By Cellina A. (Ed)
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Xm) E R m, beside the notation FXi, FXiXi for partial derivatives, we shall use the notation FI% = DI%F, (X = «(Xl' ... ,(Xm), (Xi ~ 0 integers, k = (Xl + ... + (Xm = I(XI, for partial derivatives of order k. Below we shall require a function F to be of some class C in a closed subset D of Rm. By this we shall mean that F is defined in an open subset U of Rm containing D, and that F has continuous partial derivatives FI% of orders I(XI, 0 ~ I(XI ~ r, in U. This convention, which we use here only to simplify notation, has its motivation in the result due to Whitney [1-4] that if, for a real valued function F(x) defined on a closed set K of R m, there are functions F1%' 0 ~ I(XI ~ r, also continuous on K, such that the usual Taylor polynomials centered at any point of K approximate F (as usual), then F can be extended to a function of class C on an open neighborhood U of K so that DI%F = FI% everywhere on K.
The integral J 2[1} ; x], a quadratic form in I}, is often called the accessory integral of I[ x], and also the second variation of I[ x] at x. Remark 1. ii) holds even in a slightly more general situation. Indeed, let z(a,t) = (zl, ... ,(O,t). 2). The proof is the same with F(t, a) = fo(t, x(t) + z(a, t), x'(t) + z,(a, t». t(O, t) = rJ'(t) as a -+ O. ii). Remark 2. ii) that x' be essentially bounded can be omitted under suitable assumptions. For instance, let x(t), t1 :5"; t:5"; t 2, be a given AC n-vector function, whose graph r lies in A and let r6 = [(x, Yllt 1 :5"; t:5"; t 2, Iy - x(tll :5"; <5].
We may not repeat every time the need for the identification mentioned above (Cf. i, ii, iii)). e. e. ). This will be referred to as the DuBois-Reymond necessary condition, or the equation (Eo). 4 for a proof. For instance, for n = 1, I[ x] = Sl>(x'Z + x 2)dt, fo = x'z + x 2, and (Eo) reduces to (d/dt)(x Z - x'Z) = 0, or x 2 - X'Z = c, a costant. Here M(t) = XZ(t) - X'2(t). 7 A. (c) The AC 2n-vector function (x(t), A(t)), t E [t 1,t2], x(t) = A(t) = (Ab ... 3) dAi dt (Xl, ... ,xn), -oH ox i ' for i = 1, ...
Methods Of Nonconvex Analysis by Cellina A. (Ed)