On Darboux-Type Differential Inclusions with Uncertainty

In this article we prove the existence results for solutions of the Darboux-type problems in fuzzy partial differential inclusions with local conditions of integral types. We present two problems involving open and closed level sets of a given fuzzy mapping. In the first case fuzzy differential inclusion has been transformed into an equivalent Darboux-type problem for partial differential equations and then using the Tychonoff fixed point theorem we prove the existence result for this crisp case. For the second case we use Nadler’s fixed point theorem and selection theorem of Kuratowski-Ryll-Nardzewski to find the solution of given differential inclusions problem. We furnish an example to validate our results.

On Some Boundary Value Problems for an Ultrahyperbolic Equation

The correct formulation of a characteristic problem and a Darboux type problem in the special weighted functional spaces for an ultrahyperbolic equation is investigated.

Spatial Problem of Darboux Type for One Model Equation of Third Order

For a hyperbolic type model equation of third order a Darboux type problem is investigated in a dihedral angle. It is shown that there exists a real number ρ0 such that for α > ρ0 the problem under consideration is uniquely solvable in the Frechet space. In the case where the coefficients are constants, Bochner's method is developed in multidimensional domains, and used to prove the uniquely solvability of the problem both in Frechet and in Banach spaces.

On a Darboux Type Multidimensional Problem for Second-Order Hyperbolic Systems

The correct formulation of a Darboux type multidimensional problem for second-order hyperbolic systems is investigated. The correct formulation of such a problem in the Sobolev space is proved for temporal type surfaces on which the boundary conditions of a Darboux type problem are given.

On A Darboux Problem for A Third Order Hyperbolic Equation with Multiple Characteristics

A Darboux type problem for a model hyperbolic equation of the third order with multiple characteristics is considered in the case of two independent variables. The Banach space , α ≥ 0, is introduced where the problem under consideration is investigated. The real number α 0 is found such that for α > α 0 the problem is solved uniquely and for α < α 0 it is normally solvable in Hausdorff's sense. In the class of uniqueness an estimate of the solution of the problem is obtained which ensures stability of the solution.