Suzuki type multivalued contractions in C^*-algebra valued metric spaces with an application

1.   Introduction

The classical Banach contraction principle is proved in metric spaces. One natural way to improve this result is to enlarge the class of spaces. With this idea in mind, several authors generalized the notion of metric spaces which saw the evolution of some new notions (see ^{[1,2,3,4,5,6,7,8,9,10,11]}). In 2008, Suzuki ^[12] established a new type of contraction mappings and studied the existence and uniqueness of fixed point theorems, which is a genuine extension of the Banach contraction principle. Later on, many researchers have been worked on this contraction mapping for single-valued mappings as well as multivalued mappings. One of the initial results was introduced by Nadler ^[13] in the line of research of multivalued mappings. Later on, the domain of fixed points of multivalued functions was developed into a very rich and fruitful theory.

In 2014, Ma et al. ^[14] introduce the class of $C^{*}$-algebra valued metric spaces (in short $C^{*}$-AVMS), wherein the range set $\mathbb{R}$ is replaced by an unital $C^{*}$-algebra, which is more generalized than the class of metric spaces and proved some related fixed point results. Later on, many researchers extended this class by considering some generalized class (see ^{[2,4,10,15,16,17]}).

Inspired by the preceding observations, we prove a fixed point result via Suzuki type contraction multivalued mapping in $C^{*}$-algebra valued metric spaces and give the application in Fredholm type integral equation.

Throughout the paper, $\mathcal{C}$ denotes an unital $C^{*}$-algebra. An element $0_{\mathcal{C}}\in\mathcal{C}$ is known as zero element in $\mathcal{C}$ and if $0_{\mathcal{C}}\preccurlyeq a\in\mathcal{C}$ then $a$ is called positive element in $\mathcal{C}$. Also, $\mathcal{C}_{+} = \{a\in\mathcal{C}; a\succcurlyeq0_{\mathbb{A}}\}$. Moreover, if $a = a^{*}$ and $\sigma(a) = \{\lambda\in \mathbb{R}:\lambda I-a\hbox { is non-invertible }\}\subseteq[0, \infty)$. The partial ordering on $\mathcal{C}$ can be defined as follows: $a\preccurlyeq b$ if and only if $0_\mathcal{C}\preccurlyeq b-a$.

In 2014, Ma et al. ^[14] introduced the following definition:

Definition 1.1. Let $A\neq\emptyset$ and a mapping $d: A \times A \rightarrow \mathcal{C}$ satisfies the following (for all $a, b, c\in A$):

$(i)$ $d(a, b) \succcurlyeq 0_\mathcal{C}$ and $d(a, b) = 0_\mathcal{C}$ iff $a = b$;

$(ii)$ $d(a, b) = d(b, a)$;

$(iii)$ $d(a, b)\preccurlyeq d(a, c)+d(c, b)$.

Then the mapping $d$ is known as $C^*$-algebra valued metric on $A$ and $(A, \mathcal{C}, d)$ is known as $C^*$-algebra valued metric space.

Definition 1.2. ^[14] Let $(A, \mathcal{C}, d)$ be $C^*$-algebra valued metric space, $a\in A$ and $\{a_{n}\}$ a sequence in $X$.

1. $\{a_{n}\}$ is called convergent (with respect to $\mathcal{C}$), if for given $\epsilon \succ0_{\mathcal{C}}$, there exists $k\in\mathbb{N}$ such that $d(a_n, a)\prec \epsilon$, for all $n > k$. Equivalently, $\lim\limits_{n\to\infty}d(a_n, a) = 0_{\mathcal{C}}$.

2. $\{a_n\}$ is called Cauchy sequence (with respect to $\mathcal{C}$), if for given $\epsilon \succ0_{\mathcal{C}}$, there exists $k\in\mathbb{N}$ such that $d(a_n, a_{m})\prec \epsilon$, for all $n, m > k$. Equivalently, $\lim\limits_{n\to\infty}d(a_{n}, a_{m}) = 0_{\mathcal{C}}$.

3. $(A, \mathcal{C}, d)$ is called complete with respect to $\mathcal{C}$, if every Cauchy in $A$ is convergent to some point $a$ in $A$.

Let $(A, \mathcal{C}, d)$ be a $C^*$-algebra valued metric space. The set

is called open ball of radius $0_{\mathcal{C}}\prec\epsilon\in\mathcal{C}$ and at center $a\in A$. Similarly, the set

is called closed ball of radius $0_{\mathcal{C}}\prec\epsilon\in\mathcal{C}$ and at center $a\in A$. The set of open balls

forms a basis of some topology $\tau_{d}$ on $A$.

Definition 1.3. The max function on $\mathbb{A}$ ($C^*$-algebra) with the partial order relation $'\preccurlyeq'$ is defined by (for all $a, b\in\mathbb{A}_{+}$):

The family $\mathcal{CB^{\mathcal{C}}}(A)$ stands for all nonempty, closed and bounded subsets of $(A, \mathcal{C}, d)$. Moreover, for $M, N\in\mathcal{CB^{\mathcal{C}}}(A)$ and $x\in A,$ we define:

Define $C^*$-algebra valued Hausdorff metric $H_{\mathcal{C}}:\mathcal{CB^{\mathcal{C}}}(A)\times\mathcal{CB^{\mathcal{C}}}(A)\to\mathcal{C}$ by:

Remark 1.1. Let $(A, \mathcal{C}, d)$ be a $C^*$-algebra valued metric space and $M$ a nonempty subset of $A$, then

where, $\overline{M}$ denotes the closure of $M$ with respect to $C^*$-algebra valued metric $A.$ Also, $M$ is closed in $(A, \mathcal{C}, d)$ if and only if $M = \overline{M}.$

Proposition 1.1. Let $(A, \mathcal{C}, d)$ be a $C^*$-algebra valued metric space. For $M, N, L\in\mathcal{CB^{A}}(A)$, we have the following:

(i) $\delta_{\mathcal{P}}(M, M) = diam(M);$

(ii) $\delta_{\mathcal{P}}(M, N) = 0_{\mathcal{C}}\; \Rightarrow\; M\subseteq N$;

(iii) $N\subset L\Rightarrow\delta_{\mathcal{P}}(M, L)\preccurlyeq\delta_{\mathcal{P}}(M, N)$;

(iv) $\delta_{\mathcal{C}}(M\cup N, L) = \max\{\delta_{\mathcal{C}}(M, L), \delta_{\mathcal{C}}(N, L)\}$.

Proof. $(i)$ Suppose $M\in\mathcal{CB^{A}}(A).$ Then by the definition of $\delta_{\mathcal{C}}$, we have

$(ii)$ Suppose $M, N\in\mathcal{CB^{A}}(A)$ such that $\delta_{\mathcal{C}}(M, N) = 0_{\mathcal{C}}$. Then

which implies that $\delta_{\mathcal{C}}(M, N) = 0_{\mathcal{C}}.$ Therefore, $dist_{\mathcal{C}}(a, N) = 0_{\mathcal{C}}$ for all $a\in M$ implies that '$a$' is in the closure of $N$ for all $a\in M$. Since $N$ is closed, so $M\subseteq N.$

$(iii)$ Suppose $M, N, L\in\mathcal{CB^{A}}(A)$ such that $N\subseteq L$. Then

Thus

$(iv)$ Suppose $M, N, L\in\mathcal{CB^{A}}(A)$. Then

2.   Fixed point results

In this section, firstly we define following notions which are needed in our subsequent discussions.

Consider, $\mathcal{O}_{\mathcal{C}} = \{h\in \mathcal{C}; \; 0\leq\|h\| < 1\}$ and $\mathcal{O}^{'}_{\mathcal{C}} = \{h\in \mathcal{C}; \; 0\leq\|h\|\leq1\}.$

Next, let $\xi:\mathcal{O}_{\mathcal{C}}\to\mathcal{O}^{'}_{\mathcal{C}}$ be the non-increasing function defined by

Now, we present our main result as follows:

Theorem 2.1. Let $(A, \mathcal{C}, d)$ be complete $C^*$-algebra valued metric space and $f:X\to \mathcal{CB^{A}}(A)$. Suppose that there exists $h\in\mathcal{O}_{\mathcal{C}}$ such that $f$ satisfies the following:

for all $a, b \in A$, where $\xi$ is defined by (2.1) and

Then $f$ has a unique fixed point.

Proof. Consider $h_{1}\in\mathcal{O}_{\mathcal{C}}$ such that $0\leq \|h\| < \|h_{1}\| < 1$. Let $a_{1}\in A$ and $a_{2}\in fa_{1}$ be arbitrary points. Since $a_{2}\in fa_{1}$, then $d(a_{2}, fa_{2})\preccurlyeq H_{\mathcal{C}}(fa_{1}, fa_{2})$ and $\|\xi(h)\| < 1$,

Hence the assumption (2.2) yielding thereby,

Assume that $\max\{d(a_{1}, a_{2}), dist_{\mathcal{C}}(a_{2}, fa_{2})\} = dist_{\mathcal{C}}(a_{2}, fa_{2})$, then we have

a contradiction. Thus we have $dist_{\mathcal{C}}(a_{2}, fa_{2})\preccurlyeq h^{*}d(a_{1}, a_{2})h$. Hence there exists $a_{3}\in fa_{2}$ such that $d(a_{2}, a_{3})\preccurlyeq h^{*}_{1}d(a_{1}, a_{2})h_{1}$. By proceeding with this procedure, we can construct a sequence $\{a_{n}\}$ in $A$ such that $a_{n+1}\in fa_{n}$ with

and

Thus

Hereby, we presume that $\{a_{n}\}$ is a Cauchy sequence. Since $A$ is complete $C^{*}$-algebra valued metric space, so there is some point $z\in A$ such that $\lim\limits_{n\to\infty}a_{n} = z.$

Now, we shall show that

As $\lim\limits_{n\to\infty}a_{n} = z$, there exists $N_{0}\in\mathbb{N}$ such that

Therefore, we have

yielding thereby

Since $a_{n+1}\in fa_{n}$, then

Hence from (2.4), we have

On making limit as $n\to\infty$, we obtain (2.3).

To show that $z\in fz$. First, we take the case $\xi(h) = I$ for $0\leq\|h\|\leq\frac{1}{2}$. Let on contrary that $z\notin fz$. Now, let $u\in fz$ such that

Since $u\in fz$ implies that $u\neq z$ then from (2.3), we have

Also, since $\xi(h)^{*}dist_{\mathcal{C}}(z, fz)\xi(h)\preccurlyeq dist_{\mathcal{C}}(z, fz)\preccurlyeq d(z, u)$, then in view of condition (2.2), we have

Hence

Thus $\|dist_{\mathcal{C}}(u, fu)\|\leq \|h^{*}d(z, u)h\| < \|d(z, u)\|$. Therefore (2.5) gives arise

Therefore, we obtain

a contradiction. Therefore, $dist_{\mathcal{C}}(z, fz) = 0_{\mathcal{C}}$, which deduce that $z$ is a fixed point $f$.

Now, we take the case $\frac{1}{2}\leq\|h\|\leq1.$ Now, we prove

for all $a\in A$. If $a = z$, then above inequality holds. Hence we let $a\neq z$. Then, for each $n\in\mathbb{N}$, there exists a sequence $b_{n}\in fa$ such that

Now, by using (2.3), for all $n\mathbb{N}$, we have

Assume that $d(a, z)\succcurlyeq dist_{\mathcal{C}}(a, fa)$, then

On making limit as $n\to\infty$, we obtain $dist_{\mathcal{C}}(a, fa)\preccurlyeq\left(1+\|h\|^{2}\right)d(a, z)$. Hence

and from (2.2), we have (2.6). If $d(a, z)\prec dist_{\mathcal{C}}(a, fa),$ then

so that

On making limit as $n\to\infty$, we have $\xi(h)^{*}dist_{\mathcal{C}}(a, fa)\xi(h)\preccurlyeq d(a, z).$ Hence from (2.2), again we have (2.6).

Finally, from (2.6), we have

yielding thereby

a contraction. Hence $dist_{\mathcal{C}}(z, fz) = 0_{\mathcal{C}}$ implies that $z\in fz.$

For uniqueness, suppose there are $z, w\in A$ so that $z\in fz$ and $w\in fw.$ Thus by conditions (2.2), we have

a contraction. Hence $d(z, w) = 0_{\mathcal{C}}$ implies that $z = w.$ This completes the proof.

Example 2.1. Suppose $A = \big\{0, \frac{1}{10}, \frac{1}{5}\big\}$. The $C^{*}$-algebra valued metric $d:A\times A\to \mathcal{C}$ is defined by

Then $(A, \mathcal{C}, d)$ is a complete $C^{*}$-algebra valued metric space. Note that $\{0\}$ and $\big\{\frac{1}{10}\big\}$ are bounded sets in $(A, \mathcal{C}, d)$. In fact, if $a\in\big\{0, \frac{1}{10}, \frac{1}{5}\big\}$ then

Hence $\{0\}$ is closed. Next,

Hence $\big\{\frac{1}{10}\big\}$ is also closed. Now, define $f:A\rightarrow \mathcal{CB^{\mathcal{C}}}(A)$ by:

To prove the contractive condition $(i)$ of Theorem 2.1, we need the following:

Case 1. Let $a = 0$, then

For $a = 0$ or $b = \frac{1}{10}$, we have

For $a = 0$ or $b = \frac{1}{5}$, we have

Case 2. Let $a = \frac{1}{5}$. Then

this implies that

Hence the contractive condition $(i)$ of Theorem 2.1 is satisfied. Observe that, the mapping f has a unique fixed point (namely $a = 0$).

Corollary 2.1. The conclusions of Theorem 2.1 remain true if the contractive condition (2.2) is replaced by any one of the following:

assume that there exists $h\in\mathcal{O}_{\mathcal{C}}$ such that $\xi(h)^{*}dist_{\mathcal{C}}(a, fa)\xi(h)\preccurlyeq d(a, b)$ implies

(ⅰ) $H_{\mathcal{C}}(fa, fb)\preccurlyeq h^{*}d(a, b)h;$

(ⅱ) $H_{\mathcal{C}}(fa, fb)\preccurlyeq h^{*}\max\big\{d(a, b), dist_{\mathcal{C}}(a, fa)\big\}h;$

(ⅲ) $H_{\mathcal{C}}(fa, fb)\preccurlyeq h^{*}\max\left\{d(a, b), dist_{\mathcal{C}}(a, fa), dist_{\mathcal{C}}(b, fb)\right\}h;$

(ⅳ) $H_{\mathcal{C}}(fa, fb)\preccurlyeq h^{*}\max\Big\{d(a, b), \frac{dist_{\mathcal{C}}(a, fa)+dist_{\mathcal{C}}(b, fb)}{2}, \frac{dist_{\mathcal{C}}(a, fb)+dist_{\mathcal{C}}(b, fa)}{2}\Big\}h.$

for all $a, b \in A$, where $\xi$ is defined as in Theorem 2.1. Then $f$ has a unique fixed point.

The following corollary can be obtain from $(iii)$ of Corollary 2.1:

Corollary 2.2. Let $(A, \mathcal{C}, d)$ be complete $C^*$-algebra valued metric space and $f:A\to \mathcal{CB^{A}}(A)$. Suppose that there exists $h\in\mathcal{O}_{\mathcal{C}}$ such that $f$ satisfies the following:

for all $a, b \in A$, where $\gamma = (1/3)h$ and $\xi$ is defined as in Theorem 2.1. Then $f$ has a unique fixed point.

Now, we are presenting following corollary, by considering $f$ as a single-valued mapping:

Corollary 2.3. Let $(A, \mathcal{C}, d)$ be complete $C^*$-algebra valued metric space and $f:A\to A$. Suppose that there exists $h\in\mathcal{O}_{\mathcal{C}}$ such that $f$ satisfies the following:

for all $a, b \in A$ and $\xi$ is defined as in Theorem 2.1. Then $f$ has a unique fixed point.

3.   Applications

Now, we provide the following system of Fredholm integral equations to examine the existence and uniqueness of solution in support of Corollary 2.3.

where, $G:E\times E\times\mathbb{R}\to\mathbb{R}$, $l\in L^{\infty}(E)$ and $E$ is a measurable set.

Suppose that $A = L^\infty(E)$, $H = L^2(E)$ and $L(H) = \mathcal{C}$. Assume that $\phi_{u}$ is a multiplicative operator defined on $H$, that is, $\pi_u : H \rightarrow H$ such that

Define $d:A\times A\rightarrow \mathcal{C}$ by:

Hence $(A, \mathcal{C}, d)$ is a complete $C^*$-algebra valued $b$-metric space.

Now, we present our following theorem.

Theorem 3.1. Suppose that (for all $a, b\in A$)

(1) there exist a continuous function $\psi:E\times E \rightarrow \mathbb{R}$ and $k \in (0, 1)$ such that

implies that

$for\; all\; x, y\in E$.

(2) $\sup_{x\in E}\int_{E} \mid\psi(x, y)\mid dy \leq 1$.

Then the integral equation (3.1) has a unique solution in $A$.

Proof. Define $f:A \rightarrow A$ by:

Set $h = kI$, then $h\in \mathcal{C}$. For any $u\in H$, we have

where,

Since $\|h\| < 1$, so all the requirements of Corollary 2.3 are satisfied. Therefore, $f$ has a unique fixed point, means that Equation (3.1) has a unique solution.

4.   Conclusions

As $C^{*}$-algebra valued metric space is a relatively new addition to the existing literature. Many researchers proved fixed point theorems in such space in several directions. This note proved multivalued fixed point theorems in $C^{*}$-algebra valued metric spaces wherein we generalized the Suzuki fixed point theorem ^[12]. An example is also adopted to highlight the realized improvements in our newly proved result. Finally, we apply Theorem 2.1 to examine the existence and uniqueness of the solution for a system of Fredholm integral equation.

Acknowledgments

This research is supported by Deanship of Scientific Research, Prince Sattam bin Abdulaziz University, Al-Kharj, Saudi Arabia.

Conflict of interest

The authors declare that they have no competing interests.