#### Abstract

In this article, we are generalizing the concept of control fuzzy metric spaces by introducing orthogonal control fuzzy metric spaces. We prove some fixed point results in this setting. We provide nontrivial examples to show the validity of our main results and the introduced concepts. An application to fuzzy integral equations is also included. Our results generalize and improve several developments from the existing literature.

#### 1. Introduction and Preliminaries

Many authors studied fixed point theory explicitly, introduced and popularized lot of spaces, and made the area of fixed point theory more fascinating. In this connectedness, Bakhtin  and Czerwik  provided a generalization of metric spaces, named as a b-metric space. Zadeh  originated fuzzy sets. The formulation of metric spaces and fuzzy sets, named as fuzzy metric spaces, helped many authors in various ways. Nădăban  originated fuzzy b-metric spaces. Many authors  worked in fuzzy b-metric spaces.

In , the authors introduced the concept of an extended fuzzy b-metric space as a generalization of fuzzy b-metric spaces. The work  originates the concept of controlled metric type spaces (see also ). Recently, in , the notion of controlled-type metric spaces has been generalized by a formulation of controlled fuzzy metric spaces, which are also generalizations of extended fuzzy b-metric spaces.

Eshaghi et al.  introduced the notion of an orthogonal set and proved the Banach fixed point result. Many of the authors  continued working on orthogonal spaces. In this article, we are generalizing the concept of control fuzzy metric spaces . Namely, we initiate the notion of orthogonal control fuzzy metric spaces.

Let us first recall some basic definitions related to this manuscript.

Definition 1 (see ). A 4-tuple is called a fuzzy b-metric space if is an arbitrary (nonempty) set, is a continuous t-norm, and is a fuzzy set on satisfying the following conditions, for all and for a given real number:(B1)(B2) if and only if (B3)(B4)(B5) is continuous

Definition 2 (see ). A 4-tuple is called an extended fuzzy b-metric space if is a (nonempty) set, where is a continuous t-norm, and is a fuzzy set on , satisfying the following conditions, for all : is continuous

Definition 3 (see ). A 4-tuple is called a control fuzzy metric space if is a (nonempty) set, where is a continuous t-norm and is a fuzzy set on , satisfying the following conditions, for all : is continuous

Definition 4 (see ). Let be a set and let and , for some , be the orbit of. A function is said to be -orbitally lower semicontinuous at if such that then we get

#### 2. Main Results

In this section, we introduce orthogonal control fuzzy metric spaces and prove some fixed point results.

Definition 5. A 4-tuple is called an orthogonal control fuzzy metric space if is an (nonempty) orthogonal set, where is a continuous t-norm and is a fuzzy set on , satisfying the following conditions: such that , , and is continuous, such that and Now, we show that the following are equivalent:(i)(ii)

Proof. (ii) ⟹ (i)Similarly, we can easily prove (i)⇒ (ii).

Example 1. Let , where and . Define a binary relation ⊥ by . Given aswith a continuous t-norm defined by Given asThen, is an orthogonal control fuzzy metric space, but it is not a control fuzzy metric space.

Proof. , and are obvious. Here, we prove :
() , such that , , and
We have the following cases to prove .

Case 1. If , then . Also, and
This implies

Case 2. If then and clearly, . This implies

Case 3. If , , and , then and clearly,This implies

Case 4. If , , and, then we have the following cases:(1) and (2) and (3) and (4) and (5) and (6) and (7)(8)

Proof of (1). If and , thenObserve that . This implies .
On the contrary,Observe that ; then,Note that then,This impliesSimilarly, (2)–(8) are easily satisfied.
Now, we show that is not a control fuzzy metric space. Let . Also, let , , and then,On the contrary,This impliesThis fails .

Example 2. Let , where and . Define a binary relation ⊥ by . Define byfor all with a continuous t-norm defined by Given asThen, is an orthogonal control fuzzy metric space, but it is not a control fuzzy metric space.

Proof. First, we show that is an orthogonal control fuzzy metric space. , and are obvious. Here, we prove and :
() :() : , such that , , and :Now, we show that is not a control fuzzy metric space. Indeed,This impliesNow, let; then, and This implies thatTaking , we get a contradiction.

Remark 1. Every control fuzzy metric space is an orthogonal control fuzzy metric space, but the converse is not true.

Remark 2. Note that Example 2 also holds for the t-norm

Definition 6. Let be an orthogonal control fuzzy metric space. Then, a sequence is said to be G-convergent to , where if and only if for any and for all .

Definition 7. Let be an orthogonal control fuzzy metric space. Then, a sequence is said to be a G-Cauchy sequence with if and only if for all and .

Definition 8. Let be an orthogonal control fuzzy metric space; then, it is G-complete if and only if every G-Cauchy sequence is convergent.

Definition 9. is ⊥-continuous at in an orthogonal control fuzzy metric space if for each orthogonal sequence in so that if exists and is finite for all , then again exists and is finite for all . Furthermore, is ⊥-continuous if is ⊥-continuous at each Also, is ⊥-preserving if ; hence, .

Remark 3. It is not necessary that the limit of a convergent sequence will be unique in an orthogonal control fuzzy metric space.
For this, take a sequence defined by for each integer , and define an orthogonal control fuzzy metric space as in Example 2 with . Also, in particular, take ; then,for all Observe that the sequence converges to all with .

Remark 4. It is not necessary that the convergent sequence will be a Cauchy sequence in an orthogonal control fuzzy metric space.
For this, take a sequence defined by for each integer , and define an orthogonal control fuzzy metric space as in Example 2 with . Also, in particular, take ; then,for all Observe that the sequence converges to all with . However, does not exist.
Mihet  introduced a control function . We generalize it as follows.

Definition 10. Let be the class of all mappings such that is orthogonal continuous, nondecreasing, and If

Theorem 10. Let be an orthogonal G-complete control fuzzy metric space with such thatfor all Suppose that is an ⊥-continuous, ⊥-contraction, and ⊥-preserving mapping so thatfor all where Also, assume that, for every exist and are finite. Then, has a unique fixed point in . Furthermore,

Proof. Since is an orthogonal G-complete control fuzzy metric space, there exists such thatThis yields that. ConsiderIf , then is a fixed point of Suppose that for all Since is ⊥-preserving, is an orthogonal sequence. Since is an ⊥-contraction, we haveNow, from, we haveNow, taking limit as in (33), in (32) together with (26), we havefor all and . Thus, is an orthogonal G-Cauchy sequence in . The completeness of implies the existence of such thatfor all . Now, since is an⊥-continuous mapping, one writes . For and from , we haveTaking in (36) and using (35), we get for all , that is, .
Now, for uniqueness, let be another fixed point for and let there exist such that. We can obtainSince is an ⊥-preserving, this implies thatFrom (27), we can deriveWe can writefor all . By taking limit as , we get , for all ; hence, .

Corollary 1. Let be an orthogonal G-complete control fuzzy metric space. Let be ⊥-contraction and ⊥-preserving. Also, assume that if is an O-sequence with , then for all . Therefore, has a unique fixed point . Furthermore, , for all and .

Proof. We can prove alike as in the proof of Theorem 1 that is a G-Cauchy sequence and converges to . Hence, for all . We get from (26) thatThen, we can writeTaking limit as , we get , and hence, . The rest of proof is similar as in Theorem 1.

Theorem 2. Let be an orthogonal G-complete control fuzzy metric space with so thatfor all . If is an ⊥-contraction and ⊥-preserving and satisfiesfor all , then. Furthermore, is a fixed point of if and only if is -orbitally lower semicontinuous at .

Proof. Since is an orthogonal G-complete control fuzzy metric space, there exists such thatThis says that . ConsiderIf , then is a fixed point of . Suppose that for all . Since is ⊥-preserving, is an orthogonal sequence. Since is an ⊥-contraction, we haveNow, from , we have