Volume 2, Issue 5

This paper continues the investigation of weight problems in Orlicz classes for maximal functions and singular integrals defined on homogeneous type spaces considered in [Gogatishvili and Kokilashvili, Georguian Math. J. 2: 361–384, 1995].

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.

A method of finding the so-called Liouville's type formulas for the number of representations of integers by quadratic forms is developed.

Statements connected with the so-called block-orthonormalized systems are given. The convergence and summability almost everywhere by the ( c , 1) method with respect to such systems are considered. In particular, the well-known theorems of Menshov-Rademacher and Kacmarz on the convergence and (c,1)-summability almost everywhere of orthogonal series are generalized.

Two integral operators P α and for analytic functions in the open unit disk are introduced. The object of the present paper is to derive some properties of integral operators P α and .

Certain fixed point theorems are established for nonlinear semigroups of Lipschitzian mappings defined on nonconvex domains in Hilbert and Banach spaces. Some known results are thus generalized.