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We prove existence theorems for first order boundary value problems on (0, ∞), of the formu + F ( · , u) = f , P u(0) = ξ, where the function F = F (t, u) has a t-independent limit F ∞ (u) at infinity and P is a given projection. The right-hand side f is in L p ((0, ∞), R N ) and the solutions u are sought in W 1,p ((0, ∞), R N ), so that they tend to 0 at infinity. By using a degree for Fredholm mappings of index zero, we reduce the existence question to finding a priori bounds for thedoi:10.12775/tmna.2005.005 fatcat:6zt3qw5wu5eq3cgimauksylcgy