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Common fixed point results via simulation type functions in non-Archimedean modular metric spaces and applications

Common fixed point results via simulation type functions in non-Archimedean modular metric spaces... Topol. Algebra Appl. 2022; 10:13–24 Research Article Open Access Mahpeyker Öztürk* and Ekber Girgin Common xed point results via simulation type functions in non-Archimedean modular metric spaces and applications https://doi.org/10.1515/taa-2020-0109 Received 29 September, 2021; accepted 7 January, 2022 Abstract: In this study, we demonstrate the existence and uniqueness of common xed points of a generalized (α, β)− simulation contraction on a non-Archimedean modular metric space. We achieve some consequences in non-Archimedean modular metric spaces as an application, using the structure of a directed graph. Even- tually, we contemplate the existence of solutions to a class of functional equations standing up dynamic programming with the help of our outcomes. Keywords: Non-Archimedean modular metric space, Simulation function, Cyclic (α, β)-admissible pair MSC: 47H10, 54H25, 37C25 1 Introduction Subsequently, the letters R, R , and N will specify the set of all real numbers, the set of all nonnegative real numbers, and the set of all positive integer numbers, respectively. In 2010, Chistyakov [1, 2] set up a new structure named modular metric, which is an extension of metric and a linear modular. Denition 1.1 [1, 3] Let M be non-void set and δ : 0,∞ ×M×M ! 0,∞ be a function fullling the following ( ) [ ] statements: for all λ, μ > 0 and p, q, k 2 M, δ . p = q if and only if δ p, q = 0; ( ) 1 λ δ . δ p, q = δ p, q ; ( ) ( ) 2 λ λ δ . δ p, q ≤ δ p, k + δ k, q . ( ) ( ) μ ( ) 3 λ+μ λ So, δ is entitled modular metric in M, and in the present case, M is a modular metric space. If we substitute (δ ) by δ . δ p, p = 0 for all λ > 0 and p 2 M, ( ) 4 λ then δ is said to be a pseudomodular metric on M. A modular metric δ on M is named regular if the following weaker version of (δ ) is provided for some λ > 0; δ . p = q if and only if δ p, q = 0. ( ) 5 λ As well, δ is a convex function if, for λ, μ > 0 and p, q, k 2 M, the inequality holds: λ μ δ . δ p, q ≤ δ p, k + δ k, q . ( ) ( ) μ ( ) 6 λ+μ λ λ+μ λ+μ For all λ, μ > 0 and p, q, k 2 M , if the following property δ . δ p, q ≤ δ p, k + δ k, q ( ) ( ) μ ( ) 7 maxfλ,μg λ *Corresponding Author: Mahpeyker Öztürk: Department of Mathematics, Sakarya University, Sakarya, Turkey, E-mail: mahpeykero@sakarya.edu.tr Ekber Girgin: Department of Mathematics, Sakarya University, Sakarya, Turkey, E-mail: girginekber@gmail.com Open Access. © 2022 Mahpeyker Öztürk and Ekber Girgin, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. 14 Ë Mahpeyker Öztürk and Ekber Girgin is put in a place of (δ ), then, we attain a new space with some new features. This new structure M is named a non-Archimedean modular metric space. For simplicity, throughout the study, we will write: δ p, q = δ λ, p, q ( ) ( ) for all λ > 0 and p, q 2 M. Denition 1.2 [1, 2] Presume that the function δ be a pseudomodular on M and δ 2 M be a xed element. The hereinbelow two sets M and M are called modular spaces (around δ ). δ 0 M = M (p ) = fp 2 M : δ (p, p ) as λ ! ∞g 0 0 δ δ λ and * * M = M (p ) = fp 2 M : 9λ = λ p > 0 such that δ p, p < ∞g . 0 ( ) ( 0) δ δ λ It is noticeable that M  M , but this inclusion may be proper in general. Presume that from [1, 2], δ is modular on M; we derive that the modular space M can be equipped with a (nontrivial) metric, induced by δ and presented by: d p, q = inffλ > 0 : δ p, q < λg ( ) ( ) δ λ for all p, q 2 M . Note that if δ is a convex modular on M, then according to [1, 2], the two modular spaces coincide, i.e., * * M = M , and this common set can be endowed with the metric d given by: δ δ δ d p, q = inffλ > 0 : δ p, q < 1,8p, q 2 M g , ( ) ( ) δ λ δ which is called Luxemburg distance. Denition 1.3 [4] Let M be a modular metric space, S be a subset of M and p be a sequence in M . ( n) δ δ δ n2N Then, i. A sequence p is called δ−convergent to p 2 M if and only if δ p , p ! 0 as n ! ∞ for all λ > 0, ( n) ( n ) δ λ n2N p is said to be the δ−limit of (p ). ii. A sequence p is called δ−Cauchy if δ p , p ! 0, as m, n ! ∞ for all λ > 0. ( n) ( n m) n2N iii. A subset S is called δ−complete if any δ−Cauchy sequence in S is δ−convergent to the point of S. Denition 1.4 [4] Let M be a modular metric space. The mapping K : M ! M is δ−continuous if δ δ δ δ p , p ! 0, provided to δ Kp , Kp ! 0 as n ! ∞. ( n ) ( n ) λ λ Denition 1.5 [5] A simulation function is a mapping ζ : 0,∞ ! R admitting the following features: [ ) ζ . ζ 0, 0 = 0, ( ) ζ . ζ p, q < p − q, for all p, q > 0, ( ) ζ . iffq g andfp g are sequences in 0,∞ such that lim q = lim p = l and l 2 0,∞ , then n n ( ) n n ( ) n!∞ n!∞ lim sup ζ q , p < 0. ( n n) n!∞ Argoubi et al.[6] modied the denition of simulation function and so introduced it as: Denition 1.6 A simulation function is a mapping ζ : 0,∞ ! R satisfying the followings: [ ) i. ζ p, q < p − q, for all p, q > 0, ( ) ii. iffq g andfp g are sequences in 0,∞ such that lim q = lim p > 0, and q < p , then lim sup ζ q , p < n n ( ) n n n n ( n n) n!∞ n!∞ n!∞ Example 1.7 [6] The function ζ : 0,∞ × 0,∞ ! R is a simulation function specied by c [ ) [ ) 1 if q, p = 0, 0 , ( ) ( ) ζ q, p = c ( ) cp − q otherwise, where c 2 0, 1 . ( ) One may point to [5]-[12] for some detailed information about simulation functions. Denition 1.8 [13] Let G : 0,∞ ! R is called C−class function when it has continuity property and admits [ ) the following features: Common xed point results via simulation type functions Ë 15 i. G p, q ≤ p, ( ) ii. G p, q = p implies that either p = 0 or q = 0, for all p, q 2 0,∞ . ( ) [ ) Denition 1.9 [14] Let ζ : 0,∞ ! R be a function admitting the following features: [ ) a. ζ q, p < G p, q for all t, s > 0, where G : 0,∞ ! R is a C−class function, ( ) ( ) [ ) b. iffq g andfp g are sequences in 0,∞ such that lim q = lim p > 0, and q < p , then lim sup ζ q , p < n n ( ) n n n n ( n n) n!∞ n!∞ n!∞ C . Then ζ is named a C −simulation function. Denition 1.10 [14] A mapping G : 0,∞ ! R has the property C , if there exists a C ≥ 0 such that [ ) G G i. G p, q > C implies p > q, ( ) ii. G q, q ≤ C for all q 2 0,∞ . ( ) [ ) Let T denotes the family of all C −simulation functions ζ : 0,∞ ! R. [ ) G G Recently, Alizadeh et al. [15] dened the concept of cyclic α, β −admissible mapping as indicated below: ( ) Denition 1.11 Let M be a non-void set, K be a self-mapping on M and α, β : M ! 0,∞ be two mappings. [ ) K is named a cyclic α, β −admissible mapping if ( ) (i) for some p 2 M, α p ≥ 1 implies β Kp ≥ 1; ( ) ( ) (ii) for some p 2 M, β p ≥ 1 implies α Kp ≥ 1. ( ) ( ) Besides, they demonstrated xed point theorems by using the structure of cyclic α, β −admissible mapping. ( ) Example 1.12 [15] Let K : R ! R be a function described as Kp = − p + p . Presume that α, β : R ! R are furnishing mappings for all p, q 2 R such that p −q α (p) = e and β (q) = e . Then K is a cyclic α, β −admissible mapping. ( ) The generalized cyclic α, β −admissible mappings have been generalized by Latif et al. [16], as follows. ( ) Denition 1.13 Let M be a non-void set, K, L be two self-mappings on M and α, β : M ! 0,∞ be two [ ) mappings. (K, L) is called a cyclic α, β −admissible pair if ( ) (i) for some p 2 M α p ≥ 1 implies β Kp ≥ 1; ( ) ( ) (ii) for some p 2 M β p ≥ 1 implies α Lp ≥ 1. ( ) ( ) Remark 1.16 In the above denition, if we acquire K = L, then K is a cyclic α, β -admissible mapping. ( ) Throughout this study, δ will be used as a convex and regular function. 2 Main results Initially, this section aims to innovate the generalized (α, β)−simulation contraction. Besides, common xed point results are procured satisfying generalized (α, β)−simulation contraction in non-Archimedean modular metric spaces. Denition 2.1 Let M be a non-Archimedean modular metric space, K, L : M ! M be two self-mappings δ δ δ and α, β : M ! [0,∞) be two functions. The pair (K, L) is named a generalized (α, β)−simulation contrac- tion if there exists ζ 2 T such that α p β q ≥ 1 ) ζ δ Kp, Lq , S p, q ≥ C , (1) ( ) ( ) ( ( ) ( )) λ G n o δ p,Lq +δ q,Kp ( ) ( ) λ λ where S (p, q) = max δ (p, q) , δ (p, Kp) , δ (q, Lq) , , for all p, q 2 M . λ λ λ δ Theorem 2.2 Let M be a δ−complete non-Archimedean modular metric space and K and L are generalized α, β −simulation contraction. Presume that the following circumstances hold: ( ) i. (K, L) is a cyclic α, β −admissible pair, ( ) ii. there exists p 2 M such that α p ≥ 1, 0 ( 0) iii. K or L is a δ−continuous mapping, iv. if fp g is a sequence in M such that p ! p and α p ≥ 1, β p ≥ 1 for all n 2 N, then α p ≥ 1 n n ( 2n) ( 2n−1) ( ) and β p ≥ 1. ( ) 16 Ë Mahpeyker Öztürk and Ekber Girgin Then K and L hold a common xed point. Additionally, if α p β q ≥ 1 for all p, q 2 Fix K, L , then K and ( ) ( ) ( ) L hold a unique common xed point. Proof Let p 2 M be such that α p ≥ 1. We will construct a sequencefp g in M by ( ) n 0 0 δ δ p = Kp , 2n+2 2n+1 (2) p = Lp , 2n+1 2n for all n 2 N. Also, as K, L is a cyclic α, β −admissible pair and α p ≥ 1, then ( ) ( ) ( ) β p = β Kp ≥ 1 ( ) ( ) 1 0 which implies α p = α Lp ≥ 1. ( ) ( ) 2 1 By proceeding with this process, we get α (p ) ≥ 1 and β (p ) ≥ 1 for all n 2 N. Thus, α (p ) β (p ) ≥ 1 2n 2n+1 2n 2n+1 for all n 2 N. From (1), we have C ≤ ζ δ Kp , Lp , S p , p ( ( ) ( )) G λ 2n 2n+1 2n 2n+1 = ζ δ p , p , S p , p ( ( ) ( )) λ 2n+1 2n+2 2n 2n+1 < G S p , p , δ p , p . ( ( ) ( )) 2n 2n+1 λ 2n+1 2n+2 Further, using (i) of Denition 1.10, we achieve δ p , p < S p , p , (3) ( ) ( ) λ 2n+1 2n+2 2n 2n+1 where S p , p = maxfδ p , p , δ p , Kp , δ p , Lp , ( ) ( ) ( ) ( ) 2n 2n+1 λ 2n 2n+1 λ 2n 2n λ 2n+1 2n+1 p ,Lp + p ,Kp ( 2n 2n+1) ( 2n+1 2n) = maxfδ p , p , δ p , p , δ p , p , ( ) ( ) ( ) λ 2n 2n+1 λ 2n 2n+1 λ 2n+1 2n+2 δ (p ,p )+δ (p ,p ) λ 2n 2n+2 λ 2n+1 2n+1 = maxfδ p , p , δ p , p , δ p , p , ( ) ( ) ( ) λ 2n 2n+1 λ 2n 2n+1 λ 2n+1 2n+2 δ (p ,p )+δ (p ,p ) maxfλ,λg 2n 2n+2 λ 2n+1 2n+1 ≤ maxfδ p , p , δ p , p , ( ) ( ) λ 2n 2n+1 λ 2n+1 2n+2 δ (p ,p )+δ (p ,p ) λ 2n 2n+1 λ 2n+1 2n+2 = maxfδ (p , p ) , δ (p , p )g . 2n 2n+1 2n+1 2n+2 λ λ If maxfδ p , p , δ p , p g = δ p , p ( 2n 2n+1) ( 2n+1 2n+2) ( 2n+1 2n+2) λ λ λ for each n 2 N, then from (3), we have a contradiction. So, S p , p = δ p , p for all n 2 N. ( ) ( ) 2n 2n+1 λ 2n 2n+1 Consequently, (3) gives δ p , p < δ p , p . (4) ( ) ( ) λ 2n+1 2n+2 λ 2n 2n+1 Hence for all n 2 N, continuing this way we get δ p , p < δ p , p . ( 2n 2n+1) ( 2n−1 2n−2) λ λ So fδ p , p g is a monotonically decreasing sequence of nonnegative real numbers, and hence there ( 2n 2n+1) exists l ≥ 0 such that lim δ p , p = l. (5) ( 2n 2n+1) n!∞ Assume that l > 0. Then using (1) and (b) of the Denition 1.9, we have C ≤ lim sup ζ δ Kp , Lp , S p , p ( ( ) ( )) G λ 2n 2n+1 2n 2n+1 n!∞ = lim sup ζ δ p , p , δ p , p < C , ( ( 2n+1 2n+2) ( 2n 2n+1)) λ λ G n!∞ Common xed point results via simulation type functions Ë 17 which is a contradiction and so l = 0. Now, we show that fp g is a δ−Cauchy sequence. It is sucient to show that fp g is a δ−Cauchy se- 2n quence. Assume on the contrary. There exists ε > 0 such that we can nd two subsequencesfm g andfn g k k of positive integers satisfying n > m ≥ k such the following inequalities hold: k k δ p , p ≥ ε, and δ p , p < ε. (6) λ 2n 2m λ 2n −1 2m k k k k From (6) and (δ ), it follows that ε ≤ δ p , p = δ p , p 2n 2m 2n 2m λ maxfλ,λg k k k k (7) ≤ δ p , p + δ p , p 2n 2n −1 2n −1 2m λ λ k k k k < ε + δ p , p . 2n 2n −1 k k On taking limit as k ! ∞ in above relation, we obtain that lim δ p , p = ε. (8) 2n 2m λ k k k!∞ Also, we have α p β p ≥ 1 and by (1), 2n 2m −1 k k C ≤ ζ δ Kp , Lp , S p , p 2n 2m −1 2n 2m −1 G λ k k k k (9) = ζ δ p , p , S p , p , 2n +1 2m 2n 2m −1 k k k k where S p , p = max δ p , p , δ p , Kp , 2n 2m −1 λ 2n 2m −1 λ 2n 2n k k k k k k δ p ,Lp +δ p ,Kp λ( 2n 2m −1) λ( 2m −1 2n ) k k k k δ p , Lp , 2m −1 2m −1 λ k k (10) = max δ p , p , δ p , p , 2n 2m −1 2n 2n +1 λ λ k k k k δ p ,p +δ p ,p λ( 2n 2m ) λ( 2m −1 2n +1) k k k k δ p , p , . λ 2m −1 2m k k Also, from (6) and (δ ), it follows that δ p , p = δ p , p λ 2n +1 2m maxfλ,λg 2n +1 2m k k k k ≤ δ p , p + δ p , p λ 2n +1 2n λ 2n 2m k k k k = δ p , p + δ p , p λ 2n +1 2n maxfλ,λg 2n 2m k k k k (11) ≤ δ p , p + δ p , p λ 2n +1 2n λ 2n 2n −1 k k k k +δ p , p λ 2n −1 2m k k < δ p , p + δ p , p + ε, λ 2n +1 2n λ 2n 2n −1 k k k k δ p , p = δ p , p λ 2m −1 2n +1 maxfλ,λg 2m −1 2n +1 k k k k ≤ δ p , p + δ p , p λ 2m −1 2m λ 2m 2n +1 k k k k = δ p , p + δ p , p λ 2m −1 2m maxfλ,λg 2m 2n +1 k k k k ≤ δ p , p + δ p , p λ 2m −1 2m λ 2m 2n −1 k k k k ≤ +δ p , p λ 2n −1 2n +1 k k = δ p , p + δ p , p (12) λ 2m −1 2m λ 2m 2n −1 k k k k +δ p , p maxfλ,λg 2n −1 2n +1 k k ≤ δ p , p + δ p , p λ 2m −1 2m λ 2n −1 2m k k k k +δ p , p + δ p , p λ 2n −1 2n λ 2n 2n +1 k k k k < ε + δ p , p + δ p , p λ 2m −1 2m λ 2n −1 2n k k k k +δ p , p , λ 2n 2n +1 k k 18 Ë Mahpeyker Öztürk and Ekber Girgin and δ p , p = δ p , p 2n 2m −1 2n 2m −1 λ k k maxfλ,λg k k ≤ δ p , p + δ p , p 2n 2n −1 2n −1 2m −1 λ k k λ k k = δ p , p + δ p , p 2n 2n −1 2n −1 2m −1 λ k k maxfλ,λg k k (13) ≤ δ p , p + δ p , p 2n 2n −1 2n −1 2m λ k k λ k k +δ p , p 2m 2m −1 λ k k < ε + δ p , p + δ p , p . 2n 2n −1 2m 2m −1 λ k k λ k k Moreover, because (K, L) is a cyclic (α, β)−admissible pair we attain α p β p ≥ 1. Then by (1), 2n 2m −1 k k C ≤ ζ δ Kp , Lp , S p , p G 2n 2m −1 2n 2m −1 λ k k k k (14) = ζ δ p , p , S p , p 2n +1 2m 2n 2m −1 λ k k k k < G S p , p , δ p , p . 2n 2m −1 2n +1 2m k k λ k k Thence by (i) of Denition 1.10, we gain δ p , p < S p , p . (15) 2n +1 2m 2n 2m −1 λ k k k k Taking the limit as k ! ∞ and by using (6),(8), (10), (11), (12), (13), we procure lim δ p , p = lim S p , p = ε. (16) λ 2n +1 2m 2n 2m −1 k k k k k!∞ k!∞ Taking s = δ p , p , t = lim S p , p and using (13), (15) and (b) of Denition 1.9, we n 2n +1 2m n 2n 2m −1 k k k k k!∞ obtain C ≤ lim sup ζ δ Kp , Lp , S p , p G 2n 2m −1 2n 2m −1 λ k k k k k!∞ = lim sup ζ δ p , p , S p , p < C . λ 2n +1 2m 2n 2m −1 G k k k k k!∞ The attained last inequality causes a contradiction. Accordinglyfp g is a δ−Cauchy sequence. 2n As M is a δ−complete non-Archimedean modular metric space, there exists u 2 M such that δ p , u ! 0 ( n ) δ δ λ as n ! ∞. Now we present u is a common xed point of K and L. From (iii), we will presume that K is a δ−continuous mapping. Since δ p , j ! 0 as n ! ∞ and K is a δ−continuous mapping, we get ( 2n ) δ p , j = δ Kp , Kj ! 0, ( 2n+1 ) ( 2n+1 ) λ λ as n ! ∞. Hence p ! Kj as n ! ∞. But p ! j as n ! ∞. By the uniqueness of the limit, we achieve 2n+1 2n+1 Kj = j. From (iv), we have β j ≥ 1. This implies that α p β j ≥ 1 for all n 2 N. Next, we will demonstrate that j ( ) ( 2n) ( ) is a xed point of L. We presume that j ≠ Kj, that is δ j, Kj > 0. By K and L are generalized (α, β)−simulation ( ) contraction, we attain C ≤ ζ (δ (Kp , Lj) , S (p , j)) < G (S (p , j) , δ (Kp , Lj)) (17) 2n 2n 2n 2n G λ λ and then by (i) of Denition 1.10, we procure δ (Kp , Lj) < S (p , j) . (18) 2n 2n Taking limit as n ! ∞ in (18), we have δ (j, Lj) < δ (j, Lj), λ λ which is a contradiction. Thus j = Lj and hence j is a common xed point K and L. Ultimately, we will demonstrate the uniqueness of a common xed point of K and L. We assume that s is another common xed point, that is, δ (j, s) ≠ 0. From hypothesis, we gain α (j) ≥ 1 and β (s) ≥ 1 and thus α (j) β (s) ≥ 1. By using (1), C ≤ ζ δ Kj, Ls , S j, s < G S j, s , δ Kj, Ls (19) ( ( ) ( )) ( ( ) ( )) G λ λ Common xed point results via simulation type functions Ë 19 and from (i) of Denition 1.10, we have δ Kj, Ls < S j, s ( ) ( ) n o δ j,Ls +δ s,Kj λ( ) λ( ) = max δ (j, s) , δ (j, Kj) , δ (s, Ls) , λ λ λ n o δ j,s +δ s,j ( ) ( ) λ λ = max δ (j, s) , δ (j, Kj) , δ (s, Ls) , λ λ λ = maxfδ j, s , 0g ( ) = δ j, s , ( ) which is a contradiction. Hence j = s. 3 Consequences This section aims to acquaint some consequences from the main theorem. Corollary 3.1 Let M be a δ−complete non-Archimedean modular metric space and K, L : M ! M be two δ δ mappings. Presume that there exists k 2 0, 1 such that ( ) p, q 2 M with α p β q ≥ 1 ) κ ( ) ( ) n o δ (p,Lq)+δ (q,Kp) λ λ δ Kp, Lq ≤ k max δ p, q , δ p, Kp , δ q, Lq , . ( ) ( ) ( ) ( ) λ λ λ λ Suppose that the following circumstances hold: i. (K, L) is a cyclic α, β −admissible pair, ( ) ii. there exists p 2 M such that α p ≥ 1, 0 ( 0) iii. K or L is a δ−continuous mapping, iv. iffp g is a sequence in M such that p ! ξ and α p ≥ 1 and β p ≥ 1 for all n 2 N, then α p ≥ 1 n n ( 2n) ( 2n−1) ( ) and β p ≥ 1. ( ) Then, K and L admit a common xed point. Corollary 3.2 Let M be a complete non-Archimedean modular metric space and K, L : M ! M be two δ δ δ mappings. Assume that there exists a continuous function φ : 0,∞ ! 0,∞ with φ t = 0 if and only if [ ) [ ) ( ) t = 0 such that p, q 2 M with α p β q ≥ 1 ) ( ) ( ) n o δ p,Lq +δ q,Kp ( ) ( ) λ λ δ (Kp, Lq) ≤ max δ (p, q) , δ (p, Kp) , δ (q, Lq) , λ λ λ λ n o δ p,Lq +δ q,Kp ( ) ( ) λ λ −φ max δ (p, q) , δ (p, Kp) , δ (q, Lq) , . λ λ λ Suppose that the following conditions hold: i. (K, L) is a cyclic (α, β)−admissible pair, ii. there exists p 2 M such that α (p ) ≥ 1, 0 0 iii. K or L is a δ−continuous mapping, iv. iffp g is a sequence in M such that p ! p and α (p ) ≥ 1 and β (p ) ≥ 1 for all n 2 N, then α (p) ≥ 1 n n 2n 2n−1 and β (p) ≥ 1. Then, K and L admit a common xed point. We will replace the condition of cyclic (α, β)-admissible pair with cyclic (α, β)-admissible mapping in our main result, that is, we will take K = L. Then, we have the following results. Corollary 3.3 Let M be a complete non-Archimedean modular metric space and K : M ! M be a given δ δ δ mapping. Assume that there exists ζ 2 T such that p, q 2 M with α p β q ≥ 1 ) ( ) ( ) n o δ p,Kq +δ q,Kp λ( ) λ( ) ζ δ (Kp, Lq) , max δ (p, q) , δ (p, Kp) , δ (q, Lq) , ≥ C . λ λ λ λ Suppose that the following conditions hold: i. K is a cyclic α, β −admissible mapping; ( ) 20 Ë Mahpeyker Öztürk and Ekber Girgin ii. there exists p 2 M such that α p ≥ 1 and β p ≥ 1; ( ) ( ) 0 0 0 iii. V is a δ-continuous mapping, or; iv. iffp g is a sequence in M such that p ! p and β p ≥ 1 for all n 2 N, then β p ≥ 1. n n ( n) ( ) Then, K admits a xed point. Corollary 3.4 Let M be a complete non-Archimedean modular metric space and K : M ! M be a given δ δ δ mapping. Assume that there exists ζ 2 T such that α p β q ≥ 1 ) ζ δ Kp, Kq , δ p, q ≥ C , ( ) ( ) ( ( ) ( )) λ λ G p, q 2 M . Suppose that the following conditions hold: i. K is a cyclic α, β −admissible mapping; ( ) ii. there exists p 2 M such that α p ≥ 1 and β p ≥ 1; ( ) ( ) 0 0 0 iii. K is a δ-continuous mapping, or; iv. iffp g is a sequence in M such that p ! ξ and β p ≥ 1 for all n 2 N, then β p ≥ 1. n n ( n) ( ) Then, K admits a xed point. Example 3.5 Let M = R, δ p, q = jp − qj for all p 2 M . Dene the mapping K : M ! M as follows: ( ) δ δ δ δ , if p 2 0, 1 [ ] Kp = p , if p > 1. and α, β : M ! 0, +∞ are dened by [ ) 1, p 2 [0, 1] α (p) = β (p) = 0, otherwise. p.q If we consider the functions ζ (p, q) = q−p, G (q, p) = q−p and C = a , a > 2, then all the hypotheses 3 12 of Corollary 3.4 are satised and p = 0 is a unique xed point of K. p p p p  p p 1 6− 5 On the other hand, δ K 6, K 5 = > = δ 6, 5 , λ > 0. Thus K is not a Banach contraction λ λ λ λ with respect to M . 4 Common xed points on non-Archimedean modular metric spaces with a directed graph Let M be a non-Archimedean modular metric space and Δ = f(p, p) : p 2 Mg denotes the diagonal of M × M . Let Z be a directed graph such that the set V(Z) of its vertices coincides with M and J(Z) be the set δ δ of edges of the graph such that Δ  J(Z). Let M be a non-Archimedean modular metric space endowed with a graph Z and K, L : M ! M . We set δ δ δ M = fp 2 M : p, Kp 2 J Z and Kp, LKp 2 J Z g . ( ) ( ) ( ) ( ) KL Denition 4.1 Let M be a non-Archimedean modular metric space endowed with a directed graph Z and (K, L) be a cyclic (α, β) admissible pair. We say that K and L are generalized (α, β)-Z-simulation contraction if there exists ζ 2 T such that α (p) β (η) ≥ 1 ) ζ (δ (Kp, Lq) , S (p, q)) ≥ C , (20) n o κ p,Lq +δ q,Lp λ( ) λ( ) where S (p, q) = max δ (p, q) , δ (p, Kp) , δ (q, Lq) , , for all (p, q) 2 J (Z). λ λ λ Theorem 4.2 Let M be a δ-complete non-Archimedean modular metric space endowed with a directed graph Z. K and L are generalized α, β -Z-simulation contraction. Suppose that the following conditions hold: ( ) i. there exists p 2 M such that α(p ) ≥ 1; 0 0 KL ii. iffp g is a sequence in M such that p ! p, p , p 2 J Z for all n ≥ 0; n n ( 2n 2n+1) ( ) Common xed point results via simulation type functions Ë 21 ii . L is δ-continuous, p, p 2 J Z for all k ≥ 0 and α(p ) ≥ 1 then α(p) ≥ 1, or; a ( ) 2n +1 2n ii . K is δ-continuous, p , p 2 J Z for all k ≥ 0 and β(p ) ≥ 1, then β(p) ≥ 1; ( ) b 2n 2n+1 iii. there is a sequencefp g in M , such that n2N δ p , Kp 2 J Z ) p , Kp 2 J Z ( ) ( ) ( ) ( ) 2n 2n 2n+2 2n+2 and p , Lp 2 J Z ) p , Lp 2 J Z ; ( ) ( ) ( ) ( ) 2n+1 2n+1 2n+3 2n+3 Then K and L hold a common xed point. Proof Let p be a given point in M , then p , Kp 2 J Z and Kp , LKp 2 J Z , that is, p , p 2 0 ( 0 0) ( ) ( 0 0) ( ) ( 0 1) KL J Z and p , Lp = p , p 2 J Z . From (iii), it follows that p , Kp 2 J Z and p , Lp 2 J Z . ( ) ( 1 1) ( 1 2) ( ) ( 2 2) ( ) ( 3 3) ( ) That is, p , p 2 J Z and p , p 2 J Z . Continuing this way, we can obtain a sequence fp g in ( 2 3) ( ) ( 3 4) ( ) n M such that p , Kp 2 J Z and p , Lp 2 J Z for all n 2 N. Also, p , p 2 J Z and ( 2n 2n) ( ) ( 2n+1 2n+1) ( ) ( 2n 2n+1) ( ) p , p 2 J Z for all n 2 N. Using arguments as in the proof of the Theorem 2.2, we obtain that ( 2n+1 2n+2) ( ) δ p , p ! 0 as n ! ∞, (21) ( n n+1) andfp g is δ-Cauchy sequence. Since M is a δ-complete, there exists a p 2 M such that δ δ δ p , p ! 0 as n ! ∞. * * Now we show that p is a common xed point. As δ p , p ! 0 and p , p 2 J Z , so there n ( ) ( ) λ 2n 2n+1 exists a subsequence p of fp g such that from (ii) either L is δ-continuous, p , p 2 J Z ( ) 2n 2n 2n +1 k k * * * and α p ≥ 1 or K is δ-continuous, p , p 2 J Z and β p ≥ 1. Assume that L is δ-continuous, ( ) 2n * * p , p 2 J Z and α p ≥ 1. Thus ( ) 2n +1 * * δ Lp , Lp = δ p , Lp ! 0 as k ! ∞. 2n +1 2n +2 λ k λ k * * * This implies that Lp = p . From Denition 4.1, we have α p β p ≥ 1 2n +1 * * C ≤ ζ δ Kp , Lp , S p , p G λ 2n +1 2n +1 k k (22) * * < G S p , p , δ Kp , Lp 2n +1 λ 2n +1 k k and then by (i) of Denition 1.10, we get * * δ Kp , Lp < S p , p . (23) λ 2n +1 2n +1 k k where * * * * S p , p = maxfδ p , p , δ p , Kp , δ p , Lp , 2n +1 λ 2n +1 λ λ 2n +1 2n +1 k k k k * * δ p , Lp , δ p , Kp . λ 2n +1 λ 2n +1 k k On taking the limit as k ! ∞ on both sides of inequality (23), we have * * * * δ p , Kp < δ p , Kp , (24) λ λ * * * a contradiction. Thus Kp = p , and hence p is a common xed point. Similarly, the result follows if we * * suppose that K is δ-continuous, p , p 2 J Z and β p ≥ 1. ( ) 2n We note that Theorem 4.2 does not guarantee the uniqueness of a common xed point. To obtain the uniqueness, additional assumptions as given in the following theorem are needed. Theorem 4.3 Let M be a δ-complete non-Archimedean modular metric space endowed with a directed graph Z. K and L are generalized α, β -Z-simulation contraction. Suppose that the following conditions hold: ( ) i. there exists p 2 M such that α(p ) ≥ 1; 0 KL 0 ii. iffp g is a sequence in M such that p ! p, p , p 2 J Z for all n ≥ 0; n n ( ) ( ) 2n 2n+1 ii . L is δ-continuous, p, p 2 J Z for all k ≥ 0 and α(p ) ≥ 1 then α(p) ≥ 1, or; a ( ) 2n +1 2n ii . K is δ-continuous, p , p 2 J Z for all k ≥ 0 and β(p ) ≥ 1, then β(p) ≥ 1; ( ) b 2n 2n+1 k 22 Ë Mahpeyker Öztürk and Ekber Girgin iii. there is a sequencefp g in M , such that n2N δ p , Kp 2 J Z ) p , Kp 2 J Z ( ) ( ) ( ) ( ) 2n 2n 2n+2 2n+2 and p , Lp 2 J Z ) p , Lp 2 J Z ; ( ) ( ) ( ) ( ) 2n+1 2n+1 2n+3 2n+3 * * * * * * iv. there exist p , q 2 Fix(K, L) such that p , q 2 J Z , α(p ) ≥ 1 and β(q ) ≥ 1. ( ) Then K and L hold a unique common xed point. * * * * * * Proof Let p , q be common xed points of K and L, then p , q 2 J Z . Also, since α(p ) ≥ 1 and β(q ) ≥ 1. ( ) Thus, by Denition 4.1, we get that * * * * * * * * C ≤ ζ δ Kp , Lq , S p , q < G S p , q , δ Kp , Lq (25) G λ λ and from (i) of the Denition 1.10, we have * * * * δ Kp , Lq < S p , q * * = max δ p , q , 0 * * = δ p , q . * * which is a contradiction. Hence p = q . Example 4.4 Let M = [0, 1], δ (p, q) = jp − qj for all p 2 M and E (G) = f(p, q) : p, q 2 [0, 1]g . Dene δ λ δ the mappings K, L : M ! M as follows: p p Kp = , Lp = , p 2 M . 2 3 and α, β : M ! [0, +∞) are dened by 1, p 2 0, 1 ( ) α p = β p = ( ) ( ) 0, otherwise. p.q If we consider the functions ζ p, q = q−p, G q, p = q−p and C = , a > 2, then all the hypotheses ( ) ( ) a 4 24 of Theorem 4.3 are satised and p = 0 is a unique common xed point. 5 Application to dynamic programming The purpose of this section is to present an application on dynamic programming. Throughout this section, we assume that A and B are Banach spaces, W  A and D  B is a decision space. Now, consider the following system of functional equations: P (p) = supfs (p, q) + χ (p, q, P (τ (p, q)))g , (26) q2D Q p = supfs p, q + ς p, q, Q τ p, q g , (27) ( ) ( ) ( ( ( ))) q2D where τ : W × D ! W , r : W × D ! R, χ : W × D × R ! R, ς : W × D × R ! R. Let M = B(W) denotes the space of all bounded real-valued function on W. Consider the metric given by δ (m, n) = supjm (p) − n (p)j , for all m, n 2 A and λ > 0. p2W Then M is a δ−complete non-Archimedean modular metric space. Now, take the mappings E, F : M ! M as δ δ Em p = supfs p, q + χ p, q, m τ p, q g , (28) ( ) ( ) ( ( ( ))) q2D Common xed point results via simulation type functions Ë 23 and Fm p = supfs p, q + ς p, q, m τ p, q g , (29) ( ) ( ) ( ( ( ))) q2D where p 2 W and m 2 M . If the functions s, χ, τ are bounded, then E and F are well-dened. Theorem 5.1 Suppose that there exists k 2 [0, 1) such that for each (p, q) 2 W × D and m , m 2 M , the 1 2 inequality jχ (p, q, m (τ (p, q))) − ς (p, q, m (τ (p, q)))j ≤ kjm − m j , 1 2 1 2 holds. Then, E and F admit a common xed point in M . Proof Let ε > 0 be an arbitrary real number, p 2 W and m , m 2 M . Then by (28) and (29), there exist 1 2 δ q , q 2 D such that 1 2 Em p < s p, q + χ p, q , m τ p, q + ε, (30) ( ) ( ) ( ( ( ))) 1 1 1 1 1 Fm p < s p, q + ς p, q , m τ p, q + ε, (31) ( ) ( ) ( ( ( ))) 2 2 2 2 2 Em (p) ≥ s (p, q ) + χ (p, q , m (τ (p, q ))) , (32) 1 2 2 1 2 and Fm p ≥ s p, q + ς p, q , m τ p, q . (33) ( ) ( ) ( ( ( ))) 2 1 1 2 1 Then from (30) and (33), it follows easily that Em p − Fm p ≤ χ p, q , m τ p, q + ε − ς p, q , m τ p, q ( ) ( ) ( ( ( ))) ( ( ( ))) 1 2 1 1 1 1 2 1 ≤ jχ p, q , m τ p, q − ς p, q , m τ p, q j + ε ( ( ( ))) ( ( ( ))) 1 1 1 1 2 1 ≤ kjm − m j + ε. 1 2 Similarly, from (31) and (32), we have Fm p − Em p ≤ kjm − m j + ε. ( ) ( ) 2 1 1 2 Thus, we deduce from above inequalities that jEm p − Fm p j ≤ kjm − m j + ε. (34) ( ) ( ) 1 2 1 2 Since the inequality (34) is valid for any p 2 W, then 1 1 δ Em , Fm = jEm − Fm j ≤ kjm − m j + ε = kδ m , m + ε, ( ) ( ) λ 1 2 1 2 1 2 λ 1 2 δ δ also ε > 0 is arbitrary, so δ Em , Fm ≤ kδ m , m . ( ) ( ) λ 1 2 λ 1 2 Now, taking ζ a, b = kb − a, G b, a = b − a, then for C = 0, S p, q = δ p, q , and α p β q = 1 ( ) ( ) ( ) ( ) ( ) ( ) G λ in Theorem 2.2. Hence all conditions of Theorem 2.2 are satised. Therefore E and F have a common xed point, that is, the functional equation (26) and (27) have a common solution in M . Acknowledgments: The authors would like to thank the referees for the valuable comments and suggestions on this manuscript. References [1] V. V. Chistyakov, Modular metric spaces, I: Basic concepts, Nonlinear Anal., 72 (2010), 1-14. [2] V. V. Chistyakov, Modular metric spaces, II: Application to superposition operators, Nonlinear Anal., 72 (2010), 15-30. [3] M. Paknazar, M.A. Kutbi, M. Demma, P. Salimi, On non-Archimedean modular metric space and some nonlinear contraction mappings, Available online: https://pdfs.semanticscholar.org/ (accessed on 30 October 2019). 24 Ë Mahpeyker Öztürk and Ekber Girgin [4] C. Mongkolkeha, W. Sintunavarat, P. Kumam, Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory Appl., 2011: 93 (2011), https://doi.org/10.1186/1687-1812-2011-93. [5] F. Khojasteh, S. Shukla, S. Radenovic, A new approach to the study of xed point theorems via simulation functions, Filomat, 29 (2015), no. 6, 1189-1194. [6] H. Argoubi, B. Samet, C. Vetro, Nonlinear contractions involving simulation functions in a metric space with a partial order, J. Nonlinear Sci. Appl., 8 (2015), 1082-1094. [7] M. Abbas, A. Latif, Y. Suleiman Fixed points for cyclic R−contractions and solution of nonlinear Volterra integro-dierantial equations, Fixed Point Theory Appl., 2016:61 (2016), https://doi.org/10.1186/s13663-016-0552-1. [8] M. Asadi, M. Gabeleh, C. Vetro, A new approach to the generalization of Darbo’s xed point problem by using simulation functions with application to integral equations, Result Math., 78:86 (2019), https://doi.org/10.1007/s00025-019-1010-2. [9] M. Asadi, M. Azhini, E. Karapınar, H. Monfared, Simulation functions over M−metric spaces, East Asian Math. Journal, 33(2015), no. 5, 559-570. [10] M. Monfared, M. Asadi, M. Gabeleh, A. Farajzadeh, New generalization of Darbo’s xed point theorem via α−admissible simulation functions with application, Sahand Communications in Mathematical Analysis, 17(2020), no. 2, 161-171. [11] B. Samet, Best proximity point results in partially ordered metric spaces via simulation functions, Fixed Point Theory Appl., 2015:232 (2015), https://doi.org/10.1186/s13663-015-0484-1. [12] F. Tchier, C. Vetro, F. Vetro, Best approximation and variational inequality problems involving a simulation function, Fixed Point Theory Appl., 2016:26 (2016), https://doi.org/10.1186/s13663-016-0512-9. [13] A.H. Ansari, Note on φ − ψ−contractive type mappings and related xed point, in roceedings of the Second Regional Con- ference on Mathematical Sciences and Applications, (Tonekabon, Iran), 2014 September, Payame Noor University, (2014), 377–380. [14] S. Radenovic, S. Chandok, Simulation type functions and coincidence point results, Filomat, 32 (2018), no. 1, 141-147. [15] S. Alizadeh, F. Moradlou, P. Salimi, Some xed point results for α, β − ψ, ϕ −contractive mappings, Filomat, 28 (2014), ( ) ( ) no. 3, 635-647. [16] A.Latif, A.H. Ansari, Fixed points and functional equation problems via cyclic admissible generalized contractive type map- pings , J. Nonlinear Sci. Appl., 9 (2016), 1129-1142. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Topological Algebra and its Applications de Gruyter

Common fixed point results via simulation type functions in non-Archimedean modular metric spaces and applications

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Topol. Algebra Appl. 2022; 10:13–24 Research Article Open Access Mahpeyker Öztürk* and Ekber Girgin Common xed point results via simulation type functions in non-Archimedean modular metric spaces and applications https://doi.org/10.1515/taa-2020-0109 Received 29 September, 2021; accepted 7 January, 2022 Abstract: In this study, we demonstrate the existence and uniqueness of common xed points of a generalized (α, β)− simulation contraction on a non-Archimedean modular metric space. We achieve some consequences in non-Archimedean modular metric spaces as an application, using the structure of a directed graph. Even- tually, we contemplate the existence of solutions to a class of functional equations standing up dynamic programming with the help of our outcomes. Keywords: Non-Archimedean modular metric space, Simulation function, Cyclic (α, β)-admissible pair MSC: 47H10, 54H25, 37C25 1 Introduction Subsequently, the letters R, R , and N will specify the set of all real numbers, the set of all nonnegative real numbers, and the set of all positive integer numbers, respectively. In 2010, Chistyakov [1, 2] set up a new structure named modular metric, which is an extension of metric and a linear modular. Denition 1.1 [1, 3] Let M be non-void set and δ : 0,∞ ×M×M ! 0,∞ be a function fullling the following ( ) [ ] statements: for all λ, μ > 0 and p, q, k 2 M, δ . p = q if and only if δ p, q = 0; ( ) 1 λ δ . δ p, q = δ p, q ; ( ) ( ) 2 λ λ δ . δ p, q ≤ δ p, k + δ k, q . ( ) ( ) μ ( ) 3 λ+μ λ So, δ is entitled modular metric in M, and in the present case, M is a modular metric space. If we substitute (δ ) by δ . δ p, p = 0 for all λ > 0 and p 2 M, ( ) 4 λ then δ is said to be a pseudomodular metric on M. A modular metric δ on M is named regular if the following weaker version of (δ ) is provided for some λ > 0; δ . p = q if and only if δ p, q = 0. ( ) 5 λ As well, δ is a convex function if, for λ, μ > 0 and p, q, k 2 M, the inequality holds: λ μ δ . δ p, q ≤ δ p, k + δ k, q . ( ) ( ) μ ( ) 6 λ+μ λ λ+μ λ+μ For all λ, μ > 0 and p, q, k 2 M , if the following property δ . δ p, q ≤ δ p, k + δ k, q ( ) ( ) μ ( ) 7 maxfλ,μg λ *Corresponding Author: Mahpeyker Öztürk: Department of Mathematics, Sakarya University, Sakarya, Turkey, E-mail: mahpeykero@sakarya.edu.tr Ekber Girgin: Department of Mathematics, Sakarya University, Sakarya, Turkey, E-mail: girginekber@gmail.com Open Access. © 2022 Mahpeyker Öztürk and Ekber Girgin, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. 14 Ë Mahpeyker Öztürk and Ekber Girgin is put in a place of (δ ), then, we attain a new space with some new features. This new structure M is named a non-Archimedean modular metric space. For simplicity, throughout the study, we will write: δ p, q = δ λ, p, q ( ) ( ) for all λ > 0 and p, q 2 M. Denition 1.2 [1, 2] Presume that the function δ be a pseudomodular on M and δ 2 M be a xed element. The hereinbelow two sets M and M are called modular spaces (around δ ). δ 0 M = M (p ) = fp 2 M : δ (p, p ) as λ ! ∞g 0 0 δ δ λ and * * M = M (p ) = fp 2 M : 9λ = λ p > 0 such that δ p, p < ∞g . 0 ( ) ( 0) δ δ λ It is noticeable that M  M , but this inclusion may be proper in general. Presume that from [1, 2], δ is modular on M; we derive that the modular space M can be equipped with a (nontrivial) metric, induced by δ and presented by: d p, q = inffλ > 0 : δ p, q < λg ( ) ( ) δ λ for all p, q 2 M . Note that if δ is a convex modular on M, then according to [1, 2], the two modular spaces coincide, i.e., * * M = M , and this common set can be endowed with the metric d given by: δ δ δ d p, q = inffλ > 0 : δ p, q < 1,8p, q 2 M g , ( ) ( ) δ λ δ which is called Luxemburg distance. Denition 1.3 [4] Let M be a modular metric space, S be a subset of M and p be a sequence in M . ( n) δ δ δ n2N Then, i. A sequence p is called δ−convergent to p 2 M if and only if δ p , p ! 0 as n ! ∞ for all λ > 0, ( n) ( n ) δ λ n2N p is said to be the δ−limit of (p ). ii. A sequence p is called δ−Cauchy if δ p , p ! 0, as m, n ! ∞ for all λ > 0. ( n) ( n m) n2N iii. A subset S is called δ−complete if any δ−Cauchy sequence in S is δ−convergent to the point of S. Denition 1.4 [4] Let M be a modular metric space. The mapping K : M ! M is δ−continuous if δ δ δ δ p , p ! 0, provided to δ Kp , Kp ! 0 as n ! ∞. ( n ) ( n ) λ λ Denition 1.5 [5] A simulation function is a mapping ζ : 0,∞ ! R admitting the following features: [ ) ζ . ζ 0, 0 = 0, ( ) ζ . ζ p, q < p − q, for all p, q > 0, ( ) ζ . iffq g andfp g are sequences in 0,∞ such that lim q = lim p = l and l 2 0,∞ , then n n ( ) n n ( ) n!∞ n!∞ lim sup ζ q , p < 0. ( n n) n!∞ Argoubi et al.[6] modied the denition of simulation function and so introduced it as: Denition 1.6 A simulation function is a mapping ζ : 0,∞ ! R satisfying the followings: [ ) i. ζ p, q < p − q, for all p, q > 0, ( ) ii. iffq g andfp g are sequences in 0,∞ such that lim q = lim p > 0, and q < p , then lim sup ζ q , p < n n ( ) n n n n ( n n) n!∞ n!∞ n!∞ Example 1.7 [6] The function ζ : 0,∞ × 0,∞ ! R is a simulation function specied by c [ ) [ ) 1 if q, p = 0, 0 , ( ) ( ) ζ q, p = c ( ) cp − q otherwise, where c 2 0, 1 . ( ) One may point to [5]-[12] for some detailed information about simulation functions. Denition 1.8 [13] Let G : 0,∞ ! R is called C−class function when it has continuity property and admits [ ) the following features: Common xed point results via simulation type functions Ë 15 i. G p, q ≤ p, ( ) ii. G p, q = p implies that either p = 0 or q = 0, for all p, q 2 0,∞ . ( ) [ ) Denition 1.9 [14] Let ζ : 0,∞ ! R be a function admitting the following features: [ ) a. ζ q, p < G p, q for all t, s > 0, where G : 0,∞ ! R is a C−class function, ( ) ( ) [ ) b. iffq g andfp g are sequences in 0,∞ such that lim q = lim p > 0, and q < p , then lim sup ζ q , p < n n ( ) n n n n ( n n) n!∞ n!∞ n!∞ C . Then ζ is named a C −simulation function. Denition 1.10 [14] A mapping G : 0,∞ ! R has the property C , if there exists a C ≥ 0 such that [ ) G G i. G p, q > C implies p > q, ( ) ii. G q, q ≤ C for all q 2 0,∞ . ( ) [ ) Let T denotes the family of all C −simulation functions ζ : 0,∞ ! R. [ ) G G Recently, Alizadeh et al. [15] dened the concept of cyclic α, β −admissible mapping as indicated below: ( ) Denition 1.11 Let M be a non-void set, K be a self-mapping on M and α, β : M ! 0,∞ be two mappings. [ ) K is named a cyclic α, β −admissible mapping if ( ) (i) for some p 2 M, α p ≥ 1 implies β Kp ≥ 1; ( ) ( ) (ii) for some p 2 M, β p ≥ 1 implies α Kp ≥ 1. ( ) ( ) Besides, they demonstrated xed point theorems by using the structure of cyclic α, β −admissible mapping. ( ) Example 1.12 [15] Let K : R ! R be a function described as Kp = − p + p . Presume that α, β : R ! R are furnishing mappings for all p, q 2 R such that p −q α (p) = e and β (q) = e . Then K is a cyclic α, β −admissible mapping. ( ) The generalized cyclic α, β −admissible mappings have been generalized by Latif et al. [16], as follows. ( ) Denition 1.13 Let M be a non-void set, K, L be two self-mappings on M and α, β : M ! 0,∞ be two [ ) mappings. (K, L) is called a cyclic α, β −admissible pair if ( ) (i) for some p 2 M α p ≥ 1 implies β Kp ≥ 1; ( ) ( ) (ii) for some p 2 M β p ≥ 1 implies α Lp ≥ 1. ( ) ( ) Remark 1.16 In the above denition, if we acquire K = L, then K is a cyclic α, β -admissible mapping. ( ) Throughout this study, δ will be used as a convex and regular function. 2 Main results Initially, this section aims to innovate the generalized (α, β)−simulation contraction. Besides, common xed point results are procured satisfying generalized (α, β)−simulation contraction in non-Archimedean modular metric spaces. Denition 2.1 Let M be a non-Archimedean modular metric space, K, L : M ! M be two self-mappings δ δ δ and α, β : M ! [0,∞) be two functions. The pair (K, L) is named a generalized (α, β)−simulation contrac- tion if there exists ζ 2 T such that α p β q ≥ 1 ) ζ δ Kp, Lq , S p, q ≥ C , (1) ( ) ( ) ( ( ) ( )) λ G n o δ p,Lq +δ q,Kp ( ) ( ) λ λ where S (p, q) = max δ (p, q) , δ (p, Kp) , δ (q, Lq) , , for all p, q 2 M . λ λ λ δ Theorem 2.2 Let M be a δ−complete non-Archimedean modular metric space and K and L are generalized α, β −simulation contraction. Presume that the following circumstances hold: ( ) i. (K, L) is a cyclic α, β −admissible pair, ( ) ii. there exists p 2 M such that α p ≥ 1, 0 ( 0) iii. K or L is a δ−continuous mapping, iv. if fp g is a sequence in M such that p ! p and α p ≥ 1, β p ≥ 1 for all n 2 N, then α p ≥ 1 n n ( 2n) ( 2n−1) ( ) and β p ≥ 1. ( ) 16 Ë Mahpeyker Öztürk and Ekber Girgin Then K and L hold a common xed point. Additionally, if α p β q ≥ 1 for all p, q 2 Fix K, L , then K and ( ) ( ) ( ) L hold a unique common xed point. Proof Let p 2 M be such that α p ≥ 1. We will construct a sequencefp g in M by ( ) n 0 0 δ δ p = Kp , 2n+2 2n+1 (2) p = Lp , 2n+1 2n for all n 2 N. Also, as K, L is a cyclic α, β −admissible pair and α p ≥ 1, then ( ) ( ) ( ) β p = β Kp ≥ 1 ( ) ( ) 1 0 which implies α p = α Lp ≥ 1. ( ) ( ) 2 1 By proceeding with this process, we get α (p ) ≥ 1 and β (p ) ≥ 1 for all n 2 N. Thus, α (p ) β (p ) ≥ 1 2n 2n+1 2n 2n+1 for all n 2 N. From (1), we have C ≤ ζ δ Kp , Lp , S p , p ( ( ) ( )) G λ 2n 2n+1 2n 2n+1 = ζ δ p , p , S p , p ( ( ) ( )) λ 2n+1 2n+2 2n 2n+1 < G S p , p , δ p , p . ( ( ) ( )) 2n 2n+1 λ 2n+1 2n+2 Further, using (i) of Denition 1.10, we achieve δ p , p < S p , p , (3) ( ) ( ) λ 2n+1 2n+2 2n 2n+1 where S p , p = maxfδ p , p , δ p , Kp , δ p , Lp , ( ) ( ) ( ) ( ) 2n 2n+1 λ 2n 2n+1 λ 2n 2n λ 2n+1 2n+1 p ,Lp + p ,Kp ( 2n 2n+1) ( 2n+1 2n) = maxfδ p , p , δ p , p , δ p , p , ( ) ( ) ( ) λ 2n 2n+1 λ 2n 2n+1 λ 2n+1 2n+2 δ (p ,p )+δ (p ,p ) λ 2n 2n+2 λ 2n+1 2n+1 = maxfδ p , p , δ p , p , δ p , p , ( ) ( ) ( ) λ 2n 2n+1 λ 2n 2n+1 λ 2n+1 2n+2 δ (p ,p )+δ (p ,p ) maxfλ,λg 2n 2n+2 λ 2n+1 2n+1 ≤ maxfδ p , p , δ p , p , ( ) ( ) λ 2n 2n+1 λ 2n+1 2n+2 δ (p ,p )+δ (p ,p ) λ 2n 2n+1 λ 2n+1 2n+2 = maxfδ (p , p ) , δ (p , p )g . 2n 2n+1 2n+1 2n+2 λ λ If maxfδ p , p , δ p , p g = δ p , p ( 2n 2n+1) ( 2n+1 2n+2) ( 2n+1 2n+2) λ λ λ for each n 2 N, then from (3), we have a contradiction. So, S p , p = δ p , p for all n 2 N. ( ) ( ) 2n 2n+1 λ 2n 2n+1 Consequently, (3) gives δ p , p < δ p , p . (4) ( ) ( ) λ 2n+1 2n+2 λ 2n 2n+1 Hence for all n 2 N, continuing this way we get δ p , p < δ p , p . ( 2n 2n+1) ( 2n−1 2n−2) λ λ So fδ p , p g is a monotonically decreasing sequence of nonnegative real numbers, and hence there ( 2n 2n+1) exists l ≥ 0 such that lim δ p , p = l. (5) ( 2n 2n+1) n!∞ Assume that l > 0. Then using (1) and (b) of the Denition 1.9, we have C ≤ lim sup ζ δ Kp , Lp , S p , p ( ( ) ( )) G λ 2n 2n+1 2n 2n+1 n!∞ = lim sup ζ δ p , p , δ p , p < C , ( ( 2n+1 2n+2) ( 2n 2n+1)) λ λ G n!∞ Common xed point results via simulation type functions Ë 17 which is a contradiction and so l = 0. Now, we show that fp g is a δ−Cauchy sequence. It is sucient to show that fp g is a δ−Cauchy se- 2n quence. Assume on the contrary. There exists ε > 0 such that we can nd two subsequencesfm g andfn g k k of positive integers satisfying n > m ≥ k such the following inequalities hold: k k δ p , p ≥ ε, and δ p , p < ε. (6) λ 2n 2m λ 2n −1 2m k k k k From (6) and (δ ), it follows that ε ≤ δ p , p = δ p , p 2n 2m 2n 2m λ maxfλ,λg k k k k (7) ≤ δ p , p + δ p , p 2n 2n −1 2n −1 2m λ λ k k k k < ε + δ p , p . 2n 2n −1 k k On taking limit as k ! ∞ in above relation, we obtain that lim δ p , p = ε. (8) 2n 2m λ k k k!∞ Also, we have α p β p ≥ 1 and by (1), 2n 2m −1 k k C ≤ ζ δ Kp , Lp , S p , p 2n 2m −1 2n 2m −1 G λ k k k k (9) = ζ δ p , p , S p , p , 2n +1 2m 2n 2m −1 k k k k where S p , p = max δ p , p , δ p , Kp , 2n 2m −1 λ 2n 2m −1 λ 2n 2n k k k k k k δ p ,Lp +δ p ,Kp λ( 2n 2m −1) λ( 2m −1 2n ) k k k k δ p , Lp , 2m −1 2m −1 λ k k (10) = max δ p , p , δ p , p , 2n 2m −1 2n 2n +1 λ λ k k k k δ p ,p +δ p ,p λ( 2n 2m ) λ( 2m −1 2n +1) k k k k δ p , p , . λ 2m −1 2m k k Also, from (6) and (δ ), it follows that δ p , p = δ p , p λ 2n +1 2m maxfλ,λg 2n +1 2m k k k k ≤ δ p , p + δ p , p λ 2n +1 2n λ 2n 2m k k k k = δ p , p + δ p , p λ 2n +1 2n maxfλ,λg 2n 2m k k k k (11) ≤ δ p , p + δ p , p λ 2n +1 2n λ 2n 2n −1 k k k k +δ p , p λ 2n −1 2m k k < δ p , p + δ p , p + ε, λ 2n +1 2n λ 2n 2n −1 k k k k δ p , p = δ p , p λ 2m −1 2n +1 maxfλ,λg 2m −1 2n +1 k k k k ≤ δ p , p + δ p , p λ 2m −1 2m λ 2m 2n +1 k k k k = δ p , p + δ p , p λ 2m −1 2m maxfλ,λg 2m 2n +1 k k k k ≤ δ p , p + δ p , p λ 2m −1 2m λ 2m 2n −1 k k k k ≤ +δ p , p λ 2n −1 2n +1 k k = δ p , p + δ p , p (12) λ 2m −1 2m λ 2m 2n −1 k k k k +δ p , p maxfλ,λg 2n −1 2n +1 k k ≤ δ p , p + δ p , p λ 2m −1 2m λ 2n −1 2m k k k k +δ p , p + δ p , p λ 2n −1 2n λ 2n 2n +1 k k k k < ε + δ p , p + δ p , p λ 2m −1 2m λ 2n −1 2n k k k k +δ p , p , λ 2n 2n +1 k k 18 Ë Mahpeyker Öztürk and Ekber Girgin and δ p , p = δ p , p 2n 2m −1 2n 2m −1 λ k k maxfλ,λg k k ≤ δ p , p + δ p , p 2n 2n −1 2n −1 2m −1 λ k k λ k k = δ p , p + δ p , p 2n 2n −1 2n −1 2m −1 λ k k maxfλ,λg k k (13) ≤ δ p , p + δ p , p 2n 2n −1 2n −1 2m λ k k λ k k +δ p , p 2m 2m −1 λ k k < ε + δ p , p + δ p , p . 2n 2n −1 2m 2m −1 λ k k λ k k Moreover, because (K, L) is a cyclic (α, β)−admissible pair we attain α p β p ≥ 1. Then by (1), 2n 2m −1 k k C ≤ ζ δ Kp , Lp , S p , p G 2n 2m −1 2n 2m −1 λ k k k k (14) = ζ δ p , p , S p , p 2n +1 2m 2n 2m −1 λ k k k k < G S p , p , δ p , p . 2n 2m −1 2n +1 2m k k λ k k Thence by (i) of Denition 1.10, we gain δ p , p < S p , p . (15) 2n +1 2m 2n 2m −1 λ k k k k Taking the limit as k ! ∞ and by using (6),(8), (10), (11), (12), (13), we procure lim δ p , p = lim S p , p = ε. (16) λ 2n +1 2m 2n 2m −1 k k k k k!∞ k!∞ Taking s = δ p , p , t = lim S p , p and using (13), (15) and (b) of Denition 1.9, we n 2n +1 2m n 2n 2m −1 k k k k k!∞ obtain C ≤ lim sup ζ δ Kp , Lp , S p , p G 2n 2m −1 2n 2m −1 λ k k k k k!∞ = lim sup ζ δ p , p , S p , p < C . λ 2n +1 2m 2n 2m −1 G k k k k k!∞ The attained last inequality causes a contradiction. Accordinglyfp g is a δ−Cauchy sequence. 2n As M is a δ−complete non-Archimedean modular metric space, there exists u 2 M such that δ p , u ! 0 ( n ) δ δ λ as n ! ∞. Now we present u is a common xed point of K and L. From (iii), we will presume that K is a δ−continuous mapping. Since δ p , j ! 0 as n ! ∞ and K is a δ−continuous mapping, we get ( 2n ) δ p , j = δ Kp , Kj ! 0, ( 2n+1 ) ( 2n+1 ) λ λ as n ! ∞. Hence p ! Kj as n ! ∞. But p ! j as n ! ∞. By the uniqueness of the limit, we achieve 2n+1 2n+1 Kj = j. From (iv), we have β j ≥ 1. This implies that α p β j ≥ 1 for all n 2 N. Next, we will demonstrate that j ( ) ( 2n) ( ) is a xed point of L. We presume that j ≠ Kj, that is δ j, Kj > 0. By K and L are generalized (α, β)−simulation ( ) contraction, we attain C ≤ ζ (δ (Kp , Lj) , S (p , j)) < G (S (p , j) , δ (Kp , Lj)) (17) 2n 2n 2n 2n G λ λ and then by (i) of Denition 1.10, we procure δ (Kp , Lj) < S (p , j) . (18) 2n 2n Taking limit as n ! ∞ in (18), we have δ (j, Lj) < δ (j, Lj), λ λ which is a contradiction. Thus j = Lj and hence j is a common xed point K and L. Ultimately, we will demonstrate the uniqueness of a common xed point of K and L. We assume that s is another common xed point, that is, δ (j, s) ≠ 0. From hypothesis, we gain α (j) ≥ 1 and β (s) ≥ 1 and thus α (j) β (s) ≥ 1. By using (1), C ≤ ζ δ Kj, Ls , S j, s < G S j, s , δ Kj, Ls (19) ( ( ) ( )) ( ( ) ( )) G λ λ Common xed point results via simulation type functions Ë 19 and from (i) of Denition 1.10, we have δ Kj, Ls < S j, s ( ) ( ) n o δ j,Ls +δ s,Kj λ( ) λ( ) = max δ (j, s) , δ (j, Kj) , δ (s, Ls) , λ λ λ n o δ j,s +δ s,j ( ) ( ) λ λ = max δ (j, s) , δ (j, Kj) , δ (s, Ls) , λ λ λ = maxfδ j, s , 0g ( ) = δ j, s , ( ) which is a contradiction. Hence j = s. 3 Consequences This section aims to acquaint some consequences from the main theorem. Corollary 3.1 Let M be a δ−complete non-Archimedean modular metric space and K, L : M ! M be two δ δ mappings. Presume that there exists k 2 0, 1 such that ( ) p, q 2 M with α p β q ≥ 1 ) κ ( ) ( ) n o δ (p,Lq)+δ (q,Kp) λ λ δ Kp, Lq ≤ k max δ p, q , δ p, Kp , δ q, Lq , . ( ) ( ) ( ) ( ) λ λ λ λ Suppose that the following circumstances hold: i. (K, L) is a cyclic α, β −admissible pair, ( ) ii. there exists p 2 M such that α p ≥ 1, 0 ( 0) iii. K or L is a δ−continuous mapping, iv. iffp g is a sequence in M such that p ! ξ and α p ≥ 1 and β p ≥ 1 for all n 2 N, then α p ≥ 1 n n ( 2n) ( 2n−1) ( ) and β p ≥ 1. ( ) Then, K and L admit a common xed point. Corollary 3.2 Let M be a complete non-Archimedean modular metric space and K, L : M ! M be two δ δ δ mappings. Assume that there exists a continuous function φ : 0,∞ ! 0,∞ with φ t = 0 if and only if [ ) [ ) ( ) t = 0 such that p, q 2 M with α p β q ≥ 1 ) ( ) ( ) n o δ p,Lq +δ q,Kp ( ) ( ) λ λ δ (Kp, Lq) ≤ max δ (p, q) , δ (p, Kp) , δ (q, Lq) , λ λ λ λ n o δ p,Lq +δ q,Kp ( ) ( ) λ λ −φ max δ (p, q) , δ (p, Kp) , δ (q, Lq) , . λ λ λ Suppose that the following conditions hold: i. (K, L) is a cyclic (α, β)−admissible pair, ii. there exists p 2 M such that α (p ) ≥ 1, 0 0 iii. K or L is a δ−continuous mapping, iv. iffp g is a sequence in M such that p ! p and α (p ) ≥ 1 and β (p ) ≥ 1 for all n 2 N, then α (p) ≥ 1 n n 2n 2n−1 and β (p) ≥ 1. Then, K and L admit a common xed point. We will replace the condition of cyclic (α, β)-admissible pair with cyclic (α, β)-admissible mapping in our main result, that is, we will take K = L. Then, we have the following results. Corollary 3.3 Let M be a complete non-Archimedean modular metric space and K : M ! M be a given δ δ δ mapping. Assume that there exists ζ 2 T such that p, q 2 M with α p β q ≥ 1 ) ( ) ( ) n o δ p,Kq +δ q,Kp λ( ) λ( ) ζ δ (Kp, Lq) , max δ (p, q) , δ (p, Kp) , δ (q, Lq) , ≥ C . λ λ λ λ Suppose that the following conditions hold: i. K is a cyclic α, β −admissible mapping; ( ) 20 Ë Mahpeyker Öztürk and Ekber Girgin ii. there exists p 2 M such that α p ≥ 1 and β p ≥ 1; ( ) ( ) 0 0 0 iii. V is a δ-continuous mapping, or; iv. iffp g is a sequence in M such that p ! p and β p ≥ 1 for all n 2 N, then β p ≥ 1. n n ( n) ( ) Then, K admits a xed point. Corollary 3.4 Let M be a complete non-Archimedean modular metric space and K : M ! M be a given δ δ δ mapping. Assume that there exists ζ 2 T such that α p β q ≥ 1 ) ζ δ Kp, Kq , δ p, q ≥ C , ( ) ( ) ( ( ) ( )) λ λ G p, q 2 M . Suppose that the following conditions hold: i. K is a cyclic α, β −admissible mapping; ( ) ii. there exists p 2 M such that α p ≥ 1 and β p ≥ 1; ( ) ( ) 0 0 0 iii. K is a δ-continuous mapping, or; iv. iffp g is a sequence in M such that p ! ξ and β p ≥ 1 for all n 2 N, then β p ≥ 1. n n ( n) ( ) Then, K admits a xed point. Example 3.5 Let M = R, δ p, q = jp − qj for all p 2 M . Dene the mapping K : M ! M as follows: ( ) δ δ δ δ , if p 2 0, 1 [ ] Kp = p , if p > 1. and α, β : M ! 0, +∞ are dened by [ ) 1, p 2 [0, 1] α (p) = β (p) = 0, otherwise. p.q If we consider the functions ζ (p, q) = q−p, G (q, p) = q−p and C = a , a > 2, then all the hypotheses 3 12 of Corollary 3.4 are satised and p = 0 is a unique xed point of K. p p p p  p p 1 6− 5 On the other hand, δ K 6, K 5 = > = δ 6, 5 , λ > 0. Thus K is not a Banach contraction λ λ λ λ with respect to M . 4 Common xed points on non-Archimedean modular metric spaces with a directed graph Let M be a non-Archimedean modular metric space and Δ = f(p, p) : p 2 Mg denotes the diagonal of M × M . Let Z be a directed graph such that the set V(Z) of its vertices coincides with M and J(Z) be the set δ δ of edges of the graph such that Δ  J(Z). Let M be a non-Archimedean modular metric space endowed with a graph Z and K, L : M ! M . We set δ δ δ M = fp 2 M : p, Kp 2 J Z and Kp, LKp 2 J Z g . ( ) ( ) ( ) ( ) KL Denition 4.1 Let M be a non-Archimedean modular metric space endowed with a directed graph Z and (K, L) be a cyclic (α, β) admissible pair. We say that K and L are generalized (α, β)-Z-simulation contraction if there exists ζ 2 T such that α (p) β (η) ≥ 1 ) ζ (δ (Kp, Lq) , S (p, q)) ≥ C , (20) n o κ p,Lq +δ q,Lp λ( ) λ( ) where S (p, q) = max δ (p, q) , δ (p, Kp) , δ (q, Lq) , , for all (p, q) 2 J (Z). λ λ λ Theorem 4.2 Let M be a δ-complete non-Archimedean modular metric space endowed with a directed graph Z. K and L are generalized α, β -Z-simulation contraction. Suppose that the following conditions hold: ( ) i. there exists p 2 M such that α(p ) ≥ 1; 0 0 KL ii. iffp g is a sequence in M such that p ! p, p , p 2 J Z for all n ≥ 0; n n ( 2n 2n+1) ( ) Common xed point results via simulation type functions Ë 21 ii . L is δ-continuous, p, p 2 J Z for all k ≥ 0 and α(p ) ≥ 1 then α(p) ≥ 1, or; a ( ) 2n +1 2n ii . K is δ-continuous, p , p 2 J Z for all k ≥ 0 and β(p ) ≥ 1, then β(p) ≥ 1; ( ) b 2n 2n+1 iii. there is a sequencefp g in M , such that n2N δ p , Kp 2 J Z ) p , Kp 2 J Z ( ) ( ) ( ) ( ) 2n 2n 2n+2 2n+2 and p , Lp 2 J Z ) p , Lp 2 J Z ; ( ) ( ) ( ) ( ) 2n+1 2n+1 2n+3 2n+3 Then K and L hold a common xed point. Proof Let p be a given point in M , then p , Kp 2 J Z and Kp , LKp 2 J Z , that is, p , p 2 0 ( 0 0) ( ) ( 0 0) ( ) ( 0 1) KL J Z and p , Lp = p , p 2 J Z . From (iii), it follows that p , Kp 2 J Z and p , Lp 2 J Z . ( ) ( 1 1) ( 1 2) ( ) ( 2 2) ( ) ( 3 3) ( ) That is, p , p 2 J Z and p , p 2 J Z . Continuing this way, we can obtain a sequence fp g in ( 2 3) ( ) ( 3 4) ( ) n M such that p , Kp 2 J Z and p , Lp 2 J Z for all n 2 N. Also, p , p 2 J Z and ( 2n 2n) ( ) ( 2n+1 2n+1) ( ) ( 2n 2n+1) ( ) p , p 2 J Z for all n 2 N. Using arguments as in the proof of the Theorem 2.2, we obtain that ( 2n+1 2n+2) ( ) δ p , p ! 0 as n ! ∞, (21) ( n n+1) andfp g is δ-Cauchy sequence. Since M is a δ-complete, there exists a p 2 M such that δ δ δ p , p ! 0 as n ! ∞. * * Now we show that p is a common xed point. As δ p , p ! 0 and p , p 2 J Z , so there n ( ) ( ) λ 2n 2n+1 exists a subsequence p of fp g such that from (ii) either L is δ-continuous, p , p 2 J Z ( ) 2n 2n 2n +1 k k * * * and α p ≥ 1 or K is δ-continuous, p , p 2 J Z and β p ≥ 1. Assume that L is δ-continuous, ( ) 2n * * p , p 2 J Z and α p ≥ 1. Thus ( ) 2n +1 * * δ Lp , Lp = δ p , Lp ! 0 as k ! ∞. 2n +1 2n +2 λ k λ k * * * This implies that Lp = p . From Denition 4.1, we have α p β p ≥ 1 2n +1 * * C ≤ ζ δ Kp , Lp , S p , p G λ 2n +1 2n +1 k k (22) * * < G S p , p , δ Kp , Lp 2n +1 λ 2n +1 k k and then by (i) of Denition 1.10, we get * * δ Kp , Lp < S p , p . (23) λ 2n +1 2n +1 k k where * * * * S p , p = maxfδ p , p , δ p , Kp , δ p , Lp , 2n +1 λ 2n +1 λ λ 2n +1 2n +1 k k k k * * δ p , Lp , δ p , Kp . λ 2n +1 λ 2n +1 k k On taking the limit as k ! ∞ on both sides of inequality (23), we have * * * * δ p , Kp < δ p , Kp , (24) λ λ * * * a contradiction. Thus Kp = p , and hence p is a common xed point. Similarly, the result follows if we * * suppose that K is δ-continuous, p , p 2 J Z and β p ≥ 1. ( ) 2n We note that Theorem 4.2 does not guarantee the uniqueness of a common xed point. To obtain the uniqueness, additional assumptions as given in the following theorem are needed. Theorem 4.3 Let M be a δ-complete non-Archimedean modular metric space endowed with a directed graph Z. K and L are generalized α, β -Z-simulation contraction. Suppose that the following conditions hold: ( ) i. there exists p 2 M such that α(p ) ≥ 1; 0 KL 0 ii. iffp g is a sequence in M such that p ! p, p , p 2 J Z for all n ≥ 0; n n ( ) ( ) 2n 2n+1 ii . L is δ-continuous, p, p 2 J Z for all k ≥ 0 and α(p ) ≥ 1 then α(p) ≥ 1, or; a ( ) 2n +1 2n ii . K is δ-continuous, p , p 2 J Z for all k ≥ 0 and β(p ) ≥ 1, then β(p) ≥ 1; ( ) b 2n 2n+1 k 22 Ë Mahpeyker Öztürk and Ekber Girgin iii. there is a sequencefp g in M , such that n2N δ p , Kp 2 J Z ) p , Kp 2 J Z ( ) ( ) ( ) ( ) 2n 2n 2n+2 2n+2 and p , Lp 2 J Z ) p , Lp 2 J Z ; ( ) ( ) ( ) ( ) 2n+1 2n+1 2n+3 2n+3 * * * * * * iv. there exist p , q 2 Fix(K, L) such that p , q 2 J Z , α(p ) ≥ 1 and β(q ) ≥ 1. ( ) Then K and L hold a unique common xed point. * * * * * * Proof Let p , q be common xed points of K and L, then p , q 2 J Z . Also, since α(p ) ≥ 1 and β(q ) ≥ 1. ( ) Thus, by Denition 4.1, we get that * * * * * * * * C ≤ ζ δ Kp , Lq , S p , q < G S p , q , δ Kp , Lq (25) G λ λ and from (i) of the Denition 1.10, we have * * * * δ Kp , Lq < S p , q * * = max δ p , q , 0 * * = δ p , q . * * which is a contradiction. Hence p = q . Example 4.4 Let M = [0, 1], δ (p, q) = jp − qj for all p 2 M and E (G) = f(p, q) : p, q 2 [0, 1]g . Dene δ λ δ the mappings K, L : M ! M as follows: p p Kp = , Lp = , p 2 M . 2 3 and α, β : M ! [0, +∞) are dened by 1, p 2 0, 1 ( ) α p = β p = ( ) ( ) 0, otherwise. p.q If we consider the functions ζ p, q = q−p, G q, p = q−p and C = , a > 2, then all the hypotheses ( ) ( ) a 4 24 of Theorem 4.3 are satised and p = 0 is a unique common xed point. 5 Application to dynamic programming The purpose of this section is to present an application on dynamic programming. Throughout this section, we assume that A and B are Banach spaces, W  A and D  B is a decision space. Now, consider the following system of functional equations: P (p) = supfs (p, q) + χ (p, q, P (τ (p, q)))g , (26) q2D Q p = supfs p, q + ς p, q, Q τ p, q g , (27) ( ) ( ) ( ( ( ))) q2D where τ : W × D ! W , r : W × D ! R, χ : W × D × R ! R, ς : W × D × R ! R. Let M = B(W) denotes the space of all bounded real-valued function on W. Consider the metric given by δ (m, n) = supjm (p) − n (p)j , for all m, n 2 A and λ > 0. p2W Then M is a δ−complete non-Archimedean modular metric space. Now, take the mappings E, F : M ! M as δ δ Em p = supfs p, q + χ p, q, m τ p, q g , (28) ( ) ( ) ( ( ( ))) q2D Common xed point results via simulation type functions Ë 23 and Fm p = supfs p, q + ς p, q, m τ p, q g , (29) ( ) ( ) ( ( ( ))) q2D where p 2 W and m 2 M . If the functions s, χ, τ are bounded, then E and F are well-dened. Theorem 5.1 Suppose that there exists k 2 [0, 1) such that for each (p, q) 2 W × D and m , m 2 M , the 1 2 inequality jχ (p, q, m (τ (p, q))) − ς (p, q, m (τ (p, q)))j ≤ kjm − m j , 1 2 1 2 holds. Then, E and F admit a common xed point in M . Proof Let ε > 0 be an arbitrary real number, p 2 W and m , m 2 M . Then by (28) and (29), there exist 1 2 δ q , q 2 D such that 1 2 Em p < s p, q + χ p, q , m τ p, q + ε, (30) ( ) ( ) ( ( ( ))) 1 1 1 1 1 Fm p < s p, q + ς p, q , m τ p, q + ε, (31) ( ) ( ) ( ( ( ))) 2 2 2 2 2 Em (p) ≥ s (p, q ) + χ (p, q , m (τ (p, q ))) , (32) 1 2 2 1 2 and Fm p ≥ s p, q + ς p, q , m τ p, q . (33) ( ) ( ) ( ( ( ))) 2 1 1 2 1 Then from (30) and (33), it follows easily that Em p − Fm p ≤ χ p, q , m τ p, q + ε − ς p, q , m τ p, q ( ) ( ) ( ( ( ))) ( ( ( ))) 1 2 1 1 1 1 2 1 ≤ jχ p, q , m τ p, q − ς p, q , m τ p, q j + ε ( ( ( ))) ( ( ( ))) 1 1 1 1 2 1 ≤ kjm − m j + ε. 1 2 Similarly, from (31) and (32), we have Fm p − Em p ≤ kjm − m j + ε. ( ) ( ) 2 1 1 2 Thus, we deduce from above inequalities that jEm p − Fm p j ≤ kjm − m j + ε. (34) ( ) ( ) 1 2 1 2 Since the inequality (34) is valid for any p 2 W, then 1 1 δ Em , Fm = jEm − Fm j ≤ kjm − m j + ε = kδ m , m + ε, ( ) ( ) λ 1 2 1 2 1 2 λ 1 2 δ δ also ε > 0 is arbitrary, so δ Em , Fm ≤ kδ m , m . 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Journal

Topological Algebra and its Applications – de Gruyter

Published: Jan 1, 2022

Keywords: Non-Archimedean modular metric space; Simulation function; Cyclic ( α , β )-admissible pair; 47H10; 54H25; 37C25

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