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Lyapunov functions and convergence to steady state for differential equations of fractional order
Authors:Vicente Vergara  Rico Zacher
Institution:1. Departamento de Matemática y C.C., Facultad de Ciencias, University of Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
2. Institut für Mathematik, Martin-Luther-Universit?t Halle-Wittenberg, Theodor-Lieser-Strasse 5, 06120, Halle, Germany
Abstract:We study the asymptotic behaviour, as t → ∞, of bounded solutions to certain integro-differential equations in finite dimensions which include differential equations of fractional order between 0 and 2. We derive appropriate Lyapunov functions for these equations and prove that any global bounded solution converges to a steady state of a related equation, if the nonlinear potential $$\mathcal E$$ occurring in the equation satisfies the Łojasiewicz inequality.
Keywords:Integro-differential equations  Fractional derivative  Gradient system  Lyapunov function  Convergence to steady state  Ł  ojasiewicz inequality
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