# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Some stability and boundedness results for the solutions of certain fourth order differential equations. (English) Zbl 1095.34536
Summary: Sufficient conditions are established for the asymptotic stability of the zero solution of the equation $$x^{(4)}+a(\ddot x,\dddot x)\dddot x+ b(x,\dot x)\ddot x+c(\dot x)+d(x) =p(t,x,\dot x,\ddot x,\dddot x)\tag 1$$ with $p\equiv 0$ and for the boundedness of all solutions of equation (1) with $p\ne 0$.

##### MSC:
 34D20 Stability of ODE 34C11 Qualitative theory of solutions of ODE: growth, boundedness
Full Text:
##### References:
 [1] Barbashin E. A.: Liapunov’s Function. : Science. 1970. [2] Chukwu E. N.: On the boundedness of a certain fourth-order differential equation. J. London Math. Soc. (2) 11, 3 (1975), 313-324. · Zbl 0343.34025 · doi:10.1112/jlms/s2-11.3.313 [3] Ezeilo J. O. C. , Tejumola H. O.: On the boundedness and the stability properties of solutions of certain fourth order differential equation. Ann. Mat. Pure. Apl. (IV) 95 (1973), 131-145. · Zbl 0281.34048 · doi:10.1007/BF02410712 [4] Hu C. Y.: The stability in the large for certain fourth order differential equations. Ann. Differential Equations 8, 4 (1992), 422-428. · Zbl 0776.34040 [5] Ponzo P. J.: On the stability of certain nonlinear differential equations. IEEE Trans. Automatic Control 10 (1965), 470-472. [6] Reissig R., Sansone G., Conti R.: Nonlinear Differential Equations of Higher Order. : Noordhoff, Groningen. 1974. · Zbl 0275.34001 [7] Tunç C.: A note on the stability and boundedness results of solutions of certain fourth order differential equations. Appl. Math. Comp. 155, 3 (2004), 837-843. · Zbl 1069.34054 · doi:10.1016/S0096-3003(03)00819-1 [8] Wu X., Xiong K.: Remarks on stability results for the solutions of certain fourth-order autonomous differential equations. Internat. J. Control. 69, 2 (1998), 353-360. · Zbl 0934.34039 · doi:10.1080/002071798222893