Linearized oscillation theory for a nonlinear equation with a distributed delay

We obtain linearized oscillation theorems for the equation with distributed delays (1)[email protected]?(t)[email protected]?k=1mr"k(t)@!"-"~^tf"k(x(s))d"sR"k(t,s)=0. The results are applied to logistic, Lasota-Wazewska and Nicholson's blowflies equations with a distributed delay. In addition, the ''Mean Value Theorem'' is proved which claims that a solution of (1) also satisfies the linear equation with a variable concentrated delay [email protected]?(t)+(@?k=1mr"k(t)f"k^'(@x"k(t)))x(g(t))=0.

[1]  Fabien Crauste,et al.  Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics , 2005, 0904.2491.

[2]  Wan-Tong Li,et al.  Nonoscillation and Oscillation Theory for Functional Differential Equations , 2004 .

[3]  Linearized oscillations for difference equations , 1992 .

[4]  C. Philos,et al.  Oscillations in Higher-Order Neutral Differential Equations , 1993, Canadian Journal of Mathematics.

[5]  István Györi,et al.  On the existence of rapidly oscillatory solutions in the Nicholson blowflies equation , 2002 .

[6]  Gail S. K. Wolkowicz,et al.  Global Asymptotic Behavior of a Chemostat Model with Two Perfectly Complementary Resources and Distributed Delay , 2000, SIAM J. Appl. Math..

[7]  J. Yu,et al.  A Linearized Oscillation Result for Odd-Order Neutral Delay Differential Equations , 1994 .

[8]  Nonoscillations in retarded systems , 2005 .

[9]  C. Qian,et al.  Linearized oscillations in nonautonomous delay differential equations , 1993 .

[10]  V. Volterra.  Fluctuations in the Abundance of a Species considered Mathematically , 1926, Nature.

[11]  Santiago Schnell,et al.  Mechanism Equivalence in Enzyme–Substrate Reactions: Distributed Differential Delay in Enzyme Kinetics , 2004 .

[12]  C. Qian,et al.  Linearized oscillations for odd-order neutral delay differential equations , 1990 .

[13]  M. Mackey,et al.  Sufficient conditions for stability of linear differential equations with distributed delay , 2001 .

[14]  W. Ames.  Mathematics in Science and Engineering , 1999 .

[15]  Linearized oscillation theory for a nonlinear nonautonomous delay differential equation , 2003 .

[16]  Eric P. Braverman.  NICHOLSON’S BLOWFLIES EQUATION WITH A DISTRIBUTED DELAY , 2022 .

[17]  N. Shkil’.  Construction of formal particular solutions of a system of linear differential equations with delayed argument , 1974 .

[18]  L. Berezansky,et al.  Oscillation of Equations with Distributed Delays , 2001 .

[19]  Xianhua Tang.  Asymptotic behavior of a differential equation with distributed delays , 2005 .

[20]  Sanyi Tang,et al.  Global Qualitative Analysis for a Ratio-Dependent Predator–Prey Model with Delay☆☆☆ , 2002 .

[21]  Rogelio Lozano,et al.  On the passivity of linear delay systems , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[22]  Yang Kuang,et al.  Geometric Stability Switch Criteria in Delay Differential Systems with Delay Dependent Parameters , 2002, SIAM J. Math. Anal..

[23]  Mark Kot,et al.  Elements of Mathematical Ecology: Frontmatter , 2001 .

[24]  L. Berezansky,et al.  On stability of some linear and nonlinear delay differential equations , 2006 .

[25]  G. Webb,et al.  A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay , 2003, Journal of mathematical biology.

[26]  K. Gopalsamy.  Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .

[27]  D M Bortz,et al.  Incorporation of variability into the modeling of viral delays in HIV infection dynamics. , 2003, Mathematical biosciences.

[28]  Mihály Pituk.  Linearized oscillation in a nonautonomous scalar delay differential equation , 2006, Appl. Math. Lett..

[29]  Nonexistence of periodic solutions in delayed Lotka---Volterra systems , 2002 .

[30]  G. E. Hutchinson,et al.  CIRCULAR CAUSAL SYSTEMS IN ECOLOGY , 1948, Annals of the New York Academy of Sciences.

[31]  Wan-Tong Li,et al.  Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response , 2004 .

[32]  F. Smithies.  Linear Operators , 2019, Nature.

[33]  G. Ladas,et al.  Oscillation Theory of Delay Differential Equations: With Applications , 1992 .

[34]  Oscillation properties of a logistic equation with distributed delay , 2003 .

[35]  Jack K. Hale,et al.  Oscillation and Dynamics in Delay Equations , 1992 .

[36]  F. Brauer,et al.  Mathematical Models in Population Biology and Epidemiology , 2001 .

[37]  Jean-Pierre Richard,et al.  Time-delay systems: an overview of some recent advances and open problems , 2003, Autom..

[38]  Direction and stability of bifurcating periodic solutions of a chemostat model with two distributed delays , 2004 .

[39]  P. Bressloff,et al.  Symmetry and phase-locking in a ring of pulse-coupled oscillators with distributed delays , 1999 .

[40]  G. Ladas,et al.  Linearized oscillations in population dynamics , 1987 .

[41]  Asymptotic properties, nonoscillation, and stability for scalar first order linear autonomous neutral delay differential equations. , 2004 .

[42]  T. A. Burton,et al.  Fixed points, Volterra equations, and Becker’s resolvent , 2005 .

[43]  J. Schwartz,et al.  Linear Operators. Part I: General Theory. , 1960 .

[44]  Stefan Vandewalle,et al.  An Analysis of Delay-Dependent Stability for Ordinary and Partial Differential Equations with Fixed and Distributed Delays , 2004, SIAM J. Sci. Comput..

[45]  G. Ladas,et al.  Linearized oscillations in population dynamics. , 1987, Bulletin of mathematical biology.

[46]  George Seifert,et al.  Oscillation Theory of Delay Differential Equations (I. Györi and G. Ladas); Oscillation Theory for Neutral Differential Equations with Delay (D. D. Bainov and D. P. Mishev) , 1993, SIAM Rev..

[47]  Graeme C. Wake,et al.  Analysis of a class of distributed delay logistic differential equations , 2003 .

[48]  Xianhua Tang,et al.  Oscillation of first order delay differential equations with distributed delay , 2004 .

[49]  Singh M. Nayan,et al.  On Fixed Points , 1981 .

[50]  M. Gil’.  Boundedness of solutions of nonlinear differential delay equations with positive green functions and the Aizerman-Myshkis problem , 2002 .

[51]  Qingkai Kong,et al.  Oscillation Theory for Functional Di erential Equations , 1994 .

[52]  D. Bainov,et al.  Necessary and sufficient conditions for oscillation of the solutions of linear functional differential equations of neutral type with distributed delay , 1990 .

[53]  V. Volterra.  Fluctuations in the Abundance of a Species considered Mathematically , 1926 .

[54]  Constantin Corduneanu,et al.  Functional Equations with Causal Operators , 2002 .

[55]  Eduardo Liz,et al.  Attractivity properties of infinite delay Mackey-Glass type equations , 2002, Differential and Integral Equations.

[56]  Permanence extinction and global asymptotic stability in a stage structured system with distributed delays , 2005 .