时间模上一类二阶泛函动态方程振荡的充分条件
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摘要
研究了一类时间模上二阶Emden-Fowler型变时滞的中立型泛函动态方程{a(t)φ([x(t)+p(t) g (x(τ(t)))]~Δ)}~Δ+q_1(t) f_1(φ_1(x(δ_1(t))))+q_2(t) f_2(φ_2(x(δ_2(t))))=0的振荡性,其中,φ(u)=|u|~(α-1)u(α>0),φ_1(u)=|u|~(β-1)u(β>0),φ_2(u)=|u|~(γ-1)u(γ>0)。利用时间模上的有关理论和广义黎卡提变换技术,并借助各种不等式,得到了该方程振荡的一些新的充分条件,推广并丰富了一些已有结果。最后,给出了一些有趣的实例以说明文中的结果。
This paper is concerned with oscillatory behavior of the following second-order Emden-Fowler variable delay neutral functional dynamic equation { a( t)φ( [ x( t)+ p( t) g( x( τ( t))) ]~Δ) }~Δ+ q_1( t) f_1( φ_1( x( δ_1( t))))+q_2( t) f_2( φ_2( x( δ_2( t))))= 0 on a time scaleT, where φ( u)= |u|~(α-1) u(α >0), φ_1( u)= |u|~(β-1) u(β >0) and φ_2( u)=|u|~(γ-1) u(γ >0). By using the time scales theory and the Riccati transformation as well as the inequality technique, we establish some new sufficient conditions of oscillation for the equation. Our results deal with some cases not covered by the existing results in the literature. Finally, some interesting examples are given to illustrate the versatility of our results.
This paper is concerned with oscillatory behavior of the following second-order Emden-Fowler variable delay neutral functional dynamic equation { a( t)φ( [ x( t)+ p( t) g( x( τ( t))) ]~Δ) }~Δ+ q_1( t) f_1( φ_1( x( δ_1( t))))+q_2( t) f_2( φ_2( x( δ_2( t))))= 0 on a time scaleT, where φ( u)= |u|~(α-1) u(α >0), φ_1( u)= |u|~(β-1) u(β >0) and φ_2( u)=|u|~(γ-1) u(γ >0). By using the time scales theory and the Riccati transformation as well as the inequality technique, we establish some new sufficient conditions of oscillation for the equation. Our results deal with some cases not covered by the existing results in the literature. Finally, some interesting examples are given to illustrate the versatility of our results.
引文
[1] BOHNER M, PETERSON A.Dynamic Equations on Time Scales-An Introduction with Applications[M]. Boston:Birkh?user Basel, 2001.
[2] AGARWAL R P, BOHNER M, LI W T.Nonoscillation and Oscillation:Theory for Functional Differential Equations[M]. New York:Monographs and Textbooks in Pure and Applied Mathematics,2004.
[3] AGARWAL R P, O’ REGAN D, SAKER S H.Philos-type oscillation criteria for second-order half linear dynamic equations[J].The Rocky Mountain Journal of Mathematics, 2007, 37(4):1085-1104.
[4] GRACE S R, BOHNER M, AGARWAL R P. On the oscillation of second-order half-linear dynamic equations[J].Journal of Difference Equations and Applications, 2009, 15(5):451-460.
[5] SAKER S H. Oscillation criteria of second order halflinear dynamic equations on time scales[J].Journal of Computational and Applied Mathematics, 2005,177(2):375-387.
[6] HAN Z L, LI T X, SUN S R, et al. Oscillation criteria of second order half-linear dynamic equations on time scales[J].The Glob Journal of Science Frontier Research, 2010, 10:46-51.
[7] HASSAN T S. Oscillation criteria for half-linear dynamic equations on time scales[J].Journal of Mathematical Analysis and Applications, 2008, 345(1):176-185.
[8] SAKER S H, GRACE S R. Oscillation criteria for quasi-linear functional dynamic equations on time scales[J].Mathematica Slovaca, 2012, 62(3):501-524.
[9] GüVENILIR A F, NIZIGIYIMANA F. Oscillation criteria for second-order quasi-linear delay dynamic equations on time scales[J].Advances in Difference Equations, 2014, 2014:45.
[10]杨甲山.时间测度链上动力方程振动性的进展[J].安徽大学学报(自然科学版), 2018, 42(1):26-37.YANG J S. Advances of oscillation of dynamic equations on time scales[J].Journal of Anhui University(Natural Science Edition), 2018, 42(1):26-37.
[11]韩振来,孙书荣,张承慧.时间尺度上二阶中立型时滞动力方程的振动性[J].中山大学学报(自然科学版), 2010, 49(5):21-24.HAN Z L, SUN S R, ZHANG C H.Oscillation of second-order neutral delay dynamic equations on time scales[J].Acta Scientiarum Naturalium Universitatis Sunyatseni, 2010, 49(5):21-24.
[12]杨甲山,方彬.时间测度链上一类二阶非线性时滞阻尼动力方程的振动性分析[J].应用数学,2017,30(1):16-26.YANG J S, FANG B. Oscillation analysis of certain second-order nonlinear delay damped dynamic equations on time scales[J].Mathematica Applicata,2017,30(1):16-26.
[13]杨甲山.时间尺度上二阶Emden-Fowler型延迟动态方程的振动性[J].振动与冲击, 2018, 37(16):154-161.YANG J S. Oscillation for a class of second-order Emden-Fowler-type delay dynamic equations on time scales[J].Journal of Vibration and Shock, 2018, 37(16):154-161.
[14]杨甲山,李同兴.时间模上一类二阶阻尼EmdenFowler型动态方程的振荡性[J].数学物理学报,2018, 38A(1):134-155.YANG J S, LI T X. Oscillation for a class of secondorder damped Emden-Fowler dynamic equations on time scales[J].Acta Mathematica Scientia, 2018, 38A(1):134-155.
[15]张全信,高丽.时间尺度上具阻尼项的二阶半线性时滞动力方程的振动准则[J].中国科学(数学), 2010, 40(7):673-682.ZHANG Q X, GAO L. Oscillation criteria for secondorder half-linear delay dynamic equations with damping on time scales[J].Scientia Sinica Mathematica, 2010,40(7):673-682.
[16]张全信,高丽,刘守华.时间尺度上具阻尼项的二阶半线性时滞动力方程的振动准则(II)[J].中国科学(数学),2011, 41(10):885-896.ZHANG Q X, GAO L, LIU S H. Oscillation criteria for second-order half-linear delay dynamic equations with damping on time scales[J].Scientia Sinica Mathematica, 2011, 41(10):885-896.
[17]孙一冰,韩振来,孙书荣,等.时间尺度上一类二阶具阻尼项的半线性中立型时滞动力方程的振动性[J].应用数学学报, 2013, 36(3):480-494.SUN Y B,HAN Z L, SUN S R, et al. Oscillation of a class of second order half-linear neutral delay dynamic equations with damping on time scales[J].Acta Mathematicae Applicatae Sinica, 2013, 36(3):480-494.
[18]杨甲山.时间模上一类二阶非线性延迟动力系统的振动性分析[J].应用数学学报, 2018, 41(3):388-402.YANG J S. Oscillation analysis of second-order nonlinear delay dynamic equations on time scales[J].Acta Mathematicae Applicatae Sinica, 2018, 41(3):388-402.
[19]杨甲山.二阶Emden-Fowler型非线性变时滞微分方程的振荡准则[J].浙江大学学报(理学版),2017,44(2):144-149,160.YANG J S. Oscillation criteria of second-order Emden-Fowler nonlinear variable delay differential equations[J].Journal of Zhejiang University(Science Edition), 2017,44(2):144-149,160.
[2] AGARWAL R P, BOHNER M, LI W T.Nonoscillation and Oscillation:Theory for Functional Differential Equations[M]. New York:Monographs and Textbooks in Pure and Applied Mathematics,2004.
[3] AGARWAL R P, O’ REGAN D, SAKER S H.Philos-type oscillation criteria for second-order half linear dynamic equations[J].The Rocky Mountain Journal of Mathematics, 2007, 37(4):1085-1104.
[4] GRACE S R, BOHNER M, AGARWAL R P. On the oscillation of second-order half-linear dynamic equations[J].Journal of Difference Equations and Applications, 2009, 15(5):451-460.
[5] SAKER S H. Oscillation criteria of second order halflinear dynamic equations on time scales[J].Journal of Computational and Applied Mathematics, 2005,177(2):375-387.
[6] HAN Z L, LI T X, SUN S R, et al. Oscillation criteria of second order half-linear dynamic equations on time scales[J].The Glob Journal of Science Frontier Research, 2010, 10:46-51.
[7] HASSAN T S. Oscillation criteria for half-linear dynamic equations on time scales[J].Journal of Mathematical Analysis and Applications, 2008, 345(1):176-185.
[8] SAKER S H, GRACE S R. Oscillation criteria for quasi-linear functional dynamic equations on time scales[J].Mathematica Slovaca, 2012, 62(3):501-524.
[9] GüVENILIR A F, NIZIGIYIMANA F. Oscillation criteria for second-order quasi-linear delay dynamic equations on time scales[J].Advances in Difference Equations, 2014, 2014:45.
[10]杨甲山.时间测度链上动力方程振动性的进展[J].安徽大学学报(自然科学版), 2018, 42(1):26-37.YANG J S. Advances of oscillation of dynamic equations on time scales[J].Journal of Anhui University(Natural Science Edition), 2018, 42(1):26-37.
[11]韩振来,孙书荣,张承慧.时间尺度上二阶中立型时滞动力方程的振动性[J].中山大学学报(自然科学版), 2010, 49(5):21-24.HAN Z L, SUN S R, ZHANG C H.Oscillation of second-order neutral delay dynamic equations on time scales[J].Acta Scientiarum Naturalium Universitatis Sunyatseni, 2010, 49(5):21-24.
[12]杨甲山,方彬.时间测度链上一类二阶非线性时滞阻尼动力方程的振动性分析[J].应用数学,2017,30(1):16-26.YANG J S, FANG B. Oscillation analysis of certain second-order nonlinear delay damped dynamic equations on time scales[J].Mathematica Applicata,2017,30(1):16-26.
[13]杨甲山.时间尺度上二阶Emden-Fowler型延迟动态方程的振动性[J].振动与冲击, 2018, 37(16):154-161.YANG J S. Oscillation for a class of second-order Emden-Fowler-type delay dynamic equations on time scales[J].Journal of Vibration and Shock, 2018, 37(16):154-161.
[14]杨甲山,李同兴.时间模上一类二阶阻尼EmdenFowler型动态方程的振荡性[J].数学物理学报,2018, 38A(1):134-155.YANG J S, LI T X. Oscillation for a class of secondorder damped Emden-Fowler dynamic equations on time scales[J].Acta Mathematica Scientia, 2018, 38A(1):134-155.
[15]张全信,高丽.时间尺度上具阻尼项的二阶半线性时滞动力方程的振动准则[J].中国科学(数学), 2010, 40(7):673-682.ZHANG Q X, GAO L. Oscillation criteria for secondorder half-linear delay dynamic equations with damping on time scales[J].Scientia Sinica Mathematica, 2010,40(7):673-682.
[16]张全信,高丽,刘守华.时间尺度上具阻尼项的二阶半线性时滞动力方程的振动准则(II)[J].中国科学(数学),2011, 41(10):885-896.ZHANG Q X, GAO L, LIU S H. Oscillation criteria for second-order half-linear delay dynamic equations with damping on time scales[J].Scientia Sinica Mathematica, 2011, 41(10):885-896.
[17]孙一冰,韩振来,孙书荣,等.时间尺度上一类二阶具阻尼项的半线性中立型时滞动力方程的振动性[J].应用数学学报, 2013, 36(3):480-494.SUN Y B,HAN Z L, SUN S R, et al. Oscillation of a class of second order half-linear neutral delay dynamic equations with damping on time scales[J].Acta Mathematicae Applicatae Sinica, 2013, 36(3):480-494.
[18]杨甲山.时间模上一类二阶非线性延迟动力系统的振动性分析[J].应用数学学报, 2018, 41(3):388-402.YANG J S. Oscillation analysis of second-order nonlinear delay dynamic equations on time scales[J].Acta Mathematicae Applicatae Sinica, 2018, 41(3):388-402.
[19]杨甲山.二阶Emden-Fowler型非线性变时滞微分方程的振荡准则[J].浙江大学学报(理学版),2017,44(2):144-149,160.YANG J S. Oscillation criteria of second-order Emden-Fowler nonlinear variable delay differential equations[J].Journal of Zhejiang University(Science Edition), 2017,44(2):144-149,160.