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On general properties of n-th order retarded functional differential equations
Benevieri, Pierluigi
Calamai, Alessandro
Furi, Massimo
Pera, Maria Patrizia
2017
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e-ISSN
2464-8728
Abstract
Consider the second order RFDE (retarded functional differential equation) x’’ (t) = f(t, xt), where f is a continuous realvalued function defined on the Banach space R x C1([−r, 0],R). The weak assumption of continuity on f (due to the strong topology of C1([−r, 0],R)) makes not convenient to transform this equation into a first order RFDE of the type z’ (t) = g(t, zt). In fact, in this case, the associated R2-valued function g could be discontinuous (with the C0- topology) and, in addition, not necessarily defined on the whole space R x C([−r, 0],R2). Consequently, in spite of what happens for ODEs, the classical results regarding existence, uniqueness, and continuous dependence on data for first order RFDEs could not apply. Motivated by this obstruction, we provide results regarding general properties, such as existence, uniqueness, continuous dependence on data and continuation of solutions of RFDEs of the type x(n)(t) = f(t, xt), where f is an Rk-valued continuous function on the Banach space R x C(n−1)([−r, 0],Rk). Actually, for the sake of generality, our investigation will be carried out in the case of infinite delay.
Part of
49 (2017)
Source
P. Benevieri, A. Calamai et al., "On general properties of n-th order retarded functional differential equations", in: "Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics", 49 (2017), Trieste, EUT Edizioni Università di Trieste, 2017, pp. 73-93
Languages
en
Rights
Attribution-NonCommercial-NoDerivatives 4.0 Internazionale
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