Showing papers in "Journal of Differential Equations in 1970"

TL;DR: In this paper, the authors give sufficient conditions for the stability and instability of solutions of a large class of equations (1.1) in terms of functions similar to those occurring in the application of the second method of Liapunov to ordinary and functional differential equations of retarded type.

TL;DR: In this paper, it was shown that for a nonlinear one-dimensional differential delay equation a(t) = F;(t, x1(.)) (DDE) one can frequently determine (almost by inspection) if the 0 solution is asymptotically stable and give a region of attraction.

TL;DR: In this article, it was shown that the RODE (2.2) holds for all ω ∈ Ω, by restricting Ω to a subset of full probability if necessary, and that g is infinitely often continuously differentiable in its variables, although k-times continuous differentiable with k sufficiently large would suffice.

TL;DR: In this article, the uniqueness of solutions of a particular k-point boundary value problem implies the existence of solutions for any assignment of the boundary values cij, where 2 < k < n, x, < k, 0 < i < mi 1.

TL;DR: In this article, the conditions given in the first section below, which are suf%cient for existence of periodic solutions, are known to be important for a different type of boundary value problem.

TL;DR: In this article, the authors established a sufficient condition such that the equation Au + g(u) = h has a weak solution u with generalized Dirichlet data zero where A is any self-adjoint strongly and uniformly elliptic linear partial differential operator with real-valued, reasonably smooth coefficients defined on a bounded open set Omega of E superscript n.

TL;DR: In this paper, an invariance principle similar to the theorem of LaSalle for autonomous dynamical systems was proposed for infinite dimensional spaces suitable for partial differential equations, which is a generalization of Liapunov's theorem for dynamical system invariance.

TL;DR: Particle motions in magnetic fields, as related to charged particle trapping in Van Allen belt are discussed in this article, where the authors present a model of particle motion in magnetic field.

TL;DR: In this paper, the authors studied bounded quadratic systems, which have all of their trajectories bounded for t > 0, and showed that any such system will have a rest point in the plane and by translating the origin to that rest point it will have the form

TL;DR: In this paper, the basic theory of existence, uniqueness, and continuation for neutral functional differential equations is presented, based on the theory of uniqueness and continuation of functional differential equation (FDE).

TL;DR: In this paper, the concept of a differential game with a finite duration T for the system (l.l), (1.2) with payoff 1.7; y = y(t), and z = z(t) are the control functions of the players y and x.

TL;DR: In this paper, the authors introduce the concepts of stable, weak attraction, attraction, strong attraction, and saddle and nonsaddle sets for arbitrary subsets of the phase space of a dynamical system defined on a Hausdorff space.