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rec-jank-jmaa.tex
\documentclass[12pt]{article} \begin{document} \title{Referee's report on the paper ``Differential-algebraic systems with maxima'' by T.~Jankowski} \maketitle The paper is devoted to the solvability analysis of the two-point boundary-value problems for differential-algebraic systems of equations with ``maxima.'' As the main tool, the author uses Samoilenko's numerical-analytic method, whose scheme is ajusted appropriately to be able to consider differential-algebraic systems. The convergence of approximate solutions is proved on the assumption that the right-hand side of the differential equation is globally Lipschitzian with a sufficiently small Lipschitz constant. This is a more or less standard ``parallel'' to the well-known things (see, e.g., the survey ``The theory of the numerical-analytic method: Achievements and new trends of development'', Ukrain. Math. J. 50 (1998), 116-135, by Ronto, Samoilenko, and Trofimchuk). It is interesting, however, that the unique solution is shown to be approximated by the Seidel-type iterations. Also, having dicscussed equations with ``maxima'', the author extends the techniques developed to more general equations with argument deviations. As possible disadvantages of the paper, we mention, e.g., the following things. (1) The problem setting seems to be a little unnatural, because the assumptions made on the ``algebraic'' are such that this latter one, theoretically, can be disposed of. It would be good to provide some evidence for doing this... (2) The notation is very clumsy. For example, a constant parameter, for an unknown reason, is denoted by $\bar{k}_0$ (?!) The symbol used to denote operator $A$ is simultaneously used to denote a matrix in the boundary condition, etc. (3) The manuscript contains some misprints. It would be nice if the author could eliminate these disadvantages, which make the paper a little bit difficult to read; in this case, the paper could be recommended for publication in JMAA. Referee\\ Academician A.~M.~Samoilenko Institute of Mathematics NAS of Ukraine Director \end{document}