Raghib M. Abu-Saris

A self-invertibility condition for global periodicity of difference equations

Abstract Given a non-degenerate interval of real numbers D , and a continuous function f : D k → D with k ≥ 2 , we consider a k th-order difference equation of the form y n + 1 = f ( y n , … , y n − k + 1 ) ; n = 0 , 1 , 2 , … . We develop an easy-to-apply necessary condition so that all solutions of the above-mentioned equation are periodic of the same period.

On globally attracting two-cycles of second order difference equations

We establish necessary and sufficient conditions so that all solutions of a second order autonomous difference equation are attracted to period-2 solutions. In addition, we prove that difference equations enjoying this property posses invariants or first integrals.

A global convergence criterion for higher order nonlinear difference equations with applications

We develop a sufficient condition for the attractivity of all solutions of the difference equation to a periodic cycle of period p (not necessarily minimal). We utilize the established condition and prove that every positive solution of the fifth-order rational difference equation converges to a periodic solution of period six.

A NECESSARY AND SUFFICIENT CONDITION FOR THE EXISTENCE Of A UNIQUE SOLUTION OF A DISCRETE BOUNDARY VALUE PROBLEM

A kth-order linear difference equation with constant coefficients subject to boundary conditions is considered. A necessary and sufficient condition for the existence of a unique solution for such a boundary value problem is established. The condition established answers a fundamental question for well-posedness and can be easily applied using a simple and computationally tractable algorithm that does not require finding the roots of the associated characteristic equation.

Bifurcation in Deterministic Discrete Dynamical Systems: Advances in Theory and Applications

1Department of Health Informatics, College of Public Health and Health Informatics, King Saud Bin Abdulaziz University for Health Sciences, Riyadh 11481, Saudi Arabia 2Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 17551, Al-Ain, UAE 3Department of Mathematics, National Tsing Hua University, Taiwan 4Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA

A note on the attractivity of period-four solutions of a third-order rational difference equation

We prove that every positive solution of the third-order rational difference equation: converges to a period-four solution.

On Solving Non-Autonomous Linear Difference Equations with Applications

An explicit formula is established for the general solution of the homogeneous non-autonomous linear difference equation: The formula developed is then used to characterize globally periodic linear difference equations with constant coefficients.

Generalized Exponential Vandermonde Determinant and Hermite Multipoint Discrete Boundary Value Problem

We introduce a determinant that encompasses the classical Vandermonde determinant, the generalized Vandermonde determinant, and the recently introduced exponential Vandermonde determinant when the exponents are nonnegative integers. An explicit factorization of such a determinant will be established. This factorization enables us to develop a computationally tractable necessary and sufficient condition for the existence of a unique solution of a Hermite (

Characterization and Transformation of Invariants

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Deterministic Discrete Dynamical Systems: Advances in Regular and Chaotic Behavior with Applications

-point) discrete boundary value problem.