Almost automorphic delayed differential equations and Lasota-Wazewska model

Aníbal Coronel; Christopher Maulén; Manuel Pinto; Daniel Sepúlveda

doi:10.3934/dcds.2017083

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April 2017, 37(4): 1959-1977. doi: 10.3934/dcds.2017083

Almost automorphic delayed differential equations and Lasota-Wazewska model

Aníbal Coronel ^1,, , Christopher Maulén ^2, , Manuel Pinto ^2, and Daniel Sepúlveda ^3,

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

* Corresponding author: acoronel@ubiobio.cl

Received March 2016 Revised November 2016 Published December 2016

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.

Keywords: Abstract delay differential equations, almost automorphic, exponential dichotomy, ergodicity, evolution operator.

Mathematics Subject Classification: Primary:43A60, 65L03;Secondary:92D25.

Citation: Aníbal Coronel, Christopher Maulén, Manuel Pinto, Daniel Sepúlveda. Almost automorphic delayed differential equations and Lasota-Wazewska model. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1959-1977. doi: 10.3934/dcds.2017083

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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