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We study the existence of positive solutions on the half-line $[0,\infty)$for the nonlinear second order differential equation \[\bigl(a(t)x^{\prime}\bigr)^{\prime}+b(t)F(x)=0,\quad t\geq0, \] satisfyingDirichlet type conditions, say $x(0)=0$, $\lim_{t\rightarrow\infty}x(t)=0$. Thefunction $b$ is allowed to change sign and the nonlinearity $F$ is assumed tobe asymptotically linear in a neighborhood of zero and infinity. Our resultscover also the cases in which $b$ is a periodic function for large $t$ or it isunbounded from below.

A Dirichlet problem on the half-line for nonlinear equations with indefinite weight