#### Abstract

In this paper, we introduce an ordered implicit relation and investigate some new fixed point theorems in a cone rectangular metric space subject to this relation. Some examples are presented as illustrations. We obtain a homotopy result as an application. Our results generalize and extend several fixed point results in literature.

#### 1. Introduction

Many authors generalized the classical concept of metric space, by changing the metric conditions partially. Branciari [1] introduced rectangular metric space (RMS), where the triangular inequality condition of metric space was replaced by rectangular inequality. He also proved an analog of the Banach contraction principle in rectangular metric spaces. Azam and Arshad [2] mentioned some necessary conditions to get a unique fixed point for Kannan-type mappings in this context. Later, Karapinar et al. [3] investigated some fixed points for contractions on rectangular metric spaces. On the other hand, Di Bari and Vetro [4] used -weakly contractive condition to give an extension of the results in [3]. Subsequently, a number of authors were engrossed in rectangular metric spaces and proved the existence and uniqueness of fixed point theorems for certain types of mappings [2–6].

The significance of the Banach contraction principle lies in the fact that it is a very essential tool to check the existence of solutions for differential equations, integral equations, matrix equations, and functional equations made by mathematical models of real-world problems. There has been a tendency for consistent theorists to improve both the underlying space and the contractive condition (explicit type) used by Banach [7] under the effect of one of the structures like order metric structure [8, 9], graphic metric structure [10, 11], multivalued mapping structure [12–14], -admissible mapping structure [15], comparison functions, and auxiliary functions. The process of developing new fixed point theorems in the complete metric spaces is in progress under various new restrictions. In this regard, we can find very nice results by Debnath et al. that appeared in [10, 12, 14].

Later on, Popa [16] introduced self-mappings satisfying implicit relation and obtained fixed points, under the effect of these functions. Popa [16–18] obtained some fixed point theorems in metric spaces. However, scrutiny into the fixed points of self-mappings satisfying implicit relations in order metric structure was made by Beg and Butt [19, 20], and some common fixed point theorems were established by Berinde and Vetro [21, 22] and Sedghi et al. [23]. Huang and Zhang [24] introduced cone metric by replacing real numbers with ordering Banach spaces and established a convergence criterion for sequences in cone metric space to generalize Banach fixed point theorem. Huang and Zhang [24] considered the concept of normal cone for their drawn outcomes; however, Rezapour and Hamlbarani [25] left the normality condition in some results by Huang. Many authors have investigated fixed point theorems and common fixed point theorems of self-mappings for normal and nonnormal cones in cone metric spaces (see [26–29]).

Azam et al. [5] introduced the notion of a cone rectangular metric space and proved the Banach contraction principle in this context. In 2012, Rashwan [6] extended this idea as a continuation, which improved the results in [5]. The appealing nature of these spaces has enticed scrutiny into fixed point theorems for various contractions on cone rectangular metric spaces (see [5, 6]).

In this paper, we continue the study initiated by Azam et al. [5] subject to an ordered implicit relation. Since every cone metric space is a cone rectangular metric space but not conversely, therefore we prefer to establish results in cone rectangular metric spaces. These results are supported by some examples and an application in homotopy theory.

#### 2. Preliminaries

*Definition 1. **A binary relation**over a set**defines a partial order if**has the following axioms:*(1)reflexive(2)antisymmetric(3)transitiveA set having partial order is known as a partially ordered set denoted by .

In the present article, stands for a real Banach space. Now, we present some definitions and relevant results, which will be required in the sequel.

*Definition 2 (see [24]). **A subset**of**is called a cone if and only if the following conditions are satisfied:*(1) is closed, nonempty, and (2), for all and such that (3)Given , define the partial order with respect to as follows:

if and only if for all (5)

represents that but , while stands for (interior of ).

*Definition 3 (see [24]). **The cone**is called normal if, for all*, *there exists**such that*

Throughout this paper, we assume and a partial order with respect to the cone defined in . If , then and are identical; otherwise, they are different.

*Definition 4 (see [24]). **Let**be a nonempty set, and**satisfies the following:*

(d1) , and if and only if

(d2)

(d3) ,

Then, is called a cone metric on , and is then known as a cone metric space.

*Example 1 (see [5]). **Let*, *, and**. Define**by*where is a constant. Then, defines a cone metric on .

Proposition 5 (see [5]). * Consider a cone metric space, with cone. Then, for ,we have*(1)

*(2)*

*If**and*,*then**(3)*

*If**for each*,*then*

*If**and*,*then*Surely, cone metric space “being space” generalizes metric space, because in cone metric space, the range of a metric function is an ordered vector space instead of real numbers. Although the set of real numbers is an ordered vector space, we can find many significant ordered vector spaces in the literature (see [25, 27, 28, 30]). In Theorem 1.4 and Lemma 2.1 that appeared in [31, 32], respectively, the authors developed a metric depending on a given cone metric and proved that a complete cone metric space is always a complete metric space, and then, this relationship between metric and cone metric led them to say that every contraction mapping in a cone metric space is essentially contraction mapping in a metric space.

This paper addresses the fixed point results in the cone rectangular metric spaces. We know that every metric is a rectangular metric but rectangular metric needs not to be a metric (see [1–3, 33]). Also, we know that every cone metric is a cone rectangular metric but conversely does not hold in general (see [5, 6, 29]). In view of observations given in [5, 29], we infer that results in this paper are independent of what authors investigated in [31, 32]. The implicit relation and hence the contractive condition employed are even new and original in the rectangular metric space. The theorems in this paper are new in rectangular metric space, but we choose the cone rectangular metric space for the sake of the generality of our results.

*Definition 6 (see [24]). **Let A mapping**is said to be a cone rectangular metric if for all**the following conditions are satisfied:*

(dR1) and if and only if

(dR2)

(dR3) for all distinct

The cone rectangular metric space is denoted by .

*Example 2 (see [5]). **Let*, , *. Take**, and define*One can easily check that is a cone rectangular metric space, but not a cone metric space, since .

*Definition 7 (see [1]). **Let**be a real Banach space*, *be a cone rectangular metric space and**with*.
(1)A sequence in is called a Cauchy sequence, if there exists a natural number such that for all (2)The sequence is said to be convergent if there exists an such that for all and (3)The is called complete if every Cauchy sequence converges in

#### 3. Ordered Implicit Relations

Many authors have used implicit relations to establish fixed point results and have applied these results to solve nonlinear functional equations (see [19–22, 34, 35]).

In this section, we define a new ordered implicit relation and explain it with an example. In the next section, we use it along with some other assumptions to develop some new fixed point theorems in the cone rectangular metric space.

Let be a real Banach space and be the space of all bounded linear operators with the usual norm defined in that is .

In this section, generalizing the idea of [16], we define the following notion:

Let be a continuous operator which satisfies the conditions given below:

() and

() If , then there exists an order preserving operator with such that and for all or if then and for all

() whenever

Let .

*Example 3. **Let**be the partial order with respect to cone**as defined in Section**2**and let**be a real Banach space. For all*, *and**, define**, by*

Then, the operator :

Let and , then and . Now, we show that . Given that and and by Definition 2 (2), we have

Thus, .

Let be such that . If then, we have

So

By Definition 2 (2), we have for either or or

For (6), if and , then . Thus, there exists defined by ( is a scalar) such that . Now, if and , then, . So, there exists defined by ( is a scalar) such that , for some . For if both and , then we get an absurdity.

For (7), if and , then . Thus, there exists defined by ( is a scalar) such that . Now, if and , then, . So, there exists defined by ( is a scalar) such that . For if both and , then we get an absurdity. Similar arguments hold for (8).

Let be such that and consider, then , which holds whenever .

Similarly, the operators defined by (1)(2)(3)(4)(5) are members of

The following remark is essential in the sequel.

*Remark 8. **If* ,*the Neumann series**converges if**and diverges otherwise. Also, if*, *then there exists**such that**and*.

#### 4. New Results

Recently, Popa [16] has employed implicit type contractive condition on self-mapping to obtain some fixed point theorems. Ran and Reurings [9] have presented an analog of Banach fixed point theorem for monotone self-mappings in an ordered metric space. Huang and Zhang [24] introduced the idea of cone metric spaces and obtained analogs of Banach fixed point theorem, Kannan fixed point theorem, and Chatterjea fixed point theorem in cone metric spaces. In this section, we prove some fixed point results for ordered implicit relations in a cone rectangular metric space which improves the results in [9, 16, 24]. We derive these results under two different partial orders: one defined in underlying set and the other in real Banach space.

Theorem 9. * Letbe a complete cone rectangular metric space andbe a cone. Let. If there exist, identity operatorandsuch that, for all comparable elementsand
*(1)

*there exists such that*(2)

*for all , implies*(3)

*for every ,*(4)

*for a sequence with whose all sequential terms are comparable, we have for all and*

*Then, .*

*Proof. *Let be as assumed in (1). We construct a sequence by starting with . Then, . By assumption (2), we have , . For and , we have by (9).
that is,
By (dR3), we have
and so we rewrite (11) employing condition () as follows:
and thus, by , there exists an order preserving operator with such that
Now, put and in (9) to have
that is,
By (dR3), we have
and implies
By , there exists with such that
By continuing this pattern, we can construct a sequence such that with , and
For with , we have by Remark 8Since , so, as . Thus, which implies that is a Cauchy sequence in . Since is a complete cone rectangular metric space, so, there exists such that as . Equivalently, there exists a natural number such that
We claim that
We assume against our claim that
By (dR3), (9), and assumption (3), we have
Thus, , which is an absurdity. Hence, for each , we have
Assume that . As and by the assumption (4), we have for all and then by (9), we get
Letting and in view of assumption (4) and (26), we have
By , we have
This is a contradiction to . Thus, . Hence, . It follows from (dR1) that .

Theorem 10. * Letbe a complete cone rectangular metric space andbe a self-mapping on. If for all comparable elements, there exist, identity operatorandsuch thatand
*(1)

*there exists such that*(2)

*for any , implies*(3)

*for every ,*(4)

*for a sequence with whose all sequential terms are comparable, we have for all and*

Then, has a fixed point in .

*Proof. *Let be in as assumed in (1). Define the sequence by for all . Since and by assumption (2) , repeated application of assumption (2) gives us either or for each . By (30), we have for and By (dR3), we have
and then using , we obtain
By , there exists an order preserving operator with such that
Using , take and in (30), we have
By (dR3), and , we get
By continuing the pattern, we construct a sequence such that
Hence, by the same reasoning as in the proof of Theorem 9, we have .

Theorem 11. * Letbe a complete cone rectangular metric space andbe a monotone self-mapping on. If for all comparable elements, there exist, identity operatorandsuch thatand
*(1)

*there exists such that either or*(2)

*for every ,*(3)

*for a sequence with whose all sequential terms are comparable, we have for all and*

Then, has a fixed point in .

*Proof. *This proof follows the same pattern as given in the previous two proofs, so, we omit it.

*Remark 12. *(1)In Theorem 9, Theorem 10, and Theorem 11, uniqueness of the fixed point of can be attained by assuming that for every pair of elements , there exists either an upper bound or lower bound of (2)The cone is taken as nonnormal in the above theorems

Theorem 13. *Let**be a complete cone rectangular metric space and**be a monotone self-mapping on*. *If for all comparable elements*, *there exist*, *identity operator**and**such that**Moreover, if
*(1)*there exists such that or *(2)*for a sequence with whose all sequential terms are comparable, we have for all and **Then, there exists such that .*

*Proof. *Define for all and , then with also define implicit relation by
The proof follows by the application of Theorem 9.

The following examples illustrate the main theorem.

*Example 4. **Let**be a real Banach space. With*, *then**is a real Banach space. Define the partial order**on**by*Define , then, it is a cone in .

Let , define order on by and define such that
Define the function by