HTML

ePub

PDF

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 dr : 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) dr (x, y) = 0 if and only if x = y,

(ii) dr (x, y) = dr (y, x),

(iii) dr (x, y) ≤ dr (x, u) + dr (u, v) + dr (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. (Salimi et al, 2013) 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.

Let Ψ be the family of all functions ψ : [0, +∞) → [0, +∞) such that ψ is nondecreasing and for each t > 0. Obviously, if ψ ∈ Ψ, then ψ (t) < t for each t > 0.

Definition 2. (Salimi et al, 2013) Let T be a self mapping on a metric space (X, d_r) and let α, η : X × X → [0, +∞) be two functions. It is called an α−admissible mapping with respect to η if α (x, y) ≥ η (x, y) implies that α (T x, T y) ≥ η (T x, T y) for all x, y ∈ X.

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 ∈ 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 x₀ ∈ X such that α (x₀, T x₀) ≥ 1. Define the sequence {x_n} by x_n = Tⁿx₀. Then

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

Theorem 1. Let .X, d.. be a Hausdorff and complete rectangular metric space, and let T : . → X be an α.admissible mapping with respect to η. Assume that there exists a continuous function ψ ∈ Ψ such that

where

Also, suppose that the following assertions are hold:

1. there exists x₀ ∈ X such that α (x₀, Tx₀) ≥ η (x₀, Tx₀),

2. for all x, y, z ∈ X,(α (x, y) ≥ η (x, y) and α (y, z) ≥ η (y, z)) implies α (x, z) ≥ η (x, z),

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, Ta) ≥ η (a, Ta) 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, dr) 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

where

Also, suppose that the following assertions are hold:

1. there exists x₀ ∈ X such that α (x₀, Tx₀) ≥ 1,

2. for all x, y, z ∈ X (α (x, y) ≥ 1 and α (y, z) ≥ 1) implies α (x, z) ≥ 1,

3. either T is continuous or (X, d_r) is α−regular.

Then T has a periodic point a ∈ X and if α (a, Ta) ≥ 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

where

Also, suppose that the following assertions are hold:

1. there exists x₀ ∈ X such that 1 ≥ η (x₀, T x₀),

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

where

Also, suppose that the following assertions are hold:

1. there exists x₀ ∈ X such that α (x₀, Tx₀) ≥ η (x₀, Tx₀),

2. for all x, y, z ∈ X (α (x, y) ≥ η (x, y) and α (y, z) ≥ η (y, z)) implies α (x, z) ≥ η (x, z),

3. either T is continuous or (X, d_r) is α−regular.

Then T has a periodic point a ∈ X and if α (a, Ta) ≥ η (a, Ta) 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 $\ne$ x_m for n $\ne$ 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 → +∞ :

Lemma 3. Let , T⁰x₀ = x₀ be a Picard sequence in a rectangular metric space (X, d_r) induced by the mapping T : X → X and the initial point x₀ ∈ X. If d_r(x_n, x_n+1) < d_r(x_n−1, x_n) for all n ∈ then x_n $\ne$ x_m whenever n $\ne$ m.

Proof. Let x_n = x_m for some n, m ∈ with n < m. Then x_n+1 = Tx_n = Tx_m = x_m+1. Further, we get

which is a contradiction.

In some proofs, we will also use and 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 lim_n→+∞ d_r (q_n, q) = 0. Then lim_n→+∞ d_r (q_n, p) = d_r(q, p) for all p ∈ X. In particular, {q_n} does not converge to p if p $\ne$ q.

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 ∈ . 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 : X → X be a triangular α−admissible mapping. Assume that there exists continuous function ψ ∈ Ψ such that

where

Also, suppose that the following assertions are hold:

1. there exists x₀ ∈ X such that α (x₀,Tx₀) ≥ 1

2. either T is continuous or (X, d) α−regular

Then T has a fixed point.

Morover, if for all x, y ∈ F(T) implies α (x, y) ≥ 1 then the fixed point is unique.

Proof. Given x₀ ∈ X such that

Define a sequence {x_n} in X by x_n = Tx_n−1 = Tⁿx₀ for all n ∈ . If x_k+1 = x_k for some k ∈, then Tx_k = x_k, i.e., x_k is a fixed point of T and the proof is finished. From now on, suppose that x_n $\ne$x_n+1 for all 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

Now from (3) follows

If max {d_r(x_n−1,x_n), d_r(x_n,x_n+1)} = d_r(x_n,x_n+1), we get a contradiction. Indeed, (4) implies

Therefore, we get that d_r(x_n,x_n+1) < d_r(x_n−1,x_n). This means that there exists limn_→+∞d_r(x_n,x_n+1) = d_r*r ≥ 0. If d_r* > 0, then from (3) follows

which is a contradiction. Hence lim_n→+∞d_r(x_n,x_n+1) = 0.

Further, we will also show that lim_n→+∞d_r(x_n,x_n+2) = 0. Firstly, we have that α(x_n−1, x_n) ≥ 1, i.e., α (x_n−1, x_n+1) ≥ 1, because T is a triangular α−admissible mapping. Therefore,

where

that is,

The last relation follows from the fact that d_r(x_n−1, x_n) → 0 as n → +∞. Hence, for some n1 ∈ , we have that

whenever n ≥ n₁. 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) x = x_n(k) , y = x_m(k), we obtain

where

Now, taking in (5) the limit as

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 x_n+1 = Tx_n → Tx* as n → +∞. Let Tx* $\ne$ x*. Since d_r(x_n, x_n+1) < d_r(x_n−1, x_n) for all n ∈∪ {0}, then, according to Lemma 7, we have that all x_n are distinct. Therefore, there exists n₂ ∈ such that x*, Tx* {X_n}_n≥n2 Further, by (iii) folows:

whenever n ≥ n₂, taking the limit, we obtain d_r(x*, Tx*) = 0, i.e. x* = Tx*, 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 ∈ , 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(x*, Tx*) ≤ ψ (d_r(x*, Tx*)) < d_r(x*, Tx*) if x*$\ne$ Tx*, 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

Hence, (7) becomes

which is a contradiction. The proof of Theorem 9 is complete.

Remark 1. In the proof of the case 2 on Page 96, the authors used the fact that the rectangular metric d_r (see the 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 consequences 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

Also, suppose that the following assertions hold:

1. there exists x₀ ∈ X such that α (x₀, Tx₀) ≥ 1,

2. either T is continuous or (X, d_r) is α−regular.

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,

Also, suppose that the following assertions hold:

1. there exists x₀ ∈ X such that α (x₀, Tx₀) ≥ 1,

2. either T is continuous or (X, d_r) is α−regular

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

Also, suppose that the following assertions hold:

1. there exists x₀ ∈ X such that α (x₀, Tx₀) ≥ 1

2. either T is continuous or (X, d_r) is α−regular

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 k₁ ∈ [0, 1) such that for all x, y ∈ X holds:

- Ćirić 2: Ćirić’s generalized contraction of second order: there exists k₂ ∈ [0, 1) such that for all x, y ∈ X holds:

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 k₃ ∈ [0, 1) such that for all x, y ∈ X holds:

Since,

and

it follows that (11) implies (12) and (12) implies (13). In (Ćirić, 2003), Ćirić proved the following result:

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 , T⁰x = 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

where M(x, y)is one of the sets:

An open problem

A suggestion for further research - it is logicalnatural 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 acquiredobtained results.

An application to chemical sciences

Consider a diffusing substance placed in an absorbing medium between parallel walls such that δ1, δ2 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 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] × → is the Green’s function which is continuous and is given by

Suppose that 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 . Obviously (X, d_r) is a complete rectangular metric space.

Let the operator Ξ : X → X be defined by

Then 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 → 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 ∈ X and r ∈ I, the mapping Ξ : X → X is well defined. Also Ξ is triangular α-admissible.

Since .

Hence for all, ∈ X, we obtain

where

Taking , we obtain

Hence, all the hypotheses of Theorem 2 are contented. We conclude that Ξ has a unique fixed point 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:

where R are the given continuous functions.

Let X be the set C[a, b] of real continuous functions defined on [a, b] and let d_r : X × X → [0, ∞) be equipped with the metric defined by

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:

(1);

2) Suppose that for all x, y ∈,

Then integral equation (4) has a solution.

Proof. Employing conditions (1) − (2) along with inequality (4), we have

Taking ψ(M(u1, u2)) = 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) has a unique solution which belongs to X = C[a, b].

References

Alsulami, H.H., Chandok, S., Taoudi, M-A. & Erhan, I.M. 2015. Some fixed point theorems for (α, ψ)-rational type contractive mappings. Fixed Point Theory and Applications, 2015(art.number:97). Available at: https://doi.org/10.1186/s13663-015-0332-3.

Banach, S. 1922. Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales. Fundamenta Mathematicae, 3, pp.133-181 (in French). Available at: https://doi.org/10.4064/fm-3-1-133-181.

Budhia, L., Kir, M., Gopal, D. & Kiziltunc, H. 2017. New fixed point results in rectangular metric space and application to fractional calculus. Tbilisi Mathematical Journal, 10(1), pp.91-104. Available at: https://doi.org/10.1515/tmj-2017-0006.

Collaco, P. & Silva, J.C.E. 1997. A complete comparison of 25 contraction conditions. Nonlinear Analysis: Theory, Methods & Applications, 30(1), pp.471-476. Available at: https://doi.org/10.1016/S0362-546X(97)00353-2.

Ćirić, Lj. 2003. Some Recent Results in Metrical Fixed Point Theory. Belgrade: University of Belgrade.

Kadelburg, Z. & Radenović, S. 2014a. Fixed point results in generalized metric spaces without Hausdorff property. Mathematical Sciences, 8(art.number:125). Available at: https://doi.org/10.1007/s40096-014-0125-6.

Kadelburg, Z. & Radenović, S. 2014b. On generalized metric spaces: A survey. TWMS Journal of Pure and Applied Mathematics, 5(1), pp.3-13 [online]. Available at: http://www.twmsj.az/Files/Contents%20V.5,%20%20N.1,%20%202014/pp.3-13.pdf [Accessed: 15 November 2020].

Karapınar, E. 2014. Discusion on α-ψ Contractions on Generalized Metric Spaces. Abstract and Applied Analysis, 2014(art.ID:962784). Available at: https://doi.org/10.1155/2014/962784.

Karapınar, E., Kummam, P. & Salimi, P. 2013. On α-ψ-Meir-Keeler contractive mappings. Fixed Point Theory and Applications, art.number:94. Available at: https://doi.org/10.1186/1687-1812-2013-94.

Kirk, W.A. & Shahzad, N. 2014. Fixed Point Theory in Distance Spaces. Springer International Publishing Switzerland. ISBN: 978-3-319-10927-5.

Mohammadi, B. & Rezapour, Sh. 2013. On modified α-ψ-contractions. Journal of Advanced Mathematical Studies, 6(2), pp.162-166 [online].

Radenović, S., Kadelburg, Z., Jandrlić, D. & Jandrlić, A. 2012. Some results on weakly contractive maps. Bulletin of the Iranian Mathematical Society, 38(3), pp.625-645 [online]. Available at: http://bims.iranjournals.ir/article_229.html [Accessed: 15 March 2018].

Rhoades, B.E. 1977. A comparison of various definitions of contractive mappings. Trans. Amer. Math. Soc., 226, pp.257-290. Available at: https://doi.org/10.1090/S0002-9947-1977-0433430-4.

Samet, B., Vetro, C. & Vetro, P. 2012. Fixed point theorem for α-ψ-contractive type mappings. Nonlinear Analysis: Theory, Methods & Applications, 75(4), pp.2154-2165. Available at: https://doi.org/10.1016/j.na.2011.10.014.

Salimi, P., Latif, A. & Hussain, N. 2013. Modified α-ψ-contractive mappings with applications. Fixed Point Theory and Applications, art.number:151. Available at: https://doi.org/10.1186/1687-1812-2013-151.

Author notes

zoran.mitrovic@etf.unibl.org

Additional information

FIELD: Mathematics

ARTICLE TYPE: Original scientific paper

Alternative link

https://scindeks.ceon.rs/article.aspx?artid=0042-84692101008Y (html)

https://scindeks-clanci.ceon.rs/data/pdf/0042-8469/2021/0042-84692101008Y.pdf (pdf)

https://www.elibrary.ru/item.asp?id=44584147 (pdf)

http://www.vtg.mod.gov.rs/archive/2021/military-technical-courier-1-2021.pdf (pdf)

IR