Ambrosetti–Prodi Periodic Problem Under Local Coercivity Conditions

Abstract

Abstract In this paper we focus on the periodic boundary value problem associated with the Liénard differential equation x ′′ + f ⁢ ( x ) ⁢ x ′ + g ⁢ ( t , x ) = s {x^{\prime\prime}+f(x)x^{\prime}+g(t,x)=s} , where s is a real parameter, f and g are continuous functions and g is T-periodic in the variable t. The classical framework of Fabry, Mawhin and Nkashama, related to the Ambrosetti–Prodi periodic problem, is modified to include conditions without uniformity, in order to achieve the same multiplicity result under local coercivity conditions on g. Analogous results are also obtained for Neumann boundary conditions.