Difference equations associated with fully nonlinear boundary value problems for second order ordinary differential equations

We study the continuous problem y"=f(x,y,y'), xc[0,1], 0=G((y(0),y(1)),(y'(0), y'(1))), and its discrete approximation (y(k+1)-2y(k)+y(k-1))/h(2) =f(t(k), y(k), v(k)), k = 1,..., n-1, 0 = G((y(0), y(n)), (v(1), v(n))), where f and G = (g(0), g(1)) are continuous and fully nonlinear, h = 1/n, v(k) = (y(k) - y(k-1))/h, for k =1,..., n, and t(k) = kh, for k = 0,...,n. We assume there exist strict lower and strict upper solutions and impose additional conditions on f and G which are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. We show that the discrete approximation also has solutions which approximate solutions of the continuous problem and converge to the solution of the continuous problem when it is unique, as the grid size goes to 0. Homotopy methods can be used to compute the solution of the discrete approximation. Our results were motivated by those of Gaines.

Authors' institutional affiliations in Australian intellectual and developmental disabilities journals: a comparison of two decades

The generation of new knowledge through research can contribute significantly to the improvement of services for individuals with intellectual and developmental disabilities. This study extends a previous study by Sigafoos, Roberts, and Couzens [Aust. N.Z.J. Dev. Disabil. 17 (1991) 331] by examining research productivity in intellectual and developmental disability in Australian journals for 1990-1999. Institutions that published research articles on intellectual and developmental disabilities in Australian journals in the 1990s were identified by noting the affiliations of authors. The most productive institutions were primarily universities in Australia and the United States of America. Publication trends in the decade of the 1990s are compared with trends of the previous decade (1980-1990). (C) 2003 Published by Elsevier Ltd.