Volume 70, Issue 2

It is known that every relatively pseudocomplemented lattice is residuated and, moreover, it is distributive. Unfortunately, non-distributive lattices with a unary operation satisfying properties similar to relative pseudocomplementation cannot be converted in residuated ones. The aim of our paper is to introduce a more general concept of a relatively residuated lattice in such a way that also non-modular sectionally pseudocomplemented lattices are included. We derive several properties of relatively residuated lattices which are similar to those known for residuated ones and extend our results to posets.

For a class 𝓜 of monomorphisms of a category, mathematicians usually use different types of essentiality. Essentiality is an important notion closely related to injectivity. Banaschewski defines and gives sufficient conditions on a category 𝓐 and a subclass 𝓜 of its monomorphisms under which 𝓜-injectivity well - behaves with respect to the notions such as 𝓜-absolute retract and 𝓜-essentialness. In this paper, 𝓐 is taken to be the category of acts over a semigroup S and 𝓜 sd to be the class of strongly s -dense monomorphisms. We study essentiality with respect to strongly s -dense monomorphisms of acts. Depending on a class 𝓜 of morphisms of a category 𝓐, In some literatures, three different types of essentialness are considered. Each has its own benefits in regards with the behavior of 𝓜-injectivity. We will show that these three different definitions of essentiality with respect to the class of strongly s -dense monomorphisms are equivalent. Also, the existence and the explicit description of a strongly s -dense injective hull for any given act which is equivalent to the maximal such essential extension and minimal strongly s -dense injective extension with respect to strongly s -dense monomorphism is investigated. At last we conclude that strongly s -dense injectivity well behaves in the category Act-S .

Given a self-equivalence of a global function field, its wild set is the set of points where the self-equivalence fails to preserve parity of valuation. In this paper we describe structure of finite wild sets.

We associate to some simplest quartic fields a family of elliptic curves that has rank at least three over ℚ( m ). It is given by the equation Em:y2=x3− 3636m4+48m2+2536m4− 48m2+25x.$$\begin{array}{} \displaystyle E_m:y^2=x^3-36\left(36m^4+48m^2+25\right)\left(36m^4-48m^2+25\right)x. \end{array}$$ Employing canonical heights we show the rank is in fact at least three for all m . Moreover, we get a parametrized infinite family of rank at least four. Further, the integral points on the curve E m are discussed and we determine all the integral points on the original quartic model when the rank is three. Previous work in this setting studied the elliptic curves associated with simplest quartic fields of ranks at most two along with their integral points (see [2, 3]).

In this paper, two new analogues of the Hilbert matrix with four-parameters have been introduced. Explicit formulæ are derived for the LU -decompositions and their inverses, and the inverse matrices of these analogue matrices.

A function f : X → ℝ defined on a topological space X is called returning if for any point x ∈ X there exists a positive real number M x such that for every path-connected subset C x ⊂ X containing the point x and any y ∈ C x ∖ { x } there exists a point z ∈ C x ∖ { x , y } such that | f ( z )| ≤ max{ M x , | f ( y )|}. A topological space X is called path-inductive if a subset U ⊂ X is open if and only if for any path γ : [0, 1] → X the preimage γ –1 ( U ) is open in [0, 1]. The class of path-inductive spaces includes all first-countable locally path-connected spaces and all sequential locally contractible spaces. We prove that a function f : X → ℝ defined on a path-inductive space X is continuous if and only if it is returning and has closed graph. This implies that a (weakly) Świątkowski function f : ℝ → ℝ is continuous if and only if it has closed graph, which answers a problem of Maliszewski, inscribed to Lviv Scottish Book.

In the paper some properties of sets of points of approximate continuity and ϱ -upper continuity are presented. We will show that for every Lebesgue measurable set E ⊂ ℝ there exists a function f : ℝ → ℝ which is approximately ( ϱ -upper) continuous exactly at points from E . We also study properties of sets of points at which real function has Denjoy property. Some other related topics are discussed.

The main aim of this paper is to study the fifth Hankel determinant for the class of functions with bounded turnings. The results are also investigated for 2-fold symmetric and 4-fold symmetric functions.

In this paper a new class of well-posed boundary value problems for the biharmonic equation is studied. The considered problems are nonlocal boundary value problems of Bitsadze- -Samarskii type. These problems are solved by reducing them to Dirichlet and Neumann type problems. Theorems on existence and uniqueness of the solution are proved and exact solvability conditions of the considered problems are found. In addition, the integral representations of solutions are obtained.

In this paper, we study the existence and global exponential stability of positive piecewise pseudo almost periodic solutions for the impulsive Lasota-Wazewska model with multiply time-varying delays when coefficients are piecewise pseudo almost periodic. Under proper conditions, by using the Gronwall’s inequation, we establish some criteria to ensure that the solution of this model stability exponentially to a positive piecewise pseudo almost periodic solution. Moreover, an example and its numerical simulation are given to illustrate the theoretical results.

In this paper we discuss the existence and asymptotic behavior of strongly increasing solutions of quasilinear ordinary differential equations of the form D(αn,αn−1,…,α1)x=p(t)|x|β sgn x,t≥a.$$\begin{array}{} \displaystyle D(\alpha_n, \alpha_{n-1}, \dots, \alpha_1)x = p(t)|x|^{\beta}\text{ sgn } x, \quad t \geq a. \end{array} $$(1.1) It will be shown that there is an explicit difference between the cases α 1 α 2 ⋯ α n ≥ β and α 1 α 2 ⋯ α n < β for the structure of the totality of strongly increasing solutions of (1.1).

Neutral differential equations are one of the most important extensions of classical ordinary differential equations and aim to give a better explanation for modeling phenomena where ordinary differential equations are insufficient. Naturally, all the questions studied in the scope of ordinary differential equations attracted the attention also for neutral differential equations. In this paper we study the oscillatory properties of second order half-linear neutral differential equations. We present oscillation criteria derived using a new approach. This approach allows us to reduce common restrictions on the deviations in arguments which are present in the currently known results of this type.

If in the right-hand sides of given differential equations occur discontinuities in the state variables, then the natural notion of a solution is the one in the sense of Filippov. In our paper, we will consider this type of solutions for vector Dirichlet problems. The obtained theorems deal with the existence and localization of Filippov solutions, under effective growth restrictions. Two illustrative examples are supplied.

We study the existence of infinitely many positive homoclinic solutions to a second-order difference equation on integers with p k -Laplacian. To achieve our goal we use the critical point theory and the general variational principle of Ricceri.

The goal of this paper is to propose a modification of Lupaş-Jain operators based on a function ρ having some properties. Primarily, the convergence of given operators in weighted spaces is discussed. Then, order of approximation via weighted modulus of continuity is computed for these operators. Further, Voronovskaya type theorem in quantitative form is taken into consideration, as well. Ultimately, some graphical results that illustrate the convergence of investigated operators to f are given.

Z. D. Mitrović and S. Radenović introduced in [ The Banach and Reich contractions in b v ( s )- metric spaces , J. Fixed Point Theory Appl. 19 (2017), 3087–3095] a new class of generalized metric spaces and proved some fixed point theorems in this framework. The purpose of this paper is to consider other kinds of contractive mappings in b v ( s )-metric spaces, and show how the work in the new settings differs from the one in standard metric and b -metric spaces. Examples show the usefulness of the obtained results.

In this paper, we present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert-Schmidt norm of matrices. We also give some refinements of the following Heron type inequality for unitarily invariant norm |||⋅||| and A , B , X ∈ M n (ℂ): |||Aν XB1− ν +A1− ν XBν 2|||≤ (4r0− 1)|||A12XB12|||+2(1− 2r0)|||(1− α )A12XB12+α (AX+XB2)|||,$$\begin{array}{} \begin{split} \displaystyle \Big|\Big|\Big|\frac{A^\nu XB^{1-\nu}+A^{1-\nu}XB^\nu}{2}\Big|\Big|\Big| \leq &(4r_0-1)|||A^{\frac{1}{2}}XB^{\frac{1}{2}}||| \\ &+2(1-2r_0)\Big|\Big|\Big|(1-\alpha)A^{\frac{1}{2}}XB^{\frac{1}{2}} +\alpha\Big(\frac{AX+XB}{2}\Big)\Big|\Big|\Big|, \end{split} \end{array}$$ where 14≤ ν ≤ 34,α ∈ [12,∞ )$\begin{array}{} \displaystyle \frac{1}{4}\leq \nu \leq \frac{3}{4}, \alpha \in [\frac{1}{2},\infty ) \end{array}$ and r 0 = min{ ν , 1 – ν }.

In this work, the discussion centers on introducing non-triangular metric as a generalization of JS -metric, which may lessen the frantic pace of working on generalizing the triangle inequality by omitting it from the original ordinary metric’s definition. Then, we present S -operators as a new sort of mappings included some properties in order to characterize some known contractions. After that, we graft non-triangular metric onto S -operators in order to make the worthwhile fixed point results facile to prove and conceive.

The question as to the number of sets obtainable from a given subset of a topological space using the operators derived by composing members of the set { b , i , ∨, ∧}, where b , i , ∨ and ∧ denote the closure operator, the interior operator, the binary operators corresponding to union and intersection, respectively, is called the Kuratowski { b , i , ∨, ∧}-problem. This problem has been solved independently by Sherman [21] and, Gardner and Jackson [13], where the resulting 34 plus identity operators were depicted in the Hasse diagram. In this paper we investigate the sets of fixed points of these operators. We show that there are at most 23 such families of subsets. Twelve of them are the topology, the family of all closed subsets plus, well known generalizations of open sets, plus the families of their complements. Each of the other 11 families forms a complete complemented lattice under the operations of join, meet and negation defined according to a uniform procedure. Two of them are the well known Boolean algebras formed by the regular open sets and regular closed sets, any of the others in general need not be a Boolean algebras.

The problem of iterated partial summations is solved for some discrete distributions defined on finite supports. The power method, usually used as a computational approach to the problem of finding matrix eigenvalues and eigenvectors, is in some cases an effective tool to prove the existence of the limit distribution, which is then expressed as a solution of a system of linear equations. Some examples are presented.

If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {3 6 }, {4 4 } and {6 3 } on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {3 3 , 4 2 }, {3 2 , 4 1 , 3 1 , 4 1 }, {3 1 , 6 1 , 3 1 , 6 1 }, {3 4 , 6 1 }, {4 1 , 8 2 }, {3 1 , 12 2 }, {4 1 , 6 1 , 12 1 } and {3 1 , 4 1 , 6 1 , 4 1 } on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.