121170503
Solution of the stability problem for a class of generalized volterra prey-predator systems☆
where ei,pij, and qj are real constants, ui are real-valued continuous functions of t for t > 0, and zii = dui/dt. Both equations are required for i = 1, 2,..., m. Condition (lb) is here considered to be a restriction on the vector e = (ei) rather than on the matrix p = (pii) as is more usual. The hypothesis ~~(0) > 0, qi > 0 is appropriate in applications but as far as the mathematical development is concerned, it could be replaced by qiui(0) > 0. A discussion of historical background can be found in the papers [4,5], to which the present paper is a sequel. Suffice it to say here that the chief landmarks are: Volterra’s book [6], La Salle’s theorem [ 1, 21, and Krikorian’s analysis of the case n = 3 [3]. References [4,5] provide useful
123447202
Parabolic differential equations and inequalities with several time variables
On presente une theorie des inegalites differentielles lineaires correspondant a l'equation parabolique: Σ i=1 m d i (t,x)u td i=Σ j,k=1 n a jk (t,x)u xd j xd k+Σ j=1 n b j (t,x)u xd j+C(t,x)u
119084957
On Max Muller's existence-comparison theorem for infinite systems of ordinary differential equations
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.
119881926
Fehlerabschätzungen und Eindeutigkeitssätze für gewöhnliche und partielle Differentialgleichungen
124539695
The line method for parabolic differential equations problems in boundary layer theory and existence of periodic solutions
The (longitudinal) line method for parabolic differential equations is considered, where the space variable only is discretized. An initial-boundary value problem for a parabolic differential equation is transformed into an initial value problem for a system of ordinary differential equations of first order. Using the theory of ordinary and parabolic differential inequalities, constructive existence proofs for nonlinear problems by means of the line method can be given. Recent work on several problems in boundary layer theory along these lines is described. The main part of this exposition deals with periodic solutions of the equation ut=uxx+f(t,x,u) (periodic means periodic in t, f is assumed to be periodic). Under suitable conditions on f, there exists exactly one periodic solution u. It is shown that for the corresponding line method approximation, there exists also exactly one periodic solution. The approximations converge uniformly to u, their t-derivatives converge uniformly to ut, their first and second order differences in the x-direction converge uniformly to ux and uxx, respectively.
1300115
Flexible precision control and dynamic data structures for programming mathematical and numerical algorithms