Narrow Results

We prove that the controlled continuity, induced by controlled convergence, and continuity in norm are equivalent for linear operators defined on the space of Henstock integrable vector-valued functions.

The Schrödinger equation is fundamental in quantum mechanics as it makes it possible to determine the wave function from energies and to use this function in the mean calculation of variables, for example, as the most likely position of a group of one or more massive particles. In this paper, we present a survey on some theories involving the Schrödinger equation and the Feynman path integral. We also consider a Feynman–Kac-type formula, as introduced by Patrick Muldowney, with the Henstock integral in the description of the expectation of random walks of a particle. It is well known that the non-absolute integral defined by R. Henstock “fixes” the defects of the Feynman integral. Possible applications where the potential in the Schrödinger equation can be highly oscillating, discontinuous or delayed are mentioned in the end of the paper.

In the present paper, we consider an Hammerstein integral inclusion, where the set-valued integral involved is of Henstock-type. An existence result is obtained via Mönch's fixed point theorem, imposing a condition using a measure of noncompactness, as well as some uniform integrability conditions appropriate to Henstock integral. The Henstock integral is more general than classical integrals, therefore our result extends a large number of existence results in literature, given in single- or set-valued setting.