Abstract
In this paper, the asymptotic behavior of solutions of a class of non-autonomous third order neutral delay differential equations is studied. Results on the stability, boundedness, and the square integrability of solutions and their derivatives are obtained. An example to illustrate the results is included.
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Baculíkovà, B., Džurina, J.: On the asymptotic behavior of a class of third order nonlinear neutral differential equations. Cent. Eur. J. Math. 8, 1091–1103 (2010)
Burton, T.A.: Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Mathematics in Science and Engineering, vol. 178. Academic Press, Orlando (1985)
Das, P., Misra, N.: A necessary and sufficient condition for the solution of a functional differential equation to be oscillatory or tend to zero. J. Math. Anal. Appl. 204, 78–87 (1997)
Dorociaková, B.: Some nonoscillatory properties of third order differential equations of neutral type. Tatra Mt. Math. Publ. 38, 71–76 (2007)
Došlá, Z., Liška, P.: Oscillation of third-order nonlinear neutral differential equations. Appl. Math. Lett. 56, 42–48 (2016)
Došlá, Z., Liška, P.: Comparison theorems for third-order neutral differential equations. Electron. J. Differ. Equ. 2016(38), 1–13 (2016)
Graef, J.R., Beldjerd, D., Remili, M.: On stability, ultimate boundedness, and existence of periodic solutions of certain third order differential equations with delay. PanAmer. Math. J. 25, 82–94 (2015)
Graef, J.R., Oudjedi, L.D., Remili, M.: Stability and square integrability of solutions of nonlinear third order differential equations. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 22, 313–324 (2015)
Graef, J.R., Oudjedi, L.D., Remili, M.: Stability and square integrability of solutions to third order neutral delay differential equations. Tatra Mt. Math. Publ. 71, 81–97 (2018)
Graef, J.R., Tunç, C.: Global asymptotic stability and boundedness of certain multi-delay functional differential equations of third order. Math. Methods Appl. Sci. 38, 3747–3752 (2015)
Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)
Mihalíková, B., Kostiková, E.: Boundedness and oscillation of third order neutral differential equations. Tatra Mt. Math. Publ. 43, 137–144 (2009)
Omeike, M.O.: New results on the stability of solution of some non-autonomous delay differential equations of the third order. Differ. Equ. Control Proc. N. 1(2010), 18–29 (2010)
Omeike, M.O.: New results on the asymptotic behavior of a third-order nonlinear differential equation. Differ. Equ. Appl. 2, 39–51 (2010)
Oudjedi, L.D., Beldjerd, D., Remili, M.: On the stability of solutions for non-autonomous delay differential equations of third-order. Differ. Equ. Control Proc. N. 1, 22–34 (2014)
Remili, M., Beldjerd, D.: On the asymptotic behavior of the solutions of third order delay differential equations. Rend. Circ. Mat. Palermo 63, 447–455 (2014)
Remili, M., Beldjerd, D.: A boundedness and stability results for a kind of third order delay differential equations. Appl. Appl. Math. 10, 772–782 (2015)
Remili, M., Beldjerd, D.: On ultimate boundedness and existence of periodic solutions of kind of third order delay differential equations. Acta Univ. M. Belii Ser. Math. 1–15 (2016)
Remili, M., Beldjerd, D.: Stability and ultimate boundedness of solutions of some third order differential equations with delay. J. Assoc. Arab Univ. Basic Appl. Sci. 23, 90–95 (2017)
Remili, M., Oudjedi, L.D.: Boundedness and stability in third order nonlinear differential equations with bounded delay. An. Univ. Oradea Fasc. Mat. XXII I, 135–143 (2016)
Remili, M., Oudjedi, L.D.: Boundedness and stability in third order nonlinear differential equations with multiple deviating arguments. Arch. Math. (Brno) 52, 79–90 (2016)
Remili, M., Oudjedi, L.D.: Stability and boundedness of the solutions of non autonomous third order differential equations with delay, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 53, 139–147 (2014)
Remili, M., Oudjedi, L.D.: Stability of the solutions of nonlinear third order differential equations with multiple deviating arguments. Acta Univ. Sapientiae Math. 8, 150–165 (2016)
Remili, M., Oudjedi, L.D.: On asymptotic stability of solutions to third order nonlinear delay differential equation. Filomat 30, 3217–3226 (2016)
Remili, M., Oudjedi, L.D., Beldjerd, D.: On the qualitative behaviors of solutions to a kind of nonlinear third order differential equation with delay. Comm. Appl. Anal. 20, 53–64 (2016)
Remili, M., Oudjedi, L.D.: Uniform stability and boundedness of a kind of third order delay differential equations. Bull. Comput. Appl. Math. 2, 25–35 (2014)
Remili, M., Oudjedi, L.D.: Uniform ultimate boundedness and asymptotic behaviour of third order nonlinear delay differential equation. Afr. Mat. 27, 1227–1237 (2016)
Tian, Y.-Z., Cai, Y.-L., Fu, Y.-L., Li, T.-X.: Oscillation and asymptotic behavior of third-order neutral differential equations with distributed deviating arguments. Adv. Difference Equ. 2015(267), 14 (2015)
Tunç, C.: On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument. Nonlinear Dyn. 57, 97–106 (2009)
Tunç, C.: Bound of solutions to third-order nonlinear differential equations with bounded delay. J. Franklin Inst. 347, 415–425 (2010)
Tunç, C.: Stability and bounded of solutions to non-autonomous delay differential equations of third order. Nonlinear Dyn. 62, 945–953 (2010)
Tunç, C.: Some stability and boundedness conditions for non-autonomous differential equations with deviating arguments. Electron. J. Qual. Theory Differ. Equ. 2010(1), 1–12 (2010)
Li, T.-X., Zhang, C.-H., Xing, G.-J.: Oscillation of third-order neutral delay differential equations. Abstr. Appl. Anal. 2012, 11 (2012). Article ID 569201
Jianshe, Yu.: Asymptotic stability for a class of nonautonomous neutral differential equations. Chin. Ann. Math. Ser. B 18, 449–456 (1997)
Acknowledgements
J. R. Graef’s research was supported in part by a University of Tennessee at Chattanooga SimCenter—Center of Excellence in Applied Computational Science and Engineering (CEACSE) grant.
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Khatir, A.M., Graef, J.R. & Remili, M. Stability, boundedness, and square integrability of solutions to third order neutral differential equations with delay. Rend. Circ. Mat. Palermo, II. Ser 69, 823–836 (2020). https://doi.org/10.1007/s12215-019-00438-9
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DOI: https://doi.org/10.1007/s12215-019-00438-9