Oscillation Of Nonlinear Dynamic Equations Research Articles

THE CRITERIA FOR OSCILLATION OF TWO-DIMENSIONAL NEUTRAL DELAY DYNAMICAL SYSTEMS ON TIME SCALES

The neutral delay model has many applications in biology, ecology, man-made neural networks, mechanics and engineering. The main purpose of this paper is to establish new oscillation criteria for the two-dimensional neutral delay dynamical system [Formula: see text] on a time scale [Formula: see text], where [Formula: see text] and [Formula: see text] are positive rd-continuous functions, [Formula: see text], are in accordance with the definition given by S. H. Saker, Oscillation of nonlinear dynamic equations on time scales, Appl. Math. Comput. 148 (2004) 81–91.

Oscillation theorems for second order nonlinear dynamic equations

Some new criteria for the oscillation of nonlinear dynamic equations of the form $$\bigl(a(t)(x^{\Delta}(t))^{\alpha}\bigr)^{\Delta}+f(t,x^{\sigma}(t))=0$$ on a time scale \(\mathbb{T}\) are established.

Oscillation of second-order nonlinear delay dynamic equations on time scales

In this paper, first, we establish the equivalence of the oscillation of the nonlinear dynamic equations x ΔΔ ( t ) + p ( t ) f ( x ( t − τ ) ) = 0 and x ΔΔ ( t ) + p ( t ) ( f o x σ ) = 0 , on time scales, from which we obtain some oscillation criteria and comparison theorems for the first equation. Next, we obtain some new oscillation criteria for second-order linear dynamic equations on time scales.

Boundedness and oscillation for nonlinear dynamic equations on a time scale

We obtain some boundedness and oscillation criteria for solutions to the nonlinear dynamic equation \[ ( p ( t ) x Δ ( t ) ) Δ + q ( t ) ( f ∘ x σ ) = 0 , (p(t)x^{\Delta }(t))^{\Delta }+q(t)(f\circ x^{\sigma })=0, \] on time scales. In particular, no explicit sign assumptions are made with respect to the coefficient q ( t ) q(t) . We illustrate the results by several examples, including a nonlinear Emden–Fowler dynamic equation.

Oscillation of nonlinear dynamic equations on time scales

By means of Riccati transformation techniques, we establish some new oscillation criteria for the second order nonlinear dynamic equation (p(t)x Δ (t)) Δ +q(t)f(x(σ(t)))=0, t∈[a,b], on time scales in terms of the coefficients and the graininess function. We illustrate our results with a nontrivial example.