We consider a general class of problems of the minimization of convex integral functionals subject to linear constraints. Using Fenchel duality, we prove the equality of the values of the minimization problem and its associated dual problem. This equality is a variational criterion for the existence of a solution to a large class of inverse problems entering the class of generalized Fredholm integral equations. In particular, our abstract results are applied to marginal problems for stochastic processes. Such problems naturally arise from the probabilistic approaches to quantum mechanics.