We study Ricci solitons and $$\ast$$ -Ricci solitons on generalized Sasakian-space-forms (GSSF) $$M^{2n+1} (f_1, f_2, f_3)$$ with parallel $$\ast$$ -Ricci tensor. We find that if GSSF $$M^{2n+1} (f_1, f_2, f_3)$$ with Kenmotsu metric admits a Ricci soliton or a $$\ast$$ -Ricci soliton, then $$f_1=-1$$ and $$f_2=f_3=0$$ . Moreover, the Ricci soliton is expanding, and the $$\ast$$ -Ricci soliton is steady. Further, we provide some examples.

On a uniform grid on the real axis $$\mathbb R$$ , we study the Yanenko–Stechkin–Subbotin problem of extremal function interpolation in the mean in the space $$L_1(\mathbb R)$$ of two-way real sequences with the least value of the norm of a linear formally s This problem is considered for the class of sequences for which the generalized finite differences of order $$n$$ corresponding to the operator $$\mathcal L_n$$ are bounded in the space $$l_1$$ . In this paper, the least value of the norm is calculated exactly if the grid step $$h$$ and the averaging step $$h_1$$ of the function to be interpolated in the mean are related by the inequalities $$h<h_1\le 2h$$ . The paper is a continuation of the research by Yu. N. Subbotin and the author in this problem, initiated by Yu. N. Subbotin in 1965. The result obtained is new, in particular, for the $$n$$ -times differentiation operator $$\mathcal L_n(D)=D^n$$ .

For a certain class of second-order hyperbolic systems, a mixed problem with Dirichlet and Poincaré boundary conditions is studied. In the linear case, an explicit representation of a soultion of this problem is given and questions related to its uniqueness and existence are studied depending on the character of nonlinearities in the system.

By adopting the method of upper and lower solutions, this article shows the blow-up rate of the unique nonnegative viscosity solution $$l(t)$$ of the boundary value problem $$(u'(t))^{2}u''(t) =b(t)f(u(t)), \quad u(t)>0, \quad t>0, \qquad u(0)=\infty, \quad u(\infty)=0,$$ where $$b\in C^{1}(0,\infty)$$ , which is positive and nondecreasing on $$(0,\infty)$$ (and may vanish at zero).

We present a new approach to the definition of Grand and Small discrete Lebesgue spaces. Previously, we developed such an approach in the continuous case. The fundamental difference of our studies is that we base the definition of norms on extrapolation theory, and our approach permits including the extreme cases $$p=1$$ and $$p=\infty$$ into consideration, which is the main result of the study presented here. The Small space for $$p=\infty$$ is realized as the sum $$l^{s}(\xi)\equiv \sum _{p \in [2, \infty)} \xi(p) l^{p}$$ of spaces with a certain fundamental function $$\xi$$ , and the Grand space $$l^{g}(\zeta)$$ for $$p=1$$ is given by the product $$l^{g}(\zeta)\equiv \bigcap_{p \in (1, 2]} \zeta(p) l^{p}$$ of spaces with a certain fundamental function $$\zeta$$ . As one of the main results, we show that if the function $$\xi$$ satisfies the $$\Delta_2$$ (doubling) condition, then the space $$l^{s}(\xi)$$ coincides, up to norm equivalence, with the discrete Lorentz space $$\lambda(\psi)$$ , where $$\psi(k) \simeq \xi(1/\ln k)$$ . We also show that if the function $$\zeta$$ satisfies the $$\Delta_2$$ condition, then the space $$l^{g}(\zeta)$$ coincides, up to norm equivalence, with the discrete Marcinkiewicz space $$m(\psi)$$ , where $$\psi(k) \simeq k/(\zeta(1/\ln k))$$ . We expect that our general new construction of the norms in Grand and Small discrete Lebesgue spaces will imply further studies of the spaces and operators in these spaces in such a general setting.

An operator $$T$$ in a Banach space $$X$$ is said to be recurrent if the set $$\{x\in X:\ x\in \overline{O(T,Tx)}\}$$ is dense in $$X$$ . The operator $$T$$ is said to be weakly sequentially recurrent if the set $$\{x\in X:\ x\in \overline{O(T,Tx)}^w\}$$ is weakly dense in $$X$$ . Costakis et al. [Complex Anal. Oper. Theory 8 (8), 1601–1643] ask if $$T\oplus T$$ should be recurrent whenever so is $$T$$ . This question has been answered negatively by Grivaux et al. [arXiv: 2212.03652]. In this paper, we prove the existence of an operator $$T$$ weakly sequentially recurrent such that $$T\oplus T$$ is not.

We generalize Romanoff’s theorem. Also, we obtain a result on sums related to Euler’s totient function.

The paper studies the regularity properties of images of Fomin differentiable measures under mappings in Sobolev classes. The obtained results generalize recently obtained bounds in Gaussian case to more abstract settings.

We obtain a power series representation of the Weierstrass sigma function arising in the theory of elliptic functions and derive a differential–recurrence equation as well as closed-form expressions for its coefficients. These results are obtained using the Weierstrass system of partial differential equations satisfied by the sigma function.