120744449
An Elementary Proof of the Cauchy-Kowalevsky Theorem
(1) DtP u F( t, xi 9 ... Xn u, ... . D (k + loal 1. It states that if for t = 0 initial data DI u = (k (0 < k < p) are prescribed, where the (k are analytic functions of x = (xi,. . , xn) in a neighborhood of x0 = (xo , ... , xx), and if the function F is analytic in a neighborhood of the point (0, xo,... , k (0, xO),...), then there exists a unique solution u(t, x) in a neighborhood of (0, x0), analytic in t and x and satisfying the initial conditions. The problem is called Cauchy's problem, the initial conditions Cauchy data. It was first solved by Cauchy [2], and, in a more general and simplified way, by Sophie v. Kowalevsky [9]. It represents the analogue for partial differential equations of the basic initial value problem for an ordinary differential equation u (p)( t)=(t, u, u... ,u(P-1) ) with given initial data u(0),. .. , u(p 1) (0). Obviously, this analogy motivated Cauchy to consider the problem. As a preliminary step of the proof found, e.g., in [3; p. 39-56], the equation (or system) is first converted into a more convenient quasilinear first order system
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.
120951143
Verallgemeinerte Laplace-Operatoren und Potentiale
1300115
Flexible precision control and dynamic data structures for programming mathematical and numerical algorithms