By Gianni Dal Maso

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**Extra info for An introduction to G-convergence**

**Example text**

Here the matrix B(t) and the vector-valued function g(t) are of bounded variation, B(t) is continuous, and the derivatives are understood in the sense of the theory of distributions. The integral equations (54) y(t) = y(~) + fat (dB(s)) y(s) + g(t) - g(a) (where the integral is understood in the Stieltjes sense) equivalent to (53) and several more general equations were considered earlier [83], [841. In [82] existence and uniqueness of a solution with initial data y(a) = Yo are proved, the fundamental matrix is shown to be continuous and of bounded variation.

IT the functions (65) are assumed to be solutions of equation (58), then for all values of Ie all the solutions exist both for t < 0 and for t > o. The choice of such a definition of a solution corresponds to the case where in (61) elc -1-1e 'Y = Ie(elc - 1) . Note that 'Y -+ 1/2 as Ie -+ o. Using these arguments, one may come to the following conclusion. ) is equal to 1, and the numbers are small. Then in the case " ~ P. the solution is close to the function (62), and in the case " = Q-to the function (65).

Z("')J. by the formula (18) k The coefficients Co = bn • Ci Co, Cl, • .. 2, .... -2 - Clan-I, •.. -lJ i = 1,2, .... H the function z = 0 for t < 0, and z E em for t > 0, m is the same as in (12), then y(t) for t > 0 can be found as a. solution of equation (12) with the initial data y(+O) = [1IJ, 11'(+0) = [11'],'" , Equations . Discontinuous only in t 22 Chapter 1 where [y], [y'], ... are given by formulae (17) or (18) in which [z] = z(+O), [z'] = z'(+O}, .... Now let the coefficients ai,bi in (12) depend on t,z(t) E L 1 (loc), that is, the function z(t) is Lebesgue-integrable on each finite interval contained in its domain of definition.