| Title:
|
Boundedness criteria for a class of second order nonlinear differential equations with delay (English) |
| Author:
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Adams, Daniel O. |
| Author:
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Omeike, Mathew O. |
| Author:
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Osinuga, Idowu A. |
| Author:
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Badmus, Biodun S. |
| Language:
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English |
| Journal:
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Mathematica Bohemica |
| ISSN:
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0862-7959 (print) |
| ISSN:
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2464-7136 (online) |
| Volume:
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148 |
| Issue:
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3 |
| Year:
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2023 |
| Pages:
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303-327 |
| Summary lang:
|
English |
| . |
| Category:
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math |
| . |
| Summary:
|
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-Krasovski\v ı functional to establish our results. This work extends and improve on some results in the literature. (English) |
| Keyword:
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boundedness |
| Keyword:
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nonlinear |
| Keyword:
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differential equation of third order |
| Keyword:
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integral inequality |
| MSC:
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34C11 |
| MSC:
|
34C12 |
| MSC:
|
34K12 |
| idZBL:
|
Zbl 07729579 |
| idMR:
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MR4628615 |
| DOI:
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10.21136/MB.2022.0166-21 |
| . |
| Date available:
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2023-08-11T14:14:56Z |
| Last updated:
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2023-09-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151762 |
| . |
| Reference:
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[1] Adams, D. O., Olutimo, A. L.: 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 |
| Reference:
|
[2] Ademola, A. T., Moyo, S., Ogundare, B. S., Ogundiran, M. O., Adesina, O. A.: 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. Zbl 1414.34053, MR 3935484 |
| Reference:
|
[3] Afuwape, A. U., Omeike, M. O.: On the stability and boundedness of solutions of a kind of third order delay differential equations.Appl. Math. Comput. 200 (2008), 444-451. Zbl 1316.34070, MR 2421659, 10.1016/j.amc.2007.11.037 |
| Reference:
|
[4] Antosiewicz, H. A.: On nonlinear differential equations of the second order with integrable forcing term.J. Lond. Math. Soc. 30 (1955), 64-67. Zbl 0064.08404, MR 0065752, 10.1112/jlms/s1-30.1.64 |
| Reference:
|
[5] Athanassov, Z. S.: Boundedness criteria for solutions of certain second order nonlinear differential equations.J. Math. Anal. Appl. 123 (1987), 461-479. Zbl 0642.34031, MR 0883702, 10.1016/0022-247X(87)90324-6 |
| Reference:
|
[6] Bellman, R., Cooke, K. L.: Differential-Difference Equations.Mathematics in Science and Engineering 6. Academic Press, New York (1963). Zbl 0105.06402, MR 0147745 |
| Reference:
|
[7] Bihari, I.: Researches of the boundedness and stability of the solutions of non-linear differential equations.Acta Math. Acad. Sci. Hung. 8 (1957), 261-278. Zbl 0097.29301, MR 0094516, 10.1007/BF02020315 |
| Reference:
|
[8] Burton, T. A.: The generalized Lienard equation.J. SIAM Control, Ser. A 3 (1965), 223-230. Zbl 0135.30201, MR 0190462, 10.1137/0303018 |
| Reference:
|
[9] Burton, T. A.: Stability and Periodic Solutions of Ordinary and Functional Differential Equations.Mathematics in Science and Engineering 178. Academic Press, Orlando (1985). Zbl 0635.34001, MR 0837654, 10.1016/s0076-5392(09)x6019-4 |
| Reference:
|
[10] Burton, T. A., Grimmer, R. C.: Stability properties of $(r(t)u^\prime)^\prime + a(t)f(u)g(u^\prime) = 0$.Monastsh. Math. 74 (1970), 211-222. Zbl 0195.09804, MR 0262613, 10.1007/BF01303441 |
| Reference:
|
[11] Burton, T. A., Hatvani, L.: Stability theorems for nonautonomous functional differential equations by Liapunov functionals.Tohoku Math. J., II. Ser. 41 (1989), 65-104. Zbl 0677.34060, MR 0985304, 10.2748/tmj/1178227868 |
| Reference:
|
[12] Burton, T. A., Hering, R. H.: Liapunov theory for functional differential equations.Rocky Mt. J. Math. 24 (1994), 3-17. Zbl 0806.34067, MR 1270024, 10.1216/rmjm/1181072449 |
| Reference:
|
[13] Burton, T. A., Makay, G.: Asymptotic stability for functional differential equations.Acta Math. Hung. 65 (1994), 243-251. Zbl 0805.34068, MR 1281434, 10.1007/BF01875152 |
| Reference:
|
[14] Driver, R. D.: Ordinary and Delay Differential Equations.Applied Mathematical Sciences 20. Springer, New York (1977). Zbl 0374.34001, MR 0477368, 10.1007/978-1-4684-9467-9 |
| Reference:
|
[15] Dvořáková, S.: The Qualitative and Numerical Analysis of Nonlinear Delay Differential Equations: Doctoral Thesis.Brno University of Technology, Brno (2011). |
| Reference:
|
[16] \`El'sgol'ts, L. \`E.: Introduction to the Theory of Differential Equations with Deviating Arguments.McLaughin Holden-Day, San Francisco (1966). Zbl 0133.33502, MR 0192154 |
| Reference:
|
[17] Èl'sgol'ts, L. È., Norkin, S. B.: Introduction to the Theory and Application of Differential Equations with Deviating Arguments.Mathematics in Science and Engineering 105. Academic Press, New York (1973). Zbl 0287.34073, MR 0352647, 10.1016/s0076-5392(08)x6170-3 |
| Reference:
|
[18] Gabsi, H., Ardjouni, A., Djoudi, A.: New technique in asymptotic stability for third-order nonlinear delay differential equations.Math. Eng. Sci. Aerospace 9 (2018), 315-330. MR 4088058 |
| Reference:
|
[19] Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics.Mathematics and its Applications 74. Kluwer Academic, Dordrecht (1992). Zbl 0752.34039, MR 1163190, 10.1007/978-94-015-7920-9 |
| Reference:
|
[20] Graef, J. R., Spikes, P. W.: Asymptotic behavior of solutions of a second order nonlinear differential equation.J. Differ. Equations 17 (1975), 461-476. Zbl 0298.34028, MR 0361275, 10.1016/0022-0396(75)90056-X |
| Reference:
|
[21] Graef, J. R., Spikes, P. W.: Continuability, boundedness, and convergence to zero of solutions of a perturbed nonlinear ordinary differential equation.Czech. Math. J. 45 (1995), 663-683. Zbl 0851.34050, MR 1354925, 10.21136/CMJ.1995.128549 |
| Reference:
|
[22] Hale, J. K.: Theory of Functional Differential Equations.Applied Mathematical Sciences 3. Springer, New York (1977). Zbl 0352.34001, MR 0508721, 10.1007/978-1-4612-9892-2 |
| Reference:
|
[23] Hildebrandt, T. H.: Introduction to the Theory of Integration.Pure and Applied Mathematics 13. Academic Press, New York (1963). Zbl 0112.28302, MR 0154957 |
| Reference:
|
[24] Jones, G. S.: Fundamental inequalities for discrete and discontinuous functional equations.J. Soc. Ind. Appl. Math. 12 (1964), 43-57. Zbl 0154.05702, MR 0162069, 10.1137/0112004 |
| Reference:
|
[25] Kolmanovskii, V., Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations.Mathematics and its Applications 463. Klumer Academic, Dordrecht (1999). Zbl 0917.34001, MR 1680144, 10.1007/978-94-017-1965-0 |
| Reference:
|
[26] Kolmanovskii, V. B., Nosov, V. R.: Stability of Functional Differential Equations.Mathematics in Science and Engineering 180. Academic Press, London (1986). Zbl 0593.34070, MR 0860947 |
| Reference:
|
[27] Krasovskii, N. N.: Stability of Motion: Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay.Stanford University Press, Stanford (1963). Zbl 0109.06001, MR 0147744 |
| Reference:
|
[28] Lalli, B. S.: On the boundedness of solutions of certain second-order differential equations.J. Math. Anal. Appl. 25 (1969), 182-188. MR 0239184, 10.1016/0022-247X(69)90221-2 |
| Reference:
|
[29] Legatos, G. G.: Contribution to the qualitative theory of ordinary differential equations.Bull. Soc. Math. Grèce, N. Ser. 2 (1961), 1-44 Greek. Zbl 0107.29202, MR 0140770 |
| Reference:
|
[30] Mahmoud, A. M., Tunç, C.: Asymptotic stability of solutions of a kind of third-order stochastic differential equations with delays.Miskolc Math. Notes 20 (2019), 381-393. Zbl 1438.34255, MR 3986654, 10.18514/MMN.2019.2800 |
| Reference:
|
[31] V., J. E. Nápoles: A note on the qualitative behaviour of some second order nonlinear differential equations.Divulg. Mat. 10 (2002), 91-99. Zbl 1039.34030, MR 1946903 |
| Reference:
|
[32] Ogundare, B. S., Ademola, A. T., Ogundiran, M. O., Adesina, O. A.: 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. Zbl 1387.34096, MR 3712445, 10.1007/s11565-016-0262-y |
| Reference:
|
[33] Olehnik, S. N.: The boundedness of solutions of a second-order differential equation.Differ. Equations 9 (1973), 1530-1534. Zbl 0313.34031, MR 0333345 |
| Reference:
|
[34] Olutimo, A. L., Adams, D. O.: On the stability and boundedness of solutions of certain non-autonomous delay differential equation of third order.Appl. Math. 7 (2016), 457-467. 10.4236/am.2016.76041 |
| Reference:
|
[35] Omeike, M. O.: 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. Zbl 1476.34152, MR 2766411 |
| Reference:
|
[36] Omeike, M. O., Adeyanju, A. A., Adams, D. O.: Stability and boundedness of solutions of certain vector delay differential equations.J. Niger. Math. Soc. 37 (2018), 77-87. Zbl 1474.34504, MR 3853844 |
| Reference:
|
[37] Opial, Z.: 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 French. Zbl 0089.07002, MR 0113009, 10.4064/ap-8-1-65-69 |
| Reference:
|
[38] Peng, Q.: Qualitative analysis for a class of second-order nonlinear system with delay.Appl. Math. Mech., Engl. Ed. 22 (2001), 842-845. Zbl 0988.34060, MR 1853095, 10.1023/A:1016373806172 |
| Reference:
|
[39] Rao, M. Rama Mohana: Ordinary Differential Equations: Theory and Applications.Affiliated East-West Press, New Delhi (1980). Zbl 0482.34001, MR 0587850 |
| Reference:
|
[40] Remili, M., Beldjerd, D.: A boundedness and stability results for a kind of third order delay differential equations.Appl. Appl. Math. 10 (2015), 772-782. Zbl 1331.34135, MR 3447611 |
| Reference:
|
[41] Remili, M., Beldjerd, D.: 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. MR 3752693, 10.1016/j.jaubas.2016.05.002 |
| Reference:
|
[42] Tejumola, H. O.: 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. Zbl 0235.34081, MR 0306619 |
| Reference:
|
[43] Tunç, C.: 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. Zbl 1364.34107, MR 2683011 |
| Reference:
|
[44] Tunç, C.: On the stability and boundedness of solutions of nonlinear third order differential equations with delay.Filomat 24 (2010), 1-10. Zbl 1299.34244, MR 2791725, 10.2298/FIL1003001T |
| Reference:
|
[45] Tunç, C.: 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. Zbl 1248.34109, MR 2922501 |
| Reference:
|
[46] Tunç, C.: Stability and boundedness of solutions of non-autonomous differential equations of second order.J. Comput. Anal. Appl. 13 (2011), 1067-1074. Zbl 1227.34054, MR 2789545 |
| Reference:
|
[47] Tunç, C.: Stability to vector Liénard equation with constant deviating argument.Nonlinear Dyn. 73 (2013), 1245-1251. Zbl 1281.34102, MR 3083777, 10.1007/s11071-012-0704-8 |
| Reference:
|
[48] Tunç, C.: 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. Zbl 1306.34113, MR 3207028, 10.1007/s13370-012-126-2 |
| Reference:
|
[49] Tunç, C.: Global stability and boundedness of solutions to differential equations of third order with multiple delays.Dyn. Syst. Appl. 24 (2015), 467-478. Zbl 1335.34117, MR 3445827 |
| Reference:
|
[50] Tunç, C.: Stability and boundedness in differential systems of third order with variable delay.Proyecciones 35 (2016), 317-338. Zbl 1384.34082, MR 3552443, 10.4067/S0716-09172016000300008 |
| Reference:
|
[51] Tunç, C.: On the properties of solutions for a system of nonlinear differential equations of second order.Int. J. Math. Comput. Sci. 14 (2019), 519-534. Zbl 1417.34122, MR 3923306 |
| Reference:
|
[52] Tunç, C., Erdur, S.: New qualitative results for solutions of functional differential equations of second order.Discrete Dyn. Nat. Soc. 2018 (2018), Article ID 3151742, 13 pages. Zbl 1417.34166, MR 3866998, 10.1155/2018/3151742 |
| Reference:
|
[53] Tunç, C., Tunç, O.: On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order.J. Adv. Research 7 (2016), 165-168. 10.1016/j.jare.2015.04.005 |
| Reference:
|
[54] Tunç, C., Tunç, O.: 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. 10.1016/j.jaubas.2016.12.004 |
| Reference:
|
[55] Tunç, C., Tunç, O.: Qualitative analysis for a variable delay system of differential equations of second order.J. Taibah Univ. Sci. 13 (2019), 468-477. 10.1080/16583655.2019.1595359 |
| Reference:
|
[56] Willett, D. W., Wong, J. S. W.: 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. Zbl 0173.34703, MR 0208091, 10.1137/0114087 |
| Reference:
|
[57] Willett, D. W., Wong, J. S. W.: Some properties of the solutions of $(p(t)x^\prime)^\prime + q(t)f(x)=0$.J. Math. Anal. Appl. 23 (1968), 15-24. Zbl 0165.40803, MR 0226117, 10.1016/0022-247X(68)90112-1 |
| Reference:
|
[58] Wong, J. S. W.: 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. Zbl 0143.31803, MR 0203167, 10.1137/0114017 |
| Reference:
|
[59] Wong, J. S. W., Burton, T. A.: Some properties of solutions of $u^{\prime\prime} + a(t)f(u)g(u^\prime) = 0$. II.Monatsh. Math. 69 (1965), 368-374. Zbl 0142.06402, MR 0186885, 10.1007/BF01297623 |
| Reference:
|
[60] Yao, H., Wang, J.: Globally asymptotic stability of a kind of third-order delay differential system.Int. J. Nonlinear Sci. 10 (2010), 82-87. Zbl 1235.34198, MR 2721073 |
| Reference:
|
[61] Yoshizawa, T.: Stability Theory by Lyapunov's Second Method.Publications of the Mathematical Society of Japan 9. Mathematical Society of Japan, Tokyo (1966). Zbl 0144.10802, MR 0208086 |
| Reference:
|
[62] Zarghamee, M. S., Mehri, B.: A note on boundedness of solutions of certain second-order differential equations.J. Math. Anal. Appl. 31 (1970), 504-508. Zbl 0229.34031, MR 0265686, 10.1016/0022-247X(70)90003-X |
| Reference:
|
[63] Zhang, B.: On the retarded Liénard equation.Proc. Am. Math. Soc. 115 (1992), 779-785. Zbl 0756.34075, MR 1094508, 10.1090/S0002-9939-1992-1094508-1 |
| Reference:
|
[64] Zhang, B.: Necessary and sufficient conditions for boundedness and oscillation in the retarded Liénard equation.J. Math. Anal. Appl. 200 (1996), 453-473. Zbl 0855.34090, MR 1391161, 10.1006/jmaa.1996.0216 |
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