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Two fixed point results on F-metric spaces

Two fixed point results on F-metric spaces Topol. Algebra Appl. 2022; 10:61–67 Research Article Open Access Ishak Altun* and Ali Erduran Two xed point results on F-metric spaces https://doi.org/10.1515/taa-2022-0114 Received 10 March, 2022; accepted 30 May, 2022 Abstract: This paper presents two xed point results for both single valued and multivalued mappings on F-metric spaces, which is introduced by Jleli and Samet [1]. First, we consider the concept of P-contraction for single valued mappings. Then using the Feng-Liu’s technique, we present a xed point result for F-closed set valued mappings on F-metric spaces. Keywords: Fixed point, F-metric space, P-contraction, multivalued mapping MSC: 54H25, 47H10 1 Introduction and Preliminaries The eld of xed point theory is an important branch of nonlinear analysis, because it oers powerful tools for solving various problems including integral and dierential equations. The main two research directions are related, on the one hand, to obtain more general results by generalizing contraction conditions for single valued or multivalued mappings, and on the other to obtain new results by relaxing the structure of space. In recent years, mathematicians sought to nd new generalization of metric space. For example, Czer- wik [2] introduced the notion of b-metric by considering the following condition instead of the triangular inequality: for every x, y 2 X and s ≥ 1, d(x, y) ≤ s d(x, z) + d(z, y) . In [3] Fagin et al introduced the notion of s-relaxed metric by using the following condition instead of the triangular inequality: for every x, z, x , x , ··· , x and s ≥ 1, 1 2 n d(x, z) ≤ s d(x, x ) + d(x , x ) + ··· + d(x , z) . 1 1 2 n−1 Mustafa and Sims [4] introduced the notion of G-metric, which is a mapping dened on the product set X × X × X, and satisfying certain conditions. Mathews [5] introduced partial metric space. While the distance of a point to itself is zero on metric space, the distance of a point to itself is not always zero on partial metric space. Therefore, it is interesting conclu- sion. Jleli and Samet [6] replaced triangle inequality of metric to following condition: there exists a coecient s > 0 such that if lim d(x , x) = 0 and y 2 X then d(x, y) ≤ s[lim supd(x , y)] n n n!∞ n!∞ and hence they introduced JS-metric space. Very recently, Jleli and Samet [1] introduced F-metric structure on a nonempty set X. Then, they show that every metric and every s-relaxed metric is an F-metric but the converse is not true. To show this, they provide an example of F-metric that it is not an s-relaxed metric (see Example 2.4 in [1]). *Corresponding Author: Ishak Altun: Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey, E-mail: ishakaltun@yahoo.com Ali Erduran: Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey, E-mail: ali.erduran1@yahoo.com Open Access. © 2022 Ishak Altun and Ali Erduran, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License. 62 Ë Ishak Altun and Ali Erduran Now we recall the F-metric structure. Let F be the set of functions f : (0,∞) ! R satisfying the following conditions: (F ) f is non decreasing, i.e, 0 < s < t ) f (s) ≤ f (t), (F ) For every sequenceft g  (0,∞), we have lim t = 0 ) lim f (t ) = −∞. n n n!∞ n!∞ Denition 1 ([1]). Let X be a nonempty set, and let D : X × X ! [0,∞) be a given mapping. Suppose that there exists (f , ) 2 F × [0,∞) such that (D )(x, y) 2 X × X, D(x, y) = 0 , x = y. (D )D(x, y) = D(y, x), for all (x, y) 2 X × X. (D )For every (x, y) 2 X × X, for every N 2 N, N ≥ 2, and for every (u )  X with (u , u ) = (x, y), we have 3 1 i i=1 N N−1 D(x, y) > 0 ) f D(x, y) ≤ f D(u , u ) + . i i+1 i=1 Then D is said to be an F-metric on X, and the pair (X, D) is said to be an F-metric space. We can easily observe that every metric on X is an F-metric. Following example shows that the converse is not true. Example 1 ([1]). Let X = N and D : X × X ! [0,∞) be the mapping dened by (x − y) if (x, y) 2 [0, 3] × [0, 3] D(x, y) = jx − yj if (x, y) 2 / [0, 3] × [0, 3] for all (x, y) 2 X × X. Then D is an F-metric but not a metric on X. Let (X, D) be an F-metric space. A subset Y of X is said to be F-open if for every x 2 Y , there is some r > 0 such that B(x, r)  Y , where B(x, r) = fy 2 X : D(x, y) < rg. A subset Z of X is F-closed if XZ is F-open. It can be easily seen that the collection of all F-open subsets, say  , of X is a topology, which is rst countable, on X. Proposition 1 ([1]). Let (X, D) be an F-metric space. Then, for any nonempty subset A of X, the following state- ments are equivalent: (i) A is F-closed (ii) For any sequencefx g  A, we have lim D(x , x) = 0, x 2 X ) x 2 A. n!∞ Let (X, D) be an F-metric space,fx g be a sequence in X and x 2 X. Iffx g converges to x with respect to  , n n then we say that fx g is F-convergent to x. In this case x is said to be limit of fx g. Therefore we have fx g n n n is F-convergent to x if and only if lim D(x , x) = 0. By the routine calculation we can see that the limit of n!∞ F-convergent sequence in an F-metric space is unique. Denition 2 ([1]). Let (X, D) be an F-metric space andfx g be a sequence in X. (i) We say thatfx g is F-Cauchy, if lim D(x , x ) = 0. n m n!∞ (ii) We say that (X, D) is F-complete, if every F-Cauchy sequence in X is F-convergent to certain element in X. After the introduction of F-metric space, Jleli and Samet [1] proved the following xed point result, which is actually F-metric version of Banach xed point theorem. Two xed point results on F-metric spaces Ë 63 Theorem 1. Let (X, D) be an F-complete F-metric space and T : X ! X be a mapping. If there exists k 2 (0, 1) such that D(Tx, Ty) ≤ kD(x, y) (1.1) for all x, y 2 X, then T has a unique xed point in X. Moreover, every Picard sequence F-converges to the xed point of T. Proposition 2 ([1]). Let (X, D) be an F-metric space. Then for any nonempty subset A of X, we have x 2 A, r > 0 ) B(x, r)\ A ≠ ;, where A is the closure of A with respect to  . Proposition 3. Let (X, D) be an F-metric space, A  X and x 2 X. Then x 2 A , D(x, A) = 0, where D(x, A) = inf D(x, y). y2A Proof. x 2 A , B(x, r)\ A ≠ ; for all r > 0 , D(x, x ) < r for all r > 0 and some x 2 A r r , inffD(x, y) : y 2 Ag = 0 , D(x, A) = 0. If a subset A of an F-metric space (X, D) is compact with respect to  , then it is said to F-compact set. Jleli and Samet [1] proved that a sunset A is F-compact if and only if it is sequentially F-compact, that is, every sequence in A has an F-convergent subsequence in A. For further information and recent results about F- metric space we refer to [7–11]. In this paper, we present two xed point results for both single valued and multivalued mappings on F-metric spaces. 2 Single valued maps In this section we present a xed point theorem for single-valued P-contraction mappings on F-metric space. For this, we will use the concept of E-contraction in metric space, rst introduced by Popescu (see the ref- erences of [12, 13]) and later developed by Fulga and Proca [12, 13]. We prefer to use this contraction as P- contraction to cite Popescu. See [14, 15] for more studies about this new idea. As the parallel manner on met- ric space, a self mapping T of an F-metric space (X, d) is called P-contraction, if for all x, y 2 X the following inequality holds: D(Tx, Ty) ≤ k D(x, y) + D(x, Tx) − D(y, Ty) (2.1) for some k 2 (0, 1). Theorem 2. (X, D) be an F-complete F-metric space, and T : X ! X be a continuous P-contraction mapping. Then T has a unique xed point in X. Proof. Let x 2 X be an arbitrary point and dene a sequence fx g by x = Tx for all n 2 N. If there 0 n n+1 n exists n 2 N such that x = x then x is a xed point of T. Suppose x ≠ x for all n 2 N and hence 0 n n +1 n n n+1 0 0 0 D(x , x ) > 0. Now we have n n+1 D(x , x ) = D(Tx , Tx ) n+1 n+2 n n+1 ≤ k D(x , x ) + D(x , Tx ) − D(x , Tx ) n n+1 n n n+1 n+1 = k D(x , x ) + D(x , x ) − D(x , x ) (2.2) n n n+1 n+1 n+1 n+2 64 Ë Ishak Altun and Ali Erduran for all n 2 N. If D(x , x ) ≤ D(x , x ) for some n, then from (2.2) n+1 n+1 n+2 D(x , x ) ≤ k D(x , x ) + D(x , x ) − D(x , x ) n n n+1 n+2 n+1 n+1 n+1 n+2 ≤ k D(x , x ) + D(x , x ) − D(x , x ) n n n+1 n+1 n+2 n+1 = kD(x , x ), n+1 n+2 which is a contradiction. Hence D(x , x ) > D(x , x ) for all n 2 N and so from (2.2) we get n+1 n+1 n+2 D(x , x ) ≤ k D(x , x ) + D(x , x ) − D(x , x ) n+1 n+2 n n+1 n n+1 n+1 n+2 ≤ k D(x , x ) + D(x , x ) − D(x , x ) n n+1 n n+1 n+1 n+2 = 2kD(x , x ) − kD(x , x ). n n+1 n+1 n+2 Thus we have 2k D(x , x ) ≤ D(x , x ) n+1 n+2 n+1 1 + k 2k for all n 2 N. Let = , then 0 <  < 1 and so we get 1+k D(x , x ) ≤ D(x , x ) n+1 n+2 n n+1 ≤  D(x , x ) n−1 n n+1 ≤  D(x , x ) 0 1 for all n 2 N. Hence we have for m, n 2 N with m > n, m−1 m−1 X X D(x , x ) ≤  D(x , x ) i i+1 0 1 i=n i=n ≤ D(x , x ). 0 1 1 − Since 0 <  < 1, for all  > 0, there exists n 2 N such that for all n ≥ n , 0 0 0 < D(x , x ) < . 0 1 1 − Now let (f , ) 2 F× [0,∞) be such that (D ) is satised. Let " > 0 be xed, then by (F ) there exists  > 0 such 3 2 that 0 < t <  ) f (t) < f (") − . Considering  as  we get f D(x , x ) < f (") − . 0 1 1 − By (F ) we have m−1 X n f D(x , x ) ≤ f D(x , x ) < f (") − i i+1 0 1 1 − i=n for all m, n 2 N with m > n ≥ n . Using (D ) and this last inequality we get for m > n ≥ n , 0 3 0 m−1 D(x , x ) > 0 ) f (D(x , x ) ≤ f D(x , x ) + < f ("), n m n m i i+1 i=n which implies by (F ) that D(x , x ) < " for m > n ≥ n . Thereforefx g is F-Cauchy. Since (X, D) is F-complete n m n 1 0 there exists z 2 X such that fx g is F-convergent to z, that is, lim D(x , z) = 0. Since T is continuous, n n!∞ n which is equivalent to sequential continuous, and then lim D(Tx , Tz) = lim D(x , Tz) = 0. The n!∞ n n!∞ n+1 Two xed point results on F-metric spaces Ë 65 uniqueness of the limit we have z = Tz. Now suppose w is another xed point of T, then Tw = w and D(z, w) > 0. Hence D(z, w) = D(Tz, Tw) ≤ k D(z, w) + D(z, Tz) − D(w, Tw) = kD(z, w), which is a contradiction. Therefore the xed point of T is unique. Now we present an illustrative example. Example 2. Let X = f1, 2, 3,···g and D : X × X ! [0,∞) be the mapping dened by (x − y) if x, y 2 f1, 2g D(x, y) = . 2jx − yj if otherwise Then (X, D) is an F-metric space which is also F-complete. Note that, since D(x, y) ≥ 1 for all x, y 2 X with x ≠ y, then every F-Cauchy sequence in X should be eventually constant and hence they should be F-convergent. For the same reason the induced topology  is discrete topology, and so every self mapping on X are continuous. Now dene a mapping T : X ! X by if x is even Tx = . : x+1 if x is odd Then by the routine calculation we can show that T is P-contraction mapping with k ≥ . Hence by Theorem 2, T has a unique xed point. Remark 1. Since the inequality (2.1) is more general then (1.1), than we can take notice Theorem 1 is a special case of Theorem 2. 3 Multivalued maps In this section, we present Feng-Liu [16] type xed point result for multivalued mapping on F-metric spaces. Here, we will use the following notations: Let (X, D) be an F-metric space, T : X ! N(X) be a mapping, where N(X) denotes the collection of all nonempty subsets of X, and x 2 X. For a constant b 2 (0, 1), dene the set I  X as I = fy 2 Tx : bD(x, y) ≤ D(x, Tx)g. Let (X, ) be a topological space, x 2 X and f : X ! R be a function. Then f is called lower semi- continuous at x , if for anyfx g  X with x ! x implies f (x) ≤ lim inf f (x ). If f is lower semi-continuous n n n 0 0 n!∞ at every point of X, then it is said to be lower semi-continuous on X. Theorem 3. Let (X, D) be an F-complete F-metric space, T : X ! C(X) (C(X) is the collection of all nonempty F-closed subsets of X) be a multivalued mapping. If there exists a constant c 2 (0, 1) such that for any x 2 X, there is y 2 I satisfying D(y, Ty) ≤ cD(x, y) then T has a xed point in X provided c < b and f (x) = D(x, Tx) is lower semi continuous. Proof. Since Tx 2 C(X) for any x 2 X, then there exists y 2 Tx such that bD(x, y) ≤ D(x, Tx). 66 Ë Ishak Altun and Ali Erduran x 0 Hence I ≠ ;. For any initial point x 2 X, there exist x 2 I such that 0 1 D(x , Tx ) ≤ cD(x , x ) 1 1 0 1 Similarly, for x 2 X there exist x 2 I such that 1 2 D(x , Tx ) ≤ cD(x , x ). 2 2 1 2 Continuing this process, we can construct a sequencefx g in X such that, for all n 2 N, x 2 Tx , n n+1 n D(x , Tx ) ≤ cD(x , x ) and bD(x , x ) ≤ D(x , Tx ). (3.1) n+1 n+1 n n+1 n n+1 n n From (3.1), we have D(x , Tx ) ≤ D(x , Tx ) (3.2) n+1 n+1 n n for all n 2 N. On the other hand, thanks to denition of sequencefx g we get n=0 D(x , x ) ≤ D(x , x ) (3.3) n n n+1 n−1 for all n 2 N. By using the inequalities (3.2) and (3.3) we have D(x , x ) ≤ D(x , x ), n = 0, 1, 2,··· n+1 0 1 D(x , Tx ) ≤ D(x , Tx ), n = 0, 1, 2,··· . n n 0 0 Then for m, n 2 N, m > n, we have m−1 D(x , x ) = D(x , x ) + D(x , x ) + ··· + D(x , x ) n n+1 n+1 n+2 m−1 m i i+1 i=n n n+1 m−1 c c c ≤ D(x , x ) + D(x , x ) + ··· + D(x , x ) 0 1 0 1 0 1 n+1 m−1 b b b n m−n−1 = a (1 + a + ··· + a )D(x , x ) (for a = ) 0 1 ≤ D(x , x ). 0 1 1 − a Since lim D(x , x ) = 0, then for every  > 0, there exist N 2 N such that 0 1 1−a n!∞ 0 < D(x , x ) < , n ≥ N. (3.4) 0 1 1 − a Now, let (f , ) 2 F × [0,∞) be such that (D ) is satised. Let " > 0 be xed. By (F ), there exist  > 0 such 3 2 that 0 < t <  ) f (t) < f (") − . (3.5) Hence by (3.5) and (F ), we get m−1 f D(x , x ) ≤ f D(x , x ) < f (") − (for m > n ≥ N). (3.6) i i+1 0 1 1 − a i=1 Using (D ) and (3.6), we obtain for m > n ≥ N, D(x x ) > 0 ) f (D(x , x )) ≤ f (") n, m n m which implies by (F ) that D(x , x ) < ". n m ∞ ∞ Therefore fx g is an F-Cauchy sequence. Since (X, D) is F-complete, there exist x 2 X such that fx g n n n=0 n=0 is F-convergent to x. i.e. lim D(x , x) = 0. n!∞ Now since f (x) = D(x, Tx) is lower semi-continuous and x ! x, we have, from (3.2) 0 ≤ D(x, Tx) = f (x) ≤ lim inff (x ) = lim infD(x , Tx ) = 0. n n n n!∞ n!∞ Therefore D(x, Tx) = 0. Since Tx is F-closed, then we get x 2 Tx. Two xed point results on F-metric spaces Ë 67 Acknowledgement: The authors are thankful to the referees for making valuable suggestions leading to the better presentations of the paper Conict of interest: The authors declare that they have no conict of interest References [1] M. Jleli, B. Samet, On a new generalization of metric spaces, J. Fixed Point Theory Appl., (2018), 20:128. [2] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Univ. Ostrav., (1993), 1 (1), 5-11. [3] R. Fagin, R. Kumar, D. Sivakumar, Comparing top k list. SIAM J. Discret. Math., (2003), 17 (1), 134-160. [4] Z. Mustafa, B. Sims, A new approach to generelized metric spaces, J. Nonlinear Convex Anal., (2006), 7 (2), 289-297. [5] S.G. Mathews, Partial metric topology, In Proceedings of the 8th Summer Conference on General Topology and Applications, Annals of the Newyork Academy of Science, (1994), vol 728, pp 183-197. [6] M. Jleli, B. Samet, A generalized metric space and related xed point theorems, Fixed Point Theory Appl., (2015), 14 2015. [7] H. Aydi, E. Karapınar, Z.D. Mitrović, T. Rashid., A remark on “Existence and uniqueness for a neutral dierential problem with unbounded delay via xed point results F-metric spaces”, RACSAM, (2019), 113 (4), 3197-3206. [8] O. Alqahtani, E. Karapınar, P. Shahi, Common xed point results in function weighted metric spaces, Journal of Inequalities and Applications, (2019), 1, 1-9. [9] A. Bera, H. Garai, B. Damjanović, A. Chanda, Some interesting results on F-metric spaces, Filomat (2019), 33 (10), 3257-3268. [10] E.L. Ghasab, H. Majani, E. Karapınar, G.S. Rad, New xed point results in F-quasi-metric spaces and an application, Advances in Mathematical Physics, (2020), Article ID 9452350. [11] A. Hussain., F. Jarad, E. Karapınar, A study of symmetric contractions with an application to generalized fractional dierential equations, Advances in Dierence Equations, (2021), 1, 1-27. [12] A. Fulga, A. Proca, A new generalization of Wardowski xed point theorem in complete metric spaces, Advances in the Theory of Nonlinear Analysis and its Applications, (2017), 1 (1), 57-63. [13] A. Fulga, A. Proca, Fixed points for ' -Geraghty contractions, J. Nonlinear Sci. Appl., (2017), 10, 5125-513. [14] I. Altun, G. Durmaz, M. Olgun, P-contractive mappings on metric spaces, Journal of Nonlinear Functional Analysis, (2018), 1-7. [15] E. Karapınar, A. Fulga, H. Aydi, Study on Pata E-contractions, Advances in Dierence Equations, (2020), 1, 1-15. [16] Y. Feng, S. Lui, Fixed point theorems for multivalued contractive and multivalued Carsiti type mappings, J. Math. Anal. Appl., (2006), 317, 103-112. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Topological Algebra and its Applications de Gruyter

Two fixed point results on F-metric spaces

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Topol. Algebra Appl. 2022; 10:61–67 Research Article Open Access Ishak Altun* and Ali Erduran Two xed point results on F-metric spaces https://doi.org/10.1515/taa-2022-0114 Received 10 March, 2022; accepted 30 May, 2022 Abstract: This paper presents two xed point results for both single valued and multivalued mappings on F-metric spaces, which is introduced by Jleli and Samet [1]. First, we consider the concept of P-contraction for single valued mappings. Then using the Feng-Liu’s technique, we present a xed point result for F-closed set valued mappings on F-metric spaces. Keywords: Fixed point, F-metric space, P-contraction, multivalued mapping MSC: 54H25, 47H10 1 Introduction and Preliminaries The eld of xed point theory is an important branch of nonlinear analysis, because it oers powerful tools for solving various problems including integral and dierential equations. The main two research directions are related, on the one hand, to obtain more general results by generalizing contraction conditions for single valued or multivalued mappings, and on the other to obtain new results by relaxing the structure of space. In recent years, mathematicians sought to nd new generalization of metric space. For example, Czer- wik [2] introduced the notion of b-metric by considering the following condition instead of the triangular inequality: for every x, y 2 X and s ≥ 1, d(x, y) ≤ s d(x, z) + d(z, y) . In [3] Fagin et al introduced the notion of s-relaxed metric by using the following condition instead of the triangular inequality: for every x, z, x , x , ··· , x and s ≥ 1, 1 2 n d(x, z) ≤ s d(x, x ) + d(x , x ) + ··· + d(x , z) . 1 1 2 n−1 Mustafa and Sims [4] introduced the notion of G-metric, which is a mapping dened on the product set X × X × X, and satisfying certain conditions. Mathews [5] introduced partial metric space. While the distance of a point to itself is zero on metric space, the distance of a point to itself is not always zero on partial metric space. Therefore, it is interesting conclu- sion. Jleli and Samet [6] replaced triangle inequality of metric to following condition: there exists a coecient s > 0 such that if lim d(x , x) = 0 and y 2 X then d(x, y) ≤ s[lim supd(x , y)] n n n!∞ n!∞ and hence they introduced JS-metric space. Very recently, Jleli and Samet [1] introduced F-metric structure on a nonempty set X. Then, they show that every metric and every s-relaxed metric is an F-metric but the converse is not true. To show this, they provide an example of F-metric that it is not an s-relaxed metric (see Example 2.4 in [1]). *Corresponding Author: Ishak Altun: Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey, E-mail: ishakaltun@yahoo.com Ali Erduran: Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey, E-mail: ali.erduran1@yahoo.com Open Access. © 2022 Ishak Altun and Ali Erduran, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License. 62 Ë Ishak Altun and Ali Erduran Now we recall the F-metric structure. Let F be the set of functions f : (0,∞) ! R satisfying the following conditions: (F ) f is non decreasing, i.e, 0 < s < t ) f (s) ≤ f (t), (F ) For every sequenceft g  (0,∞), we have lim t = 0 ) lim f (t ) = −∞. n n n!∞ n!∞ Denition 1 ([1]). Let X be a nonempty set, and let D : X × X ! [0,∞) be a given mapping. Suppose that there exists (f , ) 2 F × [0,∞) such that (D )(x, y) 2 X × X, D(x, y) = 0 , x = y. (D )D(x, y) = D(y, x), for all (x, y) 2 X × X. (D )For every (x, y) 2 X × X, for every N 2 N, N ≥ 2, and for every (u )  X with (u , u ) = (x, y), we have 3 1 i i=1 N N−1 D(x, y) > 0 ) f D(x, y) ≤ f D(u , u ) + . i i+1 i=1 Then D is said to be an F-metric on X, and the pair (X, D) is said to be an F-metric space. We can easily observe that every metric on X is an F-metric. Following example shows that the converse is not true. Example 1 ([1]). Let X = N and D : X × X ! [0,∞) be the mapping dened by (x − y) if (x, y) 2 [0, 3] × [0, 3] D(x, y) = jx − yj if (x, y) 2 / [0, 3] × [0, 3] for all (x, y) 2 X × X. Then D is an F-metric but not a metric on X. Let (X, D) be an F-metric space. A subset Y of X is said to be F-open if for every x 2 Y , there is some r > 0 such that B(x, r)  Y , where B(x, r) = fy 2 X : D(x, y) < rg. A subset Z of X is F-closed if XZ is F-open. It can be easily seen that the collection of all F-open subsets, say  , of X is a topology, which is rst countable, on X. Proposition 1 ([1]). Let (X, D) be an F-metric space. Then, for any nonempty subset A of X, the following state- ments are equivalent: (i) A is F-closed (ii) For any sequencefx g  A, we have lim D(x , x) = 0, x 2 X ) x 2 A. n!∞ Let (X, D) be an F-metric space,fx g be a sequence in X and x 2 X. Iffx g converges to x with respect to  , n n then we say that fx g is F-convergent to x. In this case x is said to be limit of fx g. Therefore we have fx g n n n is F-convergent to x if and only if lim D(x , x) = 0. By the routine calculation we can see that the limit of n!∞ F-convergent sequence in an F-metric space is unique. Denition 2 ([1]). Let (X, D) be an F-metric space andfx g be a sequence in X. (i) We say thatfx g is F-Cauchy, if lim D(x , x ) = 0. n m n!∞ (ii) We say that (X, D) is F-complete, if every F-Cauchy sequence in X is F-convergent to certain element in X. After the introduction of F-metric space, Jleli and Samet [1] proved the following xed point result, which is actually F-metric version of Banach xed point theorem. Two xed point results on F-metric spaces Ë 63 Theorem 1. Let (X, D) be an F-complete F-metric space and T : X ! X be a mapping. If there exists k 2 (0, 1) such that D(Tx, Ty) ≤ kD(x, y) (1.1) for all x, y 2 X, then T has a unique xed point in X. Moreover, every Picard sequence F-converges to the xed point of T. Proposition 2 ([1]). Let (X, D) be an F-metric space. Then for any nonempty subset A of X, we have x 2 A, r > 0 ) B(x, r)\ A ≠ ;, where A is the closure of A with respect to  . Proposition 3. Let (X, D) be an F-metric space, A  X and x 2 X. Then x 2 A , D(x, A) = 0, where D(x, A) = inf D(x, y). y2A Proof. x 2 A , B(x, r)\ A ≠ ; for all r > 0 , D(x, x ) < r for all r > 0 and some x 2 A r r , inffD(x, y) : y 2 Ag = 0 , D(x, A) = 0. If a subset A of an F-metric space (X, D) is compact with respect to  , then it is said to F-compact set. Jleli and Samet [1] proved that a sunset A is F-compact if and only if it is sequentially F-compact, that is, every sequence in A has an F-convergent subsequence in A. For further information and recent results about F- metric space we refer to [7–11]. In this paper, we present two xed point results for both single valued and multivalued mappings on F-metric spaces. 2 Single valued maps In this section we present a xed point theorem for single-valued P-contraction mappings on F-metric space. For this, we will use the concept of E-contraction in metric space, rst introduced by Popescu (see the ref- erences of [12, 13]) and later developed by Fulga and Proca [12, 13]. We prefer to use this contraction as P- contraction to cite Popescu. See [14, 15] for more studies about this new idea. As the parallel manner on met- ric space, a self mapping T of an F-metric space (X, d) is called P-contraction, if for all x, y 2 X the following inequality holds: D(Tx, Ty) ≤ k D(x, y) + D(x, Tx) − D(y, Ty) (2.1) for some k 2 (0, 1). Theorem 2. (X, D) be an F-complete F-metric space, and T : X ! X be a continuous P-contraction mapping. Then T has a unique xed point in X. Proof. Let x 2 X be an arbitrary point and dene a sequence fx g by x = Tx for all n 2 N. If there 0 n n+1 n exists n 2 N such that x = x then x is a xed point of T. Suppose x ≠ x for all n 2 N and hence 0 n n +1 n n n+1 0 0 0 D(x , x ) > 0. Now we have n n+1 D(x , x ) = D(Tx , Tx ) n+1 n+2 n n+1 ≤ k D(x , x ) + D(x , Tx ) − D(x , Tx ) n n+1 n n n+1 n+1 = k D(x , x ) + D(x , x ) − D(x , x ) (2.2) n n n+1 n+1 n+1 n+2 64 Ë Ishak Altun and Ali Erduran for all n 2 N. If D(x , x ) ≤ D(x , x ) for some n, then from (2.2) n+1 n+1 n+2 D(x , x ) ≤ k D(x , x ) + D(x , x ) − D(x , x ) n n n+1 n+2 n+1 n+1 n+1 n+2 ≤ k D(x , x ) + D(x , x ) − D(x , x ) n n n+1 n+1 n+2 n+1 = kD(x , x ), n+1 n+2 which is a contradiction. Hence D(x , x ) > D(x , x ) for all n 2 N and so from (2.2) we get n+1 n+1 n+2 D(x , x ) ≤ k D(x , x ) + D(x , x ) − D(x , x ) n+1 n+2 n n+1 n n+1 n+1 n+2 ≤ k D(x , x ) + D(x , x ) − D(x , x ) n n+1 n n+1 n+1 n+2 = 2kD(x , x ) − kD(x , x ). n n+1 n+1 n+2 Thus we have 2k D(x , x ) ≤ D(x , x ) n+1 n+2 n+1 1 + k 2k for all n 2 N. Let = , then 0 <  < 1 and so we get 1+k D(x , x ) ≤ D(x , x ) n+1 n+2 n n+1 ≤  D(x , x ) n−1 n n+1 ≤  D(x , x ) 0 1 for all n 2 N. Hence we have for m, n 2 N with m > n, m−1 m−1 X X D(x , x ) ≤  D(x , x ) i i+1 0 1 i=n i=n ≤ D(x , x ). 0 1 1 − Since 0 <  < 1, for all  > 0, there exists n 2 N such that for all n ≥ n , 0 0 0 < D(x , x ) < . 0 1 1 − Now let (f , ) 2 F× [0,∞) be such that (D ) is satised. Let " > 0 be xed, then by (F ) there exists  > 0 such 3 2 that 0 < t <  ) f (t) < f (") − . Considering  as  we get f D(x , x ) < f (") − . 0 1 1 − By (F ) we have m−1 X n f D(x , x ) ≤ f D(x , x ) < f (") − i i+1 0 1 1 − i=n for all m, n 2 N with m > n ≥ n . Using (D ) and this last inequality we get for m > n ≥ n , 0 3 0 m−1 D(x , x ) > 0 ) f (D(x , x ) ≤ f D(x , x ) + < f ("), n m n m i i+1 i=n which implies by (F ) that D(x , x ) < " for m > n ≥ n . Thereforefx g is F-Cauchy. Since (X, D) is F-complete n m n 1 0 there exists z 2 X such that fx g is F-convergent to z, that is, lim D(x , z) = 0. Since T is continuous, n n!∞ n which is equivalent to sequential continuous, and then lim D(Tx , Tz) = lim D(x , Tz) = 0. The n!∞ n n!∞ n+1 Two xed point results on F-metric spaces Ë 65 uniqueness of the limit we have z = Tz. Now suppose w is another xed point of T, then Tw = w and D(z, w) > 0. Hence D(z, w) = D(Tz, Tw) ≤ k D(z, w) + D(z, Tz) − D(w, Tw) = kD(z, w), which is a contradiction. Therefore the xed point of T is unique. Now we present an illustrative example. Example 2. Let X = f1, 2, 3,···g and D : X × X ! [0,∞) be the mapping dened by (x − y) if x, y 2 f1, 2g D(x, y) = . 2jx − yj if otherwise Then (X, D) is an F-metric space which is also F-complete. Note that, since D(x, y) ≥ 1 for all x, y 2 X with x ≠ y, then every F-Cauchy sequence in X should be eventually constant and hence they should be F-convergent. For the same reason the induced topology  is discrete topology, and so every self mapping on X are continuous. Now dene a mapping T : X ! X by if x is even Tx = . : x+1 if x is odd Then by the routine calculation we can show that T is P-contraction mapping with k ≥ . Hence by Theorem 2, T has a unique xed point. Remark 1. Since the inequality (2.1) is more general then (1.1), than we can take notice Theorem 1 is a special case of Theorem 2. 3 Multivalued maps In this section, we present Feng-Liu [16] type xed point result for multivalued mapping on F-metric spaces. Here, we will use the following notations: Let (X, D) be an F-metric space, T : X ! N(X) be a mapping, where N(X) denotes the collection of all nonempty subsets of X, and x 2 X. For a constant b 2 (0, 1), dene the set I  X as I = fy 2 Tx : bD(x, y) ≤ D(x, Tx)g. Let (X, ) be a topological space, x 2 X and f : X ! R be a function. Then f is called lower semi- continuous at x , if for anyfx g  X with x ! x implies f (x) ≤ lim inf f (x ). If f is lower semi-continuous n n n 0 0 n!∞ at every point of X, then it is said to be lower semi-continuous on X. Theorem 3. Let (X, D) be an F-complete F-metric space, T : X ! C(X) (C(X) is the collection of all nonempty F-closed subsets of X) be a multivalued mapping. If there exists a constant c 2 (0, 1) such that for any x 2 X, there is y 2 I satisfying D(y, Ty) ≤ cD(x, y) then T has a xed point in X provided c < b and f (x) = D(x, Tx) is lower semi continuous. Proof. Since Tx 2 C(X) for any x 2 X, then there exists y 2 Tx such that bD(x, y) ≤ D(x, Tx). 66 Ë Ishak Altun and Ali Erduran x 0 Hence I ≠ ;. For any initial point x 2 X, there exist x 2 I such that 0 1 D(x , Tx ) ≤ cD(x , x ) 1 1 0 1 Similarly, for x 2 X there exist x 2 I such that 1 2 D(x , Tx ) ≤ cD(x , x ). 2 2 1 2 Continuing this process, we can construct a sequencefx g in X such that, for all n 2 N, x 2 Tx , n n+1 n D(x , Tx ) ≤ cD(x , x ) and bD(x , x ) ≤ D(x , Tx ). (3.1) n+1 n+1 n n+1 n n+1 n n From (3.1), we have D(x , Tx ) ≤ D(x , Tx ) (3.2) n+1 n+1 n n for all n 2 N. On the other hand, thanks to denition of sequencefx g we get n=0 D(x , x ) ≤ D(x , x ) (3.3) n n n+1 n−1 for all n 2 N. By using the inequalities (3.2) and (3.3) we have D(x , x ) ≤ D(x , x ), n = 0, 1, 2,··· n+1 0 1 D(x , Tx ) ≤ D(x , Tx ), n = 0, 1, 2,··· . n n 0 0 Then for m, n 2 N, m > n, we have m−1 D(x , x ) = D(x , x ) + D(x , x ) + ··· + D(x , x ) n n+1 n+1 n+2 m−1 m i i+1 i=n n n+1 m−1 c c c ≤ D(x , x ) + D(x , x ) + ··· + D(x , x ) 0 1 0 1 0 1 n+1 m−1 b b b n m−n−1 = a (1 + a + ··· + a )D(x , x ) (for a = ) 0 1 ≤ D(x , x ). 0 1 1 − a Since lim D(x , x ) = 0, then for every  > 0, there exist N 2 N such that 0 1 1−a n!∞ 0 < D(x , x ) < , n ≥ N. (3.4) 0 1 1 − a Now, let (f , ) 2 F × [0,∞) be such that (D ) is satised. Let " > 0 be xed. By (F ), there exist  > 0 such 3 2 that 0 < t <  ) f (t) < f (") − . (3.5) Hence by (3.5) and (F ), we get m−1 f D(x , x ) ≤ f D(x , x ) < f (") − (for m > n ≥ N). (3.6) i i+1 0 1 1 − a i=1 Using (D ) and (3.6), we obtain for m > n ≥ N, D(x x ) > 0 ) f (D(x , x )) ≤ f (") n, m n m which implies by (F ) that D(x , x ) < ". n m ∞ ∞ Therefore fx g is an F-Cauchy sequence. Since (X, D) is F-complete, there exist x 2 X such that fx g n n n=0 n=0 is F-convergent to x. i.e. lim D(x , x) = 0. n!∞ Now since f (x) = D(x, Tx) is lower semi-continuous and x ! x, we have, from (3.2) 0 ≤ D(x, Tx) = f (x) ≤ lim inff (x ) = lim infD(x , Tx ) = 0. n n n n!∞ n!∞ Therefore D(x, Tx) = 0. Since Tx is F-closed, then we get x 2 Tx. Two xed point results on F-metric spaces Ë 67 Acknowledgement: The authors are thankful to the referees for making valuable suggestions leading to the better presentations of the paper Conict of interest: The authors declare that they have no conict of interest References [1] M. Jleli, B. Samet, On a new generalization of metric spaces, J. Fixed Point Theory Appl., (2018), 20:128. [2] S. 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Journal

Topological Algebra and its Applications – de Gruyter

Published: Jan 1, 2022

Keywords: Fixed point; F -metric space; P -contraction; multivalued mapping; 54H25; 47H10

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