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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

 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,
 $m\not = 0, 1$
.
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.
Solution of the system (69)
Solution of the system (70), (71)
Solution of the system (73)
Solution of the system (74)
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