On Exponential Decay and the Markus–Yamabe Conjecture in Infinite Dimensions with Applications to the Cima System
- 162 Downloads
In this work we extend to infinite dimensions a previous result of Lyascenko given in finite dimensions for a class of Lipschitz functions. In particular, this extension substitutes the class of Lipschitz functions by a class of Hölder functions. Then we consider applications of these results to analyze its relation with the Markus–Yamabe Conjecture in infinite dimensions. We discuss in detail a celebrated example of Cima et al. ( Adv Math 131(2): 453–457, 1997) of a nonlinear system in dimension three whose Jacobian has a unique eigenvalue \(-1\) of multiplicity 3, and yet an explicit unbounded solution as \(t \rightarrow \infty \) exists. We also present explicit solutions of the same equation that tends to 0 as \(t \rightarrow \infty \). Then we look at this conjecture by considering not the properties of the whole system, but instead the properties of some solutions. Finally we present an application in an infinite dimensional Hilbert space, where we use different techniques to study the local and global asymptotic stabilities.
KeywordsMarkus-Yamabe conjecture in infinite dimensions Extension of a result of Lyascenko Cima system Global asymptotic stability
Hildebrando M. Rodrigues was partially supported by FAPESP Grant 2015/19165-5. Marco A. Teixeira was partially supported by FAPESP Grant 2012/18780-0 and CNPq Grant 300596/2009-0. Marcio Gameiro was partially supported by FAPESP Grants 2013/07460-7, 2016/08704-5, and 2016/21032-6, and by CNPq Grants 305860/2013-5 and 310740/2016-9, Brazil.
- 2.Rodrigues, H.M.: Invariância para sistemas de equações diferenciais com retardamento e aplicações. Universidade de São Paulo, Tese de Mestrado (1970)Google Scholar
- 12.Coppel, W.A.: Dichotomies in Stability Theory. Lecture Notes in Mathematics, Vol. 629. Springer, Berlin (1978)Google Scholar
- 16.Henry, Daniel: Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics. vol. 840. Springer, Berlin (1981)Google Scholar
- 18.Silva, Elves A.B., Teixeira, Marco A.: Global injectivity and asymptotic stability via minimax method. In: Progress in Nonlinear Analysis (Tianjin, 1999), Nankai Ser. Pure Appl. Math. Theoret. Phys., vol. 6, pp. 339–358. World Sci. Publ., River Edge (2000)Google Scholar
- 26.Daleckiĭ, Ju. L., Kreĭn, M.G.: Stability of solutions of differential equations in Banach space. American Mathematical Society, Providence, R.I., Translated from the Russian by S. Smith, Translations of Mathematical Monographs, vol. 43 (1974)Google Scholar