Previous |  Up |  Next

Article

Keywords:
Third order; nonlinear differential equation; uniform stability; uniform ultimate boundedness; periodic solutions
Summary:
In this paper, criteria are established for uniform stability, uniform ultimate boundedness and existence of periodic solutions for third order nonlinear ordinary differential equations. In the investigation Lyapunov’s second method is used by constructing a complete Lyapunov function to obtain our results. The results obtained in this investigation complement and extend many existing results in the literature.
References:
[1] Ademola, A. T., Arawomo, P. O.: On the asymptotic behaviour of solutions of certain differential equations of the third order. Proyecciones Journal of Mathematics 33, 1 (2014), 111–132. MR 3162791 | Zbl 1302.34076
[2] Ademola, A. T., Arawomo, P. O.: Generalization of some qualitative behaviour of solutions of third order nonlinear differential equations. Differential Equations and Control Processes N 1 (2012), 97–113. MR 2977146
[3] Ademola, A. T., Arawomo, P. O.: Asymptotic behaviour of solutions of third order nonlinear differential equations. Acta Univ. Sapientiae, Mathematica 3, 2 (2011), 197–211. MR 2915835 | Zbl 1260.34100
[4] Ademola, A. T., Arawomo, P. O.: Boundedness and asymptotic behaviour of solutions of a nonlinear differential equation of the third order. Afr. Mat. 23 (2012), 261–271. DOI 10.1007/s13370-011-0034-x | MR 2958973 | Zbl 1266.34051
[5] Ademola, A. T., Arawomo, P. O.: Boundedness and stability of solutions of some nonlinear differential equations of the third-order. The Pacific Journal of Science and Technology 10, 2 (2009), 187–193. MR 2881138
[6] Ademola, A. T., Arawomo, P. O.: Stability and ultimate boundedness of solutions to certain third-order differential equations. Applied Mathematics E-Notes 10 (2010), 61–69. MR 2606838 | Zbl 1194.34103
[7] Ademola, A. T., Arawomo, P. O.: Stability and uniform ultimate boundedness of solutions of a third-order differential equation. International Journal of Applied Mathematics 23, 1 (2010), 11–22. MR 2643291 | Zbl 1201.34054
[8] Ademola, A. T., Arawomo, P. O.: Stability and uniform ultimate boundedness of solutions of some third order differential equations. Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 27 (2011), 51–59. MR 2813591 | Zbl 1240.34259
[9] Ademola, A. T., Arawomo, P. O.: Stability, boundedness and asymptotic behaviour of solutions of certain nonlinear differential equations of the third order. Kragujevac J. Math. 35, 3 (2011), 431–445. MR 2881138 | Zbl 1265.34191
[10] Ademola, A. T., Ogundiran, M. O., Arawomo, P. O., Adesina, O. A.: Boundedness results for a certain third-order nonlinear differential equations. Appl. Math. Comput. 216 (2010), 3044–3049. DOI 10.1016/j.amc.2010.04.022 | MR 2653118
[11] Afuwape, A. U., Adesina, O. A.: On the bounds for mean-values of solutions to certain third-order nonlinear differential equations. Fasciculi Mathematici 36 (2005), 5–14. MR 2223637 | Zbl 1127.34019
[12] Andres, J.: Boundedness results for solutions of the equation $\dddot{x}+a\ddot{x}+g(x)\dot{x}+h(x)=p(t)$ without the hypothesis $h(x)x\ge 0$ for $|x|>R$. Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 80, 8 (1986), 533–539. MR 0976947
[13] Antoisewicz, H. A.: On nonlinear differential equations of the second order with integrable forcing term. J. London Math. Soc. 30 (1955), 64–67. DOI 10.1112/jlms/s1-30.1.64
[14] Babaev, E. V., Hefferlin, R.: Concept of chemical periodicity: from Mendeleev Table to Molecular Hyper-periodicity patterns. In: Research Studies Press, London, 1996, 24–81.
[15] Bereketoǧlu, H., Györi, I.: On the boundedness of solutions of a third-order nonlinear differential equation. Dynam. Systems Appl. 6, 2 (1997), 263–270. MR 1461442
[16] Callahan, J., Cox, D., Hoffman, K., O’Shea, D., Pollatsek, H., Senechal, L., L.: Calculus in Context. The Five College Calculus Project, Five Colleges, 2008.
[17] Chukwu, E. N.: On boundedness of solutions of third order differential equations. Ann. Mat. Pura. Appl. 104, 4 (1975), 123–149. DOI 10.1007/BF02417013 | MR 0377180
[18] Ezeilo, J. O. C.: A note on a boundedness theorem for some third-order differential equations. J. London Math. Soc. 36 (1961), 439–444. DOI 10.1112/jlms/s1-36.1.439 | MR 0136820 | Zbl 0104.06501
[19] Ezeilo, J. O. C.: An elementary proof of a boundedness theorem for a certain third-order differential equation. J. London Math. Soc. 38 (1963), 11–16. DOI 10.1112/jlms/s1-38.1.11 | MR 0166450 | Zbl 0116.06902
[20] Ezeilo, J. O. C.: A boundedness theorem for a certain third-order differential equation. Proc. London Math. Soc. 13, 3 (1963), 99–124. MR 0142850 | Zbl 0116.06902
[21] Hara, T.: On the uniform ultimate boundedness of solutions of certain third-order differential equations. J. Math. Anal. Appl. 80, 2 (1981), 533–544. DOI 10.1016/0022-247X(81)90122-0 | MR 0614848
[22] Mehri, B., Niksirat, M. A.: On the existence of periodic solutions for certain differential equations. J. Comp. & Appl. Math. 174 (2005), 239–249. DOI 10.1016/j.cam.2004.04.011 | MR 2106439 | Zbl 1069.34064
[23] Mehri, B., Shadman, D.: Boundedness of solutions of certain third-order differential equation. Math. Inequal. Appl., 4 (1999), 545–549. MR 1717047 | Zbl 0943.34022
[24] Mehri, B., Shadman, D.: On the existence of periodic solutions of a certain class of second order nonlinear differential equation. Math. Inequal. Appl. 1, 3 (1998), 431–436. MR 1629412
[25] Mehri, B., Shadman, D.: Periodic solutions of a certain non-linear third order differential equation. Scientia Iranica 11 (2004), 181–184. MR 2106566
[26] Mehri, B.: Periodic solution for certain non linear third-order differential equation. Indian J. Pure Appl. Math. 21, 3 (1990), 203–210. MR 1044260 | Zbl 0708.34030
[27] Minhós, F.: Periodic solutions for a third order differential equation under conditions on the potential. Portugaliae Mathematica 55, 4 (1998), 475–484. MR 1672255 | Zbl 0923.34045
[28] LaSalle, J., Lefschetz, S.: Stability by Liapunov’s direct method with applications. Academic Press, New York–London, 1961. MR 0132876
[29] Ogundare, B. S.: On the boundedness and stability results for the solutions of certain third-order nonlinear differential equations. Kragujevac J. Math. 29 (2006), 37–48. MR 2288487
[30] Omeike, M. O.: New result in the ultimate boundedness of solutions of a third-order nonlinear ordinary differential equation. J. Inequal. Pure and Appl. Math. 9, 1 (2008), Art. 15, 1–8. MR 2391282 | Zbl 1173.34321
[31] Reissig, R., Sansone, G., Conti, R.: Nonlinear Differential Equations of Higher Order. Noordhoff International Publishing, Leyeden, 1974. MR 0344556
[32] Rouche, N., Habets, N., Laloy, M.: Stability Theory by Liapunov’s Direct Method. Applied Mathematical Sciences 22, Spriger-Verlag, New York–Heidelberg–Berlin, 1977. MR 0450715 | Zbl 0364.34022
[33] Shadman, D., Mehri, B.: On the periodic solutions of certain nonlinear third order differential equations. Z Angew. Math. Mech. 75, 2 (1995), 164–166. DOI 10.1002/zamm.19950750218 | MR 1319334 | Zbl 0828.34029
[34] Swick, K. E.: On the boundedness and the stability of solutions for some non-autonomous differential equations of the third order. J. London Math. Soc. 44 (1969), 347–359. DOI 10.1112/jlms/s1-44.1.347 | MR 0236482
[35] Tejumola, H. O.: A note on the boundedness of solutions of some nonlinear differential equations of the third-order. Ghana J. of Science 11, 2 (1970), 117–118.
[36] Tejumola, H. O.: A note on the boundedness and the stability of solutions of certain third-order differential equations. Ann. Math. Pura. Appl. 92 (1972), 65–75. DOI 10.1007/BF02417936 | MR 0318615 | Zbl 0242.34046
[37] Tunç, C., Çmar, I.: On the existence of periodic solutions to nonlinear differential equations of second order. Differ. Uravn. Protsessy Upr. (Differential equations and control processes), 3 (2008), 1–6. MR 2515107
[38] Tunç, C.: Boundedness of solutions of a third-order nonlinear differential equation. J. Inequal. Pure and Appl. Math. 6, 1 (2005), 1–6. MR 2122950 | Zbl 1082.34514
[39] Tunç, C.: On existence of periodic solutions to certain nonlinear third order differential equations. Proyecciones Journal of Mathematics 28, 2 (2009), 125–132. DOI 10.4067/S0716-09172009000200002 | MR 2529760
[40] Tunç, C.: Some new results on the boundedness of solutions of a certain nonlinear differential equation of third-order. International J. of Nonlinear Science 7, 2 (2009), 246–256. MR 2496738 | Zbl 1371.34045
[41] Yoshizawa, T.: Liapunov’s function and boundedness of solutions. Funkcialaj Ekvacioj 2 (1958), 71–103. MR 0114981
[42] Yoshizawa, T.: Stability Theory by Liapunov’s Second Method. The Mathematical Society of Japan, 1966. MR 0208086
[43] Yoshizawa, T.: Stability Theory and Existence of Periodic Solutions and Almost Periodic Solutions. Spriger-Verlag, New York–Heidelberg–Berlin, 1975. MR 0466797
Partner of
EuDML logo