Nonuniform exponential stability for evolution families on the real line

Morariu, Claudia Isabela; Preda, Petre

doi:10.1515/auom-2015-0013

Abstract

The purpose of the present paper is to investigate the problem of nonuniform exponential stability of evolution families on the real line using the input-output technique known in the literature as the Perron method for the study of exponential stability. In this manuscript we describe an evolution family on the real line and we present sufficient conditions for the nonuniform exponential stability of an evolution family on the real line that does not have exponential growth.

References

[1] Barreira, L., Valls, C., Admissibility for nonuniform exponential contractions, Journal of Differential Equations 249 (2010), 2889-2904.10.1016/j.jde.2010.06.010
Search in Google Scholar Back to article
[2] Barreira, L., Valls, C., Nonuniform exponential dichotomies and admissibility, Discrete and Continuous Dynamical Systems 30 (2011), 39-53.10.3934/dcds.2011.30.39
Search in Google Scholar Back to article
[3] Chicone, C., Latushkin, Y., Evolution semigroups in dynamical systems and differential equations, Mathematical Surveys and Monographs, vol. 70, Providence, RO: American Mathematical Society, 1999.10.1090/surv/070
Search in Google Scholar Back to article
[4] Coppel, W.A., Dichotomies in Stability Theory, Lecture Notes in Math., vol. 629, Springer-Verlag, New York, 1978.10.1007/BFb0067780
Search in Google Scholar Back to article
[5] Daleckij, J.L., Krein, M.G., Stability of Differential Equations in Banach Space, Amer. Math. Soc., Providence, RI, 1974.
Search in Google Scholar Back to article
[6] Hartman, P. Ordinary Differential Equations, Wiley, New York/London/Sidney, 1964.
Search in Google Scholar Back to article
[7] Huy, N.T., Invariant manifolds of admissibile classes for semi-linear evolution equations, J. Differential Equations 246 (2009), 1820-1844.10.1016/j.jde.2008.10.010
Search in Google Scholar Back to article
[8] Latushkin, Y., Randolph, T., Schnaubelt, R., Exponential dichotomy and mild solutions of non-autonomous equations in Banach spaces, J. Dynam. Differential Equations 3 (1998), 489-510.10.1023/A:1022609414870
Search in Google Scholar Back to article
[9] Levitan, B.M., Zhikov, V.V., Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge, 1982.
Search in Google Scholar Back to article
[10] Marin, M., Florea, O., On temporal behaviour of solutions in Thermoelasticity of porous micropolar bodies, An. Sti. Univ. Ovidius Constanţa, Vol. 22, issue 1, (2014) 169-188.10.2478/auom-2014-0014
Search in Google Scholar Back to article
[11] Massera, J.L., Schäffer, J.J., Linear Differential Equations and Function Spaces, Academic Press, New York, 1966.
Search in Google Scholar Back to article
[12] Van Minh, N., Räbiger, F., Schnaubelt, R., Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half-line, Integral Equations Operator Theory 32 (1998), 332-353.10.1007/BF01203774
Search in Google Scholar Back to article
[13] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.10.1007/978-1-4612-5561-1
Search in Google Scholar Back to article
[14] Perron, O., Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), 703-728.10.1007/BF01194662
Search in Google Scholar Back to article
[15] Sasu, A.L., Sasu, B. Exponential dichotomy on the real line and admissibility of function spaces, Integr. equ. oper. theory 54 (2006), 113-130.10.1007/s00020-004-1347-z
Search in Google Scholar Back to article
[16] Sasu, A.L., Integral equations on function spaces and dichotomy on the real line, Integr. equ. oper. theory 58 (2007), 133-152.10.1007/s00020-006-1478-5
Search in Google Scholar Back to article
[17] Sharma, K., Marin, M., Reflection and transmission of waves from imperfect boundary between two heat conducting micropolar thermoelastic solids, An. Sti. Univ. Ovidius Constanţa, Vol. 22, issue 2, (2014) 151-175.10.2478/auom-2014-0040
Search in Google Scholar Back to article

Nonuniform exponential stability for evolution families on the real line

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