The paper deals with the existence of viable solutions to the differential inclusion ẍ(t) ∈ f(t,x(t)) + ext F(t,x(t)), where f is a single-valued map and ext F(t,x) stands for the extreme points of a continuous, convex and noncompact set-valued mapping F with nonempty interior.
We show the existence result of viable solutions to the second-order differential inclusion ẍ(t) ∈ F(t,x(t),ẋ(t)), x(0) = x₀, ẋ(0) = y₀, x(t) ∈ K on [0,T], where K is a closed subset of a separable Banach space E and F(·,·,·) is a closed multifunction, integrably bounded, measurable with respect to the first argument and Lipschitz continuous with respect to the third argument.
Let (X,T) be a paracompact space, Y a complete metric space, a lower semicontinuous multifunction with nonempty closed values. We prove that if is a (stronger than T) topology on X satisfying a compatibility property, then F admits a -continuous selection. If Y is separable, then there exists a sequence of -continuous selections such that for all x ∈ X. Given a Banach space E, the above result is then used to construct directionally continuous selections on arbitrary subsets of ℝ × E.
In the current work a generalization of the famous Weyl-Kodaira inversion formulas for the case of self-adjoint differential vector-operators is proved. A formula for spectral resolutions over an analytical defining set of solutions is discussed. The article is the first part of the planned two-part survey on the structural spectral theory of self-adjoint differential vector-operators in matrix Hilbert spaces.
We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.
We investigate the quadratic homogeneous holomorphic vector fields on that are semicomplete, this is, those whose solutions are single-valued in their maximal definition domain. To a generic quadratic vector field we rationally associate some complex numbers that turn out to be integers in the semicomplete case, thus showing that the linear equivalence classes of semicomplete vector fields are contained in some sort of lattice in the space of linear equivalence classes of quadratic ones. We prove...
This article continues earlier work of the author on non-linear systems of ordinary differential equations, published in Asymptotic Analysis 15 (1997), MR no. 98g:34015b. There, a completely formal theory was presented, while here we are concerned with a semi-formal approach: Solutions of non-linear systems of ordinary meromorphic differential equations are represented as, in general divergent, power series in several free parameters. The coefficients, aside from an exponential polynomial, a general...
We study the Hamilton-Jacobi equation of the minimal time function in a domain which contains the target set. We generalize the results of Clarke and Nour [J. Convex Anal., 2004], where the target set is taken to be a single point. As an application, we give necessary and sufficient conditions for the existence of solutions to eikonal equations.