))𝑢^Δ2()(𝜎(𝑠)(𝑡𝑢𝑠,𝑡+𝑥,𝜎))^Δ1Δ2(𝑠,𝑡)=𝑥^Δ1Δ2(𝑠,𝑡)𝑢(𝑠,𝑡)+𝑢^Δ1Δ2(𝑠,𝑡)𝑥(𝜎(𝑠),𝜎(𝑡))=𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−𝛽₂(𝑠,𝑡)|𝑥(𝜎(𝑠),𝜎(𝑡))|^𝛽. (6) Integrating the left hand side of (6) over𝑡from𝑐to𝑑and then over𝑠from𝑎to𝑏, we get ^𝑏_𝑎^𝑑_𝑐(𝑥(𝑠,𝑡)𝑢(𝑠,𝑡))^Δ1Δ2=Δ𝑡Δ𝑠_𝑏𝑎(𝑥(𝑠,𝑑)𝑢(𝑠,𝑑))^Δ1Δ𝑠−^𝑏_𝑎(𝑥(𝑠,𝑐)𝑢(𝑠,𝑐))^Δ1()𝑢()()𝑢()()𝑢()()𝑢().Δ𝑠=𝑥𝑏,𝑑𝑏,𝑑−𝑥𝑎,𝑑𝑎,𝑑−𝑥𝑏,𝑐𝑏,𝑐+𝑥𝑎,𝑐𝑎,𝑐 (7) Noting that𝑥(𝑎,𝑡)=𝑥(𝑏,𝑡)=0, we have ^𝑏_𝑎^𝑑_𝑐𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾=Δ𝑡Δ𝑠_{𝑏𝑎𝑑𝑐}𝛽₂(𝑠,𝑡)|𝑥(𝜎(𝑠),𝜎(𝑡))|^𝛽Δ𝑡Δ𝑠. (8) On the other hand, integrating the first equation of (1) over𝑡from𝑐to𝜏₂and then over𝑠from𝑎to𝜏₁, we get ^𝜏1_𝑎^𝜏2_𝑐𝑥^Δ1Δ2()=𝑠,𝑡Δ𝑡Δ𝑠_{𝜏1𝑎𝜏2𝑐}𝛼₁(𝑠,𝑡)𝑥(𝜎(𝑠),𝜎(𝑡))+𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢(𝑠,𝑡)Δ𝑡Δ𝑠. (9) By the boundary conditions on𝑥, it is elementary to verify that ^𝜏1_𝑎^𝜏2_𝑐(𝑥(𝑠,𝑡))^Δ1Δ2Δ𝑡Δ𝑠=𝑥𝜏₁,𝜏₂, (10) and so 𝑥𝜏₁,𝜏₂=^𝜏1_𝑎^𝜏2_𝑐𝛼₁(𝑠,𝑡)𝑥(𝜎(𝑠),𝜎(𝑡))+𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢(𝑠,𝑡)Δ𝑡Δ𝑠. (11) Hence, 𝑥𝜏₁,𝜏₂≤^𝜏1_𝑎^𝜏2_𝑐𝛼₁(𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡)|,𝜎))|Δ𝑡Δ𝑠_{𝜏1𝑎𝜏2𝑐}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−1Δ𝑡Δ𝑠. (12) By similar arguments, we easily get 𝑥𝜏₁,𝜏₂≤^𝑏_𝜏1^𝑑_𝜏2𝛼₁(𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡)|,𝜎))|Δ𝑡Δ𝑠_{𝑏𝜏1𝑑𝜏2}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−1Δ𝑡Δ𝑠. (13) Summing (12) and (13) and by Hölder's inequality with indices𝛾and𝛼, we obtain 2𝑥𝜏₁,𝜏₂≤^𝑏_𝑎^𝑑_𝑐𝛼₁(×|𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡),𝜎))|Δ𝑡Δ𝑠_{𝑏𝑎𝑑𝑐}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−1≤Δ𝑡Δ𝑠_{𝑏𝑎𝑑𝑐}𝛼₁(×|𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡),𝜎))|Δ𝑡Δ𝑠_{𝑏𝑎𝑑𝑐}𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾×^𝑏_𝑎^𝑑_𝑐𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾Δ𝑡Δ𝑠^1/𝛼. (14)
In view of (8), we have2𝑥𝜏₁,𝜏₂≤^𝑏_𝑎^𝑑_𝑐𝛼₁(𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡)|,𝜎))|Δ𝑡Δ𝑠_{𝑏𝑎𝑑𝑐}𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾×^𝑏_𝑎^𝑑_𝑐𝛽₂(𝑠,𝑡)|𝑥(𝜎(𝑠),𝜎(𝑡))|^𝛽Δ𝑡Δ𝑠^1/𝛼, (15) and so 2≤^𝑏_𝑎^𝑑_𝑐𝛼₁(𝑠,𝑡)Δ𝑡Δ𝑠+𝑀^{𝛽/𝛼−1𝑏}_𝑎^𝑑_𝑐𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾⋅^𝑏_𝑎^𝑑_𝑐𝛽⁺₂(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛼. (16) The proof is complete.

Remark 2. It is interesting to note that when𝕋=ℝTheorem 1 reduces to Theorem2.1of [3].

Theorem 3. If the nonlinear dynamic system (1) has a real solution(𝑥(𝑠,𝑡),𝑢(𝑠,𝑡))which is not identically zero on[𝑎,𝑏]_𝕋×[𝑐,𝑑]_𝕋satisfying𝑥(𝑠,𝑐)=𝑥(𝑠,𝑑)=0,𝑥(𝑎,𝑡)𝑥(𝜎(𝑎),𝑡)<0,𝑥(𝑏,𝑡)𝑥(𝜎(𝑏),𝑡)<0, and𝑥^Δ1(𝑠,𝜎(𝑡))𝑢^Δ2(𝑠,𝑡)+𝑥^Δ2(𝜎(𝑠),𝑡)𝑢^Δ1(𝑠,𝑡)=0for all(𝑠,𝑡)∈[𝑎,𝑏]_𝕋×[𝑐,𝑑]_𝕋, where𝑎,𝑏,𝑐,𝑑∈𝕋with𝑎<𝑏,𝑐<𝑑, then 2≤_𝑎^{𝜎(𝑏)𝑑}_𝑐𝛼₁(𝑠,𝑡)Δ𝑡Δ𝑠+𝑀^{𝛽/𝛼−1}_𝑎^{𝜎(𝑏)𝑑}_𝑐𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾⋅_𝑎^{𝜎(𝑏)𝑑}_𝑐𝛽⁺₂(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛼, (17) where1/𝛼+1/𝛾=1and𝑀,𝛽⁺₂(𝑠,𝑡)are as defined in Theorem 1.

Proof. Choose(𝜏₁,𝜏₂)such that|𝑥(𝜏₁,𝜏₂)|=𝑀. Note that𝑀>0. From (6) and _𝑎^{𝜎(𝑏)𝑑}_𝑐(𝑥(𝑠,𝑡)𝑢(𝑠,𝑡))^Δ1Δ2(𝜎(𝑏))𝑢(𝜎(𝑏))()𝑢()(𝜎(𝑏))𝑢(𝜎(𝑏))()𝑢()Δ𝑡Δ𝑠=𝑥,𝑑,𝑑−𝑥𝑎,𝑑𝑎,𝑑−𝑥,𝑐,𝑐+𝑥𝑎,𝑐𝑎,𝑐=0, (18) we have _𝑎^{𝜎(𝑏)𝑑}_𝑐𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾=Δ𝑡Δ𝑠_{𝑎𝜎(𝑏)𝑑𝑐}𝛽₂(𝑠,𝑡)|𝑥(𝜎(𝑠),𝜎(𝑡))|^𝛽Δ𝑡Δ𝑠. (19) By the boundary conditions on𝑥, it is elementary to check that ^𝜏1_𝑎^𝜏2_𝑐(𝑥(𝑠,𝑡))^Δ1Δ2Δ𝑡Δ𝑠=𝑥𝜏₁,𝜏₂−𝑥𝑎,𝜏₂. (20) So 𝑥𝜏₁,𝜏₂=𝑥𝑎,𝜏₂+^𝜏1_𝑎^𝜏2_𝑐𝛼₁()𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡,𝜎))Δ𝑡Δ𝑠_{𝜏1𝑎𝜏2𝑐}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢(𝑠,𝑡)Δ𝑡Δ𝑠. (21) For the fixed𝜏₂, by𝑥(𝑎,𝜏₂)𝑥(𝜎(𝑎),𝜏₂)<0, there exists𝜉∈(0,1)such that (1−𝜉)𝑥𝑎,𝜏₂+𝜉𝑥𝜎(𝑎),𝜏₂=0. (22) Integrating the first equation of (1) over𝑡from𝑐to𝜏₂, we obtain 𝑥^Δ1𝑠,𝜏₂−𝑥^Δ1()=𝑠,𝑐_𝜏2𝑐𝛼₁()𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡,𝜎))Δ𝑡_𝜏2𝑐𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢(𝑠,𝑡)Δ𝑡. (23) Multiplying (23) by𝜇(𝑠)and noting that𝑥(𝑠,𝑐)=0, we get 𝑥𝜎(𝑠),𝜏₂−𝑥𝑠,𝜏₂=𝜇(𝑠)^𝜏2_𝑐𝛼₁()𝑥(𝜎(𝑠)(𝑡(𝑠)𝑠,𝑡,𝜎))Δ𝑡+𝜇^𝜏2_𝑐𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢(𝑠,𝑡)Δ𝑡. (24) Letting𝑠=𝑎, we get 𝑥𝜎(𝑎),𝜏₂−𝑥𝑎,𝜏₂=𝜇(𝑎)^𝜏2_𝑐𝛼₁()𝑥(𝜎(𝑎)(𝑡(𝑎)𝑎,𝑡,𝜎))Δ𝑡+𝜇^𝜏2_𝑐𝛽₁(𝑎,𝑡)|𝑢(𝑎,𝑡)|^𝛾−2𝑢(𝑎,𝑡)Δ𝑡. (25) By (22) and (25), we obtain 𝑥𝑎,𝜏₂=−𝜉𝜇(𝑎)^𝜏2_𝑐𝛼₁()𝑥(𝜎(𝑎)(𝑡(𝑎)𝑎,𝑡,𝜎))Δ𝑡−𝜉𝜇^𝜏2_𝑐𝛽₁(𝑎,𝑡)|𝑢(𝑎,𝑡)|^𝛾−2𝑢(𝑎,𝑡)Δ𝑡. (26) Substituting (26) into (21), we get 𝑥𝜏₁,𝜏₂=−𝜉𝜇(𝑎)^𝜏2_𝑐𝛼₁()𝑥(𝜎(𝑎)(𝑡(𝑎)𝑎,𝑡,𝜎))Δ𝑡−𝜉𝜇^𝜏2_𝑐𝛽₁(𝑎,𝑡)|𝑢(𝑎,𝑡)|^𝛾−2𝑢()+𝑎,𝑡Δ𝑡_{𝜏1𝑎𝜏2𝑐}𝛼₁()𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡,𝜎))Δ𝑡Δ𝑠_{𝜏1𝑎𝜏2𝑐}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢()(𝑎)𝑠,𝑡Δ𝑡Δ𝑠=−𝜉𝜇^𝜏2_𝑐𝛼₁()𝑥(𝜎(𝑎)(𝑡(𝑎)𝑎,𝑡,𝜎))Δ𝑡+𝜇^𝜏2_𝑐𝛼₁()𝑥(𝜎(𝑎)(𝑡(𝑎)𝑎,𝑡,𝜎))Δ𝑡−𝜉𝜇^𝜏2_𝑐𝛽₁(𝑎,𝑡)|𝑢(𝑎,𝑡)|^𝛾−2𝑢()(𝑎)𝑎,𝑡Δ𝑡+𝜇^𝜏2_𝑐𝛽₁(𝑎,𝑡)|𝑢(𝑎,𝑡)|^𝛾−2𝑢()+𝑎,𝑡Δ𝑡_{𝜏1𝜎(𝑎)𝜏2𝑐}𝛼₁()𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡,𝜎))Δ𝑡Δ𝑠_{𝜏1𝜎(𝑎)𝜏2𝑐}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢()=()𝜇(𝑎)𝑠,𝑡Δ𝑡Δ𝑠1−𝜉^𝜏2_𝑐𝛼₁()𝑥(𝜎(𝑎)(𝑡+()𝜇(𝑎)𝑎,𝑡,𝜎))Δ𝑡1−𝜉^𝜏2_𝑐𝛽₁(𝑎,𝑡)|𝑢(𝑎,𝑡)|^𝛾−2𝑢()+𝑎,𝑡Δ𝑡_{𝜏1𝜎(𝑎)𝜏2𝑐}𝛼₁()𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡,𝜎))Δ𝑡Δ𝑠_{𝜏1𝜎(𝑎)𝜏2𝑐}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢(𝑠,𝑡)Δ𝑡Δ𝑠. (27) Immitating the arguments from (21) to (27) step by step, we have 𝑥𝜏₁,𝜏₂=𝑥𝑏,𝜏₂+^𝑏_𝜏1^𝑑_𝜏2𝛼₁()𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡,𝜎))Δ𝑡Δ𝑠_{𝑏𝜏1𝑑𝜏2}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢(𝑠,𝑡)Δ𝑡Δ𝑠. (28) By𝑥(𝑏,𝜏₂)𝑥(𝜎(𝑏),𝜏₂)<0, there exists𝜂∈(0,1)such that 1−𝜂𝑥𝑏,𝜏₂+𝜂𝑥𝜎(𝑏),𝜏₂=0. (29) Integrating the first equation of (1) over𝑡from𝜏₂to𝑑and using𝑥(𝑠,𝑑)=0, we get 𝑥𝑏,𝜏₂=𝜂𝜇(𝑏)^𝑑_𝜏2𝛼₁()𝑥(𝜎(𝑏)(𝑡(𝑏)𝑏,𝑡,𝜎))Δ𝑡+𝜂𝜇^𝑑_𝜏2𝛽₁(𝑏,𝑡)|𝑢(𝑏,𝑡)|^𝛾−2𝑢(𝑏,𝑡)Δ𝑡, (30) and, hence, 𝑥𝜏₁,𝜏₂=𝜂𝜇(𝑏)^𝑑_𝜏2𝛼₁()𝑥(𝜎(𝑏)(𝑡(𝑏)𝑏,𝑡,𝜎))Δ𝑡+𝜂𝜇^𝑑_𝜏2𝛽₁(𝑏,𝑡)|𝑢(𝑏,𝑡)|^𝛾−2𝑢()+𝑏,𝑡Δ𝑡_{𝑏𝜏1𝑑𝜏2}𝛼₁()𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡,𝜎))Δ𝑡Δ𝑠_{𝑏𝜏1𝑑𝜏2}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢(𝑠,𝑡)Δ𝑡Δ𝑠. (31) Let 𝛼₁()=()𝛼𝑠,𝑡1−𝜉₁()𝛼𝑎,𝑡,𝑠=𝑎,₁(𝑠,𝑡),𝜎(𝑎)≤𝑠<𝑏,𝜂𝛼₁()𝛽𝑏,𝑡,𝑠=𝑏,₁()=()𝛽𝑠,𝑡1−𝜉₁()𝛽𝑎,𝑡,𝑠=𝑎,₁(𝑠,𝑡),𝜎(𝑎)≤𝑠<𝑏,𝜂𝛽₁(𝑏,𝑡),𝑠=𝑏. (32) Then, (27) can be written as 𝑥𝜏₁,𝜏₂=^𝜏1_𝑎^𝜏2_𝑐𝛼₁()𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡,𝜎))Δ𝑡Δ𝑠_{𝜏1𝑎𝜏2𝑐}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢(𝑠,𝑡)Δ𝑡Δ𝑠, (33) and (31) can be written as 𝑥𝜏₁,𝜏₂=_𝜏^𝜎(𝑏)₁^𝑑_𝜏2𝛼₁()𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡,𝜎))Δ𝑡Δ𝑠_{𝜏𝜎(𝑏)1𝑑𝜏2}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢(𝑠,𝑡)Δ𝑡Δ𝑠. (34) It follows from (33) and (34) that 𝑥𝜏₁,𝜏₂≤^𝜏1_𝑎^𝜏2_𝑐𝛼₁(𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡)|,𝜎))|Δ𝑡Δ𝑠_{𝜏1𝑎𝜏2𝑐}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−1𝑥𝜏Δ𝑡Δ𝑠,₁,𝜏₂≤_𝜏^𝜎(𝑏)₁^𝑑_𝜏2𝛼₁(𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡)|,𝜎))|Δ𝑡Δ𝑠_{𝜏𝜎(𝑏)1𝑑𝜏2}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−1Δ𝑡Δ𝑠. (35) Using (19),|𝛼₁(𝑠,𝑡)|≤|𝛼₁(𝑠,𝑡)|,𝛽₁(𝑠,𝑡)≤𝛽₁(𝑠,𝑡), and Hölder's inequality, we have 2𝑥𝜏₁,𝜏₂≤_𝑎^{𝜎(𝑏)𝑑}_𝑐𝛼₁(𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡)|,𝜎))|Δ𝑡Δ𝑠_{𝑎𝜎(𝑏)𝑑𝑐}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−1≤Δ𝑡Δ𝑠_{𝑎𝜎(𝑏)𝑑𝑐}𝛼₁(𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡)|,𝜎))|Δ𝑡Δ𝑠_{𝑎𝜎(𝑏)𝑑𝑐}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−1≤Δ𝑡Δ𝑠_{𝑎𝜎(𝑏)𝑑𝑐}𝛼₁(𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡)|,𝜎))|Δ𝑡Δ𝑠_{𝑎𝜎(𝑏)𝑑𝑐}𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾×_𝑎^{𝜎(𝑏)𝑑}_𝑐𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾Δ𝑡Δ𝑠^1/𝛼≤_𝑎^{𝜎(𝑏)𝑑}_𝑐𝛼₁(𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡)|,𝜎))|Δ𝑡Δ𝑠_{𝑎𝜎(𝑏)𝑑𝑐}𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾×_𝑎^{𝜎(𝑏)𝑑}_𝑐𝛽₂(𝑠,𝑡)|𝑥(𝜎(𝑠),𝜎(𝑡))|^𝛽Δ𝑡Δ𝑠^1/𝛼≤𝑀_𝑎^{𝜎(𝑏)𝑑}_𝑐𝛼₁(𝑠,𝑡)Δ𝑡Δ𝑠+𝑀^𝛽/𝛼_𝑎^{𝜎(𝑏)𝑑}_𝑐𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾×_𝑎^{𝜎(𝑏)𝑑}_𝑐𝛽⁺₂(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛼. (36) Therefore, 2≤_𝑎^{𝜎(𝑏)𝑑}_𝑐𝛼₁(𝑠,𝑡)Δ𝑡Δ𝑠+𝑀^{𝛽/𝛼−1}_𝑎^{𝜎(𝑏)𝑑}_𝑐𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾⋅_𝑎^{𝜎(𝑏)𝑑}_𝑐𝛽⁺₂(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛼. (37) The proof is complete.

Theorem 4. Suppose that𝛽and𝛾are conjugate exponents; that is,1/𝛽+1/𝛾=1. If the nonlinear dynamic system (1) has a real solution(𝑥(𝑠,𝑡),𝑢(𝑠,𝑡))which is not identically zero on[𝑎,𝑏]_𝕋×[𝑐,𝑑]_𝕋such that𝑥(𝑎,𝑡)=𝑥(𝑠,𝑐)=𝑥(𝑠,𝑑)=0and𝑥^Δ1(𝑠,𝜎(𝑡))𝑢^Δ2(𝑠,𝑡)+𝑥^Δ2(𝜎(𝑠),𝑡)𝑢^Δ1(𝑠,𝑡)=0for all(𝑠,𝑡)∈[𝑎,𝑏]_𝕋×[𝑐,𝑑]_𝕋, where𝑎,𝑏,𝑐,𝑑∈𝕋with𝑎<𝑏,𝑐<𝑑, then 1≤_𝑎^{𝜌(𝑏)𝑑}_𝑐𝛼₁(𝑠,𝑡)Δ𝑡Δ𝑠+_𝑎^{𝜌(𝑏)𝑑}_𝑐𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾⋅_𝑎^{𝜌(𝑏)𝑑}_𝑐𝛽⁺₂(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛽. (38)

Proof. By (6) and the conditions𝑥(𝑠,𝑐)=𝑥(𝑠,𝑑)=0for all𝑠∈[𝑎,𝑏]_𝕋, we have _𝑎^{𝜌(𝑏)𝑑}_𝑐(𝑥(𝑠,𝑡)𝑢(𝑠,𝑡))^Δ1Δ2𝜌(𝑏)𝑢𝜌(𝑏)()𝑢()𝜌(𝑏)𝑢𝜌(𝑏)()𝑢()Δ𝑡Δ𝑠=𝑥,𝑑,𝑑−𝑥𝑎,𝑑𝑎,𝑑−𝑥,𝑐,𝑐+𝑥𝑎,𝑐𝑎,𝑐=0. (39) So _𝑎^{𝜌(𝑏)𝑑}_𝑐𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾=Δ𝑡Δ𝑠_{𝑎𝜌(𝑏)𝑑𝑐}𝛽₂(𝑠,𝑡)|𝑥(𝜎(𝑠),𝜎(𝑡))|^𝛽Δ𝑡Δ𝑠. (40) Fix(𝜏₁,𝜏₂)such that|𝑥(𝜏₁,𝜏₂)|=max{|𝑥(𝑠,𝑡)|∶𝜎(𝑎)≤𝑠<𝑏,𝜎(𝑐)≤𝑡<𝑑}>0. Integrating the first equation of (1) over𝑡from𝑐to𝜏₂and then over𝑠from𝑎to𝜏₁, we get 𝑥𝜏₁,𝜏₂=^𝜏1_𝑎^𝜏2_𝑐𝛼₁()𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡,𝜎))Δ𝑡Δ𝑠_{𝜏1𝑎𝜏2𝑐}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢(𝑠,𝑡)Δ𝑡Δ𝑠, (41) and so by Hölder's inequality with indices𝛾and𝛽we have 𝑥𝜏₁,𝜏₂≤_𝑎^{𝜌(𝑏)𝑑}_𝑐𝛼₁(𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡)|,𝜎))|Δ𝑡Δ𝑠_{𝑎𝜌(𝑏)𝑑𝑐}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−1Δ𝑡Δ𝑠≤_𝑎^{𝜌(𝑏)𝑑}_𝑐𝛼₁(𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡)|,𝜎))|Δ𝑡Δ𝑠_{𝑎𝜌(𝑏)𝑑𝑐}𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾⋅_𝑎^{𝜌(𝑏)𝑑}_𝑐𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾Δ𝑡Δ𝑠^1/𝛽≤𝑥𝜏₁,𝜏₂⋅_𝑎^{𝜌(𝑏)𝑑}_𝑐𝛼₁(+𝑠,𝑡)Δ𝑡Δ𝑠_{𝑎𝜌(𝑏)𝑑𝑐}𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾⋅_𝑎^{𝜌(𝑏)𝑑}_𝑐𝛽₂(𝑠,𝑡)|𝑥(𝜎(𝑠),𝜎(𝑡))|^𝛽Δ𝑡Δ𝑠^1/𝛽. (42)
Hence, 1≤_𝑎^{𝜌(𝑏)𝑑}_𝑐𝛼₁(+𝑠,𝑡)Δ𝑡Δ𝑠_{𝑎𝜌(𝑏)𝑑𝑐}𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾⋅_𝑎^{𝜌(𝑏)𝑑}_𝑐𝛽⁺₂(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛽. (43) The proof is complete.

Theorem 5. Suppose that𝛽and𝛾are conjugate exponents. If the nonlinear dynamic system (1) has a real solution(𝑥(𝑠,𝑡),𝑢(𝑠,𝑡))which is not identically zero on[𝑎,𝑏]_𝕋×[𝑐,𝑑]_𝕋such that𝑥(𝑏,𝑡)=𝑥(𝑠,𝑐)=𝑥(𝑠,𝑑)=0and𝑥^Δ1(𝑠,𝜎(𝑡))𝑢^Δ2(𝑠,𝑡)+𝑥^Δ2(𝜎(𝑠),𝑡)𝑢^Δ1(𝑠,𝑡)=0for all(𝑠,𝑡)∈[𝑎,𝑏]_𝕋×[𝑐,𝑑]_𝕋, where𝑎,𝑏,𝑐,𝑑∈𝕋with𝑎<𝑏,𝑐<𝑑, then 1≤^𝑏_𝜎(𝑎)^𝑑_𝑐𝛼₁(𝑠,𝑡)Δ𝑡Δ𝑠+^𝑏_𝜎(𝑎)^𝑑_𝑐𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾⋅^𝑏_𝜎(𝑎)^𝑑_𝑐𝛽⁺₂(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛽. (44)

Proof. By (6) and the assumption that𝑥(𝑠,𝑐)=𝑥(𝑠,𝑑)=0for all𝑠∈[𝑎,𝑏]_𝕋, we get ^𝑏_𝜎(𝑎)^𝑑_𝑐(𝑥(𝑠,𝑡)𝑢(𝑠,𝑡))^Δ1Δ2()𝑢()(𝜎(𝑎))𝑢(𝜎(𝑎))()𝑢()(𝜎(𝑎))𝑢(𝜎(𝑎))Δ𝑡Δ𝑠=𝑥𝑏,𝑑𝑏,𝑑−𝑥,𝑑,𝑑−𝑥𝑏,𝑐𝑏,𝑐+𝑥,𝑐,𝑐=0. (45) So ^𝑏_𝜎(𝑎)^𝑑_𝑐𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾=Δ𝑡Δ𝑠_{𝑏𝜎(𝑎)𝑑𝑐}𝛽₂(𝑠,𝑡)|𝑥(𝜎(𝑠),𝜎(𝑡))|^𝛽Δ𝑡Δ𝑠. (46) Fix(𝜏₁,𝜏₂)such that|𝑥(𝜏₁,𝜏₂)|=max{|𝑥(𝑠,𝑡)|∶𝜎(𝑎)≤𝑠≤𝜌(𝑏),𝜎(𝑡)≤𝑡≤𝜌(𝑏)}>0. Integrating the first equation of (1) over𝑡from𝜏₂to𝑑and then over𝑠from𝜏₁to𝑏, we get 𝑥𝜏₁,𝜏₂=^𝑏_𝜏1^𝑑_𝜏2𝛼₁()𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡,𝜎))Δ𝑡Δ𝑠_{𝑏𝜏1𝑑𝜏2}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢(𝑠,𝑡)Δ𝑡Δ𝑠, (47) and so by Hölder's inequality with indices𝛾and𝛽we have 𝑥𝜏₁,𝜏₂≤^𝑏_𝜎(𝑎)^𝑑_𝑐𝛼₁(𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡)|,𝜎))|Δ𝑡Δ𝑠_{𝑏𝜎(𝑎)𝑑𝑐}𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−1≤Δ𝑡Δ𝑠_{𝑏𝜎(𝑎)𝑑𝑐}𝛼₁(𝑥(𝜎(𝑠)(𝑡+𝑠,𝑡)|,𝜎))|Δ𝑡Δ𝑠_{𝑏𝜎(𝑎)𝑑𝑐}𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾⋅^𝑏_𝜎(𝑎)^𝑑_𝑐𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾Δ𝑡Δ𝑠^1/𝛽≤𝑥𝜏₁,𝜏₂^𝑏_𝜎(𝑎)^𝑑_𝑐𝛼₁(+𝑠,𝑡)Δ𝑡Δ𝑠_{𝑏𝜎(𝑎)𝑑𝑐}𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾⋅^𝑏_𝜎(𝑎)^𝑑_𝑐𝛽₂(𝑠,𝑡)|𝑥(𝜎(𝑠),𝜎(𝑡)|^𝛽Δ𝑡Δ𝑠^1/𝛽. (48) Hence, 1≤^𝑏_𝜎(𝑎)^𝑑_𝑐𝛼₁(+𝑠,𝑡)Δ𝑡Δ𝑠_{𝑏𝜎(𝑎)𝑑𝑐}𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾⋅^𝑏_𝜎(𝑎)^𝑑_𝑐𝛽⁺₂(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛽. (49) The proof is complete.

Remark 6. Analogously, we can also consider the cases (i)𝑥(𝑎,𝑡)=𝑥(𝑏,𝑡)=𝑥(𝑠,𝑑)=0and (ii)𝑥(𝑎,𝑡)=𝑥(𝑏,𝑡)=𝑥(𝑠,𝑐)=0. Similar results to those in Theorem 4 and Theorem 5 can easily be arrived at. The detailed proofs are omitted here.

Next, we exhibit an application of our results. Consider the following special case of (1): 𝑥^Δ1Δ2(𝑠,𝑡)=𝛽₁(𝑠,𝑡)|𝑢(𝑠,𝑡)|^𝛾−2𝑢(),𝑢𝑠,𝑡_Δ1Δ2(𝑠,𝑡)=−𝛽₂(𝑠,𝑡)|𝑥(𝜎(𝑠),𝜎(𝑡))|^𝛽−2𝑥(𝜎(𝑠),𝜎(𝑡)). (50)

Definition 7. A nontrivial solution(𝑥(𝑠,𝑡),𝑢(𝑠,𝑡))of the dynamic equation (50) defined on[𝑠₀,∞)_𝕋×[𝑡₀,∞)_𝕋is said to be proper if sup|𝑥(𝑠,𝑡)|+|𝑢(𝑠,𝑡)|∶𝑠∈[𝑎,∞)_𝕋,𝑡∈[𝑐,∞)_𝕋>0 (51) for all𝑎≥𝑠₀,  𝑐≥𝑡₀.A proper solution(𝑥(𝑠,𝑡),𝑢(𝑠,𝑡))of the dynamic system (50) is called weakly oscillatory if at least one argument has a sequence of zeros diverging to∞.

Theorem 8. Assume that𝑢(𝜏,𝑡)is bounded on[𝑡₀,∞)_𝕋for a fixed𝜏,𝑢(𝑠,𝜂)is bounded on[𝑠₀,∞)_𝕋for a fixed𝜂, ^∞_𝑠0^∞_𝑡0𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠<∞,_{∞𝑠0∞𝑡0}𝛽₂(𝑠,𝑡)Δ𝑡Δ𝑠<∞, (52) then every weakly oscillatory proper solution of (50) is bounded on[𝑠₀,∞)_𝕋×[𝑡₀,∞)_𝕋.

Proof. Let(𝑥(𝑠,𝑡),𝑢(𝑠,𝑡))be a nontrivial weakly oscillatory proper solution of (50) on[𝑠₀,∞)_𝕋×[𝑡₀,∞)_𝕋,and𝑥(𝑠,𝑡)have a sequence of zeros diverging to∞. Suppose thatlimsup|𝑥(𝑠,𝑡)|=∞; then, for any positive constant𝑀₀, there exists{𝑆_𝑛,𝑇_𝑛}such that|𝑥(𝑆_𝑛,𝑇_𝑛)|>𝑀₀,𝑛=1,2,….Since𝑥(𝑠,𝑡)is an oscillatory solution, there exists𝐼∶=(𝑎,𝑏)_𝕋×(𝑐,𝑑)_𝕋such that𝑥(𝑎,𝑡)=𝑥(𝑏,𝑡)=𝑥(𝑠,𝑐)=𝑥(𝑠,𝑑)=0,|𝑥(𝑠,𝑡)|>0on𝐼, and|𝑥(𝜏₁,𝜏₂)|=max{|𝑥(𝑠,𝑡)|∶𝜎(𝑎)≤𝑠<𝑑,𝜎(𝑐)≤𝑡<𝑑}∶=𝑀>𝑀₀. From (52), we can choose𝑎,𝑐sufficiently large such that ^∞_𝑎^∞_𝑐𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠<𝑀^{(𝛼−𝛽)/(𝛼−1)},_{∞𝑎∞𝑐}𝛽₂(𝑠,𝑡)Δ𝑡Δ𝑠<1. (53) By Theorem 1, we have 2≤𝑀^{𝛽/𝛼−1𝑏}_𝑎^𝑑_𝑐𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛾⋅∫^𝑏_𝑎∫^𝑑_𝑐𝛽⁺₂(𝑠,𝑡)Δ𝑡Δ𝑠^1/𝛼, (54) and so 2^𝛼≤𝑀^{𝛽−𝛼𝑏}_𝑎^𝑑_𝑐𝛽₁(𝑠,𝑡)Δ𝑡Δ𝑠^𝛼−1⋅^𝑏_𝑎^𝑑_𝑐𝛽⁺₂(𝑠,𝑡)Δ𝑡Δ𝑠≤𝑀^𝛽−𝛼⋅𝑀^𝛼−𝛽=1, (55) which contradicts𝛼>1. Hence, there exists a positive constant𝐾such that|𝑥(𝑠,𝑡)|≤𝐾.
Integrating the second equation of the dynamic equation (50) over𝑡from𝜏₂to𝑡and then over𝑠from𝜏₁to𝑠, respectively, we get ^𝑠_𝜏1^𝑡_𝜏2𝑢^Δ1Δ2()()𝜏𝑠,𝑡Δ𝑡Δ𝑠=𝑢𝑠,𝑡−𝑢₁,𝑡−𝑢𝑠,𝜏₂+𝑢𝜏₁,𝜏₂=−^𝑠_𝜏1^𝑡_𝜏2𝛽₂(𝑠,𝑡)|𝑥(𝜎(𝑠),𝜎(𝑡))|^𝛽−2𝑥(𝜎(𝑠),𝜎(𝑡))Δ𝑡Δ𝑠. (56) Since𝑢(𝜏,𝑡)is bounded on[𝑡₀,∞)_𝕋for a fixed𝜏and𝑢(𝑠,𝜂)is bounded on[𝑠₀,∞)_𝕋for a fixed𝜂, we get |𝑢(𝑠,𝑡)|≤𝑢𝜏₁,𝑡+𝑢𝑠,𝜏₂−𝑢𝜏₁,𝜏₂+^𝑠_𝜏1^𝑡_𝜏2𝛽₂(𝑠,𝑡)|𝑥(𝜎(𝑠),𝜎(𝑡))|^𝛽−1≤𝑢𝜏Δ𝑡Δ𝑠₁,𝑡+𝑢𝑠,𝜏₂−𝑢𝜏₁,𝜏₂+𝐾^𝛽−1∞_𝜏1^∞_𝜏2𝛽₂(𝑠,𝑡)Δ𝑡Δ𝑠. (57)
Since∫^∞_𝑠0∫^∞_𝑡0|𝛽₂(𝑠,𝑡)|Δ𝑡Δ𝑠<∞,|𝑢(𝑠,𝑡)|is bounded. Solimsup{|𝑥(𝑠,𝑡)|+𝑢(𝑠,𝑡)|}is bounded on[𝑠₀,∞)_𝕋×[𝑡₀,∞)_𝕋.The proof is complete.

Acknowledgments

The first author's research was supported by NNSF of China (11071054) and Natural Science Foundation of Hebei Province (A2011205012). The corresponding author's research was partially supported by an HKU URG Grant.