123036278
Functional differential equations of the cauchy-kowalevsky type
with initial values u(0, z) = ~b(z) prescribed for t = 0. In the classical setting, t is complex, z = (z I . . . . . zn)eC", u = (u 1 . . . . . urn), and us = 8u/Sz is the m by n Jordan matrix. In the customary treatment of the problem, which goes back to Cauchy, the solution is obtained as a power series in t and z~. The recursion formulae for the coefficients produce a uniqueness proof, while the convergence proof for the power series by the method of majorants leads to existence. In 1941, M. Nagumo [2] discovered a completely different proof. He transformed the problem into an equivalent equation v = F(v)for v = ut and showed by means of the Schauder fixed point theorem that there exists a solution v. One important consequence of his proof is the fact that t is allowed to be a real variable. More precisely, there are two cases, the (classical) complex case, where all variables are complex and all functions involved are holomorphic in all their variables, and the real case, where t is real, z~C" and where the functions u and f a r e continuous in all variables and holomorphic in all variables save t. Starting with a paper by Ovsjannikov [31 in 1965, generalized or abstract versions of the C K theorem developed, which have basic ideas in common with
120951143
Verallgemeinerte Laplace-Operatoren und Potentiale
122832071
THERE IS AN ELEMENTARY PROOF OF PEANO'S EXISTENCE THEOREM.
Abstract : An elementary, constructive proof of Peano's existence theorem is given. The maximal solution is constructed as a limit of a decreasing sequence of approximate solutions. (Author)
123298075
Eindeutigkeitssätze für gewöhnliche, parabolische und hyperbolische Differentialgleichungen
119881926
Fehlerabschätzungen und Eindeutigkeitssätze für gewöhnliche und partielle Differentialgleichungen
122666640
Remarks on a paper by F. Browder about contraction
THROUGHOUT this note, X denotes a complete metric space with distance function d, and T is a map from X into itself. Powers of T are defined by Tax = x and T”+ lx = T(T”x), n 2 0. Occasionally, we use the notation xk = Tkx, in particular x0 = x, x1 = TX, for the sake of brevity. The set {x, TX, T'x, . . . , T”x,. . .} is called an orbit (starting at x) and denoted by O(x). The set 0(x, y) is the union of the two orbits starting at x and y, 0(x, y) = {xk, yk: k 3 O}. The diameter diam A of a nonempty subset A of X is the supremum of the numbers d(x, y), where x and y vary in A. The letter 4 denotes a contractive gauge function, i.e. a continuous, increasing function from R,, the non-negative reals, into R, which satisfies 4(s) 0.
1300115
Flexible precision control and dynamic data structures for programming mathematical and numerical algorithms