Advances in Pure and Applied Mathematics

APAM - ISSN 1869-6090 - © ISTE Ltd

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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.

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.

The purpose of this paper is to give some properties for the so-called ε-pseudo weakly demicompact linear operators acting on Banach spaces. Some sufficient conditions on the entries of an unbounded 2 × 2 block operator matrix $$$\mathcal{L}_{0}$$$ ensuring its ε-pseudo weak demicompactness are provided. In addition, we develop, in the bounded case, the class of ε-pseudo Fredholm perturbation to investigate the essential pseudo-spectra of $$$\mathcal{L}_{0}$$$. The results are formulated in terms of some denseness conditions on the topological dual space.

In this paper, we conduct a mathematical analysis of a tumor growth model with treatments. The model consists of a system that describes the evolution of metastatic tumors and the number of cells present in the primary tumor. The former evolution is described by a transport equation, and the latter by an ordinary differential equation of Gompertzian type. The two dynamics are coupled through a nonlocal boundary condition that takes into account the tumor colonization rate. We prove an existence result where the main difficulty is to handle the coupling and to take into account the time discontinuities generated by treatment terms. The proof is based on a Banach fixed point theorem in a suitable functional space. We also develop a computational code based on the method of characteristics and present numerical tests that highlight the effects of different therapies.

In this paper, we reconsider the notion of a Weyl p-almost automorphic function introduced by S. Abbas [1] in 2012 and propose several new ways for introduction of the class of Weyl p-almost automorphic functions (1 ⩽ p < ∞). We first analyze the introduced classes of Weyl p-almost automorphic functions of type 1, jointly Weyl p-almost automorphic functions and Weyl p-almost automorphic functions of type 2 in the one-dimensional setting. After that, we introduce and analyze generalizations of these classes in the multi-dimensional setting, working with general Lebesgue spaces with variable exponents. We provide several illustrative examples and applications to the abstract Volterra integro-differential equations.

In this paper, we investigate the existence of at least three weak solutions for a class of nonlocal elliptic equations with Navier boundary value conditions. The proof of our result uses the basic theory and critical point theory of variable exponential Lebesgue Sobolev spaces. Moreover a generalization of Corollary 1.1 in [21] is obtained.

The main purpose of this paper is to study unital ring homomorphisms of associative rings $$$\varphi : R\rightarrow S$$$ satisfying one of the following conditions: (a) the unit-preserving property: $$$\varphi(R^{\times})=S^{\times}$$$ and (b) the inverse unit-preserving property: $$$\varphi^{-1}(S^{\times})=R^{\times}$$$.
We establish the relationship between these two conditions. Several characterizations of such conditions are settled. An application to the index of unit groups of rings $$$R\subset S$$$ having a nonzero common ideal is given.