generally leads to a functional differential equation for f(x) and an associated inequality. It is only occasionally that the equation forf(x) reduces to a differential equation but it may sometimes reduce to a simple functional equation or difference equation. The analysis is easily extended by replacing the substitution z = x + r by some other substitution z = s(x) which transforms the range ...

The exact exchange potential in time-dependent density-functional theory is defined as an orbital functional through the time-dependent optimized effective potential (TDOEP) method. We numerically solve the TDOEP integral equation for the real-time nonlinear intersubband electron dynamics in a semiconductor quantum well with two occupied subbands. It is found that memory effects become signific...

This note studies the martingale property of a nonnegative, continuous local martingale Z, given as a nonanticipative functional of a solution to a stochastic differential equation. The condition states that Z is a (uniformly integrable) martingale if and only if an integral test of a related functional holds.

Various singular integral inequalities play an important role in the development of the theory of differential equations, functional differential equations, and integral equations. For example, Henry 1 proposed a linear integral inequality with singular kernel to investigate some qualitative properties for a parabolic differential equation, and Sano and Kunimatsu 2 gave a modified version of He...

We study the distribution of the exponential functional I(ξ, η) = ∫∞ 0 exp(ξt−)dηt, where ξ and η are independent Lévy processes. In the general setting, using the theory of Markov processes and Schwartz distributions, we prove that the law of this exponential functional satisfies an integral equation, which generalizes Proposition 2.1 in [9]. In the special case when η is a Brownian motion wit...