Mathematica Bohemica, Vol. 143, No. 4, pp. 377-389, 2018


Boundedness and square integrability of solutions of certain third-order differential equations

Djamila Beldjerd, Moussadek Remili

Received March 13, 2017.   First published January 2, 2018.

Abstract:  We establish some new sufficient conditions which guarantee the boundedness and square integrability of solutions of certain third order differential equation. Example is included to illustrate the results. By this work, we extend and improve some results in the literature.
Keywords:  third-order differential equation; boundedness; square integrability
Classification MSC:  34C11, 34D05


References:
[1] A. U. Afuwape: Further ultimate boundedness results for a third-order non-linear system of differential equations. Boll. Unione Mat. Ital., VI. Ser., C, Anal. Funz. Appl. 4 (1985), 347-361. MR 0805225 | Zbl 0592.34024
[2] A. U. Afuwape, M. O. Omeike: Further ultimate boundedness of solutions of some system of third order nonlinear ordinary differential equations. Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 43 (2004),7-20. MR 2124598 | Zbl 1068.34052
[3] L. È. Èl'sgol'ts: Introduction to the Theory of Differential Equations With Deviating Arguments. Translated from the Russian by Robert J. McLaughlin. Holden-Day, San Francisco-London-Amsterdam (1966). MR 0192154 | Zbl 0133.33502
[4] J. R. Graef, D. Beldjerd, M. Remili: On stability, ultimate boundedness, and existence of periodic solutions of certain third order differential equations with delay. Panam. Math. J. 25 (2015), 82-94. MR 3364326 | Zbl 1331.34145
[5] J. R. Graef, L. D. Oudjedi, M. Remili: Stability and square integrability of solutions of nonlinear third order differential equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 22 (2015), 313-324. MR 3405345 | Zbl 1326.34088
[6] M. Omeike: Stability and boundedness of solutions of some non-autonomous delay differential equation of the third order. An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 55 (2009), 49-58. MR 2510712 | Zbl 1199.34390
[7] M. O. Omeike: New results on the asymptotic behavior of a third-order nonlinear differential equation. Differ. Equ. Appl. 2 (2010), 39-51. DOI 10.7153/dea-02-04 | MR 2654750 | Zbl 1198.34085
[8] M. O. Omeike: New results on the stability of solution of some non-autonomous delay differential equations of the third order. Differ. Uravn. Protsessy Upr. (2010), 18-29. MR 2766411
[9] C. Qian: On global stability of third-order nonlinear differential equations. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 42 (2000), 651-661. DOI 10.1016/S0362-546X(99)00120-0 | MR 1776296 | Zbl 0969.34048
[10] C. Qian: Asymptotic behavior of a third-order nonlinear differential equation. J. Math. Anal. Appl. 284 (2003), 191-205. DOI 10.1016/S0022-247X(03)00302-0 | MR 1996127 | Zbl 1054.34078
[11] R. Reissig, G. Sansone, R. Conti: Non-linear Differential Equations of Higher Order. Monographs and Textbooks on Pure and Applied Mathematics. Noordhoff International Publishing, Leyden (1974). MR 0344556 | Zbl 0275.34001
[12] M. Remili, D. Beldjerd: On the asymptotic behavior of the solutions of third order delay differential equations. Rend. Circ. Mat. Palermo (2) 63 (2014), 447-455. DOI 10.1007/s12215-014-0169-3 | MR 3298595 | Zbl 1321.34097
[13] M. Remili, D. Beldjerd: A boundedness and stability results for a kind of third order delay differential equations. Appl. Appl. Math. 10 (2015), 772-782. MR 3447611 | Zbl 1331.34135
[14] M. Remili, D. Beldjerd: On ultimate boundedness and existence of periodic solutions of kind of third order delay differential equations. Acta Univ. M. Belii, Ser. Math. 24 (2016), 43-57. MR 3492940 | Zbl 06699479
[15] M. Remili, D. Beldjerd: Stability and ultimate boundedness of solutions of some third order differential equations with delay. Journal of the Association of Arab Universities for Basic and Applied Sciences 23 (2017), 90-95.
[16] M. Remili, D. L. Oudjedi: Stability and boundedness of the solutions of non autonomous third order differential equations with delay. Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 53 (2014), 139-147. MR 3331011 | Zbl 1317.34157
[14] M. Remili, L. D. Oudjedi: Uniform stability and boundedness of a kind of third order delay differential equations. Bull. Comput. Appl. Math. 2 (2014), 25-35.
[18] M. Remili, L. D. Oudjedi: Boundedness and stability in third order nonlinear differential equations with bounded delay. An. Univ. Oradea, Fasc. Mat. 23 (2016), 135-143. MR 3496022 | Zbl 1349.34303
[19] M. Remili, L. D. Oudjedi: Boundedness and stability in third order nonlinear differential equations with multiple deviating arguments. Arch. Math., Brno 52 (2016), 79-90. DOI 10.5817/AM2016-2-79 | MR 3535630 | Zbl 06644060
[20] M. Remili, L. D. Oudjedi: Stability of the solutions of nonlinear third order differential equations with multiple deviating arguments. Acta Univ. Sapientiae, Math. 8 (2016), 150-165. DOI 10.1515/ausm-2016-0009 | MR 3535706 | Zbl 1351.34087
[21] A. I. Sadek: On the stability of solutions of some non-autonomous delay differential equations of the third order. Asymptotic Anal. 43 (2005), 1-7. MR 2148124 | Zbl 1075.34075
[22] C. Tunç: On asymptotic stability of solutions to third order nonlinear differential equations with retarded argument. Commun. Appl. Anal. 11 (2007), 515-528. MR 2368199 | Zbl 1139.34054
[23] C. Tunç: Stability and boundedness of solutions of nonlinear differential equations of third-order with delay. Differ. Uravn. Protsessy Upr. (2007), 13 pages. MR 2384532
[24] C. Tunç: On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument. Nonlinear Dyn. 57 (2009), 97-106. DOI 10.1007/s11071-008-9423-6 | MR 2511159 | Zbl 1176.34064
[25] C. Tunç: Some stability and boundedness conditions for non-autonomous differential equations with deviating arguments. Electron. J. Qual. Theory Differ. Equ. 2010 (2010), paper No. 1, 12 pages. MR 2577154 | Zbl 1201.34123
[26] C. Tunç: The boundedness of solutions to nonlinear third order differential equations. Nonlinear Dyn. Syst. Theory 10 (2010), 97-102. MR 2643196 | Zbl 1300.34077
[27] C. Tunç, M. Gözen: Stability and uniform boundedness in multidelay functional differential equations of third order. Abstr. Appl. Anal. 2013 (2013), Article ID 248717, 7 pages. DOI 10.1155/2013/248717 | MR 3055944 | Zbl 1276.34058
[28] Y. Zhu: On stability, boundedness and existence of periodic solution of a kind of third order nonlinear delay differential system. Ann. Differ. Equations 8 (1992), 249-259. MR 1190138 | Zbl 0758.34072

Affiliations:   Djamila Beldjerd, Moussadek Remili (corresponding author), Department of Mathematics, University of Oran 1 Ahmed Ben Bella, 31000 Oran, Algeria, e-mail: dj.beldjerd@gmail.com, remilimous@gmail.com


 
PDF available at: