View of COMMON FIXED POINTS IN POLISH SPACES

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VOLUME: 08, Special Issue 08, (IC-RAAPAMAA-2021) Paper id-IJIERM-VIII-

VIII, October 2021

18 COMMON FIXED POINTS IN POLISH SPACES

Monika Parashar

Assistant Professor, Acropolis Institute of Management Studies and Research Dr. Bhawna Somani

Assistant Professor, Acropolis Institute of Management Studies and Research

Abstract- The purpose of this paper is to obtain a new common fixed point theorem for three continuous random multi-valued operators using measurable mappings under generalized distance function in Polish space.

Keywords: Random multi-valued operator, Random fixed point, measurable mapping, Asymptotically T-Regular mappings

1. INTRODUCTION

Random fixed point theorems are stochastic generalization of classical fixed point theorem and are required for the theory of random equations. The study was initiated with the work of Spacek [64] and Hans [29] proved random fixed point theorem for contraction mapping on polish spaces. On the other hand Nadler [49] prove the multi-valued version of Banach Contraction Principle in Complete metric spaces. Sehgal, V.M.

and Singh [57] proved random approximation and random fixed point theorem for a set valued Mapping. Lin [46] studied random approximations and random fixed point theorems for non selfmaps. Beg and Azam [8] proves some results on fixed point of asymptotically regular multi-valued mappings. Sessa, Rhoades and Khan [59] proved common fixed point of compatible mappings in a metric space and Banach spaces. Jungck [39] proved common fixed points for commuting and compatible maps.

Kaneko and Sessa [42] proved results on fixed point theorem for compitable, multi- valued and single valued maps. Badshah and Singh [3] proved common fixed point of commuting mappings. Beg and Shahzad [11] studied random fixed points of random multi-valued operators on metric spaces. Itoh [37] studied random fixed point theorem for multi- valued contraction mapping. Beg and Azam [7] studied fixed points of multi- valued locally contractive mappings.

Badshah and Syyed [2], Badshah and Gagrani, proved common fixed points of random multi-valued operators on Polish spaces. Beg and Shahzad [10] proved results for random fixed points of random multi-valued operators on Polish spaces.

2. PRELIMINARIES

Let (𝑋, d) be a Polish space, that is a separable complete metric space and (Ω, ∑) be a measurable space. Let 2^𝑋 be a family of all subsets of X and CB(X) denote the family of all non-empty bounded closed subsets of 𝑋.

A mapping𝑇: Ω× 𝑋 →2^𝑋 is called measurable, if for any open subset C of 𝑋 𝑇⁻¹(𝐶) = {𝜔 𝜖 Ω: 𝑇(𝜔) ∩ 𝐶 ≠ 𝜙} 𝜖 a A mapping 𝜉:Ω⟶ 𝑋 is said to be e measurable selector of measurable mapping T: Ω→2^𝑋 if 𝜉 is measurable and if for any 𝜖 𝑋 , 𝑓(. , 𝑋) is measurable.

A mapping T: Ω× 𝑋 → 𝐶𝐵(𝑋) is called random multi-valued operator if for every 𝑥 𝜖 𝑋 𝑇(. , 𝑋) is measurable.

A measurable mapping 𝜉:Ω⟶ 𝑋 is called random fixed point of a random multi- valued operator 𝑇: Ω× 𝑋 → 𝐶𝐵(𝑋)(𝑓:Ω× 𝑋 → 𝑋 ),𝜔 𝜖 Ω,

𝜉(𝜔) 𝜖 𝑇(𝜔, 𝜉(𝜔))( 𝑓(𝜔, 𝜉(𝜔) =𝜉(𝜔)) Let T: Ω× 𝑋 → 𝐶𝐵(𝑋) be a random operator and {𝜉_𝑛} a sequence of measurable mappings

𝜉_𝑛:Ω⟶ 𝑋 . The sequence {𝜉𝑛} is said to be asymptotically T- regular if

𝑑(𝜉_𝑛(𝜔), 𝑇(𝜔,𝜉_𝑛(𝜔)) → 0

Theorem 1: Let X be a Polish Space. Let 𝐴, 𝐵, 𝑆 : Ω× 𝑋 → 𝐶𝐵(𝑋) be any three

continuous random multivalued

operators. If there exist measurable mappings 𝛼, 𝛽:Ω→ (0,1) such that

𝐴𝐵 = 𝐵𝐴, 𝑆𝐴 = 𝐴𝑆 𝑎𝑛𝑑 𝐵(𝑋)𝐴(𝑋) 𝑎𝑛𝑑 𝑆(𝑋) ⊂ 𝐴(𝑋) And 𝑑(𝐵(𝜔, 𝑥), 𝑆(𝜔, 𝑦)) ≤ 𝛼(𝜔) (1)

⦋𝑑(𝐴(𝜔,𝑦),𝑆(𝜔,𝑥))⦌² 𝑑(𝐴(𝜔,𝑥),𝐵(𝜔,𝑦))+𝑑(𝐴(𝜔,𝑥),𝑆(𝜔,𝑦))+ 𝛽(𝜔) (𝑑(𝐴(𝜔, 𝑥), 𝐴(𝑤, 𝑦))

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VIII, October 2021

19 For each, 𝑦 є 𝑋 , 𝜔 є Ω and α, β є 𝑅⁺ with

α+ β < 1, there exist a unique random fixed point of A, B and S.

Proof: Let 𝜉₀ 𝑎𝑛𝑑 𝜉₁ be two measurable mappings such that 𝐵𝜉0(𝜔) = 𝐴𝜉1(𝜔) and let 𝜉2 be a measurable mappings such that 𝑆𝜉1(𝜔) = 𝐴𝜉₂(𝜔).

In general, we choose 𝜉2𝑛+1(𝜔) = 𝐴𝜉_2𝑛+2(𝜔) such that

𝐵𝜉_2𝑛(𝜔) = 𝐴𝜉_2𝑛+1(𝜔) 𝑎𝑛𝑑 𝑆𝜉_2𝑛+1(𝜔) = 𝐴𝜉_2𝑛+2(𝜔)

We can do this since equation (1) holds, using equations (2) and (3), we have 𝑑 (𝐴𝜉2𝑛+1(𝜔) , 𝐴𝜉2𝑛+2(𝜔) ) =

d(𝐵𝜉2𝑛(𝜔), 𝑆𝜉2𝑛+1(𝜔))

≤ 𝛼(𝜔)[𝑑(𝐴𝜉2𝑛+1(𝜔), 𝑆𝜉2𝑛+1(𝜔)]² 𝑑(𝐴𝜉_2𝑛(𝜔), 𝐵𝜉2𝑛+1(𝜔)) + 𝑑(𝐴𝜉_2𝑛(𝜔), 𝑆𝜉_2𝑛+1(𝜔))

+ 𝛽(𝜔)𝑑(𝐴𝜉_2𝑛(𝜔), 𝐵𝜉_2𝑛+1(𝜔)))

= 𝛼(𝜔)[𝑑(𝐴𝜉_2𝑛+1(𝜔), 𝐴𝜉_2𝑛+1(𝜔)]² 𝑑(𝐴𝜉_2𝑛(𝜔), 𝐴𝜉_2𝑛+1(𝜔)) +

𝑑(𝐴𝜉_2𝑛(𝜔), 𝐴𝜉_2𝑛+1(𝜔))

+ 𝛽(𝜔)𝑑(𝐴𝜉2𝑛(𝜔), 𝐴𝜉2𝑛+1(𝜔)) (3)

= 𝛼(𝜔)𝑑(𝐴𝜉_2𝑛+1(𝜔), 𝐴𝜉2𝑛+1(𝜔)) 𝑑(𝐴𝜉_2𝑛(𝜔), 𝐵𝜉_2𝑛+1(𝜔))

+ 𝛽(𝜔)𝑑(𝐴𝜉_2𝑛(𝜔), 𝐴𝜉_2𝑛+1(𝜔))

𝑑(𝐴𝜉2𝑛+1(𝜔),𝐴𝜉2𝑛+2(𝜔) ≤ 𝛼(𝜔) 𝑑(𝐴𝜉_2𝑛+1(𝜔), 𝐴𝜉_2𝑛+1(𝜔))+

𝛽(𝜔)𝑑(𝐴𝜉_2𝑛(𝜔), 𝐴𝜉_2𝑛+1(𝜔))

(1 − 𝛼(𝜔)) 𝑑 (𝐴𝜉2𝑛+1(𝜔), 𝐴𝜉_2𝑛+2(𝜔) ) ≤ 𝛽(𝜔)𝑑(𝐴𝜉_2𝑛(𝜔), 𝐴𝜉_2𝑛+1(𝜔))

𝑑(𝐴𝜉2𝑛+1(𝜔),𝐴𝜉2𝑛+2(𝜔) ) ≤

𝛽(𝜔)

(1−𝛼(𝜔)) 𝑑(𝐴𝜉_2𝑛(𝜔), 𝐴𝜉_2𝑛+1(𝜔)) 𝑑(𝐴𝜉2𝑛+1(𝜔),𝐴𝜉2𝑛+2(𝜔)) ≤ 𝑘𝑑(𝐴𝜉_2𝑛(𝜔), 𝐴𝜉_2𝑛+1(𝜔))

𝐵(𝐴𝜉_2𝑛(𝜔)) → 𝐵𝜉(𝜔), 𝑆(𝐴𝜉2𝑛+1(𝜔)) → 𝑆𝜉(𝜔) 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝐵(𝐴𝜉2𝑛(𝜔)) = 𝐴(𝐵𝜉2𝑛(𝜔)), 𝑆(𝐴𝜉_2𝑛+1(𝜔)) = 𝐵(𝑆𝜉_2𝑛+1(𝜔)) (4)

𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 = 0,1,2,3 … …

𝑇𝑎𝑘𝑖𝑛𝑔 𝑛 →∞ 𝑤𝑒 ℎ𝑎𝑣𝑒

𝐵𝜉(𝜔) = 𝑆𝜉(𝜔) = 𝐴𝜉(𝜔) = 𝜉(𝜔) (5) 𝐴(𝐴𝜉(𝜔)) = 𝐴(𝐵𝜉(𝜔)) = 𝐵(𝐴𝜉(𝜔)) =

𝐴(𝑆𝜉(𝜔)) = 𝑆(𝐵𝜉(𝜔)) = 𝑆(𝑆𝜉(𝜔)) = 𝑆(𝐴𝜉(𝜔)) (6) Therefore we have

𝑑(𝐵𝜉(𝜔), 𝑆(𝐵𝜉(𝜔))

≤ 𝛼(𝜔) [𝑑(𝐴𝜉(𝜔), 𝑆(𝐵𝜉(𝜔))]²

𝑑(𝐴𝜉(𝜔), 𝐵𝜉(𝜔)) + 𝑑 (𝐴𝜉(𝜔), 𝑆(𝐵𝜉(𝜔)))

+ 𝛽(𝜔)𝑑(𝐴𝜉(𝜔), 𝐴𝜉(𝜔))

≤ 𝛼(𝜔) [𝑑(𝐴𝜉(𝜔), 𝐴𝜉(𝜔))]²

𝑑(𝐴𝜉(𝜔), 𝐴𝜉(𝜔)) + 𝑑(𝐴𝜉(𝜔), 𝐴𝜉(𝜔))

+ 𝛽(𝜔)𝑑(𝐴𝜉(𝜔), 𝐴𝜉(𝜔))

It implies that 𝑑(𝐴𝜉(𝜔), 𝑆(𝐵𝜉(𝜔)) = 0

And so 𝐵𝜉(𝜔) = 𝑆(𝐵𝜉(𝜔)) (7) Therefore 𝐵𝜉(𝜔) is a common fixed point

of A, B and S.

[From (6) and (7)]

3. UNIQUENESS

Let 𝜉1(𝜔) 𝑎𝑛𝑑 𝜉2(𝜔) be two common fixed

points of Ω × 𝑋such that 𝐵 𝜉₁(𝜔) = 𝑆 𝜉1(𝜔) = 𝐴 𝜉1(𝜔) = 𝜉1(𝜔)

And 𝐵𝜉₂(𝜔) = 𝑆𝜉₂(𝜔) = 𝐴𝜉₂(𝜔) = 𝜉₂(𝜔) (8) Then

𝑑( 𝜉₁(𝜔), 𝜉₂(𝜔)) = 𝑑(𝐵 𝜉1(𝜔), 𝑆𝜉₂(𝜔))

≤ 𝛼(𝜔) [𝑑(𝐴𝜉₂(𝜔), 𝑆𝜉2(𝜔))]²

𝑑(𝐴 𝜉₁(𝜔), 𝐵𝜉2(𝜔)) + 𝑑(𝐴 𝜉1(𝜔), 𝑆𝜉2(𝜔)) + 𝛽(𝜔)𝑑(𝐴 𝜉₁(𝜔), 𝐴 𝜉₂(𝜔))

= 𝛼(𝜔) [𝑑(𝜉₂(𝜔), 𝜉₂(𝜔))]²

𝑑(𝜉₁(𝜔), 𝜉₂(𝜔)) + 𝑑( 𝜉₁(𝜔), 𝜉₂(𝜔)) + 𝛽(𝜔)𝑑( 𝜉₁(𝜔), 𝜉2(𝜔))

= 𝛽(𝜔)𝑑( 𝜉1(𝜔), 𝜉2(𝜔)) i.e. 𝑑( 𝜉1(𝜔), 𝜉₂(𝜔)) ≤

𝛽(𝜔)𝑑( 𝜉₁(𝜔), 𝜉2(𝜔))

⟹ (1 − 𝛽(𝜔)) 𝑑( 𝜉₁(𝜔), 𝜉₂(𝜔)) ≤ 0 [𝑆𝑖𝑛𝑐𝑒(1 − 𝛽(𝜔)) ≠ 0]

⟹ 𝑑( 𝜉₁(𝜔), 𝜉₂(𝜔))=0

⟹ 𝜉1(𝜔) = 𝜉₂(𝜔)

Hence 𝜉1(𝜔) = 𝜉2(𝜔)

This proves the uniqueness of common fixed points.

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