Loading [MathJax]/jax/element/mml/optable/MathOperators.js
Research article

A note on the preconditioned tensor splitting iterative method for solving strong M-tensor systems

  • Received: 19 October 2021 Revised: 26 January 2022 Accepted: 06 February 2022 Published: 09 February 2022
  • MSC : 15A69, 65F10

  • In this note, we present a new preconditioner for solving the multi-linear systems, which arise from many practical problems and are different from the traditional linear systems. Based on the analysis of the spectral radius, we give new comparison results between some preconditioned tensor splitting iterative methods. Numerical examples are given to demonstrate the efficiency of the proposed preconditioned method.

    Citation: Qingbing Liu, Aimin Xu, Shuhua Yin, Zhe Tu. A note on the preconditioned tensor splitting iterative method for solving strong M-tensor systems[J]. AIMS Mathematics, 2022, 7(4): 7177-7186. doi: 10.3934/math.2022400

    Related Papers:

    [1] Mustafa Mudhesh, Hasanen A. Hammad, Eskandar Ameer, Muhammad Arshad, Fahd Jarad . Novel results on fixed-point methodologies for hybrid contraction mappings in Mb-metric spaces with an application. AIMS Mathematics, 2023, 8(1): 1530-1549. doi: 10.3934/math.2023077
    [2] Muhammad Tariq, Muhammad Arshad, Mujahid Abbas, Eskandar Ameer, Saber Mansour, Hassen Aydi . A relation theoretic m-metric fixed point algorithm and related applications. AIMS Mathematics, 2023, 8(8): 19504-19525. doi: 10.3934/math.2023995
    [3] Huaping Huang, Bessem Samet . Two fixed point theorems in complete metric spaces. AIMS Mathematics, 2024, 9(11): 30612-30637. doi: 10.3934/math.20241478
    [4] Hanadi Zahed, Zhenhua Ma, Jamshaid Ahmad . On fixed point results in F-metric spaces with applications. AIMS Mathematics, 2023, 8(7): 16887-16905. doi: 10.3934/math.2023863
    [5] Umar Ishtiaq, Fahad Jahangeer, Doha A. Kattan, Manuel De la Sen . Generalized common best proximity point results in fuzzy multiplicative metric spaces. AIMS Mathematics, 2023, 8(11): 25454-25476. doi: 10.3934/math.20231299
    [6] Leyla Sağ Dönmez, Abdurrahman Büyükkaya, Mahpeyker Öztürk . Fixed-point results via αji-(DC(PˆE))-contractions in partial -metric spaces. AIMS Mathematics, 2023, 8(10): 23674-23706. doi: 10.3934/math.20231204
    [7] Jamshaid Ahmad, Abdullah Shoaib, Irshad Ayoob, Nabil Mlaiki . Common fixed points for (κGm)-contractions with applications. AIMS Mathematics, 2024, 9(6): 15949-15965. doi: 10.3934/math.2024772
    [8] Muhammad Waseem Asghar, Mujahid Abbas, Cyril Dennis Enyi, McSylvester Ejighikeme Omaba . Iterative approximation of fixed points of generalized αm-nonexpansive mappings in modular spaces. AIMS Mathematics, 2023, 8(11): 26922-26944. doi: 10.3934/math.20231378
    [9] Tahair Rasham, Muhammad Nazam, Hassen Aydi, Abdullah Shoaib, Choonkil Park, Jung Rye Lee . Hybrid pair of multivalued mappings in modular-like metric spaces and applications. AIMS Mathematics, 2022, 7(6): 10582-10595. doi: 10.3934/math.2022590
    [10] Amer Hassan Albargi, Jamshaid Ahmad . Fixed point results of fuzzy mappings with applications. AIMS Mathematics, 2023, 8(5): 11572-11588. doi: 10.3934/math.2023586
  • In this note, we present a new preconditioner for solving the multi-linear systems, which arise from many practical problems and are different from the traditional linear systems. Based on the analysis of the spectral radius, we give new comparison results between some preconditioned tensor splitting iterative methods. Numerical examples are given to demonstrate the efficiency of the proposed preconditioned method.



    In 1922, S. Banach [15] provided the concept of Contraction theorem in the context of metric space. After, Nadler [28] introduced the concept of set-valued mapping in the module of Hausdroff metric space which is one of the potential generalizations of a Contraction theorem. Let (X,d) is a complete metric space and a mapping T:XCB(X) satisfying

    H(T(x),T(y))γd(x,y)

    for all x,yX, where 0γ<1, H is a Hausdorff with respect to metric d and CB(X)={SX:S is closed and bounded subset of X equipped with a metric d}. Then T has a fixed point in X.

    In the recent past, Matthews [26] initiate the concept of partial metric spaces which is the classical extension of a metric space. After that, many researchers generalized some related results in the frame of partial metric spaces. Recently, Asadi et al. [4] introduced the notion of an M-metric space which is the one of interesting generalizations of a partial metric space. Later on, Samet et al. [33] introduced the class of mappings which known as (α,ψ)-contractive mapping. The notion of (α,ψ) -contractive mapping has been generalized in metric spaces (see more [10,12,14,17,19,25,29,30,32]).

    Throughout this manuscript, we denote the set of all positive integers by N and the set of real numbers by R. Let us recall some basic concept of an M-metric space as follows:

    Definition 1.1. [4] Let m:X×XR+be a mapping on nonempty set X is said to be an M-metric if for any x,y,z in X, the following conditions hold:

    (i) m(x,x)=m(y,y)=m(x,y) if and only if x=y;

    (ii) mxym(x,y);

    (iii) m(x,y)=m(y,x);

    (iv) m(x,y)mxy(m(x,z)mxz)+(m(z,y)mz,y) for all x,y,zX. Then a pair (X,m) is called M-metric space. Where

    mxy=min{m(x,x),m(y,y)}

    and

    Mxy=max{m(x,x),m(y,y)}.

    Remark 1.2. [4] For any x,y,z in M-metric space X, we have

    (i) 0Mxy+mxy=m(x,x)+m(y,y);

    (ii) 0Mxymxy=|m(x,x)m(y,y)|;

    (iii) Mxymxy(Mxzmxz)+(Mzymzy).

    Example 1.3. [4] Let (X,m) be an M-metric space. Define mw, ms:X×XR+ by:

    (i)

    mw(x,y)=m(x,y)2mx,y+Mx,y,

    (ii)

    ms={m(x,y)mx,y, if xy0, if x=y.

    Then mw and ms are ordinary metrics. Note that, every metric is a partial metric and every partial metric is an M-metric. However, the converse does not hold in general. Clearly every M-metric on X generates a T0 topology τm on X whose base is the family of open M -balls

    {Bm(x,ϵ):xX, ϵ>0},

    where

    Bm(x,ϵ)={yX:m(x,y)<mxy+ϵ}

    for all xX, ε>0. (see more [3,4,23]).

    Definition 1.4. [4] Let (X,m) be an M-metric space. Then,

    (i) A sequence {xn} in (X,m) is said to be converges to a point x in X with respect to τm if and only if

    limn(m(xn,x)mxnx)=0.

    (ii) Furthermore, {xn} is said to be an M-Cauchy sequence in (X,m) if and only if

    limn,m(m(xn,xm)mxnxm), and limn,m(Mxn,xmmxnxm)

    exist (and are finite).

    (iii) An M-metric space (X,m) is said to be complete if every M-Cauchy sequence {xn} in (X,m) converges with respect to τm to a point xX such that

    limnm(xn,x)mxnx=0, and limn(Mxn,xmxnx)=0.

    Lemma 1.5. [4] Let (X,m) be an M-metric space. Then:

    (i) {xn} is an M-Cauchy sequence in (X,m) if and only if {xn} is a Cauchy sequence in a metric space (X,mw).

    (ii) An M-metric space (X,m) is complete if and only if the metric space (X,mw) is complete. Moreover,

    limnmw(xn,x)=0 if and only if (limn(m(xn,x)mxnx)=0, limn(Mxnxmxnx)=0).

    Lemma 1.6. [4] Suppose that {xn} convergesto x and {yn} converges to y as n approaches to in M-metric space (X,m). Then we have

    limn(m(xn,yn)mxnyn)=m(x,y)mxy.

    Lemma 1.7. [4] Suppose that {xn} converges to xas n approaches to in M-metric space (X,m).Then we have

    limn(m(xn,y)mxny)=m(x,y)mxy for all yX.

    Lemma 1.8. [4] Suppose that {xn} converges to xand {xn} converges to y as n approaches to in M-metric space (X,m). Then m(x,y)=mxymoreover if m(x,x)= m(y,y), then x=y.

    Definition 1.9. Let α:X×X[0,). A mapping T:XX is said to be an α-admissible mapping if for all x,yX

    α(x,y)1α(T(x),T(y))1.

    Let Ψ be the family of the (c)-comparison functions ψ:R+{0}R+{0} which satisfy the following properties:

    (i) ψ is nondecreasing,

    (ii) n=0ψn(t)< for all t>0, where ψn is the n-iterate of ψ (see [7,8,10,11]).

    Definition 1.10. [33] Let (X,d) be a metric space and α:X×X[0,). A mapping T:XX is called (α,ψ)-contractive mapping if for all x,yX, we have

    α(x,y)d(T(x),T(x))ψ(d(x,y)),

    where ψΨ.

    A subset K of an M-metric space X is called bounded if for all xK, there exist yX and r>0 such that xBm(y,r). Let ¯K denote the closure of K. The set K is closed in X if and only if ¯K=K.

    Definition 1.11. [31] Define Hm:CBm(X)×CBm(X)[0,) by

    Hm(K,L)=max{m(K,L),m(L,K)},

    where

    m(x,L)=inf{m(x,y):yL} andm(L,K)=sup{m(x,L):xK}

    Lemma 1.12. [31] Let F be any nonempty set in M-metric space (X,m), then

    x¯F if and only if m(x,F)=supaF{mxa}.

    Proposition 1.13. [31] Let A,B,CCBm(X), then

    (i) m(A,A)=supxA{supyAmxy},

    (ii) (m(A,B)supxAsupyBmxy)(m(A,C)infxAinfzCmxz)+

    (m(C,B)infzCinfyBmzy).

    Proposition 1.14. [31] Let A,B,CCBm(X) followingare hold

    (i) Hm(A,A)=m(A,A)=supxA{supyAmxy},

    (ii) Hm(A,B)=Hm(B,A),

    (iii) Hm(A,B)supxAsupyAmxy)Hm(A,C)+Hm(B,C)infxAinfzCmxzinfzCinfyBmzy.

    Lemma 1.15. [31] Let A,BCBm(X) and h>1.Then for each xA, there exist at the least one yB such that

    m(x,y)hHm(A,B).

    Lemma 1.16. [31] Let A,BCBm(X) and l>0.Then for each xA, there exist at least one yB such that

    m(x,y)Hm(A,B)+l.

    Theorem 1.17. [31] Let (X,m) be a complete M-metric space and T:XCBm(X). Assume that there exist h(0,1) such that

    Hm(T(x),T(y))hm(x,y), (1.1)

    for all x,yX. Then T has a fixed point.

    Proposition 1.18. [31] Let T:XCBm(X) be a set-valued mapping satisfying (1.1) for all x,y inan M-metric space X. If zT(z) for some z in Xsuch that m(x,x)=0 for xT(z).

    We start with the following definition:

    Definition 2.1. Assume that Ψ is a family of non-decreasing functions ϕM:R+R+ such that

    (i) +nϕnM(x)< for every x>0 where ϕnM is a nth-iterate of ϕM,

    (ii) ϕM(x+y)ϕM(x)+ϕM(y) for all x,yR+,

    (iii) ϕM(x)<x, for each x>0.

    Remark 2.2. If αn|n= =0 is a convergent series with positive terms then there exists a monotonic sequence (βn)|n= such that βn|n== and αnβn|n==0 converges.

    Definition 2.3. Let (X,m) be an M-metric pace. A self mapping T:XX is called (α,ϕM)-contraction if there exist two functions α:X×X[0,) and ϕMΨ such that

    α(x,y)m(T(x),T(y))ϕM(m(x,y)),

    for all x,yX.

    Definition 2.4. Let (X,m) be an M-metric space. A set-valued mapping T:XCBm(X) is said to be (α,ϕM)-contraction if for all x,yX, we have

    α(x,y)Hm(T(x),T(x))ϕM(m(x,y)), (2.1)

    where ϕMΨ and α:X×X[0,).

    A mapping T is called α-admissible if

    α(x,y)1α(a1,b1)1

    for each a1T(x) and b1T(y).

    Theorem 2.5. Let (X,m) be a complete M-metric space.Suppose that (α,ϕM) contraction and α-admissible mapping T:XCBm(X)satisfies the following conditions:

    (i) there exist x0X such that α(x0,a1)1 for each a1T(x0),

    (ii) if {xn}X is a sequence such that α(xn,xn+1)1 for all n and {xn}xX as n, then α(xn,x)1 for all nN. Then T has a fixed point.

    Proof. Let x1T(x0) then by the hypothesis (i) α(x0,x1)1. From Lemma 1.16, there exist x2T(x1) such that

    m(x1,x2)Hm(T(x0),T(x1))+ϕM(m(x0,x1)).

    Similarly, there exist x3T(x2) such that

    m(x2,x3)Hm(T(x1),T(x2))+ϕ2M(m(x0,x1)).

    Following the similar arguments, we obtain a sequence {xn}X such that there exist xn+1T(xn) satisfying the following inequality

    m(xn,xn+1)Hm(T(xn1),T(xn))+ϕnM(m(x0,x1)).

    Since T is α-admissible, therefore α(x0,x1)1α(x1,x2)1. Using mathematical induction, we get

    α(xn,xn+1)1. (2.2)

    By (2.1) and (2.2), we have

    m(xn,xn+1)Hm(T(xn1),T(xn))+ϕnM(m(x0,x1))α(xn,xn+1)Hm(T(xn1),T(xn))+ϕnM(m(x0,x1))ϕM(m(xn1,xn))+ϕnM(m(x0,x1))=ϕM[(m(xn1,xn))+ϕn1M(m(x0,x1))]ϕM[Hm(T(xn2),T(xn1))+ϕn1M(m(x0,x1))]ϕM[α(xn1,xn)Hm(T(xn1),T(xn))+ϕn1M(m(x0,x1))]ϕM[ϕM(m(xn2,xn1))+ϕn1M(m(x0,x1))+ϕn1M(m(x0,x1))]ϕ2M(m(xn2,xn1))+2ϕnM(m(x0,x1))....
    m(xn,xn+1)ϕnM(m(x0,x1))+nϕnM(m(x0,x1))m(xn,xn+1)(n+1)ϕnM(m(x0,x1)).

    Let us assume that ϵ>0, then there exist n0N such that

    nn0(n+1)ϕnM(m(x0,x1))<ϵ.

    By the Remarks (1.2) and (2.2), we get

    limnm(xn,xn+1)=0.

    Using the above inequality and (m2), we deduce that

    limnm(xn,xn)=limnmin{m(xn,xn),m(xn+1,xn+1)}=limnmxnxn+1limnm(xn,xn+1)=0.

    Owing to limit, we have limnm(xn,xn)=0,

    limn,mmxnxm=0.

    Now, we prove that {xn} is M-Cauchy in X. For m,n in N with m>n and using the triangle inequality of an M-metric we get

    m(xn,xm)mxnxmm(xn,xn+1)mxnxn+1+m(xn+1,xm)mxn+1xmm(xn,xn+1)mxnxn+1+m(xn+1,xn+2)mxn+1xn+1+m(xn+2,xm)mxn+2xmm(xn,xn+1)mxnxn+1+m(xn+1,xn+2)mxn+1xn+2++m(xm1,xm)mxm1xmm(xn,xn+1)+m(xn+1,xn+2)++m(xm1,xm)=m1r=nm(xr,xr+1)m1r=n(r+1)ϕrM(m(x0,x1))m1rn0(r+1)ϕrM(m(x0,x1))m1rn0(r+1)ϕrM(m(x0,x1))<ϵ.

    m(xn,xm)mxnxm0, as n, we obtain limm,n(Mxnxmmxnxm)=0. Thus {xn} is a M-Cauchy sequence in X. Since (X,m) is M-complete, there exist xX such that

    limn(m(xn,x)mxnx)=0 andlimn(Mxnxmxnx)=0.

    Also, limnm(xn,xn)=0 gives that

    limnm(xn,x)=0 and limnMxnx=0, (2.3)
    limn{max(m(xn,x),m(x,x))}=0,

    which implies that m(x,x)=0 and hence we obtain mxT(x)=0. By using (2.1) and (2.3) with

    limnα(xn,x)1.

    Thus,

    limnHm(T(xn),T(x))limnϕM(m(xn,x))limnm(xn,x).
    limnHm(T(xn),T(x))=0. (2.4)

    Now from (2.3), (2.4), and xn+1T(xn), we have

    m(xn+1,T(x))Hm(T(xn),T(x))=0.

    Taking limit as n and using (2.4), we obtain that

    limnm(xn+1,T(x))=0. (2.5)

    Since mxn+1T(x)m(xn+1,T(x)) which gives

    limnmxn+1T(x)=0. (2.6)

    Using the condition (m4), we obtain

    m(x,T(x))supyT(x)mxym(x,T(x))mx,T(x)m(x,xn+1)mxxn+1+m(xn+1,T((x))mxn+1T(x).

    Applying limit as n and using (2.3) and (2.6), we have

    m(x,T(x))supyT(x)mxy. (2.7)

    From (m2), mxym(xy) for each yT(x) which implies that

    mxym(x,y)0.

    Hence,

    sup{mxym(x,y):yT(x)}0.

    Then

    supyT(x)mxyinfyT(x)m(x,y)0.

    Thus

    supyT(x)mxym(x,T(x)). (2.8)

    Now, from (2.7) and (2.8), we obtain

    m(T(x),x)=supyT(x)mxy.

    Consequently, owing to Lemma (1.12), we have x¯T(x)=T(x).

    Corollary 2.6. Let (X,m) be a complete M-metric space and anself mapping T:XX an α-admissible and (α,ϕM)-contraction mapping. Assume that thefollowing properties hold:

    (i) there exists x0X such that α(x0,T(x0))1,

    (ii) either T is continuous or for any sequence {xn}X with α(xn,xn+1)1 for all nN and {xn}x as n , we have α(xn,x)1 for all nN. Then T has a fixed point.

    Some fixed point results in ordered M-metric space.

    Definition 2.7. Let (X,) be a partially ordered set. A sequence {xn}X is said to be non-decreasing if xnxn+1 for all n.

    Definition 2.8. [16] Let F and G be two nonempty subsets of partially ordered set (X,). The relation between F and G is defined as follows: F1G if for every xF, there exists yG such that xy.

    Definition 2.9. Let (X,m,) be a partially ordered set on M-metric. A set-valued mapping T:XCBm(X) is said to be ordered (α,ϕM)-contraction if for all x,yX, with xy we have

    Hm(T(x),T(y))ϕM(m(x,y))

    where ϕMΨ. Suppose that α:X×X[0,) is defined by

    α(x,y)={1     if Tx1Ty0       otherwise.

    A mapping T is called α-admissible if

    α(x,y)1α(a1,b1)1,

    for each a1T(x) and b1T(y).

    Theorem 2.10. Let (X,m,) be a partially orderedcomplete M-metric space and T:XCBm(X) an α-admissible ordered (α,ϕM)-contraction mapping satisfying the following conditions:

    (i) there exist x0X such that {x0}1{T(x0)}, α(x0,a1)1 for each a1T(x0),

    (ii) for every x,yX, xy implies T(x)1T(y),

    (iii) If {xn}X is a non-decreasing sequence such that xnxn+1 for all n and {xn}xX as n gives xnx for all nN. Then T has a fixed point.

    Proof. By assumption (i) there exist x1T(x0) such that x0x1 and α(x0,x1)1. By hypothesis (ii), T(x0)1T(x1). Let us assume that there exist x2T(x1) such that x1x2 and we have the following

    m(x1,x2)Hm(T(x0),T(x1))+ϕM(m(x0,x1)).

    In the same way, there exist x3T(x2) such that x2x3 and

    m(x2,x3)Hm(T(x1),T(x2))+ϕ2M(m(x0,x1)).

    Following the similar arguments, we have a sequence {xn}X  and xn+1T(xn) for all n0 satisfying x0x1x2x3...xnxn+1. The proof is complete follows the arguments given in Theorem 2.5.

    Example 2.11. Let X=[16,1] be endowed with an M -metric given by m(x,y)=x+y2. Define T:XCBm(X) by

    T(x)={{12x+16,14}, if x=16{x2,x3},  if 14x13{23,56},  if 12x1.

    Define a mapping α:X×X[0,) by

    α(x,y)={1     if x,y[14,13]0       otherwise.

    Let ϕM:R+R+ be given by ϕM(t)=1710 where ϕMΨ, for x,yX. If x=16, y=14 then m(x,y)=524, and

    Hm(T(x),T(y))=Hm({312,14},{18,112})=max(m({312,14},{18,112}),m({18,112},{312,14}))=max{316,212}=316ϕM(t)m(x,y).

    If x=13, y=12 then m(x,y)=512, and

    Hm(T(x),T(y))=Hm({16,19},{23,1})=max(m({16,19},{23,1}),m({23,1},{16,19}))=max{1736,718}=1736ϕM(t)m(x,y).

    If x=16, y=1, then m(x,y)=712 and

    Hm(T(x),T(y))=Hm({312,14},{23,56})=max(m({312,14},{23,56}),m({23,56},{312,14}))=max{1124,1324}=1324ϕM(t)m(x,y).

    In all cases, T is (α,ϕM)-contraction mapping. If x0=13, then T(x0)={x2,x3}.Therefore α(x0,a1)1 for every a1T(x0). Let x,yX be such that α(x,y)1, then x,y[x2,x3] and T(x)={x2,x3} and T(y)= {x2,x3} which implies that α(a1,b1)1 for every a1T(x) and b1T(x). Hence T is α-admissble.

    Let {xn}X be a sequence such that α(xn,xn+1)1 for all n in N and xn converges to x as n converges to , then xn[x2,x3]. By definition of α -admissblity, therefore x[x2,x3] and hence α(xn,x)1. Thus all the conditions of Theorem 2.3 are satisfied. Moreover, T has a fixed point.

    Example 2.12. Let X={(0,0),(0,15),(18,0)} be the subset of R2 with order defined as: For (x1,y1),(x2,y2)X, (x1,y1)(x2,y2) if and only if x1x2, y1y2. Let m:X×XR+ be defined by

    m((x1,y1),(x2,y2))=|x1+x22|+|y1+y22|, for x=(x1,y1), y=(x2,y2)X.

    Then (X,m) is a complete M-metric space. Let T:XCBm(X) be defined by

    T(x)={{(0,0)}, if x=(0,0),{(0,0),(18,0)},  if x(0,15){(0,0)},  if x(18,0).

    Define a mapping α:X×X[0,) by

    α(x,y)={1     if x,yX0       otherwise.

    Let ϕM:R+R+ be given by ϕM(t)=12. Obviously, ϕMΨ. For x,yX,

    if x=(0,15) and y=(0,0), then Hm(T(x),T(y))=0 and m(x,y)=110 gives that

    Hm(T(x),T(y))=Hm({(0,0),(18,0)},{(0,0)})=max(m({(0,0),(18,0)},{(0,0)}),m({(0,0)},{(0,0),(18,0)}))=max{0,0}=0ϕM(t)m(x,y).

    If x=(18,0) and y=(0,0) then Hm(T(x),T(y))=0, and m(x,y)=116 implies that

    Hm(T(x),T(y))ϕM(t)m(x,y).

    If x=(0,0) and y=(0,0) then Hm(T(x),T(y))=0, and m(x,y)=0 gives

    Hm(T(x),T(y))ϕM(t)m(x,y).

    If x=(0,15) and y=(0,15) then Hm(T(x),T(y))=0, and m(x,y)=15 implies that

    Hm(T(x),T(y))ϕM(t)m(x,y).

    If x=(0,18) and y=(0,18) then Hm(T(x),T(y))=0, and m(x,y)=18 gives that

    Hm(T(x),T(y))ϕM(t)m(x,y).

    Thus all the condition of Theorem 2.10 satisfied. Moreover, (0,0) is the fixed point of T.

    In this section, we present an application of our result in homotopy theory. We use the fixed point theorem proved for set-valued (α,ϕM)-contraction mapping in the previous section, to establish the result in homotopy theory. For further study in this direction, we refer to [6,35].

    Theorem 3.1. Suppose that (X,m) is a complete M-metricspace and A and B are closed and open subsets of X respectively, suchthat AB. For a,bR, let T:B×[a,b]CBm(X) be aset-valued mapping satisfying the following conditions:

    (i) xT(y,t) for each yB/Aand t[a,b],

    (ii) there exist ϕMΨ and α:X×X[0,) such that

    α(x,y)Hm(T(x,t),T(y,t))ϕM(m(x,y)),

    for each pair (x,y)B×B and t[a,b],

    (iii) there exist a continuous function Ω:[a,b]R such that for each s,t[a,b] and xB, we get

    Hm(T(x,s),T(y,t))ϕM|Ω(s)Ω(t)|,

    (iv) if xT(x,t),then T(x,t)={x},

    (v) there exist x0 in X such that x0T(x0,t),

    (vi) a function :[0,)[0,) defined by (x)=xϕM(x) is strictly increasing and continuous if T(.,t has a fixed point in B for some t^{\intercal }\in \left[a, b\right], then T\left(., t\right) has afixed point in A for all t\in \left[a, b\right]. Moreover, for a fixed t\in \left[a, b\right] , fixed point is unique provided that \phi_{M}\left(t\right) = \frac{1}{2}t where t > 0.

    Proof. Define a mapping \alpha _{\ast }:X\times X\rightarrow \left[0, \infty \right) by

    \alpha _{\ast }\left( x, y\right) = \left \{ \begin{array}{l} 1 \ \ \ \ \text{ if }x\in T\left( x, t\right) , \ y\in T\left( y, t\right) \\ \ \\ 0 \ \ \ \ \ \ \ \text{otherwise.} \end{array} \right.

    We show that T is \alpha _{\ast } -admissible. Note that \alpha _{\ast }\left(x, y\right) \geq 1 implies that x\in T\left(x, t\right) and y\in T\left(y, t\right) for all t\in \left[a, b\right] . By hypothesis \left(iv\right), T\left(x, t\right) = \left \{ x\right \} and T\left(y, t\right) = \left \{ y\right \}. It follows that T is \alpha _{\ast } -admissible. By hypothesis \left(v\right), there exist x_{0}\in X such that x_{0}\in \left(x_{0}, t\right) for all t , that is \alpha _{\ast }\left(x_{0}, x_{0}\right) \geq 1 . Suppose that \alpha _{\ast }\left(x_{n}, x_{n+1}\right) \geq 1 for all n and x_{n} converges to q as n approaches to \infty and x_{n}\in T\left(x_{n}, t\right) and x_{n+1}\in T\left(x_{n+1}, t\right) for all n and t\in \left[a, b\right] which implies that q\in T\left(q, t\right) and thus \alpha _{\ast }\left(x_{n}, q\right) \geq 1. Set

    D = \left \{ t\in \left[ a, b\right] : \ x\in T\left( x, t\right) \text{ for }x\in A\right \} .

    So T\left(., t^{\intercal }\right) has a fixed point in B for some t^{\intercal }\in \left[a, b\right] , there exist x\in B such that x\in T\left(x, t\right). By hypothesis \left(i\right) x\in T\left(x, t\right) for t\in \left[a, b\right] and x\in A so D\neq \phi . Now we now prove that D is open and close in \left[a, b\right]. Let t_{0}\in D and x_{0}\in A with x_{0}\in T\left(x_{0}, t_{0}\right). Since A is open subset of X , \overline{B_{m}\left(x_{0}, r\right) } \subseteq A for some r > 0 . For \epsilon = r+m_{xx_{0}}-\phi \left(r+m_{xx_{0}}\right) and a continuous function \Omega on \left[a, b \right] , there exist \delta > 0 such that

    \phi _{M}\left \vert \Omega \left( t\right) -\Omega \left( t_{0}\right) \right \vert < \epsilon \text{ for all }t\in \left( t_{0}-\delta , t_{0}+\delta \right) .

    If t\in \left(t_{0}-\delta, t_{0}+\delta \right) for x\in B_{m}\left(x_{0}, r\right) = \left \{ x\in X:m\left(x_{0}\, , x\right) \leq m_{x_{0}x}+r\right \} and l\in T\left(x, t\right), we obtain

    \begin{eqnarray*} m\left( l, x_{0}\right) & = &m\left( T\left( x, t\right) , x_{0}\right) \\ & = &H_{m}\left( T\left( x, t\right) , T\left( x_{0}, t_{0}\right) \right) . \end{eqnarray*}

    Using the condition \left(iii\right) of Proposition 1.13 and Proposition 1.18, we have

    \begin{equation} m\left( l, x_{0}\right) \leq H_{m}\left( T\left( x, t\right) , T\left( x_{0}, t_{0}\right) \right) +H_{m}\left( T\left( x, t\right) , T\left( x_{0}, t_{0}\right) \right) \end{equation} (2.9)

    as x\in T\left(x_{0}, t_{0}\right) and x\in B_{m}\left(x_{0}, r\right) \subseteq A\subseteq B , t_{0}\in \left[a, b\right] with \alpha _{\ast }\left(x_{0}, x_{0}\right) \geq 1. By hypothesis \left(ii\right) , \left(iii\right) and \left(2.9\right)

    \begin{eqnarray*} m\left( l, x_{0}\right) &\leq &\phi _{M}\left \vert \Omega \left( t\right) -\Omega \left( t_{0}\right) \right \vert +\alpha _{\ast }\left( x_{0}, x_{0}\right) H_{m}\left( T\left( x, t\right) , T\left( x_{0}, t_{0}\right) \right) \\ &\leq &\phi _{M}\left \vert \Omega \left( t\right) -\Omega \left( t_{0}\right) \right \vert +\phi _{M}\left( m\left( x, x_{0}\right) \right) \\ &\leq &\phi _{M}\left( \epsilon \right) +\phi _{M}\left( m_{xx_{0}}+r\right) \\ &\leq &\phi _{M}\left( r+m_{xx_{0}}-\phi _{M}\left( r+m_{xx_{0}}\right) \right) +\phi _{M}\left( m_{xx_{0}}+r\right) \\ & < &r+m_{xx_{0}}-\phi _{M}\left( r+m_{xx_{0}}\right) +\phi _{M}\left( m_{xx_{0}}+r\right) = r+m_{xx_{0}}. \end{eqnarray*}

    Hence l\in \overline{B_{m}\left(x_{0}, r\right) } and thus for each fixed t\in \left(t_{0}-\delta, t_{0}+\delta \right), we obtain T\left(x, t\right) \subset \overline{B_{m}\left(x_{0}, r\right) } therefore T: \overline{B_{m}\left(x_{0}, r\right) }\rightarrow CB_{m}\left(\overline{ B_{m}\left(x_{0}, r\right) }\right) satisfies all the assumption of Theorem \left(3.1\right) and T\left(., t\right) has a fixed point \overline{B_{m}\left(x_{0}, r\right) } = B_{m}\left(x_{0}, r\right) \subset B . But by assumption of \left(i\right) this fixed point belongs to A . So \left(t_{0}-\delta, t_{0}+\delta \right) \subseteq D, thus D is open in \left[a, b\right]. Next we prove that D is closed. Let a sequence \left \{ t_{n}\right \} \in D with t_{n} converges to t_{0}\in \left[a, b \right] as n approaches to \infty. We will prove that t_{0} is in D .

    Using the definition of D, there exist \left \{ t_{n}\right \} in A such that x_{n}\in T\left(x_{n}, t_{n}\right) for all n . Using Assumption \left(iii\right) \left(v\right), and the condition \left(iii\right) of Proposition 1.13, and an outcome of the Proposition 1.18, we have

    \begin{eqnarray*} m\left( x_{n}, x_{m}\right) &\leq &H_{m}\left( T\left( x_{n}, t_{n}\right) , T\left( x_{m}, t_{m}\right) \right) \\ &\leq &H_{m}\left( T\left( x_{n}, t_{n}\right) , T\left( x_{n}, t_{m}\right) \right) +H_{m}\left( T\left( x_{n}, t_{m}\right) , T\left( x_{m}, t_{m}\right) \right) \\ &\leq &\phi _{M}\left \vert \Omega \left( t_{n}\right) -\Omega \left( t_{m}\right) \right \vert +\alpha _{\ast }\left( x_{n}, x_{m}\right) H_{m}\left( T\left( x_{n}, t_{m}\right) , T\left( x_{m}, t_{m}\right) \right) \\ &\leq &\phi _{M}\left \vert \Omega \left( t_{n}\right) -\Omega \left( t_{m}\right) \right \vert +\phi _{M}\left( m\left( x_{n}, x_{m}\right) \right) \\ &\Rightarrow & \\ m\left( x_{n}, x_{m}\right) -\phi _{M}\left( m\left( x_{n}, x_{m}\right) \right) &\leq &\phi _{M}\left \vert \Omega \left( t_{n}\right) -\Omega \left( t_{m}\right) \right \vert \\ &\Rightarrow & \\ \Re \left( m\left( x_{n}, x_{m}\right) \right) &\leq &\phi _{M}\left \vert \Omega \left( t_{n}\right) -\Omega \left( t_{m}\right) \right \vert \\ \Re \left( m\left( x_{n}, x_{m}\right) \right) \, & < &\left \vert \Omega \left( t_{n}\right) -\Omega \left( t_{m}\right) \right \vert \\ m\left( x_{n}, x_{m}\right) & < &\frac{1}{\Re }\left \vert \Omega \left( t_{n}\right) -\Omega \left( t_{m}\right) \right \vert . \end{eqnarray*}

    So, continuity of \frac{1}{\Re }, \Re and convergence of \left \{ t_{n}\right \}, taking the limit as m, n\rightarrow \infty in the last inequality, we obtain that

    \lim\limits_{m, n\rightarrow \infty }m\left( x_{n}, x_{m}\right) = 0.

    Sine m_{x_{n}x_{m}}\leq m\left(x_{n}, x_{m}\right), therefore

    \lim\limits_{m, n\rightarrow \infty }m_{x_{n}x_{m}} = 0.

    Thus, we have \lim_{n\rightarrow \infty }m\left(x_{n}, x_{n}\right) = 0 = \lim_{m\rightarrow \infty }m\left(x_{m}, x_{m}\right) . Also,

    \lim\limits_{m, n\rightarrow \infty }\left( m\left( x_{n}, x_{m}\right) -m_{x_{n}x_{m}}\right) = 0, \ \lim\limits_{m, n\rightarrow \infty }\left( M_{x_{n}x_{m}}-m_{x_{n}x_{m}}\right) .

    Hence \left \{ x_{n}\right \} is an M -Cauchy sequence. Using Definition 1.4, there exist x^{\ast } in X such that

    \lim\limits_{n\rightarrow \infty }\left( m\left( x_{n}, x^{\ast }\right) -m_{x_{n}x^{\ast }}\right) = 0\text{ and }\lim\limits_{n\rightarrow \infty }\left( M_{x_{n}x^{\ast }}-m_{x_{n}x^{\ast }}\right) = 0.

    As \lim_{n\rightarrow \infty }m\left(x_{n}, x_{n}\right) = 0 , therefore

    \lim\limits_{n\rightarrow \infty }m\left( x_{n}, x^{\ast }\right) = 0\text{ and } \lim\limits_{n\rightarrow \infty }M_{x_{n}x^{\ast }} = 0.

    Thus, we have m\left(x, x^{\ast }\right) = 0. We now show that x^{\ast }\in T\left(x^{\ast }, t^{^{\ast }}\right). Note that

    \begin{eqnarray*} m\left( x_{n}, T\left( x^{\ast }, t^{^{\ast }}\right) \right) &\leq &H_{m}\left( T\left( x_{n}, t_{n}\right) , T\left( x^{\ast }, t^{^{\ast }}\right) \right) \\ &\leq &H_{m}\left( T\left( x_{n}, t_{n}\right) , T\left( x_{n}, t^{^{\ast }}\right) \right) +H_{m}\left( T\left( x_{n}, t^{^{\ast }}\right) , T\left( x^{\ast }, t^{^{\ast }}\right) \right) \\ &\leq &\phi _{M}\left \vert \Omega \left( t_{n}\right) -\Omega \left( t^{^{\ast }}\right) \right \vert +\alpha _{\ast }\left( x_{n}, t^{^{\ast }}\right) H_{m}\left( T\left( x_{n}, t^{^{\ast }}\right) , T\left( x^{\ast }, t^{^{\ast }}\right) \right) \\ &\leq &\phi _{M}\left \vert \Omega \left( t_{n}\right) -\Omega \left( t^{^{\ast }}\right) \right \vert +\phi _{M}\left( m\left( x_{n}, t^{^{\ast }}\right) \right) . \end{eqnarray*}

    Applying the limit n\rightarrow \infty in the above inequality, we have

    \lim\limits_{n\rightarrow \infty }m\left( x_{n}, T\left( x^{\ast }, t^{^{\ast }}\right) \right) = 0.

    Hence

    \begin{equation} \lim\limits_{n\rightarrow \infty }m\left( x_{n}, T\left( x^{\ast }, t^{^{\ast }}\right) \right) = 0. \end{equation} (2.10)

    Since m\left(x^{\ast }, x^{\ast }\right) = 0, we obtain

    \begin{equation} \sup\limits_{y\in T\left( x^{\ast }, t^{^{\ast }}\right) }m_{x^{\ast }y} = \sup\limits_{y\in T\left( x^{\ast }, t^{^{\ast }}\right) }\min \left \{ m\left( x^{\ast }, x^{\ast }\right) , m\left( y, y\right) \right \} = 0. \end{equation} (2.11)

    From above two inequalities, we get

    m\left( x^{\ast }, T\left( x^{\ast }, t^{^{\ast }}\right) \right) = \sup\limits_{y\in T\left( x^{\ast }, t^{^{\ast }}\right) }m_{x^{\ast }y}.

    Thus using Lemma 1.12 we get x^{\ast }\in T\left(x^{\ast }, t^{^{\ast }}\right). Hence x^{\ast }\in A. Thus x^{\ast }\in D and D is closed in \left[a, b\right], D = \left[a, b\right] and D is open and close in \left[a, b\right]. Thus T\left(., t\right) has a fixed point in A for all t\in \left[a, b\right]. For uniqueness, t\in \left[a, b\right] is arbitrary fixed point, then there exist x\in A such that x\in T\left(x, t\right) . Assume that y is an other point of T\left(x, t\right) , then by applying condition 4, we obtain

    \begin{eqnarray*} m\left( x, y\right) & = &H_{m}\left( T\left( x, t\right) , T\left( y, t\right) \right) \\ &\leq &\alpha _{M}\left( x, y\right) H_{m}\left( T\left( x, t\right) , T\left( y, t\right) \right) \leq \phi _{M}\left( m\left( x, y\right) \right) . \end{eqnarray*}

    For \phi _{M}\left(t\right) = \frac{1}{2}t, where t > 0, the uniqueness follows.

    In this section we will apply the previous theoretical results to show the existence of solution for some integral equation. For related results (see [13,20]). We see for non-negative solution of \left(3.1\right) in X = C\left(\left[0, \delta \right], \mathbb{R} \right). Let X = C\left(\left[0, \delta \right], \mathbb{R} \right) be a set of continuous real valued functions defined on \left[0, \delta \right] which is endowed with a complete M -metric given by

    m\left( x, y\right) = \sup\limits_{t\in \left[ 0, \delta \right] }\left( \left \vert \frac{x\left( t\right) +x\left( t\right) }{2}\right \vert \right) \text{ for all }x, y\in X.

    Consider an integral equation

    \begin{equation} v_{1}\left( t\right) = \rho \left( t\right) +\int_{0}^{\delta }h\left( t, s\right) J\left( s, v_{1}\left( s\right) \right) ds\text{ for all }0\leq t\leq \delta . \end{equation} (3.1)

    Define g:X\rightarrow X by

    g\left( x\right) \left( t\right) = \rho \left( t\right) +\int_{0}^{\delta }h\left( t, s\right) J\left( s, x\left( s\right) \right) ds

    where

    (i) for \delta > 0 , \ \left(a\right) J:\left[0, \delta \right] \times \mathbb{R} \rightarrow \mathbb{R}, \left(b\right) h:\left[0, \delta \right] \times \left[0, \delta \right] \rightarrow \left[0, \infty \right), \left(c\right) \rho : \left[0, \delta \right] \rightarrow \mathbb{R} are all continuous functions

    (ii) Assume that \sigma :X\times X\rightarrow \mathbb{R} is a function with the following properties,

    (iii) \sigma \left(x, y\right) \geq 0 implies that \sigma \left(T\left(x\right), T\left(y\right) \right) \geq 0,

    (iv) there exist x_{0}\in X such that \sigma \left(x_{0}, T\left(x_{0}\right) \right) \geq 0,

    (v) if \left \{ x_{n}\right \} \in X is a sequence such that \sigma \left(x_{n}, x_{n+1}\right) \geq 0 for all n\in \mathbb{N} and x_{n}\rightarrow x as n\rightarrow \infty, then \sigma \left(x, T\left(x\right) \right) \geq 0

    (vi)

    \sup\limits_{t\in \left[ 0, \delta \right] }\int_{0}^{\delta }h\left( t, s\right) ds\leq 1

    where t\in \left[0, \delta \right] , s\in \mathbb{R},

    \left(vii\right) there exist \phi _{M}\in \Psi , \sigma \left(y, T\left(y\right) \right) \geq 1 and \sigma \left(x, T\left(x\right) \right) \geq 1 such that for each t\in \left[0, \delta \right], we have

    \begin{equation} |J\left( s, x\left( t\right) \right) +J\left( s, y\left( t\right) \right) |\leq \phi _{M}\left( \left \vert x+y\right \vert \right) . \end{equation} (3.3)

    Theorem 4.1. Under the assumptions \left(i\right) -\left(vii\right) theintegral Eq \left(3.1\right) has a solution in \left \{ X = C\left(\left[0, \delta \right], \mathbb{R} \right) \mathit{\text{ for all }}t\in \left[0, \delta \right] \right \}.

    Proof. Using the condition \left(vii\right) , we obtain that

    \begin{eqnarray*} m\left( g\left( x\right) , g\left( y\right) \right) & = &\left \vert \frac{ g\left( x\right) \left( t\right) +g\left( y\right) \left( t\right) }{2} \right \vert = \left \vert \int_{0}^{\delta }h\left( t, s\right) \left[ \frac{ J\left( s, x\left( s\right) \right) +J\left( s, y\left( s\right) \right) }{2} \right] ds\right \vert \\ &\leq &\int_{0}^{\delta }h\left( t, s\right) \left \vert \frac{J\left( s, x\left( s\right) \right) +J\left( s, y\left( s\right) \right) }{2} \right \vert ds \\ &\leq &\int_{0}^{\delta }h\left( t, s\right) \left[ \phi _{M}\left \vert \frac{ x\left( s\right) +y\left( s\right) }{2}\right \vert \right] ds \\ &\leq &\left( \sup\limits_{t\in \left[ 0, \delta \right] }\int_{0}^{\delta }h\left( t, s\right) ds\right) \left( \phi _{M}\left \vert \frac{x\left( s\right) +y\left( s\right) }{2}\right \vert \right) \\ &\leq &\phi _{M}\left( \left \vert \frac{x\left( s\right) +y\left( s\right) }{ 2}\right \vert \right) \end{eqnarray*}
    m\left( g\left( x\right) , g\left( y\right) \right) \leq \phi \left( m\left( x, y\right) \right)

    Define \alpha _{\ast }:X\times X\rightarrow \left[0, +\infty \right) by

    \alpha _{\ast }\left( x, y\right) = \left \{ \begin{array}{l} 1 \ \ \ \ \text{ if }\sigma \left( x, y\right) \geq 0 \\ \ \\ 0 \ \ \ \ \ \ \ \text{otherwise} \end{array} \right.

    which implies that

    m\left( g\left( x\right) , g\left( y\right) \right) \leq \phi _{M}\left( m\left( x, y\right) \right) .

    Hence all the assumption of the Corollary 2.6 are satisfied, the mapping g has a fixed point in X = C\left(\left[0, \delta \right], \mathbb{R} \right) which is the solution of integral Eq \left(3.1\right).

    In this study we develop some set-valued fixed point results based on \left(\alpha _{\ast }, \phi _{M}\right) -contraction mappings in the context of M -metric space and ordered M -metric space. Also, we give examples and applications to the existence of solution of functional equations and homotopy theory.

    The authors declare that they have no competing interests.



    [1] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302–1324. https://doi.org/10.1016/j.jsc.2005.05.007 doi: 10.1016/j.jsc.2005.05.007
    [2] K. Pearson, Essentially positive tensors, Int. J. Algebra., 4 (2010), 421–427.
    [3] D. Liu, W. Li, S. Vong, The tensor splitting with application to solve multi-linear systems, J. Comput. Appl. Math., 330 (2018), 75–94. https://doi.org/10.1016/j.cam.2017.08.009 doi: 10.1016/j.cam.2017.08.009
    [4] W. Li, D. Liu, S. Vong, Comparison results for splitting iterations for solving multi-linear system, Appl. Numer. Math., 134 (2018), 105–121. https://doi.org/10.1016/j.apnum.2018.07.009 doi: 10.1016/j.apnum.2018.07.009
    [5] L. Cui, M. Li, Y. Song, Preconditioned tensor splitting iterations method for solving multi-linear systems, Appl. Math. Lett., 96 (2019), 89–94. https://doi.org/10.1016/j.aml.2019.04.019 doi: 10.1016/j.aml.2019.04.019
    [6] W. Ding, L. Qi, Y. Wei, \mathcal{M}-tensors and nonsingular \mathcal{M}-tensors, Linear Algebra Appl., 439 (2013), 3264–3278. https://doi.org/10.1016/j.laa.2013.08.038 doi: 10.1016/j.laa.2013.08.038
    [7] Q. Liu, J. Huang, S. Zeng, Convergence analysis of the two preconditioned iterative methods for \mathcal{M}-matrix linear systems, J. Comput. Appl. Math., 281 (2015), 49–57. https://doi.org/10.1016/j.cam.2014.11.034 doi: 10.1016/j.cam.2014.11.034
    [8] K. C. Chang, K. Pearson, T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507–520. https://dx.doi.org/10.4310/CMS.2008.v6.n2.a12 doi: 10.4310/CMS.2008.v6.n2.a12
    [9] W. Ding, Y. Wei, Solving multi-linear system with \mathcal{M}-tensors, J. Sci. Comput., 68 (2016), 689–715. https://doi.org/10.1007/s10915-015-0156-7 doi: 10.1007/s10915-015-0156-7
    [10] L. Zhang, L. Qi, G. Zhou, \mathcal{M}-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437–452. https://doi.org/10.1137/130915339 doi: 10.1137/130915339
    [11] Z. Luo, L. Qi, N. Xiu, The sparsest solutions to Z-tensor complementarity problems, Optim. Lett., 11 (2017), 471–482. https://doi.org/10.1007/s11590-016-1013-9 doi: 10.1007/s11590-016-1013-9
    [12] M. Ng, L. Qi, G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090–1099. https://doi.org/10.1137/09074838X doi: 10.1137/09074838X
    [13] Y. Matsuno, Exact solutions for the nonlinear Klein-Gordon and Liouville equations in four-dimensional Euclidean space, J. Math. Phys., 28 (1987), 2317–2322. https://doi.org/10.1063/1.527764 doi: 10.1063/1.527764
    [14] D. Zwillinger, Handbook of differential equations, 3 Eds., Boston: Academic Press Inc, 1997.
    [15] D. Kressner, C. Tobler, Krylov subspace methods for linear systems with tensor product structure, SIAM J. Matrix Anal. Appl., 31 (2010), 1688–1714. https://doi.org/10.1137/090756843 doi: 10.1137/090756843
    [16] C. Tobler, Low-rank Tensor methods for linear systems and eigenvalue problems, Ph.D. thesis, 2012.
  • This article has been cited by:

    1. Amjad Ali, Muhammad Arshad, Eskandar Ameer, Asim Asiri, Certain new iteration of hybrid operators with contractive M -dynamic relations, 2023, 8, 2473-6988, 20576, 10.3934/math.20231049
    2. Muhammad Tariq, Muhammad Arshad, Mujahid Abbas, Eskandar Ameer, Saber Mansour, Hassen Aydi, A relation theoretic m-metric fixed point algorithm and related applications, 2023, 8, 2473-6988, 19504, 10.3934/math.2023995
    3. Imo Kalu Agwu, Naeem Saleem, Umar Isthiaq, A new modified mixed-type Ishikawa iteration scheme with error for common fixed points of enriched strictly pseudocontractive self mappings and ΦΓ-enriched Lipschitzian self mappings in uniformly convex Banach spaces, 2025, 26, 1989-4147, 1, 10.4995/agt.2025.17595
    4. Muhammad Tariq, Sabeur Mansour, Mujahid Abbas, Abdullah Assiry, A Solution to the Non-Cooperative Equilibrium Problem for Two and Three Players Using the Fixed-Point Technique, 2025, 17, 2073-8994, 544, 10.3390/sym17040544
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1952) PDF downloads(82) Cited by(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog