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

Irina Astashova; Josef Diblík; Evgeniya Korobko

doi:10.3934/dcdss.2021133

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Preface: Special issue on advances in partial differential equations

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 and 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.
[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.
[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.
[4]	I. V. Astashova, On asymptotical behavior of solutions to a quasi-linear second order differential equations, Funct. Differ. Equ., 16 (2009), 93-115.
[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.
[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.
[7]	F. V. Atkinson, On second-order non-linear oscillations, Pacific J. Math., 5 (1955), 643-647. doi: 10.2140/pjm.1955.5.643.
[8]	R. Bellman, Stability Theory of Differential Equations, Dover Publications, Inc., New York, 2008.
[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.
[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.
[11]	M. Bohner and S. G. Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer, Cham, 2016. doi: 10.1007/978-3-319-47620-9.
[12]	M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. doi: 10.1007/978-0-8176-8230-9.
[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.
[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.
[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.
[16]	J. Diblík, Asymptotic behavior of solutions of discrete equations, Funct. Differ. Equ., 11 (2004), 37-48.
[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.
[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.
[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.
[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.
[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.
[22]	S. N. Elaydi, An Introduction to Difference Equations, 3$^{rd}$ edition, Undergraduate Texts in Mathematics, Springer, New York, 2005.
[23]	R. Emden, Gaskugeln: Anwendungen der mechanischen Wärmetheorie auf Kosmologie und Meteorologischen Probleme, Teubner, Leipzig and Berlin, 1907. doi: 10.1007/BF01736734.
[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.
[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.
[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.
[27]	H. Goenner and P. Havas, Exact solutions of the generalized Lane–Emden equation, J. Math. Phys., 41 (2000), 7029-7042.
[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
[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.
[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.
[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.
[32]	C. M. Khalique, The Lane–Emden–Fowler equation and its generalizations - Lie symmetry analysis, Astrophysics, I. Kucuk (Ed.), 7 (2012), 131–148.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[41]	M. A. Radin, Difference Equations for Scientists and Engineering: Interdisciplinary Difference Equations, World Scientific Publishing, Singapore, 2019. doi: 10.1142/11349.

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.
[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.
[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.
[4]	I. V. Astashova, On asymptotical behavior of solutions to a quasi-linear second order differential equations, Funct. Differ. Equ., 16 (2009), 93-115.
[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.
[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.
[7]	F. V. Atkinson, On second-order non-linear oscillations, Pacific J. Math., 5 (1955), 643-647. doi: 10.2140/pjm.1955.5.643.
[8]	R. Bellman, Stability Theory of Differential Equations, Dover Publications, Inc., New York, 2008.
[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.
[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.
[11]	M. Bohner and S. G. Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer, Cham, 2016. doi: 10.1007/978-3-319-47620-9.
[12]	M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. doi: 10.1007/978-0-8176-8230-9.
[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.
[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.
[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.
[16]	J. Diblík, Asymptotic behavior of solutions of discrete equations, Funct. Differ. Equ., 11 (2004), 37-48.
[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.
[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.
[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.
[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.
[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.
[22]	S. N. Elaydi, An Introduction to Difference Equations, 3$^{rd}$ edition, Undergraduate Texts in Mathematics, Springer, New York, 2005.
[23]	R. Emden, Gaskugeln: Anwendungen der mechanischen Wärmetheorie auf Kosmologie und Meteorologischen Probleme, Teubner, Leipzig and Berlin, 1907. doi: 10.1007/BF01736734.
[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.
[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.
[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.
[27]	H. Goenner and P. Havas, Exact solutions of the generalized Lane–Emden equation, J. Math. Phys., 41 (2000), 7029-7042.
[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
[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.
[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.
[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.
[32]	C. M. Khalique, The Lane–Emden–Fowler equation and its generalizations - Lie symmetry analysis, Astrophysics, I. Kucuk (Ed.), 7 (2012), 131–148.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[41]	M. A. Radin, Difference Equations for Scientists and Engineering: Interdisciplinary Difference Equations, World Scientific Publishing, Singapore, 2019. doi: 10.1142/11349.

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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