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We offer sufficient conditions for the asymptotic stability of the equilibrium point of linear neutral differential systems. An application of our results to a family of artificial neural networks of neutral type is also illustrated.

We investigate the oscillatory behavior of all solutions of a new class of first order nonlinear neutral difference equations. Several explicit oscillation criteria are established. Our main results are supported by illustrative examples.

Some new criteria for the oscillation of fourth order nonlinear difference equations of the formΔ(1a3(n)(Δ1a2(n)(Δ1a1(n)(Δx(n))α1)α2)α3)+δq(n)f(x[g(n)])=0, where δ=±1 are investigated.

and (B) (a(t) w(x>i)* + Q(t, x) = Pjt, x, 4. Problems of these types for less general equations have been studied by Onose [ 51, Kamanev [3] and Wong [ 71 for the equation 2 + q(t)./-(x) = 0, and Coles 1 l] for the equation 2 + Q(f, x) = 0. We are concerned with only continuable solutions of (A) and (B), which exist on some half line [ 1,, co 1. A solution x(t) of (A) (resp. (B)) will be called nonoscillatory if there exists a t, 2 t, such that x(t) # 0 for t >, t, , and x(t) will be called oscillatory if it has no last zero. Our main results are given in Section 2 (Theorems 2.1-2.3), which generalize some of the results established by Coles [ 11, Graef, Rankin and Spikes [2 1, and Kamanev [3]. In Section 3 we give some illustrative examples.

New oscillation criteria are given for a second order nonlinear functional differential equation ( α ( t ) χ ( t ))′ + δp ( t ) χ ′[ σ ( t )] − q ( t )| χ [ g ( t )]| λ sgn χ [ g ( t )] = 0, where δ = ±1 and λ > 0.

In this paper, sufficient conditions are established so that all bounded solutions of the linear system of difference equations (−1) m+1 δ m y i (n)+ σ j=1 N q ij y i (n−T ji )=0, m ⩾1, i=1,…N to be oscillatory, where qij are real numbers and τjj are positive integers. We shall also study the oscillatory behavior of all bounded solutions of the linear systems of neutral difference equations of the form (−1) m+1 δ m y i (n)+ σ j=1 N q ij y i (n−T ji )=0, m ⩾1, i=1,…N where c is real number and σ and τ are positive integers.

The asymptotic and oscillatory behavior of solutions of damped nonlinear second order differential equations with deviating arguments of the type (a(t) ψ(x(t)) x (t)) . + p(t) x (t) + q(t) + q(t)f(x[g(t)]) = 0 ( . = d/dt) is studied. Criteria for oscillation of all solutions when the damping coefficient “p” is of constant sign on [t0, ∞) are established. Results on the asymptotic and oscillatory behavior of solutions of the damped-forced equation (a(t)ψ(x(t)) x (t)) . + p(t) x (t) + q(t)f(x[g(t)]) = e(t) , where q is allowed to change signs in [t0, ∞), are also presented. Some of the results of this paper extend, improve, and correlate a number of existing criteria.

Some new integral conditions for the oscillation of the nonlinear nth order delay differential equation Open image in new window are established.

We study the asymptotic properties of the solutions of a class of even-order damped differential equations with p-Laplacian like operators, delayed and advanced arguments. We present new theorems that improve and complement related contributions reported in the literature. Several examples are provided to illustrate the practicability, maneuverability, and efficiency of the results obtained. An open problem is proposed.

The oscillation behavior of solutions for higher-order delay dynamic equations of neutral type is investigated by making use of comparison with second-order dynamic equa- tions. The method can be utilized to study other types of higher-order equations on time scales as well.

Sufficient conditions are derived for all solutions of a class of third-order nonlinear differential equations with a superlinear neutral term to be either oscillatory or convergent to zero asymptotically. Examples illustrating the results are included and some suggestions for further research are indicated.