Integrability and asymptotic behaviour of a differential-difference matrix equation

Asymptotic Behaviour of a Difference Equation with Complex-valued Coefficients

Abstract. The asymptotic behaviour for solutions of a difference equation ∆zn = f(n, zn), where the complex-valued function f(n, z) is in some meaning close to a holomorphic function h, and of a Riccati difference equation is studied using a Lyapunov function method. The paper is motivated by papers on the asymptotic behaviour of the solutions of differential equations with complex-valued right...

Difference integrability conditions for parameterized linear difference and differential equations

This paper is devoted to integrability conditions for systems of linear difference and differential equations with difference parameters. It is shown that such a system is difference isomonodromic if and only if it is difference isomonodromic with respect to each parameter separately. Due to this result, it is no longer necessary to solve non-linear difference equations to verify isomonodromici...

Oscillatory and Asymptotic Behaviour of a Homogeneous Neutral Delay Difference Equation of Second Order

In this paper we find sufficient conditions for every solution of the neutral delay difference equation ∆(rn∆(yn − pnyn−m)) + qnG(yn−k) = 0 to oscillate or to tend to zero or ±∞ as n → ∞, where ∆ is the forward difference operator given by ∆xn = xn+1−xn, pn, qn, and rn are infinite sequences of real numbers with qn ≥ 0, rn > 0. Different ranges of {pn} are considered. This paper improves,genera...

Asymptotic behaviour for a semilinear nonlocal equation

We study the semilinear nonlocal equation ut = J∗u− u− u in the whole R . First, we prove the global well-posedness for initial conditions u(x, 0) = u0(x) ∈ L(R ) ∩ L∞(RN ). Next, we obtain the long time behavior of the solutions. We show that different behaviours are possible depending on the exponent p and the kernel J : finite time extinction for p < 1, faster than exponential decay for the ...

Differential Equation of Sensitivity Matrix for Final Velocity Constraint