An approach of Banach algebra in fuzzy metric spaces with an application

1.   Introduction

In 1965, the theory of fuzzy sets was introduced by Zadeh ^[1]. Many authors generalized this theory in different directions for different purposes and still, it's playing a very important role in current research. By using this theory, Kramosil et al. ^[2] introduced the concept of a fuzzy metric space (FM-space) which performs the probabilistic metric space approach to the fuzzy setting and they proved that every metric induces an FM. In the sense of Kramosil et al. ^[2], Grabiec ^[3] proved two fixed point theorems (FP-theorems), that is, "Banach and Edelstein contraction theorems for self-mappings on complete and compact FM-spaces, respectively". Later on, the modified form of the FM-space was given by George and Vermani ^[4]. Gregory and Sapena ^[5] established some FP-theorems on FM-spaces. After that, Rodriguez-Lopez and Romaguera ^[6] gave the idea of a Hausdorff metric on a given FM-space by using the concept of George and Veeramani ^[4] on the nonempty compact subsets. In 2011, Kiany et al. ^[7] established some FP and end point theorems by using set-valued fuzzy-contractive on complete FM-spaces. Recently, Shamas et al. ^[8,9] established some unique FP-theorems on complete FM-spaces with integral and differential types of applications. Some more related results in the context of metric spaces and FM-spaces, can be found in (e.g., see ^{[11,12,13,14,15,16,17,18,19,20,21,22]} and the references are therein).

In 2007, Huang and Zhang ^[23] introduced the notion of a cone metric space which modifies the concept of metric spaces by using the Banach space instead of real numbers as a range set and proved some FP-theorems for nonlinear contractive type mappings with the normality of cone condition. Du ^[24] presented a note on cone metric fixed point theory and its equivalence. Later on, Cakalli et al. ^[25] presented the concept that any topological vector space valued cone metric space is metrizable and proved some results in topological vector space valued cone metric spaces. In 2013, Liu et al. ^[26] presented the new concept of a cone metric space (CM-space) over Banach algebras by the replacement of Banach algebra instead of Banach space and proved some FP-theorems by using generalized Lipschitz mappings with weaker and natural conditions of the generalized Lipschitz constant of spectral radius. In 2016, Yan et al. ^[27] proved the result on partially order CM-spaces over Banach algebras for FP and CFP.

The notion of a fuzzy cone metric space (FCM-space) was introduced by Oner et al. ^[28]. They proved some basic properties and a "fuzzy cone Banach contraction theorem" which is stated as: "A self-mapping on a complete FCM-space in which fuzzy cone contractive sequences are Cauchy has a unique FP". Later on, Oner et al. ^[29] defined a closed ball and pre-compact in an FCM-space, and established a Baire's theorem on a complete FCM-space. In 2017, Rehman and Li ^[30] extended and improved a "fuzzy cone Banach contraction theorem" and proved some generalized FP-theorems in complete FCM-spaces without the assumption of "fuzzy cone contractive sequences are Cauchy". After that, Jabeen et al. ^[31] proved some CFP-theorems on FCM-spaces with an application. Chen et al. ^[32], Priyobarta et al. ^[33] and Talha et al. ^[34,35] proved some FP and coupled FP-results in the context of FCM-spaces with different types of applications.

In this paper, we present the new concept of an FM-space over a Banach algebra $\mathcal{A}$ and prove some basic topological properties and a Hausdorff metric on the said space. In previous literature, a mapping FM is represented by $M:U\times U \times (0, \infty)\to [0, 1]$ where $U$ is a non-empty set. Oner et al. ^[28] replaced $(0, \infty)$ by cone condition and defined a mapping FCM as: $M_c:U\times U \times int(P)\to [0, 1]$, where $P$ is a cone of a real Banach space $E$. Now, in this paper, we use a Banach algebra $\mathcal{A}$ instead of $(0, \infty)$ in an FM-space and present the new concept of an FM-space over a Banach algebra $\mathcal{A}$, this FM mapping over a Banach algebra $\mathcal{A}$ can be written as: $M_\mathcal{A}:U\times U \times \mathcal{A}\to [0, 1]$. By using this new concept, we present some basic properties and a FP-theorem for self-mappings on a G-complete FM-space over the Banach algebra $\mathcal{A}$. This new concept will play a very important role in the fixed point theory. Moreover, we present a supportive integral type application to validate our work. By using this new concept, one can prove some more topological properties and fixed point results in FM-space over a Banach algebra $\mathcal{A}$ with different types of applications.

2.   Preliminaries

In this section, we shall present the basic helpful concepts related to our main results.

Let $\mathcal{A}$ represent a real Banach space in which the operation of multiplication is defined by;

(i) $u(v w) = (u v) w$,

(ii) $(u + v) w = u w + v w$ and $u(v + w) = u v + v w$,

(iii) $\beta(u v) = (\beta u) v = u(\beta v)$,

(iv) $\|u v\|\leq \|u\| \|v\|$.

for all $u, v, w\in\mathcal{A}$.

Let $e^*\in \mathcal{A}$ be the unit multiplicative identity of $\mathcal{A}$ such that $v e^* = e^* v = v$, for all $v\in \mathcal{A}$ and an element $v\in\mathcal{A}$ is said to be invertible if there is $z\in\mathcal{A}$ such that $v z = z v = e$. The inverse of $v$ is denoted by $v^{-1}$. For more details, we refer the readers to ^[36].

Proposition 2.1. ^[36] Let $\mathcal{A}$ be a Banach algebra with the identity element $e^*$ and $v\in\mathcal{A}$. If the spectral radius $\varrho(v) < 1$, that is,

then $(e^* -v)$ is invertible. Therefore, we have

Lemma 2.2. ^[37] Let $u, v$ be any two vectors in a Banach algebra $\mathcal{A}$. If they commute with each other, then the following hold:

(i) $\varrho(u v)\leq \varrho(u) \varrho(v)$;

(ii) $\varrho(u+ v)\leq \varrho(u)+ \varrho(v)$;

(iii) $|\varrho(u)- \varrho(v)|\leq \varrho(u-v)$.

Lemma 2.3. ^[37] Let $z$ be any vector in a Banach algebra $\mathcal{A}$. If $\varrho(z)\in(0, 1)$, then we have

Definition 2.4. ^[38] A binary operation $\ast:[0, 1]\times[0, 1]\rightarrow [0, 1]$ is called a continuous $t$-norm, if

(i) $\ast$ is commutative, associative and is continuous,

(ii) for all $\alpha_1, \beta_1, \alpha_2, \beta_2\in[0, 1]$, $1\ast \alpha_1 = \alpha_1, \ \alpha_1\ast \beta_1\le \alpha_2\ast \beta_2$, whenever $\alpha_1\le \alpha_2$ and $\beta_1\le \beta_2$.

The following are the basic three conditions of continuous $t$-norm (see ^[38]):

(i) The minimum $t$-norm is: $\alpha_1\ast \beta_1 = \min\{\alpha_1, \beta_1\}$.

(ii) The product $t$-norm is: $\alpha_1\ast \beta_1 = \alpha_1 \beta_1$.

(iii) The Lukasiewicz $t$-norm is: $\alpha_1\ast \beta_1 = \max\{0, \alpha_1+\beta_1-1\}$.

Definition 2.5. ^[4] A 3-tuple $(U, M, \ast)$ is said to be an FM-space if $U$ is an arbitrary set, $\ast$ is a continuous $t$-norm and $M$ is a fuzzy set on $U^2\times (0, \infty)$, satisfying the conditions:

(F-1) $M(\zeta_1, \zeta_2, t) > 0$ and $M(\zeta_1, \zeta_2, t) = 1$ if and only if $\zeta_1 = \zeta_2$,

(F-2) $M(\zeta_1, \zeta_2, t) = M(\zeta_2, \zeta_1, t)$,

(F-3) $M(\zeta_1, \zeta_2, t)\ast M(\zeta_2, \zeta_3, s)\le M(\zeta_1, \zeta_3, t+s)$,

(F-4) $M(\zeta_1, \zeta_2, .):(0, \infty)\to [0, 1]$ is continuous,

for all $\zeta_1, \zeta_2, \zeta_3\in U$ and $t, s > 0$.

Definition 2.6. ^[4] Let $(U, M, \ast)$ be an FM-space, let $\zeta\in U$ and $(\zeta_k)$ be a sequence in $U$. Then

(i) $(\zeta_k)$ is said to converge to $\zeta$ if for any $t > 0$ and $\rho\in (0, 1)$, there is $k_1\in\mathbb{N}$ such that $M(\zeta_k, \zeta, t) > 1-\rho$, for all $k\geq k_1$. We can write this $\lim\limits_{k\to\infty}\zeta_k = \zeta$.

(ii) $(\zeta_k)$ is said to be a Cauchy sequence if for any $t > 0$ and $\rho\in(0, 1)$, there is $k_1\in\mathbb{N}$ such that $M(\zeta_k, \zeta_m, t) > 1-\rho$, for all $k, m\geq k_1$.

(iii) $(U, M, *)$ is complete if every Cauchy sequence is convergent in $U$.

Note: In the sense of Gregori et al. ^[5], a sequence $(\zeta_k)$ in an FM-space is said to be G-Cauchy if $\lim\limits_{k\to\infty}M(\zeta_k, \zeta_{k+l}, t) = 1$ for $t > 0$ and an FM-space $(U, M, \ast)$ is called G-complete if every G-Cauchy sequence is convergent.

Definition 2.7. ^[6] Let $(U, M, \ast)$ be an FM-space. $M$ is said to be continuous on $U^2\times(0, \infty)$, if

whenever if ${(\zeta_k, v_k, t_k)}\in U^2\times(0, \infty)\to (\zeta_1, v_1, t)\in U^2\times(0, \infty)$, we have

Remark 2.8. ^[4] Let $(U, M, \ast)$ be an FM-space. $\mathcal{T} = \{B\subset U: \zeta_1\in B, \ \text{if and only if there are}\ \ \ t > 0\ \text{and}\ \rho\in(0, 1)\ \text{such that}\ \mathcal{B}(\zeta_1, \rho, t)\subset B\}$ is a topology on $U$.

As a further study in this paper, we introduce the new concept of an FM-space over a Banach algebra $\mathcal{A}$. We present some topological properties and a fuzzy Banach contraction theorem over the Banach algebra $\mathcal{A}$. Moreover, a supportive application of integral is given at the end to validate our work.

3.   Main results

Our first definition is as follows:

Definition 3.1. A 4-tuple $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ is called an FM-space over $\mathcal{A}$ if $U$ is an arbitrary set, $\mathcal{A}$ is a Banach algebra, $\ast$ is a continuous $t$-norm and $M_\mathcal{A}$ is a fuzzy set on $U\times U\times\mathcal{A}$, satisfying the following conditions:

(FB-1) $M_\mathcal{A}(\zeta_1, \zeta_2, t) > 0$, and $M_\mathcal{A}(\zeta_1, \zeta_2, t) = 1$ if and only if $\zeta_1 = \zeta_2$;

(FB-2) $M_\mathcal{A}(\zeta_1, \zeta_2, t) = M_\mathcal{A}(\zeta_2, \zeta_1, t)$;

(FB-3) $M_\mathcal{A}(\zeta_1, \zeta_2, t)\ast M_\mathcal{A}(\zeta_2, \zeta_3, s)\le M_\mathcal{A}(\zeta_1, \zeta_3, t+s)$;

(FB-4) $M_\mathcal{A}(\zeta_1, \zeta_2, .):\mathcal{A}\longrightarrow [0, 1]$ is continuous

for all $\zeta_1, \zeta_2, \zeta_3\in U$ and $s, t\in\mathcal{A}$. Then the 4-tuple $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ is said to be an FM-space over $\mathcal{A}$.

Example 3.2. Let $U = \mathbb{R}$ and $\ast$ be a continuous $t$-norm, which is defined by $a\ast b = ab$, for all $a, b\in[0, 1]$ and $M_\mathcal{A}: U\times U\times\mathcal{A}\to [0, 1]$ be defined by

for all $\zeta_1, \zeta_2\in U$ and $t\in\mathcal{A}$, $M_\mathcal{A}$ is an FM-space over $\mathcal{A}$.

Proof.

(FB-1) For all $\zeta_1, \zeta_2\in U$ and $t\in\mathcal{A}$, $M_\mathcal{A}(\zeta_1, \zeta_2, t) > 0$, where $\zeta_1\neq \zeta_2$. Assume that $\zeta_2 = \zeta_1$. Then $\|\zeta_1- \zeta_2\| = 0$ and hence we get

So $M_\mathcal{A}(\zeta_1, \zeta_2, t) = 1$. Conversely, suppose that $M_\mathcal{A}(\zeta_1, \zeta_2, t) = 1$. Then $\left(e^{\|\zeta_1- \zeta_2\|t^{-1}}\right)^{-1} = 1$, and so $e^{\|\zeta_1- \zeta_2\|t^{-1}} = e^{0}$. Due to the same base, we compare the power on both sides and multiplying by $t$, we have that $\|\zeta_1- \zeta_2\|t^{-1}t = 0$, and so $\|\zeta_1\| = \| \zeta_2\|.$ Hence, it is proved that $M_\mathcal{A}(\zeta_1, \zeta_2, t) = 1$ if and only if $\zeta_1 = \zeta_2$.

(FB-2) If $\|\zeta_1- \zeta_2\| = \|\zeta_2- \zeta_1\|$, for all $\zeta_1, \zeta_2\in U$ and $t\in\mathcal{A}$, then

This implies that

(FB-3) We will prove that $M_\mathcal{A}(\zeta_1, \zeta_2, s+t)\geq M_\mathcal{A}(\zeta_1, \zeta_3, s)\ast M_\mathcal{A}(\zeta_3, \zeta_2, t)$, for all $\zeta_1, \zeta_2, \zeta_3\in U$ and $s, t\in\mathcal{A}$. Since

Thus we have

Since $e^{\zeta}$ is an increasing function for $0 < \zeta\in U$, we have

Hence, it is proved that $M_\mathcal{A}(\zeta_1, \zeta_2, s+t)\geq M_\mathcal{A}(\zeta_2, \zeta_3, s)\ast M_\mathcal{A}(\zeta_3, \zeta_2, t)$.

(FB-4) Let us take a sequence $(t_k)$ in $\mathcal{A}$ such that $t_k\to t\in\mathcal{A}$. That is, $\lim\limits_{k}\|t_k-t\| = 0$.

Without loss of the generality, fix $\zeta_1, \zeta_2\in U$. Since the function $e^{\zeta_1}$ is continuous on $\mathbb{R}$, we have

with respect to the usual metric. Therefore, $M_\mathcal{A}(\zeta_1, \zeta_2, .): \mathcal{\mathcal{A}}\to [0, 1]$ is continuous.

Hence, it is proved that the four-tuple $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ is an FM-space over $\mathcal{A}$.

Proposition 3.3. Let $(U_1, M_{\mathcal{A}_1}, \mathcal{A}, \ast)$ and $(U_2, M_{\mathcal{A}_2}, \mathcal{A}, \ast)$ be two FM-spaces over $\mathcal{A}$. For $(\zeta_1, \zeta_2), (v_1, v_2)\in U_1\times U_2$ and $t\in\mathcal{A}$ such that

$M_\mathcal{A}$ is an FM-space over $\mathcal{A}$ on $U_1\times U_2$.

Proof.

(FB-1) Since $M_{\mathcal{A}_1}(\zeta_1, v_1, t) > 0$ and $M_{\mathcal{A}_2}(\zeta_2, v_2, t) > 0$,

Therefore, we have that

Further, we suppose that for $t\in\mathcal{A}$, $(\zeta_1, v_1, t) = (\zeta_2, v_2, t)$. This implies that $\zeta_1 = v_1$ and $\zeta_2 = v_2$, for $t\in\mathcal{A}$. Hence, we get

It follows that,

Conversely, suppose that $M_\mathcal{A}((\zeta_1, \zeta_2), (v_1, v_2), t) = 1$, for $t\in\mathcal{A}$. Then

So

It follows that

Thus, we get that $\zeta_1 = \zeta_2$ and $v_1 = v_2$. Therefore, $(\zeta_1, \zeta_2, t) = (v_1, v_2, t)$ for $t\in \mathcal{A}$.

(FB-2) We will prove that $M_\mathcal{A}((\zeta_1, \zeta_2), (v_1, v_2), t) = M_\mathcal{A}((v_1, v_2), (\zeta_1, \zeta_2), t)$, for $t\in\mathcal{A}$. Now, we observe that

It follows that for all $(\zeta_1, \zeta_2), (v_1, v_2)\in U_1\times U_2$ and $t\in\mathcal{A}$,

(FB-3) Since $(U_1, M_{\mathcal{A}_1}, \mathcal{A}, \ast)$ and $(U_2, M_{\mathcal{A}_2}, \mathcal{A}, \ast)$ are FM-spaces over $\mathcal{A}$, we have that

and

for all $(\zeta_1, \zeta_2), (v_1, v_2), (w_1, w_2)\in U_1\times U_2$ and for $s, t\in\mathcal{A}$. Therefore, we have

(FB-4) Note that $M_{\mathcal{A}_1}(\zeta_1, v_1, t)$ and $M_{\mathcal{A}_2}(\zeta_2, v_2, t)$ are continuous with resprct to $t\in\mathcal{A}$ and $\ast$ is also continuous. Therefore, it follows that

is also continuous at $t\in\mathcal{A}$.

Definition 3.4. Let $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ be an FM-space over $\mathcal{A}$. For every $t > 0$, in $\mathcal{A}$, the open ball $\mathcal{B}(\zeta_0, \rho, t)$ with center at $\zeta_0\in U$ and radius $\rho \in (0, 1)$ is given as

Definition 3.5. A subset $V$ of $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ is said be open if for any given point $v\in V$, there are $\rho\in(0, 1)$ and $t\in\mathcal{A}$ such that

Theorem 3.6. Every open ball in $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ is an open set.

Proof. Suppose that $\mathcal{B}(\zeta_0, \rho, t)$ is an open ball in a $(U, M_\mathcal{A}, \mathcal{A}, \ast)$. Then $\zeta\in\mathcal{B}(\zeta_0, \zeta, t)$ implies that $M_\mathcal{A}(\zeta_0, \zeta, t) > 1-\rho$. Let us take $t^*\in(0, t)$ in $\mathcal{A}$ such that

Let $r^* = M_\mathcal{A}(\zeta_0, \zeta, t^*) > 1-\rho\ \Rightarrow \ \rho^* > 1-\rho$. Then we can choose a point $s\in(0, 1)$ such that

Now, for given $\rho^*$ and $s$ such that $\rho^* > 1-s$, we take another point $\rho'\in(0, 1)$ such that $\rho^*\ast \rho'\geq 1-s$.

Now, by considering the ball $\mathcal{B}(\zeta, 1-\rho', t-t^*)$, we claim that

Again, we choose a point $\zeta_1\in\mathcal{B}(\zeta, 1-\rho', t-t^*)$ such that $M_\mathcal{A}(\zeta, \zeta_1, t-t^*) > \rho'$. Then we have

Therefore, we get that $\zeta_1\in\mathcal{B}(\zeta_0, \rho, t)$ and hence

Thus, the proof is completed.

Theorem 3.7. Let $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ be an FM-space over $\mathcal{A}$. If we define

then $\mathcal{T}_{\mathcal{A}}$ is a topology on $U$.

Proof.

i) If $\zeta_1\in\emptyset$, then $\emptyset = \mathcal{B}(\zeta_1, \rho, t)\subseteq\emptyset$. Hence, $\emptyset\in\mathcal{T}_{\mathcal{A}}$. So for any $\zeta_1\in U$, $0 < \rho < 1$ and $t\in\mathcal{A}$ such that $\mathcal{B}(\zeta_1, \rho, t)\subseteq U$, we have $U\in\mathcal{T}_{\mathcal{A}}$.

ii) Let $A_k\in \mathcal{T}_{\mathcal{A}}$ for every $k\in I$ and $\zeta_1\in\bigcup_{k\in I}\subset A_k$. Then there exists $k_0\in I$ such that $\zeta_1\in A_{k_0}$. So there exist $t\in\mathcal{A}$ and $\rho\in(0, 1)$ such that $\mathcal{B}(\zeta_1, \rho, t)\subset A_{k_0}$. Since $A_{k_0}\subset\bigcup_{k\in I}$, $\mathcal{B}(\zeta_1, \rho, t)\subset\bigcup_{k\in I}A_k$. Thus, $\bigcup_{k\in I}A_k\in\mathcal{T}_{\mathcal{A}}$.

iii) Suppose that $A, B\in\mathcal{T}_{\mathcal{A}}$ and $\zeta_1\in A\cap B$. Then, $\zeta_1\in A$ and $\zeta_1\in B$ and so there are $t_1, t_2\in\mathcal{A}$ and $0 < \rho_1, \rho_2 < 1$ such that $\mathcal{B}(\zeta_1, \rho_1, t_1)\subset A$ and $\mathcal{B}(\zeta_1, \rho_2, t_2)\subset B$. By choosing $t\in\mathcal{A}$ such that $t < (t_1, t_2)$ and taking $\rho = \min\{\rho_1, \rho_2\}$, we have

Thus, $A\cap B\in\mathcal{T}_{\mathcal{A}}$. Hence, it is proved that $\mathcal{T}_{\mathcal{A}}$ is a topology on $U$.

Theorem 3.8. Every FM-space over $\mathcal{A}$ is Hausdorff.

Proof. Let $\zeta_1, \zeta_2$ be any two distinct points of $U$. Then from the definition of an FM over $\mathcal{A}$, $0 < M_\mathcal{A}(\zeta_1, \zeta_2, t) < 1$ for $t\in\mathcal{A}$. We can say that $M_\mathcal{A}(\zeta_1, \zeta_2, t) = \rho$ for some $\rho\in(0, 1)$. For each $\rho_0\in(\rho, 1)$ there exists $\rho_1\in(0, 1)$ such that $\rho_1\ast \rho_1 > \rho_0$. Now, we consider the sets $\mathcal{B}(\zeta_1, 1-\rho_1, t/2)$ and $\mathcal{B}(\zeta_2, 1-\rho_1, t/2)$. Then we have

for $t\in\mathcal{A}$. Suppose that

for $t\in\mathcal{A}$. Then there exists $y\in \mathcal{B}\left(u_1, 1-\rho_1, \frac{t}{2}\right)\cap \mathcal{B}\left(\zeta_2, 1-\rho_1, \frac{t}{2}\right)$. Therefore, we have $M_\mathcal{A}(\zeta_1, y, t/2) > 1-(1-\rho_1) = \rho_1$ and $M_\mathcal{A}(\zeta_2, y, t/2) > 1-(1-\rho_1) = \rho_1$ for $t\in\mathcal{A}$. From Definition 3.1 (fmB3),

for $t\in\mathcal{A}$. Then $\rho_1\ast \rho_1 < \rho$, which implies that $\rho < \rho_0 < \rho$. This is a contradiction.

Thus $\mathcal{B}\left(\zeta_1, 1- \rho_1, \frac{t}{2}\right)\cap \mathcal{B}\left(\zeta_2, 1- \rho_1, \frac{t}{2}\right) = \emptyset\ \text{ for }\ t\in\mathcal{A}.$

Definition 3.9. Let $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ be an FM-space over $\mathcal{A}$. A subset $B$ of $U$ is called $F$-bounded over $\mathcal{A}$ if there are $t\in\mathcal{A}$ and $\rho\in (0, 1)$, such that $M_\mathcal{A}(\zeta_1, \zeta_2, t) > 1-\rho$ for all $\zeta_1, \zeta_2\in B$.

Theorem 3.10. Let $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ be an FM-space over $\mathcal{A}$. Then every compact set in $U$ is closed and $F$-bounded over $\mathcal{A}$.

Proof. Let $B$ be a compact subset of $U$, $t\in\mathcal{A}$ and $\rho\in(0, 1)$. Since $\{\mathcal{B}(\zeta, \rho, t):\zeta\in B\}$ is an open cover of $B$, there are $\zeta_1, \zeta_2, ..., \zeta_j\in B$ such that

For any $\zeta, v\in B$, there exist $1\leq k, l\leq j$ such that $\zeta\in \mathcal{B}(\zeta_k, \rho, t)$ and $v\in \mathcal{B}(\zeta_k, \rho, t)$. Then we can write

for $t\in\mathcal{A}$. Let us take $\gamma = \min\{M_\mathcal{A}(\zeta_k, \zeta_l, t):1\leq k, l\leq j\}$. Then we have

Let $t^* = 3t$ and choose a point $\rho^*\in (0, 1)$ such that $(1-\rho)\ast\gamma\ast(1-\rho) > 1-\rho^*$. Then, for any $\zeta, v\in B$, we have that $M_\mathcal{A}(\zeta, v, t^*) > 1-\rho^*$, and hence $B$ is $F$-bounded over $\mathcal{A}$. On the other hand, from Theorem 3.8, every FM-space over a Banach algebra $\mathcal{A}$ is Hausdorff and every compact subset of a Hausdorff space is closed, that is, $B$ is closed.

Definition 3.11. Let $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ be an FM-space over $\mathcal{A}$ and $(\zeta_k)$ be a sequence in $U$. Then, $(\zeta_k)$ is said to converge to a point $\zeta\in U$, if for any $t\in\mathcal{A}$ and $\rho\in(0, 1)$, there is $k_0\in \mathbb{N}$ such that $M_\mathcal{A}(\zeta_k, \zeta, t) > 1-\rho$, for all $k\geq k_0$. We can write this as,

Theorem 3.12. Let $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ be an FM-space over $\mathcal{A}$. Then a sequence $(\zeta_k)$ in $U$ converges to a point $\zeta\in U$ if and only if $M_\mathcal{A}(\zeta_k, \zeta, t)\to 1, \ \mathit{\text{as}}\ k\to\infty$, for each $t\in\mathcal{A}$.

Proof. Assume that a sequence $(\zeta_k)$ converges to a point $\zeta\in U$. Then for each $t\in\mathcal{A}$ and $\rho\in (0, 1)$, there is $k_0\in \mathbb{N}$ such that $M_\mathcal{A}(\zeta_k, \zeta, t) > 1-\rho$, for all $k\geq k_0$. Hence,

Conversely, suppose that $M_\mathcal{A}(\zeta_k, \zeta, t)\to 1$ as $k\to\infty$. Then, for each $t\in\mathcal{A}$ and $\rho\in(0, 1)$, there is $k_0\in \mathbb{N}$ such that $\rho > 1-M_\mathcal{A}(\zeta_k, \zeta, t)$, for all $k\geq k_0$, which implies that $M_\mathcal{A}(\zeta_k, \zeta, t) > 1-\rho$. Hence, $\zeta_k\to \zeta\in U$ as $k\to\infty$.

Definition 3.13. Let $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ be an FM-space over $\mathcal{A}$ and $(\zeta_k)$ be a sequence in $U$. Then

(1) $(\zeta_k)$ is said to be a Cauchy sequence if for any $t\in \mathcal{A}$ and $\rho\in(0, 1)$, there is $k_0\in\mathbb{N}$ such that $M_\mathcal{A}(\zeta_k, \zeta_m, t) > 1-\rho$, for all $k, m\geq k_0$;

(2) $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ is complete if every Cauchy sequence is convergent in $U$.

Definition 3.14. A sequence $(\zeta_k)$ in $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ is called $G_\mathcal{A}$-Cauchy sequence if $\lim\limits_{k\to\infty}M_\mathcal{A}(\zeta_k, \zeta_{k+l}, t) = 1$, for all $k, m\geq k_0$ and $t\in\mathcal{A}$. $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ is $G_\mathcal{A}$-complete if every $G_\mathcal{A}$-Cauchy sequence is convergent in $U$.

Lemma 3.15. Let $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ be an FM-space over $\mathcal{A}$. Then $M_\mathcal{A}(\zeta_1, \zeta_2, .): \mathcal{A}\to [0, 1]$ is non-decreasing for all $\zeta_1, \zeta_2 \in U$.

Proof. On the contrary, suppose that $M_\mathcal{A}(\zeta_1, \zeta_2, t) > M_\mathcal{A}(\zeta_1, \zeta_2, s)$, for $s, t\in \mathcal{A}$ with $s > t$. Then

By Definition 3.1 (2), i.e., $M_\mathcal{A}(\zeta_2, \zeta_2, s - t) = 1$, we get that

which is a contradiction.

Theorem 3.16. Let $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ be a $G_\mathcal{A}$-complete FM-space over $\mathcal{A}$. Let a mapping $T:U\to U$ be a mapping satisfying

for all $\zeta, v\in U$, $\alpha\in(0, 1)$ and $t\in\mathcal{A}$. Then $T$ has a unique fixe point (FP).

Proof. Fix $\zeta_0\in U$ and construct an iterative sequence in $U$ such that $\zeta_{k+1} = T\zeta_k$ with $k\geq 0$, for $t\in\mathcal{A}$. Then, by (3.1), for $t\in\mathcal{A}$,

Then we get

Similarly,

This implies that

Now, from (3.2), (3.3), and by induction, for $t\in \mathcal{A}$, we have

Hence we get that

Now, for any positive integer $\ell$ and for $t\in\mathcal{A}$, from (3.4), we have

Hence it is proved that $(\zeta_k)$ is a $G_\mathcal{A}$-Cauchy sequence. Since $(U, M_\mathcal{A}, \mathcal{A}, \ast)$ is $G_\mathcal{A}$-complete, there is $z\in U$ such that

Now, from (3.1), (3.2) and (3.5), we have for all $t\in\mathcal{A}$,

Hence we get that $M_\mathcal{A}(z, Tz, t) = 1$, i.e., $Tz = z$.

For uniqueness, assume that $w$ is another FP of the mapping $T$ in $U$. Then, from (3.1) and (3.2), we have for all $t\in\mathcal{A}$,

Hence $M_\mathcal{A}(z, w, t) = 1\ \Rightarrow \ z = w$.

4.   Application

This section deals with an application of a nonlinear integral equation (NIE) to support our work. Let $U = C([0, \rho], \mathbb{R})$ be the space of $\mathbb{R}$-valued continuous functions on $[0, \rho]$, where $0 < \rho\in \mathbb{R}$. The NIE is

where $\kappa, s\in [0, \rho]$ and $P :[0, \rho]\times[0, \rho]\times \mathbb{R} \to \mathbb{R}.$ The induced metric $m:U\times U\to \mathbb{R}$ is defined by

The binary operation $\ast$ is defined as $\delta \ast \varrho = \delta\varrho$ for all $\delta, \varrho \in [0, \rho]$. An FM over $\mathcal{A}$ $M_{\mathcal{A}}:U \times U\times \mathcal{A}\to [0, 1]$ is defined by

for all $\zeta, \nu\in U$ and $t\in \mathcal{A}$. Then easily it can be verified that $(U, M_{\mathcal{A}}, {\mathcal{A}}, \ast)$ is a complete FM space over a Banach algebra $\mathcal{A}$.

Theorem 4.1. Assume that there exist $A_{\zeta}, A_{\nu}\in U$ such that

where $\kappa\in [0, \rho]$. If there is $\beta\in (0, 1)$ such that

for all $\zeta, \nu\in U$. Then the NIE (4.1) has a unique solution in $U$.

Proof. We define an integral operator $T:U\to U$ by

for all $\zeta, \nu, \hbar\in U$. Now, from (4.2) and (4.4), for $t\in \mathcal{A}$, we have

This implies that

for all $\zeta, \nu\in U$ and $t\in \mathcal{A}$. This means that the inequality (3.1) of Theorem 3.16 is satisfied. Thus, the integral operator $T$ satisfies all the conditions of Theorem 3.16 and $T$ has a unique FP, i.e., (4.1) has a unique solution in $U$.

5.   Conclusions

In this paper, we discussed a new concept of an FM-space over a Banach algebra $\mathcal{A}$, i.e., four-tuple $(U, M_\mathcal{A}, \mathcal{A}, \ast)$. We proved some basic properties and a Hausdorff metric in the FM-space over $\mathcal{A}$. We proved a "Banach contraction principle for fixed point" on the FM-space over $\mathcal{A}$. Moreover, we presented an application of a nonlinear integral equation. One can use this concept to present some more properties, FP and CFP theorems for different contractive type mappings in FM-space over $\mathcal{A}$ with different types of integral operators.

Conflict of interest

The authors declare that they have no competing interests.