Mohammad Saleh

Dynamics of a higher order rational difference equation

Abstract In this paper, we will investigate a nonlinear rational difference equation of higher order. Our concentration is on invariant intervals, periodic character, the character of semicycles and global asymptotic stability of all positive solutions of x n + 1 = β x n + γ x n - k Bx n + Cx n - k , n = 0 , 1 , … It is worth to mention that our results solve the open problem proposed by Kulenvic and Ladas in their monograph [Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002].

On the rational difference equation y n +1 = A + y n / y n-k

We find conditions for the global asymptotic stability of the unique negative equilibrium y- = 1 + A of the equation yn+1 = A + yn/yn-k, where y-k, y-k+1,...,y0, A ∈ R, and k ∈ {1,2,3,4,...}.

On the difference equation yn+1=A+ynyn-k with A < 0

Abstract We find conditions for the global asymptotic stability of the unique negative equilibrium y ¯ = 1 + A of the equation (0.1) y n + 1 = A + y n y n - k , where y−k,xa0y−k+1,xa0…xa0,xa0y0xa0∈xa0(0,xa0∞), A

On weakly projective and weakly injective modules

The purpose of this paper is to further the study of weakly injective and weakly projective modules as a generalization of injective and projective modules. For a locally q.f.d. module M, there exists a module K ∈ �(M) such that K ⊕ N is weakly injective in �(M), for any N ∈ �(M). Similarly, if M is projective and right perfect in �(M), then there exists a module K ∈ �(M) such that K ⊕ N is weakly projective in �(M), for any N ∈ �(M). Consequently, over a right perfect ring every module is a direct summand of a weakly projective module. For some classes M of modules in �(M), we study when direct sums of modules from M satisfy property P in �(M). In particular, we get characterizations of locally countably thick modules, a generalization of locally q.f.d. modules.

A note on tightness

The purpose of this note is to prove a results of Jain and Lopez-Permouth under a weaker conditions replacing R -weak injectivity by R -tightness and even getting a simpler proof.