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Kernel sections and (almost) periodic solutions of a non-autonomous parabolic PDE with a discrete state-dependent delay
1. | Department of Mathematics and Systems Science, College of Science, National University of Defense Technology, Changsha, 410073, China, China |
References:
[1] |
W. Aiello, H. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869.
doi: doi:10.1137/0152048. |
[2] |
Y. Cao, J. Fan and T. Gard, The effects of state-dependent time delay on a state-structured population grouwth model, Nonlinear Anal. TMA, 19 (1992), 95-105.
doi: doi:10.1016/0362-546X(92)90113-S. |
[3] |
F. Hartung, Linearized stability in periodic functional differential equations with state-dependent delays, J. Comput. Anal. Math., 174 (2005), 201-211.
doi: doi:10.1016/j.cam.2004.04.006. |
[4] |
R. Torrejón, Positive almost periodic solutions of a state-dependent delay nonlinear integral equation, Nonlinear Anal. TMA, 20 (1993), 1383-1416.
doi: doi:10.1016/0362-546X(93)90167-Q. |
[5] |
H. Walther, The solution manifold and $C^1$-smoothness of solution operators for differential equations with state dependent delay, J. Differential Equations, 195 (2003), 46-65.
doi: doi:10.1016/j.jde.2003.07.001. |
[6] |
E. Hernóndez, A. Prokopczyk and L. Ladeira, A note on partial functional differential equations with state-dependent delay, Nonlinear Anal. RWA, 7 (2006), 510-519.
doi: doi:10.1016/j.nonrwa.2005.03.014. |
[7] |
E. Hernóndez, M. Mckibben and H. Henriquez, Existence results for partial neutral functional differential equations with state-dependent delay, Math. Comput. Modelling, 49 (2009), 1260-1267.
doi: doi:10.1016/j.mcm.2008.07.011. |
[8] |
E. Hernóndez, M. Pierri and G. Goncalves, Existence results for an impulsive abstact partial differenrial equation with state-dependent delay, Comput. Math. Appl., 52 (2006), 411-420.
doi: doi:10.1016/j.camwa.2006.03.022. |
[9] |
E. Hernóndez, R. Sakthivel and S. T. Aki, Existence results for impulsive evolution equation with state-dependent delay, E. J. Differential Equations, 2008 (2008), 1-11. |
[10] |
A. Rezounenko and J. Wu, A non-local PDE model for population dynamics with state-selective : Local theory and global attractors, J. Comput. Appl. Math., 190 (2006), 99-113.
doi: doi:10.1016/j.cam.2005.01.047. |
[11] |
A. Rezounenko, Differential equations with discrete state dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Anal. TMA, 70 (2009), 3978-3986.
doi: doi:10.1016/j.na.2008.08.006. |
[12] |
A. Rezounenko, Partial differential equations with disctete and distributed state-dependent delays, J. Math. Anal. Appl., 326 (2007), 1031-1045.
doi: doi:10.1016/j.jmaa.2006.03.049. |
[13] |
A. Rezounenko, On a class of P.D.E.s with nonlinear distirbuted in space and time state-dependent delay terms, Mathematics Methods in the Applied sciences, 31 (2008), 1569-1585.
doi: doi:10.1002/mma.986. |
[14] |
A. Rezounenko, Non-linear partial differential equations with discrete state-dependent delays in a metric space, Nonlinear Anal., 73 (2010), 1707-1714.
doi: doi:10.1016/j.na.2010.05.005. |
[15] |
V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics," AMS Providence, Rhode Island, 2001. |
[16] |
J. Wu, "Theory and Applications of Partial Functional Differential Equations," Springer-verlag, New York, 1996. |
[17] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Defferential Equations," Applied Mathematical Sciences, vol.44, Springer, New York, Berlin, 1983. |
[18] |
M. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289.
doi: doi:10.1126/science.267326. |
[19] |
K. Gopalsamy, M. Kulenovi and G. Ladas, Oscillations and global attractivity in models of hematopoiesis, J. Dynamics and Differential Equations, 2 (1990), 117-132.
doi: doi:10.1007/BF01057415. |
[20] |
K. Gopalsamy, S. Trofimchuk and N. Bantsur, A note on global attractivity in models of hematopoiesis, Ukrainian Mathematical J., 50 (1998), 3-12.
doi: doi:10.1007/BF02514684. |
[21] |
X. Wang and Z. Li, Dynamics for a class of general Hematopoiesis model with periodic coefficients, Appl. Math. and Comput., 186 (2007), 460-468.
doi: doi:10.1016/j.amc.2006.07.109. |
[22] |
X. Wang and Z. Li, Global attractivity, oscillation and Hopf bifurcation for a class of diffusive hematopoiesis models, Cent. Eur. J. Math., 5 (2007), 397-414.
doi: doi:10.2478/s11533-006-0042-5. |
show all references
References:
[1] |
W. Aiello, H. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869.
doi: doi:10.1137/0152048. |
[2] |
Y. Cao, J. Fan and T. Gard, The effects of state-dependent time delay on a state-structured population grouwth model, Nonlinear Anal. TMA, 19 (1992), 95-105.
doi: doi:10.1016/0362-546X(92)90113-S. |
[3] |
F. Hartung, Linearized stability in periodic functional differential equations with state-dependent delays, J. Comput. Anal. Math., 174 (2005), 201-211.
doi: doi:10.1016/j.cam.2004.04.006. |
[4] |
R. Torrejón, Positive almost periodic solutions of a state-dependent delay nonlinear integral equation, Nonlinear Anal. TMA, 20 (1993), 1383-1416.
doi: doi:10.1016/0362-546X(93)90167-Q. |
[5] |
H. Walther, The solution manifold and $C^1$-smoothness of solution operators for differential equations with state dependent delay, J. Differential Equations, 195 (2003), 46-65.
doi: doi:10.1016/j.jde.2003.07.001. |
[6] |
E. Hernóndez, A. Prokopczyk and L. Ladeira, A note on partial functional differential equations with state-dependent delay, Nonlinear Anal. RWA, 7 (2006), 510-519.
doi: doi:10.1016/j.nonrwa.2005.03.014. |
[7] |
E. Hernóndez, M. Mckibben and H. Henriquez, Existence results for partial neutral functional differential equations with state-dependent delay, Math. Comput. Modelling, 49 (2009), 1260-1267.
doi: doi:10.1016/j.mcm.2008.07.011. |
[8] |
E. Hernóndez, M. Pierri and G. Goncalves, Existence results for an impulsive abstact partial differenrial equation with state-dependent delay, Comput. Math. Appl., 52 (2006), 411-420.
doi: doi:10.1016/j.camwa.2006.03.022. |
[9] |
E. Hernóndez, R. Sakthivel and S. T. Aki, Existence results for impulsive evolution equation with state-dependent delay, E. J. Differential Equations, 2008 (2008), 1-11. |
[10] |
A. Rezounenko and J. Wu, A non-local PDE model for population dynamics with state-selective : Local theory and global attractors, J. Comput. Appl. Math., 190 (2006), 99-113.
doi: doi:10.1016/j.cam.2005.01.047. |
[11] |
A. Rezounenko, Differential equations with discrete state dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Anal. TMA, 70 (2009), 3978-3986.
doi: doi:10.1016/j.na.2008.08.006. |
[12] |
A. Rezounenko, Partial differential equations with disctete and distributed state-dependent delays, J. Math. Anal. Appl., 326 (2007), 1031-1045.
doi: doi:10.1016/j.jmaa.2006.03.049. |
[13] |
A. Rezounenko, On a class of P.D.E.s with nonlinear distirbuted in space and time state-dependent delay terms, Mathematics Methods in the Applied sciences, 31 (2008), 1569-1585.
doi: doi:10.1002/mma.986. |
[14] |
A. Rezounenko, Non-linear partial differential equations with discrete state-dependent delays in a metric space, Nonlinear Anal., 73 (2010), 1707-1714.
doi: doi:10.1016/j.na.2010.05.005. |
[15] |
V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics," AMS Providence, Rhode Island, 2001. |
[16] |
J. Wu, "Theory and Applications of Partial Functional Differential Equations," Springer-verlag, New York, 1996. |
[17] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Defferential Equations," Applied Mathematical Sciences, vol.44, Springer, New York, Berlin, 1983. |
[18] |
M. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289.
doi: doi:10.1126/science.267326. |
[19] |
K. Gopalsamy, M. Kulenovi and G. Ladas, Oscillations and global attractivity in models of hematopoiesis, J. Dynamics and Differential Equations, 2 (1990), 117-132.
doi: doi:10.1007/BF01057415. |
[20] |
K. Gopalsamy, S. Trofimchuk and N. Bantsur, A note on global attractivity in models of hematopoiesis, Ukrainian Mathematical J., 50 (1998), 3-12.
doi: doi:10.1007/BF02514684. |
[21] |
X. Wang and Z. Li, Dynamics for a class of general Hematopoiesis model with periodic coefficients, Appl. Math. and Comput., 186 (2007), 460-468.
doi: doi:10.1016/j.amc.2006.07.109. |
[22] |
X. Wang and Z. Li, Global attractivity, oscillation and Hopf bifurcation for a class of diffusive hematopoiesis models, Cent. Eur. J. Math., 5 (2007), 397-414.
doi: doi:10.2478/s11533-006-0042-5. |
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