The Boundary Integral Approach to Static and Dynamic Contact Problems (H. Antes and P. D. Panagiotopoulos)

enable a comparison to associated autonomous equations or inequalities. While GL deals exclusively with first-order equations, it also deals with first-order systems of n dimensions, and so nth-order scalar equations can be dealt with as first-order systems in the usual way. On the other hand, BM had an extensive treatment of nth-order equations, with special emphasis on the n 2 case. It has only a short chapter on first-order systems. For general first-order systems each book includes certain definitions ofoscillatory (and nonoscillatory) solution. Both books are quite self-contained; detailed proofs are given for most results. Although there are some misprints in each, they are fairly easy to spot and correct. Both books draw on a very large bibliography of publications, many of which involve the book’s authors. There are 163 references listed in BM and an even slightly longer list appears in GL. There is an obvious overlap in these, but the list in BM contains a number of papers in Russian not referred to in GL. The reviewer finds one of the definitions of a real-valued oscillatory solution given in BM to be very satisfying. According to this definition, the solution is oscillatory if it is not eventually identically zero, but has arbitrarily large zeros. In GL the more standard definition is used; the solution is to only have arbitrarily large zeros. To the reviewer, solutions that are identically constant-valued from some time on should not be regarded as oscillatory. In physical processes evolving in time, one really never "sees" constant-valued behavior. In his work, the reviewer has used the definition that there exists an increasing sequence to -oo as n -oo such that (-1)nx(tn) > 0, 1, 2, 3 An even more realistic definition might be that in addition to such a sequence tn, there exist a positive number b such that (1)x (tn) > b for n, 1, 2, 3 a study of the existence of such solutions might be of great interest to workers in applied areas. The definition of an oscillatory solution of the parabolic and hyperbolic partial differential equations considered in BM also seems to the reviewer somewhat unsatisfying. Such solutions, defined on a set 2 x R+, where f2 is a bounded domain in R are said to be oscillatory if each set x [to, cx), to > 0, contains a zero of the solution u(x, t). A more interesting and useful definition would be that there exists an x0 f2 and an increasing sequence t,, -oo as n oo such that (1)u (x0, t,,) > 0 for n 1, 2, 3 However, getting conditions for the existence of such more restrictively oscillating solutions would probably be much more difficult. For the definition in BM as stated above, the so-called averaged solution v(t) fn u(x, t)dx is shown to have certain oscillatory properties. This would clearly not be a feasible method for getting oscillatory solutions in the more restricted sense mentioned above. Both books mention a number of open problems; a larger and more explicit list appears in GL. Besides some open problems suggested above involving some of the more restrictive definitions ofoscillatory solutions mentioned above, one more might be added. For a first-order scalar linear delay differential equations on a finite delay interval [-r, 0], do all nontrivial solutions oscillate in the sense of the definition in BM, if the associated characteristic equation has no real roots?