The paper studies a functional equation and inequality which arise in dynamic programming of multi-stage decision processes. The authors are interested in the solvability of and iterative approximations for the solutions to the functional equation and inequality.
Let \(S\subset X\) and \(D\subset Y\) be the state and decision spaces, respectively, where \(X\) and \(Y\) are real Banach spaces. The authors study the functional equation and inequality stated below, where \(opt\) denotes \(sup\) or \(inf\), \(\lambda \) is a constant in \([0,1],\) \(x\) and \(y\), respectively, represent the state and decision vectors, \(f\left( x\right) \) denotes the optimal return function with initial state \(x\), and \(a\) and \(b\) denote transformation processes. \[ \begin{split} f\left( x\right) =\lambda \text{ }opt_{y\in D}\{u(x,y)+A(x,y,f(a(x,y))\}\\ +(1-\lambda )opt_{y\in D}\{v(x,y)+B(x,y,f(b(x,y))\},\forall x\in S \end{split}\tag{1} \]
\[ \begin{split} f\left( x\right) \geq \lambda \text{ }opt_{y\in D}\{u(x,y)+A(x,y,f(a(x,y))\}\\ +(1-\lambda )opt_{y\in D}\{v(x,y)+B(x,y,f(b(x,y))\},\forall x\in S \end{split}\tag{2} \] Let \(F\) denote the set of functions \(f\) from \(S\) to \(\mathbb{R}\equiv ]-\infty ,+\infty [ .\) Define the sets \(B\left( S\right),BC\left( S\right) ,BB\left( S\right) \) as follows: \[ \begin{aligned} & B\left( S\right) \equiv \left\{ f\in F:f\text{ }is\text{ }bounded\right\}\\ & BC\left( S\right) \equiv \{f\in B\left( S\right) :f\text{ }is\text{ } continuous\} \\ & BB\left( S\right) \equiv \{f\in F:f\text{ }is\text{ }bounded\text{ }on \text{ }each\text{ }bounded\text{ }subset\text{ }of\text{ }S\}. \end{aligned} \] Define the norm \(\|\cdot \|_{1}\)on \(B\left( S\right) \) and \(BC\left( S\right) \) by \(\|w\|_{1}\equiv \sup_{x\in S}|w\left( x\right)|\). Then \(\left( B\left( S\right) ,\|\cdot\|_{1}\right) \) and \(\left( BC\left( S\right) ,\|\cdot\|_{1}\right) \) are Banach spaces.
Also, denoting by \(\mathbb{N}\) the set of positive integers, define, \(\forall \left( k,f,g\right) \in\mathbb{N}\times BB\left( S\right) \times BB\left(S\right)\), \[ \begin{aligned} &\overline{B}\left( 0,k\right) \equiv \left\{ x\in S:\|x\|\leq k\right\} \\ & d_{k}\left( f,g\right) \equiv \sup \left\{ \mid f\left( x\right) -g\left(x\right) \mid :x\in \overline{B}\left( 0,k\right) \right\} \\ & d\left( f,g\right) \equiv \sum_{k=1}^{\infty }\frac{1}{2^{k}}\cdot \frac{d_{k}\left( f,g\right) }{1+d_{k}\left( f,g\right) } \end{aligned} \] Some preliminary lemmas are proved to establish that \(\left( BB\left(S\right) ,d\right) \) is a complete metric space induced by the countable family of pseudo-metrics \(\{d_{k}\}_{k\in\mathbb{N}}\).
Using several fixed-point theorems due to Krasnoselskii, Boyd–Wong and Liu, the paper goes on to prove theorems establishing the existence and/or uniqueness and iterative approximations of solutions for the functional equation \((1)\) in the Banach spaces \(BC(S)\) and \(B(S)\) and the complete metric space \(BB(S)\), respectively. Examples are provided to illustrate the usefulness of the theorems.
In the final section, utilizing the monotone iterative method, it goes on to prove a couple of theorems regarding the solvability and iterative approximations of the functional inequality \((2)\) in the complete metric space \((BB(S),d),\) and provide some illustrative examples.
The functional equation studied in the paper includes the functional equations in some of the earlier literature as special cases. The results of the paper extend, improve and unify some earlier results in the literature.