Heteroclinic Solutions for Classical and Singular φ-Laplacian Non-Autonomous Differential Equations

In this paper, we consider the second order discontinuous differential equation in the real line, (a (t,u) phi(u'))' = f (t,u,ut), a.e.t is an element of R, u(-infinity) = v(-), u(+proportional to) = v(+), with phi an increasing homeomorphism such that phi(0) = 0 and phi(R) = R, a is an element of C(R-2, R) with a(t,x) > 0 for (t, x) is an element of R-2, f: R-3 -> R a L-1-Caratheodory function and v(-), v(+) is an element of R such that v(-) < v(+). The existence and localization of heteroclinic connections is obtained assuming a Nagumo-type condition on the real line and without asymptotic conditions on the nonlinearities phi and f. To the best of our knowledge, this result is even new when phi(y) = y, that is for equation(a (t, u(t)) u'(t))' = f (t, u(t), u' (t)), a.e.t is an element of R. Moreover, these results can be applied to classical and singular 0 -Laplacian equations and to the mean curvature operator.