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Abstract: A multiplicative calculus dealing with real valued functions is extended to a multiplicative type complex calculus (MCC) dealing with complex valued functions. Some fundamental theorems and concepts of the classical calculus are interpreted from the view point of the MCC and the analogies between them are given. Also new notations for the MCC are defined. The MCC is distinguished from the classical calculus by calling the classical calculus as the additive type complex calculus (ACC). [ABSTRACT FROM AUTHOR]

Abstract: In this paper, some necessary and sufficient conditions for the oscillation of a differential equation with fractional delay and piecewise constant argument are obtained. [Copyright &y& Elsevier]

Abstract: The principle of this paper is to deal with a new existence theory for positive periodic solutions to a kind of nonautonomous functional differential equations with impulse actions at fixed moments. Easily verifiable sufficient criteria are established. The approach is based on the fixed-point theorem in cones. The paper extends some previous results and obtains some new results. [Copyright &y& Elsevier]

Abstract: We find conditions for the solution to a first-order differential equations to be: bounded by an exponential function, uniformly stable, and uniformly asymptotically stable. Estimates are obtained using Liapunov functionals (energy functionals). Our assumptions are weaker than those in the references, and results are extended to systems of equations and to nonlinear equations. [Copyright &y& Elsevier]

Abstract: By using averaging functions, new interval oscillation criteria are established for the second-order functional differential equation,that are different from most known ones in the sense that they are based on information only on a sequence of subintervals of [t 0, ∞], rather than on the whole half-line. Our results can be applied to three cases: ordinary, delay, and advance differential equations. In the case of half-linear functional differential equations, our criteria implies that the τ(t) ≤ t delay and Gt(t) ≥ t advance cases do not affect the oscillation. In particular, several examples are given to illustrate the importance of our results. [Copyright &y& Elsevier]

Abstract: In this paper, the author investigates two-species nonautonomous delay diffusive preypredator models with Michaelis-Menten functional response. Sufficient conditions are derived for permanence of the species. Further, the author also establishs the sufficient conditions that guarantee the arbitrary positive periodic solution of the systems is global attractive. [Copyright &y& Elsevier]

Abstract: This paper discusses the antiperiodic boundary value problem for first-order impulsive ordinary differential equations. We establish several existence results by using the Leray-Schauder alternative, the lower and upper solution method and the monotone iterative technique. [Copyright &y& Elsevier]

This paper is concerned with the limit circle and limit point criteria of second-order linear difference equations. A sufficient and necessary condition, a sufficient and necessary condition subject to a certain restriction, and several sufficient conditions are established. These results improve and extend some previous results. [Copyright &y& Elsevier]

For the second-order delay differential equation y″(t)+a(t)y′(t)+∑lower limit i=1, upper limit n pi(t)y(gi(t))=0, t>0, t≠tk with impulsive conditions y(tκ+)-y(tκ−)=bκy(tκ−), y′(tκ+)-y′(tκ−)=bκy′(tκ−), an explicit necessary and sufficient condition for all bounded solutions to be oscillatory is obtained by the comparison theorem on bounded oscillation of the impulsive differential equation with the corresponding nonimpulsive differential equation. [Copyright &y& Elsevier]

In recent years, many authors demonstrated the usefulness of fractional calculus operators in the derivation of (explicit) particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. In particular, by making use of the operators of fractional calculus based upon the Cauchy-Goursat integral formula, a certain family of nearly-simple harmonic vibration equations was considered systematically in a series of papers which appeared quite recently. The main object of the present sequel to all these earlier works is to investigate some relationships between such families of fractional differential equations and certain classes of ordinary differential equations. A preliminary interpretation of this family of nearly-simple harmonic vibration equations by means of an example involving linear electric circuits is also provided. [Copyright &y& Elsevier]

In this paper, we consider the approximate solution of the following problem. Given ω⊂RN (1 ⪯ N ⪯ 3), find u(x, t) such that ∂tb(u)−Δu+Pe▽u=0, in Q&z.tbnd;ω×(0,T), u=0, on ∑&z.tbnd;∂ω×(0,T), u(x,0)=u0(x), ∀x∈ω, where the function b(s) is a monotonically increasing function satisfying 0 ⪯ b′ ⪯ ∞ To solve this problem, we introduce a new nonstandard time discretization scheme. We prove stability and convergence results. [Copyright &y& Elsevier]

In this paper, the existence of at least three positive solutions for the boundary value problem (BVP) of second-order functional differential equation with the form y″(t) + f(t, yt) = 0, for t ∊ [0,1], αy(t) -βy′(t) =η(t), for t ∊ [−τ,0], −γy(t) + Δy′(t) = ζ(t), for t ∊ [1, 1 + a], is studied. Moreover, we investigate the existence of at least three partially symmetric positive solutions for the above BVP with αΔ = βγ. [Copyright &y& Elsevier]