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Characteristic exponents and some applications to differential equations. (English) Zbl 0535.34036

The authors consider continuous, possibly nonlinear, functions A from a real Banach space E into itself such that \(A(0)=0\). For such functions they define upper and lower characteristic exponents which are generalizations of the Lozinskij logarithmic norm. The upper and lower characteristic exponents are related to the stability properties of the differential equation (1) \(x'=A(x)\). For example, they prove that if the upper characteristic exponent is negative, then the zero solution of (1) is asymptotically stable. The characteristic exponents are characterized in terms of some properties of the solutions of (1).
Reviewer: A.Hausrath

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
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[1] Fred Brauer, Perturbations of nonlinear systems of differential equations, J. Math. Anal. Appl. 14 (1966), 198 – 206. · Zbl 0156.09805 · doi:10.1016/0022-247X(66)90021-7
[2] Stavros N. Busenberg and Linda K. Jaderberg, Decay characteristics for differential equations without linear terms, J. Differential Equations 18 (1975), 87 – 102. · Zbl 0297.34006 · doi:10.1016/0022-0396(75)90082-0
[3] W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Co., Boston, Mass., 1965. · Zbl 0154.09301
[4] Germund Dahlquist, Stability and error bounds in the numerical integration of ordinary differential equations, Kungl. Tekn. Högsk. Handl. Stockholm. No. 130 (1959), 87. · Zbl 0085.33401
[5] Ju. L. Dalec\(^{\prime}\)kiĭ and M. G. Kreĭn, Stability of solutions of differential equations in Banach space, American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by S. Smith; Translations of Mathematical Monographs, Vol. 43.
[6] Massimo Furi and Alfonso Vignoli, Spectrum for nonlinear maps and bifurcation in the non differentiable case, Ann. Mat. Pura Appl. (4) 113 (1977), 265 – 285 (English, with Italian summary). · Zbl 0366.47028 · doi:10.1007/BF02418377
[7] V. Lakshmikantham and S. Leela, Differential and integral inequalities, theory and applications, Vol. I, Academic Press, New York, 1969. · Zbl 0177.12403
[8] S. M. Lozinskii, Error estimates for numerical integration of ordinary differential equations. I, Izv. Vysš. Učebn. Zaved. Matem., No. 5, (6) (1958), 52-90. (Russian)
[9] R. H. Martin Jr., A bound for solutions of Volterra-Stieltjes integral equations, Proc. Amer. Math. Soc. 23 (1969), 506 – 512. · Zbl 0201.43603
[10] R. H. Martin Jr., Bounds for solutions of a class of nonlinear differential equations., J. Differential Equations 8 (1970), 416 – 430. · Zbl 0213.36802 · doi:10.1016/0022-0396(70)90015-X
[11] T. Ważewski, Sur la limitation des intégrales des systèmes d’équations différentielles linéaires ordinaires, Studia Math. 10 (1948), 48 – 59 (French). · Zbl 0036.05703
[12] Aurel Wintner, Asymptotic integration constants, Amer. J. Math. 68 (1946), 553 – 559. · Zbl 0063.08295 · doi:10.2307/2371784
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