Existence of nonoscillatory solutions of higher-order neutral delay difference equations with variable coefficients

In this paper, we consider the following higher-order neutral delay difference equations with positive and negative coefficients: Δ^m(x_n + cx_n−k) + p_nx_n−r − q_nx_n−l = 0, n≥n₀, where c ϵ R, m ⩾ 1, k ⩾ 1, r, l ⩾ 0 are integers, and {p_n}^∞_n=n0 and {q_n}_n=n0^∞ are sequences of nonnegative real numbers. We obtain the global results (with respect to c) which are some sufficient conditions for the existences of nonoscillatory solutions.     

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.     

Existence of nonoscillatory solutions of first-order linear neutral delay differential equations

In this paper, the existence of nonoscillatory solutions of the first-order neutral delay differential equations with variable coefficients and delays are studied. Some new sufficient conditions are given. In particular, conditions given in this paper are weaker than those known, so the results in this paper have wider application than the existing ones.     

The persistence of nonoscillatory solutions of difference equations under impulsive perturbations

We obtain a sufficient condition for the persistence of nonoscillatory solutions of the difference equation with continuous variable, ,under the impulsive perturbations, x(t_k+τ)−x(t_k)=I_k(x(t_k)),kN(1),     

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.     

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 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.     

Asymptotic behavior of positive solutions of sublinear differential equations of Emden-Fowler type

This paper is concerned with asymptotic analysis of positive solutions of the second-order nonlinear differential equation

(A)     

Asymptotic behavior of solutions of asymptotically constant coefficient systems of linear differential equations

Sufficient conditions are given for the n × n system y'=(A+P(t))y to have a solution such that as t → ∞, where λ is an eigenvalue of the constant matrix A and v is an associated eigenvector. The integrability conditions on P allow conditional convergence and the o(1) terms are specified precisely.     

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.     

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.     

Multiple solutions for higher-order difference equations

This paper establishes the existence of twin nonnegative solutions to (i) conjugate, and (ii) (n,p), higher-order discrete boundary value problems.     

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).     

Asymptotic stability and boundedness of non-linear difference equations†

The problem of asymptotic stability and boundedness of non-linear difference equations is considered, In this paper we shall show some results which guarantee asymptotic stability for the trivial solution. Wo wish to present results which provide explicit regions of asymptotic stability. The results replace these words, 1 sufficiently small', by explicit estimates which are simple.

The concept of asymptotic stability to difference equations can be quite useful in the design of system response to arbitrary initial conditions.     

Asymptotic properties of solutions to discrete Coulomb equations

We construct two finite-difference models for the Coulomb differential equation which arises in the quantum mechanics analysis of the scattering of two charged point particles. These difference equations correspond to the standard and Mickens-Ramadhani schemes for the Coulomb equation. Our major goal is to determine the first two terms in the asymptotic solutions and compare them to the corresponding solutions of the Coulomb differential equation. In particular, the form of the anomalous phase term is examined.     

Asymptotic behavior and blow up of solutions for semilinear parabolic equations at critical energy level

In this paper we study the initial boundary value problem of semilinear parabolic equations with semilinear term f(u). By using the family of potential wells method we prove that if f(u) satisfies some conditions, J(u₀) ≤ d and I(u₀) > 0, then the solution decays to zero exponentially as t → ∞. On the other hand, if J(u₀) ≤ d, I(u₀) < 0, then the solution blows up in finite time.