424 This work is licensed under a Creative Commons Attribution 4.0 International License IHJPAS.37 (2) 2024 Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq PISSN: 1609-4042, EISSN: 2521-3407 Wisam Mohammed Mukhlif 1* , Sabah Hassan Malih2 and Shrooq bahjat Smeein3 1Department of Mathematics, College of Education for Pure Science (Ibn- ALHaitham), University of Baghdad, City Baghdad, Iraq. 2Department of Mathematics, College of Education for Pure Science (Ibn- ALHaitham), University of Baghdad, City Baghdad, Iraq. 3Information Department, Section Mathematics Sultanate of Oman. *Corresponding Author. Abstract In this paper, we introduce a new one-step iteration process in Banach space and prove the existence of a common random fixed point of three non-expansive multivalued random operators through strong and weak convergences of an iterative process. The necessary and sufficient condition for the convergence of a sequence of measurable functions to a random fixed point of non-expansive multivalued random operators in uniformly convex Banach spaces is also established. Our random iteration scheme includes new random multivalued iterations as special cases. The results obtained in this paper are an extension and refinement of previously known results. A new random iterative scheme for approximating random common fixed points of three random non-expansive multivalued random operators is defined and we have proved weak and strong convergence theorems in a uniformly convex Banach space. Keywords: common fixed points, random operators, one-step iteration, Banach spaces. 1. Introduction The random fixed point theories are a generalizations of the classical fixed point theories. In the 1950s the Probability School in Prague presented a study on random fixed point theory [1].On other hand,[2] obtain common random fixed point of two multivalued random operator .Recently, S. H. Khan et al.[3]introduce a new one-step iterative process to find the common random fixed point of two multivalued Non-expansive random operator. ,the nonlinear random systems have appeared in the literature (see [4-17]) . Let 𝛷 be a separable banach space . Let𝛢 is subset of 𝛷is called proximinal if βˆ€π‘’ ∈ 𝛷, βˆƒπ‘˜ ∈ 𝛢 βˆ‹ 𝑑(𝑒, π‘˜) = inf{‖𝑒 βˆ’ 𝑣‖ ∢ 𝑣 ∈ 𝛢 } = 𝑑(𝑒, 𝛢). We denote the set of all subsets bounded proximinal of𝛢 by 𝛱(𝛢) [18] . let 𝐢𝐡(𝛢) be be all closed boundedsubsets of 𝛢. And let His Hausdorff induced by the metric space 𝑑 of 𝛷, implies H(πœ› , Ξ²) = π‘šπ‘Žπ‘₯{𝑠𝑒𝑝 𝑑(𝑒, Ξ²)π‘’βˆˆπœ› , 𝑠𝑒𝑝 𝑑(𝑣, πœ› )π‘£βˆˆΞ² } for every πœ› , Ξ² ∈ 𝐢𝐡(𝛢). A multivalued random operator 𝐡: 𝛹 Γ— 𝛢 β†’ 𝛱(𝛢)is called contraction if there isπ‘˜(Ο±) ∈ [0,1) and Received: 13 May 2023 Accepted: 22 June 2023 Published: 20 April 2024 Common Fixed Points of Three Multivalued Nonexpansive Random Operators for One Step Iterative Scheme doi.org/10.30526/37.2.3486 https://creativecommons.org/licenses/by/4.0/ https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042 https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407 https://orcid.org/0009-0009-8635-2210 mailto:wassoalraqy@gmail.com https://orcid.org/0000-0002-1600-3631 mailto:sabah.h.m@ihcoedu.uobaghdad.edu.iq https://orcid.org/0009-0002-9351-4176 mailto:shrooq.bahjat@utas.edu.om IHJPAS.37 (2) 2024 425 for each Ο± ∈ Ξ¨ βˆ‹ H( 𝐡(Ο±, 𝑒), 𝐡(Ο±, 𝑣) ) ≀ π‘˜(Ο±)‖𝑒 βˆ’ 𝑣‖for all 𝑒, 𝑣 ∈ 𝛢 and 𝐡 is said to be nonexpansive random operator if for each Ο± ∈ Ξ¨ H( 𝐡(Ο±, 𝑒), 𝐡(Ο±, 𝑣)) ≀ ‖𝑒 βˆ’ 𝑣‖, a point Ο‘(Ο±) is called random fixed point of 𝐡 if Ο‘(Ο±) ∈ 𝐡(Ο‘(Ο±)). Study the fixed point of results paper [22-30] under multivalued non-expansive random operator. We will give some definition … Definition 1.1.[19] : A separable Banach space 𝛷 is said to satisfy opials condition if the sequence {𝑆𝑛} in 𝛷, 𝑆𝑛 β†’ 𝑒 implies that lim π‘›β†’βˆž 𝑠𝑒𝑝‖𝑆𝑛 βˆ’ 𝑒‖ < lim π‘›β†’βˆž 𝑠𝑒𝑝‖𝑆𝑛 βˆ’ 𝑣‖ for all 𝑣 ∈ 𝛷 , 𝑣 β‰  𝑒. Definition 1.2. Let Ξ₯ subset of 𝛷, and β„›, Ξ– , 𝐡 ∢ Ξ¨ Γ— Ξ₯ β†’ 𝛱(Ξ₯)Three multivalued nonexpansive random operator are satisfy the condition (𝐴′) if βˆƒa non-decreasing function πœ’: [0, ∞) β†’ [0, ∞) with πœ’(0) = 0, πœ’(π‘Ÿ) > 0, βˆ€π‘Ÿ ∈ (0, ∞) βˆ‹ either 𝑑(𝑒(𝜚), 𝐡 𝑒(𝜚)) β‰₯ πœ’(𝑑(𝑒(𝜚), 𝐹))or 𝑑(𝑒(𝜚), Ζ𝑒(𝜚)) β‰₯ πœ’(𝑑(𝑒(𝜚), 𝐹))or 𝑑(𝑒(𝜚), R𝑒(𝜚)) β‰₯ πœ’(𝑑(𝑒(𝜚), 𝐹)) For all 𝑒(𝜚) ∈ Ξ₯ Definition 1.3 let 𝛢 be a non empty subset of a separable banach space 𝛷satisfying opials condition and 𝐡 ∢ Ξ¨ Γ— 𝛢 β†’ 𝛱(𝛢) be amultivalued random operator is said to be demiclosed at 𝜁(Ο±)) if {𝑆𝑛(𝜚)} and { ΢𝑛(𝜚)} are two sequences βˆ‹ {𝑆𝑛(𝜚)} converges weak to 𝑆(𝜚)also {𝐡((Ο±), 𝑆𝑛(𝜚)} converges to 𝜁(𝜚) imply that 𝑆(𝜚) ∈ 𝛢 and 𝜁(𝜚) πœ– 𝐡(𝑆(𝜚)) for each Ο± ∈ 𝛹, then 𝐼 βˆ’ 𝐡 is dimiclosed with respect to 0 . Lemma 1.4[20] Let {𝑆𝑛}, {πœ‚π‘›)} , {πœŒπ‘›} be a sequence in uniformly convex banach space 𝛷. Let {𝛼𝑛}, {πœ…π‘›} , {πœŽπ‘›} are sequence in [0,1] with 𝛼𝑛 + πœ…π‘› + πœŽπ‘› = 1 , lim π‘›β†’βˆž 𝑠𝑒𝑝‖𝑆𝑛‖ = π‘ž , lim π‘›β†’βˆž π‘ π‘’π‘β€–πœ‚π‘›β€– = π‘ž , lim π‘›β†’βˆž π‘ π‘’π‘β€–πœŒπ‘›β€– = π‘ž and lim π‘›β†’βˆž ‖𝛼𝑛𝑆𝑛 + πœ…π‘›πœ‚π‘› + πœŽπ‘›πœŒπ‘›β€– = π‘ž if lim π‘›β†’βˆž inf 𝛼𝑛 > 0, lim π‘›β†’βˆž inf πœ…π‘› > 0, lim π‘›β†’βˆž inf πœŽπ‘› > 0, Then lim π‘›β†’βˆž ‖𝑆𝑛 βˆ’ πœ‚π‘›β€– = lim π‘›β†’βˆž ‖𝑆𝑛 βˆ’ πœŒπ‘›β€– = lim π‘›β†’βˆž β€–πœŒπ‘› βˆ’ πœ‚π‘›β€– = 0. 2. Main Results IDefinition2.1 Let 𝛷 be a separable Banach space and Ξ₯ β‰  βˆ… closed subset and convex .we can be written 𝐹 = 𝐹 (𝐡) ∩ 𝐹(β„›) ∩ 𝐹( Ξ–) the set of all common random fixed point of the β„›, Ξ– , 𝐡 ∢ Ξ¨ Γ— Ξ₯ β†’ 𝛱(Ξ₯) be three multivalued non-expansive random operator with common random fixed point Ο‘(Ο±) . The iterations are as follows :- 𝑆0(𝜚)πœ–π›Ά 𝑆𝑛+1(𝜚) = π›Όπ‘›πœ‚π‘›(Ο±) + πœ…π‘›πœŒπ‘›(𝜚) + πœŽπ‘›πœ‰π‘›(𝜚) , 𝑛 ∈ 𝑁 (2.1) where πœ‚π‘›(𝜚) ∈ 𝐡(𝑆𝑛(Ο±)) and πœŒπ‘›(𝜚) ∈ Ξ–(𝑆𝑛(Ο±)) such that β€–πœ‚π‘›(𝜚) βˆ’ πœ‚π‘›+1(𝜚)β€– ≀ Ξ— (𝐡(𝑆𝑛(Ο±)), 𝐡(𝑆𝑛+1(Ο±))) + πœ‡π‘›and β€–πœŒπ‘›(𝜚) βˆ’ πœŒπ‘›+1(𝜚)β€– ≀ Ξ— (Ξ–(𝑆𝑛(Ο±)), Ξ–(𝑆𝑛+1(Ο±))) + πœ‡π‘› and πœ‰π‘›(𝜚) ∈ 𝑅(𝑆𝑛(Ο±))β€–πœ‰π‘›(𝜚) βˆ’ πœ‰π‘›+1(𝜚)β€– ≀ Ξ— (β„›(𝑆𝑛(Ο±)), β„›(𝑆𝑛+1(Ο±))) + πœ‡π‘› and {𝛼𝑛}, {πœ…π‘›}, { πœŽπ‘›} are sequence in (0,1) satisfying 𝛼𝑛 + πœ…π‘› + πœŽπ‘› = 1 . Lemma 2.2 Let 𝛷,Ξ₯ β‰  βˆ… ,and β„›, Ξ– , 𝐡 ∢ Ξ¨ Γ— Ξ₯ β†’ 𝛱(Ξ₯), Let {𝑆𝑛(𝜚)}issequence defined in 2.1. If 𝐹(𝐡) β‰  βˆ… and 𝐡ϑ(Ο±) = β„›Ο‘(Ο±) = Ξ–Ο‘(Ο±) = {Ο‘(Ο±)} for any Ο‘(Ο±) ∈ 𝐹 then π‘™π‘–π‘š π‘›β†’βˆž β€Šβˆ₯βˆ₯π‘₯𝑛 βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ exists βˆ€Ο‘(Ο±) ∈ 𝐹. IHJPAS.37 (2) 2024 426 Proof. let𝐹 β‰  βˆ…. Let Ο‘(Ο±) ∈ 𝐹 . Then βˆ₯βˆ₯𝑆𝑛+1(𝜚) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ = βˆ₯βˆ₯π›Όπ‘›πœ‚π‘›(𝜚) + πœ…π‘›πœŒπ‘›(𝜚) + πœŽπ‘›πœ‰π‘›(Ο±) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ = βˆ₯βˆ₯𝛼𝑛(πœ‚π‘›(𝜚) βˆ’ Ο‘(Ο±)) + πœ…π‘›(πœŒπ‘›(𝜚) βˆ’ Ο‘(Ο±)) + πœŽπ‘›(πœ‰π‘›(Ο±) βˆ’ Ο‘(Ο±))βˆ₯βˆ₯ β©½ 𝛼𝑛βˆ₯βˆ₯πœ‚π‘›(𝜚) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ + πœ…π‘›βˆ₯βˆ₯πœŒπ‘›(𝜚) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ + πœŽπ‘›βˆ₯βˆ₯πœ‰π‘›(Ο±) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ β©½ 𝛼𝑛𝑑 (πœ‚π‘›(𝜚), 𝐡(Ο‘(Ο±))) + πœ…π‘›π‘‘ (πœŒπ‘›(𝜚), Ξ–(Ο‘(Ο±))) + πœŽπ‘›π‘‘ (πœ‰π‘›(Ο±), 𝑅(Ο‘(Ο±))) β©½ 𝛼𝑛𝐻(𝐡𝑆𝑛(𝜚), 𝐡ϑ(Ο±)) + πœ…π‘›π» (Ξ– (𝑆𝑛(𝜚)), Ξ–(Ο‘(Ο±))) +πœŽπ‘›π» (𝑅(𝑆𝑛(𝜚)), 𝑅(Ο‘(Ο±))) β©½ 𝛼𝑛βˆ₯βˆ₯𝑆𝑛(𝜚) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ + πœ…π‘›βˆ₯βˆ₯𝑆𝑛(𝜚) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ + πœŽπ‘›βˆ₯βˆ₯𝑆𝑛(𝜚) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ = βˆ₯βˆ₯𝑆𝑛(𝜚) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯. (2.2) Thus π‘™π‘–π‘š π‘›β†’βˆž β€Šβˆ₯βˆ₯𝑆𝑛(𝜚) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ exists for all Ο‘(Ο±) ∈ 𝐹 . Definition 2.3 Letβ„›, Ξ– , 𝐡 ∢ Ξ¨ Γ— Ξ₯ β†’ 𝛱(Ξ₯) are said to satisfy condition ( 𝐢′) if 𝑑 ((𝜚, 𝑒), 𝑣(𝜚)) ≀ 𝑑(𝑀(𝜚), 𝑣(𝜚)) for 𝑣(𝜚) ∈ Ξ–(𝜚, 𝑒), 𝑀(𝜚) ∈ β„›(𝜚, 𝑒). Lemma2.4.Let 𝛷and Ξ₯andLet β„›, Ξ– , 𝐡 ∢ Ξ¨ Γ— Ξ₯ β†’ 𝛱(Ξ₯)satisfying conditionn(𝐢′ )and {𝑆𝑛(𝜚)} be sequence defined in (2.1). If 𝐹 β‰  βˆ… and 𝐡ϑ(Ο±) = β„›Ο‘(Ο±) = Ξ–Ο‘(Ο±) = {Ο‘(Ο±)}for any Ο‘(Ο±) ∈ 𝐹 then π‘™π‘–π‘š πš€β†’βˆž β€Šπ‘‘ (𝑆𝑛(𝜚), 𝐡(𝑆𝑛(𝜚))) = π‘™π‘–π‘š π‘›β†’βˆž β€Šπ‘‘ (𝑆𝑛(𝜚), Ξ–(𝑆𝑛(𝜚))) = 0 = π‘™π‘–π‘š π‘›β†’βˆž β€Šπ‘‘ (𝑆𝑛(𝜚), β„›(𝑆𝑛(𝜚))) Proof. By Lemma (2.2), π‘™π‘–π‘š π‘›β†’βˆž β€Šβˆ₯βˆ₯𝑆𝑛(𝜚) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯exists. We suppose that π‘™π‘–π‘š π‘›β†’βˆž β€Šβˆ₯βˆ₯𝑆𝑛(𝜚) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ = 𝑐 for 𝑐 β‰₯ 0. Also have𝐡, S, β„› arenonexpansive random operator and 𝐹 β‰  βˆ…, we haveβˆ₯βˆ₯πœ‚π‘›(𝜚) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ = 𝑑 (πœ‚π‘›(𝜚), Ξ–(Ο‘(Ο±))) β©½ 𝐻 (𝐡(𝑆𝑛(𝜚)), 𝐡(Ο‘(Ο±))) β©½ βˆ₯βˆ₯𝑆𝑛(𝜚) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯. Take limsup for both side, get π‘™π‘–π‘šβ€†π‘ π‘’π‘ π‘›β†’βˆž β€Šβˆ₯βˆ₯πœ‚π‘›(𝜚) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ β©½ 𝑐 Similarly, π‘™π‘–π‘šβ€†π‘ π‘’π‘ π‘›β†’βˆž β€Šβˆ₯βˆ₯πœŒπ‘›(𝜚) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ β©½ 𝑐 and , π‘™π‘–π‘šβ€†π‘ π‘’π‘ π‘›β†’βˆž β€Šβˆ₯βˆ₯πœ‰π‘›(Ο±) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ β©½ 𝑐 as, π‘™π‘–π‘š π‘›β†’βˆž β€Šβˆ₯βˆ₯𝑆𝑛+1(𝜚) βˆ’ Ο‘(Ο±)βˆ₯βˆ₯ = 𝑐 That mean π‘™π‘–π‘š π‘›β†’βˆž β€Šβˆ₯βˆ₯𝛼𝑛(πœ‚π‘›(𝜚) βˆ’ Ο‘(Ο±)) + πœ…π‘›(πœŒπ‘›(𝜚) βˆ’ Ο‘(Ο±)) + πœŽπ‘›(πœ‰π‘›(Ο±) βˆ’ Ο‘(Ο±))βˆ₯βˆ₯ = 𝑐 Applying lemma (1.4), we get lim π‘›β†’βˆž β€–πœ‚π‘›(𝜚) βˆ’ πœŒπ‘›(𝜚)β€– = 0 . But from the condition (𝐢′) we obtain that 𝑑(𝑆𝑛(𝜚), πœ‚π‘›(𝜚)) ≀ 𝑑(πœŒπ‘›(𝜚), πœ‚π‘›(𝜚)), lim sup π‘›β†’βˆž 𝑑(𝑆𝑛(𝜚), πœ‚π‘›(𝜚)) ≀ 0 . That is, IHJPAS.37 (2) 2024 427 lim π‘›β†’βˆž ‖𝑆𝑛(𝜚) βˆ’ πœ‚π‘›(𝜚)β€– = 0 . (2.3) Also from lemma (1.4) and (2.2) we obtain ‖𝑆𝑛(𝜚) βˆ’ πœŒπ‘›(𝜚)β€– ≀ ‖𝑆𝑛(𝜚) βˆ’ πœ‚π‘›(𝜚)β€– + β€–πœ‚π‘›(𝜚) βˆ’ πœŒπ‘›(𝜚)β€– implies that lim π‘›β†’βˆž ‖𝑆𝑛(𝜚) βˆ’ πœŒπ‘›(𝜚)β€– = 0 . (2.4) Also from lemma (1.4) and (2.3) we have ‖𝑆𝑛(𝜚) βˆ’ πœ‰π‘›(Ο±)β€– ≀ ‖𝑆𝑛(𝜚) βˆ’ πœŒπ‘›(𝜚)β€– + β€–πœŒπ‘›(𝜚) βˆ’ πœ‰π‘›(Ο±)β€– implies that lim π‘›β†’βˆž ‖𝑆𝑛(𝜚) βˆ’ πœ‰π‘›(Ο±)β€– = 0 . (2.5) Now, we get 𝑑 (𝑆𝑛(𝜚), 𝐡(𝑆𝑛(𝜚))) ≀ 𝑑(𝑆𝑛(𝜚), πœ‚π‘›(𝜚)), also, 𝑑 (𝑆𝑛(𝜚), Ξ–(𝑆𝑛(𝜚))) ≀ 𝑑(𝑆𝑛(𝜚), πœŒπ‘›(𝜚)), and 𝑑 (𝑆𝑛(𝜚), β„›(𝑆𝑛(𝜚))) ≀ 𝑑(𝑆𝑛(𝜚), πœ‰π‘›(Ο±)), we gives 𝑑 (𝑆𝑛(𝜚), 𝐡(𝑆𝑛(𝜚))) β†’ 0, 𝑑 (𝑆𝑛(𝜚), Ξ–(𝑆𝑛(𝜚))) β†’ 0 π‘Žπ‘›π‘‘ 𝑑 (𝑆𝑛(𝜚), β„›(𝑆𝑛(𝜚))) β†’ 0 as n β†’ ∞. Theorem 2.5. Let Ξ₯ and let𝛷 satisfying the Opial’scondition. and β„›, Ξ– , 𝐡 ∢ Ξ¨ Γ— Ξ₯ β†’ 𝛱(Ξ₯).If 𝐹 β‰  βˆ… and 𝐡ϑ(Ο±) = β„›Ο‘(Ο±) = Ξ–Ο‘(Ο±) = {Ο‘(Ο±)}for any Ο‘(Ο±) ∈ 𝐹, 𝐼 βˆ’ 𝐡 , 𝐼 βˆ’ Ξ–and 𝐼 βˆ’ β„›are demi-closed to 0, then {𝑆𝑛(𝜚)} converges to a commonrandom fixed point of 𝐡, S, R and weakly. Proof:- Let Ο‘(Ο±) ∈ 𝐹. by lemma (2.2), we have lim π‘›β†’βˆž ‖𝑆𝑛(𝜚) βˆ’ Ο‘(Ο±)β€–exists. we prove that {𝑆𝑛(𝜚)}subsequentialhave a weak unique limit in F . now, let πœ”1(Ο±), πœ”2(Ο±)π‘Žπ‘›π‘‘ πœ”3(Ο±) be weak limits of the subsequences {𝑆𝑛𝑖(𝜚)}, {𝑆𝑛𝑗(𝜚)} π‘Žπ‘›π‘‘ {𝑆𝑛𝑙(𝜚)} π‘œπ‘“ {𝑆𝑛(𝜚)}, respect. By uselemma (2.4), there isπœ‚π‘›(𝜚) ∈ 𝐡(𝑆𝑛(𝜚))such that lim π‘›β†’βˆž ‖𝑆𝑛(𝜚) βˆ’ Ο‘(Ο±)β€– = 0 and 𝐼 βˆ’ 𝐡is demi-closedto 0 , therefore we getπœ”1(Ο±) ∈ π΅πœ”1(Ο±). Similarly, πœ”1(Ο±) ∈ Ξ–πœ”1(Ο±) π‘Žπ‘›π‘‘ πœ”1(Ο±) ∈ β„›πœ”1(Ο±).Again in a same way, we can prove that πœ”1(Ο±), πœ”2(Ο±), πœ”3(Ο±) ∈ 𝐹. now, we need to prove unique. For this letπœ”1(Ο±) β‰  πœ”2(Ο±) β‰  πœ”3(Ο±). Then by the Opial’s condition, lim π‘›β†’βˆž ‖𝑆𝑛(𝜚) βˆ’ πœ”1(Ο±)β€– = lim π‘›π‘–β†’βˆž ‖𝑆𝑛𝑖(𝜚) βˆ’ πœ”1(Ο±)β€– < lim π‘›π‘–β†’βˆž ‖𝑆𝑛𝑖(𝜚) βˆ’ πœ”2(Ο±)β€– = lim π‘›β†’βˆž ‖𝑆𝑛(𝜚) βˆ’ πœ”2(Ο±)β€– = lim π‘›π‘—β†’βˆž ‖𝑆𝑛𝑗(𝜚) βˆ’ πœ”2(Ο±)β€– < lim π‘›π‘—β†’βˆž ‖𝑆𝑛𝑗(𝜚) βˆ’ πœ”3(Ο±)β€– = lim π‘›β†’βˆž ‖𝑆𝑛(𝜚) βˆ’ πœ”3(Ο±)β€– IHJPAS.37 (2) 2024 428 = lim π‘›π‘™β†’βˆž ‖𝑆𝑛𝑙(𝜚) βˆ’ πœ”3(Ο±)β€– < lim π‘›π‘™β†’βˆž ‖𝑆𝑛𝑙(𝜚) βˆ’ πœ”1(Ο±)β€– = lim π‘›β†’βˆž ‖𝑆𝑛(𝜚) βˆ’ πœ”1(Ο±)β€– Hence , its a contradiction. Hence {𝑆𝑛(𝜚)}converges and weakly in 𝐹. Remark 2.6Let Ξ₯ and let𝛷 satisfying the Opial’scondition and ,β„›, Ξ– , 𝐡 ∢ Ξ¨ Γ— Ξ₯ β†’ 𝛱(Ξ₯) and {𝑆𝑛(𝜚)}be the sequence in (2.1),. If 𝐹 β‰  βˆ… and 𝐡ϑ(Ο±) = β„›Ο‘(Ο±) = Ξ–Ο‘(Ο±) = {Ο‘(Ο±)}then {𝑆𝑛(𝜚)}converges to a common-random fixed point of 𝐡, Ξ–,β„› and weakly. Theorem 2.7. Let 𝛷,Ξ₯, and {𝑆𝑛(𝜚)} and𝐡, Ξ–,,β„› defined in the Lema(2.4). and 𝐹 β‰  βˆ…and𝐡ϑ(Ο±) = β„›Ο‘(Ο±) = Ξ–Ο‘(Ο±) = {Ο‘(Ο±)} for any Ο‘(Ο±) ∈ 𝐹, then {𝑆𝑛(𝜚)} converges to a commonrandom fixed point of 𝐡, β„›, Ξ–andstronglyiff lim π‘›β†’βˆž 𝑖𝑛𝑓 𝑑(𝑆𝑛(𝜚) , 𝐹) = 0 Proof. The first direction of the proof is clear. Conversely, let lim π‘›β†’βˆž 𝑖𝑛𝑓 𝑑(𝑆𝑛(𝜚) , 𝐹) = 0. by lemma(2.2), ‖𝑆𝑛+1(𝜚) βˆ’ Ο‘(Ο±)β€– ≀ ‖𝑆𝑛(𝜚) βˆ’ Ο‘(Ο±)β€–. This gives 𝑑(𝑆𝑛+1(𝜚), 𝐹 ) ≀ 𝑑(𝑆𝑛(𝜚), 𝐹 ), so that lim π‘›β†’βˆž 𝑖𝑛𝑓 𝑑(𝑆𝑛(𝜚) , 𝐹)exists. But, by use hypothesis, lim π‘›β†’βˆž 𝑖𝑛𝑓 𝑑(𝑆𝑛(𝜚) , 𝐹) = 0. Therefore we must have lim π‘›β†’βˆž 𝑖𝑛𝑓 𝑑(𝑆𝑛(𝜚) , 𝐹) = 0. we need toprove that {𝑆𝑛(𝜚)}is a Cauchy sequence in Ξ₯. suppose Ξ΅ >0 . we have lim π‘›β†’βˆž 𝑑(𝑆𝑛(𝜚) , 𝐹) = 0 , thereis a constant 𝑛0 such that βˆ€π‘› β‰₯ 𝑛0, we have lim π‘›β†’βˆž 𝑖𝑛𝑓 𝑑(𝑆𝑛(𝜚) , 𝐹) < πœ– 4 In particular, inf{‖𝑆𝑛0(𝜚) βˆ’ Ο‘(Ο±)β€– ∢ Ο‘(Ο±) ∈ 𝐹 } < πœ– 4 . There must exist a Ο‘(Ο±)βˆ— ∈ 𝐹such that ‖𝑆𝑛0(𝜚) βˆ’ Ο‘(Ο±)βˆ—β€– < πœ– 2 Now for π‘š, 𝑛 β‰₯ 𝑛0, we have ‖𝑆𝑛+π‘š(𝜚) βˆ’ 𝑆𝑛(𝜚)β€– ≀ ‖𝑆𝑛+π‘š(𝜚) βˆ’ Ο‘(Ο±)βˆ—β€– + ‖𝑆𝑛(𝜚) βˆ’ Ο‘(Ο±)βˆ—β€– ≀ 2‖𝑆𝑛0(𝜚) βˆ’ Ο‘(Ο±)βˆ—β€– < 2 ( πœ– 2 ) = πœ– Hence {𝑆𝑛(𝜚)}is a Cauchy sequence inΞ₯of 𝛷, and therefore it must converge in Ξ₯. Let lim π‘›β†’βˆž 𝑆𝑛(𝜚) = π‘ž(Ο±) Now 𝑑(π‘ž(Ο±), π΅π‘ž(Ο±)) ≀ 𝑑(π‘ž(Ο±), 𝑆𝑛(𝜚)) + 𝑑(𝑆𝑛(𝜚), 𝐡𝑆𝑛(𝜚)) + 𝐻(𝐡𝑆𝑛(𝜚) , π΅π‘ž(Ο±)) ≀ 𝑑(π‘ž(Ο±), 𝑆𝑛(𝜚)) + 𝑑(𝑆𝑛(𝜚), πœ‚π‘›(𝜚)) + 𝑑(𝑆𝑛(𝜚), π‘ž(Ο±)) β†’ 0 π‘Žπ‘  𝑛 β†’ ∞, gives that 𝑑(π‘ž(Ο±), π΅π‘ž(Ο±)) = 0 which implies that π‘ž(Ο±) ∈ π΅π‘ž(Ο±). Similarly, 𝑑(π‘ž(Ο±), π‘†π‘ž(Ο±)) ≀ 𝑑(π‘ž(Ο±), 𝑆𝑛(𝜚)) + 𝑑(𝑆𝑛(𝜚), Ζ𝑆𝑛(𝜚)) + 𝐻(Ζ𝑆𝑛(𝜚), Ξ–π‘ž(Ο±)) ≀ 𝑑(π‘ž(Ο±), 𝑆𝑛(𝜚)) + 𝑑(𝑆𝑛(𝜚), πœŒπ‘›(𝜚)) + 𝑑(𝑆𝑛(𝜚), π‘ž(Ο±)) β†’ 0 π‘Žπ‘  𝑛 β†’ ∞, IHJPAS.37 (2) 2024 429 gives that 𝑑(π‘ž(Ο±), Ξ–π‘ž(Ο±)) = 0 which implies that π‘ž(Ο±) ∈ Ξ–π‘ž(Ο±).Similarly, 𝑑(π‘ž(Ο±), π‘…π‘ž(Ο±)) ≀ 𝑑(π‘ž(Ο±), 𝑆𝑛(𝜚)) + 𝑑 (𝑆𝑛(𝜚), ℛ𝑆𝑛(𝜚) ) + 𝐻(ℛ𝑆𝑛(𝜚), β„›π‘ž(Ο±)) ≀ 𝑑(π‘ž(Ο±), 𝑆𝑛(𝜚)) + 𝑑(𝑆𝑛(𝜚), πœŒπ‘›(𝜚)) + 𝑑(𝑆𝑛(𝜚), π‘ž(Ο±)) β†’ 0 π‘Žπ‘  𝑛 β†’ ∞, implies thatπ‘ž(Ο±) ∈ β„›π‘ž(Ο±). Consequently, π‘ž(Ο±) ∈ 𝐹. Nowwe can use the conditional (𝐴`) to find the strong converge of {𝑆𝑛(𝜚)}define in(2.1)., we assume that β„›, Ξ– , 𝐡 ∢ Ξ¨ Γ— Ξ₯ β†’ 𝛱(Ξ₯)satisfycondition (𝐴`). Theorem 2.8. Let Ξ₯, 𝛷 and,the sequence{𝑆𝑛(𝜚)} beas in Lemma (2.4). Let β„›, Ξ– , 𝐡 ∢ Ξ¨ Γ— Ξ₯ β†’ 𝛱(Ξ₯)satisfying condition (𝐴`). If 𝐹 β‰  βˆ… and 𝐡ϑ(Ο±) = β„›Ο‘(Ο±) = Ξ–Ο‘(Ο±) = {Ο‘(Ο±)} for any Ο‘(Ο±) ∈ 𝐹 then {𝑆𝑛(𝜚)} converges to a common- fixed point of 𝐡, Ξ–,β„› and strongly. Proof :since By used lemma(2.4), have lim π‘›β†’βˆž ‖𝑆𝑛(𝜚) βˆ’ 𝐹‖exists βˆ€Ο‘(Ο±) ∈ 𝐹. let 𝑐 β‰₯ 0. If 𝑐 = 0, clear that. let𝑐 > 0. Now ‖𝑆𝑛+1(𝜚) βˆ’ Ο‘(Ο±)β€– ≀ ‖𝑆𝑛(𝜚) βˆ’ Ο‘(Ο±)β€–gives inf Ο‘(Ο±)∈𝐹 ‖𝑆𝑛+1(𝜚) βˆ’ Ο‘(Ο±)β€– ≀ inf Ο‘(Ο±)∈𝐹 ‖𝑆𝑛(𝜚) βˆ’ Ο‘(Ο±)β€–which implies that 𝑑(𝑆𝑛+1(𝜚), 𝐹 ) ≀ 𝑑(𝑆𝑛(𝜚), 𝐹 ) π‘Žπ‘›π‘‘ so lim π‘›β†’βˆž 𝑑(𝑆𝑛(𝜚) , 𝐹) exists. By condition (𝐴`)either lim π‘›β†’βˆž πœ’(𝑑(𝑆𝑛(𝜚) , 𝐹)) ≀ lim π‘›β†’βˆž 𝑑(𝑆𝑛(𝜚) , 𝐡𝑆𝑛(𝜚)) = 0 or lim π‘›β†’βˆž πœ’(𝑑(𝑆𝑛(𝜚) , 𝐹)) ≀ lim π‘›β†’βˆž 𝑑(𝑆𝑛(𝜚) , Ζ𝑆𝑛(𝜚)) = 0 or lim π‘›β†’βˆž πœ’(𝑑(𝑆𝑛(𝜚) , 𝐹)) ≀ lim π‘›β†’βˆž 𝑑(𝑆𝑛(𝜚) , 𝑅𝑆𝑛(𝜚)) = 0 In both the cases, lim π‘›β†’βˆž πœ’(𝑑(𝑆𝑛(𝜚) , 𝐹)) = 0 Since πœ’ is a non-decreasing function whereπœ’(0) = 0, lim π‘›β†’βˆž 𝑑(𝑆𝑛(𝜚), 𝐹) = 0 . 5. 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