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Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line. (English) Zbl 0977.34056
Here, nonautonomous linear evolution equations $(*)\quad u'= A(t)u,\;t\in\mathbb{R}_+,\qquad (**)\quad u'=A(t)u+f(t),\;t\in\mathbb{R}_+,$ on a Banach space $$X$$ are considered. Instead of discussing the operator $$L$$ defined by $$Lu:u'-A(t)u$$ as in [W. N. Zhang, J. Math. Anal. Appl. 191, No. 1, 180-201 (1995; Zbl 0832.34050) and D. Henry, Lecture Notes in Math., 840, Berlin etc.: Springer-Verlag (1981; Zbl 0456.35001)], etc., the authors investigate the evolution family $${\mathcal U}=\{U(t,s): t\geq s$$ in $$\mathbb{R} _+\}$$ generated by solutions to $$(*)$$ so that solutions to $$(**)$$ can be expressed as $u(t)=U(t,s) u(s)+\int_s^tU(t,\xi) f(\xi)d\xi,\;t\geq s\text{ in }\mathbb{R}_+.$ By defining an evolution semigroup ${\mathcal T}=\{T(t): t\geq 0\} \text{ such that }\bigl[T(t) v\bigr](s)= \begin{cases} U(s,s-t) v(s-t), & s\geq t,\\ U(s,0)v(0), & 0\leq s\leq t,\end{cases}$ the authors characterize exponential stability (in theorems 2.2 and 3.2), exponential expansiveness (in theorem 2.5), and exponential dichotomy (in theorems 4.3 and 4.5) of the evolution family $${\mathcal U}$$ on the half-line $$\mathbb{R}_+$$ in terms of the infinitesimal generator of $${\mathcal T}$$.

##### MSC:
 34G10 Linear differential equations in abstract spaces 34D09 Dichotomy, trichotomy of solutions to ordinary differential equations 47D06 One-parameter semigroups and linear evolution equations 47H20 Semigroups of nonlinear operators
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##### References:
 [1] [AuM] Aulbach B., Nguyen Van Minh,Semigroups and exponential stability of nonautonomous linear differential equations on the half-line, (R.P. Agrawal Ed.), Dynamical Systems and Aplications, World Scientific, Singapore 1995, pp. 45-61. · Zbl 0842.34059 [2] [BaV] Batty C., V? Quoc Phóng,Stability of individual elements under one-parameter semigroups, Trans. Amer. Math. Soc. 322 (1990), 805-818. · Zbl 0711.47023 · doi:10.2307/2001726 [3] [BeG] Ben-Artzi A., Gohberg I.,Dichotomies of systems and invertibility of linear ordinary differential operators, Oper. Theory Adv. Appl. 56 (1992), 90-119. · Zbl 0766.47021 [4] [BGK] Ben-Artzi A., Gohberg I., Kaashoek M.A.,Invertibility and dichotomy of differential operators on the half-line, J. Dyn. Differ. Equations 5 (1993), 1-36. · Zbl 0771.34011 · doi:10.1007/BF01063733 [5] [Bus] Bu?e C.,On the Perron-Bellman theorem for evolutionary processes with exponential growth in Banach spaces, preprint. [6] [Cop] Coppel W.A., ?Dichotomies in Stability Theory?, Springer-Verlag, Berlin Heidelberg, New York, 1978. · Zbl 0376.34001 [7] [DaK] Daleckii Ju. L., Krein M.G., ?Stability of Solutions of Differential Equations in Banach Spaces?, Amer. Math. Soc., Providence RI, 1974. [8] [Dat] Datko R.,Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal. 3 (1972), 428-445. · Zbl 0241.34071 · doi:10.1137/0503042 [9] [Hen] Henry D., ?Geometric Theory of Semilinear Parabolic Equations?, Springer-Verlag, Berlin, Heidelberg, New York, 1981. · Zbl 0456.35001 [10] [Kat] Kato, T., ?Perturbation Theory for Linear Operators?, Springer-Verlag, Berlin, Heidelberg, New York, 1966. [11] [LaM] Latushkin Y., Montgomery-Smith S.,Evolutionary semigroups and Lyapunov theorems in Banach spaces, J. Funct. Anal. 127 (1995), 173-197. · Zbl 0878.47024 · doi:10.1006/jfan.1995.1007 [12] [LMR1] Latushkin Y., Montgomery-Smith S., Randolph T.,Evolutionary semigroups and dichotomy of linear skew-product flows on locally compact spaces with Banach fibers, J. Diff. Eq. 125 (1996), 73-116. · Zbl 0881.47020 · doi:10.1006/jdeq.1996.0025 [13] [LMR2] Latushkin Y., Montgomery-Smith S., Randolph T.,Evolution semigroups and robust stability of evolution operators on Banach spaces, preprint. [14] [LaR] Latushkin Y., Randolph T.,Dichotomy of differential equations on Banach spaces and an algebra of weighted composition operators, Integral Equations Oper. Theory 23 (1995), 472-500. · Zbl 0839.47026 · doi:10.1007/BF01203919 [15] [LRS] Latushkin Y., Randolph T., Schnaubelt R.,Exponential dichotomy and mild solutions of nonautonomous equations in Banach spaces, to appear in J. Dynamics Diff. Equations. · Zbl 0908.34045 [16] [LeZ] Levitan B.M., Zhikov V.V., ?Almost Periodic Functions and Differential Equations?, Cambridge Univ. Press, 1982. · Zbl 0499.43005 [17] [MaS] Massera J.J., Schäffer J.J., ?Linear Differential Equations and Function Spaces?, Academic Press, New York, 1966. [18] [Mi1] Nguyen Van Minh,Semigroups and stability of nonautonomous differential equations in Banach spaces, Trans. Amer. Math. Soc. 345 (1994), 223-242. · Zbl 0820.34039 · doi:10.2307/2154602 [19] [Mi2] Nguyen Van Minh,On the proof of characterizations of the exponential dichotomy, preprint. · Zbl 0911.34054 [20] [Nee] van Neerven, J.M.A.M.,Characterization of exponential stability of a semigroup of operators in terms of its action by convolution on vector-valued function spaces over ?+, J. Diff. Eq. 124 (1996), 324-342. · Zbl 0913.47033 · doi:10.1006/jdeq.1996.0012 [21] [Nic] Nickel G.,On evolution semigroups and well-posedness of non-autonomous Cauchy problems, PhD thesis, Tübingen, 1996. [22] [Pal] Palmer K.J.,Exponential dichotomy and Fredholm operators, Proc. Amer. Math. Soc. 104 (1988), 149-156. · Zbl 0675.34006 · doi:10.1090/S0002-9939-1988-0958058-1 [23] [Paz] Pazy A., ?Semigroups of Linear Operators and Applications to Partial Differential Equations?, Springer-Verlag, Berlin, Heidelberg, New York, 1983. · Zbl 0516.47023 [24] [RRS] Räbiger F., Rhandi A., Schnaubelt R., Voigt J.,Non-autonomous Miyadera perturbations, preprint. [25] [RS1] Räbiger F., Schnaubelt R.,The spectral mapping theorem for evolution semigroups on spaces of vector valued functions, Semigroup Forum 48 (1996), 225-239. · Zbl 0897.47037 · doi:10.1007/BF02574098 [26] [RS2] Räbiger F., Schnaubelt R.,Absorption evolution families with applications to non-autonomous diffusion processes, Tübinger Berichte zur Funktionalanalysis 5 (1995/96), 335-354. [27] [Rau] Rau R.,Hyperbolic evolution semigroups onvector valued function spaces, Semigroup Forum 48 (1994), 107-118. · Zbl 0802.47043 · doi:10.1007/BF02573658 [28] [SaS] Sacker R., Sell G.,Dichotomies for linear evolutionary equations in Banach spaces, J. Diff. Eq. 113 (1994), 17-67. · Zbl 0815.34049 · doi:10.1006/jdeq.1994.1113 [29] [Sch] Schnaubelt R.,Exponential bounds and hyperbolicity of evolution families, PhD thesis, Tübingen, 1996. · Zbl 0880.47025 [30] [Tan] Tanabe H., ?Equations of Evolution?, Pitman, London, 1979. · Zbl 0417.35003 [31] [V?] V? Quôc Phóng,On the exponential stability and dichotomy of C o -semigroups, preprint. [32] [Zha] Zhang W.,The Fredholm alternative and exponential dichotomies for parabolic equations, J. Math. Anal. Appl. 191 (1995), 180-201. · Zbl 0832.34050 · doi:10.1016/S0022-247X(85)71126-2 [33] [Zhi] Zhikov V.V.,On the theory of the admissibility of pairs of function spaces, Soviet Math. Dokl. 13 (1972), 1108-1111. · Zbl 0265.34073
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