Research article

Complex-valued controlled rectangular metric type spaces and application to linear systems

  • Received: 13 March 2023 Revised: 27 April 2023 Accepted: 03 May 2023 Published: 11 May 2023
  • MSC : 47H10, 54H25

  • Fixed point theory can be generalized to cover multidisciplinary areas such as computer science; it can also be used for image authentication to ensure secure communication and detect any malicious modifications. In this article, we introduce the notion of complex-valued controlled rectangular metric-type spaces, where we prove fixed point theorems for self-mappings in such spaces. Furthermore, we present several examples and give two applications of our main results: solving linear systems of equations and finding a unique solution for an equation of the form f(x)=0.

    Citation: Fatima M. Azmi, Nabil Mlaiki, Salma Haque, Wasfi Shatanawi. Complex-valued controlled rectangular metric type spaces and application to linear systems[J]. AIMS Mathematics, 2023, 8(7): 16584-16598. doi: 10.3934/math.2023848

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  • Fixed point theory can be generalized to cover multidisciplinary areas such as computer science; it can also be used for image authentication to ensure secure communication and detect any malicious modifications. In this article, we introduce the notion of complex-valued controlled rectangular metric-type spaces, where we prove fixed point theorems for self-mappings in such spaces. Furthermore, we present several examples and give two applications of our main results: solving linear systems of equations and finding a unique solution for an equation of the form f(x)=0.



    Let Mn be the set of n×n complex matrices. Mn(Mk) is the set of n×n block matrices with each block in Mk. For AMn, the conjugate transpose of A is denoted by A. When A is Hermitian, we denote the eigenvalues of A in nonincreasing order λ1(A)λ2(A)...λn(A); see [2,7,8,9]. The singular values of A, denoted by s1(A),s2(A),...,sn(A), are the eigenvalues of the positive semi-definite matrix |A|=(AA)1/2, arranged in nonincreasing order and repeated according to multiplicity as s1(A)s2(A)...sn(A). If AMn is positive semi-definite (definite), then we write A0(A>0). Every AMn admits what is called the cartesian decomposition A=ReA+iImA, where ReA=A+A2, ImA=AA2. A matrix AMn is called accretive if ReA is positive definite. Recall that a norm |||| on Mn is unitarily invariant if ||UAV||=||A|| for any AMn and unitary matrices U,VMn. The Hilbert-Schmidt norm is defined as ||A||22=tr(AA).

    For A,B>0 and t[0,1], the weighted geometric mean of A and B is defined as follows

    AtB =A1/2(A1/2BA1/2)tA1/2.

    When t=12, A12B is called the geometric mean of A and B, which is often denoted by AB. It is known that the notion of the (weighted) geometric mean could be extended to cover all positive semi-definite matrices; see [3, Chapter 4].

    Let A,B,XMn. For 2×2 block matrix M in the form

    M=(AXXB)M2n

    with each block in Mn, its partial transpose of M is defined by

    Mτ=(AXXB).

    If M and Mτ0, then we say it is positive partial transpose (PPT). We extend the notion to accretive matrices. If

    M=(AXYB)M2n,

    and

    Mτ=(AYXC)M2n

    are both accretive, then we say that M is APT (i.e., accretive partial transpose). It is easy to see that the class of APT matrices includes the class of PPT matrices; see [6,10,13].

    Recently, many results involving the off-diagonal block of a PPT matrix and its diagonal blocks were presented; see [5,11,12]. In 2023, Alakhrass [1] presented the following two results on 2×2 block PPT matrices.

    Theorem 1.1 ([1], Theorem 3.1). Let (AXXB) be PPT and let X=U|X| be the polar decomposition of X, then

    |X|(AtB)(U(A1tB)U),t[0,1].

    Theorem 1.2 ([1], Theorem 3.2). Let (AXXB) be PPT, then for t[0,1],

    ReX(AtB)(A1tB)(AtB)+(A1tB)2,

    and

    ImX(AtB)(A1tB)(AtB)+(A1tB)2.

    By Theorem 1.1 and the fact si+j1(XY)si(X)sj(Y)(i+jn+1), the author obtained the following corollary.

    Corollary 1.3 ([1], Corollary 3.5). Let (AXXB) be PPT, then for t[0,1],

    si+j1(X)si(AtB)sj(A1tB).

    Consequently,

    s2j1(X)sj(AtB)sj(A1tB).

    A careful examination of Alakhrass' proof in Corollary 1.3 actually revealed an error. The right results are si+j1(X)si(AtB)12sj((A1tB)12) and s2j1(X)sj((AtB)12)sj((A1tB)12). Thus, in this note, we will give a correct proof of Corollary 1.3 and extend the above inequalities to the class of 2×2 block APT matrices. At the same time, some relevant results will be obtained.

    Before presenting and proving our results, we need the following several lemmas of the weighted geometric mean of two positive matrices.

    Lemma 2.1. [3, Chapter 4] Let X,YMn be positive definite, then

    1) XY=max{Z:Z=Z,(XZZY)0}.

    2) XY=X12UY12 for some unitary matrix U.

    Lemma 2.2. [4, Theorem 3] Let X,YMn be positive definite, then for every unitarily invariant norm,

    ||XtY||||X1tYt||||(1t)X+tY||.

    Now, we give a lemma that will play an important role in the later proofs.

    Lemma 2.3. Let M=(AXYB)M2n be APT, then for t[0,1],

    (ReAtReBX+Y2X+Y2ReA1tReB)

    is PPT.

    Proof: Since M is APT, we have that

    ReM=(ReAX+Y2X+Y2ReB)

    is PPT.

    Therefore, ReM0 and ReMτ0.

    By the Schur complement theorem, we have

    ReBX+Y2(ReA)1X+Y20,

    and

    ReAX+Y2(ReB)1X+Y20.

    Compute

    X+Y2(ReAtReB)1X+Y2=X+Y2((ReA)1t(ReB)1)X+Y2=(X+Y2(ReA)1X+Y2)t(X+Y2(ReB)1X+Y2)ReBtReA.

    Thus,

    (ReBtReA)X+Y2(ReAtReB)1X+Y20.

    By utilizing (ReBtReA)=ReA1tReB, we have

    (ReAtReBX+Y2X+Y2ReA1tReB)0.

    Similarly, we have

    (ReAtReBX+Y2X+Y2ReA1tReB)0.

    This completes the proof.

    First, we give the correct proof of Corollary 1.3.

    Proof: By Theorem 1.1, there exists a unitary matrix UMn such that |X|(AtB)(U(A1tB)U). Moreover, by Lemma 2.1, we have (AtB)(U(A1tB)U)=(AtB)12V(U(A1tB)12U). Now, by si+j1(AB)si(A)sj(B), we have

    si+j1(X)si+j1((AtB)(U(A1tB)U))=si+j1((AtB)12VU(A1tB)12U)si((AtB)12)sj((A1tB)12),

    which completes the proof.

    Next, we generalize Theorem 1.1 to the class of APT matrices.

    Theorem 2.4. Let M=(AXYB) be APT, then

    |X+Y2|(ReAtReB)(U(ReA1tReB)U),

    where UMn is any unitary matrix such that X+Y2=U|X+Y2|.

    Proof: Since M is an APT matrix, we know that

    (ReAtReBX+Y2X+Y2ReB1tReA)

    is PPT.

    Let W be a unitary matrix defined as W=(I00U). Thus,

    W(ReAtReBX+Y2X+Y2ReA1tReB)W=(ReAtReB|X+Y2||X+Y2|U(ReA1tReB)U)0.

    By Lemma 2.1, we have

    |X+Y2|(ReAtReB)(U(ReA1tReB)U).

    Remark 1. When M=(AXYB) is PPT in Theorem 2.4, our result is Theorem 1.1. Thus, our result is a generalization of Theorem 1.1.

    Using Theorem 2.4 and Lemma 2.2, we have the following.

    Corollary 2.5. Let M=(AXYB) be APT and let t[0,1], then for every unitarily invariant norm |||| and some unitary matrix UMn,

    ||X+Y2||||(ReAtReB)(U(ReA1tReB)U)||||(ReAtReB)+U(ReA1tReB)U2||||ReAtReB||+||ReA1tReB||2||(ReA)1t(ReB)t||+||(ReA)t(ReB)1t||2||(1t)ReA+tReB||+||tReA+(1t)ReB||2.

    Proof: The first inequality follows from Theorem 2.4. The third one is by the triangle inequality. The other conclusions hold by Lemma 2.2.

    In particular, when t=12, we have the following result.

    Corollary 2.6. Let M=(AXYB) be APT, then for every unitarily invariant norm |||| and some unitary matrix UMn,

    ||X+Y2||||(ReAReB)(U(ReAReB)U)||||(ReAReB)+U(ReAReB)U2||||ReAReB||||(ReA)12(ReB)12||||ReA+ReB2||.

    Squaring the inequalities in Corollary 2.6, we get a quick consequence.

    Corollary 2.7. If M=(AXYB) is APT, then

    tr((X+Y2)(X+Y2))tr((ReAReB)2)tr(ReAReB)tr((ReA+ReB2)2).

    Proof: Compute

    tr((X+Y2)(X+Y2))tr((ReAReB)(ReAReB))=tr((ReAReB)2)tr((ReA)(ReB))tr((ReA+ReB2)2).

    It is known that for any X,YMn and any indices i,j such that i+jn+1, we have si+j1(XY)si(X)sj(Y) (see [2, Page 75]). By utilizing this fact and Theorem 2.4, we can obtain the following result.

    Corollary 2.8. Let M=(AXYB) be APT, then for any t[0,1], we have

    si+j1(X+Y2)si((ReAtReB)12)sj((ReA1tReB)12).

    Consequently,

    s2j1(X+Y2)sj((ReAtReB)12)sj((ReA1tReB)12).

    Proof: By Lemma 2.1 and Theorem 2.4, observe that

    si+j1(X+Y2)=si+j1(|X+Y2|)si+j1((ReAtReB)(U(ReA1tReB)U))=si+j1((ReAtReB)12V(U(ReA1tReB)U)12)si((ReAtReB)12V)sj((U(ReA1tReB)U)12)=si((ReAtReB)12)sj((ReA1tReB)12).

    Finally, we study the relationship between the diagonal blocks and the real part of the off-diagonal blocks of the APT matrix M.

    Theorem 2.9. Let M=(AXYB) be APT, then for all t[0,1],

    Re(X+Y2)(ReAtReB)(ReA1tReB)(ReAtReB)+(ReA1tReB)2,

    and

    Im(X+Y2)(ReAtReB)(ReA1tReB)(ReAtReB)+(ReA1tReB)2.

    Proof: Since M is APT, we have that

    ReM=(ReAX+Y2X+Y2ReB)

    is PPT.

    Therefore,

    (ReAtReBRe(X+Y2)Re(X+Y2)ReA1tReB)=12(ReAtReBX+Y2X+Y2ReA1tReB)+12(ReAtReBX+Y2X+Y2ReA1tReB)0.

    So, by Lemma 2.1, we have

    Re(X+Y2)(ReAtReB)(ReA1tReB).

    This implies the first inequality.

    Since ReM is PPT, we have

    (ReAiX+Y2iX+Y2ReB)=(I00iI)(ReM)(I00iI)0,(ReAiX+Y2iX+Y2ReB)=(I00iI)((ReM)τ)(I00iI)0.

    Thus,

    (ReAiX+Y2iX+Y2ReB)

    is PPT.

    By Lemma 2.3,

    (ReAtReBiX+Y2iX+Y2ReA1tReB)

    is also PPT.

    So,

    12(ReAtReBiX+Y2iX+Y2ReA1tReB)+12(ReAtReBiX+Y2iX+Y2ReA1tReB)0,

    which means that

    (ReAtReBIm(X+Y2)Im(X+Y2)ReA1tReB)0.

    By Lemma 2.1, we have

    Im(X+Y2)(ReAtReB)(ReA1tReB).

    This completes the proof.

    Corollary 2.10. Let (ReAX+Y2X+Y2ReB)0. If X+Y2 is Hermitian and t[0,1], then,

    X+Y2(ReAtReB)(ReA1tReB)(ReAtReB)+(ReA1tReB)2.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The work is supported by National Natural Science Foundation (grant No. 12261030), Hainan Provincial Natural Science Foundation for High-level Talents (grant No. 123RC474), Hainan Provincial Natural Science Foundation of China (grant No. 124RC503), the Hainan Provincial Graduate Innovation Research Program (grant No. Qhys2023-383 and Qhys2023-385), and the Key Laboratory of Computational Science and Application of Hainan Province.

    The authors declare that they have no conflict of interest.



    [1] S. Banach, Sur les operations dans les ensembles et leur application aux equation sitegrales, Fund. Math., 3 (1922), 133–181. https://doi.org/10.4064/FM-3-1-133-181 doi: 10.4064/FM-3-1-133-181
    [2] J. A. Gatica, W. A. Kirk, A fixed point theorem for k-set-contractions defined in a cone, Pac. J. Math., 53 (1974), 131–136.
    [3] Z. Zhang, K. Wang, On fixed point theorems of mixed monotone operators and applications, Nonlinear Anal. Theor., 70 (2009), 3279–3284. https://doi.org/10.1016/j.na.2008.04.032 doi: 10.1016/j.na.2008.04.032
    [4] J. Brzdek, M. Piszczek, Fixed points of some nonlinear operators in spaces of multifunctions and the Ulam stability, J. Fixed Point Theory Appl., 19 (2017), 2441–2448. https://doi.org/10.1007/s11784-017-0441-1 doi: 10.1007/s11784-017-0441-1
    [5] S. Nadler, Multi-valued contraction mappings, Pac. J. Math., 30 (1969), 475–488.
    [6] S. Ghaler, 2-metrishche raume und ihre topologische strukture, Math. Nachr., 26 (1963), 115–148. https://doi.org/10.1002/mana.19630260109 doi: 10.1002/mana.19630260109
    [7] I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Func. Anal. Gos. Ped. Inst. Unianowsk, 30 (1989), 26–37.
    [8] S. Czerwik, Contraction mappings in b-metric spaces, Acta Mathematica et Informatica Universitatis Ostraviensis, 1 (1993), 5–11.
    [9] M. A. Kutbi, E. Karapınar, J. Ahmad, A. Azam, Some fixed point results for multi-valued mappings in b-metric spaces, J. Inequal. Appl., 2014 (2014), 126. https://doi.org/10.1186/1029-242X-2014-126 doi: 10.1186/1029-242X-2014-126
    [10] T. Kamran, M. Samreen, Q. U. Ain, A generalization of b-metric space and some fixed point theorems, Mathematics, 5 (2017), 19. https://doi.org/10.3390/math5020019 doi: 10.3390/math5020019
    [11] E. Karapinar, A short survey on the recent fixed point results on b-metric spaces, Constructive Mathematical Analysis, 1 (2018), 15–44. https://doi.org/10.33205/cma.453034 doi: 10.33205/cma.453034
    [12] W. Shatanawi, K. Abodayeh, A. Mukheimer, Some fixed point theorems in extended b-metric spaces, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 80 (2018), 71–78.
    [13] R. George, S. Radenovic, P. K. 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
    [14] N. Mlaiki, M. Hajji, T. Abdeljawad, A new extension of the rectangular-metric spaces, Adv. Math. Phys., 2020 (2020), 8319584. https://doi.org/10.1155/2020/8319584 doi: 10.1155/2020/8319584
    [15] N. Mlaiki, H. Aydi, N. Souayah, T. Abdeljawad, Controlled metric type spaces and the related contraction principle, Mathematics, 6 (2018), 194. https://doi.org/10.3390/math6100194 doi: 10.3390/math6100194
    [16] T. Abdeljawad, N. Mlaiki, H. Aydi, N. Souayah, Double controlled metric type spaces and some fixed point results, Mathematics, 6 (2018), 320. https://doi.org/10.3390/math6120320 doi: 10.3390/math6120320
    [17] W. Shatanawi, Common fixed points for mappings under contractive conditions of (α,β,ψ)-admissibility type, Mathematics, 6 (2018), 261. https://doi.org/10.3390/math6110261 doi: 10.3390/math6110261
    [18] N. Y. Özgür, N. Mlaiki, N. Taş, N. Souayah, A new generalization of metric spaces: rectangular M-metric spaces, Math. Sci., 12 (2018), 223–233. https://doi.org/10.1007/s40096-018-0262-4 doi: 10.1007/s40096-018-0262-4
    [19] M. Zhou, N. Saleem, X. L. Liu, N. Ozgur, On two new contractions and discontinuity on fixed points, AIMS Mathematics, 7 (2022), 1628–1663. https://doi.org/10.3934/math.2022095 doi: 10.3934/math.2022095
    [20] A. Latif, R. F. A. Subaie, M. O. Ansari, Fixed points of generalized multi-valued contractive mappings in metric type spaces, J. Nonlinear Var. Anal., 6 (2022), 123–138. https://doi.org/10.23952/jnva.6.2022.1.07 doi: 10.23952/jnva.6.2022.1.07
    [21] E. Karapınar, R. P. Agarwal, S. S. Yesilkaya, C. Wang, Fixed-point results for Meir-Keeler type contractions in partial metric spaces: a survey, Mathematics, 10 (2022), 3109. https://doi.org/10.3390/math10173109 doi: 10.3390/math10173109
    [22] F. M. Azmi, New fixed point results in double controlled metric type spaces with applications, AIMS Mathematics, 8 (2023), 1592–1609. https://doi.org/10.3934/math.2023080 doi: 10.3934/math.2023080
    [23] F. M. Azmi, New contractive mappings and solutions to boundary-value problems in triple controlled metric type spaces, Symmetry, 14 (2022), 2270. https://doi.org/10.3390/sym14112270 doi: 10.3390/sym14112270
    [24] A. Azam, B. Fisher, M. Khan, Common fixed point theorems in complex-valued metric spaces, Numer. Func. Anal. Opt., 32 (2011), 243–253.
    [25] K. Rao, P. Swamy, J. Prasad, A common fixed point theorem in complex-valued b-metric spaces, Bulletin of Mathematics and Statistics Research, 1 (2013), 1–8.
    [26] N. Ullah, M. S. Shagari, A. Azam, Fixed point theorems in complex-valued extended b-metric spaces, Journal of Pure and Applied Analysis, 5 (2019), 140–163. https://doi.org/10.2478/mjpaa-2019-0011 doi: 10.2478/mjpaa-2019-0011
    [27] M. Abbas, V. Raji, T. Nazir, S. Radenovi, Common fixed point of mappings satisfying rational inequalities in ordered complex valued generalized metric spaces, Afr. Mat., 26 (2015), 17–30. https://doi.org/10.1007/s13370-013-0185-z doi: 10.1007/s13370-013-0185-z
    [28] N. Ullah, M. S. Shagari, Fixed point results in complex-valued rectangular extended b-metric spaces with applications, Mathematical Analysis and Complex Optimization, 1 (2020), 107–120. https://doi.org/10.29252/maco.1.2.11 doi: 10.29252/maco.1.2.11
    [29] N. Mlaiki, T. Abdeljawad, W. Shatanawi, H. Aydi, Y. Gaba, On complex-valued triple controlled metric spaces and applications, J. Funct. Space., 2021 (2021), 5563456. https://doi.org/10.1155/2021/5563456 doi: 10.1155/2021/5563456
    [30] M. S. Aslam, M. S. R. Chowdhury, L. Guran, M. A. Alqudah, T. Abdeljawad, Fixed point theory in complex valued controlled metric spaces with an application, AIMS Mathematics, 7 (2022), 11879–11904. https://doi.org/10.3934/math.2022663 doi: 10.3934/math.2022663
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