Time averaging for functional differential equations

Journal of Applied Mathematics, Aug 2018

We present a result on the averaging for functional differential equations on finite time intervals. The result is formulated in both classical mathematics and nonstandard analysis; its proof uses some methods of nonstandard analysis.

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Time averaging for functional differential equations

Hindawi Publishing Corporation Journal of Applied Mathematics We present a result on the averaging for functional differential equations on finite time intervals. The result is formulated in both classical mathematics and nonstandard analysis; its proof uses some methods of nonstandard analysis. 1. Introduction The idea of the method of averaging is to determine conditions in which solutions of an autonomous dynamical system can be used to approximate solutions of a more complicated time varying dynamical system. The method of averaging has become one of the most important tool ever developed for nonlinear time varying systems. Applications have been found in celestial mechanics, noise control, nonlinear oscillations, stability analysis, bifurcation theory and vibrational control, among many other fields. Although averaging of ordinary differential equations is considered a mature field?the reader may consult [ 1, 5, 8, 22, 24 ] for more references and information on the subject (see also [13, 14])?averaging of functional differential equations has only recently been developed (see [ 6, 7, 9, 11, 12, 15, 16, 19 ]). This paper aims to present a result on the averaging for functional differential equations of the form x? (t) = f t ? (1.1) on finite time intervals. The result is not new (see [10] and the references therein). However, by means of nonstandard analysis methods, we propose a new proof where all the analysis is achieved in Rd (it is not the case in [10]) which makes it more simple. The paper is organized as follows. Section 2 contains the notation and conditions required to state and prove our main result as well as the main result itself. The proof of this result is given in Section 4.2. To avoid complicating the proof unnecessarily, several subsidiary lemmas have been placed in Section 4.1. The main result is formulated in both classical mathematics and nonstandard analysis. Its proof makes use of Robinson?s nonstandard analysis (NSA) [ 21 ]. We will work in the axiomatic form IST (for internal set theory) of nonstandard analysis, given by Nelson [ 20 ]. For that, Section 3.1 is devoted to a short description of IST. Then, in Section 3.2, we present the nonstandard translate (Theorem 3.6) in the language of IST of our main result (Theorem 2.2). We recall that IST is a conservative extension of ordinary mathematics. This means that any statement of ordinary mathematics which is a theorem of IST was already a theorem of ordinary mathematics, so there is no need to translate the proof. 2. Notation, conditions, and main result Let r ? 0 be a given constant. Throughout this paper C0 = C([?r, 0], Rd) will denote the Banach space of all continuous functions from [?r, 0] into Rd with the norm ? = sup{|?(?)| : ?r ? ? ? 0}, where | ? | is a norm of Rd. Let t0 ? R and T > t0. If x(t) is a continuous function defined on [t0 ? r, T ] and t ? [t0, T ], then xt ? C0 is defined by xt(?) = x(t + ?) for ? ? [?r, 0]. The hypotheses, which are denoted by the letter H, are listed as follows. (H1) The functional f : R ?C0 ? Rd in (1.1) is continuous. (H2) The functional f is Lipschitzian in u ? C0, that is, there exists some constant k such that f ?, u1 ? f ?, u2 ? k u1 ? u2 , ?? ? R, ?u1, u2 ? C0. (H3) For all u ? C0, there exists a limit f 0(u) := lim T?? T 0 1 T (resp., the solution of (1.1)) such that yt0 = ? (resp., xt0 = ?) is denoted by y = y(?; t0, ?) (resp., x = x(?; t0, ?)) and J (resp., I) will denote its maximal interval of definition. Remark 2.1. Existence and uniqueness of solutions of (2.3) will be justified a posteriori. Indeed, we will show in Lemma 4.1 below that the function f 0 is k-Lipschitz so that existence and uniqueness are guaranteed. Under the above assumptions, we will state the main result of this paper which gives nearness of solutions of (1.1) and (2.3) on finite time intervals. Theorem 2.2. Let assumptions (H1), (H2), and (H3) hold. Let ? ? C0 and t0 ? R. Let x be the solution of (1.1) and y the solution of (2.3) with xt0 = yt0 = ?. Then for any ? > 0 and T > t0, T ? J, there exists ?0 = ?0(?, T ) > 0 such that, for ? ? (0, ?0], x is defined at least on [t0, T ] and |x(t) ? y(t)| < ? on t ? [t0, T ]. 3. Nonstandard main result 3.1. Internal set theory In IST we adjoin to ordinary mathematics (say ZFC) a new undefined unary predicate standard (st). The axioms of IST are the usual axioms of ZFC plus three others which govern the use of the new predicate. Hence, all theorems of ZFC remain valid in IST. What is new in IST is an addition, not a change. We call a formula of IST external in the case where it involves the new predicate st; otherwise, we call it internal. Thus internal formulas are the formulas of ZFC. The theory IST is a conservative extension of ZFC, that is, every internal theorem of IST is a theorem of ZFC. Some of the theorems which are proved in IST are external and can be reformulated so that they become internal. Indeed, there is a reduction a (...truncated)


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Mustapha Lakrib. Time averaging for functional differential equations, Journal of Applied Mathematics, 2003, DOI: 10.1155/S1110757X03203077