Oscillation of solutions for second-order nonlinear difference equations with nonlinear neutral term

Some Riccati type difference inequalities are given for the second-order nonlinear difference equations with nonlinear neutral term.

and using these inequalities, we obtain some oscillation criteria for the above equation.     

Positive solutions of higher-order nonlinear delay difference equations

Classification schemes for positive solutions of a class of higher-order nonlinear delay difference equations are given in terms of their asymptotic behavior, and necessary as well as sufficient conditions for the existence of these solutions are also provided.     

The oscillation of partial difference equations with continuous arguments

For the partial difference equations

and

we shall obtain sufficient conditions for the oscillation of all solutions of these equations.     

Stability of solutions of differential equations with retarded arguments

A criterion is given for the stability of trivial solutions of the differential equations with the variable coefficients and variable delays. The derivation is based on Lyapunov's direct method.     

Oscillation of nonlinear partial difference equations with delays

In this paper, we first derive a discrete Gaussian formula and then apply the formula to two classes of nonlinear (parabolic and hyperbolic) partial difference equations with delays to obtain sufficient conditions under which every solution of the two classes of nonlinear partial difference equations with delays is oscillatory.     

Oscillation in nonlinear difference equations

In this paper, we shall discuss oscillatory behavior of the solutions of difference equations, including the self-adjoint second-order linear equation and the discrete version of the nonlinear wave equation. Our work is to give sufficient conditions such that every nontrivial solution of the equations oscillates.     

Periodic solutions of a class of nonlinear difference equations via critical point method

In this paper, we obtain some new sufficient conditions for the existence of nontrivial m-periodic solutions of the following nonlinear difference equation

by using the critical point method, where f: Z × R → R is continuous in the second variable, m ≥ 2 is a given positive integer, p_n+m = p_n for any n  Z and f(t + m, z) = f(t, z) for any (t, z)  Z × R, (−1)^δ = −1 and δ > 0.     

Regularity for solutions of nonlinear evolution equations with nonlinear perturbations

The approximate controllability for the nonlinear control system with nonlinear monotone hemicontinuous and the nonlinear perturbations is studied. The existence, uniqueness, and a variation of solutions of the system are also given.     

Stability of θ-methods for advanced differential equations with piecewise continuous arguments

This paper deals with the stability analysis of numerical methods for the solution of advanced differential equations with piecewise continuous arguments. We focus on the behaviour of the one-leg θ-method and the linear θ-method in the solution of the equation x′(t) = ax(t + a₀x([t]) + a₁x([t+1]), with real a, a₀, a₁ and [·] designates the greatest-integer function. The stability regions of two θ-methods are determined. The conditions under which the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given.     

Convolutional solutions of partial difference equations

A significant part of the theory of one-dimensional linear shift-invariant systems is based on the concept of weighting function (or impulse response): the output is the convolution of the weighting function with the input. This paper introduces the concept of linear translation-invariant systems and uses this notion in studying impulse response, z-transforms, and transfer functions for multidimensional systems.     

Asymptotic and oscillatory behavior of solutions of general nonlinear difference equations of second order

The asymptotic and oscillatory behavior of solutions of some general second-order nonlinear difference equations of the form

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=0 nZ,

is studied. Oscillation criteria for their solutions, when “p_n” is of constant sign, are established. Results are also presented for the damped-forced equation

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=e_n nZ.

Examples are inserted in the text for illustrative purposes.     

Unbounded solutions of quasi-linear difference equations

We study positive increasing solutions of the nonlinear difference equation $\text{δ(a}{}_{\mathrm{n}}\text{φ}{}_{\mathrm{p}}\text{(δχ}{}_{\mathrm{n}}\text{))=b}{}_{\mathrm{n}}\text{f(χ}{}_{\mathrm{n+1}}\text{,}\text{φ}{}_{\mathrm{p}}\text{(u)=|u|}{}^{\mathrm{p-2}}\text{u,}\text{p>1}$ where {a_n}, {b_n} are positive real sequences for n ≥ 1, fR → R is continuous with uf(u) > 0 for u ≠ 0. A full characterization of limit behavior of all these solutions in terms of a_n, b_n is established. Examples, showing the essential role of used hypotheses, are also included. The tools used are the Schauder fixed-point theorem and a comparison method based on the reciprocity principle.     

Periodic solutions of neutral difference equations

For linear or convex neutral difference equations with finite delay and with infinite delay, the Massera-type theorems are established. That is, the necessary and sufficient condition for existence of a periodic solution is the existence of a bounded solution. In this way, the Massera theorem for the ordinary differential equations is extended to neutral difference equations. As immediate implications the corresponding new results for delay difference equations are obtained since the latter can be regarded as a special case of the former.     

Oscillations and nonoscillations of half-linear difference equations generated by deviating arguments

For the half-linear difference equation

where α > 0, we shall offer sufficient conditions for the oscillation of all solutions, as well as necessary and sufficient conditions for the existence of both bounded and unbounded nonoscillatory solutions. Several examples which dwell upon the importance of our results are also included.     

Three positive solutions to second-order three-point impulsive differential equations with deviating arguments

The existence of at least three positive solutions to three-point boundary conditions for second-order impulsive differential equations with deviating arguments is discussed in this paper. Deviating arguments may be of advanced or delayed types. To obtain the results the Leggett–Williams fixed point is applied [R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979), pp. 673–688.]. An example is added to verify the obtained results.