120951143
Verallgemeinerte Laplace-Operatoren und Potentiale
121877895
Ordinary differential inequalities in ordered Banach spaces
x <y+xi -=zyi for i = I,..., n. The function f (t, x) defined in a set D C J x Rn and with values in Rn, is said to be quasimonotone increasing in x (in short quasimonotone), if each component fi( t, x1 , . . . , xn) is weakly increasing in xj for i # j; more exactly, if (t, 4, (t, y) E D, x < y, xt = yi implies fi(t, x) 6 fi(t, y) (2) (i = l,..., n). (A) Let f (t, x) be quasimonotone increasing in x. Let v, w be continuous in. J and d$%rentiable in Jo and let (4 40) < 4% (b) v’ -f (t, v) < w’ -f (t, w) in Jo.
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.
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
119458260
Inequalities involving derivatives
73669991
Comparison theorems for parabolic functional inequalities.
1300115
Flexible precision control and dynamic data structures for programming mathematical and numerical algorithms