Some critical remarks on “Some new fixed point results in rectangular metric spaces with an application to fractional-order functional differential equations”

In this manuscript, we generalize, improve, and enrich recent results established by Budhia et al. [L. Budhia, H. Aydi, A.H. Ansari, D. Gopal, Some new fixed point results in rectangular metric spaces with application to fractional-order functional differential equations, Nonlinear Anal. Model. Control, 25(4):580–597, 2020]. This paper aims to provide much simpler and shorter proofs of some results in rectangular metric spaces. According to one of our recent lemmas, we show that the given contractive condition yields Cauchyness of the corresponding Picard sequence. The obtained results improve well-known comparable results in the literature. Using our new approach, we prove that a Picard sequence is Cauchy in the framework of rectangular metric spaces. Our obtained results complement and enrich several methods in the existing state-ofart. Endorsing the materiality of the presented results, we also propound an application to dynamic programming associated with the multistage process.


Introduction and preliminaries
It is well known that the Banach contraction principle [5] is one of the most essential and attractive results in nonlinear analysis and mathematical analysis in general. The whole fixed point theory is a significant subject in different fields as geometry, differential equations, informatics, physics, economics, engineering, and many others (see [8,23,25,27]). After the solutions are guaranteed, the numerical methodology has been adopted to obtain the approximated solution [28].
In 2000, generalized metric spaces were introduced by Branciari [6] in such a way that the triangle inequality is replaced by the quadrilateral inequality d(x, y) d(x, z) + d(z, u) + d(u, y) for all pairwise distinct points x, y, z, and u. Any metric space is a generalized metric space, but in general, generalized metric space might not be a metric space. Various fixed point results were established in such spaces (see [3,4,7,[9][10][11][12][13]16,26] and references therein).
In this paper, we will discuss some results recently established in [7]. Firstly, we propound some basic notions and definitions, which are necessary for the subsequent analysis. Then the function d is called a rectangular metric, and the pair (M, d) is called a rectangular metric space (in short RMS).
Notice that the definitions of convergence and the Cauchyness of sequences in rectangular metric spaces are similar to those found in the standard metric spaces. Also, a rectangular metric space (M, d) is complete if every Cauchy sequence in M is convergent.
Samet et al. [24] introduced the concept of an α−ψ-contractive mapping and proved fixed point theorems for such mappings. Karapınar [13] extended the concepts given in [24] to obtain the existence and uniqueness of a fixed point of α−ψ-contraction mappings in the setting of rectangular metric spaces. After that, Salimi et al. [23] introduced a modified α−ψ-contractive mapping and obtained some fixed point theorems in the complete metric spaces. Alsulami et al. [1] established some fixed point theorems for an α−ψ-rational-type contractive mapping in the context of rectangular metric spaces.

Definition 2. (See
If η(p, r) = 1 for all p, r ∈ M , then T is called an α-admissible mapping.
T is said to be a triangular α-admissible mapping if for all p, r, q ∈ M , the following holds: (α(p, r) 1 and α(r, q) 1) implies α(p, q) 1.
Otherwise, a rectangular metric space (M, d) is said to be an α-regular with respect to η if for any sequence p n in M such that α(p n , p n+1 ) η(p n , p n+1 ) for all n ∈ N and p n → p as n → +∞, implies α(p n , p) η(p n , p).
For more details on the triangular α-admissible mapping, see [14, pp. 1, 2]. In this paper the following results play an important role. https://www.journals.vu.lt/nonlinear-analysis Lemma 1. (See [14,Lemma 7]). Let T be a triangular α-admissible mapping. Assume that there exists p 0 ∈ M such that α(p 0 , T p 0 ) 1. Define a sequence {p n } by p n = T n p 0 . Then α(p m , p n ) 1 for all m, n ∈ N ∪ {0} with m < n.
The following definition is due to [2], where the class of C-functions is introduced.
Definition 3. A C-function F : [0, +∞) 2 → R is a continuous function such that for all x, y ∈ [0, +∞): The letter C will denote the class of all C-functions. For detailed description and examples of C-functions, we refer the reader to [2,7].
The following remark plays a significant role in the rest of this article.

Remark 1.
It is worth to mention that for each C-function, F (x, y) < x if x = 0 and y = 0.
In [7] the authors proved the following two theorems. Assume that: Then there exists x ∈ M such that T n x = x for some n ∈ N, i.e., x is a periodic point. If in addition, α(x, T x) η(x, T x) for each periodic point x, then T has a fixed point. Taking F (x, y) = x − y, the authors obtained the following corollaries. Also suppose that the following assertions are contended: Then T has a periodic point x ∈ M . If in addition, α(x, T x) η(x, T x) holds for each periodic point, then T has a fixed point. Moreover, if for all a, b ∈ F (T ), we have α(a, b) η(a, b), then the fixed point is unique.
Taking ψ(t) = t in Corollary 1, the authors obtained the following result.
Also suppose that the following assertions hold: Then T has a periodic point x ∈ M . If in addition, α(x, T x) η(x, T x) for each periodic point, then T has a fixed point. Moreover, if for all a, b ∈ F (T ), we have α(a, b) η(a, b), then the fixed point is unique. where m(p, r) is the same as in Corollary 2. Suppose also that the following hold: Then T has a periodic point x ∈ M . If in addition, α(x, T x) η(x, T x) for each periodic point, then T has a fixed point. Moreover, if for all a, b ∈ F (T ), we have α(a, b) η(a, b), then the fixed point is unique.
In the sequel the authors in [7] gave two examples, which support their obtained theoretical results. In the next example, rectangular metric space (M, d) is not Hausdorff, and the mapping T has no fixed point. So the hypothesis that (M, d) is Hausdorff does not guarantee the existence of a fixed point.
ψ d(T p, T r) F ψ m(p, r) , φ m(p, r) whenever α(p, r) η(p, r), but F (T ) is empty. However, the given example is not correct, namely, it does not satisfy all the conditions of Theorem 2 [7, Thm. 1]. It is easy to check that T is not α-admissible with respect to η. Indeed, The following two lemmas, in the setting of rectangular metric spaces, are modifications of a well-known result in metric spaces (see, e.g., [22, Lemma 2.1]). Many known proofs of fixed point results in rectangular metric spaces become much simpler and shorter using both these lemmas.
Lemma 2. (See [12].) Let (M, d) be a rectangular metric space, and let {p n } be a sequence in it with distinct elements (p n = p m for n = m). Suppose that d(p n , p n+1 ) and d(p n , p n+2 ) tend to 0 as n → +∞ and that {p n } is not a Cauchy sequence. Then there exist 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 → +∞: Proof. Let p n = p m for some n, m ∈ N with n < m, then p n+1 = T p n = T p m = p m+1 , and we acquire which is a contradiction.
In the proof of our results, the following exciting and significant proposition is used in the context of rectangular metric spaces.  Note that we neither assume that the rectangular metric space (M, d) is Hausdorff, nor that the mapping d is continuous.
The authors [1, p. 6, line 6+)] claimed that the sequence {p n } in rectangular metric space (M, d) is a Cauchy sequence if lim n→+∞ d(p n , p n+k ) = 0 for all k ∈ N. However, it is ambivalent. We rectify the proof that the sequence {p n } is Cauchy. For more details, we refer the reader to the noteworthy and informative article [20, p. 7].
Our first new result in this paper is the following. where Also, suppose that the following assertions are satisfied: (i) there exists u 0 ∈ M such that α(u 0 , T u 0 ) 1; (ii) T is continuous or (M, d) α-regular.
Then T has a fixed point. Moreover, if for all p, r ∈ F (T ), we have α(p, r) 1, then the fixed point is unique.
Define a sequence {u n } in X by u n = T u n−1 = T n u 0 for all n ∈ N. If u k+1 = u k for some k ∈ N, then T u k = u k , i.e., u k is a fixed point of T , and the proof is completed in this case. From now, suppose that u n = u n+1 for all n ∈ N ∪ {0}. Using (2) and the fact that T is an α-admissible mapping, we have α(u 1 , u 2 ) = α(T u 0 , T u 1 ) 1.
In order to prove that the sequence {u n } is a Cauchy sequence, we use Lemma 2. According to Lemma 1, α(u n(k) , u m(k) ) 1 if m(k) < n(k), then putting in (1) p = u n(k) , r = u m(k) , we obtain Now, since F, ψ and φ are continuous, taking limit k → +∞ in (5), and utilizing Remark 1, we obtain which is a contradiction. Hence the sequence {u n } is a Cauchy sequence. Since (M, d) is a complete rectangular metric space, there exists a point u * ∈ X such that u n → u * as n → +∞. If T is continuous, we get u n+1 = T u n → T u * as n → +∞. Let T u * = u * . Since d(u n , u n+1 ) < d(u n−1 , u n ) for all n ∈ N ∪ {0}, according to Lemma 3, we conclude that u n 's are distinct. Therefore, there exists n 1 ∈ N such that u * , T u * / ∈ {u n } n n1 . Further, by (iii) it follows that d(u * , T u * ) d(u * , u n ) + d(u n , u n+1 ) + d(u n+1 , T u * ) whenever n n 1 . Taking the limit in the last inequality, it follows that d(u * , T u * ) = 0, i.e., u * = T u * , which is again a contradiction.
In the case that (M, d) is α-regular, and since α(u n , u * ) 1 for all n ∈ N, from (1) we obtain the following: where m(u n , u * ) = max d(u n , u * ), d(u n , u n+1 ), d(u * , T u * ), https://www.journals.vu.lt/nonlinear-analysis Passing limit n → +∞ in (6) and using Proposition 1, the continuity of the functions F , ψ, and φ as well as Remark 1, it follows that if u * = T u * , then which is a contradiction. Hence, u * is a fixed point of T . Now, we show that the fixed point is unique if α(p, r) 1 whenever p, r ∈ Fix(T ). Indeed, in this case, by the contractive condition (1), for such possible distinct fixed points p, r, we have ψ d(p, r) = ψ d(T p, T r) F ψ m(p, r) , φ m(p, r) , where m(p, r) = max d(p, r), d(p, T p), d(r, T r), d(p, T p)d(r, T r) Hence, (7) becomes which is again a contradiction. The proof of the Theorem 3 is complete.

Remark 2.
In the proof of the main theorem [7, p. 586, line 1 − , Thm. 1, case 3], the authors used the fact that rectangular metric d (see also inequality (5) on the same page) is continuous, which is not given in the formulation of Theorem 1 in [7]. In the proof of the same theorem (p. 585), the authors also claimed that lim n→+∞ d(u n , u n+2 ) = 0, which is not correct because we do not know whether lim n→+∞ d(u n , u n+2 ) exists or not.
By taking F (x, y) = x−y in Theorem 3, we obtain the following result as a corollary. Also suppose that the following assertions hold: Then T has a fixed point. Moreover, if for all p, r ∈ F (T ), we have α(p, r) 1, then the fixed point is unique.
Taking ψ(t) = 1 for all t ∈ [0, +∞) in Corollary 4, the following useful corollary is obtained. Suppose also that the following assertions hold: Then T has a fixed point. Moreover, if for all p, r ∈ F (T ), we have α(p, r) 1, then the fixed point is unique.
Consider φ(t) = (1 − q)t for 0 < q < 1 in Corollary 5, then we obtain the following result. Also suppose that the following conditions are contended: (i) there exists p 0 ∈ M such that α(p 0 , T p 0 ) 1; (ii) T is continuous or (M, d) is α-regular.
Then T has a fixed point. Moreover, if for all p, r ∈ F (T ), we have α(p, r) 1, then the fixed point is unique.

Application to a dynamical programming
This section aims to apply our results to solve the existence and uniqueness of the solution of the dynamic programming problem. In particular, the problem of dynamic programming related to multistage process reduces to solving the existence and uniqueness of the solution of the following functional equation:  [18,19].
In fact, the distance in B(W ) is given by Obviously, T is well-defined if the functions f and G are bounded.
Theorem 4. Let T be an operator defined by (9), and suppose that the following conditions hold: (i) G : W × D × R → R and f : W × D → R are continuous and bounded; (ii) there exist F ∈ C and ψ, φ ∈ Ψ such that ψ(t) = t for all t ∈ [0, ∞), and for all h, k ∈ B(W ), x ∈ W and y ∈ D.
Then the functional equation (8) has a unique solution.
This amounts to say that Lastly, we specify α : Evidently, α(h 1 , h 2 ) = 1 and α(T (h 1 ), T (h 2 )) = 1 for all h 1 , h 2 ∈ B(W ). This endorses that T is a triangular α-admissible mapping. Hence, due to Theorem 3, T has a unique fixed point h * ∈ B(W ), that is, h * is a unique solution of the functional equation (8). This completes the proof.

Conclusion
This article is devoted to addressing some weaknesses of the main results introduced in [7]. Antithetical to the results in [7], we used only one function α : M ×M → [0, +∞) instead of two α and η as used in [8, Defs. 2.3, 3.1]. We also dropped the property of Hausdorffness of the rectangular metric space (M, d) and the continuity of the mapping d.
Using our new approach, we proved that a Picard sequence is Cauchy in the framework of rectangular metric spaces. Our obtained results complement and enrich several methods in the existing state-of-art. Thereafter, we apply our results to study a dynamic programming problem associated with a multistage process to affirm the applicability of the obtained results.