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<p/> <p>The main purpose of this paper is to study the functional equation arising in dynamic programming of multistage decision processes <inline-formula> <graphic file="1687-1812-2011-701519-i1.gif"/></inline-formula>,<inline-formula> <graphic file="1687-1812-2011-701519-i2.gif"/></inline-formula><inline-formula> <graphic file="1687-1812-2011-701519-i3.gif"/></inline-formula>. A few iterative algorithms for solving the functional equation are suggested. The existence, uniqueness and iterative approximations of solutions for the functional equation are discussed in the Banach spaces <inline-formula> <graphic file="1687-1812-2011-701519-i4.gif"/></inline-formula> and <inline-formula> <graphic file="1687-1812-2011-701519-i5.gif"/></inline-formula> and the complete metric space <inline-formula> <graphic file="1687-1812-2011-701519-i6.gif"/></inline-formula>, respectively. The properties of solutions, nonnegative solutions, and nonpositive solutions and the convergence of iterative algorithms for the functional equation and other functional equations, which are special cases of the above functional equation, are investigated in the complete metric space <inline-formula> <graphic file="1687-1812-2011-701519-i7.gif"/></inline-formula>, respectively. Eight nontrivial examples which dwell upon the importance of the results in this paper are also given.</p>