Existence results for second-order monotone differential inclusions on the positive half-line

Consider in a real Hilbert space H the differential equation (inclusion) (E): p ( t ) u ″ ( t ) + q ( t ) u ′ ( t ) ∈ A u ( t ) + f ( t ) a.e. in ( 0 , ∞ ) , with the condition (B): u ( 0 ) = x ∈ D ( A ) ¯ , where A : D ( A ) ⊂ H → H is a (possibly set-valued) maximal monotone operator whose range contains 0; p , q ∈ L ∞ ( 0 , ∞ ) , with ess inf p > 0 and q + ∈ L 1 ( 0 , ∞ ) . More than four decades ago, V. Barbu established the existence of a unique bounded solution to (E), (B), in the particular case p ≡ 1 , q ≡ 0 and f ≡ 0 . Subsequently the existence and uniqueness of bounded solutions in the homogeneous case ( f ≡ 0 ) have been further investigated by H. Brezis (1972), N. Pavel (1976), L. Véron (1974–1976), and by E.I. Poffald and S. Reich (1986) when A is an m-accretive operator in a Banach space. The non-homogeneous case has received less attention from this point of view. In this paper, we prove existence and uniqueness of bounded solutions to (E), (B) in the general case of non-constant functions p, q satisfying the mild conditions above, thus compensating for the lack of existence theory for such kind of second order problems. Note that our results open up the possibility to apply Lionsʹ method of artificial viscosity towards approximating the solutions of some nonlinear parabolic and hyperbolic problems, as shown in the last section of the paper.