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Research article

Some results in function weighted b-metric spaces

  • Received: 29 November 2022 Revised: 12 January 2023 Accepted: 18 January 2023 Published: 01 February 2023
  • MSC : 47H09, 47H10, 54H25

  • In this paper, we introduce F-b-metric space (function weighted b-metric space) as a generalization of the F-metric space (the function weighted metric space). We also propose and prove some topological properties of the F-b-metric space, the theorems of fixed point and the common fixed point for the generalized expansive mappings, and an application on dynamic programing.

    Citation: Budi Nurwahyu, Naimah Aris, Firman. Some results in function weighted b-metric spaces[J]. AIMS Mathematics, 2023, 8(4): 8274-8293. doi: 10.3934/math.2023417

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  • In this paper, we introduce F-b-metric space (function weighted b-metric space) as a generalization of the F-metric space (the function weighted metric space). We also propose and prove some topological properties of the F-b-metric space, the theorems of fixed point and the common fixed point for the generalized expansive mappings, and an application on dynamic programing.



    The idea of statistical convergence was given by Zygmund [1] in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was introduced by Steinhaus [2] and Fast [3] and then reintroduced independently by Schoenberg [4]. Over the years and under different names, statistical convergence has been discussed in the Theory of Fourier Analysis, Ergodic Theory, Number Theory, Measure Theory, Trigonometric Series, Turnpike Theory and Banach Spaces. Later on it was further investigated from the sequence spaces point of view and linked with summability theory by Bilalov and Nazarova [5], Braha et al. [6], Cinar et al. [7], Colak [8], Connor [9], Et et al. ([10,11,12,13,14]), Fridy [15], Isik et al. ([16,17,18]), Kayan et al. [19], Kucukaslan et al. ([20,21]), Mohiuddine et al. [22], Nuray [23], Nuray and Aydın [24], Salat [25], Sengul et al. ([26,27,28,29]), Srivastava et al. ([30,31]) and many others.

    The idea of statistical convergence depends upon the density of subsets of the set N of natural numbers. The density of a subset E of N is defined by

    δ(E)=limn1nnk=1χE(k),

    provided that the limit exists, where χE is the characteristic function of the set E. It is clear that any finite subset of N has zero natural density and that

    δ(Ec)=1δ(E).

    A sequence x=(xk)kN is said to be statistically convergent to L if, for every ε>0, we have

    δ({kN:|xkL|ε})=0.

    In this case, we write \newline

    xkstatLaskorSlimkxk=L.

    In 1932, Agnew [32] introduced the concept of deferred Cesaro mean of real (or complex) valued sequences x=(xk) defined by

    (Dp,qx)n=1qnpnqnk=pn+1xk,n=1,2,3,

    where p=(pn) and q=(qn) are the sequences of non-negative integers satisfying

    pn<qnandlimnqn=. (1)

    Let K be a subset of N and denote the set {k:k(pn,qn],kK} by Kp,q(n).

    Deferred density of K is defined by

    δp,q(K)=limn1(qnpn)|Kp,q(n)|, provided the limit exists

    where, vertical bars indicate the cardinality of the enclosed set Kp,q(n). If qn=n, pn=0, then the deferred density coincides with natural density of K.

    A real valued sequence x=(xk) is said to be deferred statistically convergent to L, if for each ε>0

    limn1(qnpn)|{k(pn,qn]:|xkL|ε}|=0.

    In this case we write Sp,q-limxk=L. If qn=n, pn=0, for all nN, then deferred statistical convergence coincides with usual statistical convergence [20].

    In this section, we give some inclusion relations between statistical convergence of order α, deferred strong Cesàro summability of order α and deferred statistical convergence of order α in general metric spaces.

    Definition 1. Let (X,d) be a metric space, (pn) and (qn) be two sequences as above and 0<α1. A metric valued sequence x=(xk) is said to be Sd,αp,q-convergent (or deferred d-statistically convergent of order α) to x0 if there is x0X such that

    limn1(qnpn)α|{k(pn,qn]:xkBε(x0)}|=0,

    where Bε(x0)={xX:d(x,x0)<ε} is the open ball of radius ε and center x0. In this case we write Sd,αp,q-limxk=x0 or xkx0(Sd,αp,q). The set of all Sd,αp,q-statistically convergent sequences will be denoted by Sd,αp,q. If qn=n and pn=0, then deferred d-statistical convergence of order α coincides d -statistical convergence of order α denoted by Sd,α. In the special cases qn=n,pn=0 and α=1 then deferred d -statistical convergence of order α coincides d-statistical convergence denoted by Sd.

    Definition 2. Let (X,d) be a metric space, (pn) and (qn) be two sequences as above and 0<α1. A metric valued sequence x=(xk) is said to be strongly wd,αp,q-summable (or deferred strongly d-Ces àro summable of order α) to x0 if there is x0X such that

    limn1(qnpn)αqnk=pn+1d(xk,x0)=0.

    In this case we write wd,αp,q-limxk=x0 or xkx0(wd,αp,q). The set of all strongly wd,αp,q-summable sequences will be denoted by wd,αp,q. If qn=n and pn=0, for all nN, then deferred strong d-Cesàro summability of order α coincides strong d-Cesàro summability of order α denoted by wd,α. In the special cases qn=n,pn=0 and α=1 then deferred strong d-Cesàro summability of order α coincides strong d-Ces àro summability denoted by wd.

    Theorem 1. Let (pn) and (qn) be sequences of non-negative integers satisfying the condition (1), (X,d) be a linear metric space and x=(xk),y=(yk) be metric valued sequences, then

    (i) If Sd,αp,q-limxk=x0 and Sd,αp,q-limyk=y0, then Sd,αp,q-lim(xk+yk)=x0+y0,

    (ii)If Sd,αp,q-limxk=x0 and cC, then Sd,αp,q-lim(cxk)=cx0,

    (iii) If Sd,αp,q-limxk=x0,Sd,αp,q-limyk=y0 and x,y(X), then Sd,αp,q-lim(xkyk)=x0y0.

    Proof. Omitted.

    Theorem 2. Let (pn) and (qn) be sequences of non-negative integers satisfying the condition (1) and α and β be two real numbers such that 0<αβ1. If a sequence x=(xk) is deferred strongly d-Cesàro summable of order α to x0, then it is deferred d-statistically convergent of order β to x0, but the converse is not true.

    Proof. First part of the proof is easy, so omitted. For the converse, take X=R and choose qn=n,pn=0 (for all nN),d(x,y)=|xy| and define a sequence x=(xk) by

    xk={3n,k=n20,kn2.

    Then for every ε>0, we have

    1(qnpn)α|{k(pn,qn]:xkBε(0)}|[n]nα0, as n,

    where 12<α1, that is xk0(Sd,αp,q). At the same time, we get

    1(qnpn)αqnk=pn+1d(xk,0)[n][3n]nα1

    for α=16 and

    1(qnpn)αqnk=pn+1d(xk,0)[n][3n]nα

    for 0<α<16, i.e., xk0(wd,αp,q) for 0<α16.

    From Theorem 2 we have the following results.

    Corollary 1. ⅰ) Let (pn) and (qn) be sequences of non-negative integers satisfying the condition (1) and α be a real number such that 0<α1. If a sequence x=(xk) is deferred strongly d-Cesàro summable of order α to x0, then it is deferred d-statistically convergent of order α to x0, but the converse is not true.

    ⅱ) Let (pn) and (qn) be sequences of non-negative integers satisfying the condition (1) and α be a real number such that 0<α1. If a sequence x=(xk) is deferred strongly d-Cesàro summable of order α to x0, then it is deferred d-statistically convergent to x0, but the converse is not true.

    ⅲ) Let (pn) and (qn) be sequences of non-negative integers satisfying the condition (1). If a sequence x=(xk) is deferred strongly d-Cesàro summable to x0, then it is deferred d-statistically convergent to x0, but the converse is not true.

    Remark Even if x=(xk) is a bounded sequence in a metric space, the converse of Theorem 2 (So Corollary 1 i) and ii)) does not hold, in general. To show this we give the following example.

    Example 1. Take X=R and choose qn=n,pn=0 (for all nN),d(x,y)=|xy| and define a sequence x=(xk) by

    xk={1k,kn30,k=n3n=1,2,....

    It is clear that x and it can be shown that xSd,αwd,α for 13<α<12.

    In the special case α=1, we can give the followig result.

    Theorem 3. Let (pn) and (qn) be sequences of non-negative integers satisfying the condition (1) and x=(xk) is a bounded sequence in a metric space. If a sequence x=(xk) is deferred d-statistically convergent to x0, then it is deferred strongly d-Cesàro summable to x0.

    Proof. Let x=(xk) be deferred d-statistically convergent to x0 and ε>0 be given. Then there exists x0X such that

    limn1(qnpn)|{k(pn,qn]:xkBε(x0)}|=0,

    Since x=(xk) is a bounded sequence in a metric space X, there exists x0X and a positive real number M such that d(xk,x0)<M for all kN. So we have

    1(qnpn)qnk=pn+1d(xk,x0)=1(qnpn)qnk=pn+1d(xk,x0)εd(xk,x0)+1(qnpn)qnk=pn+1d(xk,x0)<εd(xk,x0)M(qnpn)|{k(pn,qn]:xkBε(x0)}|+ε

    Takin limit n, we get wdp,q-limxk=x0.

    Theorem 4. Let (pn) and (qn) be sequences of non-negative integers satisfying the condition (1) and α be a real number such that 0<α1. If liminfnqnpn>1, then Sd,αSd,αp,q.

    Proof. Suppose that liminfnqnpn>1; then there exists a ν>0 such that qnpn1+ν for sufficiently large n, which implies that

    (qnpnqn)α(ν1+ν)α1qαnνα(1+ν)α1(qnpn)α.

    If xkx0(Sd,α), then for every ε>0 and for sufficiently large n, we have

    1qαn|{kqn:xkBε(x0)}|1qαn|{k(pn,qn]:xkBε(x0)}|να(1+ν)α1(qnpn)α|{k(pn,qn]:xkBε(x0)}|.

    This proves the proof.

    Theorem 5. Let (pn) and (qn) be sequences of non-negative integers satisfying the condition (1) and α and β be two real numbers such that 0<αβ1. If limn(qnpn)αqβn=s>0, then Sd,αSd,βp,q.

    Proof. Let limn(qnpn)αqβn=s>0. Notice that for each ε>0 the inclusion

    {kqn:xkBε(x0)}{k(pn,qn]:xkBε(x0)}

    is satisfied and so we have the following inequality

    1qαn|{kqn:xkBε(x0)}|1qαn|{k(pn,qn]:xkBε(x0)}|1qβn|{k(pn,qn]:xkBε(x0)}|=(qnpn)αqβn1(qnpn)α|{k(pn,qn]:xkBε(x0)}|(qnpn)αqβn1(qnpn)β|{k(pn,qn]:xkBε(x0)}|.

    Therefore Sd,αSd,βp,q.

    Theorem 6. Let (pn),(qn),(pn) and (qn) be four sequences of non-negative real numbers such that

    pn<pn<qn<qn for all nN, (2)

    and α,β be fixed real numbers such that 0<αβ1, then

    (i) If

    limn(qnpn)α(qnpn)β=a>0 (3)

    then Sd,βp,qSd,αp,q,

    (ii) If

    limnqnpn(qnpn)β=1 (4)

    then Sd,αp,qSd,βp,q.

    Proof. (i) Let (3) be satisfied. For given ε>0 we have

    {k(pn,qn]:xkBε(x0)}{k(pn,qn]:xkBε(x0)},

    and so

    1(qnpn)β|{k(pn,qn]:xkBε(x0)}|(qnpn)α(qnpn)β1(qnpn)α|{k(pn,qn]:xkBε(x0)}|.

    Therefore Sd,βp,qSd,αp,q.

    (ii) Let (4) be satisfied and x=(xk) be a deferred d-statistically convergent sequence of order α to x0. Then for given ε>0, we have

    1(qnpn)β|{k(pn,qn]:xkBε(x0)}|1(qnpn)β|{k(pn,pn]:xkBε(x0)}|+1(qnpn)β|{k(qn,qn]:xkBε(x0)}|+1(qnpn)β|{k(pn,qn]:xkBε(x0)}|pnpn+qnqn(qnpn)β+1(qnpn)β|{k(pn,qn]:xkBε(x0)}|=(qnpn)(qnpn)(qnpn)β+1(qnpn)β|{k(pn,qn]:xkBε(x0)}|(qnpn)(qnpn)β(qnpn)β+1(qnpn)β|{k(pn,qn]:xkBε(x0)}|(qnpn(qnpn)β1)+1(qnpn)α|{k(pn,qn]:xkBε(x0)}|

    Therefore Sd,αp,qSd,βp,q.

    Theorem 7. Let (pn),(qn),(pn) and (qn) be four sequences of non-negative integers defined as in (2) and α,β be fixed real numbers such that 0<αβ1.

    (i) If (3) holds then wd,βp,qwd,αp,q,

    (ii) If (4) holds and x=(xk) be a bounded sequence, then wd,αp,qwd,βp,q.

    Proof.

    i) Omitted.

    ii) Suppose that wd,αp,q-limxk=x0 and (xk)(X). Then there exists some M>0 such that d(xk,x0)<M for all k, then

    1(qnpn)βqnk=pn+1d(xk,x0)=1(qnpn)β[pnk=pn+1+qnk=pn+1+qnk=qn+1]d(xk,x0)pnpn+qnqn(qnpn)βM+1(qnpn)βqnk=pn+1d(xk,x0)(qnpn)(qnpn)β(qnpn)βM+1(qnpn)αqnk=pn+1d(xk,x0)=(qnpn(qnpn)β1)M+1(qnpn)αqnk=pn+1d(xk,x0)

    Theorem 8. Let (pn),(qn),(pn) and (qn) be four sequences of non-negative integers defined as in (2) and α,β be fixed real numbers such that 0<αβ1. Then

    (i) Let (3) holds, if a sequence is strongly wd,βp,q-summable to x0, then it is Sd,αp,q-convergent to x0,

    (ii) Let (4) holds and x=(xk) be a bounded sequence in (X,d), if a sequence is Sd,αp,q-convergent to x0 then it is strongly wd,βp,q-summable to x0.

    Proof. (i) Omitted.

    (ii) Suppose that Sd,αp,q-limxk=x0 and (xk). Then there exists some M>0 such that d(xk,x0)<M for all k, then for every ε>0 we may write

    1(qnpn)βqnk=pn+1d(xk,x0)=1(qnpn)βqnpnk=qnpn+1d(xk,x0)+1(qnpn)βqnk=pn+1d(xk,x0)(qnpn)(qnpn)(qnpn)βM+1(qnpn)βqnk=pn+1d(xk,x0)(qnpn)(qnpn)β(qnpn)βM+1(qnpn)βqnk=pn+1d(xk,x0)(qnpn(qnpn)β1)M+1(qnpn)βqnk=pn+1d(xk,x0)εd(xk,x0)+1(qnpn)βqnk=pn+1d(xk,x0)<εd(xk,x0)(qnpn(qnpn)β1)M+M(qnpn)α|{k(pn,qn]:d(xk,x0)ε}|+qnpn(qnpn)βε.

    This completes the proof.

    The authors declare that they have no conflict of interests.



    [1] I. A. Bakhtin, The contraction mapping principle in quasi- metric spaces, Funct. Anal., 30 (1989), 26–37.
    [2] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inf. Univ. Ostrav., 1 (1993), 5–11.
    [3] R. George, S. Radenović, K. P. Reshma, S. Shukla, Rectangular b-metric space and contraction principles, J. Nonlinear Sci. Appl., 8 (2015), 1005–1013. https://doi.org/10.22436/jnsa.008.06.11 doi: 10.22436/jnsa.008.06.11
    [4] R. George, A. Belhenniche, S. Benahmed, Z. D. Mitrović, N. Mlaiki, L. Guran, On an open question in controlled rectangular b-metric spaces, Mathematics, 8 (2020), 2239. https://doi.org/10.3390/math8122239 doi: 10.3390/math8122239
    [5] A. Belhenniche, L. Guran, S. Benahmed, F. L. Pereira, Solving nonlinear and dynamic programming equations on extended b-metric spaces with the fixed-point technique, Fixed Point Theory Algorithms Sci. Eng., 2022 (2022), 24. https://doi.org/10.1186/s13663-022-00736-5 doi: 10.1186/s13663-022-00736-5
    [6] M. A. Khamsi, Remarks on cone metric spaces and fixed point theorems of contractive mappings, Fixed Point Theory Appl., 2010 (2010), 315398. https://doi.org/10.1155/2010/315398 doi: 10.1155/2010/315398
    [7] A. Branciari, A fixed point theorem of Banach-Caccippoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57 (2000), 31–37. https://doi.org/10.5486/PMD.2000.2133 doi: 10.5486/PMD.2000.2133
    [8] R. Fagin, R. Kumar, D. Sivakumar, Comparing top k lists, SIAM J. Discrete Math., 17 (2003), 134–160. https://doi.org/10.1137/S0895480102412856 doi: 10.1137/S0895480102412856
    [9] Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal., 7 (2006), 289–297.
    [10] X. Huang, C. Zhu, X. Wen, Fixed point theorems for expanding mappings in cone metric spaces, Math. Rep., 14 (2012), 141–148.
    [11] C. Chen, C. Zhu, Fixed point theorems for times reasonable expansive mapping, Fixed Point Theory Appl., 2008 (2008), 302617. https://doi.org/10.1155/2008/302617 doi: 10.1155/2008/302617
    [12] M. Akkouchi, A common fixed point theorem for expansive mappings under strict implicit conditions on b-metric spaces, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. Math., 50 (2011), 5–15.
    [13] J. Rashmi, R. D. Daheriya, M. Ughade, Fixed point, coincidence point and common fixed point theorems under various expansive conditions in b-metric spaces, Int. J. Sci. Innovative Math. Res., 3 (2015), 26–34.
    [14] M. Jleli, B. Samet, On a new generalization of metric spaces, J. Fixed Point Theory Appl., 20 (2018), 128. https://doi.org/10.1007/s11784-018-0606-6 doi: 10.1007/s11784-018-0606-6
    [15] M. Jleli, B. Samet, A generalized metric space and related fixed point theorems, Fixed Point Theory Algorithm Sci. Eng., 2015 (2015), 61. https://doi.org/10.1186/s13663-015-0312-7 doi: 10.1186/s13663-015-0312-7
    [16] A. Hussain, H. Al-Sulami, H. Hussain, H. Farooq, Newly fixed disc results using advanced contractions on F-metric space, J. Appl. Anal. Comput., 10 (2020), 2313–2322. https://doi.org/10.11948/20190197 doi: 10.11948/20190197
    [17] H. Işik, N. Hussain, A. R. Khan, Endpoint results for weakly contractive mappings in F-metric spaces with an application, Int. J. Nonlinear Anal. Appl., 11 (2020), 351–361. https://doi.org/10.22075/ijnaa.2020.20368.2148 doi: 10.22075/ijnaa.2020.20368.2148
    [18] S. Z. Wang, B. Y. Li, Z. M. Gao, K. Iseki, Some fixed point theorems on expansion mappings, Math. Jpn., 29 (1984), 631–636.
    [19] M. Abbas, B. E. Rhoades, Common fixed point theorems for occasionally weakly compatible mappings satisfying a generalized contractive condition, Math. Commun., 13 (2008), 295–301.
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