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Research article

The recurrence formula for the number of solutions of a equation in finite field

  • Received: 29 August 2020 Accepted: 02 December 2020 Published: 07 December 2020
  • MSC : 11D45, 11D72

  • The main purpose of this paper is using analytic methods to give a recurrence formula of the number of solutions of an equation over finite field. We use analytic methods to give a recurrence formula for the number of solutions of the above equation. And our method is based on the properties of the Gauss sum. It is worth noting that we used a novel method to simplify the steps and avoid complicated calculations.

    Citation: Yanbo Song. The recurrence formula for the number of solutions of a equation in finite field[J]. AIMS Mathematics, 2021, 6(2): 1954-1964. doi: 10.3934/math.2021119

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  • The main purpose of this paper is using analytic methods to give a recurrence formula of the number of solutions of an equation over finite field. We use analytic methods to give a recurrence formula for the number of solutions of the above equation. And our method is based on the properties of the Gauss sum. It is worth noting that we used a novel method to simplify the steps and avoid complicated calculations.


    In 1965, Zadeh [1], introduced the concept of a fuzzy set which is defined as: "a set constructed from a function having a domain is a nonempty set W and range in [0,1] is called a fuzzy set, that is, if G:W[0,1], then the set constructing from the mapping G is called a fuzzy set". Later on, the theory of fuzzy sets has been extensively developed and investigated in many directions with different types of applications. Kramosil and Michalek [2], introduced the notion of fuzzy metric spaces (FM spaces) by using the concept of fuzzy set and some more derived concepts from the one in order. They compared the FM concepts with the statistical metric space and proved that both the spaces are equivalent in some cases. After that, the modified form of the FM space was given by George and Veeramani [3] and proved that every metric induces an FM. They proved some basic properties and Baire's theorem for FM spaces.

    In 1988, Grabiec [4] proved two fixed point theorems of "Banach and Edelstein contraction mapping theorems on complete and compact FM spaces, respectively" by using the concept of Kramosil and Michalek [2]. In [5], Kiany et al. proved some fixed point results on FM spaces for set-valued contractive type mappings. Aubin and Siegel [6], Fakhar [7], Gregori and Sapena [8], Harandi [9], Hussain et al. [10], Mizoguchi and Takahashi [11], Rehman et al. [12], and Wlodarczyk et al. [13] proved some set-valued and multi-valued contractive type mapping results in different spaces.

    Bari and Vetro [14] proved fixed point theorems for a family of mappings on FM spaces. While Beg et al. [] established some invariant approximation results for fuzzy non-expansive mappings defined on FM spaces. As an application, they obtained a fixed point result on the best approximation in a fuzzy normed space. Further, they defined the strictly convex fuzzy normed space and obtained a necessary condition for the set of all t-best approximations which contained a fixed point of the arbitrary mappings. While Beg et al. [16] established some fixed point theorems on complete FM spaces for self-mappings satisfying an implicit relation. Bari and Vetro [17], Imdad and Ali [18], Hierro et al. [19], Jleli et al. [20], Li et al. [21], Pant and Chauhan [22], Lopez and Romaguera [23], Rehman et al. [24], Roldan et al. [25,26], Sadeghi et al. [27], Shamas et al. [28,29] and Som [30] proved some fixed point and common fixed point results on FM spaces by using different contractive type mappings with applications.

    In this paper, we present some unique common fixed point theorems for a pair of self-mappings on FM spaces without continuity by using "the triangular property of fuzzy metric". We use the concept of Li et al. [21] and Rehman et al. [31] and establish different contractive types of common fixed point theorems on FM spaces with illustrative examples. Further, we present weak contraction and a generalized Ćirić-contraction theorems on FM space. In addition, we present an application of fuzzy differential equations to support our work. This paper is organized as: Section 2 presents the preliminary concepts. Section 3 deals with different contractive types of unique common fixed point theorems on complete FM spaces with examples. While in Section 4, we define a generalized Ćirić-contraction and will prove a unique common fixed point theorem on complete FM spaces. Section 5, is the most important section of this paper which deals with the application of fuzzy differential equations (FDEs) to increase the validity of our work. Finally, in section 6 we discussed the conclusion.

    Definition 2.1 ([32]). An operation :[0,1]×[0,1][0,1] is known as a continuous t-norm if it holds the following;

    (1) is associative, commutative and continuous.

    (2) 1ρ1=ρ1 and ρ1ρ2ρ3ρ4, whenever ρ1ρ3 and ρ2ρ4, for each ρ1,ρ2,ρ3,ρ4[0,1].

    The basic continuous t-norms are (see [32]): The minimum, the product and the Lukasiewicz t-norms are defined respectively as following;

    ρ1ρ2=min{ρ1,ρ2},ρ1ρ2=ρ1ρ2andρ1ρ2=max{ρ1+ρ21,0}.

    Definition 2.2 ([3]). A 3-tuple (W,MF,) is said to be an FM space, if W is a nonempty arbitrary set, is a continuous t-norm and MF is a fuzzy set on W×W×(0,) satisfying the following;

    (i) MF(w,x,t)>0 and MF(w,x,t)=1 w=x.

    (ii) MF(w,x,t)=MF(x,w,t).

    (iii) MF(w,y,t)MF(y,x,s)MF(w,x,t+s).

    (iv) MF(w,x,):(0,)[0,1] is continuous.

    for all w,x,yW and t,s>0.

    Definition 2.3 ([3,8]). Let (W,MF,) be an FM space, wW and {wm} be a sequence in W. Then,

    (i) {wm} is said to be convergent to a point wW if limmMF(wm,w,t)=1 for t>0.

    (ii) {wm} is said to be a Cauchy sequence, if for each 0<ϵ<1 and t>0, there is m0N such that MF(wk,wm,t)>1ϵ,  k,mm0.

    (iii) (W,MF,) is said to be an FM space if every Cauchy sequence is convergent in W.

    (iv) {wm} is called a fuzzy contractive, if  β(0,1) so that

    1MF(wm,wm+1,t)1β(1MF(wm1,wm,t)1)for t>0, and m1.

    Definition 2.4. [17] Let (W,MF,) be an FM space. Then fuzzy metric MF is triangular if,

    1MF(w,x,t)1(1MF(w,y,t)1)+(1MF(y,x,t)1)w,xW, and t>0.

    Note: A fuzzy metric FM is triangular, if MF:W×W×(0,)[0,1] is defined by

    MF(w,x,t)=tt+|wx|w,xW, and t>0.

    Lemma 2.5 ([17]). Let (W,MF,) be an FM space. Let wW and {wm} be a sequence in W. Then wmw iff limiMF(wm,w,t)=1, for t>0.

    Definition 2.6 ([8]). Let (W,MF,) be an FM space and G:WW. Then F is called a fuzzy contraction, if  h(0,1) so that

    1MF(F1w,F1x,t)1h(1MF(w,x,t)1)w,xW, and t>0. (2.1)

    Now we present our first main result.

    Theorem 3.1. Let (W,MF,) be a complete FM space in which MF is triangular and a pair of self-mappings F1,F2:WW satisfies,

    1MF(F1w,F2x,t)1a(1MF(w,x,t)1)+b(1MF(w,F1w,t)1+1MF(x,F2x,t)1+1MF(x,F1w,t)1+1MF(w,F2x,t)1)+c(1min{MF(w,F1w,t),MF(x,F2x,t)}1)+d(1max{MF(w,F2x,t),MF(x,F1w,t)}1), (3.1)

     w,xW, t>0, and a,b,c,d[0,1). Then F1 and F2 have a common fixed point in W if (a+4b+c)<1. Moreover, if (a+2b+d)<1, then F1 and F2 have a unique common fixed point in W.

    Proof. Fix w0W and define a sequence {wm} in W such that

    w2m+1=F1w2mandw2m+2=F2w2m+1forall m0.

    Now by a view of (3.19), we have

    1MF(w2m+1,w2m+2,t)1=1MF(F1w2m,F2w2m+1,t)1a(1MF(w2m,w2m+1,t)1)+b(1MF(w2m,F1w2m,t)1+1MF(w2m+1,F2w2m+1,t)11MF(w2m+1,F1w2m,t)1+1MF(w2m,F2w2m+1,t)1)+c(1min{MF(w2m,F1w2m,t),MF(w2m+1,F2w2m+1,t)}1)+d(1max{MF(w2m,F2w2m+1,t),MF(w2m+1,F1w2m,t)}1)a(1MF(w2m,w2m+1,t)1)+b(1MF(w2m,w2m+1,t)1+1MF(w2m+1,w2m+2,t)1+1MF(w2m,w2m+2,t)1)+c(1min{MF(w2m,w2m+1,t),MF(w2m+1,w2m+2,t)}1)+d(1max{MF(w2m,w2m+2,t),1}1)

    Then, for t>0, we have

    1MF(w2m+1,w2m+2,t)1a(1MF(w2m,w2m+1,t)1)+b(1MF(w2m,w2m+1,t)1+1MF(w2m+1,w2m+2,t)1+1MF(w2m,w2m+2,t)1)+c(1min{MF(w2m,w2m+1,t),MF(w2m+1,w2m+2,t)}1) (3.2)

    Now two possibilities arise,

    (i) If MF(w2m,w2m+1,t) is a minimum term in {MF(w2m,w2m+1,t),MF(w2m+1,w2m+2,t)}, then after simplification, (3.2) can be written as;

    1MF(w2m+1,w2m+2,t)1a+2b+c12b(1MF(w2m,w2m+1,t)1)for t>0. (3.3)

    (ii) If MF(w2m+1,w2m+2,t) is a minimum term in {MF(w2m,w2m+1,t),MF(w2m+1,w2m+2,t)}, then after simplification, (3.2) can be written as;

    1MF(w2m+1,w2m+2,t)1a+2b12bc(1MF(w2m,w2m+1,t)1)for t>0. (3.4)

    Let us define β:=max{a+2b+c12b,a+2b12bc}<1, then from the above two cases, we get that

    1MF(w2m+1,w2m+2,t)1β(1MF(w2m,w2m+1,t)1)for t>0. (3.5)

    Similarly,

    1MF(w2m+2,w2m+3,t)1=1Mj(F2w2m+1,F1w2m+2,t)1a(1MF(w2m+2,w2m+1,t)1)+b(1MF(w2m+2,F1w2m+2,t)1+1MF(w2m+1,F2w2m+1,t)1+1MF(w2m+1,F1w2m+2,t)1+1MF(w2m+2,F2w2m+1,t)1)+c(1min{MF(w2m+2,F1w2m+2,t),MF(w2m+1,F2w2m+1,t)}1)+d(1max{MF(w2m+1,F1w2m+2,t),MF(w2m+2,F2w2m+1,t)}1)=a(1MF(w2m+1,w2m+2,t)1)+b(1MF(w2m+1,w2m+2,t)1+1MF(w2m+2,w2m+3,t)1+1MF(w2m+1,w2m+3,t)1)+c(1min{MF(w2m+1,w2m+2,t),MF(w2m+2,w2m+3,t)}1)+d(1max{MF(w2m+1,w2m+3,1}1)

    Then, for t>0, we have

    1MF(w2m+2,w2m+3,t)1a(1MF(w2m+1,w2m+2,t)1)+b(1MF(w2m+1,w2m+2,t)1+1MF(w2m+2,w2m+3,t)1+1MF(w2m+1,w2m+3,t)1)+c(1min{MF(w2m+1,w2m+2,t),MF(w2m+2,w2m+3,t)}1) (3.6)

    Now again there are two possibilities that arises,

    (i) If MF(w2m+1,w2m+2,t) is a minimum term in {MF(w2m+1,w2m+2,t),MF(w2m+2,w2m+3,t)}, then after simplification, (3.6) can be written as;

    1MF(w2m+2,w2m+3,t)1a+2b+c12b(1MF(w2j+1,w2m+2,t)1)for t>0. (3.7)

    (ii) If MF(w2m+2,w2m+3,t) is a minimum term in {MF(2j+1,w2m+2,t),MF(w2m+2,w2m+3,t)}, then after simplification, (3.6) can be written as;

    1MF(w2m+2,w2m+3,t)1a+2b12bc(1MF(w2m+1,w2m+2,t)1)for t>0. (3.8)

    Then

    1MF(w2m+2,w2m+3,t)1β(1MF(w2m+1,w2m+2,t)1)for t>0, (3.9)

    where β is similar as in (3.5). Then, from (3.5) and (3.9), and by induction, for t>0, we have

    1MF(w2m+2,w2m+3,t)1β(1MF(w2m+1,w2m+2,t)1)β2(1MF(w2m,w2m+1,t)1)β2m+2(1MF(w0,w1,t)1)0,as j.

    Hence, proved that {wm}m0 is a fuzzy contractive sequence in W, that is,

    limmMF(wm,wm+1,t)=1for t>0. (3.10)

    Since MF is triangular, for k>m and t>0, then we have

    1MF(wm,wk,t)1(1MF(wm,wm+1,t)1)+(1MF(wm+1,wm+2,t)1)++(1MF(wk1,wk,t)1)(βm+βm+1++βk1)(1MF(w0,w1,t)1)(βm1β)(1MF(w0,w1,t)1)0,as m,

    which shows that {wm} is a Cauchy sequence. Since, by the completeness of (W,MF,),  κW such that

    limmMF(κ,wm,t)=1for t>0. (3.11)

    Now we have to show that F1κ=κ, since MF is triangular, therefore

    1MF(κ,F1κ,t)1(1MF(κ,w2m+2,t)1)+(1MF(w2m+2,F1κ,t)1)for t>0. (3.12)

    Now by using (3.19), (3.11) and (3.10), for t>0, we have

    1MF(w2m+2,F1κ,t)1=1MF(F2w2m+1,F1κ,t)1a(1MF(w2m+1,κ,t)1)+b(1MF(w2m+1,F2w2m+1,t)1+1MF(κ,F1κ,t)11MF(κ,F2w2m+1,t)1+1MF(w2m+1,F1κ,t)1)+c(1min{MF(w2m+1,F2w2m+1,t),MF(κ,F1κ,t)}1)+d(1max{MF(w2m+1,F1κ,t),MF(κ,F2w2z+1,t)}1)=a(1MF(w2m+1,κ,t)1)+b(1MF(w2m+1,w2m+2,t)1+1MF(κ,F1κ,t)11MF(κ,w2m+2,t)1+1MF(w2m+1,F1κ,t)1)+c(1min{MF(w2m+1,w2m+2,t),MF(κ,F1κ,t)}1)+d(1max{MF(w2m+1,F1κ,t),MF(κ,w2m+2,t)}1) (2b+c)(1MF(κ,F1κ,t)1)as j.

    Then,

    limmsup(1MF(w2m+2,F1κ,t)1)(2b+c)(1MF(κ,F1κ,t)1)for t>0. (3.13)

    The above (3.13) is together with (3.11) and (3.12), we get that

    1MF(κ,F1κ,t)1(2b+c)(1MF(κ,F1κ,t)1)for t>0.

    As (2b+c)<1, where (a+4b+c)<1, therefore MF(κ,F1κ,t)=1, this implies that F1κ=κ.

    Similarly, again by triangular property of MF,

    1MF(κ,F2κ,t)1(1MF(κ,w2m+1,t)1)+(1MF(w2m+1,F2κ,t)1)for t>0. (3.14)

    Again by using (3.19), (3.10) and (3.11), similar as above, after simplification, we get

    limmsup(1MF(w2m+1,F2κ,t)1)(2b+c)(1MF(κ,F2κ,t)1)for t>0. (3.15)

    The above (3.15) is together with (3.11) and (3.14), we get that

    1MF(κ,F2κ,t)1(2b+c)(1MF(κ,F2κ,t)1)for t>0.

    As (2b+c)<1, where (a+4b+c)<1, therefore MF(κ,F2κ,t)=1, this implies that F2κ=κ. Hence proved that κ is a common fixed point of F1 and F2.

    Uniqueness: let κW be the other common fixed point of F1 and F2 such that F1κ=F2κ=κ, then again by the view of (3.19), for t>0, we have

    1MF(κ,κ,t)1=(1MF(F1κ,F2κ,t)1)a(1MF(κ,κ,t)1)+b(1MF(κ,F1κ,t)1+1MF(κ,F2κ,t)1+1MF(κ,F2κ,t)1+1Mm(κ,F1κ,t)1)+c(1min{MF(κ,F1κ,t),MF(κ,F2κ,t)}1)+d(1min{MF(κ,F2κ,t),MF(κ,F1κ,t)}1)=(a+2b+d)(1MF(κ,κ,t)1)=(a+2b+d)(1MF(F1κ,F2κ,t)1)(a+2b+d)2(1MF(κ,κ,t)1)(a+2b+d)m(1MF(κ,κ,t)1)0,as m,

    where (a+4b+d)<1. Hence we get that MF(κ,κ,t)=1, this implies that κ=κ. Thus, F1 and F2 have a unique common fixed point in W.

    If we put d=0 in Theorem 3.1, we get the following corollary;

    Corollary 3.2. Let (W,MF,) be a complete FM space in which MF is triangular and a pair of self-mappings F1,F2:WW satisfies,

    1MF(F1w,F2x,t)1a(1MF(w,x,t)1)+b(1MF(w,F1w,t)1+1MF(x,F2x,t)1+1MF(x,F1w,t)1+1MF(w,F2x,t)1)+c(1min{MF(w,F1w,t),MF(x,F2x,t)}1) (3.16)

     w,xW, t>0, and a,b,c[0,1) with (a+4b+c)<1. Then F1 and F2 have a unique common fixed point in W.

    If we put c=0 in Theorem 3.1, we get the following corollary;

    Corollary 3.3. Let (W,MF,) be a complete FM space in which MF is triangular and a pair of self-mappings F1,F2:WW satisfies,

    1MF(F1w,F2x,t)1a(1MF(w,x,t)1)+b(1MF(w,F1w,t)1+1MF(x,F2x,t)1+1MF(x,F1w,t)1+1MF(w,F2x,t)1)+d(1max{MF(w,F2x,t),MF(x,F1w,t)}1), (3.17)

     w,xW, t>0, and a,b,d[0,1) with (a+4b+d)<1. Then F1 and F2 have a unique common fixed point in W.

    If we put c=d=0 in Theorem 3.1, we get the following corollary;

    Corollary 3.4. Let (W,MF,) be a complete FM space in which MF is triangular and a pair of self-mappings F1,F2:WW satisfies,

    1MF(F1w,F2x,t)1a(1MF(w,x,t)1)+c(1min{MF(w,F1w,t),MF(x,F2x,t)}1)+d(1max{MF(w,F2x,t),MF(x,F1w,t)}1) (3.18)

     w,xW, t>0, and a,c,d[0,1). Then F1 and F2 have a common fixed point if (a+c)<1. Moreover, if (a+d)<1, then F1 and F2 have a unique common fixed point in W.

    If the mapping F1=F2 in Theorem 3.1, we get the following corollary;

    Corollary 3.5. Let (W,MF,) be a complete FM space in which MF is triangular and a self-mapping F1:WW satisfies,

    1MF(F1w,F1x,t)1a(1MF(w,x,t)1)+b(1MF(w,F1w,t)1+1MF(x,F1x,t)1+1MF(x,F1w,t)1+1MF(w,F1x,t)1)+c(1min{MF(w,F1w,t),MF(x,F1x,t)}1)+d(1max{MF(w,F1x,t),MF(x,F1w,t)}1) (3.19)

     w,xW, t>0, and a,b,c,d[0,1). Then F1 has a fixed point if (a+4b+c)<1. Moreover, if (a+4b+d)<1, then F1 has a unique fixed point in W.

    Example 3.6. Let W=[0,) and t-norm is a product continuous t-norm. Let MF:W×W×(0,)[0,1] be defined as

    MF(w,x,t)=tt+d(w,x),where d(w,x)=2|wx|3,

     w,xW and t>0. Then (W,MF,) is complete. The mappings F1,F2:WW be defined as

    F1w={2w5+110,  if w[0,1],3w4+3,  if w(1,).

    And,

    F2x={2x5+110,  if x[0,1],2x7+607,  if x(1,).

    Then we have

    1MF(F1w,F2x,t)1=2|F1wF2x|3t=4|wx|15t=25(1MF(w,x,t)1)25(1MF(w,x,t)1)+120(1MF(w,F1w,t)1+1MF(x,F2x,t)1+1MF(x,F1w,t)1+1MF(w,F2x,t)1)+27(1min{MF(w,F1w,t),MF(x,F2x,t)}1)+27(1max{MF(w,F2x,t),MF(x,F1w,t)}1).

    Hence all the conditions of Theorem 3.1 are satisfied with a=25,b=120, and c=d=27, where (a+4b+c)=3135<1 and (a+2b+d)=5570<1, the self mappings F1 and F2 have a unique common fixed point, that is, F1(12)=F2(12)=12[1,).

    In the following theorem, we use a function ψ:[0,)[0,) such that ψ(0)=0, and ψ(ξ)<ξ, for ξ>0, and prove a unique common fixed point result in FM spaces.

    Theorem 3.7. Let F1,F2:WW be a pair of self-mappings on a complete FM space (W,MF,) in which MF is triangular. Suppose that there exists a non-decreasing function ψ:[0,)[0,) with ψ(0)=0, ψ(ξ)<ξ, for ξ>0 and m=0ψm(ξ)<, ξ0 such that the following inequality holds;

    1MF(F1w,F2x,t)1ψ(1N(F1,F2,w,x,t)1)+min{(1MF(w,F1w,t)1),(1MF(x,F2x,t)1),(1MF(x,F1w,t)1),(1MF(w,F2x,t)1)}, (3.20)

    where

    1N(F1,F2,w,x,t)1=max{(1MF(w,x,t)1),(1MF(w,F1w,t)1),(1MF(x,F2x,t)1),14(1MF(x,F1w,t)1+1MF(w,F2x,t)1)},

    for all w,xW, (0,1), then F1 and F2 have a unique common fixed point in W.

    Proof. Fix w0W and define a sequence {wm} in W such that

    w2m+1=F1w2mandw2m+2=F2w2m+1forall m0.

    Now by the view of (3.20), for t>0,

    1MF(w2m+1,w2m+2,t)1=1MF(F1w2m,F2w2m+1,t)1ψ(1N(F1,F2,w2m,w2m+1,t)1)+min{(1MF(w2m,F1w2m,t)1),(1MF(w2m+1,F2w2m+1,t)1),(1MF(w2m+1,F1w2m,t)1),(1MF(w2m,F2w2m+1,t)1)}=ψ(1N(F1,F2,w2m,w2m+1,t)1)+min{(1MF(w2m,w2m+1,t)1),(1MF(w2m+1,w2m+2,t)1),(1MF(w2m+1,w2m+1,t)1),(1MF(w2m,w2m+2,t)1)},

    where

    1N(F1,F2,w2m,w2m+1,t)1=max{(1MF(w2m,w2m+1,t)1),(1MF(w2m,F1w2m,t)1),(1MF(w2m+1,F2w2m+1,t)1),14(1MF(w2m+1,F1w2m,t)1+1MF(w2m,F2w2m+1,t)1)}=max{(1MF(w2m,w2m+1,t)1),(1MF(w2m,w2m+1,t)1),(1MF(w2m+1,w2m+2,t)1),14(1MF(w2m+1,w2m+1,t)1+1MF(w2m,w2m+2,t)1)}=max{(1MF(w2m,w2m+1,t)1),(1MF(w2m+1,w2m+2,t)1)},

    which is further implies that

    1MF(w2m+1,w2m+2,t)1ψ(max{(1MF(w2m,w2m+1,t)1),(1MF(w2m+1,w2m+2,t)1)}).

    Now if,

    1MF(w2m+1,w2m+2,t)1>(1MF(w2m,w2m+1,t)1)for t>0,

    then for t>0, we have

    1MF(w2m+1,w2m+2,t)1ψ(1MF(w2m+1,w2m+2,t)1)<(1MF(w2m+1,w2m+2,t)1),

    which is a contradiction. Hence,

    1MF(w2m+1,w2m+2,t)1ψ(1MF(w2m,w2m+1,t)1)for t>0.

    Similarly, it can be shown that

    1MF(w2m+2,w2m+3,t)1ψ(1MF(w2m+1,w2m+2,t)1)for t>0.

    Thus, by induction for all m0 and t>0, we have that

    1MF(wm,wm+1,t)1ψ(1MF(wm1,wm,t)1)ψ2(1MF(wm2,wm1,t)1)ψm(1MF(w0,w1,t)1).

    Hence, for k>m and t>0,

    1MF(wm,wk,t)1(1MF(wm,wm+1,t)1)+(1MF(wm+1,wm+2,t)1)++(1MF(wk1,wk,t)1)ψm(1MF(w0,w1,t)1)+ψm+1(1MF(w0,w1,t)1)++ψk1(1MF(w0,w1,t)1)k1n=mψn(1MF(w0,w1,t)1).

    Since, m=0ψm(1MF(w0,w1,t)1)<, hence {wm} is a Cauchy sequence and from the completeness of (W,MF,), it follows that wmκW, as m. This can be written as

    limmMF(wm,κ,t)=1for t>0. (3.21)

    Now we have to show that F1κ=κ, since MF is triangular, therefore

    1MF(κ,F1κ,t)1(1MF(κ,w2m+2,t)1)+(1MF(w2m+2,F1κ,t)1)for t>0. (3.22)

    Now from (3.20), for t>0, we have

    1MF(w2m+2,F1κ,t)1=1MF(F1κ,F2w2m+1,t)1ψ(1N(F1,F2,κ,w2m+1,t)1)+min{(1MF(κ,F1κ,t)1),(1MF(w2m+1,F2w2m+1,t)1),(1MF(w2m+1,F1κ,t)1),(1MF(κ,F2w2m+1,t)1)}.

    Now we substitute the value of (1N(F1,F2,κ,w2m+1,t)1) in the above inequality and then from (3.21), for t>0, we have that

    1MF(w2m+2,F1κ,t)1ψ(max{(1MF(w2m+1,κ,t)1),(1MF(κ,F1κ,t)1),(1MF(w2m+1,F2w2m+1,t)1),14(1MF(w2m+1,F1κ,t)1+1MF(κ,F2w2m+1,t)1)})+min{(1MF(κ,F1κ,t)1),(1MF(w2m+1,F2w2m+1,t)1),(1MF(w2m+1,F1κ,t)1),(1MF(κ,F2w2m+1,t)1)}=ψ(max{(1MF(w2m+1,κ,t)1),(1MF(κ,F1κ,t)1),(1MF(w2m+1,w2m+2,t)1),14(1MF(w2m+1,F1κ,t)1+1MF(κ,w2m+2,t)1)})+min{(1MF(κ,F1κ,t)1),(1MF(w2m+1,w2m+2,t)1),(1MF(w2m+1,F1κ,t)1),(1MF(κ,w2m+2,t)1)} ψ(1MF(κ,F1κ,t)1).

    Then,

    limjsup(1MF(w2m+2,F1κ,t)1)ψ(1MF(κ,F1κ,t)1)for t>0. (3.23)

    The above (3.23) is together with (3.21) and (3.22), for t>0, we have that

    1MF(κ,F1κ,t)1ψ(1MF(κ,F1κ,t)1)<(1MF(κ,F1κ,t)1),

    which is a contradiction. Hence, MF(κ,F1κ,t)=1  F1κ=κ for t>0. Similarly, by the triangular property of MF,

    1MF(κ,F2κ,t)1(1MF(κ,w2m+1,t)1)+(1MF(w2m+1,F2κ,t)1)for t>0. (3.24)

    Again by using (3.20) and (3.21), similar as above, after simplification, we get that

    limmsup(1MF(w2m+1,F2κ,t)1)ψ(1MF(κ,F2κ,t)1)for t>0. (3.25)

    The above (3.25) is together with (3.21) and (3.24), we have that

    1MF(κ,F2κ,t)1ψ(1MF(κ,F2κ,t)1)<(1MF(κ,F2κ,t)1),

    which is a contradiction. Hence, MF(κ,F2κ,t)=1  F2κ=κ for t>0. Hence proved that κ is a common fixed point of F1 and F2.

    Uniqueness: let κW be the other common fixed point of F1 and F2 such that F1κ=F2κ=κ, then again by the view of (3.20), for t>0, we have

    1MF(κ,κ,t)1=1MF(F1κ,F2κ,t)1ψ(1N(F1,F2,κ,κ,t)1)+min{(1MF(κ,F1κ,t)1),(1MF(κ,F2κ,t)1),(1MF(κ,F1κ,t)1),(1MF(κ,F2κ,t)1)}=ψ(max{(1MF(κ,κ,t)1),(1MF(κ,F1κ,t)1),(1MF(κ,F2κ,t)1),14(1MF(κ,F1κ,t)1+1MF(κ,F2κ,t)1)})+min{(1MF(κ,F1κ,t)1),(1MF(κ,F2κ,t)1),(1MF(κ,F1κ,t)1),(1MF(κ,F2κ,t)1)}=ψ(max{(1MF(κ,κ,t)1),(1MF(κ,κ,t)1),(1MF(κ,κ,t)1),14(1MF(κ,κ,t)1+1MF(κ,κ,t)1)})+min{(1MF(κ,κ,t)1),(1MF(κ,κ,t)1),(1MF(κ,κ,t)1),(1MF(κ,κ,t)1)}=ψ(1MF(κ,κ,t)1),

    Hence, we get that

    1MF(κ,κ,t)1ψ(1MF(κ,κ,t)1)<1MF(κ,κ,t)1,for t>0,

    which is a contradiction. Hence, MF(κ,κ,t)=1  κ=κ for t>0.

    If we define a mapping ψ by ψ(ξ)=λξ in Theorem 3.7, where λ(0,1), we get the following corollary;

    Corollary 3.8. Let (W,MF,) be a complete FM space in which MF is triangular and a pair of self mappings F1,F2:WW satisfies,

    1MF(F1w,F2x,t)1λ(1N(F1,F2,w,x,t)1)+min{(1MF(w,F1w,t)1),(1MF(x,F2x,t)1),(1MF(x,F1w,t)1),(1MF(w,F2x,t)1)}, (3.26)

    where

    1N(F1,F2,w,x,t)1=max{(1MF(w,x,t)1),(1MF(w,F1w,t)1),(1MF(x,F2x,t)1),14(1MF(x,F1w,t)1+1MF(w,F2x,t)1)},

     w,xW, λ(0,1) and 0. Then F1 and F2 have a unique common fixed point in W.

    If we put =0 in Corollary 3.8, we get the following corollary;

    Corollary 3.9. Let (W,MF,) be a complete FM space in which MF is triangular and a pair of self mappings F1,F2:WW satisfies,

    1MF(F1w,F2x,t)1λmax{(1MF(w,x,t)1),(1MF(w,F1w,t)1),(1MF(x,F2x,t)1),14(1MF(x,F1w,t)1+1MF(w,F2x,t)1)} (3.27)

     w,xW, λ(0,1). Then F1 and F2 have a unique common fixed point in W.

    Definition 3.10. A self-mapping F1 will be called weakly contractive on a complete FM space (W,MF,), i.e., F1:WW, if there exists a continuous and non-decreasing function φ:[0,)[0,) such that φ(τ)=0 if and only if τ=0, limτφ(τ)= and satisfying

    1MF(w,x,t)1(1MF(w,x,t)1)φ(1MF(w,x,t)1), w,xW and t>0. (3.28)

    Theorem 3.11. Let a pair of weakly self-contractive on a complete FM space (W,MF,), that is, F1,F2:WW in which a fuzzy metric MF is triangular and satisfies,

    1MF(F1w,F2x,t)1(1MF(w,x,t)1)φ(1MF(w,x,t)1), w,xW and t>0, (3.29)

    where φ:[0,)[0,) is a continuous and monotone non-decreasing function with φ(τ)=0 if and only if τ=0 and limτφ(τ)=. Then F1 and F2 have a unique common fixed point in W.

    Proof. Fix w0W and define a sequence {wm} in W such that

    w2m+1=F1w2mandw2m+2=F2w2m+1forall m0.

    Now by view of (3.29), for t>0,

    1MF(w2m+1,w2m+2,t)1=1MF(F1w2m,F2w2m+1,t)1(1MF(w2m,w2m+1,t)1)φ(1MF(w2m,w2m,t)1)<(1MF(w2m,w2m+1,t)1). (3.30)

    Similarly, for t>0,

    1MF(w2m+2,w2m+3,t)1=1MF(F1w2m+2,F2w2m+1,t)1(1MF(w2m+2,w2m+1,t)1)φ(1MF(w2m+2,w2m,t)1)<(1MF(w2m+1,w2m+2,t)1)<(1MF(w2m,w2m+1,t)1),byusing(3.30). (3.31)

    Thus (1MF(w2m+3,w2m+2,t)1) is a monotone decreasing sequence of non-negative real numbers and convergent to some point ϱ as j. Let we denote (1MF(w2m+3,w2m+2,t)1) by ϱ2m+2. Then, we have that ϱ2m+2ϱ as m. Now we have to prove that ϱ=0. If not, then on taking m, we have

    (1MF(wm,wm+1,t)1)(1MF(wm1,wm,t)1)+φ(1MF(wm1,wm,t)1)for t>0,

    which gives that

    ϱϱφ(ϱ)<ϱ,

    a contradiction. Hence, we conclude that (1MF(wm,wm+1,t)1)=ϱm0 as m, this can be written as

    limmMF(wm,wm+1,t)=1for t>0. (3.32)

    Next we have to prove that {wm} is a Cauchy sequence. Let {m(i)} and {k(i)} be the increasing sequences of integers and there exists ε such that for all integers i and p(i),n(i)0,

    m(i)=2p(i)+1>k(i)=2n(i)>i,orm(i)=2p(i)>k(i)=2n(i)+1>i.

    This implies that

    (1MF(wk(i),wm(i),t)1)εfor t>0,(1MF(w2n(i),w2p(i)+1,t)1)ε or (1MF(w2n(i)+1,w2p(i),t)1)ε for t>0, (3.33)

    and

    (1MF(wk(i),wm(i)1,t)1)<εfor t>0,(1MF(w2n(i)1,w2p(i),t)1)<ε or (1MF(w2n(i),w2p(i)1,t)1)<ε for t>0. (3.34)

    By taking limit i on the above (3.34) and by using (3.32), we have that

    limi(1MF(wk(i),wm(i)1,t)1)=εfor t>0,limi(1MF(w2n(i)1,w2p(i),t)1)=ε or limi(1MF(w2n(i),w2p(i)1,t)1)=ε for t>0. (3.35)

    Then, from (3.33), (3.29) and (3.34), for t>0, if

    ε1MF(wk(i),wm(i),t)1=1MF(w2n(i),w2n(i)+1,t)1=1MF(F1w2p(i),F2w2n(i)1,t)1(1MF(w2n(i)1,w2p(i),t)1)φ(1MF(w2n(i)1,w2p(i),t)1).

    Now by applying limit i and from (3.35), we get

    εεφ(ε)<ε, (3.36)

    which is a contradiction. Similarly, again from (3.33), (3.29) and (3.34), for t>0, if

    ε1MF(wk(i),wm(i),t)1=1MF(w2n(i)+1,w2p(i),t)1=1MF(F1w2n(i),F2w2p(i)1,t)1(1MF(w2n(i),w2p(i)1,t)1)φ(1MF(w2n(i),w2p(i)1,t)1).

    By taking limit i and from (3.35), we get

    εεφ(ε)<ε, (3.37)

    which is also a contradiction. Therefore, in both cases, that is (3.36) and (3.37), we got a contradiction. Hence proved that {wm} is a Cauchy sequence in W. Now from the completeness of (W,MF,), it follows that wmκW, as m. This can be written as

    limmMF(wm,κ,t)=1for t>0. (3.38)

    Now we have to show that F1κ=κ. Since MF is triangular and from (3.29) and (3.38) for t>0, we have that

    1MF(κ,F1κ,t)1(1MF(κ,w2m+2,t)1)+(1MF(w2m+2,F1κ,t)1)=(1MF(κ,w2m+2,t)1)+(1MF(F2w2m+1,F1κ,t)1)(1MF(κ,w2m+2,t)1)+(1MF(w2m+1,κ,t)1)φ(1MF(w2m+1,κ,t)1) 0as j.

    Hence, MF(κ,F1κ,t)=1  F1κ=κ for t>0. Similarly, by the triangular property of MF, and again from (3.29) and (3.38) for t>0, we have that

    1MF(κ,F2κ,t)1(1MF(κ,w2m+1,t)1)+(1MF(w2m+1,F2κ,t)1)=(1MF(κ,w2m+1,t)1)+(1MF(F1w2m,F2κ,t)1)(1MF(κ,w2m+1,t)1)+(1MF(w2m,κ,t)1)φ(1MF(w2m,κ,t)1) 0as m.

    Hence, MF(κ,F2κ,t)=1  F2κ=κ for t>0, which shows that κ is a common fixed point of the mappings F1 and F2.

    Uniqueness: let κW be the other common fixed point of F1 and F2 such that F1κ=F2κ=κ, then again by the view of (3.29), for t>0, we have

    1MF(κ,κ,t)1=1MF(F1κ,F2κ,t)1(1MF(κ,κ,t)1)φ(1MF(κ,κ,t)1),

    which by the property of φ is contradiction unless MF(κ,κ,t)=1,  κ=κ. Hence proved that F1 and F2 have a unique common fixed point in W.

    Example 3.12. Let W=[0,) and t-norm is a product continuous t-norm. Let MF:W×W×(0,)[0,1] be defined as

    MF(w,x,t)=tt+|wx|,w,xW, and t>0.

    Then (W,MF,) is complete and MF is triangular. Now we define F1,F2:WW by F1(w)=F2(w)=w2+44,  w[0,1]. Further, a mapping φ:[0,)[0,) be defined as φ(τ)=τ2, for τ>0. Then, from (3.29), for t>0, we have

    1MF(F1w,F2x,t)1=|F1wF2x|t=|w2x2|4t|wx|2t=|wx|t|wx|2t=(1MF(w,x,t)1)φ(1MF(w,x,t)1),

    for all w,xW. Hence the conditions of Theorem 3.11 are satisfied and the mappings F1 and F2 have a unique common fixed point in W, that is, F1(2)=F2(2)=2W.

    In this section, we define a generalized Ćirić type of fuzzy contraction on FM spaces and present a unique common fixed point theorem for a pair of self-mappings on a complete FM space.

    Definition 4.1. Let (W,MF,) be an FM space. A self-mapping F1:WW is said to be a generalized Ćirić type fuzzy-contraction if  α(0,1) such that

    1MF(F1w,F1x,t)1αmax{(1MF(w,x,t)1),(1MF(w,F1w,t)1),(1MF(x,F1x,t)1),(1MF(x,F1w,t)1),(1MF(w,F1x,t)1),12(1MF(w,F1w,t)1+1MF(x,F1x,t)1),12(1MF(x,F1w,t)1+1MF(w,F1x,t)1)} (4.1)

     w,xW and t>0.

    In the following, we present a more generalized Ćirić type fuzzy contraction result for a pair of self-mappings to prove that a pair of self-mappings on a complete FM space have a unique common fixed point.

    Theorem 4.2. Let (W,MF,) be a complete FM space in which MF is triangular and a pair of self-mappings F1,F2:WW satisfies,

    1MF(F1w,F2x,t)1α(1MF(w,x,t)1)+βmax{(1MF(w,x,t)1),(1MF(w,F1w,t)1),(1MF(x,F2x,t)1),(1MF(x,F1w,t)1),(1MF(w,F2x,t)1),12(1MF(w,F1w,t)1+1MF(x,F2x,t)1),12(1MF(x,F1w,t)1+1MF(w,F2x,t)1)} (4.2)

    for all w,xW, t>0, α(0,1) and β0 with (α+2β)<1. Then F1 and F2 have a unique common fixed point in W.

    Proof. Fix w0W and define a sequence {wm} in W such that

    w2m+1=F1w2mandw2m+2=F2w2m+1for m0. (4.3)

    Now, from (4.2), for t>0, we have

    1M(w2m+1,w2m+2,t)1=1M(F1w2m,F2w2m+1,t)1α(1MF(w2m,w2m+1,t)1)+βmax{(1MF(w2m,w2m+1,t)1),(1MF(w2m,F1w2m,t)1),(1MF(w2m+1,F2w2m+1,t)1),(1MF(w2m+1,F1w2m,t)1),(1MF(w2m,F2w2m+1,t)1),12(1MF(w2m,F1w2m,t)1+1MF(w2m+1,F2w2m+1,t)1),12(1MF(w2m+1,F1w2m,t)1+1MF(w2m,F2w2m+1,t)1)},

    after simplification, we get that

    1M(w2m+1,w2m+2,t)1=α(1MF(w2m,w2m+1,t)1)+βmax{(1MF(w2m,w2m+1,t)1),(1MF(w2m+1,w2m+2,t)1),(1MF(w2m,w2m+2,t)1),12(1MF(w2m,w2m+1,t)1+1MF(w2m+1,w2m+2,t)1),}. (4.4)

    Then, we may have the following four cases;

    (i) If 1MF(w2m,w2m+1,t)1 is the maximum in (4.4), then after simplification for t>0, we obtain

    1M(w2m+1,w2m+2,t)1λ1(1MF(w2m,w2m+1,t)1),where λ1=α+β<1. (4.5)

    (ii) If 1MF(w2m+1,w2m+2,t)1 is the maximum in (4.4), then after simplification for t>0, we obtain

    1M(w2m+1,w2m+2,t)1λ2(1MF(w2m,w2m+1,t)1),where λ2=α1β<1. (4.6)

    (iii) If 1MF(w2m,w2m+2,t)1 is the maximum in (4.4), then after simplification for t>0, we obtain

    1M(w2m+1,w2m+2,t)1λ3(1MF(w2m,w2m+1,t)1),where λ3=α+β1β<1. (4.7)

    (iv) If 12(1MF(w2m,w2m+1,t)1+1MF(w2m+1,w2m+2,t)1) is the maximum in (4.4), then after simplification for t>0, we obtain

    1M(w2m+1,w2m+2,t)1λ4(1MF(w2m,w2m+1,t)1),where λ4=2α+β2β<1. (4.8)

    Let us define μ1:=max{λ1,λ2,λ3,λ4}<1, then from (4.5)–(4.8), we get that

    1M(w2m+1,w2m+2,t)1μ1(1MF(w2m,w2m+1,t)1)for t>0. (4.9)

    Similarly, again by the view of (4.2), for t>0, we have

    1M(w2m+2,w2m+3,t)1=1M(F1w2m+2,F2w2m+1,t)1α(1MF(w2m+2,w2m+1,t)1)+βmax{(1MF(w2m+2,w2m+1,t)1),(1MF(w2m+2,F1w2m+2,t)1),(1MF(w2m+1,F2w2m+1,t)1),(1MF(w2m+1,F1w2m+2,t)1),(1MF(w2m+2,F2w2m+1,t)1),12(1MF(w2m+2,F1w2m+2,t)1+1MF(w2m+1,F2w2m+1,t)1),12(1MF(w2m+1,F1w2m+2,t)1+1MF(w2m+2,F2w2m+1,t)1)},

    after simplification, we get that

    1MF(w2m+2,w2m+3,t)1=α(1MF(w2m+1,w2m+2,t)1)+βmax{(1MF(w2m+1,w2m+2,t)1),(1MF(w2m+2,w2m+3,t)1),(1MF(w2m+1,w2m+3,t)1),12(1MF(w2m+2,w2m+3,t)1+1MF(w2m+1,w2m+2,t)1),}. (4.10)

    Again we may have the following four cases;

    (i) If (1MF(w2m+1,w2m+2,t)1) is the maximum term in (4.10), then after simplification for t>0, we obtain

    1M(w2m+2,w2m+3,t)1λ1(1MF(w2m+1,w2m+2,t)1),where λ1=α+β<1. (4.11)

    (ii) If (1MF(w2m+2,w2m+3,t)1) is the maximum term in (4.10), then after simplification for t>0, we obtain

    1M(w2m+2,w2m+3,t)1λ2(1MF(w2m+1,w2m+2,t)1),where λ2=α1β<1. (4.12)

    (iii) If (1MF(w2m+1,w2m+3,t)1) is the maximum term in (4.10), then after simplification for t>0, we obtain

    1M(w2m+2,w2m+3,t)1λ3(1MF(w2m+1,w2m+2,t)1),where λ3=α+β1β<1. (4.13)

    (iv) If 12(1MF(w2m+2,w2m+3,t)1+1MF(w2m+1,w2m+2,t)1) is the maximum term in (4.10), then after simplification for t>0, we obtain

    1M(w2m+2,w2m+3,t)1λ4(1MF(w2m+1,w2m+2,t)1),where λ4=2α+β2β<1. (4.14)

    Hence, from (4.11)–(4.14), we get that

    1M(w2m+2,w2m+3,t)1μ1(1MF(w2m+1,w2m+2,t)1)for t>0, (4.15)

    where μ1=max{λ1,λ2,λ3,λ4}<1. Now from (4.9) and (4.15), we have that

    1M(w2m+2,w2m+3,t)1μ1(1MF(w2m+1,w2m+2,t)1)(μ1)2(1MF(w2m,w2m+1,t)1)(μ1)2m+2(1MF(w0,w1,t)1)0,as m.

    Hence proved that {wm}m0 is a fuzzy contractive sequence, therefore

    limmMF(wm,wm+1,t)=1for t>0. (4.16)

    Since MF is triangular, for k>m and t>0, then we have

    1MF(wm,wk,t)1(1MF(wm,wm+1,t)1)+(1MF(wm+1,wm+2,t)1)++(1MF(wk1,wk,t)1)((μ1)m+(μ1)m+1++(μ1)k1)(1MF(w0,w1,t)1)((μ1)m1μ1)(1MF(w0,w1,t)1)0,as m,

    which shows that {wm} is a cauchy sequence. By the completeness of (W,MF,),  κW such that

    limmMF(κ,wm,t)=1for t>0. (4.17)

    Now we have to show that F1κ=κ. Since, MF is triangular, therefore

    1MF(κ,F1κ,t)1(1MF(κ,w2m+2,t)1)+(1MF(w2m+2,F1κ,t)1)for t>0. (4.18)

    Now, by the view of (4.2), (4.16) and (4.17) for t>0, we have that

    1MF(w2m+2,F1κ,t)1=1MF(F2w2m+1,F1κ,t)1α(1MF(w2m+1,κ,t)1)+βmax{(1MF(w2m+1,κ,t)1),(1MF(κ,F1κ,t)1),(1MF(w2m+1,F2w2m+1,t)1),(1MF(w2m+1,F1κ,t)1),(1MF(κ,F2w2m+1,t)1),12(1MF(κ,F1κ,t)1+1MF(w2m+1,F2w2m+1,t)1),12(1MF(w2m+1,F1κ,t)1+1MF(κ,F2w2m+1,t)1)}=α(1MF(w2m+1,κ,t)1)+βmax{(1MF(w2m+1,κ,t)1),(1MF(κ,F1κ,t)1),(1MF(w2m+1,w2m+2,t)1),(1MF(w2m+1,F1κ,t)1),(1MF(κ,w2m+2,t)1),12(1MF(κ,F1κ,t)1+1MF(w2m+1,w2m+2,t)1),12(1MF(w2m+1,F1κ,t)1+1MF(κ,w2m+2,t)1)} βmax{1MF(κ,F1κ,t)1,12(1MF(κ,F1κ,t)1)},as j.

    Then,

    limmsup(1MF(w2m+2,F1κ,t)1)β(1MF(κ,F1κ,t)1)for t>0. (4.19)

    The above (4.19) is together with (4.18) and (4.17), we get that

    1MF(κ,F1κ,t)1β(1MF(κ,F1κ,t)1)for t>0.

    Since (1β)0, therefore we get that MF(κ,F1κ,t)=1, this implies that F1κ=κ. Similarly, we can show F2κ=κ. Hence proved that κ is a common fixed point of F1 and F2, that is, F1κ=F2κ=κ.

    Uniqueness: let κW be the other common fixed point of F1 and F2 such that F1κ=F2κ=κ, then by the view of (4.2), for t>0, we have

    1MF(κ,κ,t)1=(1MF(F1κ,F2κ,t)1)α(1MF(κ,κ,t)1)+βmax{(1MF(κ,κ,t)1),(1MF(κ,F1κ,t)1),(1MF(κ,F2κ,t)1),(1MF(κ,F1κ,t)1),(1MF(κ,F2κ,t)1),12(1MF(κ,F1κ,t)1+1MF(κ,F2κ,t)1),12(1MF(κ,F1κ,t)1+1MF(κ,F2κ,t)1)}=(α+β)(1MF(κ,κ,t)1)=(α+β)(1MF(F1κ,F2κ,t)1)(α+β)2(1MF(κ,κ,t)1)(α+β)m(1MF(κ,κ,t)1)0,as m.

    Hence we get that MF(κ,κ,t)=1, this implies that κ=κ. Thus, F1 and F2 have a unique common fixed point in W.

    If the mapping F1=F2 or one of them considers an identity map in Theorem 4.2, then we get the following corollary:

    Corollary 4.3. Let (W,MF,) be a complete FM space in which MF is triangular and a pair of self-mappings F1:WW satisfies,

    1MF(F1w,F1x,t)1α(1MF(w,x,t)1)+βmax{(1MF(w,x,t)1),(1MF(w,F1w,t)1),(1MF(x,F1x,t)1),(1MF(x,F1w,t)1),(1MF(w,F1x,t)1),12(1MF(w,F1w,t)1+1MF(x,F1x,t)1),12(1MF(x,F1w,t)1+1MF(w,F1x,t)1)} (4.20)

     w,xW, t>0, α(0,1) and β0 with (α+2β)<1. Then F1 has a unique fixed point.

    Remark 4.4. If we put β=0 in Theorem 4.2, we get "a fuzzy Banach contraction theorem for FP" on a complete FM space.

    Example 4.5. Let W=[0,1] and from Example 3.6, the mappings F1,F2:WW be defined as F1u=F2u=7u10+415 for all uW. Then, we have

    1MF(F1w,F2x,t)1=2|F1wF2x|3t=7|wx|15t=710(1MF(w,x,t)1) w,xW and t>0.

    Hence, the mappings F1 and F2 are contractive and satisfied the conditions of Theorem 4.2 with α=710, β=17. The mappings F1 and F2 have a common fixed point, that is, F1(89)=F2(89)=89[0,1].

    In this section, we present an application of fuzzy differential equations (FDEs) to support our results. Some differential equation results in different directions can be found in (see [33,34,35,36] the references are therein). From the book of Lakshmikantham et al. [38], we have the following FDEs.

    Let E be the space of all fuzzy subsets w of R where w:RI=[0,1].

    w"(s)=h(s,w(s),w(s)),sJ=[a,b],w(s1)=w1, w(s2)=w2,s1,s2J=[a,b], (5.1)

    where h:J×E×EE is a continuous function. This problem is equivalent to the integral equation

    w(s)=s2s1K(s,τ)(h(τ,w(τ),w(τ)))dτ+B(s),

    where Green's function K is given by

    K(s,τ)={(s2s)(τs1)s2s1,s1τss2,(s2τ)(ss1)s2s1,s1sτs2.

    And B(s) satisfies B"=0, B(s1)=w1, B(s2)=w2. Let here we recall some properties of K(s,τ), that are;

    s2s1K(s,τ)dτ(s2s1)28,

    and

    s2s1Ks(s,τ)dτs2s12.

    Let C=C1(J,E), is a continuous t-norm, and MF:C×C×(0,)[0,1] be defined as

    MF(w,x,t)=tt+D(w,x)where D(w,x)=|wx|, (5.2)

    for all w,xC and t>0. Then one can verify that MF is triangular and (C,MF,) is complete.

    Now, we are in the position to prove the existing result for the above boundary value problem by applying Theorem 3.1.

    Theorem 5.1. Assume that h1,h2:J×E×EE and let there exists p,q(0,1) with pq such that for all w,xC1(J,E), satisfies

    |h1(s,w,w)h2(s,x,x)|p|wx|+q|wx|. (5.3)

    Let there exists η(0,1) such that

    D(w,x)ηM(F1,F2,w,x), (5.4)

    where

    M(F1,F2,w,x)=max{|wx|,|F1ww|+|F2xx|+|F1wx|+|F2xw|min{|F1ww|,|F2xx|},max{|F1wx|,|F2xw|}}. (5.5)

    Then the integral equations

    w(s)=s2s1K(s,τ)(h1(τ,w(τ),w(τ))dτ+B(s),sJ,

    and

    x(s)=s2s1K(s,τ)(h2(τ,x(τ),x(τ))dτ+B(s),sJ,

    have a unique common solution in C1[[s1,s2],E].

    Proof. Suppose that C=[[s1,s2],E] with metric

    D(w,x)=maxs1ss2(p|w(s)x(s)|+q|w(s)x(s)|). (5.6)

    The space (C,D) is a complete metric space. Now, the operators F1,F2:CC are defined as

    F1(w)(s)=s2s1K(s,τ)(h1(τ,w(τ),w(τ))dτ+B(s),sJ,

    and

    F2(x)(s)=s2s1K(s,τ)(h2(τ,x(τ),x(τ))dτ+B(s),sJ,

    where h1,h2C(J×E×E,E), wC1(J,E), and BC(J,E). Now by the properties of K(s,τ) and by using our hypothesis,

    |F1w(s)F2x(s)|s2s1|K(s,τ)||h1(τ,w(τ),w(τ))h2(τ,x(τ),x(τ))|dτD(w,x)s2s1|K(s,τ)|dτ(s2s1)28D(w,x)D(w,x)8.

    And

    |(F1w)(s)(F2x)(s)|s2s1|Ks(s,τ)||h1(τ,w(τ),w(τ))h2(τ,x(τ),x(τ))|dτD(w,x)s2s1|Ks(s,τ)|dτs2s12D(w,x)D(w,x)2.

    Now, from the above and by the view of (5.3) and (5.6), we have that

    D(F1w,F2x)=maxs1ss2(p|F1w(s)F2x(s)|+q|(F1w)(s)(F2x)(s)|)pD(w,x)8+qD(w,x)2(58q)D(w,x)

    Now, from (5.4), we have that

    D(F1w,F2x)(58q)D(w,x)ξM(F1,F2,w,x), (5.7)

    where ξ=58qη<1. Now we apply Theorem 3.1 to get that F1 and F2 have a unique common fixed point wC, i.e., w is a solution of the BVP. We may have the following main four cases:

    1) If |wx| is the maximum term in (5.5), then M(F1,F2,w,x)=|wx|. Now from (5.2), (5.4) and (5.7), we have

    1MF(F1w,F2x,t)1=D(F1w,F2x)tξM(F1,F2,w,x)t=ξ|wx|t=ξ(1MF(w,x,t)1).

    This implies that

    1MF(F1w,F2x,t)1ξ(1MF(w,x,t)1)for t>0, (5.8)

    for all w,xC. Thus, the operators F1 and F2 satisfy the conditions of Theorem 3.1 with ξ=a and b=c=d=0 in (3.19). Then the operators F1 and F2 have a unique common fixed point wC, i.e., w is a solution of the BVP (5.1).

    2) If |F1ww|+|F2xx|+|F1wx|+|F2xw| is the maximum term in (5.5), then M(F1,F2,w,x)=|F1ww|+|F2xx|+|F1wx|+|F2xw|. Now from (5.2), (5.4) and (5.7), we have

    1MF(F1w,F2x,t)1=D(F1w,F2x)tξM(F1,F2,w,x)t=ξ|F1ww|+|F2xx|+|F1wx|+|F2xw|t=ξ(1MF(w,F1w,t)1+1MF(x,F2x,t)1+1MF(x,F1w,t)1+1MF(w,F2x,t)1).

    This implies that

    1MF(F1w,F2x,t)1ξ(1MF(w,F1w,t)1+1MF(x,F2x,t)1+1MF(x,F1w,t)1+1MF(w,F2x,t)1)for t>0, (5.9)

    for all w,xC. Thus, the operators F1 and F2 satisfy the conditions of Theorem 3.1 with ξ=b and a=c=d=0 in (3.19). Then the operators F1 and F2 have a unique common fixed point wC, i.e., w is a solution of the BVP (5.1).

    3) If min{|F1ww|,|F2xx|} is the maximum term in (5.5), then M(F1,F2,w,x)=min{|F1ww|,|F2xx|}. Now, if |F1ww| is the minimum term in {|F1ww|,|F2xx|}, then M(F1,F2,w,x)=|F1ww|. Therefore, from (5.2), (5.4) and (5.7), we have

    1MF(F1w,F2x,t)1=D(F1w,F2x)tξM(F1,F2,w,x)t=ξ|F1ww|t=ξ(1MF(w,F1w,t)1).

    This implies that

    1MF(F1w,F2x,t)1ξ(1MF(w,F1w,t)1)for t>0. (5.10)

    Similarly, if |F2xx| is the minimum term in {|F1ww|,|F2xx|}, then M(F1,F2,w,x)=|F2xx|. Therefore, again from (5.2), (5.4) and (5.7), we have

    1MF(F1w,F2x,t)1=D(F1w,F2x)tξM(F1,F2,w,x)t=ξ|F2xx|t=ξ(1MF(x,F2x,t)1).

    This implies that

    1MF(F1w,F2x,t)1ξ(1MF(x,F2x,t)1)for t>0, (5.11)

    for all w,xC. Thus, from (5.10) and (5.11) the operators F1 and F2 satisfy the conditions of Theorem 3.1 with ξ=c and a=b=d=0 in (3.19). Then the operators F1 and F2 have a unique common fixed point wC, i.e., w is a solution of the BVP (5.1).

    4) If max{|F1wx|,|F2xw|} is the maximum term in (5.5), then M(F1,F2,w,x)=max{|F1wx|,|F2xw|}. Now, if |F1wx| is the maximum term in {|F1wx|,|F2xw|}, then M(F1,F2,w,x)=|F1wx|. Therefore, from (5.2), (5.4) and (5.7), we have

    1MF(F1w,F2x,t)1=D(F1w,F2x)tξM(F1,F2,w,x)t=ξ|F1wx|t=ξ(1MF(x,F1w,t)1).

    This implies that

    1MF(F1w,F2x,t)1ξ(1MF(x,F1w,t)1)for t>0. (5.12)

    Similarly, if |F2xw| is the maximum term in {|F1wx|,|F2xw|}, then M(F1,F2,w,x)=|F2xw|. Therefore, again from (5.2), (5.4) and (5.7), we have

    1MF(F1w,F2x,t)1=D(F1w,F2x)tξM(F1,F2,w,x)t=ξ|F2xw|t=ξ(1MF(w,F2x,t)1).

    This implies that

    1MF(F1w,F2x,t)1ξ(1MF(w,F2x,t)1)for t>0. (5.13)

    for all w,xC. Thus, from (5.12) and (5.13) the operators F1 and F2 satisfy the conditions of Theorem 3.1 with ξ=d and a=b=c=0 in (3.19). Then the operators F1 and F2 have a unique common fixed point wC, i.e., w is a solution of the BVP (5.1).

    In this paper, we presented some generalized unique common fixed point theorems for a pair of self-mappings on complete FM spaces. The triangular property of fuzzy metric is used as a basic tool throughout the complete paper and proved all the results without continuity of self-mappings. We defined weak-contraction and a generalized Ćirić-contraction on FM space and proved unique common fixed point theorems. The results are supported by suitable examples and showed the uniqueness of common fixed points. In addition, we presented an application of fuzzy differential equations and proved the existing result for a unique common solution to support our main work. By using this concept, one can prove more generalized different contractive type single-valued mapping results for fixed point, common fixed point, and coincidence point on FM spaces without the continuity of self-mappings by using different types of applications such as differential equations and integral equations applications.

    The authors Dr. Thabet Abdeljawad and Dr. Nabil Mlaiki would like to thank Prince Sultan University, Riyadh, Saudi Arabia for paying the APC and for the support through the TAS research lab.

    The authors declare that they have no conflict of interest.



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