On fixed point theorems within multiplicative S-metric space

Introduction

Banach [1] contraction principle has been a very advantageous and efficacious means in nonlinear analysis. Various authors have generalized Banach contraction principle in different spaces. Singxi et al. [18] and Sastry et al. [10] studied some common fixed point theorems for different mappings on a 2-metric space. Dhage [4] proved fixed point results in D-metric space. A. E. Bashirov et al. [2] introduce a new kind of space called multiplicative metric space in the year 2008 and studied some properties of multiplicative derivatives and multiplicative intergrals.

Definition 1.1 [2]: "Let 𝒳 be a non-empty set. A multiplicative metric is a mapping on 𝑑: 𝑋 × 𝑋 → ℝ + satisfying the following axioms:

1. 𝑑(𝑢, 𝑣) ≥ 1 for all 𝑢, 𝑣 𝜖 𝒳 and 𝑑(𝑢, 𝑣) = 1 if and only if 𝑢 = 𝑣, 2. 𝑑(𝑢, 𝑣) = 𝑑(𝑣, 𝑢) for all 𝑢, 𝑣 𝜖 𝒳, 3. 𝑑(𝑢, 𝑣) ≤ 𝑑(𝑢, 𝑤). 𝑑(𝑤, 𝑣), for all 𝑢, 𝑣 ∈ 𝒳.

Then the mapping 𝑑 together with 𝒳 that is, (𝒳, 𝑑) is a multiplicative metric space." In 2012, Ozavsar and Cevikel [8] introduced the concept of multiplicative contraction mappings and proved some fixed point theorems of such mappings on a complete multiplicative metric space. Definition 1.2 [8]: "Let (𝒳, 𝑑) be a multiplicative metric space. A mapping 𝑓: 𝑋 → 𝑋 is called a multiplicative contraction if there exist a real constant 𝜆 𝜖[0,1) such that 𝑑(𝑓𝑢, 𝑓𝑣) ≤ 𝑑(𝑢, 𝑣) 𝜆 , for all 𝑢, 𝑣 ∈ 𝒳." Theorem 1.3 [8]: "Let (𝒳, 𝑑) be a multiplicative metric space and let 𝑓: 𝑋 → 𝑋 be a multiplicative contraction. If (𝒳, 𝑑) is complete then, 𝑓 has a unique fixed point." In 2012, Sedghi et. al. [11] establish the concept of 𝒮-metric space as a generalization of Gmetric space and metric space. Definition 1.4 [11]: "Let 𝒳 be a non-empty set. An 𝒮-metric on 𝑋 is a function 𝒮 ∶ 𝒳 × 𝒳 × 𝒳 → [0, ∞) satisfying the following axioms:

1. 𝒮(𝑢, 𝑣, 𝑤) = 0 if and only if 𝑢 = 𝑣 = 𝑤, 2. 𝒮(𝑢, 𝑣, 𝑤) ≤ 𝒮(𝑢, 𝑢, 𝑎) + 𝒮(𝑣, 𝑣, 𝑎) + 𝒮(𝑤, 𝑤, 𝑎), for all 𝑢, 𝑣, 𝑤 ∈ 𝒳 An 𝒮-metric space is a pair (𝒳, 𝒮) where 𝒮 is a metric on 𝒳." In the setting of 𝒮-metric space they proved several fixed and common fixed point theorems (see [5], [12]- [16]). In 2021, Naga Raju [9] introduce the concept of multiplicative S-metric space and studied its topological properties. Definition 1.5 [9]: "Let 𝒳 be a non-empty set. We say that the function 𝒮 ∶ 𝒳 × 𝒳 × 𝒳 → [0, ∞) is a multiplicative 𝒮-metric on 𝒳 iff it satisfies the following axioms:

1. 𝒮(𝑢, 𝑣, 𝑤) ≥ 1, 2. 𝒮(𝑢, 𝑣) = 1 if and only if 𝑢 = 𝑣 = 𝑤, 3. 𝒮(𝑢, 𝑣, 𝑤) ≤ 𝒮(𝑢, 𝑢, 𝑎). 𝒮(𝑣, 𝑣, 𝑎). 𝒮(𝑤, 𝑤, 𝑎), for all 𝑢, 𝑣, 𝑤, 𝑎 ∈ 𝒳.

Then the mapping (𝒳, 𝒮) is called a multiplicative 𝒮-metric space." Definition 1.6 [9]: "We say that a sequence {𝑢 𝑛 } in a multiplicative 𝒮-metric space (𝒳, 𝒮) multiplicative 𝒮-convergent to some 𝛼 ∈ 𝒳 iff for each ∈ 0 > 1, there exist 𝐻 ∈ ℕ such that 𝒮(𝑢 𝑛 , 𝑢 𝑛 , 𝑢) <∈ 0 , for all 𝑛 ≥ 𝐻." Definition 1.7 [9]: "We say that a sequence {𝑢 𝑛 } in a multiplicative 𝒮-metric space (𝒳, 𝒮) multiplicative 𝒮-Cauchy sequence in 𝒳 iff for each ∈ 0 > 1, there exist 𝐻 ∈ ℕ such that 𝒮(𝑢 𝑛 , 𝑢 𝑛 , 𝑢 𝑚 ) <∈ 0 , for all 𝑛, 𝑚 ≥ 𝐻." Definition 1.8 [9]: "We say that a multiplicative 𝒮-metric space (𝒳, 𝒮) is multiplicative 𝒮-complete iff every multiplicative 𝒮-Cauchy sequence in 𝒳 is multiplicative 𝒮-convergent in 𝒳." Definition 1.9 [9]: "Let (𝒳, 𝒮) and (℧, 𝒮 ′ ) be two multiplicative 𝒮-metric spaces. Then we say that 𝑓: 𝒳 → ℧ is multiplicative 𝒮continuous at some point 𝜃 ∈ 𝒳 iff for every 𝑟 > 1, there exists 𝜂 > 1such that 𝑓(𝐵(𝜃, 𝜂)) ⊂ 𝐵(𝑓(𝜃), 𝑟). Thus, we say that 𝑓 is multiplicative 𝒮-continuous at every point of 𝒳." Lemma 1.10 [9]: "In multiplicative 𝒮-metric space (𝒳, 𝒮) we have 𝒮(𝑢, , 𝑢, 𝑣) = 𝒮(𝑣, 𝑣, 𝑢) for all 𝑢, 𝑣 ∈ 𝒳." Theorem 1.13 [9]: "In multiplicative 𝒮-metric space (𝒳, 𝒮), {𝑢 𝑛 } is a multiplicative 𝒮-Cauchy sequence in 𝒳 iff 𝒮(𝑢 𝑛 , 𝑢 𝑛 , 𝑢 𝑚 ) → 1 as 𝑚 → ∞." Definition 1.14 [6]: "Let 𝑓 and 𝑔 be two mappings of a metric space (𝒳, 𝑑) into itself. Then 𝑓 and 𝑔 are said to be weakly compatible if they commutes at coincident points, that is, if 𝑓𝑡 = 𝑔𝑡 for some 𝑡 ∈ 𝒳 implies that 𝑓𝑔𝑡 = 𝑔𝑓𝑡." Definition 1.15 [7]: "A function ϕ ∶ ℝ + → ℝ + is called an altering distance function if the following property is satisfied:

In our result we use the following class of function.

Note that Ψ is non empty, since ψ(t) = 𝑒 𝑡 for 𝑡 ∈ [1, ∞). Thus ψ ∈ Ψ."

Remark: Clearly for ψ ∈ Ψ, ψ(t) > 1 for 𝑡 > 1 and ψ(1) need not be equal to 1. Next we claim that {𝑢 𝑛 } is a multiplicative 𝒮-Cauchy sequence. Suppose that {𝑢 𝑛 } is not a multiplicative 𝒮-Cauchy sequence.

Main

Then there exists an ∈ 0 > 1 for which we can find two subsequences of positive integers {𝑚(𝑐)} and {𝑛(𝑐)} such that for all positive integers 𝑐, 𝑛(𝑐) > 𝑚(𝑐) ≥ 𝑐 and 𝒮(𝑢 𝑚(𝑐) , 𝑢 𝑚(𝑐) , 𝑢 𝑛(𝑐) ) ≥∈ 0 .

Assume that 𝑛(𝑐) is the smallest positive integer, we get 𝑛(𝑐) > 𝑚(𝑐) ≥ 𝑐, 𝒮(𝑢 𝑛(𝑐) , 𝑢 𝑛(𝑐) , 𝑢 𝑚(𝑐) ) ≥∈ 0 and 𝒮(𝑢 𝑛(𝑐)-1 , 𝑢 𝑛(𝑐)-1 , 𝑢 𝑚(𝑐) ) <∈ 0 .

∈ 0 ≤ 𝒮(𝑢 𝑚(𝑐) , 𝑢 𝑚(𝑐) , 𝑢 𝑛(𝑐) ) = 𝒮(𝑢 𝑛(𝑐) , 𝑢 𝑛(𝑐) , 𝑢 𝑚(𝑐) )

≤ 𝒮(𝑢 𝑛(𝑐) , 𝑢 𝑛(𝑐) , 𝑢 𝑛(𝑐)-1 )𝒮(𝑢 𝑛(𝑐) , 𝑢 𝑛(𝑐) , 𝑢 𝑛(𝑐)-1 )𝒮(𝑢 𝑛(𝑐)-1 , 𝑢 𝑛(𝑐)-1 , 𝑢 𝑚(𝑐) )

Taking limit as 𝑐 → ∞ and using (2.5), we get

𝒮(𝑢 𝑚(𝑐)-1 , 𝑢 𝑚(𝑐)-1 , 𝑢 𝑛(𝑐)-1 )

≤ 𝒮(𝑢 𝑚(𝑐)-1 , 𝑢 𝑚(𝑐)-1 , 𝑢 𝑚(𝑐) ). 𝒮(𝑢 𝑚(𝑐)-1 , 𝑢 𝑚(𝑐)-1 , 𝑢 𝑚(𝑐) ). 𝒮(𝑢 𝑚(𝑐) , 𝑢 𝑚(𝑐) , 𝑢 𝑛(𝑐)-1 ).

Taking limit as 𝑐 → ∞ ad using (2.5) and (2.6), we get . 𝒮(𝑢 𝑛(𝑐) , 𝑢 𝑛(𝑐) , 𝑢 𝑚(𝑐) ),

Taking limit as 𝑐 → ∞ in above inequalities and using (2. ≤ 𝒮(𝑢 𝑚(𝑐)-1 , 𝑢 𝑚(𝑐)-1 , 𝑢 𝑚(𝑐) ). 𝒮(𝑢 𝑚(𝑐)-1 , 𝑢 𝑚(𝑐)-1 , 𝑢 𝑚(𝑐) ). 𝒮(𝑢 𝑚(𝑐) , 𝑢 𝑚(𝑐) , 𝑢 𝑛(𝑐)-1 ).

Taking limit as 𝑐 → ∞ in above inequalities and using (2.5) and (2.8), we get lim 𝑐→∞ 𝒮(𝑢 𝑛(𝑐)-1 , 𝑢 𝑛(𝑐)-1 , 𝑢 𝑚(𝑐) ) =∈ 0 .

(2.9) Similarly, we have .

Taking limit on both sides and using (2.6), (2.13) and (2.

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