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The paper concerns the investigation of nonconvex and nondifferentiableintegral functionals on general Banach spaces, which may not be reflexiveand/or separable. Considering two major subdifferentials of variationalanalysis, we derive nonsmooth versions of the Leibniz rule onsubdifferentiation under the integral sign, where the integral of thesubdifferential set-valued mappings generated by Lipschitzian integrands isunderstood in the Gelfand sense. Besides examining integration over completemeasure spaces and also over those with nonatomic measures, our specialattention is drawn to a stronger version of measure nonatomicity, known assaturation, to invoke the recent results of the Lyapunov convexity theorem typefor the Gelfand integral of the subdifferential mappings. The main results areapplied to the subdifferential study of the optimal value functions andderiving the corresponding necessary optimality conditions in nonconvexproblems of stochastic dynamic programming with discrete time on the infinitehorizon.

Subdifferentials of Nonconvex Integral Functionals in Banach Spaces with Applications to Stochastic Dynamic Programming