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Exact asymptotics of positive solutions to Dickman equation
1. | Brno University of Technology, Faculty of Civil Engineering, Department of Mathematics and Descriptive Geometry, Veveří 331/95,602 00 Brno, Czech Republic |
2. | Universidad de Los Lagos, Departamento de Ciencias Exactas, Casilla 933, Osorno, Chile |
$\dot x (t)=-\frac{1}{t}\,x(t-1),$ |
$t \to \infty $ |
$x(t)=1$ |
$0≤ t≤ 1$ |
References:
[1] |
E. Bach and R. Peralta,
Asymptotic semismoothness probabilities, Math. Comp., 65 (1996), 1701-1715.
doi: 10.1090/S0025-5718-96-00775-2. |
[2] |
R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, 1963. |
[3] |
H. Bereketoğlu and M. Pituk,
Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays, Discrete Contin. Dyn. Syst., (2003), 100-107.
|
[4] |
H. Bereketoğlu and F. Karakoç,
Asymptotic constancy for impulsive delay differential equations, Dyn. Syst. Appl., 17 (2008), 71-83.
|
[5] |
D. Broadhurst, Dickman polylogarithms and their constants, arXiv: 1004. 0519v1 [math-ph] 4 Apr 2010, 1-11. |
[6] |
N. G. de Bruijn,
On the number of positive integers $≤ x$ and free of prime factors $> y$, Indag. Math., 54 (1951), 50-60.
|
[7] |
N. G. de Bruijn,
On the number of positive integers $≤ x$ and free of prime factors $> y$. Ⅱ, Indag. Math., 28 (1966), 239-247.
|
[8] |
N. G. de Bruijn,
The asymptotic behaviour of a function occurring in the theory of primes, J. Indian Math. Soc. (N.S.), 15 (1951), 25-32.
|
[9] |
J. Diblík,
A criterion for existence of positive solutions of systems of retarded functional differential equations, Nonlinear Anal., 38 (1999), 327-339.
doi: 10.1016/S0362-546X(98)00199-0. |
[10] |
J. Diblík and N. Koksch,
Positive solutions of the equation $\dot{x}(t)=-c(t)x(t-τ )$ in the critical case, J. Math. Anal. Appl., 250 (2000), 635-659.
doi: 10.1006/jmaa.2000.7008. |
[11] |
J. Diblík and M. Růžičková,
Asymptotic behavior of solutions and positive solutions of differential delayed equations, Funct. Differ. Equ., 14 (2007), 83-105.
|
[12] |
K. Dickman,
On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astron. Fys., 22A (1930), 1-14.
|
[13] |
I. Györi and M. Pituk,
Asymptotic formulas for a scalar linear delay differential equation, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), 1-14.
|
[14] |
I. Györi and M. Pituk,
Stability criteria for linear delay differential equations, J. Differential Equations, 10 (1997), 841-852.
|
[15] |
E. Kozakiewicz,
Über das asymptotische Verhalten der nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument, Wiss. Z. Humboldt Univ. Berlin, Math. Nat. R., 13 (1964), 577-589.
|
[16] |
E. Kozakiewicz,
Über die nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument, Math. Nachr., 32 (1966), 107-113.
doi: 10.1002/mana.19660320112. |
[17] |
E. Kozakiewicz,
Zur Abschätzung des Abklingens der nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument, Wiss. Z. Humboldt Univ. Berlin, Math. Nat. R., 15 (1966), 675-676.
|
[18] |
P. Moree, Integers without large prime factors: From Ramanujan to de Bruijn, Integers, 14A (2014), Paper No. A5, 13 pp. |
[19] |
A. D. Myshkis, Linear Differential Equations with Retarded Arguments, Second edition. Izdat. "Nauka", Moscow, 1972. |
[20] |
M. Pituk and G. Röst,
Large time behavior of a linear delay differential equation with asymptotically small coefficient, Bound. Value Probl., 2014 (2014), 1-9.
doi: 10.1186/1687-2770-2014-114. |
[21] |
V. Ramaswami,
On the number of positive integers less than $x$ and free of prime divisors greated than $x^c$, Bull. Amer. Math. Soc., 55 (1949), 1122-1127.
doi: 10.1090/S0002-9904-1949-09337-0. |
[22] |
K. P. Rybakowski,
Wa_zewski's principle for retarded functional differential equations, J. Differential Equations, 36 (1980), 117-138.
doi: 10.1016/0022-0396(80)90080-7. |
[23] |
K. Soundararajan,
An asymptotic expansion related to the Dickman function, The Ramanujan Journal, 29 (2012), 25-30, arXiv:1005.3494v1.
doi: 10.1007/s11139-011-9304-3. |
[24] |
F. I. Wheeler,
Two differential-difference equations arising in number theory, Trans. Amer. Math. Soc., 318 (1990), 491-523.
doi: 10.1090/S0002-9947-1990-0963247-X. |
[25] | |
[26] |
Nieuw Archief voor Wiskunde, Problem Section, 4/14 No 3 Nov. 1996, p. 429. |
[27] |
Nieuw Archief voor Wiskunde, Problem Section, 5/9 No 2 June 2008, p. 232. |
[28] |
Nieuw Archief voor Wiskunde, Problem Section, 5/11 No 1 March 2010, p. 76. |
show all references
References:
[1] |
E. Bach and R. Peralta,
Asymptotic semismoothness probabilities, Math. Comp., 65 (1996), 1701-1715.
doi: 10.1090/S0025-5718-96-00775-2. |
[2] |
R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, 1963. |
[3] |
H. Bereketoğlu and M. Pituk,
Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays, Discrete Contin. Dyn. Syst., (2003), 100-107.
|
[4] |
H. Bereketoğlu and F. Karakoç,
Asymptotic constancy for impulsive delay differential equations, Dyn. Syst. Appl., 17 (2008), 71-83.
|
[5] |
D. Broadhurst, Dickman polylogarithms and their constants, arXiv: 1004. 0519v1 [math-ph] 4 Apr 2010, 1-11. |
[6] |
N. G. de Bruijn,
On the number of positive integers $≤ x$ and free of prime factors $> y$, Indag. Math., 54 (1951), 50-60.
|
[7] |
N. G. de Bruijn,
On the number of positive integers $≤ x$ and free of prime factors $> y$. Ⅱ, Indag. Math., 28 (1966), 239-247.
|
[8] |
N. G. de Bruijn,
The asymptotic behaviour of a function occurring in the theory of primes, J. Indian Math. Soc. (N.S.), 15 (1951), 25-32.
|
[9] |
J. Diblík,
A criterion for existence of positive solutions of systems of retarded functional differential equations, Nonlinear Anal., 38 (1999), 327-339.
doi: 10.1016/S0362-546X(98)00199-0. |
[10] |
J. Diblík and N. Koksch,
Positive solutions of the equation $\dot{x}(t)=-c(t)x(t-τ )$ in the critical case, J. Math. Anal. Appl., 250 (2000), 635-659.
doi: 10.1006/jmaa.2000.7008. |
[11] |
J. Diblík and M. Růžičková,
Asymptotic behavior of solutions and positive solutions of differential delayed equations, Funct. Differ. Equ., 14 (2007), 83-105.
|
[12] |
K. Dickman,
On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astron. Fys., 22A (1930), 1-14.
|
[13] |
I. Györi and M. Pituk,
Asymptotic formulas for a scalar linear delay differential equation, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), 1-14.
|
[14] |
I. Györi and M. Pituk,
Stability criteria for linear delay differential equations, J. Differential Equations, 10 (1997), 841-852.
|
[15] |
E. Kozakiewicz,
Über das asymptotische Verhalten der nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument, Wiss. Z. Humboldt Univ. Berlin, Math. Nat. R., 13 (1964), 577-589.
|
[16] |
E. Kozakiewicz,
Über die nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument, Math. Nachr., 32 (1966), 107-113.
doi: 10.1002/mana.19660320112. |
[17] |
E. Kozakiewicz,
Zur Abschätzung des Abklingens der nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument, Wiss. Z. Humboldt Univ. Berlin, Math. Nat. R., 15 (1966), 675-676.
|
[18] |
P. Moree, Integers without large prime factors: From Ramanujan to de Bruijn, Integers, 14A (2014), Paper No. A5, 13 pp. |
[19] |
A. D. Myshkis, Linear Differential Equations with Retarded Arguments, Second edition. Izdat. "Nauka", Moscow, 1972. |
[20] |
M. Pituk and G. Röst,
Large time behavior of a linear delay differential equation with asymptotically small coefficient, Bound. Value Probl., 2014 (2014), 1-9.
doi: 10.1186/1687-2770-2014-114. |
[21] |
V. Ramaswami,
On the number of positive integers less than $x$ and free of prime divisors greated than $x^c$, Bull. Amer. Math. Soc., 55 (1949), 1122-1127.
doi: 10.1090/S0002-9904-1949-09337-0. |
[22] |
K. P. Rybakowski,
Wa_zewski's principle for retarded functional differential equations, J. Differential Equations, 36 (1980), 117-138.
doi: 10.1016/0022-0396(80)90080-7. |
[23] |
K. Soundararajan,
An asymptotic expansion related to the Dickman function, The Ramanujan Journal, 29 (2012), 25-30, arXiv:1005.3494v1.
doi: 10.1007/s11139-011-9304-3. |
[24] |
F. I. Wheeler,
Two differential-difference equations arising in number theory, Trans. Amer. Math. Soc., 318 (1990), 491-523.
doi: 10.1090/S0002-9947-1990-0963247-X. |
[25] | |
[26] |
Nieuw Archief voor Wiskunde, Problem Section, 4/14 No 3 Nov. 1996, p. 429. |
[27] |
Nieuw Archief voor Wiskunde, Problem Section, 5/9 No 2 June 2008, p. 232. |
[28] |
Nieuw Archief voor Wiskunde, Problem Section, 5/11 No 1 March 2010, p. 76. |
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