Fixed Point Problems in Cone Rectangular Metric Spaces with Applications

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.


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 ðψ, ϕÞ contractions on rectangular metric spaces. On the other hand, Di Bari and Vetro [4] used ðψ, ϕÞ -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][3][4][5][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 ten-dency 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][13][14], α-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][17][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][27][28][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.

Preliminaries
Definition 1. A binary relation R over a set Y ≠ ϕ defines a partial order if R has the following axioms: A set having partial order R is known as a partially ordered set denoted by ðY , RÞ.
In the present article, E stands for a real Banach space. Now, we present some definitions and relevant results, which will be required in the sequel.
Definition 3 (see [24]). The cone P ⊆ E is called normal if, for all σ, ς ∈ P , there exists K > 0 such that Throughout this paper, we assume Y = ðY , RÞ and ⪯ a partial order with respect to the cone P defined in E. If Y ⊆ E, then R and ⪯ are identical; otherwise, they are different.
Proposition 5 (see [5]). Consider a cone metric space ðY , d c Þ, with cone P . Then, for u, v, w ∈ E ,we have (1) If u⪯αu and α ∈ ½0, 1Þ, then u = 0 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][2][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 d cr : Y × Y ↦ E is said to be a cone rectangular metric if for all σ, ς, ξ, υ ∈ Y the following conditions are satisfied: The cone rectangular metric space is denoted by ðY , d cr Þ.
Definition 7 (see [1]). Let E be a real Banach space, ðY , d cr Þ be a cone rectangular metric space and c ∈ E with 0 ≪ c.

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 ðE, k:kÞ be a real Banach space and BðE, EÞ be the space of all bounded linear operators T : E → E with the usual norm k: In this section, generalizing the idea of [16], we define the following notion: Let L : E 6 → E be a continuous operator which satisfies the conditions given below: , then there exists an order preserving operator S ∈ BðE, EÞ with kSk 1 < 1 such that v 1 ⪯Sðv 2 Þ and v 3 Example 3. Let ⪯ be the partial order with respect to cone P as defined in Section2and let ðE, k:kÞ be a real Banach space. For all v i ∈ Eði = 1 to 6Þ, α ∈ ð0, 1/3Þ and ð1 + αÞ/2 ≤ β ≤ 1 + α , define L : Then, the operator L ∈ G: Given that γ 5 − v 5 ∈ P and γ 6 − v 6 ∈ P and by Definition 2 (2), we have Thus For (6), if v 1 = 0 and v 2 ≠ 0, then ðβ − 2αÞv 2 ∈ P . Thus, there exists S : For if both v 1 ≠ 0 and v 3 ≠ 0, then we get an absurdity. Similar arguments hold for (8).
Similarly, the operators L : The following remark is essential in the sequel.

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.
that is, By (dR3), we have and so we rewrite (11) employing condition (L 1 ) as follows: and thus, by ðL 2 Þ, there exists an order preserving operator S ∈ BðE, EÞ with kSk 1 < 1 such that Now, put σ = σ 1 and κ = σ 2 in (9) to have By ðL 2 Þ, there exists S ∈ BðE, EÞ with kSk 1 < 1 such that By continuing this pattern, we can construct a sequence fσ n g such that σ n Rσ n+1 with σ n+1 = f ðσ n Þ, and For m, n ∈ ℕ with m > n, we have by Remark 8 Since kSk 1 < 1, so, S n → 0 as n → ∞. Thus, lim n→∞ d cr ð σ n , σ m Þ = 0 which implies that fσ n g is a Cauchy sequence in Y . Since ðY , d cr Þ is a complete cone rectangular metric space, so, there exists x * ∈ Y such that σ n → x * as n → ∞. Equivalently, there exists a natural number N 2 such that We claim that We assume against our claim that By (dR3), (9), and assumption (3), we have Thus, T 4 ðd cr ðσ n , f ðσ n ÞÞÞ ≺ 0, which is an absurdity. Hence, for each n ≥ 1, we have Assume that kd cr ðx * , f ðx * ÞÞk > 0. As σ n−1 Rσ n and by the assumption (4), we have σ n Rx * for all n ∈ ℕ and then by (9), we get Letting n → ∞ and in view of assumption (4) and (26), we have By ðL 1 Þ, we have This is a contradiction to ðL 3 Þ. Thus, kd cr ðx * , f ðx * ÞÞk = 0. Hence, d cr ðx * , f ðx * ÞÞ = 0. It follows from (dR1) that x * = f ðx * Þ. Theorem 10. Let ðY , d cr Þ be a complete cone rectangular metric space and f be a self-mapping on Y . If for all comparable elements σ, κ ∈ Y , there exist T ∈ BðE, EÞ, identity operator I : E → E and L ∈ G such that and 5
Theorem 11. Let ðY , d cr Þ be a complete cone rectangular metric space and f be a monotone self-mapping on Y . If for all comparable elements σ, κ ∈ Y , there exist T ∈ BðE, EÞ, identity operator I : E → E, and L ∈ G such that (1) there exists σ 0 ∈ Y such that either σ 0 Rf ðσ 0 Þ or f ð σ 0 ÞRσ 0 (2) for every fσ n g ⊆ Y , d cr ðσ n , σ n+1 Þ⪯Tðd cr ðσ n−1 , σ n ÞÞ (3) for a sequence fσ n g with σ n → x * whose all sequential terms are comparable, we have σ n Rx * for all n ∈ ℕ and d cr ðx * , f ðx * ÞÞ°d cr ðx * , f 2 ðx * ÞÞ Then, f has a fixed point in Y .
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 f can be attained by assuming that for every pair of elements σ, κ ∈ X, there exists either an upper bound or lower bound of σ, κ Moreover, if (1) there exists σ ∈ Y such that σ 0 Rf ðσ 0 Þ or f ðσ 0 ÞRσ 0 (2) for a sequence fσ n g with σ n → x * whose all sequential terms are comparable, we have σ n Rx * for all n ∈ ℕ and d cr ðx * , f ðx * ÞÞ ≤ d cr ðx * , f 2 ðx * ÞÞ Then, there exists x * ∈ Y such that x * = f ðx * Þ.
Proof. Define SðvÞ = qv for all v ∈ E and q ∈ ½0, 1Þ, then S ∈ BðE, EÞ with ∥S∥ 1 < 1 also define implicit relation by The proof follows by the application of Theorem 9.
The following examples illustrate the main theorem.