The aim of this work is to investigate the global stability, periodic nature, oscillation and the boundedness of solutions of the difference equation x n + 1 = A ∏ i = l k x n − 2 i − 1 B + C ∏ i = l k − 1 x n − 2 i , n = 0 , 1 , … where A , B , C are nonnegative real numbers and l , k are nonnegative integers, l k . We discuss the existence of unbounded solutions under certain conditions when l = 0 .

On the recursive sequence xn+1=A∏i=lkxn−2i−1B+C∏i=lk−1xn−2i

On positive solutions of a (k+1)th order difference equation

On some solvable systems of difference equations

In this note we give some additional information on the behavior of the solutions of the difference equation xn+1 = xn−1 1 + xn−1xn , n = 0, 1... where the initial conditions x−1, x0 are real numbers.

More on a rational recurrence relation.

On the difference equation xn = xn−2/(bn + cnxn−1xn−2)

On the recursive sequence xn+1=α+xn−1xn

This paper is devoted for a rigorous investigation of Hahn’s difference operator and the associated calculus. Hahn’s difference operator generalizes both the difference operator and Jackson’s q-difference operator. Unlike these two operators, the calculus associated with Hahn’s difference operator receives no attention. In particular, its right inverse has not been constructed before. We aim to establish a calculus of differences based on Hahn’s difference operator. We construct a right inverse of Hahn’s operator and study some of its properties. This inverse also generalizes both Norlund sums and the Jackson q-integrals. We also define families of corresponding exponential and trigonometric functions which satisfy first and second order difference equations, respectively.

Hahn Difference Operator and Associated Jackson–Nörlund Integrals

Stability of the recursive sequence xn+1 = (α - βxn)/(γ + xn-1)

In this paper, we study Hyers–Ulam stability and Hyers–Ulam–Rassias stability of first order non-linear impulsive time varying delay dynamic system on time scales, via a fixed point approach. We obtain some results of existence and uniqueness of solutions by using Picard operator. The main tools for our results are the Gronwall’s inequality on time scales, abstract Gronwall lemma and Banach contraction principle. In order to overcome difficulties arises in our considered model, we pose some conditions along with Lipchitz condition. At the end, an example is given that shows the validity of our main results.

Alaa E. Hamza

Papers

On the recursive sequence xn+1=A∏i=lkxn−2i−1B+C∏i=lk−1xn−2i

On the recursive sequence xn+1=α+xn−1xn

Hahn Difference Operator and Associated Jackson–Nörlund Integrals

Stability of the recursive sequence xn+1 = (α - βxn)/(γ + xn-1)

Stability Analysis of the First Order Non-linear Impulsive Time Varying Delay Dynamic System on Time Scales