## Abstract and Applied Analysis

### Exponential dichotomy for evolution families on the real line

#### Abstract

We give necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pair $(L^{p} (\mathbb{R}, X) , L^{q} (\mathbb{R}, X))$. We show that the admissibility of the pair $(L^{p} (\mathbb{R}, X) , L^{q} (\mathbb{R}, X))$ is equivalent to the uniform exponential dichotomy of an evolution family if and only if $p \geq q$. As applications we obtain characterizations for uniform exponential dichotomy of semigroups.

#### Article information

Source
Abstr. Appl. Anal., Volume 2006 (2006), Article ID 31641, 16 pages.

Dates
First available in Project Euclid: 30 January 2007

https://projecteuclid.org/euclid.aaa/1170202990

Digital Object Identifier
doi:10.1155/AAA/2006/31641

Mathematical Reviews number (MathSciNet)
MR2217210

Zentralblatt MATH identifier
1137.34343

#### Citation

Sasu, Adina Luminiţa. Exponential dichotomy for evolution families on the real line. Abstr. Appl. Anal. 2006 (2006), Article ID 31641, 16 pages. doi:10.1155/AAA/2006/31641. https://projecteuclid.org/euclid.aaa/1170202990

#### References

• A. Ben-Artzi and I. Gohberg, Dichotomy of systems and invertibility of linear ordinary differential operators, Operator Theory: Advances and Applications 56 (1992), 90--119.
• A. Ben-Artzi, I. Gohberg, and M. A. Kaashoek, Invertibility and dichotomy of differential operators on a half-line, Journal of Dynamics and Differential Equations 5 (1993), no. 1, 1--36.
• C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs, vol. 70, American Mathematical Society, Rhode Island, 1999.
• S.-N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces, Journal of Differential Equations 120 (1995), no. 2, 429--477.
• J. L. Dalec'kiĭ and M. G. Kreĭn, Stability of Differential Equations in Banach Space, Translations of Mathematical Monographs, vol. 43, American Mathematical Society, Rhode Island, 1974.
• G. Gühring, F. Räbiger, and W. M. Ruess, Linearized stability for semilinear non-autonomous evolution equations with applications to retarded differential equations, Differential and Integral Equations 13 (2000), no. 4--6, 503--527.
• Y. Latushkin, T. Randolph, and R. Schnaubelt, Exponential dichotomy and mild solutions of nonautonomous equations in Banach spaces, Journal of Dynamics and Differential Equations 10 (1998), no. 3, 489--510.
• M. Megan, B. Sasu, and A. L. Sasu, On nonuniform exponential dichotomy of evolution operators in Banach spaces, Integral Equations and Operator Theory 44 (2002), no. 1, 71--78.
• M. Megan, A. L. Sasu, and B. Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete and Continuous Dynamical Systems 9 (2003), no. 2, 383--397.
• --------, Perron conditions for pointwise and global exponential dichotomy of linear skew-product flows, Integral Equations and Operator Theory 50 (2004), no. 4, 489--504.
• A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer, New York, 1983.
• V. A. Pliss and G. R. Sell, Robustness of the exponential dichotomy in infinite-dimensional dynamical systems, Journal of Dynamics and Differential Equations 11 (1999), no. 3, 471--513.
• W. M. Ruess, Existence and stability of solutions to partial functional-differential equations with delay, Advances in Differential Equations 4 (1999), no. 6, 843--876.
• --------, Linearized stability for nonlinear evolution equations, Journal of Evolution Equations 3 (2003), no. 2, 361--373.
• R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, Journal of Differential Equations 113 (1994), no. 1, 17--67.
• A. L. Sasu, Integral characterizations for stability of linear skew-product semiflows, Mathematical Inequalities & Application 7 (2004), no. 4, 535--541.
• A. L. Sasu and B. Sasu, A lower bound for the stability radius of time-varying systems, Proceedings of the American Mathematical Society 132 (2004), no. 12, 3653--3659.
• B. Sasu and A. L. Sasu, Stability and stabilizability for linear systems of difference equations, Journal of Difference Equations and Applications 10 (2004), no. 12, 1085--1105.
• --------, Exponential dichotomy and $(\ell ^p ,\ell ^q )$-admissibility on the half-line, Journal of Mathematical Analysis and Applications 316 (2006), no. 2, 397--408.
• A. L. Sasu and B. Sasu, Exponential dichotomy on the real line and admissibility of function spaces, Integral Equations and Operator Theory 54 (2006), no. 1, 113--130.
• B. Sasu and A. L. Sasu, Exponential trichotomy and $p$-admissibility for evolution families on the real line, to appear in Mathematische Zeitschrift.
• S. Siegmund, Dichotomy spectrum for non-autonomous differential equations, Journal of Dynamics and Differential Equations 14 (2002), no. 1, 243--258.
• N. Van Minh and N. T. Huy, Characterizations of dichotomies of evolution equations on the half-line, Journal of Mathematical Analysis and Applications 261 (2001), no. 1, 28--44.
• N. Van Minh, F. Räbiger, and R. Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line, Integral Equations and Operator Theory 32 (1998), no. 3, 332--353.
• W. N. Zhang, The Fredholm alternative and exponential dichotomies for parabolic equations, Journal of Mathematical Analysis and Applications 191 (1995), no. 1, 180--201.