On the Growth of Nonoscillatory Solutions for Difference Equations with Deviating Argument

The half-linear difference equations with the deviating argument Δ(an|Δxn|αsgn Δxn)+bn|xn+q|αsgn xn+q=0 , q ∈ ℤ are considered. We study the role of the deviating argument q, especially as regards the growth of the nonoscillatory solutions and the oscillation. Moreover, the problem of the existence of the intermediate solutions is completely resolved for the classical half-linear equation (q = 1). Some analogies or discrepancies on the growth of the nonoscillatory solutions for the delayed and advanced equations are presented; and the coexistence with different types of nonoscillatory solutions is studied

UA12/2/1 College Heights Herald, Vol. 53, No. 23

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Global monotonicity and oscillation for second order differential equation

summary:Oscillatory properties of the second order nonlinear equation $$(r(t)x^{\prime })^{\prime }+q(t)f(x)=0$$ are investigated. In particular, criteria for the existence of at least one oscillatory solution and for the global monotonicity properties of nonoscillatory solutions are established. The possible coexistence of oscillatory and nonoscillatory solutions is studied too