CUBO, A Mathematical Journal

Non-algebraic limit cycles in Holling type III zooplankton-phytoplankton models

2021-12-24T01:50:12-03:00

We prove that for certain polynomial differential equations in the plane arising from predator-prey type III models with generalized rational functional response, any algebraic solution should be a rational function. As a consequence, limit cycles, which are unique for these dynamical systems, are necessarily trascendental ovals. We exemplify these findings by showing a numerical simulation within a system arising from zooplankton-phytoplankton dynamics.

Basic asymptotic estimates for powers of Wallis’ ratios

2021-12-24T01:50:45-03:00

For any \(a\in{\mathbb R}\), for every \(n\in{\mathbb N}\), and for \(n\)-th Wallis' ratio \(w_n:=\prod_{k=1}^n\frac{2k-1}{2k}\), the relative error \(r_{\,\!_0}(a,n):=\big(v_{\,\!_0}(a,n)-w_n^a\big)/w_n^a\) of the approximation \(w_n^a\approx v_{\,\!_0}(a,n):=(\pi n)^{-a/2} \) is estimated as \( \big|r_{\,\!_0}(a,n)\big| < \frac{1}{4n}\). The improvement \(w_n^a\approx v(a,n):=(\pi n)^{-a/2}\left(1-\frac{a}{8n}+\frac{a^2}{128n^2}\right)\) is also studied.

The structure of extended function groups

2021-12-24T01:51:12-03:00

Conformal (respectively, anticonformal) automorphisms of the Riemann sphere are provided by the Möbius (respectively, extended Möbius) transformations. A Kleinian group (respectively, an extended Kleinian group) is a discrete group of Möbius transformations (respectively, a discrete group of Möbius and extended Möbius transformations, necessarily containing extended ones).

A function group (respectively, an extended function group) is a finitely generated Kleinian group (respectively, a finitely generated extended Kleinian group) with an invariant connected component of its region of discontinuity.

A structural decomposition of function groups, in terms of the Klein- Maskit combination theorems, was provided by Maskit in the middle of the 70’s. One should expect a similar decomposition structure for extended function groups, but it seems not to be stated in the existing literature. The aim of this paper is to state and provide a proof of such a decomposition structural picture.

Entropy solution for a nonlinear parabolic problem with homogeneous Neumann boundary condition involving variable exponents

2021-12-24T01:51:31-03:00

In this paper we prove the existence and uniqueness of an entropy solution for a non-linear parabolic equation with homogeneous Neumann boundary condition and initial data in \(L^1\). By a time discretization technique we analyze the existence, uniqueness and stability questions. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.

Independent partial domination

2021-12-24T01:51:50-03:00

For \(p\in(0,1]\), a set \(S\subseteq V\) is said to \(p\)-dominate or partially dominate a graph \(G = (V, E)\) if \(\frac{|N[S]|}{|V|}\geq p\). The minimum cardinality among all \(p\)-dominating sets is called the \(p\)-domination number and it is denoted by \(\gamma_{p}(G)\). Analogously, the independent partial domination (\(i_p(G)\)) is introduced and studied here independently and in relation with the classical domination. Further, the partial independent set and the partial independence number \(\beta_p(G)\) are defined and some of their properties are presented. Finally, the partial domination chain is established as \(\gamma_p(G)\leq i_p(G)\leq \beta_p(G) \leq \Gamma_p(G)\).

Foundations of generalized Prabhakar-Hilfer fractional calculus with applications

2021-12-24T01:52:36-03:00

Here we introduce the generalized Prabhakar fractional calculus and we also combine it with the generalized Hilfer calculus. We prove that the generalized left and right side Prabhakar fractional integrals preserve continuity and we find tight upper bounds for them. We present several left and right side generalized Prabhakar fractional inequalities of Hardy, Opial and Hilbert-Pachpatte types. We apply these in the setting of generalized Hilfer calculus.

Existence and uniqueness of solutions to discrete, third-order three-point boundary value problems

2021-12-24T01:52:54-03:00

The purpose of this article is to move towards a more complete understanding of the qualitative properties of solutions to discrete boundary value problems. In particular, we introduce and develop sufficient conditions under which the existence of a unique solution for a third-order difference equation subject to three-point boundary conditions is guaranteed. Our contributions are realized in the following ways. First, we construct the corresponding Green’s function for the problem and formulate some new bounds on its summation. Second, we apply these properties to the boundary value problem by drawing on Banach’s fixed point theorem in conjunction with interesting metrics and appropriate inequalities. We discuss several examples to illustrate the nature of our advancements.

Some integral inequalities related to Wirtinger's result for \(p\)-norms

2021-12-24T01:53:13-03:00

In this paper we establish several natural consequences of some Wirtinger type integral inequalities for \(p\)-norms. Applications related to the trapezoid unweighted inequalities, of Grüss' type inequalities and reverses of Jensen's inequality are also provided.

On the periodic solutions for some retarded partial differential equations by the use of semi-Fredholm operators

2021-12-24T01:53:30-03:00

The main goal of this work is to examine the periodic dynamic behavior of some retarded periodic partial differential equations (PDE). Taking into consideration that the linear part realizes the Hille-Yosida condition, we discuss the Massera’s problem to this class of equations. Especially, we use the perturbation theory of semi-Fredholm operators and the Chow and Hale’s fixed point theorem to study the relation between the boundedness and the periodicity of solutions for some inhomogeneous linear retarded PDE. An example is also given at the end of this work to show the applicability of our theoretical results.

Extension of exton's hypergeometric function \(K_{16}\)

2021-12-24T01:53:47-03:00

The purpose of this article is to introduce an extension of Exton's hypergeometric function \(K_{16}\) by using the extended beta function given by Özergin et al. [11]. Some integral representations, generating functions, recurrence relations, transformation formulas, derivative formula and summation formulas are obtained for this extended function. Some special cases of the main results of this paper are also considered.