Weakly injective and weakly projective modules, which are extensions of the classes of quasi-injective modules and quasi-projective modules respectively, are studied. Descriptions are obtained for weakly injective and weakly projective Abelian groups, as well as for some classes of Abelian groups closely related to them.

In a Banach space, two nonlocal problems are studied for a functional-differential equation that generalizes the Euler-Poisson-Darboux equation, with the Erdelyi-Kober operator appearing in additional nonlocal conditions. By reducing the problems to operator equations, conditions for their unique solvability are established; these conditions are imposed on the operator coefficient of the equation and on the nonlocal data. The solution is expressed in terms of the Bessel and Struve operator functions introduced by the author. Examples are provided.

The aim of this paper is twofold. First, we establish the ^-transform characterizations of the Besov spaces B_p,_q (Rⁿ, {t_k}) and the Triebel-Lizorkin spaces F_p,_q (Rⁿ, {t_k}) for q = to, in the sense of Frazier and Jawerth. Second, under some suitable assumptions on a padmissible weight sequence {t_k}, we prove that Ap,q (Rⁿ, {t_k }) = Ap,q (Rⁿ,t_j), j ϵ Z, in the sense of equivalent quasi-norms, with A ϵ {B,F}. Moreover, we find a necessary and sufficient condition for the coincidence of the spaces A_p,_q (Rⁿ,t_i), i ϵ {1, 2}.

In the Tricomi problem, the value of the desired function is specified at all points of the boundary characteristic. In this paper, we study the correctness of the problem where part of the boundary characteristic is freed from the boundary condition and this missing Tricomi condition is replaced by a nonlocal condition with displacement on the internal characteristic and on the part of the boundary characteristic. On the degeneracy segment, a general conjugation condition is specified.