Introduction

One of trends in mathematical research is to refine the frameworks of the known theorems and their results. For instance, Polish mathematician Banach observed the first metric fixed point results in the setting of complete normed spaces. An immediate extension of this theorem was given by Caccioppoli who observed the characterization of the Banach fixed point theorem in the context of complete metric spaces. Afterwards, for various abstract spaces, several analogs of the Banach contraction principle have been reported. Among them, we can underline some of interesting abstract structures such as modular metric space, partial metric space, b-metric space, fuzzy metric space, probabilistic metric space, G-metric space, etc.

This paper will be restricted to the recently introduced generalization of a metric space, namely, a modular metric space. Chistyakov introduced the notion of modular metric spaces (Chistyakov, 2010a, 2010b) inspired partly by the classical linear modulars on function spaces. Informally speaking, whereas a metric on a set represents the nonnegative finite distances between any two points of the set, a modular on a set attributes a nonnegative (possibly, infinite valued) “field of (generalized) velocities”: to each “time” 𝜆>0 the absolute value of an average velocity 𝑤_𝜆(𝓍,𝓎) is associated in such a way that in order to cover the “distance” between the points 𝓍, 𝓎∈ 𝜓, it takes time 𝜆 to move from 𝓍 to 𝓎 with the velocity 𝑤_𝜆(𝓍,𝓎). But the way we approached the concept of modular metric spaces is different. Indeed, we look at these spaces as a nonlinear version of the classical modular spaces introduced by H. Nakano (Nakano, 1950) on vector spaces and modular function spaces introduced by (Musielak, 1983) and (Orlicz, 1988a, 1988b). More about modular metric spaces can be read in (Hussain et al, 2011), (Paknazar & De la Sen, 2017) and (Paknazar & De la Sen, 2020).

In the formulation given by (Khamsi, 1996) and (Kozlowski, 1988), a modular on a vector space 𝜓 is a function 𝑚 ∶ 𝜓 → [0, +∞) satisfying:

(1) 𝑚(𝓍) = 0 if and only if 𝓍 = 0,

(2) 𝑚(𝑎𝓍) = 𝑚(𝓍) for every 𝑎 ∈ 𝑅 with |𝑎| = 1,

(3) 𝑚(𝑎𝓍 + 𝑏𝓎) ≤ 𝑚(𝓍) + 𝑚(𝓎) if 𝑎, 𝑏 ≥ 0 and 𝑎 + 𝑏 = 1.

A modular 𝑚 is said to be convex if, instead of (3), it satisfies the stronger property:

(3*) 𝑚(𝑎𝓍 + 𝑏𝓎) ≤ 𝑎𝑚(𝓍) + 𝑏𝑚(𝓎) if 𝑎, 𝑏 ≥ 0 and 𝑎 + 𝑏 = 1.

Given a modular 𝑚 on 𝜓, the modular space is defined by

𝜓_𝑚 = {𝑥 ∈ 𝜓 ∶ 𝑚(𝑎𝓍) → 0 as 𝑎 → 0}.

It is possible to define a corresponding F-norm (or a norm when 𝑚 is convex) on the modular space. The Orlicz spaces 𝐿^𝜙 are examples of this construction (Rao & Ren, 2002). The modular metric approach is more natural and has not been used extensively. For more on the metric fixed point theory, the reader may consult the book (Khamsi & Kirk, 2001) and for modular function spaces (Chistyakov, 2010a, 2010b) and (Chistyakov, 2008). Some recent work in modular metric spaces has been presented in (Mongkolkeha et al, 2011) and (Padcharoen et al, 2016). It has been almost a century since several mathematicians improved, extended and enriched the classical Banach contraction principle (Banach, 1922) in different directions along with variety of applications. In the sequel, we recall some basic concepts about modular metric spaces.

Throughout this paper, ℕ will denote the set of natural numbers.

Let 𝜓 be a nonempty set. Throughout this paper, for a function

𝜔: (0, +∞) × 𝜓 × 𝜓 → [0, +∞), we write

𝜔_λ ( 𝓍, 𝓎) = 𝜔(𝜆, 𝓍, 𝓎) for all 𝜆 > 0 and 𝓍, 𝓎 ∈ 𝜓.

DEFINITION 1. (Chistyakov, 2006) Let 𝜓 be a nonempty set. A function 𝜔: (0, +∞) × 𝜓 × 𝜓 → [0, +∞) is said to be a metric modular on 𝜓 if it satisfies, for all 𝓍, 𝓎, 𝑧 ∈ 𝜓, the following conditions:

1) 𝜔_λ (𝓍, 𝓎) = 0 for all 𝜆 > 0 if and only if 𝓍 = 𝓎,

2) 𝜔_λ (𝓍, 𝓎) = 𝜔_λ (𝓎, 𝓍) for all λ > 0 ,

3) 𝜔_{λ +μ} (𝓍, 𝓎) ≤ 𝜔_λ (𝓍, 𝑧) + 𝜔_μ (𝑧, 𝓎) for all 𝜆, 𝜇 > 0.

If instead of (1) we have only the condition (1'):

𝜔_λ (𝓍, 𝓍) = 0 for all 𝓍 ∈ 𝜓 , 𝜆 > 0, then 𝜔 is said to be a pseudo modular (metric) on 𝜓.

An important property of the (metric) pseudo modular on the set 𝜓 is that the mapping 𝜆 ↦ 𝜔_λ ( 𝓍, 𝓎) is non increasing for all 𝓍, 𝓎 ∈ 𝜓.

DEFINITION 2. (Chistyakov, 2006) Let 𝜔 be a pseudo modular on 𝜓.

Fixed 𝓍₀∈𝜓. The set

𝜓_𝜔= 𝜓_𝜔(𝓍₀) = {𝓍 ∈ 𝜓 : 𝜔_λ (𝓍, 𝓍₀) → 0 as 𝜆 → +∞} is said to be a modular metric space (around 𝓍₀).

DEFINITION 3. (Padcharoen et al, 2016) Let 𝜓_𝜔 be a modular metric space.

1) The sequence {𝓍_𝜂} in 𝜓_𝜔 is said to be 𝜔-convergent to 𝓍∈𝜓_𝜔 if and only if there exists a number 𝜆 > 0, possibly depending on {𝓍_𝜂} and 𝓍, such that .

2) The sequence {𝓍_𝜂} in 𝜓_𝜔 is said to be 𝜔-Cauchy if there exists 𝜆 > 0, possibly depending on the sequence, such that 𝜔_λ (𝓍_𝑚, 𝓍_𝜂) → 0 as 𝑚, 𝜂 → +∞.

3) A subset 𝐶 of 𝜓_𝜔 is said to be 𝜔 -complete if any 𝜔-Cauchy sequence in 𝐶 is a convergent sequence and its limit is in 𝐶.

DEFINITION 4.(Mongkolkeha et al, 2011) Let 𝜔 be a metric modular on 𝜓 and 𝜓𝜔 be a modular metric space induced by 𝜔. If 𝜓𝜔 is a 𝜔- complete modular metric space and 𝒯: 𝜓_𝜔 → 𝜓_𝜔 be an arbitrary mapping, 𝒯 is called a contraction if for each 𝓍 ,𝓎∈𝜓_𝜔 and for all 𝜆 > 0 there exists 0 ≤ 𝜎 < 1 such that

𝜔_λ (𝒯𝓍,𝒯𝓎) ≤ 𝜎𝜔_λ (𝓍, 𝓎)

Mongkolkeha et al, (2011) proved that if 𝜓_𝜔 is a 𝜔 - complete modular metric space, then contraction mapping 𝒯 has a unique fixed point.

Main result

In this section, there is a generalization of the result proved by (Murthy & Vara Prasad, 2013):

Let 𝒯 be a self-map of a complete metric space 𝜓 satisfying:

Where,

Then 𝒯 has a unique fixed point in 𝜓.

Now we will generalize the above result in the setting of modular metric spaces as follows:

Theorem 1. Let (𝜓_𝜔, 𝜔) be a complete modular metric space. Let 𝒯 be a self-map of a complete modular metric space 𝜓_𝜔 satisfying:

where

Then 𝒯 has a unique common fixed point in 𝜓_𝜔.

Proof. Let 𝓍₀ ∈ 𝜓_𝜔 be an arbitrary point. Then we can find 𝓍₁ such that 𝓍₁ = 𝒯(𝓍₀). For this 𝓍₁, we can find 𝓍₂ ∈ 𝜓_𝜔 such that 𝓍₂ = 𝒯(𝓍₁).

In general, one can choose {𝓍_𝜂+1} in 𝜓_𝜔 such that

We may assume that 𝓍_𝜂 ≠ 𝓍_𝜂+1 for each 𝜂.

Since if there exists 𝜂 such that 𝓍_𝜂 = 𝓍_𝜂+1 then 𝓍_𝜂 = 𝓍_𝜂+1 = 𝒯(𝓍_𝜂),

We write 𝛼_𝜂 = 𝑑(𝓍_𝜂, 𝓍_𝜂+1).

Firstly, we prove that 𝛼𝜂 is a non - increasing sequence and converges to 0.

Case I. If 𝜂 is even, taking 𝓍 = 𝓍_2𝜂 and 𝓎 = 𝓍_2𝜂+1 in (𝐶₃), we get

[1 + 𝑝𝜔₁(𝓍_2𝜂, 𝓍_2𝜂+1)]𝜔₁² (𝒯𝓍_2𝜂,𝒯𝓍_2𝜂+1)

where

Using (1) we get

where,

Now consider 𝛼_2𝜂 = 𝜔₁(𝓍_2𝜂, 𝓍_2𝜂+1); then we have

where,

By triangular inequality and using the property of ∅, we get

and

If 𝛼_2𝜂 < 𝛼_2𝜂+1, then (𝐶₃ ) reduces to

Therefore,

Case II. In a similar way, if 𝜂 is odd, then we can obtain 𝛼_2𝜂+2 < 𝛼_2𝜂+1. It follows that the sequence {𝛼_𝜂} is decreasing.

Let, for some 𝑟 ≥ 0.

Suppose 𝑟 > 0; then from the inequality (𝐶₃) and (𝐶₄), we have

where,

Now by using (1) we get,

Using the triangular inequality and the property of ∅, and taking the limit

𝜂 → +∞, we get

Then ∅(𝑟²) ≤ 0, since 𝑟 is positive, then by the property of ∅, we get 𝑟 = 0, and we conclude that

Now we show that {𝓍_𝜂} is a Cauchy sequence. For the given 𝜖 > 0, we can find two sequences of positive integers {𝑚(𝜎)} and {𝜂(𝜎)} such that

and 𝜂(𝜎) > 𝑚(𝜎) > 𝜎.

Again using the triangular inequality, we have

We get

Taking the limits as 𝜎 → +∞ , we have

Now from the triangular inequality, we have

We get

Again, from the triangular inequality, we have

We get

On putting 𝓍 = 𝓍_𝑚(𝜎) and 𝓎 = 𝓍_𝜂(𝜎) in (𝐶₃), we get

where,

Using (1), we obtain

where,

Letting 𝜎 → +∞ and using (13) - (19), we get

a contradiction.

Thus, {𝓍𝜂 } is a Cauchy sequence in 𝜓𝜔, since (𝜓𝜔, 𝜔) is a completemodular metric space.

Now, we will prove that 𝑧 is a fixed point of 𝒯.For this, let 𝓍 = 𝓍𝜂 and 𝓎 = 𝑧 in (𝐶₃), we get

where,

Using (22) and (1), we have

where,

Hence, 𝜔12(𝑧, 𝒯𝑧) ≤0 ⟹ 𝒯𝑧 = 𝑧.

So, 𝒯 has a fixed point in 𝜓𝜔.

Uniqueness:

To show that 𝒯 can have only one common fixed point.Suppose 𝓍 ≠ 𝓎 be two fixed points of 𝒯.

Therefore, 𝓍 = 𝒯𝓍 and 𝓎 = 𝒯𝓎 from (𝐶₃) ,we have

Corollary 1. Let 𝒯 be a mapping of a complete modular metric space(𝜓𝜔, 𝜔) into itself satisfying the condition

where,

For all 𝓍, 𝓎 ∈ 𝜓 and ∅: [0, +∞) → [0, +∞) is a continuous function with∅(𝓉) = 0 ⇔ 𝓉 = 0 and ∅(𝓉) > 0 for each 𝓉 > 0. Then 𝒯 has a uniquefixed point in 𝜓𝜔.

Proof. Put 𝑝 = 0 in Theorem 1 and we have the required result.

Example 1. Let 𝜓 = ℝ. We define the mapping 𝜔: (0,1) × ℝ × ℝ →[0,1]by 𝜔λ(𝓍, 𝓎) =|𝓍−𝓎|1+λfor all 𝓍, 𝓎 ∈ ℝ and λ> 0. Then it is obvious that ℝ𝜔 isa complete modular metric space. Define 𝒯: ℝ𝜔 → ℝ𝜔 by 𝒯𝓍 =𝓍4and∅: [0, +∞) → [0, +∞) by ∅(𝓉) =𝓉3, for any values of 𝑝 > 0 and 𝓍, 𝓎 ∈ 𝜓.Then it is easy to verify the inequalities (𝐶3) and (𝐶4) hold.Hence from Theorem 1, the mapping 𝒯 has a unique fixed point 0.Moreover, it is 0 ∈ ℝ_𝜔.

Property P

In this section, we will show that the maps satisfying (𝐶3) and (𝐶4)possess the property P.

Let us denote the set of all fixed points of a self –mapping 𝒯 from X intoitself by F(𝒯), that is, F(𝒯)= { 𝑧X : 𝒯𝑧 = 𝑧 }. It is clearly that if 𝑧 is a fixedpoint of 𝒯, then it is also a fixed point of 𝒯𝑛 for each n, that is,F(𝒯)⊂F(𝒯𝑛) if F(𝒯) ≠ 𝜙. However, converse is false.

Indeed the mapping 𝒯: ℝ →ℝ defined by 𝒯𝓍 =12- 𝓍 has a unique fixedpoint, that is, F (𝒯) = {14} , but every 𝓍 is a fixed point for 𝒯2.If F(𝒯) = F(𝒯𝑛), for each n, then we say that 𝒯𝑛 has no periodicpoints.(Jeong & Rhoades, 2005) examined a number of situations in which thefixed point sets for maps and their iterates are the same. They state thata map 𝒯 has the property P if F(𝒯) = F(𝒯𝑛) for each n.

Theorem 2.

Under the condition of Theorem 1, 𝒯 has the property P.

Proof. From Theorem 1, 𝒯 has a fixed point. Therefore, 𝐹(𝒯𝑛) ≠ 𝜙 foreach 𝑛 . Fix 𝑛 > 1 and assume that 𝑝 ∈ 𝐹(𝒯𝑛).We wish to show that 𝑝 ∈ 𝐹(𝒯).Suppose that 𝑝 ≠ 𝒯𝑝.Using the inequality (𝐶3), we have

where,

where,

where,

If 𝜔1(𝒯𝑛−1𝑝, 𝑝) ≤ 𝜔1(𝑝, 𝒯𝑝) then𝜔12(𝑝, 𝒯𝑝) ≤ 𝜔12(𝑝, 𝒯𝑝) - ∅𝜔12(𝑝, 𝒯𝑝).This implies that 𝑝 = 𝒯𝑝, a contradiction.Therefore, 𝑝 ∈ 𝐹(𝒯) and 𝒯has the property P.

References

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.

Chistyakov, V.V. 2010a. Modular metric spaces, I: Basic concepts. Nonlinear Analysis: Theory, Methods and pplications, 72(1), pp.1-14. Available at: https://doi.org/10.1016/j.na.2009.04.057.

Chistyakov, V.V. 2010b. Modular metric spaces, II: Application to superposition operators. Nonlinear Analysis: Theory, Methods and Applications, 72(1), pp.15-30. Available at: https://doi.org/10.1016/j.na.2009.04.018.

Chistyakov, V.V. 2006. Metric modulars and their application. Doklady Mathematics, 73(1), pp.32-35. Available at: https://doi.org/10.1134/S106456240601008X.

Chistyakov, V.V. 2008. Modular Metric Spaces Generated by F-Modular. Folia Mathematica, 15(1), pp.3-24 [online]. Available at: http://fm.math.uni.lodz.pl/artykuly/15/01chistyakov.pdf [Accessed: 10 March 2022].

Hussain, N., Khamsi, M. & Latif, A. 2011. Banach operator pairs and common fixed points in modular function spaces. Fixed Point Theory and Applications, art.number:75. Available at: https://doi.org/10.1186/1687-1812-2011-75.

Jeong, G.S. & Rhoades, B.E. 2005. Maps for which F(𝒯) = F(𝒯 𝑛 ). Demonstratio Mathematica, 40(3), pp.671-680. Available at: https://doi.org/10.1515/dema-2007-0317.

Khamsi, M.A. 1996. A convexity property in Modular function spaces. Mathematica Japonica, 44(2), pp.269-279 [online]. Available at: http://69.13.193.156/publication/acpimfs.pdf [Accessed: 10 March 2022].

Khamsi, M.A. & Kirk, W.A. 2001. An Introduction to Metric Spaces and Fixed Point Theory. New York, NY, USA: John Wiley & Sons. Available at: ISBN: 978-0-471-41825-2.

Kozlowski, W.M. 1988. Modular Function Spaces, Monographs and Textbooks in Pure and Applied Mathematics. New York, NY, USA: Marce Dekker.

Mongkolkeha, C., Sintunavarat, W. & Kumam, P. 2011. Fixed point theorems for contraction mappings in modular metric spaces. Fixed Point Theory and Applications, art.number:93. Available at: https://doi.org/10.1186/1687-1812-2011-93.

Murthy, P.P. & Vara Prasad, K.N.V.V. 2013. Weak Contraction Condition Involving Cubic Terms of 𝑑(𝓍, 𝓎) under the Fixed Point Consideration. Journal of Mathematics, art.ID:967045. Available at: https://doi.org/10.1155/2013/967045.

Musielak, J. 1983. Orlicz Spaces and Modular Spaces. Berlin Heidelberg: Springer-Verlag. Available at: https://doi.org/10.1007/BFb0072210.

Nakano, H. 1950. Modulared semi-ordered linear spaces. Tokyo, Japan: Maruzen Co.

Orlicz, W. 1988a. Collected Papers. Part I. Warsaw Poland: PWN Polish Scientific Publishers.

Orlicz, W. 1988b. Collected Papers. Part II. Warsaw Poland: Polish Academy of Sciences.

Padcharoen, A., Gopal, D., Chaipunya, P. & Kumam, P. 2016. Fixed point and periodic point results for α-type F-contractions in modular metric spaces. Fixed Point Theory and Applications, art.number:39. Available at: https://doi.org/10.1186/s13663-016-0525-4.

Paknazar, M. & De la Sen, M. 2017. Best Proximity Point Results in Non-Archimedean Modular Metric Space. Mathematics, 5(2), art.number:23. Available at: https://doi.org/10.3390/math5020023.

Paknazar, M. & De la Sen, M. 2020. Some new approaches to modular and fuzzy metric and related best proximity results. Fuzzy Sets and Systems, 390, pp.138-159. Available at: https://doi.org/10.1016/j.fss.2019.12.012.

Rao, M.M. & Ren, Z.D. 2002. Applications Of Orlicz Spaces (1st ed.). Boca Raton, FL, USA: CRC Press. Available at: https://doi.org/10.1201/9780203910863.

Notas de autor

ljiljana.paunovic@pr.ac.rs

Información adicional

FIELD: Mathematics

ARTICLE TYPE: Original scientific paper

Enlace alternativo

https://scindeks.ceon.rs/article.aspx?artid=0042-84692203546P (html)

https://aseestant.ceon.rs/index.php/vtg/article/view/36958 (pdf)

https://doaj.org/article/0c36f16d9b2d4c14bd538086fb140714 (pdf)

https://elibrary.ru/item.asp?id=48719248 (pdf)

http://www.vtg.mod.gov.rs/archive/2022/military-technical-courier-3-2022.pdf (pdf)