We study conditions of discreteness of spectrum of the functional-differential operator $$ℒu=-u^{\text{'}\text{'}}+p\left(x\right)u\left(x\right)+{\int }_{-\infty }^{\infty }(u\left(x\right)-u\left(s\right)){\mathrm{d}}_{s}r(x,s)$$ on $(-\infty ,\infty )$. In the absence of the integral term this operator is a one-dimensional Schrödinger operator. In this paper we consider a symmetric operator with real spectrum. Conditions of discreteness are obtained in terms of the first eigenvalue of a truncated operator. We also obtain one simple condition for discreteness of spectrum.

We prove the existence of monotone solutions, of the functional differential inclusion ẋ(t) ∈ f(t,T(t)x) +F(T(t)x) in a Hilbert space, where f is a Carathéodory single-valued mapping and F is an upper semicontinuous set-valued mapping with compact values contained in the Clarke subdifferential ${\partial }_{c}V\left(x\right)$ of a uniformly regular function V.