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December  2021, 14(12): 4159-4178. doi: 10.3934/dcdss.2021133

Existence of a solution of discrete Emden-Fowler equation caused by continuous equation

Irina Astashova 1,2, , Josef Diblík 3,4,, and  Evgeniya Korobko 4,

1. 

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Department of Differential Equations, Leninskiye Gory 1, Main Building, 119991 Moscow, Russian Federation
2. 

Plekhanov Russian University of Economics, Institute of Digital Economics and Information Technologies, Department of Higher Mathematics, Stremyanny lane 36, 117997 Moscow, Russian Federation
3. 

Brno University of Technology, Faculty of Civil Engineering, Department of Mathematics and Descriptive Geometry, Veveří 331/95,602 00 Brno, Czech Republic
4. 

Brno University of Technology, Faculty of Electrical Engineering and Communication, Department of Mathematics, Technická 2848/8,616 00 Brno, Czech Republic

* Corresponding author: Josef Diblík

Received  September 2021 Revised  October 2021 Published  December 2021 Early access  October 2021

The paper studies the asymptotic behaviour of solutions to a second-order non-linear discrete equation of Emden–Fowler type
$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0 $
where
$ u\colon \{k_0, k_0+1, \dots\}\to \mathbb{R} $
 is an unknown solution,
$ \Delta^2 u(k) $
 is its second-order forward difference,
$ k_0 $
 is a fixed integer and
$ \alpha $
,
$ m $
 are real numbers,
$ m\not = 0, 1 $
.
Keywords: Discrete Emden–Fowler equation, exact solution, formal solution, asymptotic behavior.
Mathematics Subject Classification: Primary: 39A33, 39A22; Secondary: 39A12.
Citation: Irina Astashova, Josef Diblík, Evgeniya Korobko. Existence of a solution of discrete Emden-Fowler equation caused by continuous equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4159-4178. doi: 10.3934/dcdss.2021133
References:
[1]

R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods and Applications, 2$^ {nd}$ edition, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 2000. doi: 10.1201/9781420027020.  Google Scholar

[2]

E. Akin-Bohner and J. Hoffacker, Oscillation properties of an Emden–Fowler type equation on discrete time scales, J. Difference Equ. Appl., 9 (2003), 603-612.  doi: 10.1080/1023619021000053575.  Google Scholar

[3]

I. V. Astashova, Asymptotic behavior of singular solutions of Emden–Fowler type equations, Translation of Differ. Uravn., 55 (2019), 597–606, Differ. Equ., 55 (2019), 581–590, (Russian). doi: 10.1134/S001226611905001X.  Google Scholar

[4]

I. V. Astashova, On asymptotical behavior of solutions to a quasi-linear second order differential equations, Funct. Differ. Equ., 16 (2009), 93-115.   Google Scholar

[5]

I. Astashova, On asymptotic behavior of solutions to Emden–Fowler type higher-order differential equations, Math. Bohem., 4 (2015), 479-488.  doi: 10.21136/MB.2015.144464.  Google Scholar

[6]

I. V. Astashova, Uniqueness of solutions to second order Emden–Fowler type equations with general power–law nonlinearity, J. Math. Sci. (N.Y.), 255 (2021), 543-550.  doi: 10.1007/s10958-021-05391-6.  Google Scholar

[7]

F. V. Atkinson, On second-order non-linear oscillations, Pacific J. Math., 5 (1955), 643-647.  doi: 10.2140/pjm.1955.5.643.  Google Scholar

[8]

R. Bellman, Stability Theory of Differential Equations, Dover Publications, Inc., New York, 2008.  Google Scholar

[9]

M. Bhakta and P.-T. Nguen, On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures, Adv. Nonlinear Anal., 9 (2020), 1480-1503.  doi: 10.1515/anona-2020-0060.  Google Scholar

[10]

S. Bodine and D. A. Lutz, Asymptotic Integration of Differential and Difference Equations, Lecture Notes in Mathematics, 2129, Springer, Cham, 2015. doi: 10.1007/978-3-319-18248-3.  Google Scholar

[11]

M. Bohner and S. G. Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer, Cham, 2016. doi: 10.1007/978-3-319-47620-9.  Google Scholar

[12]

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. doi: 10.1007/978-0-8176-8230-9.  Google Scholar

[13]

M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser, Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0201-1.  Google Scholar

[14]

Z. Cheng and G. Huang, A Liouville theorem for the subcritical Lane–Emden system, Discrete Contin. Dyn. Syst., 39 (2019), 1359-1377.  doi: 10.3934/dcds.2019058.  Google Scholar

[15]

C. Cowan and A. Razani, Singular solutions of a Lane–Emden system, Discrete Contin. Dyn. Syst., 41 (2021), 621-656.  doi: 10.3934/dcds.2020291.  Google Scholar

[16]

J. Diblík, Asymptotic behavior of solutions of discrete equations, Funct. Differ. Equ., 11 (2004), 37-48.   Google Scholar

[17]

J. Diblík, Discrete retract principle for systems of discrete equations, Comput. Math. Appl., 42 (2001), 515-528.  doi: 10.1016/S0898-1221(01)00174-2.  Google Scholar

[18]

J. Diblík, Long-time behavior of positive solutions of a differential equation with state-dependent delay, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 31-46.  doi: 10.3934/dcdss.2020002.  Google Scholar

[19]

J. Diblík and I. Hlavičková, Asymptotic properties of solutions of the discrete analogue of the Emden–Fowler equation, Adv. Stud. Pure Math., 53 (2009), 23-32.  doi: 10.2969/aspm/05310023.  Google Scholar

[20]

J. Diblík and E. Korobko, Solutions of perturbed second-order discrete Emden–Fowler type equation with power asymptotics of solutions, Mathematics, Information Technologies and Applied Sciences, Post-Conference Proceedings of Extended Versions of Selected Papers, 2020 (2020), 30-44.   Google Scholar

[21]

J. Diblík and Z. Svoboda, Existence of strictly decreasing positive solutions of linear differential equations of neutral type, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 67-84.  doi: 10.3934/dcdss.2020004.  Google Scholar

[22]

S. N. Elaydi, An Introduction to Difference Equations, 3$^{rd}$ edition, Undergraduate Texts in Mathematics, Springer, New York, 2005.  Google Scholar

[23]

R. Emden, Gaskugeln: Anwendungen der mechanischen Wärmetheorie auf Kosmologie und Meteorologischen Probleme, Teubner, Leipzig and Berlin, 1907. doi: 10.1007/BF01736734.  Google Scholar

[24]

L. Erbe, J. Baoguo and A. Peterson, On the asymptotic behaviour of solutions of Emden–Fowler equations on time scales, Ann. Mat. Pura Appl., 191 (2012), 205-217.  doi: 10.1007/s10231-010-0179-5.  Google Scholar

[25]

R. H. Fowler, The solutions of Emden's and similar differential equations, Mon. Not. R. Astron. Soc., 91 (1930), 63-91.  doi: 10.1093/mnras/91.1.63.  Google Scholar

[26]

M. Galewski, Dependence on parameters for a discrete Emden–Fowler equation, Appl. Math. Comput., 218 (2011), 1247-1253.  doi: 10.1016/j.amc.2011.06.005.  Google Scholar

[27]

H. Goenner and P. Havas, Exact solutions of the generalized Lane–Emden equation, J. Math. Phys., 41 (2000), 7029-7042.   Google Scholar

[28]

S. Goldberg, Introduction to Difference Equations with Illustrative Examples from Economics, Psychology, and Sociology, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London 1958  Google Scholar

[29]

P. Guha, Generalized Emden–Fowler equations in noncentral curl forces and first integrals, Acta Mech, 231 (2020), 815-825.  doi: 10.1007/s00707-019-02602-9.  Google Scholar

[30]

T.-X. He, P. J.-S. Shiue, Z. Nie and M. Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electron. Res. Arch., 28 (2020), 1049-1062.  doi: 10.3934/era.2020057.  Google Scholar

[31]

X. He, K. Wang and L. Xu, Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium, Electron. Res. Arch., 28 (2020), 1503-1528.  doi: 10.3934/era.2020079.  Google Scholar

[32]

C. M. Khalique, The Lane–Emden–Fowler equation and its generalizations - Lie symmetry analysis, Astrophysics, I. Kucuk (Ed.), 7 (2012), 131–148. Google Scholar

[33]

I. T. Kiguradze and T. A. Chanturia, Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, (Russian), Mathematics and its Applications (Soviet Series), 89. Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-011-1808-8.  Google Scholar

[34]

E. Korobko, Asymptotic characterization of solutions of Emden–Fowler type difference equation, The Student Conference EEICT 2021, Faculty of Electrical Engineering and Communication. Selected papers, Brno University of Technology, (2021), 250–255. Google Scholar

[35]

E. Korobko, On solutions of a discrete equation of Emden–Fowler type, The Student Conference EEICT 2020, Faculty of Electrical Engineering and Communication, Brno University of Technology, (2020), 441–446. Google Scholar

[36]

H. J. Lane, On the theoretical temperature of the Sun, under the hypothesis of a gaseous mass maintaining its volume by its internal heat, and depending on the laws of gases as known to terrestrial experiment, American J. of Science, 148 (1870), 57-74.  doi: 10.2475/ajs.s2-50.148.57.  Google Scholar

[37]

W. T. Li and S. S. Cheng, Asymptotically linear solutions of a discrete Emden–Fowler equation, Far East J. Math. Sci., 6 (1998), 521-542.   Google Scholar

[38]

W. T. Li, X. L. Fan and C. K. Zhong, Positive solutions of discrete Emden–Fowler equation with singular nonlinear term, Dynam. Systems Appl., 9 (2000), 247-254.   Google Scholar

[39]

S. C. Mancas and H. C. Rost, Two integrable classes of Emden–Fowler equations with applications in astrophysics and cosmology, Zeitschrift f. Naturforschung A, 73 (2018), 805-814.  doi: 10.1515/zna-2018-0062.  Google Scholar

[40]

J. Migda, Asymptotic properties of solutions to difference equations of Emden–Fowler type, Electron. J. Qual. Theory Differ. Equ., (2019), 17pp. doi: 10.14232/ejqtde.2019.1.77.  Google Scholar

[41]

M. A. Radin, Difference Equations for Scientists and Engineering: Interdisciplinary Difference Equations, World Scientific Publishing, Singapore, 2019. doi: 10.1142/11349.  Google Scholar

show all references

References:
[1]

R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods and Applications, 2$^ {nd}$ edition, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 2000. doi: 10.1201/9781420027020.  Google Scholar

[2]

E. Akin-Bohner and J. Hoffacker, Oscillation properties of an Emden–Fowler type equation on discrete time scales, J. Difference Equ. Appl., 9 (2003), 603-612.  doi: 10.1080/1023619021000053575.  Google Scholar

[3]

I. V. Astashova, Asymptotic behavior of singular solutions of Emden–Fowler type equations, Translation of Differ. Uravn., 55 (2019), 597–606, Differ. Equ., 55 (2019), 581–590, (Russian). doi: 10.1134/S001226611905001X.  Google Scholar

[4]

I. V. Astashova, On asymptotical behavior of solutions to a quasi-linear second order differential equations, Funct. Differ. Equ., 16 (2009), 93-115.   Google Scholar

[5]

I. Astashova, On asymptotic behavior of solutions to Emden–Fowler type higher-order differential equations, Math. Bohem., 4 (2015), 479-488.  doi: 10.21136/MB.2015.144464.  Google Scholar

[6]

I. V. Astashova, Uniqueness of solutions to second order Emden–Fowler type equations with general power–law nonlinearity, J. Math. Sci. (N.Y.), 255 (2021), 543-550.  doi: 10.1007/s10958-021-05391-6.  Google Scholar

[7]

F. V. Atkinson, On second-order non-linear oscillations, Pacific J. Math., 5 (1955), 643-647.  doi: 10.2140/pjm.1955.5.643.  Google Scholar

[8]

R. Bellman, Stability Theory of Differential Equations, Dover Publications, Inc., New York, 2008.  Google Scholar

[9]

M. Bhakta and P.-T. Nguen, On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures, Adv. Nonlinear Anal., 9 (2020), 1480-1503.  doi: 10.1515/anona-2020-0060.  Google Scholar

[10]

S. Bodine and D. A. Lutz, Asymptotic Integration of Differential and Difference Equations, Lecture Notes in Mathematics, 2129, Springer, Cham, 2015. doi: 10.1007/978-3-319-18248-3.  Google Scholar

[11]

M. Bohner and S. G. Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer, Cham, 2016. doi: 10.1007/978-3-319-47620-9.  Google Scholar

[12]

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. doi: 10.1007/978-0-8176-8230-9.  Google Scholar

[13]

M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser, Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0201-1.  Google Scholar

[14]

Z. Cheng and G. Huang, A Liouville theorem for the subcritical Lane–Emden system, Discrete Contin. Dyn. Syst., 39 (2019), 1359-1377.  doi: 10.3934/dcds.2019058.  Google Scholar

[15]

C. Cowan and A. Razani, Singular solutions of a Lane–Emden system, Discrete Contin. Dyn. Syst., 41 (2021), 621-656.  doi: 10.3934/dcds.2020291.  Google Scholar

[16]

J. Diblík, Asymptotic behavior of solutions of discrete equations, Funct. Differ. Equ., 11 (2004), 37-48.   Google Scholar

[17]

J. Diblík, Discrete retract principle for systems of discrete equations, Comput. Math. Appl., 42 (2001), 515-528.  doi: 10.1016/S0898-1221(01)00174-2.  Google Scholar

[18]

J. Diblík, Long-time behavior of positive solutions of a differential equation with state-dependent delay, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 31-46.  doi: 10.3934/dcdss.2020002.  Google Scholar

[19]

J. Diblík and I. Hlavičková, Asymptotic properties of solutions of the discrete analogue of the Emden–Fowler equation, Adv. Stud. Pure Math., 53 (2009), 23-32.  doi: 10.2969/aspm/05310023.  Google Scholar

[20]

J. Diblík and E. Korobko, Solutions of perturbed second-order discrete Emden–Fowler type equation with power asymptotics of solutions, Mathematics, Information Technologies and Applied Sciences, Post-Conference Proceedings of Extended Versions of Selected Papers, 2020 (2020), 30-44.   Google Scholar

[21]

J. Diblík and Z. Svoboda, Existence of strictly decreasing positive solutions of linear differential equations of neutral type, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 67-84.  doi: 10.3934/dcdss.2020004.  Google Scholar

[22]

S. N. Elaydi, An Introduction to Difference Equations, 3$^{rd}$ edition, Undergraduate Texts in Mathematics, Springer, New York, 2005.  Google Scholar

[23]

R. Emden, Gaskugeln: Anwendungen der mechanischen Wärmetheorie auf Kosmologie und Meteorologischen Probleme, Teubner, Leipzig and Berlin, 1907. doi: 10.1007/BF01736734.  Google Scholar

[24]

L. Erbe, J. Baoguo and A. Peterson, On the asymptotic behaviour of solutions of Emden–Fowler equations on time scales, Ann. Mat. Pura Appl., 191 (2012), 205-217.  doi: 10.1007/s10231-010-0179-5.  Google Scholar

[25]

R. H. Fowler, The solutions of Emden's and similar differential equations, Mon. Not. R. Astron. Soc., 91 (1930), 63-91.  doi: 10.1093/mnras/91.1.63.  Google Scholar

[26]

M. Galewski, Dependence on parameters for a discrete Emden–Fowler equation, Appl. Math. Comput., 218 (2011), 1247-1253.  doi: 10.1016/j.amc.2011.06.005.  Google Scholar

[27]

H. Goenner and P. Havas, Exact solutions of the generalized Lane–Emden equation, J. Math. Phys., 41 (2000), 7029-7042.   Google Scholar

[28]

S. Goldberg, Introduction to Difference Equations with Illustrative Examples from Economics, Psychology, and Sociology, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London 1958  Google Scholar

[29]

P. Guha, Generalized Emden–Fowler equations in noncentral curl forces and first integrals, Acta Mech, 231 (2020), 815-825.  doi: 10.1007/s00707-019-02602-9.  Google Scholar

[30]

T.-X. He, P. J.-S. Shiue, Z. Nie and M. Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electron. Res. Arch., 28 (2020), 1049-1062.  doi: 10.3934/era.2020057.  Google Scholar

[31]

X. He, K. Wang and L. Xu, Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium, Electron. Res. Arch., 28 (2020), 1503-1528.  doi: 10.3934/era.2020079.  Google Scholar

[32]

C. M. Khalique, The Lane–Emden–Fowler equation and its generalizations - Lie symmetry analysis, Astrophysics, I. Kucuk (Ed.), 7 (2012), 131–148. Google Scholar

[33]

I. T. Kiguradze and T. A. Chanturia, Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, (Russian), Mathematics and its Applications (Soviet Series), 89. Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-011-1808-8.  Google Scholar

[34]

E. Korobko, Asymptotic characterization of solutions of Emden–Fowler type difference equation, The Student Conference EEICT 2021, Faculty of Electrical Engineering and Communication. Selected papers, Brno University of Technology, (2021), 250–255. Google Scholar

[35]

E. Korobko, On solutions of a discrete equation of Emden–Fowler type, The Student Conference EEICT 2020, Faculty of Electrical Engineering and Communication, Brno University of Technology, (2020), 441–446. Google Scholar

[36]

H. J. Lane, On the theoretical temperature of the Sun, under the hypothesis of a gaseous mass maintaining its volume by its internal heat, and depending on the laws of gases as known to terrestrial experiment, American J. of Science, 148 (1870), 57-74.  doi: 10.2475/ajs.s2-50.148.57.  Google Scholar

[37]

W. T. Li and S. S. Cheng, Asymptotically linear solutions of a discrete Emden–Fowler equation, Far East J. Math. Sci., 6 (1998), 521-542.   Google Scholar

[38]

W. T. Li, X. L. Fan and C. K. Zhong, Positive solutions of discrete Emden–Fowler equation with singular nonlinear term, Dynam. Systems Appl., 9 (2000), 247-254.   Google Scholar

[39]

S. C. Mancas and H. C. Rost, Two integrable classes of Emden–Fowler equations with applications in astrophysics and cosmology, Zeitschrift f. Naturforschung A, 73 (2018), 805-814.  doi: 10.1515/zna-2018-0062.  Google Scholar

[40]

J. Migda, Asymptotic properties of solutions to difference equations of Emden–Fowler type, Electron. J. Qual. Theory Differ. Equ., (2019), 17pp. doi: 10.14232/ejqtde.2019.1.77.  Google Scholar

[41]

M. A. Radin, Difference Equations for Scientists and Engineering: Interdisciplinary Difference Equations, World Scientific Publishing, Singapore, 2019. doi: 10.1142/11349.  Google Scholar

Figure 1.  Solution of the system (69)
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Figure 2.  Solution of the system (70), (71)
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Figure 3.  Solution of the system (73)
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Figure 4.  Solution of the system (74)
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Figure 5.  Summary of admissible values
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