Tyshchuk T. Coexistence periodic piecewise constant solutions of nonlinear boundary value problems for linear differential equations of the first order

Definitions of model of cycle type and weight of concave circular permutations are formulated. These notions are used to describe cycles of unimodal concave map and they also help to solve a problem of classification of continuous maps by their models of a cycle type. In the space of concave circular permutations we described a linear order relation induced by weight of concave circular permutation. The set of cycles of any continuous maps with -schema are described. Nonlinear boundary value problems for first-order linear partial differential equations are studied. Notion of generalized piecewise constant periodic solution of the problems are proposed. The studying of generalized piecewise constant periodic solutions is done by reducing the nonlinear boundary value problem to difference equation with continuous time. Boundary conditions and initial conditions provide a reduction of resulting difference equation with continuous time to difference equation with discrete argument. The notion of type of the generalized piecewise constant periodic solution is proposed and the problem of coexistence of such solutions by their types is studied. Generalized piecewise constant periodic solutions for a certain class of nonlinear boundary value problems are obtained in exact form.