zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
New result in the ultimate boundedness of solutions of a third-order nonlinear ordinary differential equation. (English) Zbl 1173.34321
The author considers the nonlinear third order ordinary differential equation $$ {x} + f(x,\dot {x},\ddot {x})\ddot {x} + g(x,\dot {x}) + h(x,\dot {x},\ddot {x}) = p(t,x,\dot {x},\ddot {x}) \tag1$$ or its equivalent system $$\dot {x} = y, \quad \dot {y} = z,$$ $$ \dot {z} = - f(x,y,z)z - g(x,y) - h(x,y,z) + p(t,x,y,z),$$ where it is assumed that $f$, $g$, $h$ and $p$ are continuous functions which depend only on the arguments displayed explicitly, the dots denote differentiation with respect to $t$ and the derivatives $\frac{\partial f(x,y,z)}{\partial x} \equiv f_x (x,y,z), \quad \frac{\partial f(x,y,z)}{\partial z} \equiv \quad f_z (x,y,z), \quad \frac{\partial h(x,y,z)}{\partial x} \equiv h_x (x,y,z), \quad \frac{\partial h(x,y,z)}{\partial y} \equiv h_y (x,y,z),\frac{\partial h(x,y,z)}{\partial z} \equiv h_z (x,y,z)$ and $\frac{\partial g(x,y)}{\partial x} \equiv g_x (x,y)$ exist and are continuous. Some sufficient conditions have been established for the ultimate boundedness of all solutions of (1). By this way, the author’s result improves the result obtained by the reviewer [{\it C. Tunç}, JIPAM, J. Inequal. Pure Appl. Math. 6, No. 1, Paper No. 3, 6 p., electronic only (2005; Zbl 1082.34514)].

34C11Qualitative theory of solutions of ODE: growth, boundedness
Full Text: EMIS EuDML