DOI: https://doi.org/10.11568/kjm.2016.24.4.723

### Boundedness in nonlinear perturbed differential systems via $t_{\infty}$-similarity

#### Abstract

This paper shows that the solutions to nonlinear perturbed differential system

$$

y'=f(t,y)+\int_{t_0}^tg(s,y(s))ds+h(t,y(t),Ty(t))

$$

have bounded properties. To show the bounded properties, we impose conditions on the perturbed part

$\int_{t_0}^tg(s,y(s))ds$, $h(t,y(t),Ty(t))$, and on the fundamental matrix of the unperturbed system $y'=f(t,y)$ using the notion of $h$-stability.

#### Keywords

#### Subject classification

34D10;34D20#### Sponsor(s)

Lee Man Seab, Mokwon Unversity, Department of Mathematics; Kim Hark Man, Chungnam University, Department of Mathematics; Jung Soon Mo, Hongik University, Department of liberal Mathematics;#### Full Text:

PDF#### References

V. M. Alekseev, An estimate for the perturbations of the solutions of ordinary differential equations, Vestn. Mosk. Univ. Ser. I. Math. Mekh. 2 (1961), 28– 36(Russian). (Google Scholar)

F. Brauer, Perturbations of nonlinear systems of differential equations, J. Math. Anal. Appl. 14 (1966), 198–206. (Google Scholar)

S. I. Choi and Y. H. Goo, h−stability and boundedness in perturbed functional differential systems, Far East J. Math. Sci(FJMS) 97(2015), 69–93. (Google Scholar)

S. I. Choi and Y . H. Goo, Boundedness in perturbed functional differential systems via t∞-similarity, Korean J. Math. 23 (2015), 269–282. (Google Scholar)

S. K. Choi and H. S. Ryu, h−stability in differential systems, Bull. Inst. Math. Acad. Sinica 21 (1993), 245–262. (Google Scholar)

S. K. Choi, N. J. Koo and H.S. Ryu, h-stability of differential systems via t∞- similarity, Bull. Korean. Math. Soc. 34 (1997), 371–383. (Google Scholar)

R. Conti, Sulla t∞-similitudine tra matricie l’equivalenza asintotica dei sistemi differenziali lineari, Rivista di Mat. Univ. Parma 8 (1957), 43–47. (Google Scholar)

Y. H. Goo, Boundedness in the perturbed functional differential systems via t∞-similarity, Far East J. Math. Sci(FJMS) 97(2015),763-780. (Google Scholar)

Y. H. Goo, Boundedness in the perturbed nonlinear differential systems, Far East J. Math. Sci(FJMS) 79(2013),205-217. (Google Scholar)

Y . H. Goo, Boundedness in the perturbed differential systems, J. Korean Soc. Math. Edu. Ser.B: Pure Appl. Math. 20 (2013), 223-232. (Google Scholar)

Y. H. Goo, Boundedness in perturbed nonlinear differential systems, J. Chungcheong Math. Soc. 26(2013), 605-613. (Google Scholar)

G. A. Hewer, Stability properties of the equation by t∞-similarity, J. Math. Anal. Appl. 41 (1973), 336–344. (Google Scholar)

V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, Academic Press, New York and London, 1969. (Google Scholar)

B.G. Pachpatte, On some retarded inequalities and applications, J. Ineq. Pure Appl. Math. 3 (2002) 1–7. (Google Scholar)

M. Pinto, Perturbations of asymptotically stable differential systems, Analysis 4 (1984), 161–175. (Google Scholar)

M. Pinto, Stability of nonlinear differential systems, Applicable Analysis 43 (1992), 1–20. (Google Scholar)

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