https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 181 Altering Point Results involving -Class Functions in Partial Metric Spaces Abdessalem Benterki LMP2M Laboratory, Department of Mathematics and Computer Science, University of Medea, Medea, Algeria Email: benterki.abdessalem@gmail.com Article History: Received: 28-01-2024 Revised: 15-04-2024 Accepted: 26-04-2024 Abstract: The purpose of this paper is to explore the existence of altering points satisfying the system of equations using the notion of -class functions. Here, and are two 0-complete partial metric spaces, and and are set-valued mappings. The key idea is to use the properties of -class functions to establish the existence of a sequence of points that converge to the altering point, thereby demonstrating the existence of such points in the first place. By analyzing the properties of and and their interaction with -class functions, we can gain insight into the nature of the altering points and the underlying spaces. The findings of this study provide a generalization and extension of various results in the existing literature, highlighting the significance of the proposed approach. Keywords: Altering point, partial metric space, set-valued mapping, -class function 2010 Mathematics Subject Classification. 54H25, 47H04. 1. Introduction Since 1922, when the celebrated Banach contraction principle was introduced, fixed point theory has fascinated many researchers as one of the most dynamic areas. Till now, it has been the fastest growing branch of mathematics, with several applications to real-world problems. There is a vast literature on fixed point theory, and it is currently a very active field of research. As a tool for proving the existence and uniqueness of solutions to different mathematical problems, fixed point theorems are essential in many theoretical and applied branches of mathematics, including optimisation problems, dynamical systems, nonlinear analysis, integral equations, partial differential equations, variational inequalities, fractals, economics, game theory, and nonlinear analysis. In many distinct abstract spaces, diverse results about fixed points, common fixed points, coincidence points, altering points, and so on for single-valued and set-valued https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 182 mappings have been investigated for various contractive conditions, and this tradition continues. Partial metric spaces are an extension of conventional metric spaces where the value of self-distance for each point does not have to be zero. Matthews first discussed partial metric spaces in [5] . Romaguera presented the concept of a 0-Cauchy sequence in a partial metric space, followed by the concept of a 0-complete partial metric space in [8]. Sahu first proposes the idea of altering points in [9] for the single-valued case in order to introduce a parallel S-iteration process to solve a system of operator equations. These points are more general than the fixed points, and they are extended by Petrusel, Yao in [7] for the set-valued case. For more information on altering points and applications to variational problems one can refer to [9,10]. Nedelcheva [6] presented one of the altering point results and extended the fixed point result mentioned in [3] for the objective of composing two set-valued mappings on complete metric spaces. The following is how this work is organized: Section 2 presents some preliminary observations and definitions. Section 3 establishes the primary conclusion by extending both Theorem 11 in [6] and Theorem 3.4 in [2] for a composition of two set-valued mappings using the idea of -class functions. Finally, we will discuss some relevant corollaries. 2. Preliminary results Let us begin by providing an overview of some of the key definitions and characteristics of partial metric spaces. A partial metric on a non-empty set is defined as a function that satisfies the following conditions for all : If satisfies the above conditions, then the pair is referred to as a partial metric space. This definition was introduced by Matthews [5]. The closed -ball with radius and as its center is denoted by and opened -ball by , where represents for convenience. The function provided by https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 183 is a metric on if is a partial metric on . Consider to be a partial metric space. Then: • If , then is said to converge to a point . • If there exists (and is finite) , a sequence is called a Cauchy sequence. is referred to as a 0-Cauchy sequence if . • In , every Cauchy sequence that converges to a point in such a way that indicates that is complete. • Regarding , every 0-Cauchy sequence in converges to a point where . A 0-complete is what this is called. • In , any 0-Cauchy sequence is a Cauchy in . • Complete partial metric spaces are all 0-complete. However, the reverse is not true. In the partial metric space , the represents the family of all closed and nonempty subsets. With and , we establish such that in conformity with the convention The following concepts will be used throughout the article. Let denotes a set-valued mapping defined in the partial metric space with closed values in the partial metric space . We define the composition of mappings and as the mapping such that for all , . The sets and in the following text represent intervals on that contain , like , , or . We also introduce , a set-valued mapping defined on for . We denote as Lemma 2.1. [2] Let be a partial metric space and let . Then, the following statements hold: 1. . 2. . https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 184 Given the partial metric , the closure of is represented here as . It is worth mentioning that is closed in if and only if . Lemma 2.2. [3] Consider and in a partial metric space . If and we can find that there is an element such that . Definition 1. [4] A Bianchini-Grandolfi gauge function on is defined as a nondecreasing function that satisfies the following condition: where denotes the -th iteration of the function and . In other words, is defined recursively as , , , and so on The associated estimate function is known as the sum (2.3), and it was observed that fulfils the functional equation shown below In [6], the author has defined as the set of pairs of increasing functions and that map from to , satisfying the following three conditions: Then the theorem mentioned in [6] reads as follows: Theorem 2.1. [6] Let and be complete partial metric spaces, and let and . Let be a constant, and let F and be two set-valued mappings such that . Let and be increasing functions such that , where is a collection of pairs of increasing functions from to , subject to the following three conditions. Assume there exists such that the following assumptions hold: Then there exist and such that and i.e., is an altering point of and . If and are single valued mappings, and , then is the unique fixed point of in , and is the unique fixed point of in . https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 185 Remark 1. The theorem mentioned above is satisfied without using the assumption . See the proof of [6, Theorem 11] for further details. For , , and where the identity function defined on , we have Corollary 2.2. Theorem 3.2 in [3]. A notion of -class functions was presented by A.H. Ansari in [1]. Definition 2. ( -class functions) [1,2] Assume that there is a continuous mapping . If satisfies these requirements, we will classify it as a -class function. and denotes the set of all -class functions on . Example 1. Here are some examples of -class functions with and , where : 1. This function satisfies and , and it is known as the harmonic mean. It is used, for example, to calculate the average rate of speed of two objects moving at different constant speeds. 2. . This function satisfies and , and it is known as the normalized power mean of order 1. It is used, for example, in statistics to calculate the average of a set of non-negative numbers. In [2], the authors presented the following collections of -class functions: Definition 3. [2] The set of functions of the -class that satisfy these criteria is called : • is non-decreasing for both and when '. • For every that are fixed, the series converges for all . The function is defined as follows, and represents the -th iteration of this function: . https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 186 Definition 4. [2] comprises a set of -class functions that adhere to the following specifications: exhibits non-decreasing behavior in and non-increasing behavior in . • For any given , the series converges for every , where the -th iteration of the function with the following recurrence relation is represented as : For more examples and properties on and one can refer to [2]. We recall a class of functions that were mentioned in [2]. One important need is satisfied by these functions, denoted as . Being more explicit, means that or is true for every and . Furthermore, we prove that is nondecreasing in the space, as proved by the following inequality: So, the Theorem [2, Theorem 3.4], which extends Theorem [3,Theorem 3.2], is as follows: Theorem 2.3. [2] Assuming is a partial metric space with and r>0 such that is a 0-complete subspace of . Let be a set- valued mapping. Consider , and satisfying one of the following conditions: • and is nondecreasing. • and for . We suppose the following two conditions are met: hence, in , there is a fixed point in . In the set , the unique fixed point of is if is a single-valued mapping and . 3. Main results We need to review certain concepts and establish some definitions before we can present our major finding. Definition 5. (Altering points). [7, 9] Let and be two partial metric spaces and let , be two set-valued mappings. The couple is called altering point of and if https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 187 There are some special cases that can be deduced from this definition, which can be summarized in the following table According to Theorem 2.3 and definitions 3, 4, and 5, we will establish new theorems with respect to set-valued mappings on 0-complete partial metric spaces that extend and generalize Theorem 2.1 for altering points of two set-valued mappings and generalized Theorem 2.3 [2,Theorem 3.4] and [3,Theorem 3.2]{benterki2016} for the fixed points of a two-set-valued mapping composition. At beginning, we set the family of pairs of an increasing function and an increasing function in first variable satisfying the following three assumptions: Remark 2. If such that then the subset will be the subset . As a consequence, we express and demonstrate our main finding as follows Theorem 3.1. Let and be partial metric spaces with and as 0- complete subspaces of and , respectively, where , , and . Consider set- valued mappings and such that . Let , on , and which satisfies at least one of the conditions: Given these conditions, we are going to suppose that the following is true: https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 188 Then there exist an altering point of and in . If and are both single-valued mappings and , then is the unique fixed point of in and is the unique fixed point of in . Proof. The proof is complete if . So we assume that . By assumption (a), we have Now, using Lemma 2.2, there exists such that Then . Now, denoting and . the proof is complete if . So we assume that and by using (b) we have with However, we also have If and , then wich implies that or . https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 189 Thus, or which is a contradiction. So, we assume that and . Then, by using Lemma 2.2, relation (3.1) and non empty of and , there exists and such that and Consequently, This means that . By induction we construct two sequences and satisfying: where Assuming that , and for all , we can conclude that if or for some , then we are finished. Otherwise, we have and . First, we demonstrate that the sequence and fulfilling (3.2) provides Indeed, let then we have https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 190 Similarly, Firstly, For the other inequalities, we will handle the following cases: Case 1: If and . We have This implies that which give or . So, we get a contradiction. Case 2: If and . We have https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 191 This implies that which give or . So, we get a contradiction. Case 3: If and . Then we get a contradiction with the decreasing of the sequences and . In the second place, we realised that and and then the inequality (3.3). Now we prove the second step of induction (3.2). By using assumption (c), inequality (3.3) and the nondecreasing of in first variable, we have If we assume that for some or , then we have which implies that or and then or which is a contradiction. So we assume that for all and , then there exists such that Moreover, if and is nondecreasing then we have else if and https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 192 On the other hand, we have If we assume that for some then we get a contradiction with the definition of and . Then we suppose that and then there exists such that And with the inequality (3.3), we have . On the other hand, is an element of the opened -ball and be an element of the opened -ball . Indeed, and We have for all integers and where https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 193 We know that is a 0-Cauchy sequence in since converges for each . As is a 0-complete subspace, converges to with respect to . This implies that Analogously, for a sequence we have Given that is convergent for each and is a 0-complete subspace, we can conclude that is a 0-Cauchy sequence in , and thus converges with respect to to a point . We assert now that . The modified triangle inequality of , is a -class function and assumption (c) give By taking the limit as approaches infinity, we get . Applying Lemma 2.1, we conclude that Analogously, we now claim that . This follows from the modified triangle inequality and assumption (b), which imply that Since , taking limit as , we obtain https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 194 This, according to Lemma 2.1, implies Assuming single-valued mappings and and , let and be two different altering points in for and , where , , , and . Then, we have: and So, by using , we get then we have which implies that or , thus or which is a contradiction and then there exist a unique altering points such that and . By consequence, we get, 4. Corollaries Related As corollaries, we have an extended version of [6,Theorem 11] (i.e. Theorem 2.1), [7, Theorem 6.3] and [9, Theorem 3.1] within the context of partial metric spaces that are 0- complete. Corollary 4.1. Let and be partial metric spaces where and are 0-complete subspaces of and respectively, for , , and . Consider set- valued mappings and such that . Let be increasing functions and . Assume the following: https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 195 Then there exist an altering point of and in . If and are both single-valued mappings and , then is the unique fixed point of in and is the unique fixed point of in . Proof. Take such that and then the subset will be the subset . Since the function does not depend on the second variable , we can choose to be non-decreasing or greater than , and then apply the Theorem 3.1. Using Theorem 3.1 with , , , and , we can establish the following fixed point theorem: Corollary 4.2. Theorem 3.4 in [2]. For and where and applying corollary 4.1 then we get Corollary 4.3. Let and ) be partial metric spaces such that and are 0-complete subspaces of and respectively, for , and . Let and are set-valued mappings such that . Suppose that the following assumptions hold: Then there exist an altering point of and in . If and are both single-valued mappings and , then is the unique fixed point of in and is the unique fixed point of in . Proof. For and the corresponding estimate function , we select the Bianchini-Grandolfi gauge function in accordance with Corollary 4.1. Then we set and apply Corollary 4.1. If , then , and we get the extended version of [7, Theorem 6.3] as follows Corollary 4.4. Let and be 0-complete partial metric spaces. Let . Let and be set-valued mappings. Suppose the following assumptions hold: https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 196 Then there exist an altering point of and in . If and are both single-valued mappings, then is the unique fixed point of in and is the unique fixed point of in . Proof. Given and such that , let be chosen such that Then, we can apply Corollary 4.3. References [1] A. H. Ansari, Note on φ − ψ-contractive type mappings and related fixed point, In: The 2nd regional conference on mathematics and applications, PNU (Vol.377380)(2014). [2] A. H. Ansari and A. Benterki and M. Rouaki, Some local fixed point results under -class functions with applications to coupled elliptic systems, Journal of Linear and Topological Algebra (JLTA), 7(3), (2018), 169-182. [3] A. Benterki, A local fixed point theorem for set-valued mappings on partial metric spaces, Applied General Topology, 17(1), (2016), 37-49. [4] R. M. Bianchini and M. Grandolfi, Transformazioni di tipo contracttivo generalizzato in uno spazio metrico, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat, 45, (1968), 212-216. [5] S. G. Matthews, Partial metric topology, Annals of the New York Academy of Sciences-Paper Edition, 728, (1994), 183-197. [6] D. K. Nedelcheva, Altering Points in Partial Metric Space, In: Proceedings of the Twenty-First International Conference on Geometry, Integrability and Quantization, (2020), 221-231. [7] A. Petrusel and G. Petrusel and J. C. Yao, Multi-valued graph contraction principle with applications, Optimization, 69 (7-8), (2020), 1541-1556. [8] S. Romaguera and A. kirk, type characterization of completeness for partial metric spaces, Fixed Point Theory Appl., 2010, (2009), Article ID 493298, 6 pages. [9] D. Sahu, Altering Points and Applications, Nonlinear Stud., 21, (2014), 349–365. [10] J. C. Yong and S. Kumar and S. M. K. Convergence analysis of parallel S-iteration process for a system of variational inequalities using altering points, J. Appl. Math. & Informatics 36(5-6), (2018), 381-396.