Solving a class of functional equations using fixed point theorems

Abstract

This paper is concerned with solvability of a class of functional equations arising in dynamic programming of multistage decision processes. Using the fixed point theorems due to Banach and Liu-Ume-Kang and iterative algorithms, some sufficient conditions which ensure the existence, uniqueness and iterative approximations of solutions for the functional equation in the Banach spaces $BC(S)$ and $B(S)$ and the complete metric space $BB(S)$ are provided. Four examples are constructed to illustrate the results presented in this paper.

MSC:49L20, 90C39.

1 Introduction

It is well know that the existence problems of solutions for various functional equations arising in dynamic programming are both of theoretical and of practical interest; for example, see [1–19] and the references cited therein. Bellman [2], Bhakta and Choudhury [5], Liu [9] and Liu et al. [11, 12, 14–16, 19] studied the existence, uniqueness and iterative approximations of solutions for the following functional equations arising in dynamic programming:

$$f(x)=\underset{y\in D}{inf}max\{u(x,y),v(x,y)f(a(x,y))\},\mathrm{\forall }x\in S,$$

(1.1)

$$f(x)=\underset{y\in D}{inf}max\{u(x,y),f(a(x,y))\},\mathrm{\forall }x\in S,$$

(1.2)

$$f(x)=\underset{y\in D}{sup}max\{u(x,y),f(a(x,y))\},\mathrm{\forall }x\in S,$$

(1.3)

$$f(x)=\underset{y\in D}{opt}opt\{u(x,y),f(a(x,y))\},\mathrm{\forall }x\in S,$$

(1.4)

$$f(x)=\underset{y\in D}{opt}\{u(x,y)max\{p(x,y),f(a(x,y))\}\},\mathrm{\forall }x\in S,$$

(1.5)

$$f(x)=\underset{y\in D}{opt}\{u(x,y)min\{p(x,y),f(a(x,y))\}\},\mathrm{\forall }x\in S$$

(1.6)

in the complete metric space $BB(S)$, where opt stands for the sup or inf.

Motivated and inspired by the results in [1–19], in this paper we introduce and study a new functional equation arising in dynamic programming of multistage decision processes as follows:

$$\begin{array}{rcl}f(x)& =& \lambda \underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),f(a(x,y))\}\}+(1-\lambda )\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),f(b(x,y))\}\\ +t(x,y)opt\{r(x,y),f(c(x,y))\}\},\mathrm{\forall }x\in S,\end{array}$$

(1.7)

where $\lambda \in [0,1]$ is a constant, x and y stand for the state and decision vectors, respectively, a, b and c denote the transformations of the processes, and $f(x)$ is the optimal return function with initial state x. Obviously, functional equation (1.7) includes functional equations (1.1)-(1.6) as special cases. Utilizing the Banach fixed point theorem and Liu-Ume-Kang fixed point theorem, some techniques in nonlinear analysis and a few iterative algorithms, we get the existence, uniqueness and iterative approximations of continuous bounded solutions, bounded solutions and solutions for functional equation (1.7) in the Banach spaces $BC(S)$ and $B(S)$ and the complete metric space $BB(S)$, respectively, and discuss some error estimates between the iterative sequences generated by the iterative algorithms and the solutions. Four nontrivial examples are given to show that the results presented in this paper are more general than those in [5, 6, 11, 12, 14–16, 19].

2 Preliminaries

Throughout this paper, we assume that $(X,\parallel \cdot \parallel )$ and $(Y,{\parallel \cdot \parallel }^{\prime })$ are real Banach spaces, $S\subset X$ is the state space, $D\subset Y$ is the decision space, ℕ denotes the set of all positive integers, ${\mathbb{N}}_0=\{0\}\cup \mathbb{N}$, $\mathbb{R}=(-\mathrm{\infty },+\mathrm{\infty })$, ${\mathbb{R}}^{+}=[0,+\mathrm{\infty })$ and ${\mathbb{R}}^{-}=(-\mathrm{\infty },0]$. Define

$$\begin{array}{c}{\mathrm{\Phi }}_1=\{\phi :\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}\text{ is nondecreasing and }\phi (t)<t\hfill \\ \phantom{{\mathrm{\Phi }}_1=}\text{ for each }t>0\},\hfill \\ {\mathrm{\Phi }}_2=\{(\phi ,\psi ):\phi \in {\mathrm{\Phi }}_1,\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}\text{ is nondecreasing and }\sum _{n=0}^{\mathrm{\infty }}\psi ({\phi }^n(t))<+\mathrm{\infty }\hfill \\ \phantom{{\mathrm{\Phi }}_2=}\text{ for each }t>0\},\hfill \\ B(S)=\{g:g:S\to \mathbb{R}\text{ is bounded}\},\hfill \\ BC(S)=\{g:g\in B(S)\text{ is continuous}\},\hfill \\ BB(S)=\{g:g:S\to \mathbb{R}\text{ is bounded on each bounded subsets of }S\}.\hfill \end{array}$$

Clearly, $(B(S),{\parallel \cdot \parallel }_1)$ and $(BC(S),{\parallel \cdot \parallel }_1)$ are Banach spaces with the norm ${\parallel g\parallel }_1={sup}_{x\in S}|g(x)|$. For each $k\in \mathbb{N}$ and $h,g\in BB(S)$, put

$$\begin{array}{c}{d}_k(h,g)=sup\{|h(x)-g(x)|:x\in \overline{B}(0,k)\},\hfill \\ d(h,g)=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{2^k}\cdot \frac{d_k(h,g)}{1+d_k(h,g)},\hfill \end{array}$$

where

$$\overline{B}(0,k)=\{x:x\in S\text{ and }\parallel x\parallel \le k\}.$$

Obviously, ${\{d_k\}}_{k\in \mathbb{N}}$ is a countable family of pseudometrics in $BB(S)$. A sequence ${\{x_k\}}_{k\in \mathbb{N}}$ in $BB(S)$ is said to converge to a point $x\in BB(S)$ if $d_k(x_n,x)\to 0$ as $n\to \mathrm{\infty }$ and ${\{x_n\}}_{n\in \mathbb{N}}$ is a Cauchy sequence if $d_k(x_n,x_m)\to 0$ as $n,m\to \mathrm{\infty }$ for each $k\in \mathbb{N}$. It is clear that $(BB(S),d)$ is a complete metric space.

Lemma 2.1 ([12])

Let E be a set, p and $q:E\to \mathbb{R}$ be mappings. If ${opt}_{y\in E}p(y)$ and ${opt}_{y\in E}q(y)$ are bounded, then

$$|\underset{y\in E}{opt}p(y)-\underset{y\in E}{opt}q(y)|\le \underset{y\in E}{sup}|p(y)-q(y)|.$$

Lemma 2.2 ([14])

Let α, β, γ and δ be in ℝ. Then

$$|opt\{\alpha ,\beta \}-opt\{\gamma ,\delta \}|\le max\{|\alpha -\gamma |,|\beta -\delta |\}.$$

Lemma 2.3 (Liu-Ume-Kang fixed point theorem [17])

Let $(G,\rho )$ be a complete metric space, ${\{{\rho }_k\}}_{k\in \mathbb{N}}$ be a countable family of pseudometrics on G such that for any different points $x,y\in G$, ${\rho }_k(x,y)>0$ for some $k\in \mathbb{N}$, and ρ be defined by

$$\rho (x,y)=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{2^k}\cdot \frac{{\rho }_k(x,y)}{1+{\rho }_k(x,y)},\mathrm{\forall }x,y\in G.$$

Assume that $T:G\to G$ satisfies that

$${\rho }_k(Tx,Ty)\le \phi ({\rho }_k(x,y)),\mathrm{\forall }(x,y,k)\in G^2\times \mathbb{N},$$

where $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is upper semicontinuous from the right on ${\mathbb{R}}^{+}$ and $\phi (t)<t$ for each $t>0$. Then T has a unique fixed point $w\in G$ and ${lim}_{n\to \mathrm{\infty }}{T}^n(x)=w$ for each $x\in G$.

3 Main results

Now we investigate the existence, uniqueness and iterative approximations of continuous bounded solutions and bounded solutions for functional equation (1.7) in the Banach spaces $BC(S)$ and $B(S)$, respectively, by using the Banach fixed point theorem and iterative algorithms.

Theorem 3.1 Let S be compact, $\lambda \in [0,1]$ and $\alpha \in [0,1)$. Let $p,q,r,u,v,t:S\times D\to \mathbb{R}$ and $a,b,c:S\times D\to S$ satisfy that

(C1) p, q and r are bounded in $S\times D$;

(C2) ${sup}_{(x,y)\in S\times D}max\{|u(x,y)|,|v(x,y)|+|t(x,y)|\}\le \alpha$;

(C3) for each $(x_0,g)\in S\times \{p,q,r,u,v,t,a,b,c\}$,

$$\underset{x\to x_0}{lim}g(x,y)=g(x_0,y)\mathit{\text{uniformly for}}y\in D.$$

Then functional equation (1.7) possesses a unique solution $w\in BC(S)$ such that

(C4) for each $w_0\in BC(S)$, the iterative sequence ${\{w_n\}}_{n\in {\mathbb{N}}_0}$ defined by

$$\begin{array}{rcl}{w}_n(x)& =& \lambda \underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),w_{n-1}(a(x,y))\}\}\\ +(1-\lambda )\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),w_{n-1}(b(x,y))\}\\ +t(x,y)opt\{r(x,y),w_{n-1}(c(x,y))\}\},\mathrm{\forall }(x,n)\in S\times \mathbb{N}\end{array}$$

(3.1)

converges to w and has the error estimate:

$$\begin{array}{r}{\parallel w_n-w\parallel }_1\le {\alpha }^n{\parallel w_0-w\parallel }_1\mathit{\text{and}}\\ {\parallel w_n-w\parallel }_1\le \frac{{\alpha }^n}{1-\alpha }{\parallel w_0-w_1\parallel }_1,\mathrm{\forall }n\in \mathbb{N}.\end{array}$$

(3.2)

Proof Define a mapping $H:BC(S)\to BC(S)$ by

$$\begin{array}{rcl}Hh(x)& =& \lambda \underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),h(a(x,y))\}\}\\ +(1-\lambda )\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),h(b(x,y))\}\\ +t(x,y)opt\{r(x,y),h(c(x,y))\}\},\mathrm{\forall }(x,h)\in S\times BC(S).\end{array}$$

(3.3)

Firstly, we show that H is a self-mapping in $BC(S)$. Let $(x_0,h)\in S\times BC(S)$ and $\epsilon >0$. It follows from (C1), (C3) and the compactness of S that there exist constants $M>0$, $\delta >0$ and ${\delta }_1>0$ such that

$$\begin{array}{c}\underset{(x,y)\in S\times D}{sup}max\{|h(x)|,\left|h(a(x,y))\right|,\left|h(b(x,y))\right|,\left|h(c(x,y))\right|,|p(x,y)|,|q(x,y)|,|r(x,y)|\}\hfill \\ \le M;\hfill \end{array}$$

(3.4)

$$\begin{array}{c}max\{|u(x,y)-u(x_0,y)|,|v(x,y)-v(x_0,y)|+|t(x,y)-t(x_0,y)|\}<\frac{\epsilon }{2M},\hfill \\ \mathrm{\forall }(x,y)\in S\times D\text{ with }\parallel x-x_0\parallel <\delta ;\hfill \end{array}$$

(3.5)

$$\begin{array}{c}max\{|p(x,y)-p(x_0,y)|,|q(x,y)-q(x_0,y)|,|r(x,y)-r(x_0,y)|\}<\frac{\epsilon }{2},\hfill \\ \mathrm{\forall }(x,y)\in S\times D\text{ with }\parallel x-x_0\parallel <\delta ;\hfill \end{array}$$

(3.6)

$$|h(x_1)-h(x_2)|<\frac{\epsilon }{2},\mathrm{\forall }{x}_1,x_2\in S\text{ with }\parallel x_1-x_2\parallel <{\delta }_1;$$

(3.7)

$$\begin{array}{c}max\{\parallel a(x,y)-a(x_0,y)\parallel ,\parallel b(x,y)-b(x_0,y)\parallel ,\parallel c(x,y)-c(x_0,y)\parallel \}<{\delta }_1,\hfill \\ \mathrm{\forall }(x,y)\in S\times D\text{ with }\parallel x-x_0\parallel <\delta .\hfill \end{array}$$

(3.8)

On account of (C2), (3.3)-(3.8), Lemmas 2.1 and 2.2, we infer that

$$\begin{array}{c}|Hh(x)-Hh(x_0)|\hfill \\ =|\lambda \underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),h(a(x,y))\}\}\hfill \\ +(1-\lambda )\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),h(b(x,y))\}+t(x,y)opt\{r(x,y),h(c(x,y))\}\}\hfill \\ -\lambda \underset{y\in D}{opt}\{u(x_0,y)opt\{p(x_0,y),h(a(x_0,y))\}\}\hfill \\ -(1-\lambda )\underset{y\in D}{opt}\{v(x_0,y)opt\{q(x_0,y),h(b(x_0,y))\}+t(x_0,y)opt\{r(x_0,y),h(c(x_0,y))\}\}|\hfill \\ \le \lambda |\underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),h(a(x,y))\}\}-\underset{y\in D}{opt}\{u(x_0,y)opt\{p(x,y),h(a(x,y))\}\}|\hfill \\ +\lambda |\underset{y\in D}{opt}\{u(x_0,y)opt\{p(x,y),h(a(x,y))\}\}-\underset{y\in D}{opt}\{u(x_0,y)opt\{p(x_0,y),h(a(x_0,y))\}\}|\hfill \\ +(1-\lambda )|\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),h(b(x,y))\}+t(x,y)opt\{r(x,y),h(c(x,y))\}\}\hfill \\ -\underset{y\in D}{opt}\{v(x_0,y)opt\{q(x,y),h(b(x,y))\}+t(x_0,y)opt\{r(x,y),h(c(x,y))\}\}|\hfill \\ +(1-\lambda )|\underset{y\in D}{opt}\{v(x_0,y)opt\{q(x,y),h(b(x,y))\}+t(x_0,y)opt\{r(x,y),h(c(x,y))\}\}\hfill \\ -\underset{y\in D}{opt}\{v(x_0,y)opt\{q(x_0,y),h(b(x_0,y))\}+t(x_0,y)opt\{r(x_0,y),h(c(x_0,y))\}\}|\hfill \\ \le \lambda \underset{y\in D}{sup}\{|u(x,y)-u(x_0,y)|max\{|p(x,y)|,\left|h(a(x,y))\right|\}\}\hfill \\ +\lambda \underset{y\in D}{sup}\left\{|u(x_0,y)||opt\{p(x,y),h(a(x,y))\}-opt\{p(x_0,y),h(a(x_0,y))\}|\right\}\hfill \\ +(1-\lambda )\underset{y\in D}{sup}\{|v(x,y)-v(x_0,y)|max\{|q(x,y)|,\left|h(b(x,y))\right|\}\hfill \\ +|t(x,y)-t(x_0,y)|max\{|r(x,y)|,\left|h(c(x,y))\right|\}\}\hfill \\ +(1-\lambda )\underset{y\in D}{sup}\{|v(x_0,y)||opt\{q(x,y),h(b(x,y))\}-opt\{q(x_0,y),h(b(x_0,y))\}|\hfill \\ +|t(x_0,y)||opt\{r(x,y),h(c(x,y))\}-opt\{r(x_0,y),h(c(x_0,y))\}|\}\hfill \\ \le \lambda M\underset{y\in D}{sup}|u(x,y)-u(x_0,y)|\hfill \\ +\lambda \alpha \underset{y\in D}{sup}max\{|p(x,y)-p(x_0,y)|,|h(a(x,y))-h(a(x_0,y))|\}\hfill \\ +(1-\lambda )M\underset{y\in D}{sup}\{|v(x,y)-v(x_0,y)|+|t(x,y)-t(x_0,y)|\}\hfill \\ +(1-\lambda )\underset{y\in D}{sup}\{(|v(x_0,y)|+|t(x_0,y)|)max\{|q(x,y)-q(x_0,y)|,\hfill \\ |h(b(x,y))-h(b(x_0,y))|,|r(x,y)-r(x_0,y)|,|h(c(x,y))-h(c(x_0,y))|\left\}\right\}\hfill \\ <\lambda M\frac{\epsilon }{2M}+\lambda \alpha \frac{\epsilon }{2}+(1-\lambda )M\frac{\epsilon }{2M}+(1-\lambda )\alpha \frac{\epsilon }{2}\hfill \\ <\epsilon ,\mathrm{\forall }x\in S\text{ with }\parallel x-x_0\parallel <\delta \hfill \end{array}$$

and

$$\begin{array}{c}|Hh(x)|\hfill \\ =|\lambda \underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),h(a(x,y))\}\}\hfill \\ +(1-\lambda )\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),h(b(x,y))\}+t(x,y)opt\{r(x,y),h(c(x,y))\}\}|\hfill \\ \le \lambda \underset{y\in D}{sup}\left\{|u(x,y)||opt\{p(x,y),h(a(x,y))\}|\right\}\hfill \\ +(1-\lambda )\underset{y\in D}{sup}\{|v(x,y)||opt\{q(x,y),h(b(x,y))\}|+|t(x,y)||opt\{r(x,y),h(c(x,y))\}|\}\hfill \\ \le \lambda \alpha \underset{y\in D}{sup}max\{|p(x,y)|,\left|h(a(x,y))\right|\}\hfill \\ +(1-\lambda )\underset{y\in D}{sup}\{[|v(x,y)|+|t(x,y)|]max\{|q(x,y)|,|r(x,y)|,\left|h(b(x,y))\right|,\left|h(c(x,y))\right|\}\}\hfill \\ \le \lambda \alpha M+(1-\lambda )\alpha M\hfill \\ =\alpha M,\mathrm{\forall }x\in S,\hfill \end{array}$$

which yields that Hh is bounded and continuous in S. That is, H maps $BC(S)$ into $BC(S)$.

Secondly, we show that H is a contraction mapping in $BC(S)$. Given $\epsilon >0$. In view of (3.3), Lemmas 2.1 and 2.2, we get that

$$\begin{array}{c}|Hh(x)-Hg(x)|\hfill \\ =|\lambda \underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),h(a(x,y))\}\}\hfill \\ +(1-\lambda )\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),h(b(x,y))\}+t(x,y)opt\{r(x,y),h(c(x,y))\}\}\hfill \\ -\lambda \underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),g(a(x,y))\}\}\hfill \\ -(1-\lambda )\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),g(b(x,y))\}+t(x,y)opt\{r(x,y),g(c(x,y))\}\}|\hfill \\ \le \lambda \underset{y\in D}{sup}\left\{|u(x,y)||opt\{p(x,y),h(a(x,y))\}-opt\{p(x,y),g(a(x,y))\}|\right\}\hfill \\ +(1-\lambda )\underset{y\in D}{sup}\{|v(x,y)||opt\{q(x,y),h(b(x,y))\}-opt\{q(x,y),g(b(x,y))\}|\hfill \\ +|t(x,y)||opt\{r(x,y),h(c(x,y))\}-opt\{r(x,y),g(c(x,y))\}|\}\hfill \\ \le \lambda \alpha \underset{y\in D}{sup}\left\{|h(a(x,y))-g(a(x,y))|\right\}+(1-\lambda )\underset{y\in D}{sup}\{[|v(x,y)|+|t(x,y)|]\hfill \\ \times max\{|h(b(x,y))-g(b(x,y))|,|h(c(x,y))-g(c(x,y))|\}\}\hfill \\ \le \alpha {\parallel h-g\parallel }_1,\mathrm{\forall }x\in S,h,g\in BC(S),\hfill \end{array}$$

which gives that

$${\parallel Hh-Hg\parallel }_1\le \alpha {\parallel h-g\parallel }_1,\mathrm{\forall }h,g\in BC(S),$$

(3.9)

that is, H is a contraction mapping in $BC(S)$. Thus the Banach fixed point theorem yields that H has a unique fixed point $w\in BC(S)$, which is a unique solution of functional equation (1.7) in $BC(S)$.

Thirdly, we show (C4). Note that

$$\begin{array}{rcl}w(x)& =& \lambda \underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),w(a(x,y))\}\}\\ +(1-\lambda )\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),w(b(x,y))\}\\ +t(x,y)opt\{r(x,y),w(c(x,y))\}\},\mathrm{\forall }x\in S,\end{array}$$

which together with (3.1), (3.3) and (3.9) yields that

$$\begin{array}{rcl}{\parallel w_n-w\parallel }_1& =& \underset{x\in S}{sup}|w_n(x)-w(x)|=\underset{x\in S}{sup}|H{w}_{n-1}(x)-Hw(x)|=\parallel H{w}_{n-1}-Hw\parallel \\ \le & \alpha {\parallel w_{n-1}-w\parallel }_1\le \cdots \le {\alpha }^n{\parallel w_0-w\parallel }_1,\mathrm{\forall }n\in \mathbb{N},\end{array}$$

(3.10)

which guarantees that the sequence ${\{w_n\}}_{n\in {\mathbb{N}}_0}$ converges to w. Similarly, we conclude that

$$\begin{array}{rcl}{\parallel w_n-w_{n+m}\parallel }_1& \le & \sum _{i=n}^{n+m-1}{\parallel w_i-w_{i+1}\parallel }_1=\sum _{i=n}^{n+m-1}{\parallel H{w}_{i-1}-H{w}_i\parallel }_1\le \sum _{i=n}^{n+m-1}\alpha {\parallel w_{i-1}-w_i\parallel }_1\\ \le & \cdots \le \sum _{i=n}^{n+m-1}{\alpha }^i{\parallel w_0-w_1\parallel }_1\le \frac{{\alpha }^n}{1-\alpha }{\parallel w_0-w_1\parallel }_1,\mathrm{\forall }(n,m)\in \mathbb{N}\times \mathbb{N}.\end{array}$$

Letting $m\to \mathrm{\infty }$ in the above inequalities, we infer that (3.2) holds. This completes the proof. □

Using the proof of Theorem 3.1, we have the following.

Theorem 3.2 Let $\alpha \in (0,1)$ and $\lambda \in [0,1]$. Let $p,q,r,u,v,t:S\times D\to \mathbb{R}$ and $a,b,c:S\times D\to S$ satisfy (C1) and (C2). Then functional equation (1.7) possesses a unique solution $w\in B(S)$ and for each $w_0\in B(S)$, the sequence ${\{w_n\}}_{n\in {\mathbb{N}}_0}$ defined by (3.1) converges to w and satisfies (C4).

Example 3.3 Consider the functional equation

$$\begin{array}{rcl}f(x)& =& \lambda \underset{y\in {\mathbb{R}}^{+}}{opt}\{\frac{cos(x^9+y^{18})}{y^2+2}opt\{\frac{x^{56}}{x^{31}+y^5+1},f\left({sin}^8(x^2-y^{35})\right)\}\}\\ +(1-\lambda )\underset{y\in {\mathbb{R}}^{+}}{opt}\{\frac{x^{2}sin(y^{14})}{2{(x+1)}^2+y^3}opt\{\frac{x^{25}ln(1+x^2)}{x^2+y^3+1},f\left(\frac{x^6}{x^2+y^2+1}\right)\}\\ +\frac{\sqrt{x+1}}{3\sqrt{x+2}+y^2}opt\{\frac{x^{10}{cos}^6(y^7-y)}{y^3+y^2+1},f\left(\frac{x^{7}y}{x^3+y^2+1}\right)\}\},\mathrm{\forall }x\in [0,40].\end{array}$$

(3.11)

Put $X=Y=\mathbb{R}$, $S=[0,40]$, $D={\mathbb{R}}^{+}$, $\lambda \in [0,1]$, $\alpha =\frac{5}{6}$. Let $p,q,r,u,v,t:S\times D\to \mathbb{R}$ and $a,b,c:S\times D\to S$ be defined by

$$\begin{array}{c}p(x,y)=\frac{x^{56}}{x^{31}+y^5+1},q(x,y)=\frac{x^{25}ln(1+x^2)}{x^2+y^3+1},r(x,y)=\frac{x^{10}{cos}^6(y^7-y)}{y^3+y^2+1},\hfill \\ u(x,y)=\frac{cos(x^9+y^{18})}{y^2+2},v(x,y)=\frac{x^{2}sin(y^{14})}{2{(x+1)}^2+y^3},t(x,y)=\frac{\sqrt{x+1}}{3\sqrt{x+2}+y^2},\hfill \\ a(x,y)={sin}^8(x^2-y^{35}),b(x,y)=\frac{x^6}{x^2+y^2+1},c(x,y)=\frac{x^{7}y}{x^3+y^2+1},\hfill \\ \mathrm{\forall }(x,y)\in S\times D.\hfill \end{array}$$

It is easy to see that the conditions of Theorem 3.1 are satisfied. It follows from Theorem 3.1 that functional equation (3.11) possesses a unique solution $w\in BC(S)$ and (C4) holds.

Example 3.4 Consider the functional equation

$$\begin{array}{rcl}f(x)& =& \lambda \underset{y\in {\mathbb{R}}^{-}}{opt}\{\frac{x{y}^2{sin}^{45}(x^9{y}^7)}{x^2+y^4+1}opt\{{sin}^3\left(x{y}^5\right),f\left(x^{59}\sqrt{x^4+y^{24}}\right)\}\}\\ +(1-\lambda )\underset{y\in {\mathbb{R}}^{-}}{opt}\{\frac{x^{3}arctan(x^{15}{y}^{21})}{2\pi x^3+y^2+1}opt\{{cos}^9(x^6{y}^3(x+y)),f\left(x^2{y}^6\right)\}\\ +\frac{2x^4-{sin}^2(xy)}{3x^4+{cos}^4(x^8{y}^6)}opt\{\frac{x^3-y^2}{{(x+1)}^4+y^2},f(x^{35}-y^{47}-xy)\}\},\mathrm{\forall }x\in {\mathbb{R}}^{+}.\end{array}$$

(3.12)

Set $X=Y=\mathbb{R}$, $S={\mathbb{R}}^{+}$, $D={\mathbb{R}}_{-}$, $\lambda \in [0,1]$, $\alpha =\frac{11}{12}$. Let $p,q,r,u,v,t:S\times D\to \mathbb{R}$ and $a,b,c:S\times D\to S$ be defined by

$$\begin{array}{c}p(x,y)={sin}^3\left(x{y}^5\right),q(x,y)={cos}^9(x^6{y}^3(x+y)),r(x,y)=\frac{x^3-y^2}{{(x+1)}^4+y^2},\hfill \\ u(x,y)=\frac{x{y}^2{sin}^{45}(x^9{y}^7)}{x^2+y^4+1},v(x,y)=\frac{x^{3}arctan(x^{15}{y}^{21})}{2\pi x^3+y^2+1},\hfill \\ t(x,y)=\frac{2x^4-{sin}^2(xy)}{3x^4+{cos}^4(x^8{y}^6)},a(x,y)=x^{59}\sqrt{x^4+y^{24}},b(x,y)=x^2{y}^6,\hfill \\ c(x,y)=x^{35}-y^{47}-xy,\mathrm{\forall }(x,y)\in S\times D.\hfill \end{array}$$

It is clear that the conditions of Theorem 3.2 are fulfilled. Thus Theorem 3.2 guarantees that functional equation (3.12) possesses a unique solution $w\in B(S)$, which satisfies (C4).

Next we prove the existence, uniqueness and iterative approximation of solutions for functional equation (1.7) in the complete metric space $BB(S)$ by using Liu-Ume-Kang fixed point theorem.

Theorem 3.5 Let $\alpha \in (0,1)$, $\lambda \in [0,1]$, $p,q,r,u,v,t:S\times D\to \mathbb{R}$ and $a,b,c:S\times D\to S$ satisfy that

(C5) p, q and r are bounded on $\overline{B}(0,k)\times D$, $\mathrm{\forall }k\in \mathbb{N}$;

(C6) ${sup}_{(x,y)\in \overline{B}(0,k)\times D}max\{|u(x,y)|,|v(x,y)|+|t(x,y)|\}\le \alpha$, $\mathrm{\forall }k\in \mathbb{N}$;

(C7) ${sup}_{(x,y)\in \overline{B}(0,k)\times D}max\{\parallel a(x,y)\parallel ,\parallel b(x,y)\parallel ,\parallel c(x,y)\parallel \}\le k$, $\mathrm{\forall }k\in \mathbb{N}$.

Then functional equation (1.7) possesses a unique solution $w\in BB(S)$ such that

(C8) for each $w_0\in BB(S)$, the sequence ${\{w_n\}}_{n\in {\mathbb{N}}_0}$ defined by

$$\begin{array}{rcl}{w}_n(x)& =& \lambda \underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),w_{n-1}(a(x,y))\}\}\\ +(1-\lambda )\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),w_{n-1}(b(x,y))\}\\ +t(x,y)opt\{r(x,y),w_{n-1}(c(x,y))\}\},\\ \mathrm{\forall }(x,k,n)\in \overline{B}(0,k)\times \mathbb{N}\times \mathbb{N}\end{array}$$

(3.13)

converges to w and has the following error estimates:

$$\begin{array}{r}{d}_k(w_n,w)\le {\alpha }^n{d}_k(w_0,w)\mathit{\text{and}}\\ d_k(w_n,w)\le \frac{{\alpha }^n}{1-\alpha }{d}_k(w_0,w_1),\mathrm{\forall }(k,n)\in \mathbb{N}\times \mathbb{N}.\end{array}$$

(3.14)

Proof Define a mapping $H:BB(S)\to BB(S)$ by

$$\begin{array}{rcl}Hh(x)& =& \lambda \underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),h(a(x,y))\}\}\\ +(1-\lambda )\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),h(b(x,y))\}\\ +t(x,y)opt\{r(x,y),h(c(x,y))\}\},\mathrm{\forall }(x,k,h)\in \overline{B}(0,k)\times \mathbb{N}\times BB(S).\end{array}$$

(3.15)

It follows from (C5) and (C7) that for each $(k,h)\in \mathbb{N}\times BB(S)$, there exist $\gamma (k)>0$ and $\beta (k,h)>0$ such that

$$\begin{array}{c}\underset{(x,y)\in \overline{B}(0,k)\times D}{sup}max\{|p(x,y)|,|q(x,y)|,|r(x,y)|\}\le \gamma (k);\hfill \\ \underset{(x,y)\in \overline{B}(0,k)\times D}{sup}max\{\left|h(a(x,y))\right|,\left|h(b(x,y))\right|,\left|h(c(x,y))\right|\}\le \beta (k,h),\hfill \end{array}$$

which together with (C6), (3.15) and Lemma 2.1 gives that

$$\begin{array}{c}|Hh(x)|\hfill \\ =|\lambda \underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),h(a(x,y))\}\}\hfill \\ +(1-\lambda )\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),h(b(x,y))\}\hfill \\ +t(x,y)opt\{r(x,y),h(c(x,y))\}\left\}\right|\hfill \\ \le \lambda \underset{y\in D}{sup}\{|u(x,y)|max\{|p(x,y)|,\left|h(a(x,y))\right|\}\}\hfill \\ +(1-\lambda )\underset{y\in D}{sup}\{|v(x,y)|max\{|q(x,y)|,\left|h(b(x,y))\right|\}\hfill \\ +|t(x,y)|max\{|r(x,y)|,\left|h(c(x,y))\right|\}\}\hfill \\ \le \lambda \alpha max\{\gamma (k),\beta (k,h)\}+(1-\lambda )max\{\gamma (k),\beta (k,h)\}\underset{y\in D}{sup}(|v(x,y)|+|t(x,y)|)\hfill \\ \le \lambda \alpha max\{\gamma (k),\beta (k,h)\}+(1-\lambda )\alpha max\{\gamma (k),\beta (k,h)\}\hfill \\ =\alpha max\{\gamma (k),\beta (k,h)\},\mathrm{\forall }(x,k,h)\in \overline{B}(0,k)\times \mathbb{N}\times BB(S),\hfill \end{array}$$

which means that H is a self-mapping in $BB(S)$. By virtue of (3.15), (C6), (C7), Lemmas 2.1 and 2.2, we get that

$$\begin{array}{c}|Hh(x)-Hg(x)|\hfill \\ =|\lambda \underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),h(a(x,y))\}\}\hfill \\ +(1-\lambda )\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),h(b(x,y))\}+t(x,y)opt\{r(x,y),h(c(x,y))\}\}\hfill \\ -\lambda \underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),g(a(x,y))\}\}\hfill \\ -(1-\lambda )\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),g(b(x,y))\}+t(x,y)opt\{r(x,y),g(c(x,y))\}\}|\hfill \\ \le \lambda \underset{y\in D}{sup}\left\{|u(x,y)||opt\{p(x,y),h(a(x,y))\}-opt\{p(x,y),g(a(x,y))\}|\right\}\hfill \\ +(1-\lambda )\underset{y\in D}{sup}\{|v(x,y)||opt\{q(x,y),h(b(x,y))\}-opt\{q(x,y),g(b(x,y))\}|\hfill \\ +|t(x,y)||opt\{r(x,y),h(c(x,y))\}-opt\{r(x,y),g(c(x,y))\}|\}\hfill \\ \le \lambda \alpha d_k(h,g)+(1-\lambda )\underset{y\in D}{sup}\{|v(x,y)|+|t(x,y)\left]\right\}{d}_k(h,g)\hfill \\ \le \alpha d_k(h,g),(x,k,h,g)\in \overline{B}(0,k)\times \mathbb{N}\times BB(S)\times BB(S),\hfill \end{array}$$

which yields that

$$d_k(Hh,Hg)\le \alpha d_k(h,g),\mathrm{\forall }(k,h,g)\in \mathbb{N}\times BB(S)\times BB(S).$$

(3.16)

Put $\phi (t)=\alpha t$ for all $t\in {\mathbb{R}}^{+}$. It follows from (3.16) and Lemma 2.3 that H has a unique fixed point $w\in BB(S)$, which is also a unique solution of functional equation (1.7). In light of (3.13), (3.15) and (3.16), we obtain that

$$\begin{array}{rcl}{d}_k(w_n,w)& =& d_k(H{w}_{n-1},Hw)\le \alpha d_k(w_{n-1},w)\le \cdots \\ \le & {\alpha }^n{d}_k(w_0,w),\mathrm{\forall }(k,n)\in \mathbb{N}\times \mathbb{N}\end{array}$$

(3.17)

and

$$\begin{array}{rcl}{d}_k(w_n,w_{n+m})& \le & \sum _{i=n}^{n+m-1}{d}_k(w_i,w_{i+1})=\sum _{i=n}^{n+m-1}{d}_k(H{w}_{i-1},H{w}_i)\\ \le & \sum _{i=n}^{n+m-1}\alpha d_k(w_{i-1},w_i)\le \sum _{i=n}^{n+m-1}{\alpha }^i{d}_k(w_0,w_1)\\ \le & \frac{{\alpha }^n}{1-\alpha }{d}_k(w_0,w_1),\mathrm{\forall }(k,n,m)\in \mathbb{N}\times \mathbb{N}\times \mathbb{N}.\end{array}$$

(3.18)

Clearly (3.17) means that ${\{w_n\}}_{n\in {\mathbb{N}}_0}$ converges to w. Thus (3.14) follows from (3.17) and (3.18) by letting $m\to \mathrm{\infty }$. This completes the proof. □

Remark 3.6 Theorem 3.5 extends Theorem 3.4 in [5] and Corollaries 2.2 and 2.3 in [11]. The example below shows that Theorem 3.5 extends substantially the corresponding results in [5, 11].

Example 3.7 Consider the functional equation

$$\begin{array}{rcl}f(x)& =& \lambda \underset{y\in {\mathbb{R}}^{+}}{opt}\{\frac{x^3{cos}^{75}(x^6{y}^{98})}{3{(x+1)}^3+y^5}opt\{\frac{x^{69}}{{(x^5-y^6)}^4+1},f\left(\frac{x^7{sin}^4(xy)}{x^6+x{y}^3+1}\right)\}\}\\ +(1-\lambda )\underset{y\in {\mathbb{R}}^{+}}{opt}\{\frac{{sin}^9(x^8{y}^3)}{x{y}^2+5}opt\{\frac{x^{120}{y}^7}{x^9{y}^7+1},f\left(\frac{x^6{y}^4{cos}^8(x^3{y}^9)}{x^5{y}^4+1}\right)\}\\ +\frac{x^{2}y}{x^4+{(y+1)}^2}opt\{\frac{x^{57}{y}^2}{x^{10}+y^4+1},f\left(\frac{ln(1+x^{15}{y}^{23})}{{(x+1)}^{14}(y^{23}+1)}\right)\}\},\mathrm{\forall }x\in {\mathbb{R}}^{+}.\end{array}$$

(3.19)

Put $X=Y=\mathbb{R}$, $S=D={\mathbb{R}}^{+}$, $\lambda \in [0,1]$ and $\alpha =\frac{7}{10}$. Let $p,q,r,u,v,t:S\times D\to \mathbb{R}$ and $a,b,c:S\times D\to S$ be defined by

$$\begin{array}{c}p(x,y)=\frac{x^{69}}{{(x^5-y^6)}^4+1},q(x,y)=\frac{x^{120}{y}^7}{x^9{y}^7+1},r(x,y)=\frac{x^{57}{y}^2}{x^{10}+y^4+1},\hfill \\ u(x,y)=\frac{x^3{cos}^{75}(x^6{y}^{98})}{3{(x+1)}^3+y^5},v(x,y)=\frac{{sin}^9(x^8{y}^3)}{x{y}^2+5},t(x,y)=\frac{x^{2}y}{x^4+{(y+1)}^2},\hfill \\ a(x,y)=\frac{x^7{sin}^4(xy)}{x^6+x{y}^3+1},b(x,y)=\frac{x^6{y}^4{cos}^8(x^3{y}^9)}{x^5{y}^4+1},\hfill \\ c(x,y)=\frac{ln(1+x^{15}{y}^{23})}{{(x+1)}^{14}(y^{23}+1)},\mathrm{\forall }(x,y)\in S\times D.\hfill \end{array}$$

It is clear that the conditions of Theorem 3.5 are satisfied. It follows from Theorem 3.5 that functional equation (3.19) possesses a unique solution $w\in BB(S)$, which satisfies (C8). But Theorem 3.4 in [5] and Corollaries 2.2 and 2.3 in [11] are unapplicable to functional equation (3.19).

Next we discuss the behaviors of solutions and iterative algorithms for functional equation (1.7) in the complete metric space $BB(S)$.

Theorem 3.8 Let $\lambda \in [0,1]$, $(\phi ,\psi )\in {\mathrm{\Phi }}_2$, $p,q,r,u,v,t:S\times D\to \mathbb{R}$ and $a,b,c:S\times D\to S$ satisfy that

(C9) ${sup}_{(x,y)\in \overline{B}(0,k)\times D}max\{|p(x,y)|,|q(x,y)|,|r(x,y)|\}\le \psi (\parallel x\parallel )$, $\mathrm{\forall }k\in \mathbb{N}$;

(C10) ${sup}_{(x,y)\in \overline{B}(0,k)\times D}max\{|u(x,y)|,|v(x,y)|+|t(x,y)|\}\le 1$, $\mathrm{\forall }k\in \mathbb{N}$;

(C11) ${sup}_{(x,y)\in \overline{B}(0,k)\times D}max\{\parallel a(x,y)\parallel ,\parallel b(x,y)\parallel ,\parallel c(x,y)\parallel \}\le \phi (\parallel x\parallel )$, $\mathrm{\forall }k\in \mathbb{N}$.

Then functional equation (1.7) possesses a solution $w\in BB(S)$ such that

(C12) for each $w_0\in BB(S)$ with $|w_0(x)|\le \psi (\parallel x\parallel )$, $\mathrm{\forall }(x,k)\in \overline{B}(0,k)\times \mathbb{N}$, the sequence ${\{w_n\}}_{n\in {\mathbb{N}}_0}$ defined by (3.13) converges to w and $d_k(w_n,w)\le {\sum }_{j=n-1}^{\mathrm{\infty }}\psi ({\phi }^j(k))$, $\mathrm{\forall }(k,n)\in \mathbb{N}\times \mathbb{N}$;

(C13) $|w(x)|\le {\sum }_{n=0}^{\mathrm{\infty }}\psi ({\phi }^n(\parallel x\parallel ))$, $\mathrm{\forall }(x,k)\in \overline{B}(0,k)\times \mathbb{N}$;

(C14) ${lim}_{n\to \mathrm{\infty }}w(x_n)=0$ for any $(x_0,k)\in \overline{B}(0,k)\times \mathbb{N}$, ${\{y_n\}}_{n\in \mathbb{N}}\subset D$ and $x_n\in \{a(x_{n-1},y_n),b(x_{n-1},y_n),c(x_{n-1},y_n)\}$, $\mathrm{\forall }n\in \mathbb{N}$;

(C15) w is a unique solution of functional equation (1.7) relative to (C14).

Proof Define a mapping $H:BB(S)\to BB(S)$ by

$$Hh(x)=\lambda Ah(x)+(1-\lambda )Bh(x),\mathrm{\forall }(x,k,h)\in \overline{B}(0,k)\times \mathbb{N}\times BB(S),$$

(3.20)

where

$$\begin{array}{r}Ah(x)=\underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),h(a(x,y))\}\},\\ Bh(x)=\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),h(b(x,y))\}+t(x,y)opt\{r(x,y),h(c(x,y))\}\},\\ \mathrm{\forall }(x,k,h)\in \overline{B}(0,k)\times \mathbb{N}\times BB(S).\end{array}$$

(3.21)

Note that (C9) and (C11) imply (C5) and (C7) by $(\phi ,\psi )\in {\mathrm{\Phi }}_2$, respectively. Similar to the proof of Theorem 3.5, by (C10) we conclude that the mapping H maps $BB(S)$ into $BB(S)$ and satisfies that

$$d_k(Hh,Hg)\le d_k(h,g),\mathrm{\forall }(h,g,k)\in BB(S)\times BB(S)\times \mathbb{N},$$

which yields that

$$\begin{array}{rcl}d(Hh,Hg)& =& \sum _{k=1}^{\mathrm{\infty }}\frac{1}{2^k}\cdot \frac{d_k(Hh,Hg)}{1+d_k(Hh,Hg)}\le \sum _{k=1}^{\mathrm{\infty }}\frac{1}{2^k}\cdot \frac{d_k(h,g)}{1+d_k(h,g)}\\ =& d(h,g),\mathrm{\forall }(h,g)\in BB(S)\times BB(S),\end{array}$$

(3.22)

that is, the mapping H is nonexpansive in $BB(S)$.

Now we show that for each $n\in {\mathbb{N}}_0$,

$$|w_n(x)|\le \sum _{j=0}^n\psi \left({\phi }^j(\parallel x\parallel )\right),\mathrm{\forall }(x,k)\in \overline{B}(0,k)\times \mathbb{N}.$$

(3.23)

It is easy to see that (3.23) holds for $n=0$. Assume that (3.23) holds for some $n\in {\mathbb{N}}_0$. In terms of (C9), (C11), (C12), (3.13), $(\phi ,\psi )\in {\mathrm{\Phi }}_2$ and Lemma 2.1, we gain that

$$\begin{array}{c}|w_{n+1}(x)|\hfill \\ =|\lambda \underset{y\in D}{opt}\{u(x,y)opt\{p(x,y),w_n(a(x,y))\}\}\hfill \\ +(1-\lambda )\underset{y\in D}{opt}\{v(x,y)opt\{q(x,y),w_n(b(x,y))\}+t(x,y)opt\{r(x,y),w_n(c(x,y))\}\}|\hfill \\ \le \lambda \underset{y\in D}{sup}\{|u(x,y)|max\{|p(x,y)|,\left|w_n(a(x,y))\right|\}\}\hfill \\ +(1-\lambda )\underset{y\in D}{sup}\{|v(x,y)|max\{|q(x,y)|,\left|w_n(b(x,y))\right|\hfill \\ +|t(x,y)|max\{|r(x,y)|,\left|w_n(c(x,y))\right|\}\}\hfill \\ \le \lambda \underset{y\in D}{sup}\{max\{\psi (\parallel x\parallel ),\sum _{j=0}^n\psi \left({\phi }^j\left(\parallel a(x,y)\parallel \right)\right)\}\}\hfill \\ +(1-\lambda )\underset{y\in D}{sup}\{|v(x,y)|max\{\psi (\parallel x\parallel ),\sum _{j=0}^n\psi \left({\phi }^j\left(\parallel b(x,y)\parallel \right)\right)\}\hfill \\ +|t(x,y)|max\{\psi (\parallel x\parallel ),\sum _{j=0}^n\psi \left({\phi }^j\left(\parallel c(x,y)\parallel \right)\right)\}\}\hfill \\ \le \lambda max\{\psi (\parallel x\parallel ),\sum _{j=0}^n\psi \left({\phi }^{j+1}(\parallel x\parallel )\right)\}\hfill \\ +(1-\lambda )\underset{y\in D}{sup}\{|v(x,y)|+|t(x,y)|\}max\{\psi (\parallel x\parallel ),\sum _{j=0}^n\psi \left({\phi }^{j+1}(\parallel x\parallel )\right)\}\}\hfill \\ \le \lambda (\psi (\parallel x\parallel )+\sum _{j=0}^n\psi \left({\phi }^{j+1}(\parallel x\parallel )\right))+(1-\lambda )(\psi (\parallel x\parallel )+\sum _{j=0}^n\psi \left({\phi }^{j+1}(\parallel x\parallel )\right))\hfill \\ =\sum _{j=0}^{n+1}\psi \left({\phi }^j(\parallel x\parallel )\right),\mathrm{\forall }(x,k)\in \overline{B}(0,k)\times \mathbb{N},\hfill \end{array}$$

that is, (3.23) is true for $n+1$. Hence (3.23) holds for each $n\in {\mathbb{N}}_0$.

Let $\epsilon >0$ and $k,n,m\in \mathbb{N}$. Assume that ${opt}_{y\in D}={sup}_{y\in D}$. It follows from (3.21) that for each $x_0\in \overline{B}(0,k)$ there exist $y,y_0,z,z_0\in D$ satisfying

$$\begin{array}{r}A{w}_{n+m-1}(x_0)-2^{-1}\epsilon <u(x_0,y)opt\{p(x_0,y),w_{n+m-1}(a(x_0,y))\},\\ A{w}_{n-1}(x_0)-2^{-1}\epsilon <u(x_0,y_0)opt\{p(x_0,y_0),w_{n-1}(a(x_0,y_0))\},\\ A{w}_{n+m-1}(x_0)\ge u(x_0,y_0)opt\{p(x_0,y_0),w_{n+m-1}(a(x_0,y_0))\},\\ A{w}_{n-1}(x_0)\ge u(x_0,y)opt\{p(x_0,y),w_{n-1}(a(x_0,y))\}\end{array}$$

(3.24)

and

$$\begin{array}{r}B{w}_{n+m-1}(x_0)-2^{-1}\epsilon <v(x_0,z)opt\{q(x_0,z),w_{n+m-1}(b(x_0,z))\}\\ \phantom{B{w}_{n+m-1}(x_0)-2^{-1}\epsilon <}+t(x_0,z)opt\{r(x_0,z),w_{n+m-1}(c(x_0,z))\},\\ B{w}_{n-1}(x_0)-2^{-1}\epsilon <v(x_0,z_0)opt\{q(x_0,z_0),w_{n-1}(b(x_0,z_0))\}\\ \phantom{B{w}_{n-1}(x_0)-2^{-1}\epsilon <}+t(x_0,z_0)opt\{r(x_0,z_0),w_{n-1}(c(x_0,z_0))\},\\ B{w}_{n+m-1}(x_0)\ge v(x_0,z_0)opt\{q(x_0,z_0),w_{n+m-1}(b(x_0,z_0))\}\\ \phantom{B{w}_{n+m-1}(x_0)\ge }+t(x_0,z_0)opt\{r(x_0,z_0),w_{n+m-1}(c(x_0,z_0))\},\\ B{w}_{n-1}(x_0)\ge v(x_0,z)opt\{q(x_0,z),w_{n-1}(b(x_0,z))\}\\ \phantom{B{w}_{n-1}(x_0)\ge }+t(x_0,z)opt\{r(x_0,z),w_{n-1}(c(x_0,z))\}.\end{array}$$

(3.25)

On account of (3.24), (3.25) and Lemma 2.2, we obtain that

$$\begin{array}{c}A{w}_{n+m-1}(x_0)-A{w}_{n-1}(x_0)\hfill \\ <u(x_0,y)opt\{p(x_0,y),w_{n+m-1}(a(x_0,y))\}-u(x_0,y)opt\{p(x_0,y),w_{n-1}(a(x_0,y))\}+2^{-1}\epsilon \hfill \\ \le |u(x_0,y)||w_{n+m-1}(a(x_0,y))-w_{n-1}(a(x_0,y))|+2^{-1}\epsilon ;\hfill \\ A{w}_{n+m-1}(x_0)-A{w}_{n-1}(x_0)\hfill \\ >u(x_0,y_0)opt\{p(x_0,y_0),w_{n+m-1}(a(x_0,y_0))\}\hfill \\ -u(x_0,y_0)opt\{p(x_0,y_0),w_{n-1}(a(x_0,y_0))\}-2^{-1}\epsilon \hfill \\ \ge -|u(x_0,y_0)||w_{n+p-1}(a(x_0,y_0))-w_{n-1}(a(x_0,y_0))|-2^{-1}\epsilon ;\hfill \\ B{w}_{n+m-1}(x_0)-B{w}_{n-1}(x_0)\hfill \\ <v(x_0,z)opt\{q(x_0,z),w_{n+m-1}(b(x_0,z))\}+t(x_0,z)opt\{r(x_0,z),w_{n+m-1}(c(x_0,z))\}\hfill \\ -v(x_0,z)opt\{q(x_0,z),w_{n-1}(b(x_0,z))\}-t(x_0,z)opt\{r(x_0,z),w_{n-1}(c(x_0,z))\}+2^{-1}\epsilon \hfill \\ \le |v(x_0,z)||w_{n+p-1}(b(x_0,z))-w_{n-1}(b(x_0,z))|\hfill \\ +|t(x_0,z)||w_{n+m-1}(c(x_0,z))-w_{n-1}(c(x_0,z))|+2^{-1}\epsilon \hfill \end{array}$$

and

$$\begin{array}{c}B{w}_{n+m-1}(x_0)-B{w}_{n-1}(x_0)\hfill \\ >v(x_0,z_0)opt\{q(x_0,z_0),w_{n+m-1}(b(x_0,z_0))\}\hfill \\ +t(x_0,z_0)opt\{r(x_0,z_0),w_{n+m-1}(c(x_0,z_0))\}\hfill \\ -v(x_0,z_0)opt\{q(x_0,z_0),w_{n-1}(b(x_0,z_0))\}\hfill \\ -t(x_0,z_0)opt\{r(x_0,z_0),w_{n-1}(c(x_0,z_0))\}-2^{-1}\epsilon \hfill \\ \ge -|v(x_0,z_0)||w_{n+m-1}(b(x_0,z_0))-w_{n-1}(b(x_0,z_0))|\hfill \\ -|t(x_0,z_0)||w_{n+m-1}(c(x_0,z_0))-w_{n-1}(c(x_0,z_0))|-2^{-1}\epsilon ,\hfill \end{array}$$

which together with (C10), (3.13), (3.20), (3.21) imply that

$$\begin{array}{c}|w_{n+m}(x_0)-w_n(x_0)|\hfill \\ =|\lambda A{w}_{n+m-1}(x_0)+(1-\lambda )B{w}_{n+m-1}(x_0)-\lambda A{w}_{n-1}(x_0)-(1-\lambda )B{w}_{n-1}(x_0)|\hfill \\ \le \lambda |A{w}_{n+m-1}(x_0)-A{w}_{n-1}(x_0)|+(1-\lambda )|B{w}_{n+m-1}(x_0)-B{w}_{n-1}(x_0)|\hfill \\ \le max\{|A{w}_{n+m-1}(x_0)-A{w}_{n-1}(x_0)|,|B{w}_{n+m-1}(x_0)-B{w}_{n-1}(x_0)|\}\hfill \\ \le max\{|u(x_0,y)||w_{n+m-1}(a(x_0,y))-w_{n-1}(a(x_0,y))|,\hfill \\ |u(x_0,y_0)||w_{n+m-1}(a(x_0,y_0))-w_{n-1}(a(x_0,y_0))|,\hfill \\ |v(x_0,z)||w_{n+m-1}(b(x_0,z))-w_{n-1}(b(x_0,z))|\hfill \\ +|t(x_0,z)||w_{n+m-1}(c(x_0,z))-w_{n-1}(c(x_0,z))|,\hfill \\ |v(x_0,z_0)||w_{n+m-1}(b(x_0,z_0))-w_{n-1}(b(x_0,z_0))|\hfill \\ +|t(x_0,z_0)||w_{n+m-1}(c(x_0,z_0))-w_{n-1}(c(x_0,z_0))|\}+2^{-1}\epsilon \hfill \\ \le max\{|u(x_0,y)|,|u(x_0,y_0)|,|v(x_0,z)|+|t(x_0,z)|,|v(x_0,z_0)|+|t(x_0,z_0)|\}\hfill \\ \times max\{|w_{n+m-1}(a(x_0,y))-w_{n-1}(a(x_0,y))|,|w_{n+m-1}(a(x_0,y_0))-w_{n-1}(a(x_0,y_0))|,\hfill \\ |w_{n+m-1}(b(x_0,z))-w_{n-1}(b(x_0,z))|,|w_{n+m-1}(c(x_0,z))-w_{n-1}(c(x_0,z))|,\hfill \\ |w_{n+m-1}(b(x_0,z_0))-w_{n-1}(b(x_0,z_0))|,|w_{n+m-1}(c(x_0,z_0))-w_{n-1}(c(x_0,z_0))|\}+2^{-1}\epsilon \hfill \\ \le max\{|w_{n+m-1}(a(x_0,y))-w_{n-1}(a(x_0,y))|,|w_{n+m-1}(a(x_0,y_0))-w_{n-1}(a(x_0,y_0))|,\hfill \\ |w_{n+m-1}(b(x_0,z))-w_{n-1}(b(x_0,z))|,|w_{n+m-1}(c(x_0,z))-w_{n-1}(c(x_0,z))|,\hfill \\ |w_{n+m-1}(b(x_0,z_0))-w_{n-1}(b(x_0,z_0))|,|w_{n+m-1}(c(x_0,z_0))-w_{n-1}(c(x_0,z_0))|\}+2^{-1}\epsilon \hfill \\ =|w_{n+m-1}(x_1)-w_{n-1}(x_1)|+2^{-1}\epsilon \hfill \end{array}$$

for some $x_1\in \{a(x_0,y_1),b(x_0,y_1),c(x_0,y_1)\}$ and $y_1\in \{y,y_0,z,z_0\}$, that is,

$$|w_{n+m}(x_0)-w_n(x_0)|<|w_{n+m-1}(x_1)-w_{n-1}(x_1)|+2^{-1}\epsilon .$$

(3.26)

Similarly, we infer that (3.26) holds for ${opt}_{y\in D}={inf}_{y\in D}$. Proceeding in this way, we conclude that for each $n\in \mathbb{N}$, there exist $y_i\in D$ and $x_i\in \{a(x_{i-1},y_i),b(x_{i-1},y_i),c(x_{i-1},y_i)\}$ for $i\in \{1,2,\dots ,n\}$ such that

$$\begin{array}{r}|w_{n+m-1}(x_1)-w_{n-1}(x_1)|\le |w_{n+m-2}(x_2)-w_{n-2}(x_2)|+2^{-2}\epsilon ,\\ |w_{n+m-2}(x_2)-w_{n-2}(x_2)|\le |w_{n+m-3}(x_3)-w_{n-3}(x_3)|+2^{-3}\epsilon ,\\ \cdots \\ |w_{m+1}(x_{n-1})-w_1(x_{n-1})|\le |w_m(x_n)-w_0(x_n)|+2^{-n}\epsilon .\end{array}$$

(3.27)

In terms of $(\phi ,\psi )\in {\mathrm{\Phi }}_2$, (C11), (3.23) and (3.27), we deduce that

$$\begin{array}{rcl}|w_{n+m}(x_0)-w_n(x_0)|& <& |w_m(x_n)-w_0(x_n)|+\epsilon \le \sum _{j=0}^m\psi \left({\phi }^j(\parallel x_n\parallel )\right)+\psi (\parallel x_n\parallel )+\epsilon \\ \le & \sum _{j=0}^m\psi ({\phi }^{j+n}(k))+\psi ({\phi }^n(k))+\epsilon \le \sum _{j=n-1}^{m+n}\psi ({\phi }^j(k))+\epsilon ,\end{array}$$

which means that

$$d_k(w_{n+m},w_n)\le \sum _{j=n-1}^{n+m}\psi ({\phi }^j(k))+\epsilon .$$

(3.28)

Letting $\epsilon \to 0^{+}$ in the above inequality, we deduce that

$$d_k(w_{n+m},w_n)\le \sum _{j=n-1}^{n+m}\psi ({\phi }^j(k)).$$

(3.29)

Notice that ${\sum }_{n=0}^{\mathrm{\infty }}\psi ({\phi }^n(t))<+\mathrm{\infty }$ for each $t>0$. Thus (3.29) means that ${\{w_n\}}_{n\in {\mathbb{N}}_0}$ is a Cauchy sequence in $(BB(S),d)$ and it converges to some $w\in BB(S)$. Letting $m\to \mathrm{\infty }$ in (3.29), we conclude immediately that

$$d_k(w_n,w)\le \sum _{j=n-1}^{\mathrm{\infty }}\psi ({\phi }^j(k)),\mathrm{\forall }(k,n)\in \mathbb{N}\times \mathbb{N}.$$

By virtue of (3.22), we infer that

$$d(Hw,w)\le d(Hw,H{w}_n)+d(w_{n+1},w)\le d(w,w_n)+d(w_{n+1},w)\to 0\text{as }n\to \mathrm{\infty },$$

which yields that $Hw=w$, that is, functional equation (1.7) possesses a solution $w\in BB(S)$.

Next we show (C13). Let $(x,k)\in \overline{B}(0,k)\times \mathbb{N}$. According to (C11), (3.23) and $(\phi ,\psi )\in {\mathrm{\Phi }}_2$, we know that

$$\begin{array}{rcl}|w(x)|& \le & |w(x)-w_n(x)|+|w_n(x)|\le d_k(w,w_n)+\sum _{j=0}^n\psi \left({\phi }^j(\parallel x\parallel )\right)\\ \to & \sum _{j=0}^{\mathrm{\infty }}\psi \left({\phi }^j(\parallel x\parallel )\right)\text{as }n\to \mathrm{\infty },\end{array}$$

that is, (C13) holds.

Next we show (C14). Given $(x_0,k)\in \overline{B}(0,k)\times \mathbb{N}$, ${\{y_n\}}_{n\in \mathbb{N}}\subset D$ and $x_n\in \{a(x_{n-1},y_n),b(x_{n-1},y_n),c(x_{n-1},y_n)\}$, $\mathrm{\forall }n\in \mathbb{N}$. It follows from (C11) and $(\phi ,\psi )\in {\mathrm{\Phi }}_2$ that

$$\begin{array}{rcl}\parallel x_n\parallel & \le & max\{\parallel a(x_{n-1},y_n)\parallel ,\parallel b(x_{n-1},y_n)\parallel ,\parallel c(x_{n-1},y_n)\parallel \}\\ \le & \phi (\parallel x_{n-1}\parallel )\le \cdots \le {\phi }^n(\parallel x_0\parallel )\le {\phi }^n(k)<k,\mathrm{\forall }n\in \mathbb{N},\end{array}$$

which together with (C11), (3.23) and $(\phi ,\psi )\in {\mathrm{\Phi }}_2$ implies that

$$\begin{array}{rcl}|w(x_n)|& \le & |w(x_n)-w_n(x_n)|+|w_n(x_n)|\le d_k(w,w_n)+\sum _{j=0}^n\psi \left({\phi }^j(\parallel x_n\parallel )\right)\\ \le & d_k(w,w_n)+\sum _{j=n}^{2n}\psi ({\phi }^j(k))\to 0\text{as }n\to \mathrm{\infty },\end{array}$$

which yields that ${lim}_{n\to \mathrm{\infty }}w(x_n)=0$.

Finally we show (C15). Suppose that functional equation (1.7) has another solution $h\in BB(S)$ that satisfies (C14). Let $\epsilon >0$ and $x_0\in S$. It follows from (3.21) that there exist $y,y_0,z,z_0\in D$ with

$$\begin{array}{c}Aw(x_0)-2^{-1}\epsilon <u(x_0,y)opt\{p(x_0,y),w(a(x_0,y))\},\hfill \\ Ah(x_0)-2^{-1}\epsilon <u(x_0,y_0)opt\{p(x_0,y_0),h(a(x_0,y_0))\},\hfill \\ Aw(x_0)>u(x_0,y_0)opt\{p(x_0,y_0),w(a(x_0,y_0))\}-2^{-1}\epsilon ,\hfill \\ Ah(x_0)>u(x_0,y)opt\{p(x_0,y),h(a(x_0,y))\}-2^{-1}\epsilon \hfill \end{array}$$

and

$$\begin{array}{c}Bw(x_0)-2^{-1}\epsilon <v(x_0,z)opt\{q(x_0,z),w(b(x_0,z))\}\hfill \\ \phantom{Bw(x_0)-2^{-1}\epsilon <}+t(x_0,z)opt\{r(x_0,z),w(c(x_0,z))\},\hfill \\ Bh(x_0)-2^{-1}\epsilon <v(x_0,z_0)opt\{q(x_0,z_0),h(b(x_0,z_0))\}\hfill \\ \phantom{Bh(x_0)-2^{-1}\epsilon <}+t(x_0,z_0)opt\{r(x_0,z_0),h(c(x_0,z_0))\},\hfill \\ Bw(x_0)>v(x_0,z_0)opt\{q(x_0,z_0),w(b(x_0,z_0))\}\hfill \\ \phantom{Bw(x_0)>}+t(x_0,z_0)opt\{r(x_0,z_0),w(c(x_0,z_0))\}-2^{-1}\epsilon ,\hfill \\ Bh(x_0)>v(x_0,z)opt\{q(x_0,z),h(b(x_0,z))\}\hfill \\ \phantom{Bh(x_0)>}+t(x_0,z)opt\{r(x_0,z),h(c(x_0,z))\}-2^{-1}\epsilon ,\hfill \end{array}$$

which together with (C11), (3.20) and (3.21) yield that there exist $x_1\in \{a(x_0,y_1),b(x_0,y_1),c(x_0,y_1)\}$ and $y_1\in \{y,y_0,z,z_0\}$ satisfying

$$\begin{array}{c}|w(x_0)-h(x_0)|\hfill \\ =|\lambda Aw(x_0)+(1-\lambda )Bw(x_0)-\lambda Ah(x_0)-(1-\lambda )Bh(x_0)|\hfill \\ \le \lambda |Aw(x_0)-Ah(x_0)|+(1-\lambda )|Bw(x_0)-Bh(x_0)|\hfill \\ \le max\{|Aw(x_0)-Ah(x_0)|,|Bw(x_0)-Bh(x_0)|\}\hfill \\ <max\{|u(x_0,y)||opt\{p(x_0,y),w(a(x_0,y))\}-opt\{p(x_0,y),h(a(x_0,y))\}|,\hfill \\ |u(x_0,y_0)||opt\{p(x_0,y_0),w(a(x_0,y_0))\}-opt\{p(x_0,y_0),h(a(x_0,y_0))\}|,\hfill \\ |v(x_0,z)opt\{q(x_0,z),w(b(x_0,z))\}+t(x_0,z)opt\{r(x_0,z),w(c(x_0,z))\}\hfill \\ -v(x_0,z)opt\{q(x_0,z),h(b(x_0,z))\}-t(x_0,z)opt\{r(x_0,z),h(c(x_0,z))\}|,\hfill \\ |v(x_0,z_0)opt\{q(x_0,z_0),w(b(x_0,z_0))\}+t(x_0,z_0)opt\{r(x_0,z_0),w(c(x_0,z_0))\}\hfill \\ -v(x_0,z_0)opt\{q(x_0,z_0),h(b(x_0,z_0))\}-t(x_0,z_0)opt\{r(x_0,z_0),h(c(x_0,z_0))\}\left|\right\}+\epsilon \hfill \\ \le max\{|u(x_0,y)|,|u(x_0,y_0)|,|v(x_0,z)|+|t(x_0,z)|,|v(x_0,z_0)|+|t(x_0,z_0)|\}\hfill \\ \times max\{|w(a(x_0,y))-h(a(x_0,y))|,|w(a(x_0,y_0))-h(a(x_0,y_0))|,\hfill \\ |w(b(x_0,z))-h(b(x_0,z))|,|w(c(x_0,z))-h(c(x_0,z))|,\hfill \\ |w(b(x_0,z_0))-h(b(x_0,z_0))|,|w(c(x_0,z_0))-h(c(x_0,z_0))|\}+\epsilon \hfill \\ \le |w(x_1)-h(x_1)|+\epsilon ,\hfill \end{array}$$

that is,

$$|w(x_0)-h(x_0)|<|w(x_1)-h(x_1)|+\epsilon .$$

(3.30)

Similarly, we infer that for each $n\in \mathbb{N}\setminus \{1\}$, there exist $x_i\in \{a(x_{i-1},y_i),b(x_{i-1},y_i),c(x_{i-1},y_i)\}$ and $y_i\in D$, $i\in \{2,3,\dots ,n\}$, such that

$$\begin{array}{r}|w(x_1)-h(x_1)|<|w(x_2)-h(x_2)|+2^{-1}\epsilon ,\\ |w(x_2)-h(x_2)|<|w(x_3)-h(x_3)|+2^{-2}\epsilon ,\\ \cdots \\ |w(x_{n-1})-h(x_{n-1})|<|w(x_n)-h(x_n)|+2^{-n+1}\epsilon .\end{array}$$

(3.31)

Using (3.30) and (3.31), we deduce that

$$|w(x_0)-h(x_0)|<|w(x_n)-h(x_n)|+2\epsilon \to 2\epsilon \text{as }n\to \mathrm{\infty }.$$

Letting $\epsilon \to 0^{+}$ in the above inequality, we infer that $w(x_0)=h(x_0)$. This completes the proof. □

Remark 3.9 Theorem 3.8 generalizes Theorem 3.5 in [5, 6, 12], Corollaries 2.2 and 2.3 in [11], Corollaries 3.1, 3.3 and 3.4 in [14], Theorems 2.3 and 2.4 in [15], Theorem 2.6 in [16] and Theorem 3.4 in [19]. The example below shows that Theorem 3.8 extends properly the corresponding results in [5, 6, 11, 12, 14–16, 19].

Example 3.10 Consider the functional equation

$$\begin{array}{rcl}f(x)& =& \lambda \underset{y\in {\mathbb{R}}^{-}}{opt}\{\frac{{sin}^{49}(x^{35}{y}^{27})}{{(x-y^2)}^4+1}opt\{\frac{x^8{y}^6}{x^2{y}^6+1},f\left(\frac{x^8{y}^2}{(2x^7+1)(y^2+1)}\right)\}\}\\ +(1-\lambda )\underset{y\in {\mathbb{R}}^{-}}{opt}\{\frac{(x^3+1){cos}^2(xy)}{x^3-y{x}^5+1}opt\{\frac{x^{21}{y}^5}{x^{13}{y}^5-1},f\left(\frac{x^7{y}^4}{\sqrt{4x^{12}{y}^8+1}}\right)\}\\ +\frac{(x^3+1){sin}^2(xy)}{x^3-y{x}^5+1}opt\{\frac{x^{32}{y}^4}{x^{26}{y}^4+{cos}^2(x^{3}y)},f\left(x{sin}^2\left(\frac{1}{\sqrt{2}+x{y}^2}\right)\right)\}\},\\ \mathrm{\forall }x\in {\mathbb{R}}^{+}.\end{array}$$

(3.32)

Let $\lambda \in [0,1]$, $X=Y=\mathbb{R}$, $S={\mathbb{R}}^{+}$, $D={\mathbb{R}}^{-}$. Let $p,q,r,u,v,t:S\times D\to \mathbb{R}$, $a,b,c:S\times D\to S$, φ and $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ be defined by

$$\begin{array}{c}p(x,y)=\frac{x^8{y}^6}{x^2{y}^6+1},q(x,y)=\frac{x^{21}{y}^5}{x^{13}{y}^5-1},r(x,y)=\frac{x^{32}{y}^4}{x^{26}{y}^4+{cos}^2(x^{3}y)},\hfill \\ u(x,y)=\frac{{sin}^{49}(x^{35}{y}^{27})}{{(x-y^2)}^4+1},v(x,y)=\frac{(x^3+1){cos}^2(xy)}{x^3-y{x}^5+1},\hfill \\ t(x,y)=\frac{(x^3+1){sin}^2(xy)}{x^3-y{x}^5+1},a(x,y)=\frac{x^8{y}^2}{(2x^7+1)(y^2+1)},\hfill \\ b(x,y)=\frac{x^7{y}^4}{\sqrt{4x^{12}{y}^8+1}},c(x,y)=x{sin}^2\left(\frac{1}{\sqrt{2}+x{y}^2}\right),\mathrm{\forall }(x,y)\in S\times D,\hfill \\ \phi (t)=\frac{t}{2},\psi (t)=max\{t^6,t^8\},\mathrm{\forall }t\in {\mathbb{R}}^{+}.\hfill \end{array}$$

Obviously, the conditions of Theorem 3.8 are satisfied. It follows from Theorem 3.8 that functional equation (3.32) possesses a unique solution $w\in BB(S)$ satisfying (C11)-(C14). However, Theorem 3.5 in [5, 6, 12], Corollaries 2.2 and 2.3 in [11], Corollaries 3.1, 3.3 and 3.4 in [14], Theorems 2.3 and 2.4 in [15], Theorem 2.6 in [16] and Theorem 3.4 in [19] are not applicable to functional equation (3.32).