and Applied Analysis 3 Now, let f̃ : R 1 → R be a continuous function, T -periodic with respect to the first variable, and consider the nth-order differential equation u n f̃ ( t, u, u′, u′′, . . . , u n−1 ) . 2.5 Lemma 2.1 see 14 . Assume that the following conditions hold. i There exists ρ > 0 such that, for each λ ∈ 0, 1 , one has that any possible T -periodic solution u of the problem u n λf̃ ...

We prove existence results for solutions of periodic boundary value problems concerning the nth-order differential equation with p-Laplacian [φ(x(n−1)(t))]′ = f (t,x(t),x′(t), . . . , x(n−1)(t)) and the boundary value conditions x(i)(0)=x(i)(T), i= 0, . . . ,n− 1. Our method is based upon the coincidence degree theory of Mawhin. It is interesting that f may be a polynomial and the degree of som...

where 01, ,l3, y and the x’s are real. It will be assumed throughout this paper that f (t, Ul , u, ,.-., un) is continuous on [CX, r] x R”. The approach taken here is similar to that of Barr and Sherman [I] and is based on the use of a solutionmatching technique that assumes existence and/or uniqueness of solutions to corresponding problems for the subintervals [cu, ,Kj and [p, y]. We refer to ...

We study the equivalences and implications between linear (or homogeneous) nth order fractional differential equations (FDEs) integral in spaces L1(a,b) C[a,b] when n≥ 2. establish equivalence of IVP FDE subject to initial condition u(i)(a)=ui for i {0,1,...,n-1} n≥2. The difficulty is that known conditions such first FDEs are not sufficient with In this article we provide additional ensure par...

Let f(z) be an analytic or meromorphic function in the closed unit disk sampled at the nth roots of unity. Based on these data, how can we approximately evaluate f(z) or f (m)(z) at a point z in the disk? How can we calculate the zeros or poles of f in the disk? These questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integ...

We address the question of computing one selected term of an algebraic power series. In characteristic zero, the best known algorithm for computing the Nth coefficient of an algebraic series uses differential equations and has arithmetic complexity quasi-linear in √ N . We show that in positive characteristic p, the complexity can be lowered to O(logN). The mathematical basis of this dramatic i...