Vol 15 - Issue 1 (January 2024)

Advances in Pure and Applied Mathematics

List of Articles

In the paper, we introduce the notion of compression of generalized slant Toeplitz operators to the Hardy space of $$$n$$$-dimensional torus $$$\mathbb{T}^n$$$. It deals with characterizations of introduced operator with specific as well as general symbols. Certain algebraic and structural properties of considered operators are also investigated. Finally, we discuss few results related to essentially $$$k^{th}$$$-order $$$\lambda$$$-slant Toeplitz operator.

In this paper, we study the commutativity of 3-prime near-rings satisfying some differential identities on Jordan ideals involving certain additive maps. Some well-known results characterizing the commutativity of 3-prime nearrings by derivations and left multipliers are extended to right multipliers and left generalized derivations. Furthermore, an example is given showing the necessity of the 3-primeness mentioned in the assumptions of our theorems is given.

This paper studies the asymptotic behavior of energy solutions to the focusing non-linear generalized Hartree equation
$$$i u_t+\Delta u=-｜x｜^{-\varrho}｜u｜^{p-2}(\mathcal J_\alpha *｜\cdot｜^{-\varrho}｜u｜^p)u,\quad \varrho>0,\quad p\geq2.$$$
Here, $$$u:=u(t,x)$$$, where the time variable is $$$t \in ℝ$$$ and the space variable is $$$x\inℝ^2$$$.
The source term is inhomogeneous because $$$\varrho > 0$$$. The convolution with the Riesz-potential $$$\mathcal J_\alpha:=C_\alpha｜\cdot｜^{\alpha-2}$$$ for certain $$$0 < \alpha < 2$$$ gives a non-local Hartree type non-linearity. Taking account of the standard scaling invariance, one considers the inter-critical regime $$$1 + \frac{2-2\varrho + \alpha}2 < p < \infty$$$. It is the purpose to prove the scattering under the ground state threshold. This naturally extends the previous work by the first author for space dimensions greater than three (Scattering Theory for a Class of Radial Focusing Inhomogeneous Hartree Equations, Potential Anal. (2021)). The main difference is due to the Sobolev embedding in two space dimensions $$$H^1(ℝ^2)\hookrightarrow L^r(ℝ^2)$$$, for all $$$2 \leq r < \infty$$$. This makes any exponent of the source term be energy subcritical, contrarily to the case of higher dimensions. The decay of the inhomogeneous term $$$｜x｜^{-\varrho}$$$ is used to avoid any radial assumption. The proof uses the method of Dodson-Murphy based on Tao’s scattering criteria and Morawetz estimates.

This article is devoted to explore various forms of transcendental entire solution of different quadratic trinomials generated by first order linear c-shift operator. We also investigate the forms of solutions of certain quadratic trinomials under linear and mixed partial differential operators. Our paper improves the results of Li-Xu [Axioms, 126(10)(2021), 1-19] in two directions. In addition, in a corollary, deducted from one of our main result, we extend a result of Zhang et al. [ Aims Math., 7(2022), 11597-11613]. A series of examples have been exhibited to justify the existence and forms of transcendental entire solution of such equations. In the last section of the paper we have put a relevant question for future research.