In this paper we consider the multipoint boundary value problem for one-dimensional p-Laplacian φ p (u ) + f (t, u) = 0, t ∈ (0, 1), subject to the boundary value conditions:where φ p (s) = |s| p−2 s, p > 1, ξ i ∈ (0, 1) with 0 < ξ 1 < ξ 2 < · · · < ξ n−2 < 1, and a i , b i satisfy a i , b i ∈ [0, ∞], 0 < n−2 i=1 a i < 1, and n−2 i=1 b i < 1. Using a fixed point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple (at least three) positive solutions to the above boundary value problem.  2005 Elsevier Inc. All rights reserved.

All solutions of a fourth-order nonlinear delay differential equation are shown to converge to zero or to oscillate. Novel Riccati type techniques involving third-order linear differential equations are employed. Implications in the deflection of elastic horizontal beams are also indicated.

For nonlinear difference equations of the formxn=F(n,xn−1,…,xn−m), it is usually difficult to find periodic solutions. In this paper, we consider a class of difference equations of the formxn=anxn−1+bnf(xn−k), where{an},  {bn}are periodic sequences andfis a nonlinear filtering function, and show how periodic solutions can be constructed. Several examples are also included to illustrate our results.

In this paper, the definitions of q-symmetric exponential function and q-symmetric gamma function are presented. By a q-symmetric exponential function, we shall illustrate the Laplace transform method and define and solve several families of linear fractional q-symmetric difference equations with constant coefficients. We also introduce a q-symmetric analogue Mittag-Leffler function and study q-symmetric Caputo fractional initial value problems. It is hoped that our work will provide foundation and motivation for further studying of fractional q-symmetric difference systems.MSC: 92B20; 68T05; 39A11; 34K13

The q-symmetric analogs of Cauchy's formulas for multiple integrals are obtained. We introduce the concepts of the fractional q-symmetric integrals and fractional q-symmetric derivatives and discuss some of their properties. By using some properties of q-symmetric fractional integrals and fractional difference operators, we study a boundary value problem with nonlocal boundary conditions. MSC: 26A33; 34B15