Earthline Journal of Mathematical Sciences

Some Novel Aspects of Quasi Variational Inequalities

2022-05-22T15:31:54+00:00

Quasi variational inequalities can be viewed as novel generalizations of the variational inequalities and variational principles, the origin of which can be traced back to Euler, Lagrange, Newton and Bernoulli's brothers. It is well known that quasi-variational inequalities are equivalent to the implicit fixed point problems. We consider this alternative equivalent fixed point formulation to suggest some new iterative methods for solving quasi-variational inequalities and related optimization problems using projection methods, Wiener-Hopf equations, dynamical systems, merit function and nonexpansive mappings. Convergence analysis of these methods is investigated under suitable conditions. Our results present a significant improvement of previously known methods for solving quasi variational inequalities and related optimization problems. Since the quasi variational inequalities include variational inequalities and complementarity problems as special cases. Results obtained in this paper continue to hold for these problems. Some special cases are discussed as applications of the main results. The implementation of these algorithms and comparison with other methods need further efforts.

Some Iterative Schemes for Solving Mixed Equilibrium Variational-like Inequalities

2022-05-29T15:54:23+00:00

Some new types of equilibrium variational-like inequalities are considered, which is called the bifunction mixed equilibrium variational-like inequalities. The auxiliary principle technique is used to construct some iterative schemes to solve these new equilibrium variational-like inequalities. Convergence of the suggested schemes is discussed under relaxed conditions. Several special cases are discussed as applications of the main results. The ideas and techniques may be starting point for future research.

On δ-Scattered Spaces

2022-05-31T14:36:04+00:00

The object of the present paper is to introduce the concept of δ-scattered spaces as a natural generalization of the concept of scattered spaces. We prove that the concept of δ-scatteredness of the space coincides with scatteredness. It is noted that scattered need not be δ-scattered in general, also I-space are comparable with δ-scattered space. We start out by giving a characterization of δ-scattered spaces. We study relationships between δ-scatteredness and with scattered, semi-scattered, α-scattered, sub maximal, irresolvable and N-scattered.

Results of Semigroup of Linear Operator Generating a Continuous Time Markov Semigroup

2022-06-16T17:03:52+00:00

In this paper, we present results of $\omega$-order preserving partial contraction mapping creating a continuous time Markov semigroup. We use Markov and irreducible operators and their integer powers to describe the evolution of a random system whose state changes at integer times, or whose state is only inspected at integer times. We concluded that a linear operator $P:\ell^{1}(X_+)\rightarrow \ell^{1}(X_+)$ is a Markov operator if its matrix satisfies $P_{x,y}\geqslant 0$ and $\sum_{x\in X_+}P_{x,y=1}$ for all $y\in X$.

Some Results on the v-Analogue of Gamma Function

2022-06-03T15:58:05+00:00

In this paper, some properties for the v-analogue of Gamma and digamma functions are investigated. Also, a celebrated Bohr-Mollerup type theorem related to the v-analogue of Gamma function is given. Furthermore, an expression for the v-digamma function is presented by using the v-analogue of beta function. Also, some limits for the v-analogue of Gamma and beta functions are given.

High Order Multi-block Boundary-value Integration Methods for Stiff ODEs

2022-06-16T16:38:43+00:00

In this paper, we present a new family of multi-block boundary value integration methods based on the Enright second derivative type-methods for differential equations. We rigorously show that this class of multi-block methods are generally $A_{k_1,k_2}$-stable for all block number by verifying through employing the Wiener-Hopf factorization of a matrix polynomial to determine the root distribution of the stability polynomial. Further more, the correct implementation procedure is as well determine by Wiener-Hopf factorization. Some numerical results are presented and a comparison is made with some existing methods. The new methods which output multi-block of solutions of the ordinary differential equations on application, and are unlike the conventional linear multistep methods which output a solution at a point or the conventional boundary value methods and multi-block methods which output a block of solutions per step. The second derivative multi-block boundary value integration methods are a new approach at obtaining very large scale integration methods for the numerical solution of differential equations.

Homomorphic Relations and Goursat Lemma

2022-06-20T16:46:10+00:00

Over the past years various authors have investigated the famous elementary result in group theory called Goursat's lemma for characterizing the subgroups of the direct product $A\times B$ of two groups $A,B$. Given a homomorphic relation $\rho = (R,A,B)$ where $A$ and $B$ are groups and $R$ is a subgroup of $A\times B.$ What can one say about the structure of $\rho$. In 1950 Riguet proved a theorem that allows us to obtain a characterization of $\rho$ induces by examining the sections of the direct factors. The purpose of this paper is two-fold. A first and more concrete aim is to provide a containment relation property between homomorphic relation. Indeed if $\rho,\sigma$ are homomorphic relations, we provide necessary and sufficient conditions for $\sigma\leq\rho$. A second and more abstract aim is to introduce a generalization of some notions in homological algebra. We define the concepts of $\theta$-exact. We also obtain some interesting results. We use these results to find a generalization of Lambek Lemma.

Aspects of Free Actions Based on Dependent Elements in Group Rings

2022-06-21T17:05:04+00:00

This paper contains two directions of work. The first one gives material related to free action (an inner derivation) mappings on a group ring R[G] which is a construction involving a group G and a ring R and the dependent elements related to those mappings in R[G]. The other direction deals with a generalization of the definition of dependent elements and free actions. We concentrate our study on dependent elements, free action mappings and those which satisfy T(x)γ=δx,x∈R[G] and some fixed γ,δ∈R[G]. In the first part we work with one dependent element. In other words, there exists an element γ∈R[G] such that T(x)γ=γx,x∈R[G]. In second one, we characterize the two elements γ,δ∈R[G] which have the property T(x)γ=δx,x∈R[G] and some fixed γ,δ∈R[G], when T is assumed to have additional properties like generalized a derivation mappings.

Subgroups Inclusions in 3-Factors Direct Product

2022-06-27T16:20:42+00:00

The aim of this paper is to use a correspondent theorem to characterize containment of a degenerate $2$-factor injective subdirect products. Namely, let $\Omega,\Lambda$ be degenerate 2-factor injective subdirect products of $ M_{1}\times M_{2}\times M_{3}$, we provide necessary and sufficient conditions for $\Omega\leq \Lambda.$ Based on a decomposition of the inclusion order on the subgroup lattice of a subdirect product as a relation product of three smaller partial orders, we induce a matrix product of three incidence matrices.