In this paper, we study the second-order nonlinear differential systems of Liénard-type x˙=1a(x)[ h(y)−F(x) ], y˙=−a(x)g(x). Necessary and sufficient conditions to ensure that all nontrivial solutions are oscil...In this paper, we study the second-order nonlinear differential systems of Liénard-type x˙=1a(x)[ h(y)−F(x) ], y˙=−a(x)g(x). Necessary and sufficient conditions to ensure that all nontrivial solutions are oscillatory are established by using a new nonlinear integral inequality. Our results substantially extend and improve previous results known in the literature.展开更多
By coincidence degree,the existence of solution to the boundary value problem of a generalized Liénard equationa(t)x'+F(x,x′)x′+g(x)=e(t), x(0)=x(2π),x′(0)=x′(2π)is proved,where a∈C 1[0,2π],a(t)>...By coincidence degree,the existence of solution to the boundary value problem of a generalized Liénard equationa(t)x'+F(x,x′)x′+g(x)=e(t), x(0)=x(2π),x′(0)=x′(2π)is proved,where a∈C 1[0,2π],a(t)>0(0≤t≤2π),a(0)=a(2π),F(x,y)=f(x)+α|y| β,α>0,β>0 are all constants,f∈C(R,R),e∈C[0,2π]. An example is given as an application.展开更多
By means of Mawhin's continuation theorem,we study a kind of lie'nard functional differential equations:x"(t)+f(x(t))x'(t)+g(t,x(t-τ(t))) = e(t).Some new results on the existence and uniqueness of pe...By means of Mawhin's continuation theorem,we study a kind of lie'nard functional differential equations:x"(t)+f(x(t))x'(t)+g(t,x(t-τ(t))) = e(t).Some new results on the existence and uniqueness of periodic solutions are obtained.展开更多
This paper further studies global asymptotic stability of the zero solusion of Liènard's equation x+f(x)x+g(x) = 0 (1) We obtain the following result: Theorem. The zero solution of equation (1) is globally as...This paper further studies global asymptotic stability of the zero solusion of Liènard's equation x+f(x)x+g(x) = 0 (1) We obtain the following result: Theorem. The zero solution of equation (1) is globally asymptoticly stable if f(x) and g(x) satisfy one of the following conditions. Condition 1: 1) xg(x)>0, for all x≠0; 2) x integral from n=0 to x f(x)dx≥0 and on any interval of x, integral from n=0 to x f(x)dx0; 3) integral from n-0 to x g(x)dx→+∞, as |x|→+∞, Condition 2: 1) xg(x)>0, for all x≠0; 2) x integral from n=0 to x f(x)dx≥0 and on any interval of x, integral from n=0 to x f(x)dx0; 3) F(x) and F(-x)(x>0) are all infinity,where F(x)= integral from n=0 to x f(x)dx. Compared with [1—3], this result further weakens condition on f(x), thus, it has more extensive working field.展开更多
This paper aims to investigate the retarded Liénard-type equation+f 1(x)+f 2(x)(t-τ)+f 3(x)2+φ(x)+g(x(t-τ))=0,where τ is a nonnegative constant, f 1,f 2,f 3,φ and g are continuous functions on R. Using Liapu...This paper aims to investigate the retarded Liénard-type equation+f 1(x)+f 2(x)(t-τ)+f 3(x)2+φ(x)+g(x(t-τ))=0,where τ is a nonnegative constant, f 1,f 2,f 3,φ and g are continuous functions on R. Using Liapunov functional method, we establish a sufficient condition on the stability and boundedness of the solutions of above equation. This will generalize the main results of reference [2].展开更多
In this study,the design of a computational heuristic based on the nonlinear Liénard model is presented using the efficiency of artificial neural networks(ANNs)along with the hybridization procedures of global an...In this study,the design of a computational heuristic based on the nonlinear Liénard model is presented using the efficiency of artificial neural networks(ANNs)along with the hybridization procedures of global and local search approaches.The global search genetic algorithm(GA)and local search sequential quadratic programming scheme(SQPS)are implemented to solve the nonlinear Liénard model.An objective function using the differential model and boundary conditions is designed and optimized by the hybrid computing strength of the GA-SQPS.The motivation of the ANN procedures along with GA-SQPS comes to present reliable,feasible and precise frameworks to tackle stiff and highly nonlinear differentialmodels.The designed procedures of ANNs along with GA-SQPS are applied for three highly nonlinear differential models.The achieved numerical outcomes on multiple trials using the designed procedures are compared to authenticate the correctness,viability and efficacy.Moreover,statistical performances based on different measures are also provided to check the reliability of the ANN along with GASQPS.展开更多
In this paper the generalized nonlinear Euler differential equation t^2k（tu＇）u＇＇＋ t（f（u） ＋ k（tu＇））u＇ ＋ g（u） = 0 is considered. Here the functions f（u）, g（u） and k（u） satisfy smoothness conditio...In this paper the generalized nonlinear Euler differential equation t^2k（tu＇）u＇＇＋ t（f（u） ＋ k（tu＇））u＇ ＋ g（u） = 0 is considered. Here the functions f（u）, g（u） and k（u） satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub（super） linearity are assumed. W＇e present some necessary and sufficient conditions and some tests for the equivalent planar system to have or fail to have property （X^＋）, which is very important for the existence of periodic solutions and oscillation theory.展开更多
G.Villar's Method^[4] is improved to prove theorems for the existence of periodic solution(limit cycles of Liènard equation and a more general class of nonlinear differential equations.The results obtained in...G.Villar's Method^[4] is improved to prove theorems for the existence of periodic solution(limit cycles of Liènard equation and a more general class of nonlinear differential equations.The results obtained in this paper conclude the corresponding theorems in [4].展开更多
文摘In this paper, we study the second-order nonlinear differential systems of Liénard-type x˙=1a(x)[ h(y)−F(x) ], y˙=−a(x)g(x). Necessary and sufficient conditions to ensure that all nontrivial solutions are oscillatory are established by using a new nonlinear integral inequality. Our results substantially extend and improve previous results known in the literature.
文摘By coincidence degree,the existence of solution to the boundary value problem of a generalized Liénard equationa(t)x'+F(x,x′)x′+g(x)=e(t), x(0)=x(2π),x′(0)=x′(2π)is proved,where a∈C 1[0,2π],a(t)>0(0≤t≤2π),a(0)=a(2π),F(x,y)=f(x)+α|y| β,α>0,β>0 are all constants,f∈C(R,R),e∈C[0,2π]. An example is given as an application.
基金Foundation item： Supported by the Anhui Natural Science Foundation（050460103） Supported by the NSF of Anhui Educational Bureau（KJ2008B247） Supported by the RSPYT of Anhui Educational Bu- reau（2008jq1111）
文摘By means of Mawhin's continuation theorem,we study a kind of lie'nard functional differential equations:x"(t)+f(x(t))x'(t)+g(t,x(t-τ(t))) = e(t).Some new results on the existence and uniqueness of periodic solutions are obtained.
文摘This paper further studies global asymptotic stability of the zero solusion of Liènard's equation x+f(x)x+g(x) = 0 (1) We obtain the following result: Theorem. The zero solution of equation (1) is globally asymptoticly stable if f(x) and g(x) satisfy one of the following conditions. Condition 1: 1) xg(x)>0, for all x≠0; 2) x integral from n=0 to x f(x)dx≥0 and on any interval of x, integral from n=0 to x f(x)dx0; 3) integral from n-0 to x g(x)dx→+∞, as |x|→+∞, Condition 2: 1) xg(x)>0, for all x≠0; 2) x integral from n=0 to x f(x)dx≥0 and on any interval of x, integral from n=0 to x f(x)dx0; 3) F(x) and F(-x)(x>0) are all infinity,where F(x)= integral from n=0 to x f(x)dx. Compared with [1—3], this result further weakens condition on f(x), thus, it has more extensive working field.
文摘This paper aims to investigate the retarded Liénard-type equation+f 1(x)+f 2(x)(t-τ)+f 3(x)2+φ(x)+g(x(t-τ))=0,where τ is a nonnegative constant, f 1,f 2,f 3,φ and g are continuous functions on R. Using Liapunov functional method, we establish a sufficient condition on the stability and boundedness of the solutions of above equation. This will generalize the main results of reference [2].
文摘In this study,the design of a computational heuristic based on the nonlinear Liénard model is presented using the efficiency of artificial neural networks(ANNs)along with the hybridization procedures of global and local search approaches.The global search genetic algorithm(GA)and local search sequential quadratic programming scheme(SQPS)are implemented to solve the nonlinear Liénard model.An objective function using the differential model and boundary conditions is designed and optimized by the hybrid computing strength of the GA-SQPS.The motivation of the ANN procedures along with GA-SQPS comes to present reliable,feasible and precise frameworks to tackle stiff and highly nonlinear differentialmodels.The designed procedures of ANNs along with GA-SQPS are applied for three highly nonlinear differential models.The achieved numerical outcomes on multiple trials using the designed procedures are compared to authenticate the correctness,viability and efficacy.Moreover,statistical performances based on different measures are also provided to check the reliability of the ANN along with GASQPS.
文摘In this paper the generalized nonlinear Euler differential equation t^2k（tu＇）u＇＇＋ t（f（u） ＋ k（tu＇））u＇ ＋ g（u） = 0 is considered. Here the functions f（u）, g（u） and k（u） satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub（super） linearity are assumed. W＇e present some necessary and sufficient conditions and some tests for the equivalent planar system to have or fail to have property （X^＋）, which is very important for the existence of periodic solutions and oscillation theory.
文摘G.Villar's Method^[4] is improved to prove theorems for the existence of periodic solution(limit cycles of Liènard equation and a more general class of nonlinear differential equations.The results obtained in this paper conclude the corresponding theorems in [4].
