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

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

* 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.

Citation: Aníbal Coronel, Christopher Maulén, Manuel Pinto, Daniel Sepúlveda. Almost automorphic delayed differential equations and Lasota-Wazewska model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1959-1977. doi: 10.3934/dcds.2017083
References:
[1]

J. BlotG. MophouG. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

S. Castillo and M. Pinto, Dichotomy and almost automorphic solution of difference system, Electron. J. Qual. Theory Differ. Equ. 32 (2013), 17 pp. Google Scholar

[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. Google Scholar

[8]

P. CieutatS. 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. Google Scholar

[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. Google Scholar

[10]

A. CoronelM. 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. Google Scholar

[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. Google Scholar

[12]

H.-S. DingT.-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. Google Scholar

[13]

T. Diagana, Pseudo Almost Periodic Functions in Banach Space Nova Science Publishers, Inc. , New York, 2007. Google Scholar

[14]

L. DuanL. 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. Google Scholar

[15]

S. FatajouN. V. MinhG. N'Guérékata and A. Pankov, Stepanov-like almost automorphic solutions for nonautonomous evolution equations, Electronic Journal of Differential Equations, 121 (2007), 1-17. Google Scholar

[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. Google Scholar

[17]

S. G. Gal and G. M. N'Guérékata, Almost automorphic fuzzy-number-valued functions, Journal of Fuzzy Mathematics, 13 (2005), 185-208. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

H. R. HenríquezM. 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. Google Scholar

[22]

H. R. HenríquezM. 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. Google Scholar

[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. Google Scholar

[24]

J. LiuG. 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. Google Scholar

[25]

E. LizC. Martínez and S. Trofimchuk, Attractivity properties of infinite delay Mackey-Glass type equations, Differential and Integral Equations, 15 (2002), 875-896. Google Scholar

[26]

N. V. MinhT. 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. Google Scholar

[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. Google Scholar

[28]

G. M. N'Guérékata, Topics in Almost Automorphy Springer-Verlag, New York, 2005. Google Scholar

[29]

G. M. N'Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces Kluwer Academic/Plenum Publishers, New York, 2001. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[35]

M. PintoV. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

W. A. Veech, Almost automorphic function on groups, American Journal of Mathematics, 87 (1965), 719-751. doi: 10.2307/2373071. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[42]

T. XiaoX. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[46]

S. Zaidman, Almost automorphic solutions of same abstract evolution equations, Instituto Lombardo, Accademia di Science e Letter, 110 (1976), 578-588. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

References:
[1]

J. BlotG. MophouG. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

S. Castillo and M. Pinto, Dichotomy and almost automorphic solution of difference system, Electron. J. Qual. Theory Differ. Equ. 32 (2013), 17 pp. Google Scholar

[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. Google Scholar

[8]

P. CieutatS. 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. Google Scholar

[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. Google Scholar

[10]

A. CoronelM. 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. Google Scholar

[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. Google Scholar

[12]

H.-S. DingT.-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. Google Scholar

[13]

T. Diagana, Pseudo Almost Periodic Functions in Banach Space Nova Science Publishers, Inc. , New York, 2007. Google Scholar

[14]

L. DuanL. 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. Google Scholar

[15]

S. FatajouN. V. MinhG. N'Guérékata and A. Pankov, Stepanov-like almost automorphic solutions for nonautonomous evolution equations, Electronic Journal of Differential Equations, 121 (2007), 1-17. Google Scholar

[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. Google Scholar

[17]

S. G. Gal and G. M. N'Guérékata, Almost automorphic fuzzy-number-valued functions, Journal of Fuzzy Mathematics, 13 (2005), 185-208. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

H. R. HenríquezM. 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. Google Scholar

[22]

H. R. HenríquezM. 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. Google Scholar

[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. Google Scholar

[24]

J. LiuG. 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. Google Scholar

[25]

E. LizC. Martínez and S. Trofimchuk, Attractivity properties of infinite delay Mackey-Glass type equations, Differential and Integral Equations, 15 (2002), 875-896. Google Scholar

[26]

N. V. MinhT. 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. Google Scholar

[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. Google Scholar

[28]

G. M. N'Guérékata, Topics in Almost Automorphy Springer-Verlag, New York, 2005. Google Scholar

[29]

G. M. N'Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces Kluwer Academic/Plenum Publishers, New York, 2001. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[35]

M. PintoV. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

W. A. Veech, Almost automorphic function on groups, American Journal of Mathematics, 87 (1965), 719-751. doi: 10.2307/2373071. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[42]

T. XiaoX. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[46]

S. Zaidman, Almost automorphic solutions of same abstract evolution equations, Instituto Lombardo, Accademia di Science e Letter, 110 (1976), 578-588. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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Eugenii Shustin. Exponential decay of oscillations in a multidimensional delay differential system. Conference Publications, 2003, 2003 (Special) : 809-816. doi: 10.3934/proc.2003.2003.809

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Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166

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