Journal of Mathematical Extension
Volume:14 Issue: 1, Winter 2020

In this paper, motivated by [F. Vetro, Filomat, 29:9 (2015), 2011-2020] we present some fixed point results for a class of nonexpansive self-mappings and multi-valued mappings in the framework of b-metric spaces. Our results generalize and improve the consequences of [ Khojasteh et al. Abstract and Applied Analysis, vol. 2014, Article ID 325840, 5 pages, 2014.] and [F. Vetro, Filomat, 29:9 (2015), 2011- 2020]. Some examples are provided to illustrate our results.

Let $\mathcal{L}$ be a Lie algebra crossed module and $\Act_{pi}(\mathcal{L})$ be a point wise inner Actor of $\mathcal{L}$. In this paper, we introduce lower and upper central series of $\mathcal{L}$ and show that if $\Act_{pi}(\frac{\mathcal{L}}{Z_j(\mathcal{L})}) / \Inn\Act(\frac{\mathcal{L}}{Z_j(\mathcal{L})}) $ is the nilpotent of class $k$, then $\Act_{pi}(\mathcal{L}) / \Inn\Act(\mathcal{L}) $ is the nilpotent of the maximum class $ j+k $. Moreover, if $ \dim(\mathcal{L}^i / (\mathcal{L}^i\cap Z_j(\mathcal{L})))\leqslant 1 $, then $ \Act_{pi}(\mathcal{L}) / \Inn\Act(\mathcal{L})$ is the nilpotent of the maximum class $ i+j-1$.

In this paper, we show that the system of difference equations \begin{equation*}x_{n}=\frac{x_{n-2}y_{n-3}}{y_{n-1}\left(a_{n}+b_{n}x_{n-2}y_{n-3} \right) },\y_{n}=\frac{y_{n-2}x_{n-3}}{x_{n-1}\left(\alpha_{n}+\beta_{n}y_{n-2}x_{n-3} \right) }, \ n\in\mathbb{N}_{0},\end{equation*}%where the sequences $\forall n\in\mathbb{N}_{0}$, $\left( a_{n}\right) ,\left( b_{n}\right) , \left( \alpha_{n}\right) , \left( \beta_{n}\right) $ and the initial values $x_{-j}, y_{-j}, j\in\{1,2,3\}$ are non-zero real numbers, can be solvedin the closed form. For the case when all the sequences $\left( a_{n}\right) ,\left( b_{n}\right) , \left( \alpha_{n}\right) , \left( \beta_{n}\right) $ are constant we describe the asymptotic behavior and periodicity of solutions of above system is also investigated.

For any positive integer m, ϕ(m) finds out how many residues of m that are co-prime to m, where ϕ is the Euler’s totient function. In this paper, we introduce the notion of totient and hyper totient numbers. We explore the potential links of totient, super totient and hyper totient numbers. Many postulates and characterizations of these numbers have been proposed with straight forward proofs. In the end, applications of these numbers in graph labeling have also been demonstrated over a family of well known graph.

The key aim of this article is to familiarize some basic results in soft bi topological spaces. The idea of soft Limit Point in soft bi topological space is introduced and related results are also discussed with respect to ordinary and soft points. Soft interior point in soft bi topological space and related results with respect to ordinary and soft points are also studied. The relation between soft   space and Soft   -Closures is discussed in soft topological space with respect to soft -open set. Also the same relation is discussed with respect to soft -open and set P-open sets. Soft neighborhood in soft bi topological space is defined and related results are studied. Soft sequences uniqueness of limit in soft  -Hausdorff space and soft  -Hausdorff space is studied. The product of soft Hausdorff spaces with respect to soft points in different soft weak open sets are also discussed. The marriage between Soft Hausdorff space and the diagonal is also planted here.

We shed some light on the Pythagorean relation for operator established in [7] and we study its relationship with the maximal numerical range. We then get some informations of maximal numerical ranges of selfadjoint operators. This allows us to show, contrary to the closure of the numerical range, the non-continuity in the sens of Hausdorff of the maximal numerical range. AMS Subject Classification: 47A12; 47A30; 47A63; 47B15; 47B47 Keywords and Phrases: Spectrum, numerical range, maximal numerical range.

For a ring R and a monoidM, we introduce J-M-Armendariz rings which are a generalization of weak M-Armendariz and J-Armendariz rings, and we investigate their properties. We prove that if R J(R) is a reduced ring and R is a J-M-Armendariz ring, then R is J-M N-Armendariz, where N is a unique product monoid. It is also shown that a nitely generated Abelian group G is torsion free if and only if there exists a ring R such that R is G-J-Armendariz.

We generalise the notion of cyclic-noncyclic pairs by introducing the concept of $m$-contraction pairs in convex metric spaces. We shall then investigate the exixtence and convergence of coincidence-best proximity points, and finally, we will provide examples to illustrate the new findings.

We investigate the existence of solution for a fractional integro-differential inclusion via the Caputo-Hadamard fractional derivation. We prove that dimension of the set of solutions for the inclusion problem isinfinite dimensional under some conditions.

The eventual stability with respect to part of variables  of a nonlinear differential equation with non-instantaneousimpulses is studied using Lyapunov like functions.  In thesedifferential equations there are impulses, which start abruptly at somepoints and their action continue on given finite intervals.Sufficient conditions for eventual stability, uniform eventual stability andeventual asymptotic uniform stability with respect to part of variables of the zero solution are established. Examples are given to illustrate the results.

In this article, singular d-homology, singular extended d-homology, relative singular d-homology and relative singular extended d-homology functors are introduced and some axioms, such as dimension axiom, homotopy axiom, excision axiom and exactness axiom, of homology theory relative to these homologies are investigated and are proved to hold.

The inverse data envelopment analysis is an inverse optimization problem, which can be used as an appropriate planning tool for management decisions. The typical DEA mainly focuses on post-operative evaluation of an organizational performance. Sometimes economic conditions such as economic prohibitions on exports or imports are imposed on a system. These prohibitions prevent decision-making units from best performance (efficiency one). In this case, if the system has the best performance (with a less than one efficiency score) then it will be considered as an efficient system. So, the efficiency frontier change’s problem must be studied. So by making change in definition of the best efficiency amount of a system, it still has the best performance. In these situations, the inefficient units can select a real pattern instead of reaching an unrealistic pattern that is presented in ideal terms to achieve the best conditions (the best efficiency value is one). So a long-term management plan can be developed. the efficiency frontier change will be expressed inputs and outputs as a coefficient of efficiency. The frontier change looks at the changes in inputs and outputs to reach the new frontier. One of the purposes of the data envelopment analysis is the investigation of input’s and output’s amounts by changing the amount of efficiency. So far, many models must be solved to calculating these changes. Efficiency frontier problem can replace a simple mathematical model with these models. All of these advantages can improve calculating input and output’s changes and RTS will be unchanged and decision maker can estimate unit’s RTS without solving any model. So a unit will be stayed MPSS by reducing inputs. In other frontier change methods some hyperplanes and extreme units had been deleting but our method transforms them on new frontier. So all extreme units and RTS can estimate easily. The efficiency frontier changes can delete some inefficient units so system’s cost will be reduced.  For this purpose, in this paper, the change in the efficiency frontier, its properties and its effect on the inverse data envelopment analysis is examined.