We define a general Franklin system of functions on R with vanishing means, generated by an admissible sequence T. A necessary and sufficient condition on T is found for the corresponding general Franklin system of functions on R with vanishing means to be an unconditional basis in the space H 1(R).

The notion of a Mazur-Ulam space, introduced by C. P. Niculescu in [6] by using the midpoints, is extended here for an arbitrary weight λ ∈ (0, 1). A similar characterization in terms of a class of isometries and their unique fixed point is obtained for the rational case λ = m/n and under more complicated conditions than that of in [6] or [7, p. 166].

The paper is devoted to the characterization of hyperidentities of the variety of weakly idempotent lattices that is nilpotent closure of the variety of lattices. The existence of a finite base for such hyperidentities is established.

The paper considers the following differential equation \(x'\left( t \right) + \sum\nolimits_{i = 1}^m {{p_i}\left( t \right)x\left( {{\tau _i}\left( t \right)} \right) = 0} \), t ≥ 0, where pi ∈ Lloc(R+;R+), τi ∈ C(R+;R), τi(t) ≤ t and \(\mathop {\lim }\limits_{t \to + \infty } \tau _i (t) = + \infty ,i = 1,...,m\). New oscillation criteria of all solutions for this equation are established.

The original proof of the Littlewood conjecture was a special case of a more general inequality of functions whose Fourier coefficients have gaps. In this article, we prove similar inequalities, but treating the Fourier transform of a function integrable on the real line, rather than on the unit circle.

The paper is devoted to the solution of two quadratic ρ-functional inequalities and the Hyers-Ulam stability of these inequalities. Using the direct method, we prove the Hyers-Ulam stability of these inequalities in non-Archimedean Banach spaces. The Hyers-Ulam stability of the quadratic ρ-functional equations associated with these inequalities is also established in non-Archimedean Banach spaces.

We characterize symbols h for which the corresponding Toeplitz operators are bounded on weighted Besov spaces B p (ω) of holomorphic functions on the polydisk for some 1 ≤ p < ∞.

We consider Dirichlet problems in a bounded domain Q ⊂ R n for a general secondorder elliptic equation with the boundary function in L 2. In the author’s previous papers necessary and sufficient conditions for the existence of an (n − 1)-dimensionally continuous solution were obtained under some natural assumptions on the coefficients of equation. Those assumptions are formulated in terms of an auxiliary operator equation in a special Hilbert space and are difficult to verify. In the present paper we obtain necessary and sufficient conditions for the existence of a solution in terms of the original problem for a more narrow class of the right-hand sides. It is shown that if, in addition, the boundary function is assumed to be in the space W 2 1/2 (∂Q), then the obtained conditions transform into solvability conditions in the space W 2 1 (Q).

In this paper we introduce the notions of an analytic curve and equivalent points on the Bohr-Riemann surfaces. By means of constructive and algebraic methods we prove that the points of the Bohr-Riemann surfaces locally have the same number of equivalent points.

In this paper 2n-th order p-Laplacian difference equations are considered. Using the critical point method, we establish various sufficient conditions for the existence and nonexistence of solutions for Dirichlet boundary value problem. Recent results in the literature are generalized and significantly complemented, as well as, some new results are obtained.