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This paper presents a new method of transforming fourth-order nonlinear semi-canonical neutral difference equations into canonical type equations. This technique reduces the set of nonoscillatory solutions to only two. By eliminating these two remaining types of nonoscillatory solutions, the authors are able to obtain new criteria for the oscillation of all solutions of the equation under investigation. Examples are provided to demonstrate the significance and novelty of the main results.

In the work, we focus on designing and analyzing a simple mathematical model of epidemic outbreaks involving vaccination in a heterogeneous population composed of two age groups. The model is based on the framework of matrix population models. It is designed to include the fundamental phenomena of interest while also making it as explicit as possible for examination using methods of real function analysis. Our aim is to examine differences between separable and non-separable mixing and answer the question, how many vaccines are needed to achieve herd immunity. Additionally, we aim to gain a better understanding of some controversies in vaccination prioritization where a superficial view could lead to misconceptions and subsequent poor decisions.

Let be a transcendental entire function of hyperorder strictly less than 1, and let be a nonzero finite complex number. We prove that if and partially share 0, 1 ignoring multiplicity (i.e., and ), then . This result is a generalization and improvement of the previous theorem due to Li and Yi.

We develop a delay differential equation model for the interactive wild and sterile mosquitoes. Different from the existing modelling studies, we assume that only those sexually active sterile mosquitoes play a role for the interactive dynamics. We consider the cases where the release amount is either constant or described by a given function of time. For the constant releases, we establish a threshold of releases to determine whether the wild mosquito suppression succeeds or fails. We study the existence and stability of the model equilibria. When the releases are described by given functions, the trivial equilibrium is no longer globally but locally uniformly asymptotically stable if the amount of releases is below the threshold whereas it is still globally uniformly asymptotically stable if the release amount is above the threshold. Numerical examples demonstrating the model dynamical features and brief discussions of our findings are also provided.

It is shown that if the equation f ( z + 1 ) n = R ( z , f ) , where R ( z ,  f ) is rational in both arguments and deg f ( R ( z , f ) ) ≠ n , has a transcendental meromorphic solution, then the equation above reduces into one out of several types of difference equations where the rational term R ( z ,  f ) takes particular forms. Solutions of these equations are presented in terms of Weierstrass or Jacobian elliptic functions, exponential type functions or functions which are solutions to a certain autonomous first-order difference equation having meromorphic solutions with preassigned asymptotic behavior. These results complement our previous work on the case deg f ( R ( z , f ) ) = n of the equation above and thus provide a complete difference analogue of Steinmetz’ generalization of Malmquist’s theorem.

In this paper, solution of the following difference equation is examined xn+1=xn−131+xn−1xn−3xn−5xn−7xn−9xn−11,{x_{n + 1}} = {{{x_{n - 13}}} \\over {1 + {x_{n - 1}}{x_{n - 3}}{x_{n - 5}}{x_{n - 7}}{x_{n - 9}}{x_{n - 11}}}, where the initial conditions are positive real numbers.

We describe a family of integrable lattice maps related to the known quad maps Q4. The integrability criterion we use is the vanishing of the algebraic entropy. The family has the advantage of being parametrized rationally: all its parameters are unconstrained.

Using a modification of the adapted Riccati transformation, we prove an oscillation criterion for generalizations of linear and half-linear Euler difference equations. Our main result complements a large number of previously known oscillation criteria about several similar generalizations of Euler difference equations. The major part of this paper is formed by the proof of the main theorem. To illustrate the fact that the presented criterion is new even for linear equations with periodic coefficients, we finish this paper with the corresponding corollary together with concrete examples of simple equations whose oscillatory properties do not follow from previously known criteria.

We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (for example, standard difference schemes) and difference equations in functions on words. On the universality side, we prove a version of strong Nullstellensatz for such difference equations under the assumption that the cardinality of the ground field is greater than the cardinality of the monoid and construct an example showing that this assumption cannot be omitted. On the undecidability side, we show that the following problems are undecidable:

Difference equations are often used to create discrete mathematical models. In this paper, we mainly study the dynamical behaviors of positive solutions of a nonlinear fuzzy difference equation: where parameters and initial value , are positive fuzzy numbers. We investigate the existence, boundedness, convergence, and asymptotic stability of the positive solutions of the fuzzy difference equation. At last, we give numerical examples to intuitively reflect the global behavior. The conclusion of the global stability of this paper can be applied directly to production practice.