New applications of the existence of solutions for equilibrium equations with Neumann type boundary condition

Using the existence of solutions for equilibrium equations with a Neumann type boundary condition as developed by Shi and Liao (J. Inequal. Appl. 2015:363, 2015), we obtain the Riesz integral representation for continuous linear maps associated with additive set-valued maps with values in the set of all closed bounded convex non-empty subsets of any Banach space, which are generalizations of integral representations for harmonic functions proved by Leng, Xu and Zhao (Comput. Math. Appl. 66:1-18, 2013). We also deduce the Riesz integral representation for set-valued maps, for the vector-valued maps of Diestel-Uhl and for the scalar-valued maps of Dunford-Schwartz.