Enguiça, Ricardo RoqueGavioli, AndreaSanchez, Luis2011-11-282011-11-282010-11-01Enguiça R, Gavioli A, Sanchez L. Solutions of second-order and fourth-order ODEs on the half-line.Nonlinear Analysis-Theory Methods & Applications. 2010; 73 (9): 2968-2979.0362-546Xhttp://hdl.handle.net/10400.21/676We start by studying the existence of positive solutions for the differential equation u '' = a(x)u - g(u), with u ''(0) = u(+infinity) = 0, where a is a positive function, and g is a power or a bounded function. In other words, we are concerned with even positive homoclinics of the differential equation. The main motivation is to check that some well-known results concerning the existence of homoclinics for the autonomous case (where a is constant) are also true for the non-autonomous equation. This also motivates us to study the analogous fourth-order boundary value problem {u((4)) - cu '' + a(x)u = vertical bar u vertical bar(p-1)u u'(0) = u'''(0) = 0, u(+infinity) = u'(+infinity) = 0 for which we also find nontrivial (and, in some instances, positive) solutions.engSecond orderFourth orderNon-autonomous equationVariational methodsUnbounded intervalsPositive solutionjournal article