# The boundedness and convergence of solutions of second-order differential equations with applications

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## Summary

For multidimensional second-order differential equations we present new conditions for the boundedness and convergence of all solutions and their derivatives, and the existence of periodic solutions. In particular, for scalar case we give the necessary and sufficient conditions for the boundedness and convergence of all solutions and their derivatives. These results are used to improve a result of Levin and Nohel for the reactor dynamics and a result of Holmes for the evolution equations derived from a conveying fluid and to give an answer to the conjecture on the continuous stirred tank proposed by Uppal etc.

## Keywords

Differential Equation Periodic Solution Evolution Equation Scalar Case Reactor Dynamic
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## References

- [1]
- [2]H. A. Antosiewicz,
*On nonlinear differential equations of the second order with integrable forcing term*, J. London Math. Soc.,**30**(1955), pp. 64–67.Google Scholar - [3]Z. Opial,
*Sur une équation différentielle non linéaire du second ordre*, Ann. Polon. Math.,**8**(1960), pp. 65–69.Google Scholar - [4]T. A. Burton,
*On the equation x+f(x)h(x)x+g(t)=e(t)*, Ann. Mat. Pura Appl.,**85**(1970), pp. 277–285.Google Scholar - [5]J. J. Levin -J. A. Nohel,
*Global asymptotic stability for nonlinear systems of differential equations and applications to reactor dynamics*, Arch. Rational Mech. Anal.,**5**(1960), pp. 194–211.Google Scholar - [6]P. J. Holmes,
*Bifurcations to divergence and flutter in flow-induced oscillations: a finite dimensional analysis*, J. Sound Vib.,**53**(1977), pp. 471–503.Google Scholar - [7]A. Uppal -W. H. Ray -A. B. Poore,
*On the dynamic behavior of continuous stirred tank reactors*, Chem. Eng. Sci.,**29**(1974), pp. 967–985.Google Scholar - [8]M. W. Hirsch,
*Stability and convergence in strongly monotone dynamical systems*, J. Reine Angew. Math.,**383**(1988), pp. 1–53.Google Scholar - [9]
- [10]J. Sugie,
*On the boundedness of solutions of generalized Liénard equation without the signum condition*, Nonlin. Anal. Th. Meth. Appl.,**11**(1987), pp. 1391–1397.Google Scholar - [11]T. Yoshizawa,
*Asymptotic behavior of solutions of a system of differential equations*, Contrib. Diff. Eqs.,**1**(1963), pp. 371–387.Google Scholar - [12]Li Hui-Qing,
*Necessary and sufficient conditions for complete stability of the zero solution of the Liénard equation*(Chinese), Acta Math. Sinica,**31**(1988), pp. 209–214.Google Scholar - [13]H. Miyagi -T. Ohshiro -K. Yamashita,
*Generalized Liapunov function for Liénard-type nonlinear systems*, Internat. J. Control,**48**(1988), pp. 805–812.Google Scholar - [14]Jiang Ji-Fa,
*The asymptotic behavior of a class of second order differential equations with applications to electrical circuit equations*, J. Math. Anal. Appl.,**149**(1990), pp. 26–37.Google Scholar - [15]Jiang Ji-Fa,
*On the asymptotic behavior of a class of nonlinear differential equations*, Nonlin. Anal. Th. Meth. Appl.,**14**(1990), pp. 453–467.Google Scholar - [16]Jiang Ji-Fa,
*Bistable systems of differential equations with time dependent voltage source E=E(t)*, to appear in Appl. Anal.Google Scholar - [17]Ding Tong-Ren,
*An existence theorem for harmonic solutions of periodically perturbed systems of Duffing's type*, Acta Scientiarum Naturalium Universitatis Pekinensis,**28**(1992), pp. 71–76.Google Scholar - [18]M. N. Nkashama,
*Periodically perturbed nonconservative systems of Liénard type*, Proc. Amer. Math. Soc.,**111**(1991), pp. 677–682.Google Scholar - [19]S. Lefschetz,
*Existence of periodic solutions for certain differential equations*, Proc. Nat. Ac. Sci. U.S.A.,**29**(1943), pp. 29–32.Google Scholar - [20]D. Graffi,
*Forced oscillations for several nonlinear circuits*, Ann. Math.,**54**(1951), pp. 262–271.Google Scholar - [21]J. P. LaSalle,
*Stability theory for ordinary differential equations*, J. Diff. Eqs.,**4**(1968), pp. 57–65.Google Scholar - [22]A. Strauss -J. A. Yorke,
*On asymptotically autonomous differential equations*, Math. Systems Theory,**1**(1967), pp. 175–182.Google Scholar - [23]M. Golubisky -D. Schaeffer,
*A theory for imperfect bifurcation via singularity theory*, Comm. Pure Appl. Math.,**32**(1979), pp. 21–98.Google Scholar - [24]R. K. Brayton -J. K. Moser,
*A theory of nonlinear networks — II*, Quart. Appl. Math.,**22**(1964), pp. 81–104.Google Scholar

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