# 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)
Bound of solutions to third-order nonlinear differential equations with bounded delay. (English) Zbl 1193.34139

Consider the delay differential equation

$\begin{array}{c}{x}^{\text{'}\text{'}\text{'}}+f\left(x,{x}^{\text{'}},{x}^{\text{'}\text{'}}\right)+g\left(x\left(t-r\left(t\right)\right),{x}^{\text{'}}\left(t\right)-r\left(t\right)\right)\hfill \\ \hfill +h\left(x\left(t-r\left(t\right)\right)\right)=p\left(t,x,{x}^{\text{'}},x\left(t-r\left(t\right)\right),{x}^{\text{'}}\left(t-r\left(t\right)\right),{x}^{\text{'}\text{'}}\right)\phantom{\rule{2.em}{0ex}}\left(*\right)\end{array}$

under the assumption that the delay satisfies

$0\le r\left(t\right)\le \alpha ,\phantom{\rule{1.em}{0ex}}{r}^{\text{'}}\left(t\right)\le \beta ,\phantom{\rule{1.em}{0ex}}\alpha >0,\phantom{\rule{1.em}{0ex}}0<\beta <1·$

The author gives additional conditions on $f$, $g$, $h$, $p$ such that the solution of the Cauchy problem of $\left(*\right)$ is uniformly bounded including its first and second derivative. The proof is based on the construction of a Lyapunov functional.

##### MSC:
 34K12 Growth, boundedness, comparison of solutions of functional-differential equations
##### References:
 [1] Burton, T. A.: Stability and periodic solutions of ordinary and functional-differential equations, mathematics in science and engineering, Stability and periodic solutions of ordinary and functional-differential equations, mathematics in science and engineering 178 (1985) · Zbl 0635.34001 [2] L.È. Èl’sgol’ts, Introduction to the theory of differential equations with deviating arguments (translated from the Russian by Robert J. McLaughlin Holden-Day, Inc., San Francisco, California, 1966). · Zbl 0133.33502 [3] L.È. Èl’sgol’ts, S.B. Norkin, Introduction to the theory and application of differential equations with deviating arguments (translated from the Russian by John L. Casti. Mathematics in Science and Engineering, vol. 105, Academic Press, New York, 1973). [4] Hale, J.: Theory of functional differential equations, (1977) [5] Hale, J.; Lunel, S. M. Verduyn: Introduction to functional-differential equations, applied mathematical sciences, Introduction to functional-differential equations, applied mathematical sciences 99 (1993) · Zbl 0787.34002 [6] Kolmanovskii, V.; Myshkis, A.: Introduction to the theory and applications of functional differential equations, (1999) [7] Kolmanovskii, V. B.; Nosov, V. R.: Stability of functional-differential equations. Mathematics in science and engineering, Stability of functional-differential equations. Mathematics in science and engineering 180 (1986) · Zbl 0593.34070 [8] N.N. Krasovskii, Stability of motion. Applications of Lyapunov’s second method to differential systems and equations with delay (translated by J. L. Brenner, Stanford University Press, Stanford, California, 1963). · Zbl 0109.06001 [9] Lyapunov, A. M.: Stability of motion. Mathematics in science and engineering, Stability of motion. Mathematics in science and engineering 30 (1966) · Zbl 0161.06303 [10] Ponzo, P. J.: On the stability of certain nonlinear differential equations, IEEE trans. Automatic control 10, 470-472 (1965) [11] Reissig, R.; Sansone, G.; Conti, R.: Non-linear differential equations of higher order, (1974) · Zbl 0275.34001 [12] Sadek, A. I.: Stability and boundedness of a kind of third-order delay differential system, Appl. math. Lett. 16, No. 5, 657-662 (2003) · Zbl 1056.34078 · doi:10.1016/S0893-9659(03)00063-6 [13] Tejumola, H. O.; Tchegnani, B.: Stability, boundedness and existence of periodic solutions of some third and fourth order nonlinear delay differential equations, J. nigerian math. Soc. 19, 9-19 (2000) [14] Tunç, C.: Boundedness of solutions of a third-order nonlinear differential equation, J. inequal. Pure appl. Math. 6, No. 1 (2005) · Zbl 1082.34514 [15] Tunç, C.: Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations, Kuwait J. Sci. eng. 32, No. 1, 39-48 (2005) · Zbl 1207.34043 [16] Tunç, C.: New results about stability and boundedness of solutions of certain non-linear third-order delay differential equations, Arab. J. Sci. eng. Sect. A sci. 31, No. 2, 185-196 (2006) · Zbl 1195.34113 [17] Tunç, C.; &scedil, H.; Evli: Stability and boundedness properties of certain second-order differential equations, J. franklin inst. 344, No. 5, 399-405 (2007) [18] Tunç, C.; Tunç, E.: On the asymptotic behavior of solutions of certain second-order differential equations, J. franklin inst. 344, No. 5, 391-398 (2007) [19] Tunç, C.: On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument, Nonlinear dynam 57, No. 1 – 2, 97-106 (2009) · Zbl 1176.34064 · doi:10.1007/s11071-008-9423-6 [20] Yoshizawa, T.: Stability theory by Liapunov’s second method, (1966) · Zbl 0144.10802 [21] Zhu, Y. F.: On stability, boundedness and existence of periodic solution of a kind of third order nonlinear delay differential system, Ann. differential equations 8, No. 2, 249-259 (1992) · Zbl 0758.34072