Strongly Unimodal Distributions of Measure on $$\boldsymbol{\mathbb{R}}_{\boldsymbol{+}}$$ with Smoothness Condition of Densities

The problem of describing the class of non-negative densities $$v$$ of finite measure distributions on $$\mathbb{R}_{+}$$ , which have the property of strong unimodality according to I.A. Ibragimov, is solved. It is based on the study of properties of integral transformation corresponding to the convolution of distributions. The approach to unimodality investigation based on the Volterra integral equation of first kind with a difference kernel is proposed. It allows to describe the class of all such densities. For this, it is used the smooth parametrization by means of $$\lambda\in(0,\infty)$$ connected with the set $$\mathfrak{D}(\mathbb{R}_{+})$$ of non-negative unimodal densities $$f$$ on $$\mathbb{R}_{+}$$ satisfying the equation. This makes it possible to select the entire class of all unimodal densities those that have the property of strong unimodality. It is based on the requirement that solutions to the equation have no some bifurcations on the parameter $$\lambda$$ .