Fixed point theorems in a new type of modular metric spaces

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1 Turkoglu and Manav Fixed Point Theory and Applications :25 R E S E A R C H Open Access Fixed point theorems in a new type of modular metric spaces Duran Turkoglu 1 and Nesrin Manav 2* * Correspondence: nesrinmanav2@gmail.com; nmanav@erzincan.edu.tr 2 Department of Mathematics, Faculty of Science & Arts, Erzincan University, Erzincan, Turkey Full list of author information is available at the end of the article Abstract In this paper, considering both a modular metric space and a generalized metric space in the sense of Jleli and Samet Fixed Point Theory Appl. 2015:61, 2015, we introduce a new concept of generalized modular metric space. Then we present some examples showing that the generalized modular metric space includes some kind of metric structures. Finally, we provide some fixed point results for both contraction and quasicontraction type mappings on generalized modular metric spaces. MSC: Primary 47H10; secondary 54H25 Keywords: Fixed point; Fatou property; Modular metric spaces; Generalized metric spaces; Quasi-contraction 1 Introduction In 1990, the fixed point theory in modular function spaces was initiated by Khamsi, Kozlowski, and Reich [10].Modularfunctionspacesareaspecialcase ofthetheoryofmodular vector spaces introduced by Nakano [13]. Modular metric spaces were introduced in [2, 3]. Fixed point theory in modular metric spaces was studied by Abdou and Khamsi [1]. Their approach was fundamentally different from the one studied in [2, 3]. In this paper, we follow the same approach as the one used in [1]. Generalizations of standard metric spaces are interesting because they allow for some deep understanding of the classical results obtained in metric spaces. One has always to be careful when coming up with a new generalization. For example, if we relax the triangle inequality, some of the classical known facts in metric spaces may become impossible to obtain. This is the case with the generalized metric distance introduced by Jleli and Samet in [6]. The authors showed that this generalization encompasses metric spaces, b-metric spaces, dislocated metric spaces, and modular vector spaces. In this paper, considering both a modular metric space and a generalized metric space in the sense of Jleli and Samet [6], we introduce a new concept of generalized modular metric space. Then we proceed to proving the Banach contraction principle BCP and Ćirić s fixed point theorem for quasicontraction mappings in this new space. To prove Ćirić s fixed point theorem in this new space, we take the contraction constant k < 1 C, where C is as given in Definition 1.1. For readers interested in metric fixed point theory, we recommend the book by Khamsi and Kirk [8], and for more details, see [5, 7, 9, 11, 12]. First, we give the definition of generalized modular metric spaces. The Authors This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Turkoglu and Manav Fixed Point Theory and Applications :25 Page 2 of 10 Definition 1.1 Let X be an abstract set. A function D :0, X X [0, ] issaid to be a regular generalized modular metric GMMonX if it satisfies the following three axioms: GMM 1 IfD λ x, y=0for some λ >0,thenx = y for all x, y X; GMM 2 D λ x, y=d λ y, x for all λ >0and x, y X; GMM 3 ThereexistsC >0such that, if x, y X X, {x n } X with lim D λ x n, x= 0 for some λ >0,then D λ x, y C lim sup D λ x n, y. The pair X, D issaid tobe ageneralized modularmetric space GMMS. It is easy to check that if there exist x, y X such that there exists {x n } X with lim D λ x n, x=0forsomeλ >0,andD λ x, y<, thenwemusthavec 1. In fact, throughout this work, we assume C 1. Let D be a GMM on X.Fixx 0 X.Thesets X D = X D x 0 ={x X : D λ x, x 0 0asλ } XD = {x X : λ = λx>0suchthatd λx, x 0 < } are called generalized modular sets. Next, we give some examples that inspired our definition of a GMMS. Example 1.1 Modular vector spaces MVS [13] Let X bealinearvectorspaceoverthe field R. A function ρ : X [0, ] is called regular modular if the following hold: 1 ρx=0if and only if x =0, 2 ραx=ρx if α =1, 3 ραx +1 αy ρx +ρy for any α [0, 1], for any x, y X.Letρ be regular modular defined on a vector space X.Theset { } X ρ = x X; lim ραx=0 α 0 is called a MVS.Let{x n } n N be a sequence in X ρ and x X ρ.iflim ρx n x=0,then {x n } n N is said to ρ-converge to x. ρ is said to satisfy the 2 -condition if there exists K 0 such that ρ2x Kρx for any x X ρ.moreover,ρ is said to satisfy the Fatou propertyfp if ρx y lim inf ρx n y, whenever {x n } ρ-converges to x for any x, y, x n X ρ.next,weshowthatamvsmaybe embedded with a GMM structure. Indeed, let X, ρ beamvs.defined :0,+ X X [0, + ]by x y D λ x, y=ρ. λ

3 Turkoglu and Manav Fixed Point Theory and Applications :25 Page 3 of 10 Then the following hold: i If D λ x, y=0for some λ >0and any x, y X,thenx = y; ii D λ x, y=d λ y, x for any λ >0and x, y X; iii If ρ satisfies the FP, then for any λ >0and {x n } such that {x n /λ} ρ-converges to x/λ, we have x y ρ lim inf λ ρ xn y λ which implies lim sup ρ xn y λ, D λ x, y lim inf D λx n, y lim sup D λ x n, y for any x, y, x n X ρ. Therefore, X, D satisfies all the properties of Definition 1.1 as claimed. Note that the constant C which appears in the property GMM 3 is equal to 1 provided the FP is satisfied by ρ. In the next example, we discuss the case of modular metric spaces. Example 1.2 Modular metric spaces MMS [2, 3] Let X be an abstract set. For a function ω :0,+ X X [0, ], we will write ωλ, x, y=ω λ x, y. The function ω :0, X X [0, ] is said to be a regular modular metricmm on X if it satisfies the following axioms: i x = y if and only if ω λ x, y=0for some λ >0; ii ω λ x, y=ω λ y, x for all λ >0and x, y M; iii ω λ+μ x, y ω λ x, z+ω μ z, y for all λ, μ >0and x, y, z X. Let ω be regular modular on X.Fixx 0 X.Thetwosets X ω = X ω x 0 ={x X : ω λ x, x 0 0asλ } Xω = X ω x 0={x X : λ = λx>0suchthatω λ x, x 0 < } are called modular spaces around arbitrarily chosen x 0. It is clear that X ω Xω,butthis inclusion may be proper in general. Let X ω be a MMS.Iflim ω λ x n, x=0forsomeλ > 0, then we may not have lim ω λ x n, x = 0 for all λ >0.Therefore,asitisdoneinMVS, we will say that ω satisfies the 2 -condition if this is the case, i.e., lim ω λ x n, x=0for some λ > 0 implies lim ω λ x n, x = 0 for all λ > 0. We will say that the sequence {x n } n N in X ω is ω-convergent to x X ω if lim ω λ x n, x=0forsomeλ >0.Themodularfunction ω is said to satisfy the FP if {x n } is such that lim ω λ x n, x=0forsomeλ >0,we have ω λ x, y lim inf ω λx n, y

4 Turkoglu and Manav Fixed Point Theory and Applications :25 Page 4 of 10 for any y X ω.letx ω be a MMS,whereω is a regular modular. Define D :0,+ X ω X ω [0, + ]by D λ x, y=ω λ x, y. Then the following hold: i If D λ x, y=0for some λ >0and x, y X ω,thenx = y; ii D λ x, y=d λ y, x for any λ >0and x, y X ω ; iii If ω satisfies the FP, then for any x X ω and {x n } X ω such that lim D λ x n, x=0for some λ >0,wehave ω λ x, y lim inf ω λx n, y lim sup ω λ x n, y for any y X ω, which implies D λ x, y lim inf D λx n, y lim sup D λ x n, y. In other words, X ω, D is a GMMS. Example 1.3 Generalized metric spaces GMS [6] Throughout the paper X is an abstract set. For a function D : X X [0, ]andx X, we will introduce the set { } CD, X, x= {x n } X; lim Dx n, x=0. According to [6], the function D : X X [0, ] is said to define a generalized metric GM on X if it satisfies the following axioms: For every x, y X X,wehaveDx, y=0 x = y; D 2 For every x, y X X,wehaveDx, y=dy, x, D 3 There exists C >0such that, if x, y X X, {x n } CD, X, x,wehave Dx, y C lim sup Dx n, y. The pair X, D is then called a GMS. Letusshowthatsuchastructuremaybeseenasa GMMS. Indeed, let X, DbeaGMS.DefineD :0,+ X X [0, + ]by D λ x, y= Dx, y. λ Clearly, if {x n } CD, X, xforsomex X,thenwehave lim D λx n, x=0 for any λ > 0. Then the following hold: i If D λ x, y=0for some λ >0and x, y X,thenx = y; ii D λ x, y=d λ y, x for any λ >0and x, y X;

5 Turkoglu and Manav Fixed Point Theory and Applications :25 Page 5 of 10 iii There exists C >0such that, if x, y X X, {x n } CD λ, X, x for some λ >0,we have D λ x, y C lim sup D λ x n, y. These properties show that X, D is a GMMS. 2 Fixed point theorems FPT in GMMS The following definition is useful to set new fixed point theory on GMMS. Definition 2.1 Let X D, DbeaGMMS. 1 The sequence {x n } n N in X D is said to be D-convergent to x X D if and only if D λ x n, x 0,as,forsomeλ >0. 2 The sequence {x n } n N in X D is said to be D-Cauchy if D λ x m, x n 0,asm, n, for some λ >0. 3 A subset C of X D is said to be D-closed if for any {x n } from C which D-converges to x, x C. 4 A subset C of X D is said to be D-complete if for any {x n } D-Cauchy sequence in C such that lim n,m D λ x n, x m =0for some λ,thereexistsapointx C such that lim n,m D λ x n, x=0. 5 A subset C of X D is said to be D-bounded if, for some λ >0,wehave δ D,λ C=sup { D λ x, y; x, y C } <. In general, if lim D λ x n, x = 0 for some λ > 0, then we may not have lim D λ x n, x = 0 for all λ > 0. Therefore, as it is done in modular function spaces, we will say that D satisfies 2 -condition if and only if lim D λ x n, x=0forsomeλ >0 implies lim D λ x n, x = 0 for all λ >0. Another question that comes into this setting is the concept of D-limit and its uniqueness. Proposition 2.1 Let X D, D be a GMMS. Let {x n } be a sequence in X D. Let x, y X D X D such that D λ x n, x 0 and D λ x n, y 0 as n for some λ >0.Then x = y. Proof Using the property GMM 3, we have D λ x, y C lim sup D λ x n, y=0, which implies from the property GMM 1 thatx = y. 3 Themainresults 3.1 The Banach contraction principle BCP in GMMS Now, we show an extension of the BCP to the setting of GMMS presented formerly. From now on, we mean 1 instead of λ for the same reason Abdou and Khmasi used in their work [1].

6 Turkoglu and Manav Fixed Point Theory and Applications :25 Page 6 of 10 Definition 3.1 Let X D, D beagmms and f : X D X D be a mapping. f is called a D- contraction mapping if there exists k 0, 1 such that f x, f y kd1 x, y for any x, y X D X D. x is said to be a fixed point of f if f x=x. Proposition 3.1 Let X D, D be a GMMS. Let f : X D X D be a D-contraction mapping. If ω 1 and ω 2 are fixed points of f and ω 1, ω 2 <, then we have ω 1 = ω 2. Proof Let ω 1, ω 2 X D be two fixed points of f such that ω 1, ω 2 <. Asf is a D- contraction, there exists k 0,1 suchthat ω 1, ω 2 = f ω1, f ω 2 k ω 1, ω 2. Since ω 1, ω 2 <, weconcludethat ω 1, ω 2 =0,whichimpliesω 1 = ω 2 from GMM 1. Let X D, D beagmms and f : X D X D be a mapping. For any x M,definetheorbit of x by Ox= { x, f x, f 2 x,... }. Set δ D,λ x=sup{d λ f n x, f t x; n, t N}, whereλ > 0. The following result may be seen as an extension of the BCP in GMMS. Theorem 3.1 Let X D, D be a GMMS. Assume that X D is D-complete. Let f : X D X D be a D-contraction mapping. Assume that δ D,1 x 0 is finite for some x 0 X D. Then {f n x 0 } D-converges to a fixed point ω of f. Moreover, if x, ω< for x X D, then {f n x} D- converges to ω. Proof Let x 0 X D be such that δ D,1 x 0 <.Then f n+p x 0, f n x 0 k n f p x 0, x 0 k n δ D,1 x 0 for any n, p N.Sincek <1,{f n x 0 } is D-Cauchy. As X D is D-complete, then there exists ω X D such that lim f n x 0, ω=0.since f n x 0, f ω k f n 1 x 0, ω ; n =1,2,..., we have lim f n x 0, f ω = 0. Proposition 2.1 implies that f ω=ω, i.e., ω is a fixed point of f.letx X D be such that x, ω<.then f n x, ω = f n x, f n ω k n x, ω for any n 1. Since k <1,wegetlim f n x, ω =0,i.e.,{f n x} D-converges to ω.

7 Turkoglu and Manav Fixed Point Theory and Applications :25 Page 7 of 10 If x, y< for any x, y X D,thenf hasatmostonefixedpoint.moreover,ifx D is D-complete and δ D,1 x< for any x X D,thenallorbitsD-converge to the unique fixed point of f.inmetricspaces,dx, yisalwaysfinite.becauseofthisreason,anycontraction will have at most one fixed point. Moreover, the orbits of the contraction are all bounded. Indeed, let f : M M be a contraction, where M is a metric space endowed with a metric distance d.wehave d f n+1 x, f n x k n d f x, x for any n N and x M, which implies by using the triangle inequality p 1 d f n+p x, f n x d f n+k+1 x, f n+k x k n+k d f x, x 1 1 k d f x, x, since k <1.Hence k=0 p 1 k=0 sup { d f n x, f t x ; n, t N } 1 1 k d f x, x < for any x M. Next, we investigate the extension of Ćirić s FPT [4]forquasicontractiontype mappings in GMMS and give a correct version of Theorem 4.3 in [6] since its proof is wrong [7]. 3.2 Ćirić quasicontraction in generalized modular metric spaces First, let us introduce the concept of quasicontraction mappings in the setting of GMMS. Definition 3.2 Let X D, D beagmms. The mapping f : X D X D is said to be a D- quasicontraction if there exists k 0, 1 such that f x, f y k max { D1 x, y, x, f x, D1 y, f y, D1 x, f y, D1 y, f x } for any x, y X D X D. Proposition 3.2 Let X D, D be a GMMS. Let f : X D X D be a D-quasicontraction mapping. If ω is a fixed point of f such that ω, ω <, then we have ω, ω =0.Moreover, if ω 1 and ω 2 are two fixed points of f such that ω 1, ω 2 <, ω 1, ω 1 <, and ω 2, ω 2 <, then we have ω 1 = ω 2. Proof Let ω be a fixed point of f,then ω, ω = f ω, f ω k max { } ω, ω, ω, f ω, D1 ω, f ω, D1 ω, f ω, D1 ω, f ω = k ω, ω. Since k <1and ω, ω <, then ω, ω =0.Letω 1, ω 2 X D be two fixed points of f such that ω 1, ω 2 <, ω 1, ω 1 <, and ω 2, ω 2 <. Sincef is a D-

8 Turkoglu and Manav Fixed Point Theory and Applications :25 Page 8 of 10 quasicontraction, there exists k <1suchthat ω 1, ω 2 = f ω1, f ω 2 k max { ω 1, ω 2, ω1, f ω 1, ω2, f ω 2, ω1, f ω 2, ω2, f ω 1 }. = k max { ω 1, ω 2, ω 1, ω 1, ω 2, ω 2 }. Since ω 1, ω 1 < and ω 2, ω 2 <,then ω 1, ω 1 = ω 2, ω 2 =0.Nowwehave ω 1, ω 2 k ω 1, ω 2. Since ω 1, ω 2 < and k <1,then ω 1, ω 2 =0. ThefollowingresultmaybeseenasanextensionofĆirić sfpt[4] for quasicontraction type mappings in GMMS. Theorem 3.2 Let X D, D be a D-complete GMMS. Let f : X D X D be a D-quasicontraction mapping. Assume that k < 1 C, where C is the constant from GMM 3, and there exists x 0 X D such that δ D,1 x 0 <. Then {f n x 0 } D-converges to some ω X D. If x 0, f ω < and ω, f ω <, then ω is a fixed point of f. Proof Let f be a D-quasicontraction, then there exists k 0, 1 such that, for all p, r, n N and x X D,wehave f n+p+1 x, f n+r+1 x k max { f n+p x, f n+r x, f n+p x, f n+p+1 x, f n+r x, f n+r+1 x, f n+p x, f n+r+1 x, f n+r x, f n+p+1 x }. Hence δ D,1 f x kδ D,1 x for any x X D.Consequently,wehave δ D,1 f n x 0 k n δ D,1 x 0 1 for any n 1. Using the above inequality, we get f n x 0, f n+t x 0 δ D,1 f n x 0 k n δ D,1 x 0 2 for every n, m N.Sinceδ D,1 x 0 < and k <1/C 1, we have lim D 1 f n x 0, f n+t x 0 =0, n,t which implies that {f n x 0 } is a D-Cauchy sequence. Since X D is D-complete, there exists ω X D such that lim f n x 0, ω=0,i.e.,{f n x 0 } D-converges to ω. Next, we assume x 0, f ω < and ω, f ω <. Using inequality 2andthepropertyGMM 3, we get ω, f n x 0 C lim sup f n x 0, f n+t x 0 Ck n δ D,1 x 0 3 t for every n, m N.

9 Turkoglu and Manav Fixed Point Theory and Applications :25 Page 9 of 10 Hence, f x0, f ω k max { x 0, ω, x0, f x 0, ω, f ω f x0, ω, x0, f ω } and, using 1, 2, 3, and k <1/C 1, we have f 2 x 0, f ω max { k 2 Cδ D,1 x 0, k ω, f ω, k 2 ω, f x0 }. Progressively, by induction, we can get f n x 0, f ω max { k n Cδ D,1 x 0, k ω, f ω, k n ω, f x0 } for every n 1. Moreover, we have lim sup f n x 0, f ω k ω, f ω, when x 0, f ω < and δ D,1 x 0 <.AgainthepropertyGMM 3 implies ω, f ω C lim sup f n x 0, f ω kc ω, f ω. Since kc <1and ω, f ω <,then ω, f ω = 0, i.e., f ω=ω. Acknowledgements The authors would like to thank Professor M.A. Khamsi for his helpful and constructive comments that greatly contributed to improving the final version of this paper. This work was supported by the TUBITAK The Scientific and Technological Research Council of Turkey. Funding We have no funding for this article. Abbreviation Not applicable. Availability of data and materials Data sharing not applicable to this article as no datasets were generated or analysed during the current study. Competing interests The authors declare that they have no competing interests. Authors contributions All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript. Author details 1 Department of Mathematics, Gazi University, Ankara, Turkey. 2 Department of Mathematics, Faculty of Science & Arts, Erzincan University, Erzincan, Turkey. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 30 October 2017 Accepted: 15 November 2018

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