Volume 6, Issue 1

R. Feynman formulated quantum mechanics in terms of integrals over spaces of paths (Feynman path integrals). But the absolute value of Feynman's integrand is not integrable. And his integrand does not generate a measure. So Lebesgue integration theory could not be used by Feynman. To establish the equivalence of his theory with the traditional formulation of quantum mechanics, Feynman gave an argument that his path integral satisfies Schrödinger's equation. This paper gives a proof of this part of Feynman's theory. To justify Feynman's and other investigators' use of the language and concepts of integration and probability theory, and to justify taking limits under the integral sign in Feynman's integral, we use R. Henstock's approach to non-absolute integration, which does not require the measure concept, and for which the absolute value of the integrand need not be integrable.

It has recently been established that any Baire class one function ƒ : [0, 1] → ℝ can be represented as the pointwise limit of a sequence of polygonal functions whose vertices lie on the graph of ƒ. Here we investigate the subclass of Baire class one functions having the additional property that for every dense subset D of [0, 1], the first coordinates of the vertices of the polygonal functions can be chosen from D .

Using the concept of the normal cone to a multifunction we define a derivative for a complex-valued multifunction of one complex variable being a natural generalization of the ordinary complex derivative for holomorphic functions. Using results obtined by Mordukhovich, we develop a full calculus and discuss openness and Lipschitzian properties. We also prove the fundamental theorem of calculus and the Taylor expansion formula. Finally we discuss analyticity of multifunctions in the context of the normal cone.

A theorem on the existence of separable supports of σ -finite Borel measures given on metric spaces with small topological weights is applied to constructions of certain analogues of special subsets of the real line.

In this paper we investigate the oscillatory character of the second order nonlinear difference equations of the forms Δ( c n –1 Δ( x n –1 + p n x σ n )) + q n ƒ( x τ n ) = 0, n = 1, 2, . . . and the corresponding nonhomogeneous equation Δ( c n –1 Δ( x n –1 + p n x σ n )) + q n ƒ( x τ n ) = r n , n = 1, 2, . . . via comparison with certain second order linear difference equations where the function ƒ is not necessarily monotonic. The results of this paper are essentially new and can be extended to more general equations.

This paper is concerned with the existence of global limit solutions for the quasi–nonlinear functional evolution problem where A ( t , ψ 1 ) and G ( t , ψ 1 , L t ψ 2 ) are defined, with respect to ψ 1 , on a subspace of the space PC ([– r , 0], X ) of all piecewise continuous functions ƒ : [– r , 0] → X . An appropriate subspace of PC ([– r , t ], X ) is the domain of definition of the nonlinear operators L t , t ∈ [0, T ]. The operators A ( t , ψ ) x are w -dissipative and Lipschitz — like in ( t , ψ ) which are more general conditions than those of Karsatos–Liu. The operators G and L t are Lipschitzian mappings on their respective domains. Moreover, we investigate the uniqueness and strong solution for such problem.

We prove here the existence of a bounded, radial solution in unbounded domain of the nonlinear elliptic problem under some asymptotic and sign condition on ƒ. Under stronger assumptions it is proved that this solution must be of constant sign. The existence of radial solutions, vanishing at ∞, of some semilinear equation is also established here.

We present a new version of first order necessary optimality conditions for a static minmax problem with inequality constraints in the parametric constraint case. These conditions, after some modification, turn out to characterize strict local minimizers of order one for the given problem.

A functional equation related to a problem of linear dependence of iterates is considered.