An Investigation of the Common Solutions for Coupled Systems of Functional Equations Arising in Dynamic Programming

Abstract

The purpose of this paper is to introduce the new notion of a specific point in the space of the bounded real-valued functions on a given non-empty set and present a result based on the existence and uniqueness of such points. As a consequence of our results, we discuss the existence of a unique common solution to coupled systems of functional equations arising in dynamic programming.

1. Introduction and Preliminaries

Banach fixed-point theorem [1], considered to be the source of metrical fixed-point theory, has been generalized by many researchers; see [2,3,4,5]. One of the most interesting generalizations of this theorem was given by Jleli and Samet [6] by introducing the notion of $\vartheta$-contraction.

Definition 1

([6]). A self-mapping Υ on a metric space $(\Lambda ,d)$ is said to be a ϑ-contraction, if there exist $\vartheta \in \Theta$, and $k\in (0,1)$ such that

$${\eta }_1,{\eta }_2\in \Lambda ,d(\mathsf{{\rm Y}}{\eta }_1,\mathsf{{\rm Y}}{\eta }_2)\ne 0\Rightarrow \vartheta \left(d(\mathsf{{\rm Y}}{\eta }_1,\mathsf{{\rm Y}}{\eta }_2)\right)\le {\left[\vartheta \left(d({\eta }_1,{\eta }_2)\right)\right]}^k,$$

where Θ is the set of functions $\vartheta :(0,+\infty )\to (1,+\infty )$ satisfying the following conditions:

(ϑ₁)

ϑ is non-decreasing;

(ϑ₂)

for each sequence $\left\{t_n\right\}\subset (0,+\infty ),$${lim}_{n\to +\infty }\vartheta \left(t_n\right)=1$ iff ${lim}_{n\to +\infty }{t}_n=0;$

(ϑ₃)

there exist $r\in (0,1)$ and $\lambda \in (0,+\infty ]$ such that ${lim}_{t\to 0^{+}}\frac{\vartheta \left(t\right)-1}{t^r}=\lambda .$

Then, Jleli and Samet [6] proved that every $\vartheta$-contraction on a complete metric space has a unique fixed point.

Here we give an example which illustrates the functions in $\Theta$.

Example 1.

Let ${\vartheta }_1,{\vartheta }_2:(0,+\infty )\to (1,+\infty )$ defined by

$${\vartheta }_1\left(t\right)=e^{\sqrt{t}},{\vartheta }_2\left(t\right)=1+\sqrt{t}.$$

Then ${\vartheta }_1,{\vartheta }_2\in \Theta$.

Throughout this paper, for a fixed non-empty set $\mathsf{\Omega }$, we use the notation $B\left(\mathsf{\Omega }\right)$ which stands for the set of all bounded real-valued functions on $\mathsf{\Omega }$. Also, unless otherwise specified, d is the sup metric on $B\left(\mathsf{\Omega }\right)$ defined by

$$d(h,k)=\underset{\eta \in \mathsf{\Omega }}{sup}|h\eta -k\eta |,$$

for all $h,k\in B\left(\mathsf{\Omega }\right)$. It is well known that $B\left(\mathsf{\Omega }\right)$, endowed with the sup metric, is a complete metric space.

Recently, Harjani et al. [7] introduced the notion of $\alpha$-coupled fixed point in the space of the bounded functions on a non-empty set as follows.

Definition 2

([7]). Let Ω be a non-empty set and $\alpha :\mathsf{\Omega }\to \mathsf{\Omega }$ be a given mapping. An element $(\eta ,\xi )\in B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right)$ is called an α-coupled fixed point of mapping $\Gamma :B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right)\to B\left(\mathsf{\Omega }\right)$ if $\Gamma (\eta ,\xi )=\eta$ and $\Gamma \left(\eta \right(\alpha ),\xi (\alpha \left)\right)=\xi$.

They also used the above concept to prove the existence and uniqueness of solutions for a coupled system of functional equations arising in dynamic programming. The purpose of this paper is to introduce the notion of $\alpha$-coupled common fixed points and present a result based on the existence and uniqueness of such points. As a consequence of our results, we discuss the existence of a unique common solution of coupled systems of functional equations arising in dynamic programming.

2. Main Theoretical Results

First, we introduce the notion of $\alpha$-coupled common fixed points as follows.

Definition 3.

Let Ω be a non-empty set and $\alpha :\mathsf{\Omega }\to \mathsf{\Omega }$ be a given mapping. An element $(\eta ,\xi )\in B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right)$ is called an α-coupled common fixed point of mappings $\Gamma ,\mathsf{{\rm Y}}:B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right)\to B\left(\mathsf{\Omega }\right)$ if $\Gamma (\eta ,\xi )=\mathsf{{\rm Y}}(\eta ,\xi )=\eta$ and $\Gamma \left(\eta \right(\alpha ),\xi (\alpha \left)\right)=\mathsf{{\rm Y}}\left(\eta \right(\alpha ),\xi (\alpha \left)\right)=\xi$.

Now, we give the main theorem of this paper.

Theorem 1.

Let Ω be a non-empty set, $\alpha :\mathsf{\Omega }\to \mathsf{\Omega }$ and $\Gamma ,\mathsf{{\rm Y}}:B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right)\to B\left(\mathsf{\Omega }\right)$ be given mappings. If there exist $\vartheta \in \Theta$ and $k\in (0,1)$ such that

$$\vartheta \left(d(\Gamma ({\eta }_1,{\eta }_2),\mathsf{{\rm Y}}({\xi }_1,{\xi }_2))\right)\le {\left[\vartheta (max\left\{d({\eta }_1,{\xi }_1),d({\eta }_2,{\xi }_2)\right\})\right]}^k,$$

(1)

for all ${\eta }_1,{\eta }_2,{\xi }_1,{\xi }_2\in B\left(\mathsf{\Omega }\right)$ with $d(\Gamma ({\eta }_1,{\eta }_2),\mathsf{{\rm Y}}({\xi }_1,{\xi }_2))>0$, then Γ and Υ have a unique α-coupled common fixed point.

Before going to the proof, we give the following lemma which will be used efficiently in the proof of Theorem 1.

Lemma 1.

Let $(\Lambda ,d)$ be a complete metric space and, σ and ϱ be self-mappings on Λ such that

$${\eta }_1,{\eta }_2\in \Lambda ,d(\sigma {\eta }_1,\varrho {\eta }_2)>0\Rightarrow \vartheta \left(d(\sigma {\eta }_1,\varrho {\eta }_2)\right)\le {\left[\vartheta \left(d({\eta }_1,{\eta }_2)\right)\right]}^k,$$

(2)

where $\vartheta \in \Theta$ and $k\in (0,1).$ Then σ and ϱ have a unique common fixed point.

Proof. 

Notice that by (2), we deduce

$$ln\left[\vartheta \left(d(\sigma {\eta }_1,\varrho {\eta }_2)\right)\right]\le kln\left[\vartheta \left(d({\eta }_1,{\eta }_2)\right)\right]<ln\left[\vartheta \left(d({\eta }_1,{\eta }_2)\right)\right],$$

which implies from $\left({\vartheta }_1\right)$ that

$$d(\sigma {\eta }_1,\varrho {\eta }_2)\le d({\eta }_1,{\eta }_2),\mathrm{for}\mathrm{all}{\eta }_1,{\eta }_2\in \Lambda .$$

(3)

First, we prove that $\xi$ is a fixed point of $\sigma$ if and only if $\xi$ is a fixed point of $\varrho$. Suppose that $\xi$ is a fixed point of $\varrho$. Also, assume that $\xi$ is not a fixed point of $\sigma$. Then, considering (3), we have

$$\begin{array}{c}\hfill 0<d(\sigma \xi ,\xi )=d(\sigma \xi ,\varrho \xi )\le d(\xi ,\xi )=0\end{array}$$

which is a contradiction, and this implies that $\sigma \xi =\xi .$ Similarly, it is easy to show that if $\xi$ is a fixed point of $\sigma ,$ then $\xi$ is a fixed point of $\varrho$.

Let ${\eta }_0\in \Lambda .$ Define the sequence $\left\{{\eta }_n\right\}$ in $\Lambda$ by ${\eta }_{2n+1}=\sigma {\eta }_{2n}$ and ${\eta }_{2n+2}=\varrho {\eta }_{2n+1}$ for all $n\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$. If ${\eta }_{2n}={\eta }_{2n+1}$ for some $n\in \mathbb{N},$ then ${\eta }_{2n}=\sigma {\eta }_{2n}.$ Thus, ${\eta }_{2n}$ is a fixed point of $\sigma$ and so $x_{2n}$ is a fixed point of $\varrho ,$ that is, ${\eta }_{2n}=\sigma {\eta }_{2n}=\varrho {\eta }_{2n}.$ Similarly, if ${\eta }_{2n+1}={\eta }_{2n+2}$ for some $n\in \mathbb{N},$ then it is easy to see that ${\eta }_{2n+1}=\sigma {\eta }_{2n+1}=\varrho {\eta }_{2n+1}.$ Hence we can assume that ${\eta }_n\ne {\eta }_{n+1}$ for all $n\in \mathbb{N}.$ Then, for $n=2m+1$, where $m\in \mathbb{N}\cup \left\{0\right\}$, using (2) we get

$$\begin{array}{cc}\hfill \vartheta \left(d({\eta }_n,{\eta }_{n+1})\right)& =\vartheta \left(d({\eta }_{2m+1},{\eta }_{2m+2})\right)\hfill \\ \hfill & =\vartheta \left(d(\sigma {\eta }_{2m},\varrho {\eta }_{2m+1})\right)\hfill \\ \hfill & \le {\left[\vartheta \left(d({\eta }_{2m},{\eta }_{2m+1})\right)\right]}^k\hfill \\ \hfill & \le {\left[\vartheta \left(d({\eta }_{2m-1},{\eta }_{2m})\right)\right]}^{k^2}\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le {\left[\vartheta \left(d({\eta }_0,{\eta }_1)\right)\right]}^{k^{2m+1}}\hfill \\ \hfill & ={\left[\vartheta \left(d({\eta }_0,{\eta }_1)\right)\right]}^{k^n}.\hfill \end{array}$$

By a similar method to above, for $n=2m,$, where $m\in \mathbb{N}\cup \left\{0\right\}$, we can again obtain

$$\vartheta \left(d({\eta }_n,{\eta }_{n+1})\right)\le {\left[\vartheta \left(d({\eta }_0,{\eta }_1)\right)\right]}^{k^n}.$$

Thus, for all $n\in \mathbb{N},$ we have

$$1<\vartheta \left(d({\eta }_n,{\eta }_{n+1})\right)\le {\left[\vartheta \left(d({\eta }_0,{\eta }_1)\right)\right]}^{k^n}.$$

(4)

Letting $n\to +\infty$ in the above equation, we get

$$\underset{n\to +\infty }{lim}\vartheta \left(d({\eta }_n,{\eta }_{n+1})\right)=1,$$

which implies by $\left({\vartheta }_2\right)$ that

$$\underset{n\to +\infty }{lim}d({\eta }_n,{\eta }_{n+1})=0.$$

(5)

Let $d_n:=d({\eta }_n,{\eta }_{n+1})$ for all $n\in \mathbb{N}.$ To prove that $\left\{{\eta }_n\right\}$ is a Cauchy sequence, let us consider condition $\left({\vartheta }_3\right).$ Then there exist $r\in (0,1)$ and $\lambda \in (0,+\infty ]$ such that

$$\underset{n\to +\infty }{lim}\frac{\vartheta \left(d_n\right)-1}{{\left(d_n\right)}^r}=\lambda .$$

(6)

Let $\delta \in (0,\lambda ).$ By the definition of limit, there exists $n_0\in \mathbb{N}$ such that

$${\left[d_n\right]}^r\le {\delta }^{-1}[\vartheta \left(d_n\right)-1],\mathrm{for}\mathrm{all}n>n_0.$$

Using (4) and the above inequality, we infer

$$n{\left[d_n\right]}^r\le {\delta }^{-1}n({\left[\vartheta \left(d_0\right)\right]}^{k^n}-1),\mathrm{for}\mathrm{all}n>n_0.$$

This implies that

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim}n{\left[d_n\right]}^r=\underset{n\to +\infty }{lim}n{\left[d({\eta }_n,{\eta }_{n+1})\right]}^r=0.\end{array}$$

Then, there exists $n_1\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill d({\eta }_n,{\eta }_{n+1})\le \frac{1}{n^{1/r}},\mathrm{for}\mathrm{all}n>n_1.\end{array}$$

(7)

Let $m>n>n_1.$ Then, using the triangular inequality and (7), we have

$$d({\eta }_n,{\eta }_m)\le \sum _{k=n}^{m-1}d({\eta }_k,{\eta }_{k+1})\le \sum _{k=n}^{m-1}\frac{1}{k^{1/r}}\le \sum _{k=n}^{\infty }\frac{1}{k^{1/r}}$$

and hence $\left\{{\eta }_n\right\}$ is a Cauchy sequence in $\Lambda .$ From the completeness of $(\Lambda ,d)$, there exists $\xi \in \Lambda$ such that ${\eta }_n\to \xi$ as $n\to +\infty .$

Now, we show that $\xi$ is a common fixed point of $\sigma$ and $\varrho .$ By considering (3), we deduce

$$\begin{array}{c}\hfill d\left({\eta }_{2n+1},\varrho \xi \right)=d\left(\sigma {\eta }_{2n},\varrho \xi \right)\le d\left({\eta }_{2n},\xi \right).\end{array}$$

Passing to limit as $n\to +\infty$ in the above inequality, we obtain $d(\xi ,\varrho \xi )=0$ and so $\xi =\varrho \xi .$ That is, $\xi$ is a fixed point of $\varrho .$ Taking into account the fact that $\xi$ is a fixed point of $\sigma$ iff $\xi$ is a fixed point of $\varrho ,$ we conclude that $\xi$ is also a fixed point of $\sigma .$

To show the uniqueness of common fixed point of $\sigma$ and $\varrho ,$ suppose that there exist ${\eta }_1,{\eta }_2\in \Lambda \mathrm{with}{\eta }_1\ne {\eta }_2$ such that ${\eta }_1=\sigma {\eta }_1=\varrho {\eta }_1$ and ${\eta }_2=\sigma {\eta }_2=\varrho {\eta }_2.$ Then, from (2), we get

$$\begin{array}{c}\hfill \vartheta \left(d({\eta }_1,{\eta }_2)\right)=\vartheta \left(d(\sigma {\eta }_1,\varrho {\eta }_2)\right)\le {\left[\vartheta \left(d({\eta }_1,{\eta }_2)\right)\right]}^k<\vartheta \left(d({\eta }_1,{\eta }_2)\right),\end{array}$$

which is a contradiction. Then $\sigma$ and $\varrho$ have one and only one common fixed point. □

Now, we are ready to present the proof of Theorem 1.

Proof. 

Define $\delta :B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right)\to [0,+\infty )$ by

$$\begin{array}{c}\hfill \delta (({\eta }_1,{\eta }_2),({\xi }_1,{\xi }_2))=max\{d({\eta }_1,{\xi }_1),d({\eta }_2,{\xi }_2)\},\mathrm{for}\mathrm{all}{\eta }_1,{\eta }_2,{\xi }_1,{\xi }_2\in B\left(\mathsf{\Omega }\right).\end{array}$$

Then, $\left(B\right(\mathsf{\Omega })\times B(\mathsf{\Omega }),\delta )$ is a complete metric space, since $\left(B\right(\mathsf{\Omega }),d)$ is complete.

Consider the mappings ${\mathsf{\Sigma }}_{\Gamma },{\mathsf{\Sigma }}_{\mathsf{{\rm Y}}}:B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right)\to B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right)$ defined by

$${\mathsf{\Sigma }}_{\Gamma }\left(U\right)=\left(\Gamma ({\eta }_1,{\eta }_2),\Gamma ({\eta }_1\left(\alpha \right),{\eta }_2\left(\alpha \right))\right)$$

and

$${\mathsf{\Sigma }}_{\mathsf{{\rm Y}}}\left(U\right)=\left(\mathsf{{\rm Y}}({\eta }_1,{\eta }_2),\mathsf{{\rm Y}}({\eta }_1\left(\alpha \right),{\eta }_2\left(\alpha \right))\right),$$

where $U=({\eta }_1,{\eta }_2).$ Then, ${\mathsf{\Sigma }}_{\Gamma }$ and ${\mathsf{\Sigma }}_{\mathsf{{\rm Y}}}$ satisfy all assumptions of Lemma 1. Indeed, taking account of $\left({\vartheta }_1\right)$ and (1), for all $U=({\eta }_1,{\eta }_2),V=({\xi }_1,{\xi }_2)\in B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right),$ we deduce

$$\begin{array}{c}\vartheta \left(\delta ({\mathsf{\Sigma }}_{\Gamma }\left(U\right),{\mathsf{\Sigma }}_{\mathsf{{\rm Y}}}\left(V\right))\right)\hfill \\ =\vartheta \left(\delta ((\Gamma ({\eta }_1,{\eta }_2),\Gamma ({\eta }_1\left(\alpha \right),{\eta }_2\left(\alpha \right))),(\mathsf{{\rm Y}}({\xi }_1,{\xi }_2),\mathsf{{\rm Y}}({\xi }_1\left(\alpha \right),{\xi }_2\left(\alpha \right))))\right)\hfill \\ =\vartheta (max\{d(\Gamma ({\eta }_1,{\eta }_2),\mathsf{{\rm Y}}({\xi }_1,{\xi }_2)),d(\Gamma ({\eta }_1\left(\alpha \right),{\eta }_2\left(\alpha \right)),\mathsf{{\rm Y}}({\xi }_1\left(\alpha \right),{\xi }_2\left(\alpha \right)))\})\hfill \\ =max\{\vartheta \left(d(\Gamma ({\eta }_1,{\eta }_2),\mathsf{{\rm Y}}({\xi }_1,{\xi }_2))\right),\vartheta \left(d(\Gamma ({\eta }_1\left(\alpha \right),{\eta }_2\left(\alpha \right)),\mathsf{{\rm Y}}({\xi }_1\left(\alpha \right),{\xi }_2\left(\alpha \right)))\right)\}\hfill \\ \le max\{{\left[\vartheta (max\{d({\eta }_1,{\xi }_1),d({\eta }_2,{\xi }_2)\})\right]}^k,\hfill \\ {\left[\vartheta (max\{d({\eta }_1\left(\alpha \right),{\xi }_1\left(\alpha \right)),d({\eta }_2\left(\alpha \right),{\xi }_2\left(\alpha \right))\})\right]}^k\}.\hfill \end{array}$$

Since

$$\begin{array}{cc}\hfill d({\eta }_1\left(\alpha \right),{\xi }_1\left(\alpha \right))& =\underset{\omega \in \mathsf{\Omega }}{sup}|{\eta }_1\left(\alpha \right)\left(\omega \right),{\xi }_1\left(\alpha \right)\left(\omega \right)|\hfill \\ \hfill & \le \underset{\omega \in \mathsf{\Omega }}{sup}|{\eta }_1\left(\omega \right),{\xi }_1\left(\omega \right)|=d({\eta }_1,{\xi }_1),\hfill \end{array}$$

and similarly $d({\eta }_2\left(\alpha \right),{\xi }_2\left(\alpha \right))\le d({\eta }_2,{\xi }_2),$ we infer that

$$\begin{array}{cc}\hfill \vartheta \left(\delta ({\mathsf{\Sigma }}_{\Gamma }\left(U\right),{\mathsf{\Sigma }}_{\mathsf{{\rm Y}}}\left(V\right))\right)& \le {\left[\vartheta (max\{d({\eta }_1,{\xi }_1),d({\eta }_2,{\xi }_2)\})\right]}^k\hfill \\ \hfill & ={\left[\vartheta \left(\delta (U,V)\right)\right]}^k.\hfill \end{array}$$

That is, ${\mathsf{\Sigma }}_{\Gamma }$ and ${\mathsf{\Sigma }}_{\mathsf{{\rm Y}}}$ satisfy the inequality (2). Therefore, by Lemma 1, there exists a unique $U^{*}=({\eta }_1^{*},{\eta }_2^{*})\in B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right)$ such that ${\mathsf{\Sigma }}_{\Gamma }\left(U^{*}\right)={\mathsf{\Sigma }}_{\mathsf{{\rm Y}}}\left(U^{*}\right)=U^{*}.$ This means that

$$\begin{array}{c}\hfill \Gamma ({\eta }_1^{*},{\eta }_2^{*})=\mathsf{{\rm Y}}({\eta }_1^{*},{\eta }_2^{*})={\eta }_1^{*}\end{array}$$

and

$$\begin{array}{c}\hfill \Gamma ({\eta }_1^{*}\left(\alpha \right),{\eta }_2^{*}\left(\alpha \right))=\mathsf{{\rm Y}}({\eta }_1^{*}\left(\alpha \right),{\eta }_2^{*}\left(\alpha \right))={\eta }_2^{*}.\end{array}$$

This finishes the proof. □

3. Application to Dynamic Programming

Consider the following coupled systems of functional equations

$$\begin{array}{c}{r}_1\left(\eta \right)={\displaystyle \underset{\xi \in \Delta }{sup}}\left\{p\left(\eta ,\xi \right)+P\left(\eta ,\xi ,r_1\left(\kappa \left(\eta ,\xi \right)\right),s_1\left(\kappa \left(\eta ,\xi \right)\right)\right)\right\}\hfill \\ s_1\left(\eta \right)={\displaystyle \underset{\xi \in \Delta }{sup}}\left\{p\left(\eta ,\xi \right)+P\left(\eta ,\xi ,r_1\left(\alpha \left(\kappa \left(\eta ,\xi \right)\right)\right),s_1\left(\alpha \left(\kappa \left(\eta ,\xi \right)\right)\right)\right)\right\}\hfill \end{array}$$

(8)

and

$$\begin{array}{c}{r}_2\left(\eta \right)={\displaystyle \underset{\xi \in \Delta }{sup}}\left\{p\left(\eta ,\xi \right)+Q\left(\eta ,\xi ,r_2\left(\kappa \left(\eta ,\xi \right)\right),s_2\left(\kappa \left(\eta ,\xi \right)\right)\right)\right\}\hfill \\ s_2\left(\eta \right)={\displaystyle \underset{\xi \in \Delta }{sup}}\left\{p\left(\eta ,\xi \right)+Q\left(\eta ,\xi ,r_2\left(\alpha \left(\kappa \left(\eta ,\xi \right)\right)\right),s_2\left(\alpha \left(\kappa \left(\eta ,\xi \right)\right)\right)\right)\right\}\hfill \end{array}$$

(9)

for all $\eta \in \mathsf{\Omega }$, which appear in the study of dynamic programming (see [8]), where $\mathsf{\Omega }$ is a state space, $\Delta$ is a decision space, $\kappa :\mathsf{\Omega }\times \Delta \to \mathsf{\Omega },$$p:\mathsf{\Omega }\times \Delta \to \mathbb{R},$$P,Q:\mathsf{\Omega }\times \Delta \times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ and $\alpha :\mathsf{\Omega }\to \mathsf{\Omega }$ are given mappings.

In this section, we discuss the existence of a unique common solution to the systems of functional Equations (8) and (9) by using the obtained results in the previous section.

Theorem 2.

Consider the systems of functional Equations (8) and (9). Assume that the following conditions are satisfied:

(i)

$p:\mathsf{\Omega }\times \Delta \to \mathbb{R}$ and $P,Q:\mathsf{\Omega }\times \Delta \times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are bounded functions;

(ii)

there exists $\beta \in (0,1)$ such that for arbitrary points $\eta \in \mathsf{\Omega },$$\xi \in \Delta$ and $h,k,h_1,k_1\in \mathbb{R}$,

$$\begin{array}{c}\hfill \left|P\left(\eta ,\xi ,h,k\right)-Q\left(\eta ,\xi ,h_1,k_1\right)\right|\le {\left[{\left[1+\sqrt{sup\left\{\left|h-h_1\right|,\left|k-k_1\right|\right\}}\right]}^{\beta }-1\right]}^2.\end{array}$$

Then Equations (8) and (9) have a unique common solution in $B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right)$.

Proof. 

First, we consider the operators $\Gamma$ and $\mathsf{{\rm Y}}$ defined on $B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right)$ as

$$\begin{array}{c}\left(\Gamma \left(r,s\right)\right)\left(\eta \right)={\displaystyle \underset{\xi \in \Delta }{sup}}\left\{p\left(\eta ,\xi \right)+P\left(\eta ,\xi ,r\left(\kappa \left(\eta ,\xi \right)\right),s\left(\kappa \left(\eta ,\xi \right)\right)\right)\right\},\\ \left(\mathsf{{\rm Y}}\left(r,s\right)\right)\left(\eta \right)={\displaystyle \underset{\xi \in \Delta }{sup}}\left\{p\left(\eta ,\xi \right)+Q\left(\eta ,\xi ,r\left(\kappa \left(\eta ,\xi \right)\right),s\left(\kappa \left(\eta ,\xi \right)\right)\right)\right\}\end{array}$$

(10)

for all $(r,s)\in B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right)$ and $\eta \in \mathsf{\Omega }$. Since functions $p,$ P and Q are bounded, then $\Gamma$ and $\mathsf{{\rm Y}}$ are well-defined.

Now we will show that $\Gamma$ and $\mathsf{{\rm Y}}$ satisfy the condition (1) in Theorem 1 with the sub metric d. Let $\left(r_1,s_1\right),\left(r_2,s_2\right)\in B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right).$ Then, by (ii), we get

$$\begin{array}{cc}d(\Gamma (\hfill & r_1,s_1),\mathsf{{\rm Y}}(r_2,s_2))\hfill \\ \hfill & ={\displaystyle \underset{\eta \in \mathsf{\Omega }}{sup}}\left|\Gamma \left(r_1,s_1\right)\left(\eta \right)-\mathsf{{\rm Y}}\left(r_2,s_2\right)\left(\eta \right)\right|\hfill \\ \hfill & ={\displaystyle \underset{\eta \in \mathsf{\Omega }}{sup}}\left|{\displaystyle \underset{\xi \in \Delta }{sup}}\left\{p\left(\eta ,\xi \right)+P\left(\eta ,\xi ,r_1\left(\kappa \left(\eta ,\xi \right)\right),s_1\left(\kappa \left(\eta ,\xi \right)\right)\right)\right\}\right.\hfill \\ \hfill & \left.-{\displaystyle \underset{\xi \in \Delta }{sup}}\left\{p\left(\eta ,\xi \right)+Q\left(\eta ,\xi ,r_2\left(\kappa \left(\eta ,\xi \right)\right),s_2\left(\kappa \left(\eta ,\xi \right)\right)\right)\right\}\right|\hfill \\ \hfill & \le {\displaystyle \underset{\eta \in \mathsf{\Omega }}{sup}}\{{\displaystyle \underset{\xi \in \Delta }{sup}}|P\left(\eta ,\xi ,r_1\left(\kappa \left(\eta ,\xi \right)\right),s_1\left(\kappa \left(\eta ,\xi \right)\right)\right)\hfill \\ \hfill & -Q(\eta ,\xi ,r_2\left(\kappa (\eta ,\xi )\right),s_2\left(\kappa (\eta ,\xi )\right))\left|\right\}\hfill \\ \hfill & \le {\displaystyle \underset{\eta \in \mathsf{\Omega }}{sup}}\left\{{\displaystyle \underset{\xi \in \Delta }{sup}}\left\{{\left[{\left[1+\sqrt{sup\left\{A,B\right\}}\right]}^{\beta }-1\right]}^2\right\}\right\},\hfill \end{array}$$

(11)

where

$$A:=\left|r_1\left(\kappa \left(\eta ,\xi \right)\right)-r_2\left(\kappa \left(\eta ,\xi \right)\right)\right|andB:=\left|s_1\left(\kappa \left(\eta ,\xi \right)\right)-s_2\left(\kappa \left(\eta ,\xi \right)\right)\right|.$$

It yields that

$$d(\Gamma (r_1,s_1),\mathsf{{\rm Y}}(r_2,s_2))\le {\left[{\left[1+\sqrt{sup\left\{d(r_1,r_2),d(s_1,s_2)\right\}}\right]}^{\beta }-1\right]}^2.$$

(12)

From the above inequality, we obtain

$$1+\sqrt{d(\Gamma (r_1,s_1),\mathsf{{\rm Y}}(r_2,s_2))}\le {\left[1+\sqrt{sup\left\{d(r_1,r_2),d(s_1,s_2)\right\}}\right]}^{\beta }.$$

(13)

By setting $\vartheta \in \Theta$ by $\vartheta \left(t\right)=1+\sqrt{t}$ for all $t>0$ and using (13), we infer

$$\vartheta \left(d(\Gamma (r_1,s_1),\mathsf{{\rm Y}}(r_2,s_2))\right)\le {\left[\vartheta (max\left\{d(r_1,r_2),d(s_1,s_2)\right\})\right]}^{\beta }$$

for all $\left(r_1,s_1\right),\left(r_2,s_2\right)\in B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right)$. This means that the condition (1) of Theorem 1 holds and consequently, $\Gamma$ and $\mathsf{{\rm Y}}$ have a unique $\alpha$-coupled common fixed point, i.e., Equations (8) and (9) have a unique common solution in $B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right)$. □

By using the same method in the proof of Theorem 2 together Theorem 1 with the function $\vartheta \in \Theta$ defined by $\vartheta \left(t\right)=e^{\sqrt{t}}$, we get the following result.

Theorem 3.

Consider the systems of functional Equations (8) and (9). Assume that the following conditions are satisfied:

(i)

$p:\mathsf{\Omega }\times \Delta \to \mathbb{R}$ and $P,Q:\mathsf{\Omega }\times \Delta \times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are bounded functions;

(ii)

there exists $\beta \in (0,+\infty )$ such that for arbitrary points $\eta \in \mathsf{\Omega },$$\xi \in \Delta$ and $h,k,h_1,k_1\in \mathbb{R}$,

$$\begin{array}{c}\hfill \left|P\left(\eta ,\xi ,h,k\right)-Q\left(\eta ,\xi ,h_1,k_1\right)\right|\le e^{-\beta }sup\left\{\left|h-h_1\right|,\left|k-k_1\right|\right\}.\end{array}$$

Then Equations (8) and (9) have a unique common solution in $B\left(\mathsf{\Omega }\right)\times B\left(\mathsf{\Omega }\right)$.

4. Conclusions

In this paper, we introduced the notion of $\alpha$-coupled common fixed points and established the existence and uniqueness of such points. We applied our results to ensure the existence of a unique common solution of coupled systems of functional equations arising in dynamic programming. We think that this new concept will be a powerful tool in searching for the existence of solutions for coupled systems of integral equations, differential equations, and also fractional integro-differential equations.