Linear difference operator with multiple variable parameters and applications to second-order differential equations

In this article, we first investigate the linear difference operator $(Ax)(t):=x(t)-\sum_{i=1}^{n}c_{i}(t)x(t- \delta _{i}(t))$(Ax)(t):=x(t)−∑i=1nci(t)x(t−δi(t)) in a continuous periodic function space. The existence condition and some properties of the inverse of the operator A are explicitly pointed out. Afterwards, as applications of properties of the operator A, we study the existence of periodic solutions for two kinds of second-order functional differential equations with this operator. One is a kind of second-order functional differential equation, by applications of Krasnoselskii’s fixed point theorem, some sufficient conditions for the existence of positive periodic solutions are established. Another one is a kind of second-order quasi-linear differential equation, we establish the existence of periodic solutions of this equation by an extension of Mawhin’s continuous theorem.

HYERS-ULAM STABILITY OF FIRST ORDER LINEAR DIFFERENCE OPERATORS ON BANACH SPACE

In this work, the Hyers-Ulam stability of first order linear difference operator TP defined by (Tpu)(n) = ∆u(n) - p(n)u(n); is studied on the Banach space X = l∞, where p(n) is a sequence of reals.

Explicit Fourth-Derivative Two-Step Linear Multistep Method for Ordinary Differential Equations

Many problems from science and engineering are modeled by Ordinary Differential Equations (ODEs) whose solutions describe the temporal evolution of the modeled processes. In most cases however, the arising equations are too complex to be solved analytically. Consequently, their solutions have to be approximated by numerical methods. In this article, we propose an explicit fourth-derivative two-step linear multistep method (FD2LMM) for ordinary differential equations. The proposed method is constructed by using the maximal order criteria which is obtained through the associated linear difference operator. The starting values used by the proposed method are obtained by suitable single-step method. The order, consistency, linear stability, and the convergence properties of the method are discussed. Numerical experiments are performed and the results are compared with those of existing methods in the literature.