Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 80 https://internationalpubls.com Approximation of Fixed Points via Picard-Abbas Hybrid Iteration Scheme for -Quasi-Nonexpansive Multivalued Mappings 1Manbhalang Chyne, 2,* Naveen Kumar 1, 2 Chandigarh University, Gharuan, Mohali, Punjab, India Article History: Received: 15-10-2024 Revised: 29-11-2024 Accepted: 10-12-2024 Abstract: We present results on stability and convergence for the Picard-Abbas iteration scheme for ρ-quasi-nonexpansive multivalued mappings within modular function spaces. Furthermore, we demonstrate an application of the iteration scheme in differential equations. Keywords: Picard-Abbas hybrid iteration; convergence; stability; quasi-nonexpansive mappings; modular function space. 1. Introduction Nakano [28] generalized the ordered spaces theory to modular spaces. Later, Musielak and Orlicz [26] further extended and generalized this theory. In modular spaces, it was Khamsi, Kozlowski, and Reich [9] who first studied the theory of fixed points. But, it was Khan and Abbas [10] who first studied the fixed point approximation for -nonexpansive multivalued mappings in modular function spaces, utilizing Mann iteration scheme. Later, by utilizing a three-step iteration scheme, Khan et al. [11] gave fixed point approximation results for -quasi-nonexpansive multivalued mappings. Okeke et al. [29] also provided significant results for these mappings, by utilizing the hybrid Picard - Krasnoselski iteration scheme. These spaces have rich structural properties, and are equipped with modular equivalents of metric and norm concepts. The modular type conditions, offer a more intuitive and verifiable framework compared to the traditional norm-based assumptions, thus contributing to their increasing popularity. This paper purposes to further the existing work by presenting new results on convergence and stability for -quasi-nonexpansive mappings within modular function spaces, utilizing the Picard Abbas-type hybrid iterative approach. In doing so, we contribute to the on-going discourse in fixed point theory, enhancing our understanding of the interplay between modular structures and iterative methods. 2. Preliminaries Consider a nontrivial -algebra on , and a of subsets of , with , . Let be an increasing sequence of sets with . Let be the space of all extended measurable functions. Let be the vector space of simple functions with support being contained within and be the characteristic function of in . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 81 https://internationalpubls.com We define , For simplicity, we write instead of . Definition 2.1. [12] A functional of a vector space ( or ) is a modular if for arbitrary , the statements below hold: (i) (ii) whenever (iii) whenever . If we replace (iii) by (iv) whenever . then the modular is convex. Definition 2.2. [12] The following set is called a modular function space, for a convex modular in : In general, is not sub-additive. We can equip the with the following F-norm: . If is a convex, then the norm is called the Luxemburg Norm on the modular space . Definition 2.3. [12] The nontrivial, even and convex function is called a regular convex function pseudomodular if (i) ; (ii) for any implies , where . Thai is, is monotone. (iii) for any such that , . That is, is orthogonally sub-additive. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 82 https://internationalpubls.com (iv) for all implies , where . That is, has Fatou property. (v) , and implies . That is is order continuous in . A set is if . A property is -almost everywhere if is . Definition 2.4. [12] A regular function pseudomodular is called a regular convex function modular if Let be the class of all non-zero regular convex function modular on . For , define and . Note that . Definition 2.5. [11] A satisfy -condition if as whenever decreases to and . holds true if is convex and satisfies -condition. Let . Let . Define . Let if and if . Definition 2.6. [8] A satisfy (UC1) if , we have . Note that for small enough. Definition 2.7. [8] A satisfy (UUC1) if , depending only upon and such that for any . Definition 2.8. [12] Let and . (i) is -convergent to if . (ii) is -Cauchy, if . (iii) is -closed if for , implies . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 83 https://internationalpubls.com (iv) is -compact if for , there is a subsequence of and such that as . (v) is closed if for which is convergent, as implies . (vi) is compact if for , a subsequence of and exist such that as . (vii) is -bounded if the -diameter of is finite, that is . -convergence does not imply -Cauchy. If satisfy the -condition, then this will be true. The -distance from to is defined by . Definition 2.9. [10] A set is called -proximinal if , such that . Let be the family of non-empty, -bounded, -proximinal subsets of and be the family of non-empty, -closed, -bounded subsets of . We define -Hausdorff distance on as . Definition 2.10. [10] A multivalued mapping is (i) -nonexpansive if . (ii) -quasi-nonexpansive if . Theorem 2.11. [6] Let satisfy the -condition. Consider the sequences and in Then implies and implies Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 84 https://internationalpubls.com A sequence is bounded away from if such that , and is bounded away from if such that . Lemma 2.12. [3, 8] Let (UUC1) be satisfied by , and be bounded away from 0 and 1. If such that , and , then . Definition 2.13. is a fixed point of if . Let be the set of all fixed points of . Definition 2.14. [11] A multivalued mapping satisfy if there exists a continuous non-decreasing function with such that . Lemma 2.15. [12] For a multivalued mapping with , the following statements are equivalent: (i) , i.e., . (ii) , i.e, for each . (iii) , i.e, . Also, , with being the set of fixed points of . Definition 2.16. [8] A set is said to possess the Vitali property if , and for any and with , there exists a subsequence of such that for every the subadditive measures are order equicontinuous. Definition 2.17. [8] The function modular is called separable if is a separable set function for each , which means that there exists a countable such that to every there corresponds a sequence of elements of with for every , where denotes the symmetric difference. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 85 https://internationalpubls.com 3. Convergence Analysis The Picard-Abbas iteration for singlevalued mappings was introduced by Chyne and Kumar [11]. We define the Picard-Abbas iteration for multivalued mapping as follows: where are real sequences in , and . Theorem 3.1. Let -condition and (UUC1) be satisfied by . Let be -bounded, - closed, and convex in . Consider a multivalued mapping with being a - quasi-nonexpansive mapping, and . If be defined by (3.1), then exists for all . Proof. Let . Using Lemma 2.15, we get Using (3.5) in (3.4), we get Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 86 https://internationalpubls.com Using (3.5) and (3.6) in (3.3), we get From (3.2) and (3.7), we get Therefore, the sequence is decreasing. Thus, exists for all . Theorem 3.2. Let -condition and (UUC1) be satisfied by . Let be -bounded, - closed, and convex in . Consider a multivalued mapping with being a - quasi-nonexpansive mapping, and . If be defined by (3.1, then . Proof. By Theorem 3.1, exists for all . Let Since , it suffices to show that . Now implies So, (3.7) gives Also, from (3.5), we get So, Similarly, we can show that and Now Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 87 https://internationalpubls.com So, Similarly, we can show that Again Therefore, (3.12), (3,13), (3.14) and Lemma 2.12 gives Let be given. There exists such that for all . Since , we have . Also, From Theorem 2.11, we get . So, From (3.12) and (3.16), we get Using (3.15) and Theorem 2.11, we get . But . Therefore, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 88 https://internationalpubls.com From (3.9) and (3.18) we get That is, . From (3.7), (3.8) and Lemma 2.12, we get . Hence, . Theorem 3.3. Let -condition and (UUC1) be satisfied by . Let be -bounded, - closed, and convex in . Consider a multivalued mapping with being a - quasi-nonexpansive mapping, and . If be defined by (3.1), then is - convergent to a fixed point of . Proof. By being compact, a subsequence of exists, such that for some . We show that . Let be arbitrary chosen from and from . Now, Using theorems (3.1) and (3.2), we get Thus, . That is, is -convergent to a fixed point of . Theorem 3.4. Let -condition and (UUC1) be satisfied by . Let be -bounded, - closed, and convex in . Consider a multivalued mapping satisfying Condition (I) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 89 https://internationalpubls.com with being a -quasi-nonexpansive mapping, and . If be defined by (3.1), then is -convergent to a fixed point of . Proof. We have shown in Theorem 3.1 that exists for all . If , there is nothing to prove. We assume . Again, from Theorem 3.1, we have . So, . Hence, exists. We show that . Using Theorem 3.1 and Condition (I), we get . That is, . Since is nondecreasing function and , we have Next, we show that is a -Cauchy sequence in . Let . Since , there is a constant such that , we have . In particular, . The must exist such that . For , we have . Therefore, is a -Cauchy in . Thus, it is convergent in . Let . Using Theorem 3.3, we get . Hence, is -convergent to a fixed point of . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 90 https://internationalpubls.com 4. Stability Analysis Here we give a stability result for the Picard-Abbas iteration. We begin by stating the stability definition as follows. Definition 4.1. [29] Let ( ) and operator . For a fixed , let be an iteration generating a sequence . Let be strongly convergent to . Let the sequence be bounded in and . (i) is -stable on if (ii) is almost -stable on if . Theorem 4.2. Let ( ) be convex and bounded. If be a multivalued mapping with being a -quasi-nonexpansive mapping, and , then the iteration (3.1) is -stable. Proof. Let . Define . Let be unique. Suppose . Using (3.1) and the convexity of , we have Thus, . Conversely, let . By (4.1), we get . Therefore, if and only if . Hence the proof. 5. Applications to differential equations Let . For an unknown function , with , consider the initial value problem (IVP) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 91 https://internationalpubls.com for fixed , and with being -quasi-nonexpansive mapping. For any define For a function , , , define Let us denote for any subdivision of . Lemma 5.1. [8] Consider a separable . Let be two Bochner-integrable -bounded functions, with . Then , we have We now prove our Theorem. Theorem 5.2. Consider a separable . Let be non-empty, -closed, -bounded, convex set having Vitali property, and a multivalued mapping with being a - quasi-nonexpansive mapping. For fixed , , we define by Then such that and defined by (5.7) is a solution of the IVP (5.1). Moreover, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 92 https://internationalpubls.com Proof. The proof follows that of ([15], Theorem 5.28), since being a -quasi-nonexpansive mapping. Corollary 5.3. Consider a separable . Let be non-empty, -closed, -bounded, convex set having Vitali property, and a multivalued mapping with being a - nonexpansive mapping. For fixed , , we define by Then such that and defined by (5.10) is a solution of the IVP (5.1). Moreover, Corollary 5.4. Consider a separable . Let be non-empty, -closed, -bounded, convex set having Vitali property, and a multivalued mapping with being a - contraction mapping. For fixed , , we define by Then such that and defined by (5.13) is a solution of the IVP (5.1). Moreover, . References [1] Abbas M., and Nazir T. (2014), A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesnik 66(2), 223–234. [2] Abbas M., and Rhoades B. E. (2009), Fixed point theorems for two new classes of multivalued mappings, Appl. Math. Lett. 22(9), 1364–1368. 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