Int. J. Anal. Appl. (2023), 21:74 Received: May 12, 2023. 2020 Mathematics Subject Classification. 47H10, 54H25. Key words and phrases. common fixed point; coincident point; graphical structures; neutrosophic metric space; unique solution; non-linear fractional differential equations. https://doi.org/10.28924/2291-8639-21-2023-74 © 2023 the author(s) ISSN: 2291-8639 1 Certain Fixed-Point Results via DS-Weak Commutativity Condition in Neutrosophic Metric Spaces With Application to Non-linear Fractional Differential Equations Umar Ishtiaq1,*, Muhammad Saeed2, Khaleel Ahmad2, Ilsa Shokat3, Manuel De la Sen4 1Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore, Pakistan 2Department of Mathematics, University of Management and Technology, Lahore, Pakistan 3Department of Mathematics, Bahauddin Zakariya University Multan, Sub Campus, Vehari, Pakistan 4Department of Electricity and Electronics, Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa, 48940 Leioa, Bizkaia, Spain *Corresponding author: umarishtiaq000@gmail.com ABSTRACT. This study demonstrates that, for the non-linear contractive conditions in Neutrosophic metric spaces, a common fixed-point theorem may be proved without requiring the continuity of any mappings. A novel commutativity requirement for mappings weaker than the compatibility of mappings is used to demonstrate the conclusion. We provide several examples to illustrate our major idea. Also, we provide an application to the non-linear fractional differential equation to show the validity of our main result. 1. INTRODUCTION AND PRELIMAINARIES There is still one significant deficiency despite the development of the computer industry and its incredible achievement in changing several fields of research. Computers are not intended to process phrases with uncertainty. Logic must be developed to handle uncertainty using methods other than the classical ones. One approach to uncertainty is fuzzy set theory, where topological structures are https://doi.org/10.28924/2291-8639-21-2023-74 2 Int. J. Anal. Appl. (2023), 21:74 fundamental building blocks for creating mathematical models that are applicable to real-world scenarios. El Naschie's [1] established a relation between topology, algebra and geometry, he showed that the approach of dimension is based on topological structure. El Naschie [2] established a pure mathematical derivation of the structure of real spacetime from quantum set theory that is exceedingly straightforward and simple to understand. This is done by combining the von Neumann- Connes dimensional function of the Klein-Penrose modular holographic boundary of the E8E8 exceptional Lie group bulk of our universe with components of the Menger-Urysohn dimensional theory and the topological theory of cobordism. The end result of a paper is a clear, concise mental image: quantum spacetime is just the border or surface of the quantum wave empty set, while the quantum wave itself is an empty set that represents the surface, or boundary, of the zero-set quantum particle. The fundamental distinction between quantum spacetime and a quantum wave is that the latter is a multi-fractal type of infinitely many empty sets with varying degrees of emptiness, whilst the former is a simple empty set. The notion of intuitionistic fuzzy metric spaces (IFMSs), which introduced and discussed by Park in [3], is useful in modeling. It is based on the notion of the intuitionistic fuzzy set (IFS) given by Atanssov [4] and the concept of fuzzy metric space (FMS) given by George and Veeramani [5]. Using the concept of IFSs, Alaca et al. [6] established the concept of FMS as it was introduced by Kramosil and Michalek [7]. Additionally, they developed the concept of Cauchy sequences in IFMS and used the notion introduced by Grabiec [8] to demonstrate the extension of the Banach contraction principle given in [9] to FMS and Edelstein [10] to IFMS. The common fixed-point theorem of Jungck [11] was extended to IFMS by Turkoglu et al. in [12]. The idea of IFMS and their applications were researched by Gregory et al. [13], Sadati and Park [14], and Rodriguez-Lopez and Ramagurea [15]. Sessa [16] developed the idea of weak commuting(w-commuting) mappings in metric spaces. Broader commutativity, or compatibility, was introduced by Jungck [11]. Mishra et al. [17] established the notion of suitable maps in the context of FMSs. Vasuki [18] established the notion of an R-weak commutativity in the context of FMSs. R-w- commuting mappings of type (A) in metric spaces were established by Pathak et al. [19], who demonstrated that these mappings are incompatible and that R-w-commuting mappings are not always R-w-commuting of type (A). R-w- commuting mappings on IFMS were defined by Jesic and Babacev [20]. Sharma and Deshpande [21] described (DS)-weak commutativity in FMSs and 3 Int. J. Anal. Appl. (2023), 21:74 established the notion of R-w-commuting mappings of type (A) in the context of FMSs. Recently, Boyd and Wong [22] and Jesic and Babacev [20] derived a common fixed-point theorem by utilizing nonlinear contractive conditions and assuming continuity for a pair of mappings on IFMS. The method of neutrosophic metric spaces (NMSs), which deals with membership, non-membership, and naturalness functions, was proposed by Kirişci and Simsek [23]. In the case of NMSs, Sowndrarajan, et al. [24] demonstrated some fixed-point results. Schweizer and Sklar worked on statistical metric spaces and Deshpande [26] derived several fixed-point results and under some interesting conditions in the context of an IFMS. In this manuscript, we will study some interesting circumstances in which continuity may be not necessary to find an existence and uniqueness of a solution. Also, we establish an application to non- linear fractional differential equations to show the validity of our main result. Now, we discuss some important notions that will be helpful for readers to understand main section. Definition 1.1: [25] A binary operation ∗:[0,1] → [0,1] is said to be continuous t-norm (CTN) if ∗ is fulfill the aforementioned requirements: (TN1) 𝜎 ∗ 𝜍 = 𝜍 ∗ 𝜎, (TN2) 𝜎 ∗ ( 𝜍 ∗ 𝜐) = (𝜎 ∗ 𝜍) ∗ 𝜐, (TN3) ∗ is continuous, (TN4) 𝜎 ∗ 1 = 1 for all 𝜎 ∈ [0,1], (TN5) 𝜎 ∗ 𝜍 ≤ 𝜐 ∗ 𝛥 whenever 𝜎 ≤ 𝜐 and 𝜍 ≤ 𝛥,for all 𝜎,𝜍,𝜐,𝛥 ∈ [0,1]. Definition 1.2: [25] A binary operation ♢:[0,1] × [0,1] → [0,1] is said to be continuous t-conorm (CTCN) if ♢ satisfies (TN1)-(TN3), (TN5) and the following condition: (TN4)* 𝜎♢0 = 𝜎 for all 𝜎 ∈ [0,1]. Definition 1.3: [6] A 5-tuple (𝛯,𝔅,𝔒,∗,♢) is called an IFMS if 𝛯 is an arbitrary set ∗ is a CTN, ♢ is CTCN and 𝔅,𝔒 are fuzzy sets on 𝛯2 × [0,∞) verifies the below conditions: (IFM1) 𝔅(𝜛,𝜔,𝜏) + 𝔒(𝜛,𝜔,𝜏) ≤ 1 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, (IFM2) 𝔅(𝜛,𝜔,0) = 0 for all 𝜛,𝜔 ∈ 𝛯, (IFM3) 𝔅(𝜛,𝜔,𝜏) = 1 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0 if and only if 𝜛 = 𝜔, (IFM4) 𝔅(𝜛,𝜔,𝜏) = 𝔅(𝜔,𝜛,𝜏) for all 𝜛,𝜔 ∈ 𝛯,𝜏 > 0, (IFM5) 𝔅(𝜛,𝜔,𝜏) ∗ 𝔅(𝜔,𝜇,𝑠) ≤ 𝔅(𝜛,𝜇,𝜏 + 𝑠) for all 𝜛,𝜔,𝜇 ∈ 𝛯 and 𝑠,𝜏 > 0, (IFM6) for all 𝜛,𝜔 ∈ 𝛯,𝔅(𝜛,𝜔, .): [0,∞) → (0,1] is left-continuous, 4 Int. J. Anal. Appl. (2023), 21:74 (IFM7) lim 𝜏→∞ 𝔅(𝜛,𝜔,𝜏) = 1 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, (IFM8) 𝔒(𝜛,𝜔,0) = 1 for all 𝜛,𝜔 ∈ 𝛯, (IFM9) 𝔒(𝜛,𝜔,𝜏) = 0 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0 iff 𝜛 = 𝜔, (IFM10) 𝔒(𝜛,𝜔,𝜏) = 𝔒 (𝜔,𝜛,𝜏)for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, (IFM11) 𝔒(𝜛,𝜔,𝜏)♢ 𝔒(𝜔,𝜇,𝑠) ≥ 𝔒(𝜛,𝜇,𝜏 + 𝑠) for all 𝜛,𝜔,𝜇 ∈ 𝛯 and 𝑠,𝜏 > 0, (IFM12) for all 𝜛,𝜔 ∈ 𝛯,𝔒(𝜛,𝜔, .) ∶ [0,∞) → [0,1] is a right-continuous, (IFM13) lim 𝜏→∞ 𝔒(𝜛,𝜔,𝜏) = 0 for all 𝜛,𝜔 ∈ 𝛯. Then (𝔅,𝔒) is called an intuitionistic fuzzy metric on 𝛯. Example 1.1: Suppose (𝛯,Δ) be a metric space. Define CTN by 𝜎 ∗ 𝜍 = min{𝜎,𝜍} and CTCN by 𝜎♢𝜍 = max{𝜎,𝜍} for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, 𝔅𝛥 (𝜛,𝜔,𝜏) = 𝜏 𝜏 + 𝛥(𝜛,𝜔) , 𝔒𝛥 (𝜛,𝜔,𝜏) = 𝛥(𝜛,𝜔) 𝜏 + 𝛥(𝜛,𝜔) . Then (𝛯,𝔅,𝔒,∗,♢) is an IFMS. Definition 1.4: [23] A 6-tuple (Ξ,𝔅,𝔒,𝔖,∗,♢) is called an NMS if 𝛯 is an arbitrary set, ∗ is a CTN, ♢ is CTCN and 𝔅,𝔒,𝔖 are neutrosophic sets on 𝛯2 × [0,∞) verifies the following conditions: (NMS1) 𝔅(𝜛,𝜔,𝜏) + 𝔒(𝜛,𝜔,𝜏) + 𝔖(𝜛,𝜔,𝜏) ≤ 3 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, (NMS2) 𝔅(𝜛,𝜔,0) = 0 for all 𝜛,𝜔 ∈ 𝛯, (NMS3) 𝔅(𝜛,𝜔,𝜏) = 1 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0 if and only if 𝜛 = 𝜔, (NMS4) 𝔅(𝜛,𝜔,𝜏) = 𝔅(𝜔,𝜛,𝜏) for all 𝜛,𝜔 ∈ 𝛯,𝜏 > 0, (NMS5) 𝔅(𝜛,𝜔,𝜏) ∗ 𝔅(𝜔,𝜇,𝑠) ≤ 𝔅(𝜛,𝜇,𝜏 + 𝑠) for all 𝜛,𝜔,𝜇 ∈ 𝛯 and 𝑠,𝜏 > 0, (NMS6) for all 𝜛,𝜔 ∈ 𝛯,𝔅(𝜛,𝜔, .):[0,∞) → (0,1] is left-continuous, (NMS7) lim 𝜏→∞ 𝔅(𝜛,𝜔,𝜏) = 1 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, (NMS8) 𝔒(𝜛,𝜔,0) = 1 for all 𝜛,𝜔 ∈ 𝛯, (NMS9) 𝔒(𝜛,𝜔,𝜏) = 0 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0 iff 𝜛 = 𝜔, (NMS10) 𝔒(𝜛,𝜔,𝜏) = 𝔒 (𝜔,𝜛,𝜏)for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, (NMS11) 𝔒(𝜛,𝜔,𝜏)♢ 𝔒(𝜔,𝜇,𝑠) ≥ 𝔒(𝜛,𝜇,𝜏 + 𝑠)for all 𝜛,𝜔,𝜇 ∈ 𝛯 and 𝑠,𝜏 > 0, (NMS12) for all 𝜛,𝜔 ∈ 𝛯,𝔒(𝜛,𝜔, . ) ∶ [0,∞) → [0,1] is a right-continuous, (NMS13) lim 𝜏→∞ 𝔒(𝜛,𝜔,𝜏) = 0 for all 𝜛,𝜔 ∈ 𝛯. (NMS14) 𝔖(𝜛,𝜔,0) = 1 for all 𝜛,𝜔 ∈ 𝛯, (NMS15) 𝔖(𝜛,𝜔,𝜏) = 0 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0 iff 𝜛 = 𝜔, (NMS16) 𝔖(𝜛,𝜔,𝜏) = 𝔖 (𝜔,𝜛,𝜏)for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, 5 Int. J. Anal. Appl. (2023), 21:74 (NMS17) 𝔖(𝜛,𝜔,𝜏)♢ 𝔖(𝜔,𝜇,𝑠) ≥ 𝔖(𝜛,𝜇,𝜏 + 𝑠) for all 𝜛,𝜔,𝜇 ∈ 𝛯 and 𝑠,𝜏 > 0, (NMS18) for all 𝜛,𝜔 ∈ 𝛯,𝔖(𝜛,𝜔, . ) ∶ [0,∞) → [0,1] is a right-continuous, (NMS19) lim 𝜏→∞ 𝔖(𝜛,𝜔,𝜏) = 0 for all 𝜛,𝜔 ∈ 𝛯. Then (𝔅,𝔒,𝔖) is called a neutrosophic metric on 𝛯. Definition 1.5: [23] A sequence {𝜛𝑛} in NMS (𝛯,𝔅,𝔒,𝔖,∗,♢) is called a convergent to a point 𝜛 ∈ 𝛯 if and only if lim 𝑛→∞ 𝔅(𝜛𝑛,𝜛,𝜏) = 1, lim 𝑛→∞ 𝔒(𝜛𝑛,𝜛,𝜏) = 0 and lim 𝑛→∞ 𝔖(𝜛𝑛,𝜛,𝜏) = 0 for each 𝜏 > 0. Definition 1.6: [23] A sequence in NMS (𝛯,𝔅,𝔒,𝔖,∗,♢) is called a Cauchy if for each > 0 and for each 𝜏 > 0,there exist 𝑛° ∈ ℕ such that 𝔅(𝜛𝑛,𝜛𝑚,𝜏) > 1 − ,𝔒(𝜛𝑛,𝜛𝑚,𝜏) < and 𝔖(𝜛𝑛,𝜛𝑚,𝜏) < for all 𝑛,𝑚 ≥ 𝑛𝑜, (𝛯,𝔅,𝔒,𝔖,∗,♢) is considered to be complete if and only if each Cauchy sequence is convergent. 2. MAIN RESULTS In this section, we show the existence and uniqueness of a fixed point by utilizing some interesting conditions with contractions. Lemma 2.1: If (𝛯,𝔅,𝔒,𝔖 ∗,♢) is an NMS and lim 𝑛→∞ 𝜛𝑛 = 𝜛, lim 𝑛→∞ 𝜔𝑛 = 𝜔 then lim 𝑛→∞ 𝔅(𝜛𝑛,𝜔𝑛,𝜏) = 𝔅(𝜛,𝜔,𝜏) , lim 𝑛→∞ 𝔒(𝜛𝑛,𝜔𝑛,𝜏) = 𝔒(𝜛,𝜔,𝜏), lim 𝑛→∞ 𝔖(𝜛𝑛,𝜔𝑛,𝜏) = 𝔖(𝜛,𝜔,𝜏). Definition 2.1: Let (𝛯,𝔅,𝔒,𝔖,∗,♢) be an NMS and 𝑄 ⊆ 𝛯. The smallest closed set that contains 𝑄 is known as closure of the set 𝑄 and is indicated by �̅�. Remark 2.1: An element 𝜛 ∈ �̅� if and only if there exist a sequence {𝜛𝑛} in 𝑄 such that 𝜛𝑛 → 𝜛. Definition 2.2: Let (𝛯,𝔅,𝔒,𝔖,∗,♢) be an NMS. Let for all 𝑟 ∈ (0,1) then a collection {𝐹𝑛}𝑛∈ℕ have neutrosophic diameter zero (ND-zero) if for each 𝜏 > 0 there exists 𝑛𝑜 ∈ ℕ such that 𝔅(𝜛,𝜔,𝜏) > 1 − 𝑟, 𝔒(𝜛,𝜔,𝜏) < 𝑟 and 𝔖(𝜛,𝜔,𝜏) < 𝑟 for all 𝜛,𝜔 ∈ 𝐹𝑛𝑜. Theorem 2.1: An NMS (𝛯,𝔅,𝔒,𝔖,∗,♢) is said to be complete if and only if each nested sequence {𝐹𝑛}𝑛∈ℕ of non-empty closed sets with (ND-zero) have non-empty intersection. Remark 2.2: An element 𝜛 is unique if 𝜛 ∈∩𝑛∈ℕ 𝐹𝑛. Definition 2.3: Let (𝛯,𝔅,𝔒,𝔖,∗,♢) be an NMS. A subset 𝑄 of 𝛯 is called a neutrosophic bounded if there exists 𝜏 > 0 and 𝑟 ∈ (0,1) such that 𝔅(𝜛,𝜔,𝜏) > 1 − 𝑟, 𝔒(𝜛,𝜔,𝜏) < 𝑟 and 𝔖(𝜛,𝜔,𝜏) < 𝑟 for all 𝜛,𝜔 ∈ 𝑄. 6 Int. J. Anal. Appl. (2023), 21:74 Definition 2.4: Let (𝛯,𝔅,𝔒,𝔖,∗,♢) be an NMS. Let the mapping 𝛿𝑄(𝜏) ∶ (0,∞) → [0,1], 𝜌𝑄(𝜏):(0,∞) → [0,1] and 𝛾𝑄(𝜏):(0,∞) → [0,1] be define as 𝛿𝑄 (𝜏) = inf 𝜛,𝜔∈𝑄 sup 𝜀<𝜏 𝔅(𝜛,𝜔, ) , 𝜌𝑄(𝜏) = sup 𝜛,𝜔∈𝑄 inf 𝜀<𝜏 𝔒(𝜛,𝜔, ) , 𝛾𝑄(𝜏) = sup 𝜛,𝜔∈𝑄 inf 𝜀<𝜏 𝔖(𝜛,𝜔, ) . The constants 𝛿𝑄 = sup𝜏>0𝛿𝑄(𝜏),𝜌𝑄 = inf𝜏>0𝜌𝑄(𝜏) and 𝛾𝑄 = inf𝜏>0𝛾𝑄(𝜏), we will call neutrosophic diameter of nearness, non-nearness and naturalness of set 𝑄. Remark 2.3: The inequalities 𝛿𝑄 ≥ 1 − 𝑟,𝜌𝑄 ≤ 𝑟 and 𝛾𝑄 ≤ 𝑟 are fulfilled, if 𝑄 is a neutrosophic bounded set. Definition 2.5: The set 𝑄 is called neutrosophic strongly bounded set (NSBS), if 𝛿𝑄 = 1, 𝜌𝑄 = 0 and 𝛾𝑄 = 0. Lemma 2.2: A mapping 𝜙:(0,∞) → (0,∞) is continuous, and non-decreasing, if it verifies 𝜙(𝜏) < 𝜏 for all 𝜏 > 0. Then lim 𝑛→∞ 𝜙𝑛 (𝜏) = 0, for all 𝜏 > 0, where 𝜙𝑛(𝜏) shows the nth iteration of 𝜙. Lemma 2.3: Suppose (𝛯,𝔅,𝔒,𝔖,∗,♢) be an NMS. A mapping 𝜙:(0,∞) → (0,∞) be a continuous, non-decreasing, if it verifies the condition 𝜙(𝜏) < 𝜏 for all 𝜏 > 0. Then the below circumstances are satisfied: (a) If for all 𝜛,𝜔 ∈ 𝛯 it satisfies that 𝔅(𝜛,𝜔,𝜙(𝜏)) ≥ 𝔅(𝜛,𝜔,𝜏) then 𝜛 = 𝜔. (b) If for all 𝜛,𝜔 ∈ 𝛯 it holds that 𝔒(𝜛,𝜔,𝜙(𝜏)) ≤ 𝔒(𝜛,𝜔,𝜏) then 𝜛 = 𝜔. (c) If for all 𝜛,𝜔 ∈ 𝛯 it holds that 𝔖(𝜛,𝜔,𝜙(𝜏)) ≤ 𝔖(𝜛,𝜔,𝜏) then 𝜛 = 𝜔. Definition 2.6: Let 𝑄 and 𝑈 are the self-mappings of 𝛯 in NMS (𝛯,𝔅,𝔒,𝔖,∗,♢). These maps are said to be compatible if for all 𝜏 > 0, lim 𝑛→∞ 𝔅(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = 1, lim 𝑛→∞ 𝔒(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = 0, lim 𝑛→∞ 𝔖(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = 0, whenever {𝜛𝑛} is a sequence in 𝛯 such that, lim 𝑛→∞ 𝑄𝜛𝑛 = lim 𝑛→∞ 𝑈𝜛𝑛 = 𝜇 for some 𝜇 ∈ 𝛯. Definition 2.7: Let 𝑄 and 𝑈 are the self-mappings of 𝛯 in NMS (𝛯,𝔅,𝔒,𝔖,∗,♢). If there exist a positive real number 𝑅 then the mappings are called (𝐷𝑆𝑄)-w-commuting at 𝜛 ∈ 𝛯 such that 𝔅(𝑄𝑈𝜛,𝑈𝑈𝜛,𝜏) ≥ 𝔅(𝑄𝜛,𝑈𝜛, 𝜏 𝑅 ), 𝔒(𝑄𝑈𝜛,𝑈𝑈𝜛,𝜏) ≤ 𝔒(𝑄𝜛,𝑈𝜛, 𝜏 𝑅 ), 7 Int. J. Anal. Appl. (2023), 21:74 𝔖(𝑄𝑈𝜛,𝑈𝑈𝜛,𝜏) ≤ 𝔖(𝑄𝜛,𝑈𝜛, 𝜏 𝑅 ). Then, 𝑄 and 𝑈 are (𝐷𝑆𝑄)-w-commuting on 𝛯 for all 𝜛 ∈ 𝛯. Definition 2.8: Suppose (𝛯,𝔅,𝔒,𝔖,∗,♢) be an NMS and 𝑄 and 𝑈 self-mappings of 𝛯. If there exists a positive real number 𝑅, then the mappings 𝑄 and 𝑈 are called (𝐷𝑆𝑈)-w-commuting at 𝜛 ∈ 𝛯 such that 𝔅(𝑈𝑄𝜛,𝑄𝑄𝜛,𝜏) ≥ 𝔅(𝑄𝜛,𝑈𝜛, 𝜏 𝑅 ), 𝔒(𝑈𝑄𝜛,𝑄𝑄𝜛,𝜏) ≤ 𝔒(𝑄𝜛,𝑈𝜛, 𝜏 𝑅 ), 𝔖(𝑈𝑄𝜛,𝑄𝑄𝜛,𝜏) ≤ 𝔖(𝑄𝜛,𝑈𝜛, 𝜏 𝑅 ). If these inequalities are fulfilled then 𝑄 and 𝑈 are (𝐷𝑆𝑈)-w-commuting on 𝛯 for all 𝜛 ∈ 𝛯. If the self– mappings 𝑄 and 𝑈 holds Definitions 2.7 and 2.8 then 𝑄 and 𝑈 are called (𝐷𝑆)-w-commuting mappings. Example 2.1: Suppose 𝛯 = [1,5] with the usual metric space 𝛥(𝜛,𝜔) = |𝜛 − 𝜔|. For each 𝜏 ∈ (0,∞). Define 𝔅(𝜛,𝜔,𝜏) = 𝜏 𝜏 + 𝛥(𝜛,𝜔) , 𝜛,𝜔 ∈ 𝛯, 𝔒(𝜛,𝜔,𝜏) = 𝛥(𝜛,𝜔) 𝜏 + 𝛥(𝜛,𝜔) , 𝜛,𝜔 ∈ 𝛯, 𝔖(𝜛,𝜔,𝜏) = 𝛥(𝜛,𝜔) 𝜏 , 𝜛,𝜔 ∈ 𝛯. Clearly, (𝛯,𝔅,𝔒,𝔖,∗,♢) be an NMS, where 𝜎 ∗ 𝜍 = 𝜎𝜍 and 𝜎♢𝜍 = min{1,𝜎 + 𝜍}. Define 𝑄,𝑈:𝛯 → 𝛯 by 𝑄(𝜛) = 𝜛 if 𝜛 ∈ [0,1), 𝑄(𝜛) = 1 4 if 𝜛 ∈ [1,5), 𝑈(𝜛) = 1 1 + 𝜛 for all 𝜛 ∈ [0,5]. Suppose the sequence {𝜛𝑛 = 3 + 1 𝑛 ∶ 𝑛 ≥ 1} in 𝛯. Then lim 𝑛→∞ 𝑄𝜛𝑛 = 1 4 , lim 𝑛→∞ 𝑈𝜛𝑛 = 1 4 but lim 𝑛→∞ 𝔅(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = 𝜏 𝜏 + |1 4 − 4 5 | ≠ 1, lim 𝑛→∞ 𝔒(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = |1 4 + 4 5 | 𝜏 + |1 4 − 4 5 | ≠ 0, 8 Int. J. Anal. Appl. (2023), 21:74 lim 𝑛→∞ 𝔖(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = |1 4 + 4 5 | 𝜏 ≠ 0. Which is shows that 𝑄 and 𝑈 are non-compatible. But we can obtain for 𝑅 ≥ 2 3 and 𝑄 and 𝑈 are (𝐷𝑆𝑄)-w-commuting at 𝜛 = 1. Example 2.2: Suppose 𝛯 = [1,10] with the usual metric space 𝛥(𝜛,𝜔) = |𝜛 − 𝜔| for each 𝜏 ∈ (0,∞). Define 𝔅(𝜛,𝜔,𝜏) = 𝜏 𝜏 + 𝛥(𝜛,𝜔) , 𝜛,𝜔 ∈ 𝛯 𝔒(𝜛,𝜔,𝜏) = 𝛥(𝜛,𝜔) 𝜏 + 𝛥(𝜛,𝜔) , 𝜛,𝜔 ∈ 𝛯, 𝔖(𝜛,𝜔,𝜏) = 𝛥(𝜛,𝜔) 𝜏 , 𝜛,𝜔 ∈ 𝛯, Clearly, (𝛯,𝔅,𝔒,𝔖,∗,♢) be an NMS where 𝜎 ∗ 𝜍 = 𝜎𝜍 and 𝜎♢𝜍 = min{1,𝜎 + 𝜍}. Define 𝑄,𝑈:𝛯 → 𝛯 by 𝑄𝜛 = 𝜛 if 1 ≤ 𝜛 ≤ 5,𝑄𝜛 = 𝜛 − 3 if 5 < 𝜛 ≤ 10, 𝑈𝜛 = 2 if 1 ≤ 𝜛 ≤ 5,𝑈𝜛 = 𝜛 − 1 2 if 5 < 𝜛 ≤ 1, Consider the sequence {𝜛𝑛 = 3 + 1 𝑛 ∶ 𝑛 ≥ 1} in 𝛯. Then lim 𝑛→∞ 𝑄𝑈𝜛 = 2, lim 𝑛→∞ 𝑈𝜛𝑛 = 2. lim 𝑛→∞ 𝔅(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = lim 𝜏 𝜏 + |2 + 1 2𝑛 − 2| = 1, lim 𝑛→∞ 𝔒(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = lim |2 + 1 2𝑛 − 2| 𝜏 + |2 + 1 2𝑛 −2| = 0, lim 𝑛→∞ 𝔖(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = lim |2 + 1 2𝑛 − 2| 𝜏 = 0. Which shows that 𝑄 and 𝑈 are compatible and (𝐷𝑆𝑄)-w-commuting for 𝑅 ≥ 1, at 𝜛 = 7. Theorem 2.2: Let (𝛯,𝔅,𝔒,𝔖,∗,♢) be a complete NMS. Let 𝜉,𝛶:𝛯 → 𝛯. Suppose 𝛶(𝛯) is a NSBS and 𝛶(𝛯) ⊆ 𝜉(𝜛) satisfying the following conditions: 𝔅(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏),≥ 𝔅(𝜉(𝜛),𝜉(𝜔),𝜏) 𝔒(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏) ≤ 𝔒(𝜉(𝜛),𝜉(𝜔),𝜏) 𝔖(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏) ≤ 𝔖(𝜉(𝜛),𝜉(𝜔),𝜏) }. (2.1) A mapping 𝜙:(0,∞) → (0,∞), is non-decreasing that verifies 𝜙(𝜏) < 𝜏 for all 𝜏 > 0. Then 𝜉 and 𝛶 have a coincidence point. If 𝜉 and 𝛶 are DS-w-commuting at coincidence point then both mapping 𝜉 and 𝛶 have a unique common fixed point. 9 Int. J. Anal. Appl. (2023), 21:74 Proof: Suppose 𝜛 ∈ 𝛯 be any element of 𝛯. Because 𝛶(𝛯) ⊆ 𝜉(𝛯), so there exist an element 𝜛 ∈ 𝛯 such that 𝛶(𝜛𝑜) = 𝜉(𝜛1). By induction {𝜛𝑛} be a sequence such that 𝛶(𝜛𝑛) = 𝜉(𝜛𝑛+1). Assume a nested sequence of non-empty closed sets defined by 𝐹 = {𝛶𝜛𝑛,𝛶𝜛𝑛+1 ,…}̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅, 𝑛 ∈ ℕ. We shall prove that the ND- zero of the family {𝐹𝑛}𝑛∈ℕ. Moreover, suppose 𝑟 ∈ (0,1) and 𝜏 > 0 be randomly picked. By 𝐹𝜚 ∈ g(𝜛̅̅ ̅̅ ̅) it follows that 𝐹𝜚 is a NSBS for any 𝜚 ∈ ℕ. This implies that there exists 𝜏o > 0 such that 𝔅(𝜛,𝜔,𝜏𝑜) > 1 − 𝑟, 𝔒(𝜛,𝜔,𝜏𝑜) < 𝑟 and 𝔖(𝜛,𝜔,𝜏𝑜) < 𝑟 for all 𝜛,𝜔 ∈ 𝐹𝜚. (1) From lim 𝑛→∞ 𝜙𝑛(𝜏𝑜) = 0 we deduce that there exists 𝑚 ∈ ℕ such that 𝜙 𝑚(𝜏𝑜) < 𝜏. Let 𝑛 = 𝑚 + 𝜚 and 𝜛,𝜔 ∈ 𝐹𝑛 be arbitrary. There exists sequence {𝛶𝜛𝑛(𝑖)},{𝛶𝜛𝑛(𝑗)} in 𝐹𝑛(𝑛(𝑖),𝑛(𝑗) ≥ 𝑛,𝑖, 𝑗 ∈ ℕ) such that lim 𝑖→∞ 𝛶𝜛𝑛(𝑖) = 𝜛 and lim 𝑗→∞ 𝛶𝜛𝑛(𝑗) = 𝜔. From (1), we have 𝔅(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙(𝜏)) ≥ 𝔅(𝜉𝜛𝑛(𝑖),𝜉𝜛𝑛(𝑗),𝜏) = 𝔅(𝛶𝜛𝑛(𝑖)−1,𝜏) 𝔒(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙(𝜏)) ≤ 𝔒(𝜉𝜛𝑛(𝑖),𝜉𝜛𝑛(𝑗),𝜏) = 𝔒(𝛶𝜛𝑛(𝑖)−1,𝛶𝜛𝑛(𝑗)−1,𝜏), and 𝔖(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙(𝜏)) ≤ 𝔖(𝜉𝜛𝑛(𝑖),𝜉𝜛𝑛(𝑗),𝜏) = 𝔖(𝛶𝜛𝑛(𝑖)−1,𝛶𝜛𝑛(𝑗)−1,𝜏). Thus, by induction, we get 𝔅(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙 𝑚(𝜏)) ≥ 𝔅(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏), 𝔒(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙 𝑚(𝜏) ≤ 𝔒(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏), and 𝔖(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙 𝑚(𝜏) ≤ 𝔖(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏). Since 𝜙𝑚(𝜏𝑜) < 𝜏 and because 𝔅(𝜛,𝜔, .) is non-decreasing and 𝔒(𝜛,𝜔, .) is non-increasing function, 𝔅(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜏) ≥ 𝔅(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙 𝑚(𝜏𝑜) ) ≥ 𝔅(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏𝑜), 𝔒(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜏) ≤ 𝔒(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙 𝑚(𝜏𝑜)) ≤ 𝔒(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏𝑜), 𝔖(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜏) ≤ 𝔖(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙 𝑚(𝜏𝑜)) ≤ 𝔖(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏𝑜)} . (2) As {𝛶𝜛𝑛(𝑖)−𝑚} and {𝛶𝜛𝑛(𝑗)−𝑚} are sequence in 𝐹𝜚 from (1), it follows that 𝔅(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏𝑜) > 1 − 𝑟 ∀ 𝑖, 𝑗 ∈ ℕ, 𝔒(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏𝑜) < 𝑟 ∀ 𝑖, 𝑗 ∈ ℕ, 𝔖(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏𝑜) < 𝑟 ∀ 𝑖, 𝑗 ∈ ℕ }. (3) 10 Int. J. Anal. Appl. (2023), 21:74 By implying (1) to (3), we examine 𝔅(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜏) > 1 − 𝑟, 𝔒(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜏) < 𝑟, 𝔖(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜏) < 𝑟 ∀ 𝑖, 𝑗 ∈ ℕ. By applying the limits as 𝑖, 𝑗 → ∞ and the Lemma 2.1. We conclude 𝔅(𝜛,𝜔,𝜏) > 1 − 𝑟, 𝔒(𝜛,𝜔,𝜏) < 𝑟 and 𝔖(𝜛,𝜔,𝜏) < 𝑟 for all 𝜛,𝜔 ∈ 𝐹𝑛. Then, {𝐹𝑛}𝑛∈ℕ has ND- zero. By using the Theorem 2.1 we examine that {𝐹𝑛}𝑛∈ℕ has non-empty intersection, which consists of exactly one point 𝜇. Since {𝐹𝑛}𝑛∈ℕ has ND- zero and 𝜇 ∈ 𝐹𝑛 for all 𝑛 ∈ ℕ then for each 𝑟 ∈ (0,1) and each 𝜏 > 0 there exists 𝑛𝑜 ∈ ℕ such that for all 𝑛 ≥ 𝑛𝑜 𝔅(𝛶𝜛𝑛,𝜇,𝜏) > 1 − 𝑟, 𝔒(𝛶𝜛𝑛,𝜇,𝜏) < 𝑟 and 𝔖(𝛶𝜛𝑛,𝜇,𝜏) < 𝑟. For all 𝑟 ∈ (0,1), it’s obviously satisfied lim 𝑛→∞ 𝔅(𝛶𝜛𝑛,𝜇,𝜏) > 1 − 𝑟 and lim 𝑛→∞ 𝔒(𝛶𝜛𝑛,𝜇,𝜏) < 𝑟. By using the limit that 𝑟 → 0, we get lim 𝑛→∞ 𝔅(𝛶𝜛𝑛,𝜇,𝜏) = 1, lim 𝑛→∞ 𝔒(𝛶𝜛𝑛,𝜇,𝜏) = 0 and lim 𝑛→∞ 𝔖(𝛶𝜛𝑛,𝜇,𝜏) = 0. That is, lim 𝑛→∞ 𝛶𝜛𝑛 = 𝜇. Sequence {𝜉𝜛𝑛} follows the condition that lim 𝑛→∞ 𝜉𝜛𝑛 = 𝜇. Since 𝛶(𝛯) ⊆ 𝜉(𝛯), there exists 𝑢 ∈ 𝛯 such that 𝜇 = 𝜉(𝑢). Then utilizing (1), we get 𝔅(𝛶(𝑢),𝛶(𝜛𝑛),𝜙(𝜏)) ≥ 𝔅(𝜉(𝑢),𝜉(𝜛𝑛),𝜏), 𝔒(𝛶(𝑢),𝛶(𝜛𝑛),𝜙(𝜏)) ≤ 𝔒(𝜉(𝑢),𝜉(𝜛𝑛),𝜏), 𝔖(𝛶(𝑢),𝛶(𝜛𝑛),𝜙(𝜏)) ≤ 𝔖(𝜉(𝑢),𝜉(𝜛𝑛),𝜏). Letting 𝑛 → ∞, we get 𝔅(𝛶(𝑢),𝜇,𝜙(𝜏)) ≥ 𝔅(𝜇,𝜇,𝜏) = 1, 𝔒(𝛶(𝑢),𝜇,𝜙(𝜏)) ≤ 𝔒(𝜇,𝜇,𝜏) = 0, 𝔖(𝛶(𝑢),𝜇,𝜙(𝜏)) ≤ 𝔖(𝜇,𝜇,𝜏) = 0. Since 𝔅(𝛶(𝑢),𝜇,𝜏 ) ≥ 𝔅(𝛶(𝑢),𝜇,𝜙(𝜏)), 𝔒(𝛶(𝑢),𝜇,𝜏) ≤ 𝔒(𝛶(𝑢),𝜇,𝜙(𝜏)), 𝔖(𝛶(𝑢),𝜇,𝜏) ≤ 𝔖(𝛶(𝑢),𝜇,𝜙(𝜏)). We get, 𝔅(𝛶(𝑢),𝜇,𝜏) = 1, 𝔒(𝛶(𝑢),𝜇,𝜏) = 0 and 𝔖(𝛶(𝑢),𝜇,𝜏) = 0 for all 𝜏 > 0. Thus 𝛶(𝑢) = 𝜇. Therefore 𝜉(𝑢) = 𝛶(𝑢) = 𝜇 which is shows that 𝑢 is coincidence point of 𝜉 and 𝛶. Since 𝜉 and 𝛶 are (𝐷𝑆)-w-commuting at coincidence point, so 𝜉 and 𝛶 are (𝐷𝑆𝜉)-w-commuting at coincidence points, such that 11 Int. J. Anal. Appl. (2023), 21:74 𝔅(𝜉𝛶𝑢,𝛶𝛶𝑢,𝜏) ≥ 𝜉((𝑢),𝛶(𝑢), 𝜏 𝑅 ), 𝔒(𝜉𝛶𝑢,𝛶𝛶𝑢,𝜏) ≤ 𝔒(𝜉(𝑢),𝛶(𝑢), 𝜏 𝑅 ) , 𝔖(𝜉𝛶𝑢,𝛶𝛶𝑢,𝜏) ≤ 𝔖(𝜉(𝑢),𝛶(𝑢), 𝜏 𝑅 ). Thus, 𝔅(𝜉𝛶𝑢,𝛶𝛶𝑢,𝜏) = 1, 𝔒(𝜉𝛶𝑢,𝛶𝛶𝑢,𝜏) = 0 and 𝔖(𝜉𝛶𝑢,𝛶𝛶𝑢,𝜏) = 0. So, 𝜉𝛶𝑢 = 𝛶𝛶𝑢 that is 𝜉𝜇 = 𝛶𝜇. Again using (1), we have 𝔅(𝛶(𝜛𝑛),𝛶(𝜇),𝜙(𝜏)) ≥ 𝔅(𝜉(𝜛𝑛),𝜉(𝜇),𝜏 ), 𝔒((𝛶𝜛𝑛),𝛶(𝜇),𝜙(𝜏)) ≤ 𝔒(𝜉(𝜛𝑛),𝜉(𝜇),𝜏 ), 𝔖((𝛶𝜛𝑛),𝛶(𝜇),𝜙(𝜏)) ≤ 𝔖(𝜉(𝜛𝑛),𝜉(𝜇),𝜏 ). Letting 𝑛 → ∞, we have 𝔅(𝜇,𝛶(𝜇),𝜙(𝜏)) ≥ 𝔅(𝜇,𝛶(𝜏),𝜏 ), 𝔒(𝜇,𝛶(𝜏),𝜙(𝜏)) ≤ 𝔒(𝜇,𝛶(𝜇),𝜏 ), 𝔖(𝜇,𝛶(𝜏),𝜙(𝜏)) ≤ 𝔖(𝜇,𝛶(𝜇),𝜏 ) for all 𝜏 > 0. Utilizing the Lemma 2.3, we have 𝛶(𝜇) = 𝜇. Thus 𝜉(𝜇) = 𝛶(𝜇) = 𝜇. Now, we show the uniqueness of 𝜇. Suppose 𝜇 ≠ 𝑤 be another fixed point, then utilizing (2.1), we get 𝔅(𝛶(𝜇),𝛶(𝑤),𝜙(𝜏)) ≥ 𝔅(𝜉(𝜇),𝜉(𝑤),𝜏), 𝔒(𝛶(𝜇),𝛶(𝑤),𝜙(𝜏)) ≤ 𝔒(𝜉(𝜇),𝜉(𝑤),𝜏 ), 𝔖(𝛶(𝜇),𝛶(𝑤),𝜙(𝜏)) ≤ 𝔖(𝜉(𝜇),𝜉(𝑤),𝜏 ) ∀ 𝜏 > 0. That is 𝔅(𝜇,𝑢,𝜙(𝜏)) ≥ 𝔅(𝜇,𝑢,𝜏), 𝔒(𝜇,𝑢,𝜙(𝜏)) ≤ 𝔒(𝜇,𝑢,𝜏), 𝔖(𝜇,𝑢,𝜙(𝜏)) ≤ 𝔖(𝜇,𝑢,𝜏) for all 𝜏 > 0. By using the Lemma 2.3, it follows that 𝜇 = 𝑤. Example 2.3: Consider 𝛯 = [0,2] with the usual metric space 𝛥(𝜛,𝜔) = |𝜛 − 𝜔| and for each 𝜏 ∈ [0,1], define 𝔅(𝜛,𝜔,𝜏) = 𝜏 𝜏 + |𝜛 − 𝜔| ,𝜛,𝜔 ∈ 𝛯, 𝔒(𝜛,𝜔,𝜏) = |𝜛 − 𝜔| 𝜏 + |𝜛 − 𝜔| ,𝜛,𝜔 ∈ 𝛯, 𝔖(𝜛,𝜔,𝜏) = |𝜛 − 𝜔| 𝜏 ,𝜛,𝜔 ∈ 𝛯. Clearly, (𝛯,𝔅,𝔒,𝔖,∗,♢) be a complete NMS where 𝜎 ∗ 𝜍 = 𝜎𝜍 and 𝜎♢𝜍 = min{𝜎,𝜍}. 12 Int. J. Anal. Appl. (2023), 21:74 Define 𝜉,𝛶:𝛯 → 𝛯 by 𝜉(𝜛) = { 1 if 𝜛 = 1 𝜛 2 + 1 if 𝜛 ≠ 1, 𝛶(𝜛) = { 1 if 𝜛 = 1 1 1 + 𝜛 if 𝜛 ≠ 1, 𝜙(𝜏) = 𝜏 2 ,𝜏 > 0. Then 𝛶(𝛯) ⊆ 𝜉(𝛯). We see that {𝜛𝑛} is decreasing sequence, so lim 𝑛→∞ 𝜉(𝜛𝑛) = lim 𝑛→∞ 𝛶(𝜛𝑛). Thus, mappings are not compatible. But both are (𝐷𝑆)-w-commuting at coincidence point. We will demonstrate that axiom (1) is also met. Clearly, ∀ 𝜛,𝜔 ∈ 𝛯. We get 1 (1 + 𝜛)(1 + 𝜔) ≤ 1. We get 𝔅(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏)) = 𝜏 𝜏 + 2( |𝜛−𝜔| (1+𝜛)(1+𝜔) ) ≥ 𝔅(𝜉(𝜛),𝜉(𝜔),𝜏), as easy to see in figure 1. Figure 1 Shows the graphical behavior of the contraction 𝔅(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏)) ≥ 𝔅(𝜉(𝜛),𝜉(𝜔),𝜏) when 𝜏 = 1. Now for second 𝔒, we have 𝔒(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏)) = 2( |𝜛−𝜔| (1+𝜛)(1+𝜔) ) 𝜏 + 2( |𝜛−𝜔| (1+𝜛)(1+𝜔) ) ≤ 𝔒(𝜉(𝜛),𝜉(𝜔),𝜏), 13 Int. J. Anal. Appl. (2023), 21:74 as easy to see in figure 2. Figure 2 Shows the graphical behavior of the contraction 𝔒(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏)) ≤ 𝔒(𝜉(𝜛),𝜉(𝜔),𝜏) when 𝜏 = 1. Now for third function 𝔖, we have 𝔖(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏)) = 2( |𝜛−𝜔| (1+𝜛)(1+𝜔) ) 𝜏 ≤ 𝔖(𝜉(𝜛),𝜉(𝜔),𝜏), as easy to see in figure 3. Figure 3 Shows the graphical behavior of the contraction 𝔖(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏)) ≤ 𝔖(𝜉(𝜛),𝜉(𝜔),𝜏) when 𝜏 = 1. 14 Int. J. Anal. Appl. (2023), 21:74 All the conditions of theorem 2.2 are fulfill. So, both mappings 𝜉(𝜛) and 𝛶(𝜛) have a unique common fixed point, which is 𝜛 = 1. Theorem 2.3: Let (𝛯,𝔅,𝔒,𝔖,∗,♢) be a complete NMS. Let 𝛶:𝛯 → 𝛯. Suppose 𝛶(𝛯) is NSBS verifying the axiom (2) with 𝔅(𝛶(𝜛),𝛶(𝜔),𝜚𝜏) ≥ 𝔅(𝜛,𝜔,𝜏), 𝔒(𝛶(𝜛),𝛶(𝜔),𝜚𝜏) ≤ 𝔒(𝜛,𝜔,𝜏), 𝔖(𝛶(𝜛),𝛶(𝜔),𝜚𝜏) ≤ 𝔖(𝜛,𝜔,𝜏) for some 𝜚 ∈ (0,1). Then 𝛶 has a unique fixed point. Remark 2.4: The hypothesis of Theorem 2.2 that is 𝛶(𝛯) is a NSBS, can be exchanged with the following one: there exists an element 𝜛𝑜 ∈ 𝛯 such that the orbit of an element 𝜛𝑜 described by 𝜙(𝜛𝑜,𝛶) = {𝜛𝑜,𝛶𝜛𝑜,𝛶 2𝜛𝑜,…} is a NSBS. This tells us the sequence used in the start of the proof of the Theorem 2.2 for 𝜉 = 1 is a sequence of Picard iterates defined by 𝜛𝑛+1 = 𝛶𝜛𝑛. 3. APPLICATION Now, we aim to apply Theorem 2.3 to obtain the existence of solution to a nonlinear fractional differential equation (NFDE) 𝐷0+ 𝑝 ℎ(𝜛) = 𝜉(𝜛,ℎ(𝜛)), 0 < 𝜛 < 1 (4) with the boundary conditions ℎ(0) + ℎ′(0) = 0,ℎ(1) + ℎ′(1) = 0, where 1 < 𝑝 ≤ 2 is a number, 𝐷0+ 𝑝 is the Caputo fractional derivative and 𝜉 ∶ [0,1] × [0,∞) → [0,∞) is a continuous function. Let 𝛯 = 𝐶([0,1],(0,1]) denote the space of all continuous functions defined on [0,1]. Let 𝜎 ∗ 𝜍 = min{𝜎,𝜍} for all 𝜎,𝜍 ∈ [0,1] and define an NMS as follows: 𝔅(ℎ(t),𝛿(𝜛),𝜏) = { 1 if ℎ(𝜏) = 𝛿(𝜛), 𝛼𝜏 𝛼𝜏 + 𝛾 max{sup𝜛𝜖[0,1] ℎ(𝜛),sup𝜛𝜖[0,1] 𝛿(𝜛)} ,otherwise, 𝔒(ℎ(t),𝛿(𝜛),𝜏) = { 0 if ℎ(𝜏) = 𝛿(𝜛), 𝛾 max{sup𝜛𝜖[0,1] ℎ(𝜛),sup𝜛𝜖[0,1] 𝛿(𝜛)} 𝛼𝜏 + 𝛾 max{sup𝜛𝜖[0,1] ℎ(𝜛),sup𝜛𝜖[0,1] 𝛿(𝜛)} ,otherwise, and 𝔖(ℎ(t),𝛿(𝜛),𝜏) = { 0 if ℎ(𝜏) = 𝛿(𝜛), 𝛾 max{sup𝜛𝜖[0,1] ℎ(𝜛),sup𝜛𝜖[0,1] 𝛿(𝜛)} 𝛼𝜏 ,otherwise. Observe that ℎ ∈ 𝛯 solves (4) whenever ℎ ∈ 𝛯 solves the below integral equation: 15 Int. J. Anal. Appl. (2023), 21:74 ℎ(𝜛) = 1 Г(𝑝) ∫ (1 − 𝑠 1 0 )𝑝−1(1 − 𝜛)𝜉(𝑠,ℎ(𝑠))𝛥𝑠 + 1 Г(𝑝 − 1) ∫ (1 − 𝑠 1 0 )𝑝−2(1 − 𝜛)𝜉(𝑠,ℎ(𝑠))𝛥𝑠 + 1 Г(𝑝) ∫ (𝜛 − 𝑠)𝑝−1𝜉(𝑠,ℎ(𝑠))𝛥𝑠. 𝜛 0 The graphical behavior of ℎ(𝜛) for different values of 𝑝 is shown in figure 4. Figure 4 Shows the graphical behavior of ℎ(𝜛) for 𝑝 = 1.1, 𝑝 = 1.4, 𝑝 = 1.7 and 𝑝 = 2. Theorem 4.2 The integral operator 𝛶:𝛯 → 𝛯 is given by 𝛶ℎ(𝜛) = 1 Г(𝑝) ∫ (1 − 𝑠 1 0 )𝑝−1(1 − 𝜛)𝜉(𝑠,ℎ(𝑠))𝛥𝑠 + 1 Г(𝑝 − 1) ∫ (1 − 𝑠 1 0 )𝑝−2(1 − 𝜛)𝜉(𝑠,ℎ(𝑠))𝛥𝑠 + 1 Г(𝑝) ∫ (𝜛 − 𝑠)𝑝−1𝜉(𝑠,ℎ(𝑠))𝛥𝑠, 𝜛 0 where 𝜉:[0,1] × [0,∞) → [0,∞) fulfilling the following criteria: max{sup𝑠∈[0,1] 𝜉 (𝑠,ℎ(𝑠)), sup𝑠∈[0,1] 𝜉(𝑠,𝛿(𝑠))} ≤ 1 4 max{ sup𝑠∈[0,1] ℎ(𝑠), sup𝑠∈[0,1] 𝛿(𝑠)} , for all ℎ,𝛿 ∈ 𝛯. Also, suppose that sup𝜛∈(0,1) 1 4 [ 1 − 𝜛 Г(𝑝 + 1) + 1 − 𝜛 Г(𝑝) + 𝜛𝑝 Г(𝑝 + 1) ] ≤ 𝜚 < 1. 16 Int. J. Anal. Appl. (2023), 21:74 Then NFDE has a unique solution in 𝛯. Proof: Let max{𝛶ℎ(𝜛),𝛶𝛿(𝜛)} = 1 − 𝜛 Г(𝑝) ∫ (1 − 𝑠)𝑝−1 max{sup𝑠∈[0,1]𝜉(𝑠,ℎ(𝑠)),sup𝑠∈[0,1] 𝜉(𝑠,𝛿(𝑠))}𝛥𝑠 1 0 + 1 − 𝜛 Г(𝑝 − 1) ∫ (1 − 𝑠)𝑝−2 1 0 max{sup𝑠∈[0,1]𝜉(𝑠,ℎ(𝑠)),sup𝑠∈[0,1] 𝜉(𝑠,𝛿(𝑠))}𝛥𝑠 + 1 Г(𝑝) ∫ (𝜛 − 𝑠)𝑝−1 max{sup𝑠∈[0,1]𝜉(𝑠,ℎ(𝑠)),sup𝑠∈[0,1] 𝜉(𝑠,𝛿(𝑠))} 𝜛 0 ≤ 1 − 𝜛 Г(𝑝) ∫ (1 − 𝑠)𝑝−1 max{sup𝑠∈[0,1] ℎ(𝑠),sup𝑠∈[0,1] 𝛿(𝑠)} 4 𝛥𝑠 1 0 + 1 − 𝜛 Г(𝑝 − 1) ∫ (1 − 𝑠)𝑝−2 max{sup𝑠∈[0,1]ℎ(𝑠),sup𝑠∈[0,1]𝛿(𝑠)} 4 𝛥𝑠 1 0 + 1 Г(𝑝) ∫ (𝜛 − 𝑠)𝑝−1 max{sup𝑠∈[0,1]ℎ(𝑠),sup𝑠∈[0,1]𝛿(𝑠)} 4 𝛥𝑠 𝜛 0 ≤ 1 4 max{sup𝑠∈[0,1] ℎ(𝑠),sup𝑠∈[0,1] 𝛿(𝑠)} ( 1 − 𝜛 Г(𝑝) ∫ (1 − 𝑠)𝑝−1 1 0 𝛥𝑠 + 1 − 𝜛 Г(𝑝 − 1) ∫ (1 − 𝑠)𝑝−2𝛥𝑠 1 0 + 1 Г(𝑝) ∫ (𝜛 − 𝑠)𝑝−1 𝜛 0 𝛥𝑠) ≤ 1 4 max{sup𝑠∈[0,1] ℎ(𝑠),sup𝑠∈[0,1] 𝛿(𝑠)} sup𝜛∈[0,1] [ 1 − 𝜛 Г(𝑝 + 1) + 1 − 𝜛 Г(𝑝) + 𝜛𝑝 Г(𝑝 + 1) ] = 𝜚 max{sup𝑠∈[0,1] ℎ(𝑠),sup𝑠∈[0,1] 𝛿(𝑠)}, where, 𝜚 = sup𝜛𝜖(0,1) 1 4 [ 1 − 𝜛 Г(𝑝 + 1) + 1 − 𝜛 Г(𝑝) + 𝜛𝑝 Г(𝑝 + 1) ]. Therefore, the above equation max{sup𝜛∈[0,1] 𝛶ℎ(𝜛),sup𝜛∈[0,1] 𝛶𝛿(𝜛)} ≤ 𝜚max{sup𝜛∈[0,1] ℎ(𝜛),sup𝜛∈[0,1] 𝛿(𝜛)} ⇒ 𝛼𝜏 + 𝛾 𝜚 max{sup𝜛∈[0,1] 𝛶ℎ(𝜛),sup𝜛∈[0,1] 𝛶𝛿(𝜛)} ≤ 𝛼𝜏 + 𝛾max{sup𝜛∈[0,1] ℎ(𝜛),sup𝜛∈[0,1] 𝛿(𝜛)} ⇒ 𝛼(𝜚𝜏) 𝛼(𝜚𝜏) + 𝛾max{sup𝜛∈[0,1] 𝛶ℎ(𝜛),sup𝜛∈[0,1] 𝛶𝛿(𝜛)} ≥ 𝛼𝜏 𝛼𝜏 + 𝛾max{sup𝜛∈[0,1] ℎ(𝜛),sup𝜛∈[0,1] 𝛿(𝜛)} ⇒ 𝔅(𝛶ℎ,𝛶𝛿,𝜚𝜏) ≥ 𝔅(ℎ,𝛿,𝜏), 17 Int. J. Anal. Appl. (2023), 21:74 and on the same technique, it is easy to deduce that 𝔒(𝛶ℎ,𝛶𝛿,𝜚𝜏) ≤ 𝔒(ℎ,𝛿,𝜏), 𝔖(𝛶ℎ,𝛶𝛿,𝜚𝜏) ≤ 𝔖(ℎ,𝛿,𝜏), for some 𝛼,𝛾 > 0. Observe that the conditions of the Theorem 2.3 are fulfilled. Resultantly, 𝛶 has a fixed-point fixed point; accordingly, the specified NFDE has a solution. 4. CONCLUSION In this manuscript, we established numerous interesting conditions in the context of NMS. We provided numerous non-trivial examples and their graphical views via computational techniques. 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