CUBO, A Mathematical Journal

Dual digraphs of finite semidistributive lattices

Andrew Craig — 2022-12-21

Dual digraphs of finite join-semidistributive lattices, meet-semidistributive lattices and semidistributive lattices are characterised. The vertices of the dual digraphs are maximal disjoint filter-ideal pairs of the lattice. The approach used here combines representations of arbitrary lattices due to Urquhart (1978) and PlošÄica (1995). The duals of finite lattices are mainly viewed as TiRS digraphs as they were presented and studied in Craig--Gouveia--Haviar (2015 and 2022). When appropriate, Urquhart's two quasi-orders on the vertices of the dual digraph are also employed. Transitive vertices are introduced and their role in the domination theory of the digraphs is studied. In particular, finite lattices with the property that in their dual TiRS digraphs the transitive vertices form a dominating set (respectively, an in-dominating set) are characterised. A characterisation of both finite meet-and join-semidistributive lattices is provided via minimal closure systems on the set of vertices of their dual digraphs.

Two nonnegative solutions for two-dimensional nonlinear wave equations

Svetlin Georgiev — 2022-12-21

We study a class of initial value problems for two-dimensional nonlinear wave equations. A new topological approach is applied to prove the existence of at least two nonnegative classical solutions. The arguments are based upon a recent theoretical result.

Existence of positive solutions for a nonlinear semipositone boundary value problems on a time scale

Saroj Panigrahi — 2022-12-21

In this paper, we are concerned with the existence of positive solution of the following semipositone boundary value problem on time scales:

\begin{align*} (\psi(t)y^\Delta (t))^\nabla + \lambda_1 g(t, \,y(t)) + \lambda_2 h(t,\,y(t)) = 0, \,t \in [\rho(c), \,\sigma(d)]_\mathbb{T}, \end{align*}

with mixed boundary conditions

\begin{align*} \alpha y(\rho(c))-\beta \psi(\rho(c)) y^\Delta(\rho(c))=0,\\ \gamma y(\sigma(d))+\delta \psi(d) y^\Delta(d)=0, \end{align*}

where \(\psi:C[\rho(c),\, \sigma(d)]_\mathbb{T}\), \(\psi(t)>0\) for all \(t \in [\rho(c),\,\sigma(d)]_\mathbb{T}\); both \(g\) and \(h : [\rho(c),\,\sigma(d)]_\mathbb{T} \times [0,\,\infty) \to \mathbb{R}\) are continuous and semipositone. We have established the existence of at least one positive solution or multiple positive solutions of the above boundary value problem by using fixed point theorem on a cone in a Banach space, when \(g\) and \(h\) are both superlinear or sublinear or one is superlinear and the other is sublinear for \(\lambda_i>0;\,i=1,\,2\) are sufficiently small.

A class of nonlocal impulsive differential equations with conformable fractional derivative

Mohamed Bouaouid — 2022-12-21

In this paper, we deal with the Duhamel formula, existence, uniqueness, and stability of mild solutions of a class of nonlocal impulsive differential equations with conformable fractional derivative. The main results are based on the semigroup theory combined with some fixed point theorems. We also give an example to illustrate the applicability of our abstract results.

On the minimum ergodic average and minimal systems

Manuel Saavedra — 2022-12-21

We prove some equivalences associated with the case when the average lower time is minimal. In addition, we characterize the minimal systems by means of the positivity of invariant measures on open sets and also the minimum ergodic averages. Finally, we show that a minimal system admits an open set whose measure is minimal with respect to a set of ergodic measures and its value can be chosen in [0, 1].

Positive solutions of nabla fractional boundary value problem

N. S. Gopal — 2022-12-21

In this article, we consider the following two-point discrete fractional boundary value problem with constant coefficient associated with Dirichlet boundary conditions.

\begin{align*}\begin{cases} -\big{(}\nabla^{\nu}_{\rho(a)}u\big{)}(t) + \lambda u(t) = f(t, u(t)), \quad t \in \mathbb{N}^{b}_{a + 2}, \\u(a) = u(b) = 0, \end{cases} \end{align*}

where \(1 < \nu < 2\), \(a,b \in \mathbb{R}\) with \(b-a\in\mathbb{N}_{3}\), \(\mathbb{N}^b_{a+2} = \{a+2,a+3, . . . ,b\}\), \(|\lambda| < 1\), \(\nabla^{\nu}_{\rho(a)}u\) denotes the \(\nu^{\text{th}}\)-order Riemann–Liouville nabla difference of \(u\) based at \(\rho(a)=a-1\), and \(f : \mathbb{N}^{b}_{a + 2} \times \mathbb{R} \rightarrow \mathbb{R}^{+}\).

We make use of Guo–Krasnosels'kiÄ and Leggett–Williams fixed-point theorems on suitable cones and under appropriate conditions on the non-linear part of the difference equation. We establish sufficient requirements for at least one, at least two, and at least three positive solutions of the considered boundary value problem. We also provide an example to demonstrate the applicability of established results.

Einstein warped product spaces on Lie groups

Buddhadev Pal — 2022-12-21

We consider a compact Lie group with bi-invariant metric, coming from the Killing form. In this paper, we study Einstein warped product space, \(M = M_1 \times_{f_1} M_2\) for the cases, \((i)\) \(M_1\) is a Lie group \((ii)\) \(M_2\) is a Lie group and \((iii)\) both \(M_1\) and \(M_2\) are Lie groups. Moreover, we obtain the conditions for an Einstein warped product of Lie groups to become a simple product manifold. Then, we characterize the warping function for generalized Robertson-Walker spacetime, \((M = I \times_{f_1} G_2, - dt^2 + f_1^2 g_2)\) whose fiber \(G_2\), being semi-simple compact Lie group of \(\dim G_2>2\), having bi-invariant metric, coming from the Killing form.

Infinitely many solutions for a nonlinear Navier problem involving the \(p\)-biharmonic operator

Filippo Cammaroto — 2022-12-21

In this paper we establish some results of existence of infinitely many solutions for an elliptic equation involving the \(p\)-biharmonic and the \(p\)-Laplacian operators coupled with Navier boundary conditions where the nonlinearities depend on two real parameters and do not satisfy any symmetric condition. The nature of the approach is variational and the main tool is an abstract result of Ricceri. The novelty in the application of this abstract tool is the use of a class of test functions which makes the assumptions on the data easier to verify.

A derivative-type operator and its application to the solvability of a nonlinear three point boundary value problem

René Erlín Castillo — 2022-12-21

In this paper we introduce an operator that can be thought as a derivative of variable order, i.e. the order of the derivative is a function. We prove several properties of this operator, for instance, we obtain a generalized Leibniz‘s formula, Rolle and Cauchy‘s mean theorems and a Taylor type polynomial. Moreover, we obtain its inverse operator. Also, with this derivative we analyze the existence of solutions of a nonlinear three-point boundary value problem of “variable order”.

Estimates for the polar derivative of a constrained polynomial on a disk

Gradimir V. MilovanoviÄ‡ — 2022-12-21

This work is a part of a recent wave of studies on inequalities which relate the uniform-norm of a univariate complex coefficient polynomial to its derivative on the unit disk in the plane. When there is a limit on the zeros of a polynomial, we develop some additional inequalities that relate the uniform-norm of the polynomial to its polar derivative. The obtained results support some recently established ErdÅ‘s-Lax and Turán-type inequalities for constrained polynomials, as well as produce a number of inequalities that are sharper than those previously known in a very large literature on this subject.