Mathematica Bohemica, Vol. 148, No. 3, pp. 303-327, 2023


Boundedness criteria for a class of second order nonlinear differential equations with delay

Daniel O. Adams, Mathew O. Omeike, Idowu A. Osinuga, Biodun S. Badmus

Received November 3, 2021.   Published online August 3, 2022.

Abstract:  We consider certain class of second order nonlinear nonautonomous delay differential equations of the form $a(t)x^{\prime\prime} + b(t)g(x,x^\prime) + c(t)h(x(t-r))m(x^\prime) = p(t,x,x^\prime)$ and $(a(t)x^\prime)^\prime+ b(t)g(x,x^\prime) + c(t)h(x(t-r))m(x^\prime) = p(t,x,x^\prime)$, where $a$, $b$, $c$, $g$, $h$, $m$ and $p$ are real valued functions which depend at most on the arguments displayed explicitly and $r$ is a positive constant. Different forms of the integral inequality method were used to investigate the boundedness of all solutions and their derivatives. Here, we do not require construction of the Lyapunov-Krasovskiǐ functional to establish our results. This work extends and improve on some results in the literature.
Keywords:  boundedness; nonlinear; differential equation of third order; integral inequality
Classification MSC:  34C11, 34C12, 34K12

PDF available at:  Institute of Mathematics CAS

References:
[1] D. O. Adams, A. L. Olutimo: Some results on the boundedness of solutions of a certain third order non-autonomous differential equations with delay. Adv. Stud. Contemp. Math., Kyungshang 29 (2019), 237-249. Zbl 1438.34234
[2] A. T. Ademola, S. Moyo, B. S. Ogundare, M. O. Ogundiran, O. A. Adesina: New conditions on the solutions of a certain third order delay differential equations with multiple deviating arguments. Differ. Uravn. Protsessy Upr. 2019 (2019), 33-69. MR 3935484 | Zbl 1414.34053
[3] A. U. Afuwape, M. O. Omeike: On the stability and boundedness of solutions of a kind of third order delay differential equations. Appl. Math. Comput. 200 (2008), 444-451. DOI 10.1016/j.amc.2007.11.037 | MR 2421659 | Zbl 1316.34070
[4] H. A. Antosiewicz: On nonlinear differential equations of the second order with integrable forcing term. J. Lond. Math. Soc. 30 (1955), 64-67. DOI 10.1112/jlms/s1-30.1.64 | MR 0065752 | Zbl 0064.08404
[5] Z. S. Athanassov: Boundedness criteria for solutions of certain second order nonlinear differential equations. J. Math. Anal. Appl. 123 (1987), 461-479. DOI 10.1016/0022-247X(87)90324-6 | MR 0883702 | Zbl 0642.34031
[6] R. Bellman, K. L. Cooke: Differential-Difference Equations. Mathematics in Science and Engineering 6. Academic Press, New York (1963). MR 0147745 | Zbl 0105.06402
[7] I. Bihari: Researches of the boundedness and stability of the solutions of non-linear differential equations. Acta Math. Acad. Sci. Hung. 8 (1957), 261-278. DOI 10.1007/BF02020315 | MR 0094516 | Zbl 0097.29301
[8] T. A. Burton: The generalized Lienard equation. J. SIAM Control, Ser. A 3 (1965), 223-230. DOI 10.1137/0303018 | MR 0190462 | Zbl 0135.30201
[9] T. A. Burton: Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Mathematics in Science and Engineering 178. Academic Press, Orlando (1985). DOI 10.1016/s0076-5392(09)x6019-4 | MR 0837654 | Zbl 0635.34001
[10] T. A. Burton, R. C. Grimmer: Stability properties of $(r(t)u^\prime)^\prime + a(t)f(u)g(u^\prime) = 0$. Monastsh. Math. 74 (1970), 211-222. DOI 10.1007/BF01303441 | MR 0262613 | Zbl 0195.09804
[11] T. A. Burton, L. Hatvani: Stability theorems for nonautonomous functional differential equations by Liapunov functionals. Tohoku Math. J., II. Ser. 41 (1989), 65-104. DOI 10.2748/tmj/1178227868 | MR 0985304 | Zbl 0677.34060
[12] T. A. Burton, R. H. Hering: Liapunov theory for functional differential equations. Rocky Mt. J. Math. 24 (1994), 3-17. DOI 10.1216/rmjm/1181072449 | MR 1270024 | Zbl 0806.34067
[13] T. A. Burton, G. Makay: Asymptotic stability for functional differential equations. Acta Math. Hung. 65 (1994), 243-251. DOI 10.1007/BF01875152 | MR 1281434 | Zbl 0805.34068
[14] R. D. Driver: Ordinary and Delay Differential Equations. Applied Mathematical Sciences 20. Springer, New York (1977). DOI 10.1007/978-1-4684-9467-9 | MR 0477368 | Zbl 0374.34001
[15] S. Dvořáková: The Qualitative and Numerical Analysis of Nonlinear Delay Differential Equations: Doctoral Thesis. Brno University of Technology, Brno (2011).
[16] L. È. Èl'sgol'ts: Introduction to the Theory of Differential Equations with Deviating Arguments. McLaughin Holden-Day, San Francisco (1966). MR 0192154 | Zbl 0133.33502
[17] L. È. Èl'sgol'ts, S. B. Norkin: Introduction to the Theory and Application of Differential Equations with Deviating Arguments. Mathematics in Science and Engineering 105. Academic Press, New York (1973). DOI 10.1016/s0076-5392(08)x6170-3 | MR 0352647 | Zbl 0287.34073
[18] H. Gabsi, A. Ardjouni, A. Djoudi: New technique in asymptotic stability for third-order nonlinear delay differential equations. Math. Eng. Sci. Aerospace 9 (2018), 315-330.
[19] K. Gopalsamy: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Mathematics and its Applications 74. Kluwer Academic, Dordrecht (1992). DOI 10.1007/978-94-015-7920-9 | MR 1163190 | Zbl 0752.34039
[20] J. R. Graef, P. W. Spikes: Asymptotic behavior of solutions of a second order nonlinear differential equation. J. Differ. Equations 17 (1975), 461-476. DOI 10.1016/0022-0396(75)90056-X | MR 0361275 | Zbl 0298.34028
[21] J. R. Graef, P. W. Spikes: Continuability, boundedness, and convergence to zero of solutions of a perturbed nonlinear ordinary differential equation. Czech. Math. J. 45 (1995), 663-683. DOI 10.21136/CMJ.1995.128549 | MR 1354925 | Zbl 0851.34050
[22] J. K. Hale: Theory of Functional Differential Equations. Applied Mathematical Sciences 3. Springer, New York (1977). DOI 10.1007/978-1-4612-9892-2 | MR 0508721 | Zbl 0352.34001
[23] T. H. Hildebrandt: Introduction to the Theory of Integration. Pure and Applied Mathematics 13. Academic Press, New York (1963). MR 0154957 | Zbl 0112.28302
[24] G. S. Jones: Fundamental inequalities for discrete and discontinuous functional equations. J. Soc. Ind. Appl. Math. 12 (1964), 43-57. DOI 10.1137/0112004 | MR 0162069 | Zbl 0154.05702
[25] V. Kolmanovskii, A. Myshkis: Introduction to the Theory and Applications of Functional Differential Equations. Mathematics and its Applications 463. Klumer Academic, Dordrecht (1999). DOI 10.1007/978-94-017-1965-0 | MR 1680144 | Zbl 0917.34001
[26] V. B. Kolmanovskii, V. R. Nosov: Stability of Functional Differential Equations. Mathematics in Science and Engineering 180. Academic Press, London (1986). MR 0860947 | Zbl 0593.34070
[27] N. N. Krasovskii: Stability of Motion: Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay. Stanford University Press, Stanford (1963). MR 0147744 | Zbl 0109.06001
[28] B. S. Lalli: On the boundedness of solutions of certain second-order differential equations. J. Math. Anal. Appl. 25 (1969), 182-188. DOI 10.1016/0022-247X(69)90221-2 | MR 0239184
[29] G. G. Legatos: Contribution to the qualitative theory of ordinary differential equations. Bull. Soc. Math. Grèce, N. Ser. 2 (1961), 1-44. (In Greek.) Zbl 0107.29202
[30] A. M. Mahmoud, C. Tunç: Asymptotic stability of solutions of a kind of third-order stochastic differential equations with delays. Miskolc Math. Notes 20 (2019), 381-393. DOI 10.18514/MMN.2019.2800 | MR 3986654 | Zbl 1438.34255
[31] J. E. Nápoles V.: A note on the qualitative behaviour of some second order nonlinear differential equations. Divulg. Mat. 10 (2002), 91-99. MR 1946903 | Zbl 1039.34030
[32] B. S. Ogundare, A. T. Ademola, M. O. Ogundiran, O. A. Adesina: On the qualitative behaviour of solutions to certain second order nonlinear differential equation with delay. Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 63 (2017), 333-351. DOI 10.1007/s11565-016-0262-y | MR 3712445 | Zbl 1387.34096
[33] S. N. Olehnik: The boundedness of solutions of a second-order differential equation. Differ. Equations 9 (1973), 1530-1534. MR 0333345 | Zbl 0313.34031
[34] A. L. Olutimo, D. O. Adams: On the stability and boundedness of solutions of certain non-autonomous delay differential equation of third order. Appl. Math. 7 (2016), 457-467. DOI 10.4236/am.2016.76041
[35] 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 (2010), 18-29. MR 2766411 | Zbl 1476.34152
[36] M. O. Omeike, A. A. Adeyanju, D. O. Adams: Stability and boundedness of solutions of certain vector delay differential equations. J. Niger. Math. Soc. 37 (2018), 77-87. MR 3853844 | Zbl 1474.34504
[37] Z. Opial: Sur les solutions de l'equation différentielle $x^{\prime\prime} + h(x)x^\prime + f(x) = e(t)$. Ann. Pol. Math. 8 (1960), 71-74. (In French.) DOI 10.4064/ap-8-1-65-69 | MR 0113009 | Zbl 0089.07002
[38] Q. Peng: Qualitative analysis for a class of second-order nonlinear system with delay. Appl. Math. Mech., Engl. Ed. 22 (2001), 842-845. DOI 10.1023/A:1016373806172 | MR 1853095 | Zbl 0988.34060
[39] M. Rama Mohana Rao: Ordinary Differential Equations: Theory and Applications. Affiliated East-West Press, New Delhi (1980). MR 0587850 | Zbl 0482.34001
[40] 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
[41] M. Remili, D. Beldjerd: Stability and ultimate boundedness of solutions of some third order differential equations with delay. J. Assoc. Arab Universit. Basic Appl. Sci. 23 (2017), 90-95. DOI 10.1016/j.jaubas.2016.05.002
[42] H. O. Tejumola: Boundedness criteria for solutions of some second-order differential equations. Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 50 (1971), 432-437. MR 0306619 | Zbl 0235.34081
[43] C. Tunç: On the stability of solutions for non-autonomous delay differential equations of third-order. Iran. J. Sci. Technol., Trans. A, Sci. 32 (2008), 261-273. MR 2683011 | Zbl 1364.34107
[44] C. Tunç: On the stability and boundedness of solutions of nonlinear third order differential equations with delay. Filomat 24 (2010), 1-10. DOI 10.2298/FIL1003001T | MR 2791725 | Zbl 1299.34244
[45] C. Tunç: On the qualitative behaviours of solutions to a kind of nonlinear third order differential equations with retarded argument. Ital. J. Pure Appl. Math. 28 (2011), 273-284. MR 2922501 | Zbl 1248.34109
[46] C. Tunç: Stability and boundedness of solutions of non-autonomous differential equations of second order. J. Comput. Anal. Appl. 13 (2011), 1067-1074. MR 2789545 | Zbl 1227.34054
[47] C. Tunç: Stability to vector Liénard equation with constant deviating argument. Nonlinear Dyn. 73 (2013), 1245-1251. DOI 10.1007/s11071-012-0704-8 | MR 3083777 | Zbl 1281.34102
[48] C. Tunç: A note on the stability and boundedness of non-autonomous differential equations of second order with a variable deviating arguments. Afrika Math. 25 (2014), 417-425. DOI 10.1007/s13370-012-126-2 | MR 3207028 | Zbl 1306.34113
[49] C. Tunç: Global stability and boundedness of solutions to differential equations of third order with multiple delays. Dyn. Syst. Appl. 24 (2015), 467-478. MR 3445827 | Zbl 1335.34117
[50] C. Tunç: Stability and boundedness in differential systems of third order with variable delay. Proyecciones 35 (2016), 317-338. DOI 10.4067/S0716-09172016000300008 | MR 3552443 | Zbl 1384.34082
[51] C. Tunç: On the properties of solutions for a system of nonlinear differential equations of second order. Int. J. Math. Comput. Sci. 14 (2019), 519-534. MR 3923306 | Zbl 1417.34122
[52] C. Tunç, S. Erdur: New qualitative results for solutions of functional differential equations of second order. Discrete Dyn. Nat. Soc. 2018 (2018), Article ID 3151742, 13 pages. DOI 10.1155/2018/3151742 | MR 3866998 | Zbl 1417.34166
[53] C. Tunç, O. Tunç: On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order. J. Adv. Research 7 (2016), 165-168. DOI 10.1016/j.jare.2015.04.005
[54] C. Tunç, O. Tunç: A note on the stability and boundedness of solutions to non-linear differential systems of second order. J. Assoc. Arab Universit. Basic Appl. Sci. 24 (2017), 169-175. DOI 10.1016/j.jaubas.2016.12.004
[55] C. Tunç, O. Tunç: Qualitative analysis for a variable delay system of differential equations of second order. J. Taibah Univ. Sci. 13 (2019), 468-477. DOI 10.1080/16583655.2019.1595359
[56] D. W. Willett, J. S. W. Wong: The boundedness of solutions of the equation $x^{\prime\prime} +f(x,x^\prime)+ g(x)=0$. SIAM J. Appl. Math. 14 (1966), 1084-1098. DOI 10.1137/0114087 | MR 0208091 | Zbl 0173.34703
[57] D. W. Willett, J. S. W. Wong: Some properties of the solutions of $(p(t)x^\prime)^\prime + q(t)f(x)=0$. J. Math. Anal. Appl. 23 (1968), 15-24. DOI 10.1016/0022-247X(68)90112-1 | MR 0226117 | Zbl 0165.40803
[58] J. S. W. Wong: Some properties of solutions of $u^{\prime\prime} + a(t)f(u)g(u^\prime) = 0$. III. SIAM J. Appl. Math. 14 (1966), 209-214. DOI 10.1137/0114017 | MR 0203167 | Zbl 0143.31803
[59] J. S. W. Wong, T. A. Burton: Some properties of solutions of $u^{\prime\prime} + a(t)f(u)g(u^\prime) = 0$. II. Monatsh. Math. 69 (1965), 368-374. DOI 10.1007/BF01297623 | MR 0186885 | Zbl 0142.06402
[60] H. Yao, J. Wang: Globally asymptotic stability of a kind of third-order delay differential system. Int. J. Nonlinear Sci. 10 (2010), 82-87. MR 2721073 | Zbl 1235.34198
[61] T. Yoshizawa: Stability Theory by Lyapunov's Second Method. Publications of the Mathematical Society of Japan 9. Mathematical Society of Japan, Tokyo (1966). MR 0208086 | Zbl 0144.10802
[62] M. S. Zarghamee, B. Mehri: A note on boundedness of solutions of certain second-order differential equations. J. Math. Anal. Appl. 31 (1970), 504-508. DOI 10.1016/0022-247X(70)90003-X | MR 0265686 | Zbl 0229.34031
[63] B. Zhang: On the retarded Liénard equation. Proc. Am. Math. Soc. 115 (1992), 779-785. DOI 10.1090/S0002-9939-1992-1094508-1 | MR 1094508 | Zbl 0756.34075
[64] B. Zhang: Necessary and sufficient conditions for boundedness and oscillation in the retarded Liénard equation. J. Math. Anal. Appl. 200 (1996), 453-473. DOI 10.1006/jmaa.1996.0216 | MR 1391161 | Zbl 0855.34090

Affiliations:   Daniel O. Adams (corresponding author), Mathew O. Omeike, Idowu A. Osinuga, Department of Mathematics, Federal University of Agriculture, Abeokuta, Nigeria e-mail: danielogic2008@yahoo.com, adamsdo@funaab.edu.ng; moomeike@yahoo.com; osinuga08@gmail.com; Biodun S. Badmus, Department of Physics, Federal University of Agriculture, Abeokuta, Nigeria, e-mail: badmusbs@yahoo.co.uk


 
PDF available at: