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Boundedness results for a certain third order nonlinear differential equation. (English) Zbl 1201.34055

The authors consider a class of third order non-linear differential equations of the form

x +f(x ¨)+g(x ˙)+h(x)=p(t,x,x ˙,x ¨),(1)

where f,g,hC(,), pC( + × 3 ,), + =[0,). The authors give some new sufficient conditions which guarantee the boundedness and uniform ultimate boundedness of the solutions (1). By defining an appropriate Lyapunov function, they prove their main results.

MSC:
34C11Qualitative theory of solutions of ODE: growth, boundedness
References:
[1]Ademola, T. A.; Ogundiran, M. O.; Arawomo, P. O.; Adesina, O. A.: Stability results for the solutions of a certain third order nonlinear differential equation, Math. sci. Res. J. 12, No. 6, 124134 (2008) · Zbl 1198.34083
[2]Afuwape, A. U.: Further ultimate boundedness results for a third order nonlinear system of differential equations, Boll. un. Mat. ital. C (6) 4, No. 1, 347-361 (1985) · Zbl 0592.34024
[3]A.U. Afuwape, Uniform ultimate boundedness results for some third order nonlinear differential equations, Int Center for Theor. Phys., IC/90/405, November, 1990, pp. 1 – 14.
[4]Afuwape, A. U.: Remarks on barbashin – ezeilo problem on third order nonlinear differential equations, J. math. Anal. appl. 317, 613-619 (2006) · Zbl 1099.34039 · doi:10.1016/j.jmaa.2005.05.068
[5]Afuwape, A. U.; Adesina, O. A.: On the bounds for mean-values of solutions to certain third order nonlinear differential equations, Fasc. math. 36, 5-14 (2005) · Zbl 1127.34019
[6]Andres, J.: Boundedness results for solutions of the equation x·+ax¨+g(x)x˙+h(x)=p(t) without the hypothesis h(x)sgnx0 for ∣x∣>R, Atti. accad. Naz. lincei rend. Cl. sci. Fis. mat. Natur. 80, No. 8, 533-539 (1986)
[7]Bereketog&caron, H.; Lu; Györi, I.: On the boundedness of solutions of a third order nonlinear differential equation, Dyn. syst. Appl. 6, No. 2, 263-270 (1997)
[8]Ezeilo, J. O. C.: A note on a boundedness theorem for some third order differential equations, J. London math. Soc. 36, 439-444 (1961) · Zbl 0104.06501 · doi:10.1112/jlms/s1-36.1.439
[9]Ezeilo, J. O. C.: An elementary proof of a boundedness theorem for a certain third order differential equation, J. London math. Soc. 38, 11-16 (1963) · Zbl 0116.06902 · doi:10.1112/jlms/s1-38.1.11
[10]Ezeilo, J. O. C.: Further results for the solutions of a third order differential equation, Proc. camb. Phil. soc. 59, 111-116 (1963) · Zbl 0115.30501
[11]Ezeilo, J. O. C.: A boundedness theorem for a certain third order differential equation, Proc. London math. Soc. 13, No. 3, 99-124 (1963) · Zbl 0108.09001 · doi:10.1112/plms/s3-13.1.99
[12]Ezeilo, J. O. C.: A generalization of a boundedness theorem for the equation x·+x¨+ϕ2(x˙)+ϕ3(x)=ψ(t,x,x˙,x¨), Atti. accad. Naz. lincei rend. Cl. sci. Fis. mat. Natur. 50, No. 13, 424-431 (1971)
[13]Ezeilo, J. O. C.; Tejumola, H. O.: Boundedness theorems for certain third order differential equations, Atti. accad. Naz. lincei rend. Cl. sci. Fis. mat. Natur. 55, 194-201 (1973) · Zbl 0295.34022
[14]Hara, T.: On the uniform ultimate boundedness of solutions of certain third order differential equations, J. math. Anal. appl. 80, 533-544 (1981) · Zbl 0484.34024 · doi:10.1016/0022-247X(81)90122-0
[15]Qian, C.: Asymptotic behavior of a third order nonlinear differential equation, J. math. Anal. appl. 284, No. 1, 191-205 (2003) · Zbl 1054.34078 · doi:10.1016/S0022-247X(03)00302-0
[16]Reissig, R.; Sansone, G.; Conti, R.: Nonlinear differential equations of higher order, (1974) · Zbl 0275.34001
[17]Rouche, N.; Habets, N.; Laloy, M.: Stability theory by Lyapunov’s direct method, Appl. math. Sci. 22 (1977) · Zbl 0364.34022
[18]Swick, K. E.: Asymptotic behavior of the solutions of certain third order differential equations, SIAM J. Appl. 19, No. 1, 96-102 (1970) · Zbl 0212.11403 · doi:10.1137/0119008
[19]Swick, K. E.: Boundedness and stability for a nonlinear third order differential equation, Atti. accad. Naz. lincei rend. Cl. sci. Fis. mat. Natur. 56, No. 20, 859-865 (1974) · Zbl 0326.34062
[20]Tejumola, H. O.: A note on the boundedness of solutions of some nonlinear differential equations of the third order, Ghana J. Of sci. 11, No. 2, 117-118 (1970)
[21]Tejumola, H. O.: Periodic boundary value problems for some fifth, forth and third order ordinary differential equations, J. nigerian math. Soc. 25, 37-46 (2006)
[22]Tunç, C.: On the asymptotic behavior of solutions of certain third order nonlinear differential equations, J. appl. Math. stoc. Anal. 1, 29-35 (2005) · Zbl 1077.34052 · doi:10.1155/JAMSA.2005.29
[23]Tunç, C.: Boundedness of solutions of a third- order nonlinear differential equation, J. inequal. Pure appl. Math. 6, No. 1, 1-6 (2005) · Zbl 1082.34514
[24]Yoshizawa, T.: Stability theory and existence of periodic solutions and almost periodic solutions, (1975)