SOME NEW OBSERVATIONS ON FIXED POINT RESULTS IN RECTANGULAR METRIC SPACES WITH APPLICATIONS TO CHEMICAL SCIENCES

Introduction/purpose: This paper considers, generalizes and improves recent results on fixed points in rectangular metric spaces. The aim of this paper is to provide much simpler and shorter proofs of some new results in rectangular metric spaces. Methods: Some standard methods from the fixed point theory in generalized metric spaces are used. Results: The obtained results improve the well-known results in the literature. The new approach has proved that the Picard sequence is Cauchy in rectangular metric spaces. The obtained results are used to prove the existence of solutions 8 V O J N O T E H N IČ K I G L A S N IK / M IL IT A R Y T E C H N IC A L C O U R IE R , 2 0 2 1 , V o l. 6 9 , Is s u e 1 to some nonlinear problems related to chemical sciences. Finally, an open question is given for generalized contractile mappings in rectangular metric spaces. Conclusions: New results are given for fixed points in rectangular metric spaces with application to some problems in chemical sciences.


Introduction and Preliminaries
It is well known that the Banach contraction principle (Banach, 1922) is one of the most important and attractive results in nonlinear analysis and mathematical analysis in general. The whole fixed point theory is a significant subject in different fields: geometry, differential equations, informatics, physics, economics, engineering, and many others. After solutions are guaranteed, numerical methodology is established to obtain the approximated solution. The fixed point of functions depends heavily on considered spaces defined using intuitive axioms. In particular, variants of generalized metric spaces are proposed, e.g. partial metric space, b-metric, partial b-metric, extended b-metric, rectangular metric, rectangular b-metric, Gmetric, G b −metric, S-metric, S b −metric, cone metric, cone b-metric, fuzzy metric, fuzzy b-metric, probabilistic metric, etc. For more details on all variants of generalized metric spaces, see (Budhia et al, 2017), (Collaco & Silva, 1997).
In this paper, we will discuss some results recently established in (Alsulami et al, 2015) and (Budhia et al, 2017). Firstly, we give the basic notion of a rectangular metric space (g.m.s or RMS by some authors).
Definition 1. Let X be a nonempty set and let d r : X × X → [0, +∞) satisfy the following conditions: for all x, y ∈ X and all distinct u, v ∈ X each of them different from x and y.
(i) d r (x, y) = 0 if and only if x = y, (ii) d r (x, y) = d r (y, x) , (iii) d r (x, y) ≤ d r (x, u) + d r (u, v) + d r (v, y) (quadrilateral inequality). Then the function d r is called a rectangular metric and the pair (X, d r ) is called a rectangular metric space (RMS for short).
Notice that the definitions of convergence and Cauchyness of the sequences in rectangular metric spaces are the same as the ones found in the standard metric spaces. Also, a rectangular metric space (X, d r ) is complete if each Cauchy sequence in it is convergent. Samet et al. (Samet et al, 2012) introduced the concept of α−ψ−contractive mappings and proved the fixed point theorems for such mappings. In (Karapınar, 2014), Karapinar gave contractive conditions to obtain the existence and uniqueness of a fixed point of α − ψ contraction mappings in rectangular metric spaces. Salimi et al.  introduced modified α − ψ contractive mappings and obtained some fixed point theorems in a complete metric space. Alsulami et al. (Alsulami et al, 2015) established some fixed point theorems for α − ψ−rational type contractive mappings in a rectangular metric space.
Definition 2.  Let T be a self mapping on a metric space (X, d r ) and let α, η : If η (x, y) = 1 for all x, y ∈ X, then T is called an α−admissible mapping. It is called a triangular α−admissible mapping if for all x, y, z ∈ X holds: (α (x, y) ≥ 1 and α (y, z) ≥ 1) implies α (x, z) ≥ 1.
Otherwise, a rectangular metric space (X, d r ) is α−regular with respect to η if for any sequence in X such that α (x n , x n+1 ) ≥ η (x n , x n+1 ) for all n ∈ N and x n → x as n → +∞, then α (x n , x) ≥ η (x n , x) .
For more details on a triangular α−admissible mapping, see (Karapınar et al, 2013), pages 1 and 2. In this paper, we will use the following result: Lemma 1. (Karapınar et al, 2013), Lemma 7. Let T be a triangular α−admissible mapping. Assume that there exists In (Budhia et al, 2017), the authors proved the following result: Theorem 1. Let (X, d r ) be a Hausdorff and complete rectangular metric space, and let T : X → X be an α−admissible mapping with respect to η. Assume that there exists a continuous function ψ ∈ Ψ such that Also, suppose that the following assertions hold: 3. either T is continuous or X is α−regular with respect to η.
Then T has a periodic point a ∈ X and if α (a, T a) ≥ η (a, T a) holds for each periodic point, then T has a fixed point. Moreover, if for all x, y ∈ F (T ) , we have α (x, y) ≥ η (x, y) , then the fixed point is unique.
Taking η (x, y) = 1 for x, y ∈ X, the authors obtained the following corollary: Corollary 1. Let (X, d r ) be a Hausdorff and complete rectangular metric space, and let T : X → X be an α−admissible mapping. Assume that there exists a continuous function ψ ∈ Ψ such that Also, suppose that the following assertions hold: Then T has a periodic point a ∈ X and if α (a, T a) ≥ 1 holds T has a fixed point. Moreover, if for all x, y ∈ F (T ) , we have α (x, y) ≥ 1, then the fixed point is unique. Further, taking α (x, y) = 1 for x, y ∈ X authors obtained the following corollary: Corollary 2. Let (X, d r ) be a Hausdorff and complete rectangular metric space, and let T : X → X be an α−admissible mapping. Assume that there exists a continuous function ψ ∈ Ψ such that Also, suppose that the following assertions hold: 1. there exists x 0 ∈ X such that 1 ≥ η (x 0 , T x 0 ) , 2. for all x, y, z ∈ X (1 ≥ η (x, y) and 1 ≥ η (y, z)) implies 1 ≥ η (x, z) , For ψ (t) = kt, 0 < k < 1 then the authors obtained Corollary 3. Let (X, d r ) be a Hausdorff and complete rectangular metric space, and let T : X → X be an α−admissible mapping with respect to η. Assume that Also, suppose that the following assertions hold: Then T has a periodic point a ∈ X and if α (a, T a) ≥ η (a, T a) holds, T has a fixed point. Moreover, if for all x, y ∈ F (T ) , we have α (x, y) ≥ η (x, y) , then the fixed point is unique.
The following two lemmas are a rectangular metric space modification of a result which is well known in the metric space, see, e.g, (Radenović et al, 2012), Lemma 2.1. Many known proofs of fixed point results in rectangular metric spaces become much more straightforward and shorter using both lemmas. Also, in the proofs of the main results in this paper, we will use both lemmas: Lemma 2. (Kadelburg & Radenović, 2014a), (Kadelburg & Radenović, 2014b) Let (X, d r ) be a rectangular metric space and let {x n } be a sequence in it with distinct elements (x n = x m for n = m). Suppose that d r (x n , x n+1 ) and d r (x n , x n+2 ) tend to 0 as n → +∞ and that {x n } is not a Cauchy sequence. Then there exists ε > 0 and two sequences {m (k)} and {n (k)} of positive integers such that n (k) > m (k) > k and the following sequences tend to ε as k → +∞ : which is a contradiction.
In some proofs, we will also use the following interesting as well as significant result in the context of rectangular metric spaces: Proposition 1. (Kirk & Shahzad, 2014), Proposition 3. Suppose that {q n } is a Cauchy sequence in a rectangular metric space (X, d r ) and suppose

Main results
In this section, we generalize and improve Theorem 2 and all its corollaries. The obtained generalizations extend the result in several directions. Namely, we will use only one function α : X × X → [0, +∞) instead of two α and η as in (Budhia et al, 2017), Definition 2.3. and Definition 3.1. This is possible according to the (Mohammadi & Rezapour, 2013), Page 2, after Theorem 1.2. Note that we assume neither that the rectangular metric space is Hausdorff, nor that the mapping d r is continuous.
The authors (Alsulami et al, 2015), page 6, line 6+, say that the sequence {x n } in a rectangular metric space (X, d r ) is a Cauchy if lim n→+∞ d r (x n , x n+k ) = 0, for all k ∈ N. However, it is well know that this claim is dubious. Therefore, we also improve the proof that the sequence {x n } is Cauchy Our first new result in this paper is the following: Theorem 2. Let (X, d r ) be a complete rectangular metric space and let T : where Also, suppose that the following assertions hold: Then T has a fixed point. Moreover, if then the fixed point is unique.
x k is a fixed point of T and the proof is finished. From now on, suppose that x n = x n+1 for all n ∈ N∪ {0} . Using (2) and the fact that T is an α−admissible mapping, we have By induction, we get In the first step, we will show that the sequence {d r (x n , x n+1 )} is nonincreasing and d r (x n , x n+1 ) → 0 as n → +∞. From (1), recall that where Further, we will also show that Since dr(xn−1,xn)dr(xn+1,xn+2) The last relation follows from the fact that d r (x n−1 , x n ) → 0 as n → +∞. Hence, for some n 1 ∈ N, we have that whenever n ≥ n 1 . Since, d r (x n−1 , x n ) → 0 as n → +∞ it is not hard to check that also d r (x n , x n+2 ) → 0 as n → +∞.
In order to prove that the sequence {x n } is a Cauchy one, we use Lemma 6. Namely, since according to Lemma 1, α x n(k) , x m(k) ≥ 1 if m (k) < n (k) , then, by putting in (1) where Now, taking in (5) the limit as k → +∞ follows which is a contradiction. The sequence {x n } is hence a Cauchy one. Since (X, d r ) is a complete rectangular metric space, there exists a point x * ∈ X such that x n → x * as n → +∞. If T is continuous, we get that , then, according to Lemma 7, we have that all x n are distinct. Therefore, there exists n 2 ∈ N such that x * , T x * / ∈ {x n } n≥n2 . Further, by (iii) follows: whenever n ≥ n 2 , taking the limit, we obtain d r (x * , T x * ) = 0, i.e. x * = T x * , which is a contradiction.
In the case that (X, d r ) is α−regular, we get the following: Since α (x n , x * ) ≥ 1 for all n ∈ N, then from (1) follows where By taking in (6) the limit as n → +∞ and by using Proposition 8 and the continuity of the function ψ, we get d r ( , which is a contradiction. Hence, x * is a fixed point of T. Now, we show that the fixed point is unique if α (x, y) ≥ 1 whenever x, y ∈ F (T ) . Indeed, in this case, by contractive condition (1), for such possible fixed points x, y we have where M (x, y) = max d r (x, y) , d r (x, T x) , d r (y, T y) , d r (x, T x) d r (y, T y) 1 + d r (x, y) , Hence, (7) becomes which is a contradiction. The proof of Theorem 9 is complete.
Remark 1. In the proof of case 2 on Page 96, the authors used the fact that the rectangular metric d r (see condition (3.12)) is continuous, which is not given in the formulation of (Budhia et al, 2017), Theorem 3.2.
By putting in (1) instead of M (x, y), one of the following sets immediately follows as a consequence of Theorem 9.
Corollary 4. Let (X, d r ) be a complete rectangular metric space and let T : X → X be a triangular α−admissible mapping. Assume that there exists a continuous function ψ ∈ Ψ such that x, y ∈ X, α (x, y) ≥ 1 implies d r (T x, T y) ≤ ψ (d r (x, y)) .
Also, suppose that the following assertions hold: Then T has a fixed point. Moreover, if for all x, y ∈ F (T ) , we have α (x, y) ≥ 1, then the fixed point is unique.
Corollary 5. Let (X, d r ) be a complete rectangular metric space and let T : X → X be a triangular α−admissible mapping. Assume that there exists a continuous function ψ ∈ Ψ such that for x, y ∈ X, α (x, y) ≥ 1 yields d r (T x, T y) ≤ ψ (max {d r (x, y) , d r (x, T x) , d r (y, T y)}) .
(9) Also, suppose that the following assertions hold: Then T has a fixed point. Moreover, if for all x, y ∈ F (T ) , we have α (x, y) ≥ 1, then the fixed point is unique. Corollary 6. Let (X, d r ) be a complete rectangular metric space and let T : X → X be a triangular α−admissible mapping. Assume that there exists a continuous function ψ ∈ Ψ such that for x, y ∈ X, α (x, y) ≥ 1 (10) Also, suppose that the following assertions hold: Then T has a fixed point. Moreover, if for all x, y ∈ F (T ) , we have α (x, y) ≥ 1, then the fixed point is unique.
In the book (Ćirić, 2003), Ćirić collected various contractive mappings in the usual metric spaces, see also (Rhoades, 1977) and (Collaco & Silva, 1997). The next three contractive conditions are well known in the existing literature: • Ćirić 1: Ćirić's generalized contraction of first order: there exists k 1 ∈ [0, 1) such that for all x, y ∈ X holds: (11) • Ćirić 2: Ćirić's generalized contraction of second order: there exists k 2 ∈ [0, 1) such that for all x, y ∈ X holds: (12) In both cases, (X, d) is a metric space, T : X → X is a given selfmapping of the set X.
In (Ćirić, 2003), Ćirić introduced one of the most generalized contractive conditions (so-called quasicontraction) in the context of a metric space as follows: • Ćirić 3: The self-mapping T : X → X on a metric space (X, d) is called a quasicontraction (in the sense of Ćirić) if there exists k 3 ∈ [0, 1) such that for all x, y ∈ X holds: Since, it follows that (11) implies (12) and (12) implies (13).
In (Ćirić, 2003), Ćirić proved the following result: Theorem 3. Each quasicontraction T on a complete metric space (X, d) has a unique fixed point (say) z. Moreover, for all x ∈ X, the sequence {T n x} +∞ n=0 , T 0 x = x converges to the fixed point z as n → +∞.

Now we can formulate the following notion and one open question:
Definition 3. Let (X, d r ) be a rectangular metric space and let α : X × X → [0, +∞) be a mapping. The mapping T : X → X is said to be a modified triangular α−admissible mapping if there exists a continuous function ψ ∈ Ψ such that x, y ∈ X, α (x, y) ≥ 1 implies d r (T x, T y) ≤ ψ (M (x, y)) , where M (x, y) is one of the sets:

An open problem
A suggestion for further research -it is logical to ask the following question: Problem 0.1. Let T be a modified triangular α−admissible mapping defined on a complete rectangular metric space (X, d r ) such that T is continuous or (X, d r ) is α−regular. Show that T has a fixed point.

Applications
In this section, we will focus on the applicability of the obtained results.
An application to chemical sciences Consider a diffusing substance placed in an absorbing medium between parallel walls such that δ 1 , δ 2 are the stipulated concentrations at walls. Moreover, let Ω(r) be the given source density and Ξ(r) be the known absorption coefficient. Then the concentration κ(r) of the substance under the aforementioned hypothesis governs the following boundary value problem Problem (1) is equivalent to the succeeding integral equation where Θ(r, ) : [0, 1] × R → R is the Green's function which is continuous and is given by Suppose that C(I, R) = X is the space of all real valued continuous functions defined on I and let X be endowed with the rectangular b-metric d r defined by where κ = sup{|κ(r)| : r ∈ I}. Obviously (X, d r ) is a complete rectangular metric space.
Let the operator Ξ : X → X be defined by Then κ * is a unique solution of (2) if and only if it is a fixed point of Ξ. The subsequent Theorem is furnished for the assertion of the existence of a fixed point of Ξ.
Theorem 4. Consider problem (2) and suppose that there exists ℘ > 0 and a continuous function Ξ( ) : I → R such that the following assertion holds: Then the integral equation (2) and, consequently, the boundary value problem (1) governing the concentration of the diffusing substance has a unique solution in X.
Proof. Clearly, for κ ∈ X and r ∈ I, the mapping Ξ : X → X is well defined. Also Ξ is triangular α-admissible. Hence for all κ, κ * ∈ X, we obtain Taking ψ(M (κ, κ * )) = 1 8 , we obtain Hence, all the hypotheses of Theorem 2 are contented. We conclude that Ξ has a unique fixed point κ in X, which guarantees that the integral equation (2) has a unique solution and, consequently, the boundary value problem (1) has a unique solution.

Application to a class of integral equations for an unknown function
We present the application of the existence of a fixed point for a generalized contraction to the following class of integral equations for an unknown function u: One can easily verify that (X, d r ) is a complete rectangular metric space. Let the self map T : X → X be defined by then u is a fixed point of T if and only it is a solution of (4). Also, we can easily check that T is triangular α-admissible. Now, we formulate the following subsequent theorem to show the existence of a solution of the underlying integral equation.
Theorem 5. Assume that the following assumptions hold:

Then integral equation (4) has a solution.
Proof. Employing conditions (1) − (2) along with inequality (4), we have . Which amounts to say that Taking ψ(M (u 1 , u 2 )) = 1 2 , the above inequality turns into Thus, all the hypotheses Theorem 2 are satisfied and we conclude that T has a unique fixed point x * in X, which amounts to say that integral equation (4)