Abstract

We study the existence of three weak solutions to the Dirichlet boundary condition for a $p(x)$-Laplacian equation. Using a variational method and the three critical point theorem, we would show the existence and multiplicity of the solutions. For this purpose, we focus on a generalized variable exponent Lebesgue-Sobolev space.

Keywords: $p(x)$-Laplacian; variational method; three solutions; Sobolev space.

MSC: 35A15, 35J35, 46E35

Abstract

Let $\Omega$ be a group with identity $e$, $\Gamma$ be a $\Omega$-graded commutative ring and $\Im$ a graded $\Gamma$-module. In this article, we introduce the concept of $gr$-$C$-$2^{A}$-secondary submodules and investigate some properties of this new class of graded submodules. A non-zero graded submodule $S$ of $\Im$ is said to be a $gr$-$C$-$2^{A}$-secondary submodule if whenever $r,s \in h(\Gamma)$, $L$ is a graded submodule of $\Im$, and $rs\,S\subseteq L$, then either $r\,S\subseteq L$ or $s\,S \subseteq L$ or $rs \in Gr(Ann_\Gamma(S))$.

Keywords: Graded classical 2-absorbing secondary submodules; graded 2-absorbing submodules; graded 2-absorbing primary submodules.

MSC: 13A02, 16W50

Abstract

In this paper, we define generalized $\varphi-$quasi contraction map which is more general than strict quadratic quasi contraction map by using an altering distance function $\varphi$ and prove the existence and uniqueness of fixed points of these maps satisfying asymptotically regular property in the setting of complete metric spaces. We extend these results to $T$-orbitally complete metric spaces. Examples are provided to illustrate our results. Our results generalize Theorem 4 of [O. Popescu, G. Stan, Some fixed point theorems for quadratic quasi contractive mappings, Symmetry, 11 (2019)].

Keywords: Fixed point; strict quadratic quasi contraction map; altering distance function; generalized $\varphi-$quasi contraction map; $T$-orbitally complete metric space.

MSC: 47H10, 54H25

Abstract

Let $\mathcal{O}_K$ be the ring of integers of a number field $K$, and let $\mathrm{Int}(\mathcal{O}_K)=\{R\in K[X] \mid R(\mathcal{O}_K)\subset \mathcal{O}_K\}$. The Pólya group is the group generated by the classes of the products of the prime ideals with the same norm. The Pólya group $\mathcal{P}_O(K)$ is trivial if and only if the $\mathcal{O}_K$-module $\mathrm{Int}(\mathcal{O}_K)$ has a regular basis if and only if $K$ is a Pólya field. In this paper, we give the structure of the first cohomology group of units of the real biquadratic number fields $K=\mathbb{Q}(\sqrt{d}_1, \sqrt{d}_2)$, where $d_1>1$ and $d_2>1$ are two square-free integers with $(d_1,d_2)=1$ and the prime $2$ is not totally ramified in $K/\mathbb{Q}$. We then determine the Pólya groups and the Pólya fields of $K$.

Keywords: Pólya fields; Pólya groups; real biquadratic fields; the first cohomology group of units; integer-valued polynomials.

MSC: 11R04, 11R16, 11R27, 13F20

Abstract

In this article, we propose a parallel viscosity iterative method for determining a common solution of a finite family of generalized equilibrium problems and a fixed point of a nonexpansive mapping in the setting of Hadamard manifolds. Under some mild conditions, we prove that the sequence generated by the proposed algorithm converges to a common solution of a finite family of generalized equilibrium problems and a fixed point problem for a nonexpansive mapping. We apply our result to solve a convex minimization problem and present a numerical example to demonstrate the performance of our method. Our results extend and improve many related results on generalized equilibrium problems from linear spaces to Hadamard manifolds.

Keywords: Generalized equilibrium problem; viscosity method; Hadamard manifold; monotone operator; Riemannian manifold.

MSC: 47J20, 47N10, 65B05, 47J26

Abstract

In this paper, we introduce the notion of almost $n$-multiplier on Banach algebras. This new notion generalizes the concept of $n$-multiplier introduced and studied in [J. Laali, M. Fozouni, $n$-multipliers and their relations with $n$-homomorphisms, Vietnam J. Math., 45 (2017), 451-457]. We gave some general results and the continuity of such maps with some examples for this new notion on Banach algebras. In particular, we generalize the celebrated theorem of Johnson to (left) $n$-multipliers on Banach algebras.

Keywords: Almost $n$-multiplier; automatic continuity; uniform Banach algebra.

MSC: 46H40, 47B48, 47A10

Abstract

This paper deals with the validation of convergence results for the newly proposed $C$-$\alpha$ non-expansive mappings in the CAT$(0)$ space setting. Before analyzing the convergence behavior, we emphasize key results related to $C$-$\alpha$ non-expansive mappings. The convergence results are obtained using the $JF$-iteration method. We then illustrate these results with non-trivial examples and compare them to other notable iterations. These comparisons are presented in both tabular and graphical forms. Finally, we discuss a variational inequality problem within the context of these mappings using the same iteration scheme.

Keywords: $C$-$\alpha$ non-expansive mappings; CAT(0) spaces; $JF$-iteration; variational inequality problem; fixed points.

MSC: 47H10, 54H25