Multivalued weakly Picard operators via simulation functions with application to functional equations

1.   Introduction and preliminaries

We pronounce $(\mathcal{N}, \wp)$ a metric space, the classes $\mathcal{P}(\mathcal{N})$ and $\mathcal{CB}(\mathcal{N})$ of all nonempty, closed and bounded subsets of $\mathcal{N}$.

If $\mathcal{U}, \mathcal{V}\in \mathcal{CB(N)}$, review the functionals:

$\mathcal{D}: \mathcal{P(N)} \times \mathcal{P(N)} \rightarrow \mathbb{R}_{+}$, $\mathcal{D}(\mathcal{W}, \mathcal{V}) = inf \{ \wp(\omega, \lambda): \omega\in \mathcal{W}, \; \lambda \in \mathcal{V} \}$, $\mathcal{W} \subset \mathcal{N}$;

$\mathcal{H}:\mathcal{P(N)} \times \mathcal{P(N)} \rightarrow \mathbb{R}_{+}, \mathcal{H(U, V)}: = max\{ \sup\limits_{u\in \mathcal{U}} \inf\limits_{v\in \mathcal{V}}\wp(u, v), \sup\limits_{v\in \mathcal{V}} \inf\limits_{u\in \mathcal{U}}\wp(u, v)\}$ - the Pompeiu- Hausdorff functional.

Lemma 1.1. ^[12] Suppose $\mathcal{U}\subseteq \mathcal{N}$ and $\xi > 1$. Then for $\mu \in \mathcal{N}$ there is $v \in \mathcal{V}$ such that $\wp(\mu, v) \leq \xi \mathcal{D}(\mu, \mathcal{V})$.

Definition 1.2. ^[16] A function $\mathcal{Q} : \mathcal{N} \to \mathcal{CB(N)}$ is called multi-valued weakly Picard operator (MWP operator) if for all $\mu \in \mathcal{N}$ and $\mu \in \mathcal{Q} \mu$, there is a sequence $(\mu_{m})$ in $\mathcal{N}$ such that (i) $\mu_{0} = \mu, \mu_{1} = \lambda$, (ii) $\mu_{n+1} \in \mathcal{Q}\mu_{m}$ $\forall\; m \geq 0$, and (iii) $(\mu_{m})$ tends to the fixed point of $\mathcal{Q}$.

Utilizing the approach of Rus ^[16], Popescu ^[13] defined a multivalued operator and named it as $(s, r)$-contractive multivalued and siglevalued operators and showed that the underlying mapping is an MWP operator. Following Popescu ^[13], Kamran ^[9] extended $(s, r)$-contractive operators to weakly $(s, r)$-contractive operators and proved that the underlying mappings are still MWP operators.

A new generalization of Banach's theorem was given by Khojasteh et al. ^[11] by considering a function $\zeta : [0, \infty) \times [0, \infty) \rightarrow \mathbb{R}$, which follows the assertions given below:

$(\zeta_{a})$ $\zeta(0, 0) = 0$,

$(\zeta_{b})$ $\zeta(\alpha, \beta) < \beta-\alpha$ for all $\alpha, \beta > 0$,

$(\zeta_{c})$ The sequences $\{\alpha_{n}\}, \{\beta_{n}\}$ in $(0, \infty)$ with $\lim\limits_{n\rightarrow \infty}\alpha_{n} = \lim\limits_{n\to \infty}\beta_{n} > 0$, then

They named such functions as simulation function and the contractive map is named as $\mathcal{Z}$-contraction. Later on, Rold$\acute{a}$n-L$\acute{o}$pez-de-Hierro et al. ^[15] replaced $(\zeta_{c})$ with $(\zeta'_{c})$,

($\zeta'_{c}$): the sequences $(\alpha_{n}), (\beta_{n})$ in $(0, \infty)$ with $\lim\limits_{n\rightarrow \infty}\alpha_{n} = \lim\limits_{n\rightarrow \infty}\beta_{n} > 0$ and $\alpha_{n} < \beta_{n}$, then

The lemma stated below is essential:

Lemma 1.3. ^[14] Suppose a sequence $(\mu_{n})$ in $\mathcal{N}$ with

If the sequence is not a Cauchy, then $\exists$ $\varepsilon > 0$ and sequences $\mu_{m(j)}$ and $\mu_{n(j)}$ such that $n(j) > m(j) > j$ and the sequences

converges to $\varepsilon^{+}$ when $k\rightarrow\infty$

The purpose of this paper is to introduce the notion of Suzuki type multivalued $\mathcal{Z}_{(\phi, \lambda)}$-contraction and to prove the fixed point and data dependence results for such contraction mapping. We give an example to show the validity of our results. Moreover, we present an application to functional equation arising in dynamical system to show the usability of our results.

2.   Fixed point results

We now give the definition of Suzuki type multivalued $\mathcal{Z}_{(\phi, \lambda)}$-contraction and obtain some fixed point results.

We start this section with the following notion:

Denote $\Phi$ the set consisting of all strictly increasing functions $\varphi : [0, 1) \rightarrow (0, 1]$. We define

Definition 2.1. A mapping $\mathcal{Q}:\mathcal{N} \rightarrow \mathcal{CB(N)}$ is called Suzuki type multivalued $\mathcal{Z}_{(\varphi, \lambda)}$-contraction with respect to $\zeta$, if there is a function $\varphi \in \Phi$ such that

implies

where $\rho \in [0, 1)$, and

Theorem 2.2. Suppose that $\mathcal{Q}:\mathcal{N} \to \mathcal{CB(N)}$ is Suzuki type multivalued $\mathcal{Z}_{(\varphi, \rho)}$-contraction, where $(\mathcal{N}, \wp)$ is a complete metric space. Then $\mathcal{Q}$ is an MWP operator.

Proof. Choose $\mu_0 \in X$, $\mu_1 \in \mathcal{Q}\mu_0$ and $\rho_{1}$ be real number such that $0 \leq \rho \leq s \leq \rho_1 < 1$. One can choose $\mu_2\in \mathcal{Q}\mu_1$ such that

Since

from (2.1) and (2.2), we have

this implies

where

If $\mathcal{R}(\mu_0, \mu_1) \leq \wp(\mu_1, \mu_2)$, then

This implies $\wp(\mu_1, \mu_2) = 0$ or $\mu_1 = \mu_2 \in \mathcal{Q}\mu_1$, i.e. $\mu_1$ is a fixed point. So, we consider the case when $\mathcal{R}(\mu_0, \mu_1) \leq \wp(\mu_0, \mu_1)$, this gives

Hence we can recursively define a sequence $(\mu_n)$ such that $\mu_{n+1} \in \mathcal{Q}\mu_n$ and

which implies

To prove that $(\mu_n)$ is Cauchy, contrary suppose that it is not, then by Lemma 1.3, there is $\epsilon > 0$ and two subsequences $m(j)$, $n(j)$ with $n(j) > m(j) > j$ such that $\wp(\mu_{m(j)}, \mu_{n(j)}) \to \epsilon^+$, $\wp(\mu_{m(j)+1}, \mu_{n(j)+1}) \to \epsilon^+$ as $j \to \infty$ also

By (2.4), we have

thus by (2.1), (2.2) and $(\zeta_{b})$, we have

therefore, we have

hence

so by $(\zeta_{c})$ and (2.2), we have

this leads to a contradiction. Hence $(\mu_n)$ is a Cauchy sequence. Completeness of $\mathcal{N}$ implies that there is $u\in \mathcal{N}$ such that $\lim\limits_{n \to \infty}\mu_n = u$. Moreover

Our claim is

for all $\mu\neq u.$ Since $\lim\limits_{n\rightarrow \infty }\wp(u_{n}, u) = 0,$ there exists an $n_{0}\in\mathbb{N}$ such that $\wp(u, u_{n}) < \frac{1}{3}\wp(u, \mu)$ for all $n\geq n_{0}.$ Note that $\varphi (s)\leq 1$ and

Thus $\varphi (s)\max \left\{ \mathcal{D}(u_{n}, \mathcal{Q}u_{n}), \frac{1}{2}\mathcal{D}(\mu, \mathcal{Q}u_{n})\right\} \leq \wp(u_{n}, \mu)$ holds for all $n\geq n_{0}$. From (2.1) and (2.2), we have

this gives

Tending $n\rightarrow \infty$ to inequality (2.10), we arrived at

If

then

and this implies that

Hence

valid $\forall$ $\mu\neq u.$ To show that $u\in \mathcal{Q}u,$ we discus two cases:

(i) If $0\leq r\leq s < \dfrac{1}{2}.$ Contrary suppose that $u\notin \mathcal{Q}u.$ Since $\mathcal{Q}u\neq \phi$, there exists $c\in \mathcal{Q}u$ such that $c\neq u$ and

Thus

As $\varphi (s)\max \left\{ \mathcal{D}(u, \mathcal{Q}u), \frac{1}{2}\mathcal{D}(c, \mathcal{Q}u)\right\} \leq \max \{\wp(u, c), \wp(c, c)\}\leq \wp(u, c),$ so by (2.1) and (2.2), we have

this gives

Thus

Since $\mathcal{D}(c, \mathcal{Q}c)\leq \mathcal{H}(\mathcal{Q}u, \mathcal{Q}c),$ therefore, we have

hence

Since $\rho < 1,$ therefore

By (2.9) and (2.14), we have

Using (2.9), (2.12), (2.13) and (2.15), we have

a contradiction. Hence $u\in \mathcal{Q}u.$

(ii) If $\dfrac{1}{2}\leq \rho \leq s < 1$, then we first show that

for all $\mu\in \mathcal{N}$ with $\mu\neq u$.For each $n\in \mathbb{N}$, there exists $\lambda_{n}\in \mathcal{Q}\mu$ such that $\wp(u, \lambda_{n}) < \mathcal{D}(u, \mathcal{Q}\mu)+\dfrac{1}{n}\wp(\mu, u).$ So we have

Now by (2.9), we have

If $\max \{\wp(u, \mu), \mathcal{D}(\mu, \mathcal{Q}\mu)\} = \wp(\mu, u),$ then from (2.17), we obtain that

which gives

Tending $n\to \infty,$ we have

By (2.1) with $\lambda = u,$ (2.16) satisfied. If $\max\{\wp(u, \mu), \mathcal{D}(\mu, \mathcal{Q}\mu)\} = \mathcal{D}(\mu, \mathcal{Q}\mu),$ then (2.9) gives

and hence $\mathcal{D}(\mu, \mathcal{Q}\mu)\leq \left(\frac{1}{1-\rho}\right) \wp(\mu, u).$ By (2.17), we obtain that

As $\dfrac{1}{2}\leq \rho < 1$ and $\rho \leq s,$ so

tending $n\to \infty$ gives

and thus (2.16) satisfied. Using $\mu = u_{n}$ in (2.16), we have

Tending $n\to \infty$ to inequality (2.18), we arrived at

which further gives that $\mathcal{D}(u, \mathcal{Q}u) = 0,$ that is, $u\in \mathcal{Q}u.$

For the case of single-valued Suzuki type $\mathcal{Z}_{(\varphi, \rho)}$-contraction, we have the following result:

Theorem 2.3. Assume that $\mathcal{Q} : \mathcal{N} \to \mathcal{N}$ be Suzuki type $\mathcal{Z}_{(\varphi, \rho)}$-contraction on a complete metric space $(\mathcal{N}, \wp)$, then $\mathcal{N}$ is an MWP operator and has a unique fixed point.

Proof. By Theorem 2.2, $\mathcal{Q}$ is an MWP operator. Let $\mu\neq \lambda$ be two fixed points of $\mathcal{Q}$, then

this implies

this gives

which is not possible, hence $\wp(\mu, \lambda) = 0$. Thus $\mathcal{Q}$ possesses a unique fixed point.

Example 2.1. Let $\mathcal{N} = \{0, 1, 2, 3\}$ and $\wp(\mu, \lambda) = |\mu-\lambda|$. Define $\mathcal{Q}: \mathcal{N}\to \mathcal{CB(N)}$ by

Now for $\mu, \lambda\in \mathcal{N}$ with $\mu\neq \lambda$, we have

So $max\{\mathcal{D}(\mu, \mathcal{Q}\mu): \mu \in \mathcal{N}\} = 2$, $max\{\mathcal{D}(\lambda, \mathcal{Q}\mu): \mu, \lambda \in \mathcal{N} \} = 1$ and $min\{\wp(\mu, \lambda): \mu, \lambda \in \mathcal{N}\} = 1$. Now for $s = 0.6, \varphi(s) = 0.4$ and

holds true for all $\mu, \lambda \in \mathcal{N}.$ Let $\zeta(\alpha, \beta) = \frac{4}{5}\beta-\alpha,$ then for $\rho = 0.7$, we have

Hence $\mathcal{Q}$ is $\mathcal{Z}_{(\varphi, 0.7)}$-contraction. Thus by Theorem 2.2, $\mathcal{Q}$ is an MWP operator and $Fix(\mathcal{Q}) = \{0, 2\}$.

3.   Data dependence

We now prove a data dependence result.

Theorem 3.1. Ssppose that $\mathcal{Q}_1$, $\mathcal{Q}_2$ are two multivalued operators on a metric space $(\mathcal{N}, \wp)$, if

1. $\mathcal{Q}_{i}$ is $\mathcal{Z}_{(\varphi, \rho_i)}$-contraction for $i = 1, 2$,

2. there exist a real number $\rho > 0$ such that $\mathcal{H}(\mathcal{Q}_1\mu, \mathcal{Q}_2\mu) \leq \rho$, $\forall$ $\mu \in \mathcal{N}.$

Then

1. $Fix(\mathcal{Q}_i) \in \mathcal{CL(N)}$, for $i = 1, 2$,

2. $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ are multivalued operators and

Proof. Theorem 2.2 gives that $Fix(\mathcal{Q}_i)\neq \phi$ for $i = 1, 2$. To prove that $Fix(\mathcal{Q})$ is closed, assume that $(\mu_{n})\subset Fix(\mathcal{Q})$ such that $\mu_{n} \to t$ as $n \to\infty$, we have

this implies

this gives

Now

letting $n \to \infty$, we have $\mathcal{D}(\omega, \mathcal{Q}\omega) = 0$, so $\omega \in Fix(\mathcal{Q})$ and hence $Fix(\mathcal{Q})$ is closed.

(b) From Theorem 2.2, it follows from (i) that $\mathcal{Q}_1$ and $\mathcal{Q}_2$ are MWP operators. Further, let $\mu_0$ be fixed point of $\mathcal{Q}_1$, then there exist $\mu_1 \in \mathcal{Q}_2\mu_0$ and a real number $q > 1$ such that

since $\mu_1 \in \mathcal{Q}_2\mu_0$, we can choose $\mu_2 \in \mathcal{Q}_2\mu_1$ such that

also

this implies

Hence

(Choosing $\mathcal{R}(\mu_0, \mu_1)\leq \wp(\mu_1, \mu_2)$ gives fixed point). Hence

Therefore, for operator $\mathcal{Q}_2$, we get an iterative sequence of successive approximations satisfying $\mu_{n+1}\in \mathcal{Q}_2\mu_n$ such that

for all $n \geq 0.$ If there exists a positive integer $N$ such that for $n \geq N$ and $l \geq 1$,

Choosing $1 < q < min\{\frac{1}{\rho_1}, \frac{1}{\rho_2}\}$ and taking limit as $n \to \infty$, we have $(\mu_n)$ is a Cauchy sequence in $\mathcal{N}$. Let $u \in \mathcal{N}$ be the limit point of $(\mu_n)$. We show that $u\in \mathcal{Q}_{2}$. Otherwise, let $N$ be the positive integer such that $\mathcal{D}(u, \mathcal{Q}_2\mu_n) > 0 = \wp(u, \mu_n),$ $\forall \; n \geq N.$ Then $\wp(\mu, \mu_{n+1}) > \wp(u, \mu_n)$ for all $n \geq N,$ a contradiction. Hence there exist a subsequence ${\mu_{n(j)}}$ such that for all $j \in \mathbb{N}$, we have

(since $\wp(\mu_{n(j)}, \mu_{n(j)+1}) \to 0$ as $j \to \infty$). Above inequality gives

implies

Thus

So $u \in Fix(\mathcal{Q}_2)$. In (2.1), letting $l \to \infty$, we get

for all $n \in \mathbb{N}$. Then

On the same lines, we have that for each $u_0 \in Fix(\mathcal{Q}_2)$, there exists $\mu \in Fix(\mathcal{Q}_1)$ such that

Hence

Taking limit as $q \searrow 1$, we get the desired result.

4.   An application

A variety of fixed point theorems have been used to explore the existence and uniqueness of solution of functional equations arising in dynamic programming. Utilizing Theorem 2.3, we explore the existence and uniqueness of solution for a nonlinear functional equation. From now on, $W_{1}(\mathbb{R})$ and $W_{2}(\mathbb{R})$ are assumed to be Banach spaces and $U_{1} \subset W_{1}$, $U_{2} \subset W_{2}$. Class of all bounded real-valued functions on $U_{1}$ is denoted by $\mathcal{B}(U_{1})$. The metric space pair $\mathcal{B}(U_{1}, \wp)$ where

is complete. Viewing $U_1$ and $U_2$ as the state and decision spaces respectively, the problem of dynamic programming reduces to the problem of solving the functional equation:

where $\eta : U_1 \times U_2 \to U_1$ is the transformation of the process and $p(\mu)$ represents the optimal return function with initial functional equation:

where $v : U_1 \times U_2 \to \mathbb{R}$ and $G : U_1\times U_2 \times \mathbb{R} \to \mathbb{R}$ are bounded functions. Let $\mathcal{Q}:\mathcal{B}(U_{1})\to \mathcal{B}(U_{1})$ be defined by:

Theorem 4.1. Assume that there exists $\rho > 0$ such that for every $(\mu, \lambda) \in U_1\times U_2,$ $f, g \in \mathcal{B}(U_1)$ and $t \in U_1$ the inequality

implies

where

Then (4.1) possesses a bounded solution.

Proof. It is obvious that $\mathcal{Q}$ is self map on $\mathcal{B}(U_1)$. Suppose $\gamma$ is an arbitrary positive real number and $f_1, f_2 \in \mathcal{B}(U_1)$. Take $\mu \in U_1$ and choose $\lambda_1, \lambda_2 \in U_2$ such that

where $\eta_i = \eta_{i}(\mu, \lambda_i)$, $i = 1, 2$. By definition of $\mathcal{Q}$, we get

If the inequality (4.2) holds with $f = f_1, g = f_2,$ then from (4.3), (4.4) and (4.6), we have

Similarly, from (4.3), (4.5) and (4.6), we have

Hence, from (4.6) and (4.7), we obtain that

since the inequality holds for any $\mu \in U_1$ and $\gamma > 0$, we get that

implies

so if we choose $\zeta(\alpha, \beta) = \frac{\beta}{\beta+1}-\alpha$, then we have

Hence, all the assertions of Theorem 2.3 are verified and therefore we get the conclusion.

Example 4.1. Let $W_{1} = \mathbb{R} = W_{2}$, $U_{1} = [1,50]$, $U_{2} = \mathbb{R}^{+}$. We define $\eta: U_{1}\times U_{2}\to U_{1}$, $v:U_{1}\times U_{2}\to\mathbb{R}$ and $G: U_{1}\times U_{2}\mathbb{R}\to \mathbb{R}$ by

and

Now for $s = 0.5$, $\varphi(s) = \frac{1}{2}$. Therefore for $\rho = 0.65$, it is clear that (4.2)-(4.3) are satisfied. Hence from Theorem 2.3 that the functional equation (4.1) possesses a bounded solution.

5.   Conclusions

In this paper we introduced a new type of set valued contraction in Suzuki sense together with simulation functions and proved some new fixed point and data dependence theorems for such class of mappings. We gave an example to prove the validity of our results. At last, we presented an application to nonlinear functional equation arising in dynamical system to show the usability of obtained results.

Acknowledgments

The author Thabet Abdeljawad would like to thank Prince Sultan University for funding this work through the research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM)(Group Number RG-DES-2017-01-17).

Conflict of interest

The authors declare no conflict of interest.