Second order ordinary differential equations with fully nonlinear two-point boundary conditions. I

Abstract

We establish existence results for two point boundary value problems for second order ordinary differential equations of the form y '' = f(x,y,y'), x epsilon [0, 1], where f is continuous and there exist lower and upper solutions. First we consider boundary conditions of the form G((y(0), y(1)); (y'(0), y'(1))) = 0, where G is continuous and fully nonlinear. We introduce compatibility conditions between G and the lower and upper solutions. Assuming these compatibility conditions hold and, in addition, f satisfies assumptions guarenteeing a'priori bounds on the derivatives of solutions we show that solutions exist. In the case the lower and upper solutions are constants one of our results is closely related to a result of Gaines and Mawhin. Secondly we consider boundary conditions of the form (y(i), y'(i)) epsilon J(i), i = 0, 1 where the J(i) are closed connected subsets of the plane. We introduce various compatibility type conditions relating the J(i) and the lower and upper solutions and show each is sufficient to construct a compatible G which defines these boundary conditions. Thus our existence results apply. Almost all the standard boundary conditions considered in the literature assuming upper and lower solutions are, or can be, de fined by compatible G and their associated existence results follow from ours; in many cases we can improve these results by deleting some of their assumptions.