1 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 [3,4,7,9–13,16,26] 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.

Definition 1. Let be a nonempty set, and let satisfy the following conditions: for all and all distinct each of them different from and ,

(i)

(ii)

(iii)

Then the function d is called a rectangular metric, and the pair (M, d) is called a rectan- gular metric space (in short RMS).

Notice that the definitions of convergence and the Cauchyness of sequences in rectan- gular 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 is convergent.

Samet et al. [24] introduced the concept of an -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 -contraction mappings in the setting of rectangular metric spaces. After that, Salimi et al. [23] introduced a modified -contractive mapping and obtained some fixed point theorems in the complete metric spaces. Alsulami et al. [1] established some fixed point theorems for an -rational-type contractive mapping in the context of rectangular metric spaces.

Let be the family of all functions such that is nonde- creasing and continuous (so-called an altering distance function) and if and only if (for more details, see [15, 28]).

Definition 2. (See [23].) Let be a self-mapping on a metric space (M, d), and let be two functions. Then is called an-admissible mapping with respect to if implies that for all

If for all, then is called an -admissible mapping.

is said to be a triangular -admissible mapping if for all , the following holds: and implies

Otherwise, a rectangular metric space (M, d) is said to be an -regular with respect to if for any sequencein such that for all and implies

For more details on the triangular -admissible mapping, see [14, pp. 1, 2]. In this paper the following results play an important role.

Lemma 1. (See [14, Lemma 7]). Let be a triangular-admissible mapping. Assume that there exists such that Define a sequence Then

The following definition is due to [2], where the class of -functions is introduced.

Definition 3. A-function is a continuous function such that for all

(i)

(ii)

The letter will denote the class of all -functions. For detailed description and examples of -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-function, and

Theorem 1. Let (M, d) be a complete Hausdorff rectangular metric space, and let T : M → M be an α-admissible mapping with respect to η. Suppose there exist F ∈ C and ψ, φ ∈ Ψ such that, for p, r ∈ M,

where

Assume that:

(i) there exists for which

(ii) for all and implies

(iii) is continuous or is -regular with respect to .

Then there exists such that for some is a periodic point. If in addition, for each periodic point x, then has a fixed point.

Theorem 2. To ensure the uniqueness of the fixed point in Theorem ., the authors add the following condition:

Taking the authors obtained the following corollaries.

Corollary 1. Let (M, d) be a Hausdorff and complete rectangular metric space, and let be an-admissible mapping with respect to. Assume that there exists such that, for,

where

Also suppose that the following assertions are contended:

(i) there exists

(ii) for all

(iii) T is continuous or M is α-regular with respect to η.

Then has a periodic point If in addition, holds for each periodic point, then has a fixed point. Moreover, if for all we have then the fixed point is unique.

Taking in Corollary 1, the authors obtained the following result.

Corollary 2. Let (M, d) be a Hausdorff and complete rectangular metric space. Let be an-admissible mapping with respect to. Assume that there exists such that, for

where

Also suppose that the following assertions hold:

(i) there exists

(ii) for all

(iii) T is continuous or M is α-regular with respect to η.

Then has a periodic point If in addition, for each periodic point, then has a fixed point. Moreover, if for all we have then the fixed point is unique.

Consider in Corollary 2.

Corollary 3. Let (M, d) be a Hausdorff and complete rectangular metric space. Let be an-admissible mapping with respect to such that, for

where is the same as in Corollary 2. Suppose also that the following hold:

(i) there exists

(ii) for all

(iii) T is continuous or M is α-regular with respect to η.

Then has a periodic point .If in addition, for each periodic point, then has a fixed point. Moreover, if for all we have 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 has no fixed point. So the hypothesis that (M, d) is Hausdorff does not guarantee the existence of a fixed point.

Example 1. LetandDefineas follows:

Then (M, d) is a complete rectangular metric space. Note that (M, d) is not Hausdorff because there exists no such that Given as

Define by

For their convenience, the authors in [7] use the following symbols:

Define the functionsasand and respectively. Using the obtained table, the authors easily checked that the following condition is valid.

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 is not -admissible with respect to. Indeed,

while

Further, is not continuous. Indeed, but

Also, is not defined, we do not know whether (M, d) is -regular.

The following two lemmas, in the setting of rectangular metric spaces, are modifica- tions 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 be a sequence in it with distinct elements Suppose that and tend to 0 as and that is not a Cauchy sequence. Then there exist and two sequences and of positive integers such that and the following sequences tend to

Lemma 3. Let be a Picard sequence in rectangular metric space (M, d) inducing by the mapping and initial point. If for all then whenever

Proof. Let forsome with , then 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.

Proposition 1. (See [17, Prop. 3].) Suppose that is a Cauchy sequence in a rectangular metric space (M, d), and suppose Then for all In particular, does not converge to

2 Some improved results

In this section, we generalize and improve Theorem 1 along with its corollaries. Obtained generalizations extend the result in several directions. It may be noted: we use only one function instead of two and as used in [8, Defs. 2.3, 3.1.]. This is possible according to the results enunciated in [21, p. 2].

Note that we neither assume that the rectangular metric space (M, d) is Hausdorff, nor that the mapping . is continuous.

The authors [1, p. 6, line 6+)] claimed that the sequence in rectangular metric space (M, d) is a Cauchy sequence if for all However, it is ambivalent. We rectify the proof that the sequence 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.

Theorem 3. Let (M, d) be a complete rectangular metric space, and let be a triangular-admissible mapping, where Assume there exist andsuch that, for

where

Also, suppose that the following assertions are satisfied:

(iv) there exists such that

(v) is continuous or .M, d. α-regular.

Then T has a fixed point. Moreover, if for all we have then the fixed point is unique.

Proof. Given such that

Define a sequence in by for all If for some , then is a fixed point of , and the proof is completed in this case.

From now, suppose that for allUsing (2) and the fact that is an -admissible mapping, we have

By induction we get

In the first step, we will show that the sequence is nonincreasing and

Where

Utilizing Remark 1 and condition (3), it follows that

If max we get a contradiction. Indeed, equation (4) implies

Therefore, we get that for all This means that there exists then from (3) and Remark 1 we get

which is a contradiction. Hence

Further, we show that Firstly, we have because is a triangular-admissible mapping. Therefore, we arrive at

Where

Since there exists such that for

And

also, we obtain

that is,

Hence, for according to Remark 1, it follows that

This amounts to say that, for

Now, we get

Hence, we have

Suppose that

Then we acquire the following:

that is,

which is a contradiction. Hence, it follows that

In order to prove that the sequenceis a Cauchy sequence, we use Lemma 2. According to Lemma 1, ifthen putting in (1) we obtain

Where

Now, since F, and are continuous, taking limit in (5), and utilizing Remark 1, we obtain

which is a contradiction. Hence the sequence is a Cauchy sequence. Since (M, d) is a complete rectangular metric space, there exists a point such that as If . is continuous, we get as Since for all according to Lemma 3, we conclude that are distinct. Therefore, there exists such that Further, by (iii) it follows that

wheneverTaking the limit in the last inequality, it follows that, which is again a contradiction.

In the case that (M, d) is .-regular, and since for all, from (1) we obtain the following:

where

Passing limit in (6) and using Proposition 1, the continuity of the functions ,,and as well as Remark 1, it follows that if then

which is a contradiction. Hence, is a fixed point of .

Now, we show that the fixed point is unique if whenever . Indeed, in this case, by the contractive condition (1), for such possible distinct fixed pointswe have

Where

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 which is not correct because we do not know whether exists or not.

By taking in Theorem 3, we obtain the following result as a corollary.

Corollary 4. Let .M, d. be a complete rectangular metric space, and let be a triangular-admissible mapping. Assume that there exist such that, for ,

where

Also suppose that the following assertions hold:

(i)there exists such that

(ii) T is continuous or (M, d) is α-regular.

Then has a fixed point. Moreover, if for all we have then the fixed point is unique.

Taking in Corollary 4, the following useful corollary is obtained.

Corollary 5. Let (M, d) be a complete rectangular metric space, and let be a triangular α-admissible mapping. Assume that there exists such that, for

where

Suppose also that the following assertions hold:

(i)there existssuch that

(ii) is continuous or (M, d) is -regular.

Then has a fixed point. Moreover, if for all we have, then the fixed point is unique.

Consider for in Corollary 5, then we obtain the following result.

Corollary 6. Let (M, d) be a complete rectangular metric space, and let be a triangular -admissible mapping such that, for

where

Also suppose that the following conditions are contended:

(i) there exists such that

(ii) T is continuous or (M, d) is α-regular.

Then has a fixed point. Moreover, if for all we have then the fixed point is unique.

3 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 program- ming related to multistage process reduces to solving the existence and uniqueness of the solution of the following functional equation:

where and We suppose thatis a state space,is a decision space, and are Banach spaces. Let denote the set of all bounded real valued functions on , and for an arbitrary define Clearly, the pair is a Banach space. For details, see [18, 19].

In fact, the distance in is given by

Define 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) are continuous and bounded;

(ii)there exits such that and

where

Then the functional equation (8) has a unique solution.

Proof. Let be an arbitrary positive number, and then there exist such that

which yields

In the same manner, we acquire

Since is arbitrary, we conclude

This amounts to say that

where

Lastly, we specify such that

Evidently for all This endorses that is a triangular-admissible mapping. Hence, due to Theorem 3, has a unique fixed point that is, is a unique solution of the functional equation (8). This completes the proof.

4 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 instead of two and 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.

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