In this study we construct new integral representations of Jost-type solutions the quadratic pencil Sturm-Liouville equation with piece-wise constant coefficient on entire real line. Our aim is to express special in form some operators which kernels related potential function equation. This problem technically diffucult due discontinous causes kernel also have a jump discontinuity.

The asymptotic behaviour of a Sturm-Liouville differential equation with coefficient λq(s), s ∈ [s0,∞) is investigated, where λ ∈ R and q(s) is a nondecreasing step function tending to ∞ as s → ∞. Let S denote the set of those λ’s for which the corresponding differential equation has a solution not tending to 0. It is proved that S is an additive group. Four examples are given with S = {0}, S =...

We prove a Filippov-Gronwall type inequality for solutions of a nonconvex secondorder differential inclusion of Sturm-Liouville type.

This paper develops a spectral expansion approach to the valuation of contingent claims when the underlying state variable follows a one-dimensional diffusion with the infinitesimal variance a(x), drift b(x) and instantaneous discount (killing) rate r(x). The Spectral Theorem for self-adjoint operators in Hilbert space yields the spectral decomposition of the contingent claim value function. Ba...