RETRACTED ARTICLE: New applications of the existence of solutions for equilibriumequations with Neumann type boundary condition

Zhaoqi Ji,Tao Liu,Hong Tian,Tanriver Ülker

doi:10.1186/s13660-017-1357-4

Abstract

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

Highlights

). We denote A by Ch(E ) the space of all continuous real-valued map defined on E and positively homogeneous
E The Riesz-Markov-Kakutani representation theorem states that, for every positive functional L on the space Cc(T) of continuous compact supported functional on a locally com-T pact Hausdorff space T, there exists a unique Borel regular measure μ on T such thatL(f ) = f dμ for all f ∈ Cc(T)
Leng, Xu and Zhao gave the integral representation for continuous functionals defined on the space C(T) of all continuous real-valued func

Summary

Introduction

). We denote A by Ch(E ) the space of all continuous real-valued map defined on E and positively homogeneous. Let T be a non-empty set, let A be an algebra consisting of subsets of T and let B(T; R) be the space of all bounded real-valued functions defined on T, endowed with the topology of uniform convergence. The space of all bounded set-valued measures defined on A with values in cfb(E).

Results

Conclusion