... FunctionalEquations Arising in Dynamic ProgrammingZeqing Liu,1Jeong Sheok Ume,2 and Shin Min Kang31Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China2Department ... over existing results in the literature.Hindawi Publishing CorporationFixed Point Theory and ApplicationsVolume 2010, Article ID 905858, 19 pagesdoi:10.1155/2010/905858 Research Article On ... in this paper extend, improve, and unifythe corresponding results according to Bellman 1, Bellman and Roosta 5, Bhakta and Choudhury 6, Bhakta and Mitra 7,Liu8, Liu and Ume 11, and...