No abstract

Abstract. We prove an existence theorem for the Darboux Problem uxy (x, y) = g (u (x, y)), u (x, 0) = u (0, y) = 0, where g is bounded and measurable.

Abstract. In this paper we present new identities for the Radić's determinant of a rectangular matrix. The results include representations of the determinant of a rectangular matrix as a sum of determinants of square matrices and description how the determinant is affected by operations on columns such as interchanging columns, reversing columns or decomposing a single column.

The existence of a local solution to the Darboux problem u_xy(x,y)=g(u(x,y)), u(x,0)=u(0,y)=0, where g is Lebesgue measurable and has at most polynomial growth, is proved

In this paper we prove that d-boundaries D d = {x : dist(x, Z) = d} of a compact Z ⊂ R 2 are closed absolutely continuous curves for d greater than some constant depending on Z. It is also shown that D d is a trajectory of solution to the Cauchy Problem of a differential equation with a discontinuous right-hand side. 2000 Mathematics Subject Classification. 57N05, 34A36. Key words and phrases. d-boundary, absolutely continuous curve, differential equation with discontinuous right-hand side.