ON THE ASYMPTOTIC BEHAVIOR OF A GENERALIZED NONLINEAR EQUATION

Year 2020, Volume: 38 Issue: 4, 2109 - 2121, 05.10.2021

Juan E. Nápoles Valdés María Nubia Quevedo Cubıllos Adrian R. Gómez Plata

Abstract

In this article we present a second-order differential equation in the framework of the derivative N, and various qualitative properties of the solutions are studied, firstly conditions are obtained under which the equation under study has a non-continuity solution at infinity. Later we study the conditions for the prolongation of the solutions and their oscillation.

Keywords

Generalized derivatives, asymptotic behavior, oscillation

References

• [1] A. Atangana, Derivative with a New Parameter Theory, Methods and Applications, Academic Press, 2016.
• [2] Baleanu, D., COMMENTS ON: Ortigueira M., Martynyuk V., Fedula M., Machado J.A.T., The failure of certain fractional calculus operators in two physical models, in Fract. Calc. Appl. Anal. 22(2)(2019), Fract. Calc. Appl. Anal., Volume 23: Issue 1, DOI: https://doi.org/10.1515/fca-2020-0012.
• [3] D. Baleanu, and A. Fernandez, On Fractional Operators and Their Classifications, Mathematics 2019, 7, 830; doi:10.3390/math7090830
• [4] Miroslav Bartuˇsek; Mariella Cecchi; Zuzana Doˇsl´a; Mauro Marin, Global monotonicity and oscillation for second order differential equation, Czechoslovak Mathematical Journal, Vol. 55 (2005), No. 1, 209–222.
• [5] I. Cınar, On Some Properties of Generalized Riesz Potentials, Intern. Math. Journal, Vol. 3,2003, no. 12, 1393-1397
• [6] E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955.
• [7] O. Dosly, N. Yamaoka, Oscillation constants for second-order ordinary differential equations related to elliptic equations with p-Laplacian, Nonlinear Analysis 113 (2015) 115-136.
• [8] R. Emden, Gaskugeln, Anwendungen der mechanischen Warmentheorie auf Kosmologie und metheorologische Probleme, Leipzig, 1907.
• [9] Alberto Fleitas, J. A. Mendez-Bermudez, Juan E. Napoles Valdes, Jos´e M. Sigarreta Almira, On fractional Li´enard-type systems, Revista Mexicana de F´ısica, 65 (6) 618–625 NOVEMBER-DECEMBER 2019.
• [10] R.H. Fowler, The solutions of Emden’s and similar differential equations, Monthly Notices of the RoyalAstronomical Society 91 (1930) 63-91.
• [11] P. M. Guzman, G. Langton, L. M. Lugo, J. Medina, J., J. E. N´apoles Vald´es, A new definition of a fractional derivative of local type. J. Math. Anal., 9:2, 88-98 (2018).
• [12] P. M. Guzman, L. M. Lugo, J. E. Napoles Valdes, On the stability of solutions of fractional non-conformable differential equations, Studia Universitatis Babes, -Bolyai Mathematica, to appear.
• [13] P. M. Guzman, J. E. Napoles Valdes, A note on the oscillatory character of some non-conformable generalized Lienard system, Advanced Mathematical Models and Applications Vol.4, No.2, 2019, pp.127-133.
• [14] P. M. Guzman, L. M. Lugo, J. E. Napoles Valdes, M. Vivas, On a New Generalized Integral Operator and Certain Operating Properties, Axioms 2020, 9, 69; doi:10.3390/axioms9020069
• [15] L. L. Helms, Introduction To Potential Theory, New York: Wiley-Interscience, 1969.
• [16] Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math., 264, 65-70 (2014).
• [17] Kilbas, A., Srivastava, M.H. and Trujillo, J. J. (2006). Theory and application on fractional differential equations. Amsterdam: North Holland.
• [18] K. S. Miller, An Introduction to Fractional Calculus and Fractional Differential Equations, J. Wiley and Sons, New York, 1993.
• [19] Francisco Martinez, Pshtiwan Othman Mohammed, Juan E. N´apoles Valdes, non-conformable Fractional Laplace Transform, Kragujevac Journal of Mathematics, Volume 46(3) (2022), Pages 341-354.
• [20] C. Martinez, M. Sanz, F. Periogo, Distributional Fractional Powers of Laplacian, Riesz Potential. Studia Mathematica 135 (3) 1999.
• [21] J. E. Napoles V., P. M. Guzman, L. M. Lugo, Some New Results on Nonconformable Fractional Calculus, Advances in Dynamical Systems and Applications, Volume 13, Number 2, pp. 167–175 (2018).
• [22] J. E. Napoles, P. M. Guzman, L. M. Lugo, A. Kashuri, The local non-conformable derivative and Mittag Leffler function, Sigma J Eng & Nat Sci 38 (2), 2020, 1007-1017.
• [23] J. E. Napoles, J. M. Rodriguez, J. M. Sigarreta, On Hermite-Hadamard type inequalities for non-conformable integral operators, Symmetry 2019, 11, 1108; doi:10.3390/sym11091108.
• [24] K. Oldham, J. Spanier, The Fractional Calculus, Theory and Applications of Differentiation and Integration of Arbitrary Order, Academic Press, USA, 1974.
• [25] I. Podlubny, Fractional Differential Equations, Academic Press, USA, 1999.
• [26] V. E. Tarasov, No violation of the Leibniz rule. No fractional derivative, Commun. Nonlinear Sci. Numer. Simulat. 18 (2013) 2945-2948.
• [27] V. E. Tarasov, Leibniz rule and fractional derivatives of power functions, J. of Computational and Nonlinear Dynamics, May 2016, Vol. 11, 031014 (1-4).
• [28] V. E. Tarasov, No nonlocality. No fractional derivative, Communications in Nonlinear Science and Numerical Simulation, 2018, 62, 157-163.
• [29] C. Tunc, A note on boundedness of solutions to a class of non-autonomous differential equations of second order. Appl. Anal. Discrete Math. 4 (2010), no. 2, 361–372.
• [30] C. Tunc, On the qualitative behaviors of a functional differential equation of second order. Appl. Appl. Math. 12 (2017), no. 2, 813–842.
• [31] C. Tunc, S.A. Mohammed, On the asymptotic analysis of bounded solutions to nonlinear differential equations of second order. Adv. Difference Equ. 2019, Paper No. 461, 19 pp.
• [32] O. Tunc, C. Tunc, On the asymptotic stability of solutions of stochastic differential delay equations of second order. Journal of Taibah University for Science. 13 (2019), no.1, 875–882.
• [33] S. Umarov, S. Steinberg, Variable order differential equations with piecewise constant order-function and diffusion with changing modes, Z. Anal. Anwend. 28 (4) (2009) 431-450.
• [34] X. J. Yang, M. A. Aty, C. Cattani, A New General Fractional-Order Derivative with Rabotnov fractionalexponential kernel applied to model the anomalous heat transfer, Thermal Science: Year 2019, Vol. 23, No. 3A, pp. 1677-1681.
• [35] D. Zhao and M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, 54: 903-917, 2017. DOI 10.1007/s10092-017-0213-8.