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On specification and measure expansiveness
Almost automorphic delayed differential equations and Lasota-Wazewska model
1. | GMA, Departamento de Ciencias, B´asicas Facultad de Ciencias, Universidad del Bío-Bío, Campus Fernando May, Chillán, Chile |
2. | Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Chile |
3. | Escuela de Matemáticas y Estadísticas, Universidad Central de Chile, Santiago, Chile |
Existence of almost automorphic solutions for abstract delayed differential equations is established. Using ergodicity, exponential dichotomy and Bi-almost automorphicity on the homogeneous part, sufficient conditions for the existence and uniqueness of almost automorphic solutions are given.
References:
[1] |
J. Blot, G. Mophou, G. M. N'Guérékata and D. Pennequin,
Weighted pseudo almost automorphic functions and applications to abstract differential equations, Nonlinear Analysis, 71 (2009), 903-909.
doi: 10.1016/j.na.2008.10.113. |
[2] |
S. Bochner,
A new approach to almost periodicity, Proceedings of the National Academic Science of the United States of America, 48 (1962), 2039-2043.
doi: 10.1073/pnas.48.12.2039. |
[3] |
S. Bochner,
Continuous mapping of almost automorphic and almost periodic functions, Proceedings of the National Academic Science of the United States of America, 52 (1964), 907-910.
doi: 10.1073/pnas.52.4.907. |
[4] |
T. Caraballo and D. N. Cheban,
Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition Ⅰ, J. Differential Equations, 246 (2009), 108-128.
doi: 10.1016/j.jde.2008.04.001. |
[5] |
T. Caraballo and D. N. Cheban,
Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition Ⅱ, J. Differential Equations, 246 (2009), 1164-1186.
doi: 10.1016/j.jde.2008.07.025. |
[6] |
S. Castillo and M. Pinto, Dichotomy and almost automorphic solution of difference system,
Electron. J. Qual. Theory Differ. Equ. 32 (2013), 17 pp. |
[7] |
A. Chávez, S. Castillo and M. Pinto, Discontinuous almost automorphic functions and almost automorphic solutions of differential equations with piecewise constant arguments, Electron. J. Differential Equations, 56 (2014), 13 pp. |
[8] |
P. Cieutat, S. Fatajou and G. M. N'Guérékata,
Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations, Applicable Analysis, 89 (2010), 11-27.
doi: 10.1080/00036810903397503. |
[9] |
C. Corduneanu,
Almost Periodic Functions
[With the collaboration of N. Gheorghiu and V. Barbu, Translated from the Romanian by Gitta Bernstein and Eugene Tomer, Interscience Tracts in Pure and Applied Mathematics, No. 22], Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1968. |
[10] |
A. Coronel, M. Pinto and D. Sepúlveda,
Weighted pseudo almost periodic functions, convolutions and abstract integral equations, J. Math. Anal. Appl., 435 (2016), 1382-1399.
doi: 10.1016/j.jmaa.2015.11.034. |
[11] |
C. Cuevas and M. Pinto,
Existence and uniqueness of pseudo-almost periodic solutions of semilinear Cauchy problems with non dense domain, Nonlinear Anal., 45 (2001), 73-83.
doi: 10.1016/S0362-546X(99)00330-2. |
[12] |
H.-S. Ding, T.-J. Xiao and J. Liang,
Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions., J. Math. Anal. Appl., 338 (2008), 141-151.
doi: 10.1016/j.jmaa.2007.05.014. |
[13] |
T. Diagana,
Pseudo Almost Periodic Functions in Banach Space Nova Science Publishers, Inc. , New York, 2007. |
[14] |
L. Duan, L. Huang and Y. Chen,
Global exponential stability of periodic solutions to a delay Lasota-Wazewska model with discontinuous harvesting, Proc. Amer. Math. Soc., 144 (2016), 561-573.
doi: 10.1090/proc12714. |
[15] |
S. Fatajou, N. V. Minh, G. N'Guérékata and A. Pankov,
Stepanov-like almost automorphic solutions for nonautonomous evolution equations, Electronic Journal of Differential Equations, 121 (2007), 1-17.
|
[16] |
C. Feng,
On the existence and uniqueness of almost periodic solutions for delay logistic equations, Appl. Math. Comput., 136 (2003), 487-494.
doi: 10.1016/S0096-3003(02)00072-3. |
[17] |
S. G. Gal and G. M. N'Guérékata,
Almost automorphic fuzzy-number-valued functions, Journal of Fuzzy Mathematics, 13 (2005), 185-208.
|
[18] |
J. A. Goldstein and G. M. N'Guérékata,
Almost automorphic solutions of semilinear evolution equations, Proc. Amer. Math. Soc., 133 (2005), 2401-2408.
doi: 10.1090/S0002-9939-05-07790-7. |
[19] |
R. C. Grimmer,
Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273 (1982), 333-349.
doi: 10.1090/S0002-9947-1982-0664046-4. |
[20] |
E. Hernández and J. P. C. dos Santos,
Asymptotically almost periodic solutions for a class of partial integrodifferential equations, Electron. J. Differential Equations, 38 (2006), 1-8.
|
[21] |
H. R. Henríquez, M. Pierri and P. Táboas,
On $S$-asymptotically $ω$-periodic functions on Banach spaces and applications, J. Math. Anal. Appl., 343 (2008), 365-382.
doi: 10.1016/j.jmaa.2008.02.023. |
[22] |
H. R. Henríquez, M. Pierri and P. Táboas,
Existence of S-asymptotically ω-periodic solutions
for abstract neutral equations, Bull. Austral. Math. Soc., 78 (2008), 365-382.
doi: 10.1017/S0004972708000713. |
[23] |
Z. C. Liang,
Asymptotically periodic solutions of a class of second order nonlinear differential equations, Proc. Amer. Math. Soc., 99 (1987), 693-699.
doi: 10.1090/S0002-9939-1987-0877042-9. |
[24] |
J. Liu, G. M. N'Guérékata and N. V. Minh,
A Massera type theorem for almost automorphic solutions of differential equations, Journal of Mathematical Analysis and Applications, 299 (2004), 587-599.
doi: 10.1016/j.jmaa.2004.05.046. |
[25] |
E. Liz, C. Martínez and S. Trofimchuk,
Attractivity properties of infinite delay Mackey-Glass type equations, Differential and Integral Equations, 15 (2002), 875-896.
|
[26] |
N. V. Minh, T. Naito and G. N'Guérékata,
A spectral countability condition for almost automorphy of solutions of differential equations, Proceedings of the American Mathematical Society, 134 (2006), 3257-3266.
doi: 10.1090/S0002-9939-06-08528-5. |
[27] |
N. V. Minh and T. T. Dat,
On the almost automorphy of bounded solutions of differential equations with piecewise constant argument, Journal of Mathematical Analysis and Applications, 326 (2007), 165-178.
doi: 10.1016/j.jmaa.2006.02.079. |
[28] |
G. M. N'Guérékata,
Topics in Almost Automorphy Springer-Verlag, New York, 2005. |
[29] |
G. M. N'Guérékata,
Almost Automorphic and Almost Periodic Functions in Abstract Spaces Kluwer Academic/Plenum Publishers, New York, 2001. |
[30] |
S. Nicola and M. Pierre,
A note on $S$-asymptotically periodic functions, Nonlinear Analysis, Real World Applications, 10 (2009), 2937-2938.
doi: 10.1016/j.nonrwa.2008.09.011. |
[31] |
M. Pinto,
Pseudo-almost periodic solutions of neutral integral and differential equations and applications, Nonlinear Analysis, 72 (2010), 4377-4383.
doi: 10.1016/j.na.2009.12.042. |
[32] |
M. Pinto and G. Robledo,
Cauchy matrix for linear almost periodic systems and some consequences, Nonlinear Anal., 74 (2011), 5426-5439.
doi: 10.1016/j.na.2011.05.027. |
[33] |
M. Pinto and G. Robledo,
Diagonalizability of nonautonomous linear systems with bounded continuous coefficients, J. Math. Anal. Appl., 407 (2013), 513-526.
doi: 10.1016/j.jmaa.2013.05.038. |
[34] |
M. Pinto and G. Robledo,
Existence and stability of almost periodic solutions in impulsive neural network models, Appl. Math. Comput., 217 (2010), 4167-4177.
doi: 10.1016/j.amc.2010.10.033. |
[35] |
M. Pinto, V. Torres and G. Robledo,
Asymptotic equivalence of almost periodic solutions for a class of perturbed almost periodic systems, Glasg. Math. J., 52 (2010), 583-592.
doi: 10.1017/S0017089510000443. |
[36] |
W. R. Utz and P. Waltman,
Asymptotically almost periodicity of solutions of a system of differential equations, Proc. Amer. Math. Soc., 18 (1967), 597-601.
doi: 10.1090/S0002-9939-1967-0212285-6. |
[37] |
W. A. Veech,
Almost automorphic functions, Proceedings of the National Academy of Science of the United states of America, 49 (1963), 462-464.
doi: 10.1073/pnas.49.4.462. |
[38] |
W. A. Veech,
Almost automorphic function on groups, American Journal of Mathematics, 87 (1965), 719-751.
doi: 10.2307/2373071. |
[39] |
M. Wazewska-Czyzewska and A. Lasota,
Mathematical problems of the red-blood cell system, Ann. Polish Math. Soc. Ser. Ⅲ, Appl. Math., 6 (1976), 23-40.
|
[40] |
F. Wei and K. Wang,
Global stability and asymptotically periodic solutions for non autonomous cooperative Lotka-Volterra diffusion system, Applied Math. and Computation, 182 (2006), 161-165.
doi: 10.1016/j.amc.2006.03.044. |
[41] |
F. Wei and K. Wang,
Asymptotically periodic solutions of N-species cooperation system with time delay, Nonlinear Analysis, Real World and Applications, 7 (2006), 591-596.
doi: 10.1016/j.nonrwa.2005.03.019. |
[42] |
T. Xiao, X. Zhu and J. Liang,
Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications, Nonlinear Analysis, 70 (2009), 4079-4085.
doi: 10.1016/j.na.2008.08.018. |
[43] |
Z. Yao,
Uniqueness and exponential stability of almost periodic positive solution for Lasota-Wazewska model with impulse and infinite delay, Math. Methods Appl. Sci., 38 (2015), 677-684.
doi: 10.1002/mma.3098. |
[44] |
T. Yoshizawa,
Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions Applied Mathematical Sciences, 14 Springer-Verlag, New York-Heidelberg, 1975. |
[45] |
H. Zhao,
Existence and global attractivity of almost periodic solutions for cellular neutral network with distributed delays, Appl. Math. Comput., 154 (2004), 683-695.
doi: 10.1016/S0096-3003(03)00743-4. |
[46] |
S. Zaidman,
Almost automorphic solutions of same abstract evolution equations, Instituto Lombardo, Accademia di Science e Letter, 110 (1976), 578-588.
|
[47] |
S. Zaidman,
Existence of asymptotically almost-periodic and of almost automorphic solutions for same classes of abstract differential equations, Annales des Science Mathématiques du Québec, 13 (1989), 79-88.
|
[48] |
M. Zaki,
Almost automorphic solutions of certain abstract differential equations, Annali di Mathematica pura et Applicata, 101 (1974), 91-114.
doi: 10.1007/BF02417100. |
[49] |
C. Zhang,
Almost Periodic Type Functions and Ergodicity, Science Press, Beijing, Kluwer Academic Publishers, Dordrecht, 2003.
doi: 10.1007/978-94-007-1073-3. |
show all references
References:
[1] |
J. Blot, G. Mophou, G. M. N'Guérékata and D. Pennequin,
Weighted pseudo almost automorphic functions and applications to abstract differential equations, Nonlinear Analysis, 71 (2009), 903-909.
doi: 10.1016/j.na.2008.10.113. |
[2] |
S. Bochner,
A new approach to almost periodicity, Proceedings of the National Academic Science of the United States of America, 48 (1962), 2039-2043.
doi: 10.1073/pnas.48.12.2039. |
[3] |
S. Bochner,
Continuous mapping of almost automorphic and almost periodic functions, Proceedings of the National Academic Science of the United States of America, 52 (1964), 907-910.
doi: 10.1073/pnas.52.4.907. |
[4] |
T. Caraballo and D. N. Cheban,
Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition Ⅰ, J. Differential Equations, 246 (2009), 108-128.
doi: 10.1016/j.jde.2008.04.001. |
[5] |
T. Caraballo and D. N. Cheban,
Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition Ⅱ, J. Differential Equations, 246 (2009), 1164-1186.
doi: 10.1016/j.jde.2008.07.025. |
[6] |
S. Castillo and M. Pinto, Dichotomy and almost automorphic solution of difference system,
Electron. J. Qual. Theory Differ. Equ. 32 (2013), 17 pp. |
[7] |
A. Chávez, S. Castillo and M. Pinto, Discontinuous almost automorphic functions and almost automorphic solutions of differential equations with piecewise constant arguments, Electron. J. Differential Equations, 56 (2014), 13 pp. |
[8] |
P. Cieutat, S. Fatajou and G. M. N'Guérékata,
Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations, Applicable Analysis, 89 (2010), 11-27.
doi: 10.1080/00036810903397503. |
[9] |
C. Corduneanu,
Almost Periodic Functions
[With the collaboration of N. Gheorghiu and V. Barbu, Translated from the Romanian by Gitta Bernstein and Eugene Tomer, Interscience Tracts in Pure and Applied Mathematics, No. 22], Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1968. |
[10] |
A. Coronel, M. Pinto and D. Sepúlveda,
Weighted pseudo almost periodic functions, convolutions and abstract integral equations, J. Math. Anal. Appl., 435 (2016), 1382-1399.
doi: 10.1016/j.jmaa.2015.11.034. |
[11] |
C. Cuevas and M. Pinto,
Existence and uniqueness of pseudo-almost periodic solutions of semilinear Cauchy problems with non dense domain, Nonlinear Anal., 45 (2001), 73-83.
doi: 10.1016/S0362-546X(99)00330-2. |
[12] |
H.-S. Ding, T.-J. Xiao and J. Liang,
Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions., J. Math. Anal. Appl., 338 (2008), 141-151.
doi: 10.1016/j.jmaa.2007.05.014. |
[13] |
T. Diagana,
Pseudo Almost Periodic Functions in Banach Space Nova Science Publishers, Inc. , New York, 2007. |
[14] |
L. Duan, L. Huang and Y. Chen,
Global exponential stability of periodic solutions to a delay Lasota-Wazewska model with discontinuous harvesting, Proc. Amer. Math. Soc., 144 (2016), 561-573.
doi: 10.1090/proc12714. |
[15] |
S. Fatajou, N. V. Minh, G. N'Guérékata and A. Pankov,
Stepanov-like almost automorphic solutions for nonautonomous evolution equations, Electronic Journal of Differential Equations, 121 (2007), 1-17.
|
[16] |
C. Feng,
On the existence and uniqueness of almost periodic solutions for delay logistic equations, Appl. Math. Comput., 136 (2003), 487-494.
doi: 10.1016/S0096-3003(02)00072-3. |
[17] |
S. G. Gal and G. M. N'Guérékata,
Almost automorphic fuzzy-number-valued functions, Journal of Fuzzy Mathematics, 13 (2005), 185-208.
|
[18] |
J. A. Goldstein and G. M. N'Guérékata,
Almost automorphic solutions of semilinear evolution equations, Proc. Amer. Math. Soc., 133 (2005), 2401-2408.
doi: 10.1090/S0002-9939-05-07790-7. |
[19] |
R. C. Grimmer,
Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273 (1982), 333-349.
doi: 10.1090/S0002-9947-1982-0664046-4. |
[20] |
E. Hernández and J. P. C. dos Santos,
Asymptotically almost periodic solutions for a class of partial integrodifferential equations, Electron. J. Differential Equations, 38 (2006), 1-8.
|
[21] |
H. R. Henríquez, M. Pierri and P. Táboas,
On $S$-asymptotically $ω$-periodic functions on Banach spaces and applications, J. Math. Anal. Appl., 343 (2008), 365-382.
doi: 10.1016/j.jmaa.2008.02.023. |
[22] |
H. R. Henríquez, M. Pierri and P. Táboas,
Existence of S-asymptotically ω-periodic solutions
for abstract neutral equations, Bull. Austral. Math. Soc., 78 (2008), 365-382.
doi: 10.1017/S0004972708000713. |
[23] |
Z. C. Liang,
Asymptotically periodic solutions of a class of second order nonlinear differential equations, Proc. Amer. Math. Soc., 99 (1987), 693-699.
doi: 10.1090/S0002-9939-1987-0877042-9. |
[24] |
J. Liu, G. M. N'Guérékata and N. V. Minh,
A Massera type theorem for almost automorphic solutions of differential equations, Journal of Mathematical Analysis and Applications, 299 (2004), 587-599.
doi: 10.1016/j.jmaa.2004.05.046. |
[25] |
E. Liz, C. Martínez and S. Trofimchuk,
Attractivity properties of infinite delay Mackey-Glass type equations, Differential and Integral Equations, 15 (2002), 875-896.
|
[26] |
N. V. Minh, T. Naito and G. N'Guérékata,
A spectral countability condition for almost automorphy of solutions of differential equations, Proceedings of the American Mathematical Society, 134 (2006), 3257-3266.
doi: 10.1090/S0002-9939-06-08528-5. |
[27] |
N. V. Minh and T. T. Dat,
On the almost automorphy of bounded solutions of differential equations with piecewise constant argument, Journal of Mathematical Analysis and Applications, 326 (2007), 165-178.
doi: 10.1016/j.jmaa.2006.02.079. |
[28] |
G. M. N'Guérékata,
Topics in Almost Automorphy Springer-Verlag, New York, 2005. |
[29] |
G. M. N'Guérékata,
Almost Automorphic and Almost Periodic Functions in Abstract Spaces Kluwer Academic/Plenum Publishers, New York, 2001. |
[30] |
S. Nicola and M. Pierre,
A note on $S$-asymptotically periodic functions, Nonlinear Analysis, Real World Applications, 10 (2009), 2937-2938.
doi: 10.1016/j.nonrwa.2008.09.011. |
[31] |
M. Pinto,
Pseudo-almost periodic solutions of neutral integral and differential equations and applications, Nonlinear Analysis, 72 (2010), 4377-4383.
doi: 10.1016/j.na.2009.12.042. |
[32] |
M. Pinto and G. Robledo,
Cauchy matrix for linear almost periodic systems and some consequences, Nonlinear Anal., 74 (2011), 5426-5439.
doi: 10.1016/j.na.2011.05.027. |
[33] |
M. Pinto and G. Robledo,
Diagonalizability of nonautonomous linear systems with bounded continuous coefficients, J. Math. Anal. Appl., 407 (2013), 513-526.
doi: 10.1016/j.jmaa.2013.05.038. |
[34] |
M. Pinto and G. Robledo,
Existence and stability of almost periodic solutions in impulsive neural network models, Appl. Math. Comput., 217 (2010), 4167-4177.
doi: 10.1016/j.amc.2010.10.033. |
[35] |
M. Pinto, V. Torres and G. Robledo,
Asymptotic equivalence of almost periodic solutions for a class of perturbed almost periodic systems, Glasg. Math. J., 52 (2010), 583-592.
doi: 10.1017/S0017089510000443. |
[36] |
W. R. Utz and P. Waltman,
Asymptotically almost periodicity of solutions of a system of differential equations, Proc. Amer. Math. Soc., 18 (1967), 597-601.
doi: 10.1090/S0002-9939-1967-0212285-6. |
[37] |
W. A. Veech,
Almost automorphic functions, Proceedings of the National Academy of Science of the United states of America, 49 (1963), 462-464.
doi: 10.1073/pnas.49.4.462. |
[38] |
W. A. Veech,
Almost automorphic function on groups, American Journal of Mathematics, 87 (1965), 719-751.
doi: 10.2307/2373071. |
[39] |
M. Wazewska-Czyzewska and A. Lasota,
Mathematical problems of the red-blood cell system, Ann. Polish Math. Soc. Ser. Ⅲ, Appl. Math., 6 (1976), 23-40.
|
[40] |
F. Wei and K. Wang,
Global stability and asymptotically periodic solutions for non autonomous cooperative Lotka-Volterra diffusion system, Applied Math. and Computation, 182 (2006), 161-165.
doi: 10.1016/j.amc.2006.03.044. |
[41] |
F. Wei and K. Wang,
Asymptotically periodic solutions of N-species cooperation system with time delay, Nonlinear Analysis, Real World and Applications, 7 (2006), 591-596.
doi: 10.1016/j.nonrwa.2005.03.019. |
[42] |
T. Xiao, X. Zhu and J. Liang,
Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications, Nonlinear Analysis, 70 (2009), 4079-4085.
doi: 10.1016/j.na.2008.08.018. |
[43] |
Z. Yao,
Uniqueness and exponential stability of almost periodic positive solution for Lasota-Wazewska model with impulse and infinite delay, Math. Methods Appl. Sci., 38 (2015), 677-684.
doi: 10.1002/mma.3098. |
[44] |
T. Yoshizawa,
Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions Applied Mathematical Sciences, 14 Springer-Verlag, New York-Heidelberg, 1975. |
[45] |
H. Zhao,
Existence and global attractivity of almost periodic solutions for cellular neutral network with distributed delays, Appl. Math. Comput., 154 (2004), 683-695.
doi: 10.1016/S0096-3003(03)00743-4. |
[46] |
S. Zaidman,
Almost automorphic solutions of same abstract evolution equations, Instituto Lombardo, Accademia di Science e Letter, 110 (1976), 578-588.
|
[47] |
S. Zaidman,
Existence of asymptotically almost-periodic and of almost automorphic solutions for same classes of abstract differential equations, Annales des Science Mathématiques du Québec, 13 (1989), 79-88.
|
[48] |
M. Zaki,
Almost automorphic solutions of certain abstract differential equations, Annali di Mathematica pura et Applicata, 101 (1974), 91-114.
doi: 10.1007/BF02417100. |
[49] |
C. Zhang,
Almost Periodic Type Functions and Ergodicity, Science Press, Beijing, Kluwer Academic Publishers, Dordrecht, 2003.
doi: 10.1007/978-94-007-1073-3. |
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