Almost periodic linear differential equations with non-separated solutions |
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Authors: | Rafael Ortega |
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Institution: | a Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain b Dipartimento di Matematica dell'Università, Via Saldini 50, 20133 Milano, Italy |
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Abstract: | A celebrated result by Favard states that, for certain almost periodic linear differential systems, the existence of a bounded solution implies the existence of an almost periodic solution. A key assumption in this result is the separation among bounded solutions. Here we prove a theorem of anti-Favard type: if there are bounded solutions which are non-separated (in a strong sense) sometimes almost periodic solutions do not exist. Strongly non-separated solutions appear when the associated homogeneous system has homoclinic solutions. This point of view unifies two fascinating examples by Zhikov-Levitan and Johnson for the scalar case. Our construction uses the ideas of Zhikov-Levitan together with the theory of characters in topological groups. |
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Keywords: | Favard's theory Almost periodic solutions Homoclinic solutions |
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