Abstract We investigate the boundedness character, the periodic nature, and the global asymptotic stability of all positive solutions of the equation in the title with positive parameters and nonnegative initial conditions.

Abstract We investigate the periodic character and the global stability of solutions of the equation y n  + 1  = ( p  +  y n  + 1 )/( qy n  +  y n  − 1 ) with positive parameters and positive initial conditions.

Abstract Consider the difference equation x n+1 = f(x n , where xn is in Rk and f : D → D is continuous where D ⊂ Rk. Suppose that I : Rk → R is a continuous invariant, that is, I( f(x) ) = I( x ) for every x ∈ D . We will show that if I attains an isolated minimum or maximum value at the equilibrium (fixed) point p of this system, then there exists a Liapunov function, namely ±(I( x ) − I( p )) and so the equilibrium p is stable. This result is then applied to some difference equations appearing in different fields of applications.

In this note, we show that the oscillation of all solutions of the equation (r(t)g (y'(t)))' + p(t)f(y(t)) = O, (E) extendible to infinity, follows from the oscillation of all solutions of the associated linear equation (r(t)x'(t))' + kp(t)x(t) = O, m where g(u)/u _ k or f'(u) >_ k, for every u ¢ 0 and some m, k > O. Using these results, we show that all solutions of the equilibrium capillary surface equation

In this paper we give a necessary and sufficient condition for the oscillation of the second order linear differential equation where p is a locally integrable function and either or where We give some applications which show how these results unify and imply some classical results in oscillation theory.

We investigate the global stability, and the periodicity of the recursive sequence where the parameters α,β and A and the initial condition x-1 and Xo are non negative real numbers.

Consider the neutral delay differential equation with positive and negative coefficients,[formula]wherep ∈ Rand[formula] Some sufficient conditions for the existence of a nonoscillatory solution of the above equation expressed in terms of ∫∞ sQi(s) ds < ∞,i = 1, 2, and certain technical conditions implying thatQ1(s) dominatesQ2(s) are obtained for values ofp ≠ ± 1.