By Roman Badora, Roman Ger (auth.), Zoltán Daróczy, Zsolt Páles (eds.)
The thought of practical equations has been built in a swift and efficient method within the moment half the 20 th Century. to start with, this is often given that the mathematical purposes raised the investigations of more moderen and more moderen varieties of useful equations. while, the self improvement of this conception was once additionally very fruitful. this is in lots of monographs that deal with and talk about some of the tools and methods. those advancements have been additionally basically inspired via a bunch jour nals, for example, by means of the Publicationes Mathematicae Debrecen (founded in 1953) and through the Aequationes Mathematicae (founded in 1968), be reason those journals released papers from the sphere of practical equa tions with ease and often. The latter magazine additionally publishes the once a year record of the foreign Symposia on practical Equations and a complete bibliography of the newest papers. while, there are periodically and typically prepared meetings in Poland and in Hungary dedicated to practical equations and inequali ties. In 2000, the thirty eighth overseas Symposium on useful Equations was once equipped by way of the Institute of arithmetic and Informatics of the college of Debrecen in Noszvaj, Hungary. The record approximately this assembly are available in Aequationes Math. sixty one (2001), 281-320.
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OC = lR or :oc = C). (2) en Solutions defined on can be considered as solutions defined on lR2n. The restriction of any solution to any linear subspace over lR is a solution over the given subspace. Especially, the restriction of solutions over are solutions over lRn. en On the characterization of Weierstrass's sigma junction h 43 (3) For any linear mapping L : ocm --+ ocn the functions gi 0 L, hi 0 L are solutions over ocm. (4) For any constants C1, C2 E ocn the shifted mappings x I-t h (x +Cl), y I-t h(y + C2), u I-t gi(U + (Cl + c2)/2), v I-t hi(V + (Cl - c2)/2) are also solutions.
Wl = 00 and W2 = 00, or 0 #- Wl E C and W2 = 00, or 0 #- Wl E C, 0 #- W2 E C and Wl/W2 ¢ llt The derivative of the Weierstrass's functions with respect to the first variable will be simply denoted by a prime. It is clear from the definition that for any complex number a#-O we have O"(az, aWl, aW2) = aO"(z,wl,W2). Weierstrass's ( function is defined as the logarithmic derivative of the sigma function: ( is a meromorphic function in the first variable, its poles are the points of n. ), Z- W W On the characterization of Weierstrass's sigma function otherwise L ((Z,WI,W2) =!
The aim of this paper is to prove that every measurable solution (It, h) of (4) which is not almost everywhere zero is in Coo = coo(JRn, C). In particular, the problem of W. Sander will be solved. On the other hand, we show that the results of M. Bonk can also be obtained using reduction to differential equations. Hence we obtain generalizations of the results of M. Bonk and the results of R. Rochberg and L. A. Rubel with a proof using only generally applicable methods. The proof of the regularity part depends on some new general regularity theorems proved by A.