Many other boundary conditions could be considered, however, since the analysis is similar we choose to omit the details. For example if B denoted y’(0) = y(1) = 0 the results of this case follow from the results when B is denoted by (i), by making a change of variable. Problems of the form (1 .l) have been studied by a variety of authors, see [l, 3, 5-7, lo-141 for example. The most advanced results to date seem to be those in [3, 7 and 111. However, in these papers f(x, y) is decreasing in y for each x E (0, 1). Here we will not require this assumption. For example we will show that y” + y-O + y” = 0, 0 < t < 1; y(0) = y’(1) = 0 with 0 < #I < 2, 0 5 (Y < 1 has a solution. Furthermore in fact in the case where f(x, y) is decreasing in y our results improve those of [3, 71 and [ 111. To show existence of a solution to (1.1) we will use the technique initiated in [lo] and [ll]. The methods in those papers rely on the topological transversality theorem [8-91, the Arzela-Ascoli theorem and apriori bounds on solutions.