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.     

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.     

Further oscillation criteria for delay partial difference equations

This paper is concerned with the partial difference equation A_m+1,n+A_m,n+1−A_m,n+p_m,nA_m−k,n−l=0,m,n=0,1,2,…,, where k and l are two positive integers, {p_m,n} is a real double sequence. Some new oscillation criteria for this equation are obtained.     

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.     

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.     

Oscillation criteria for second-order quasi-linear neutral difference equations

In this paper, we obtain some oscillation criteria for the second-order quasi-linear neutral delay difference equation

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, where > 0, τ ≥ 0, and σ ≥ 0 are constants, {a_n}, {p_n}, {q_n} are nonnegative sequences and f ε C(R,R).     

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.     

Oscillatory property of higher order nonlinear difference equations

In this paper, we study the oscillatory and asymptotic behaviour of solutions of higher order nonlinear difference equations of the form

We obtain some necessary and sufficient conditions for all bounded solutions of (*) to be oscillatory and for (*) to have a nonoscillatory solution of a special form.     

New oscillation criteria for second order linear difference equations with positive and negative coefficients

The oscillation of second order neutral difference equations with positive and negative coefficients of the form

is investigated. We obtain many new results using the comparison between both first order and second order difference equations. An example is given to show the strength of the obtained results.     

Oscillation theorems for certain neutral functional differential equations

In this paper, we shall offer sufficient conditions for the oscillation of all solutions to neutral functional differential equations of mixed type of the form

,here p and q are periodic functions.     

Oscillation theorems for second-order nonlinear neutral differential equations

The purpose of this paper is to study the oscillation of the second-order neutral differential equations of the form

(a(t)[z^′(t)]^γ)^′+q(t)x^β(σ(t))=0,     

Asymptotic behavior of certain Riccati difference equations

Matrix Riccati difference equations are investigated on the infinite index set. Under natural assumptions an existence and uniqueness theorem is proven. The existence of the asymptotic expansion of the solution and computability of its coefficients are shown, provided the coefficients of the equation have such an expansion.     

Linearized oscillations of first-order nonlinear neutral delay difference equations

In this paper, we establish an equivalence of oscillation for the first-order nonlinear neutral delay difference equation

and its corresponding linear equation

under weaker conditions on f_i.     

Oscillation criteria for second order quasilinear neutral delay differential equations on time scales

In this paper, we consider the second-order quasilinear neutral dynamic equation

(r(t)|Z^Δ(t)|^α−1Z^Δ(t))^Δ+q(t)|x(δ(t))|^β−1x(δ(t))=0,     

Non-oscillation of solutions of difference equations of third order

In this paper, sufficient conditions in terms of coefficient functions are obtained for non-oscillation of all solutions of a class of linear homogeneous third order difference equations of the form

     

New oscillation criteria for first-order delay difference equations

In this paper, some new oscillation criteria are obtained for the first-order delay difference equation

, where k,l₁,l₂,, l_s are positive integer numbers, f is a function. Our results improve the known results in the literature. And some examples are given to demonstrate the advantage of our results.     

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.     

Oscillation of certain two-dimensional nonlinear difference systems

Several new oscillation criteria for two-dimensional nonlinear difference systems are established. Examples which dwell upon the importance of our results are also included.