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Research article Special Issues

Ćirić type nonunique fixed point theorems in the frame of fuzzy metric spaces

  • The paper defines a new contractive condition within κorbitally complete fuzzy metric spaces (Θ,M,T), as well as fixed point theorems for single-valued and multi-valued function on Θ which is not necessarily continuous. The contractive condition is motivated by an idea proposed in Ćirić's paper "On some maps with a nonunique fixed points". Continuity of mapping κ is replaced by κorbitally continuity property which provides the existence of the fixed point, but not necessarily uniqueness.

    Citation: Tatjana Došenović, Dušan Rakić, Stojan Radenović, Biljana Carić. Ćirić type nonunique fixed point theorems in the frame of fuzzy metric spaces[J]. AIMS Mathematics, 2023, 8(1): 2154-2167. doi: 10.3934/math.2023111

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  • The paper defines a new contractive condition within κorbitally complete fuzzy metric spaces (Θ,M,T), as well as fixed point theorems for single-valued and multi-valued function on Θ which is not necessarily continuous. The contractive condition is motivated by an idea proposed in Ćirić's paper "On some maps with a nonunique fixed points". Continuity of mapping κ is replaced by κorbitally continuity property which provides the existence of the fixed point, but not necessarily uniqueness.



    In 1965, Zadeh [30] introduced the concept of a fuzzy set that found great application in various branches of science and engineering, including mathematics as well as fixed point theory. Unlike a classical set, where an element either belongs or does not belong to a set in the theory of fuzzy sets, the elements belong to a set with a measure that it involves a continuous transition from non-belonging to full belonging. In a fuzzy set, each element of the set is assigned a value from [0,1] and that number (fuzzy number) represents the degree of belonging to the fuzzy set. Mathematically, if a set Θ is given, the fuzzy set is a mapping A:Θ[0,1]. This mapping is called the membership function, whose value for a certain element in the set defines its degree of belonging.

    Guided by the basic postulates of probability theory, the mathematician Karl Menger in 1942 defined probabilistic metric spaces as a generalization of the concept of metric spaces. The distance between two objects is not a fixed number but is assigned to points of space appropriate to the distribution function. A further generalization is given by Kramosil and Michalek [21] in 1975 where they use the following idea-the distance function does not have to be given by the distribution function but by the fuzzy set. Now the distance function represents the degree of certainty with which the points b and l are at a distance less than t. Next, George and Veeramani [6,7], in order to obtain Pompeiu-Hausdorff metric modify the concept of fuzzy metrics defined by Kramosil and Michalek.

    In the fixed point theory, the first significant result is published by Sehgal and Bharucha-Reid [3,29] where they generalize the famous Banach's contraction principle [2] in probabilistic metric spaces using triangular norm TM. Following this result, scientists around the world publish papers that represent a generalization of Banach's contraction principle, both in probabilistic and fuzzy metric spaces. The novelty of this topic is confirmed by many recent papers, which includes the application of fixed point theory in solving various integral equations (see [19,26]). Important generalizations of Banach's contraction principle in metric spaces were given by the following mathematicians: Edelstein-Nemitskii, Boyd and Wong, Meir and Keeler, Kannan, Chatterje, Zamfirescu, Reich, Hardy and Rogers, Geraghty, Ćirić which gives one of the most general contractive conditions (quasi-contraction) and many others. Later on, the mentioned contractive conditions and fixed point theorems related to a certain contractive condition are generalized in probabilistic, fuzzy, but also in other spaces that represent the generalization of metric spaces. Especially, Ćirić's quasi-contraction as one of the strongest generalization of Banach's contraction is frequently used in recent studies [23,25,27]. As Banach's contractive condition implies the continuity of the mapping κ, Kannan's work 1968 provides an answer to the question of whether there is a contractive condition sufficient for the existence of a fixed point, but that the continuity of the mapping f does not have to be implied. The question concerning the continuity of the mapping f is also raised in the paper of Ćirić from 1974 [5], with a new contractive condition in metric space (X,ϱ):

    min{ϱ(κb,κl),ϱ(b,κb),ϱ(l,κl)}min{ϱ(b,κl),ϱ(κb,l)}qϱ(b,l),b,lX,q(0,1),

    where the existence of a fixed point is achieved by assuming that space X is κ orbitally complete and the mapping κ is orbitally continuous. Encouraged by this, in this paper we give a contractive condition within fuzzy metric spaces, where the existence of a fixed point is achieved by assuming that the mapping is f orbitally continuous. Both single-valued and multi-valued case is discussed in this paper.

    Before the main results, we look at the known definitions and results that are necessary for this research.

    Starting from the idea of the basic triangle inequality K. Menger [22] defined the term triangular norms. The first area where triangular norms played a significant role was the theory of probabilistic metric spaces. In addition, triangular norms are a significant operation in areas such as fuzzy sets and phase logic, the theory of generalized measures and the theory of nonlinear differential and differential equations. However, the original set of axioms was weak and B. Schweizer and A. Sklar [28] made changes and defined axioms for triangular norms that are still used today.

    Definition 1.1. [20] A binary operation T:[0,1]×[0,1][0,1] is a continuous triangular norm if it satisfies the following conditions:

    (t1) T is associative and commutative,

    (t2) T is continuous,

    (t3) T(p,1)=p, for all p[0,1],

    (t4) T(p,q)T(r,s) whenever pr and qs, for each p,q,r,s[0,1].

    Typical examples of a continuous t-norm are TP(p,q)=pq,TM(p,q)=min{p,q} and TL(p,q)=max{p+q1,0}. Very important class of triangular norms is given by O. Hadžić [9,11].

    Definition 1.2. [9] Let T be a triangular norm and Tn:[0,1][0,1],nN. T is a triangular norm of Hadžić-type if the family {Tn(b)}nN defined in the following way:

    T1(b)=T(b,b),Tn+1(b)=T(Tn(b),b),nN,b[0,1],

    is equi-continuous at b=1.

    Minimum triangular norm is trivial example of triangular norm of Hadžić-type, for nontrivial example readers are referred to the paper [9].

    In the book Triangular norm writen by Klement, Mesiar and Pap is pointed out very useful statement that using associativity of triangular norms every t-norm T can be extended in a unique way to an n-ary operation taking for (b1,,bn)[0,1]n the values

    T1i=1bi=b1,Tni=1bi=T(Tn1i=1bi,bn)=T(b1,b2,,bn).

    Example 1. [13] n-ary extensions of the Tmin,TL and TP t-norms:

    TM(b1,,bn)=min(b1,,bn),

    TL(b1,,bn)=max(ni=1bi(n1),0),

    TP(b1,,bn)=b1b2bn.

    It has been shown in the paper [20] that the triangular norm T can be extended to a countable infinite operation taking for any sequence (bn)nN from [0,1] the value

    T+i=1bi=limn+Tni=1bi.

    Since the sequence (Tni=1bi), nN is non-increasing and bounded from below, the limit T+i=1bi exists.

    In order to prove existence of fixed point the following condition is imposed [13,14] investigate the classes of triangular norms T and sequences (bn) from the interval [0,1] such that limn+bn=1 and

    limn+T+i=nbi=limn+T+i=1bn+i=1. (1.1)

    The next proposition concerning triangular norms of Hadžić type is proved in [13].

    Proposition 1.3. Let (bn)nN be a sequence of numbers from [0,1] such that limn+bn=1 and triangular norm T is of Hadžić type. Then limn+T+i=nbi=limn+T+i=1bn+i=1.

    Definition 1.4. ([6,7]) A 3tuple (Θ,M,T) is called a fuzzy metric space if Θ is an arbitrary (non-empty) set, T is a continuous triangular norm and M is a fuzzy set on Θ2×(0,+), satisfying the following conditions for each b,l,zΘ and p,q>0,

    (Fm-1) M(b,l,p)>0,

    (Fm-2) M(b,l,p)=1 if and only if b=l,

    (Fm-3) M(b,l,p)=M(l,b,p),

    (Fm-4) T(M(b,l,p),M(l,z,q))M(b,z,p+q),

    (Fm-5) M(b,l,):(0,+)[0,1] is continuous.

    Definition 1.5. ([6,7]) Let (Θ,M,T) be a fuzzy metric space.

    (ⅰ) A sequence {bn}nN is a Cauchy sequence in (Θ,M,T) if for every δ(0,1) there exists i0N such that M(bj,bk,p)>1δ,j,ki0,p>0.

    (ⅱ) A sequence {bn}nN converges to b in (Θ,M,T) if for every δ(0,1) there exists i0N such that M(bj,b,p)>1δ,ji0,p>0. Then, we say that {bn}nN is convergent.

    (ⅲ) A fuzzy metric space (Θ,M,T) is complete if every Cauchy sequence in (Θ,M,T) is convergent.

    Lemma 1.6. [8] M(b,l,) is non-decreasing for all b,lΘ.

    Let Υ and Ω be two nonempty subsets of Θ, define the Hausdorff–Pompeiu fuzzy metric as

    H(Υ,Ω,p)=min{infbΥE(b,Ω,p),inflΩE(l,Υ,p)},p>0,

    where E(b,B,p)=suplΩM(b,l,p).

    Definition 1.7. [10,12] Let (Θ,M,T) be a fuzzy metric space, AΘ and F:AC(A), (C(A is the set of all closed subsets of A). A mapping F is a weakly demicompact if for every sequence {bn}nN from A such that limn+M(bn,bn+1,p)=1,p>0,bn+1Fbn,nN, there exists a convergent subsequence {bnk}kN.

    Definition 1.8. [5] (ⅰ) An orbit of F:ΘC(Θ) at the point bΘ is a sequence {bn:bnFbn1}, where b0=b.

    (ⅱ) A multivalued function F is orbitally upper-semicontinuous if bnuΘ implies uFu whenever {bn}nN is an orbit of F at some bΘ.

    (ⅲ) A space Θ is Forbitally complete if every orbit of F at some bΘ which is Cauchy sequence converges in Θ.

    Lemma 1.9. [17] Let (Θ,M,T) be a fuzzy metric space and let {bn} be a sequence in Θ such that

    limp0+M(bn,bn+1,p)>0,nN, (1.2)

    and

    limn+M(bn,bn+1,p)=1,p>0. (1.3)

    If {bn} is not a Cauchy sequence in (Θ,M,T), then there exist ε(0,1),p0>0, and sequences of positive integers {lk},{ik},lk>ik>k,kN, such that the following sequences

    {M(bik,blk,p0)},{M(bik,blk+1,p0)},{M(bik1,blk,p0)},
    {M(bik1,blk+1,p0)},{M(bik+1,blk+1,p0)},

    tend to 1ε, as k+.

    Definition 1.10. [4,5] A mapping κ is called a orbitally continuous if limi+κnib=u implies limi+κκnib=κu, for every uΘ. A space Θ is called a κorbitally complete if every Cauchy sequence of the form {κnib}+i=1,bΘ converges in Θ.

    Recently [24], orbitally continuous property is used to obtain common fixed points results in Menger probabilistic metric spaces.

    Instigated by the paper [5] we give the following contractive condition.

    Theorem 2.1. Let (Θ,M,T) be κorbitally complete fuzzy metric space such that limp+M(b,l,p)=1 and let κ:ΘΘ be an orbitally continuous mapping on Θ. If κ satisfies the following condition:

    max{M(κb,κl,p),M(κb,b,p),M(κl,l,p)}+min{1M(b,κl,p),1M(κb,l,p)}M(b,l,pq), (2.1)

    for some q<1 and for all b,lΘ,p>0, and if triangular norm T satisfies the following condition:

    For arbitrary b0Θ and b1=κb0 there exists ς(q,1) such that

    limn+T+i=nM(b0,b1,pςi)=1,p>0, (2.2)

    then for each bΘ the sequence {κnb}+n=1 converges to a fixed point of κ.

    Proof. Let b0Θ is arbitrary. Now, we can construct a sequence {bn}nN{0} such that bn+1=κbn, for all nN{0}. If bn0=bn0+1 for some n0N{0} then the proof is finished. So, suppose that bnbn+1, for all nN{0}. Let nN and p>0. By (2.1) for b=bn1,l=bn, we have

    max{M(κbn1,κbn,p),M(κbn1,bn1,p),M(κbn,bn,p)}+min{1M(bn1,κbn,p),1M(κbn1,bn,p)}M(bn1,bn,pq),

    which means that

    max{M(bn,bn+1,p),M(bn,bn1,p)}M(bn1,bn,pq). (2.3)

    Since pq>p, by using Lemma 1.6, we get that

    M(bn,bn+1,p)M(bn1,bn,pq),nN,p>0. (2.4)

    In the following, we will show that the sequence {bn} is a Cauchy sequence.

    Let ϑ(q,1). Then, sum +i=1ϑi is convergent and there exists i0N such that +i=nϑi<1, for every j>i0. Let j>k>i0. Since M is nondecreasing, by (Fm-4), for every p>0 we have

    M(bj,bj+k,p)M(bj,bj+k,pj+k1s=jϑs)T(M(bj,bj+1,pϑj),M(bj+1,bj+k,pj+k1s=j+1ϑs))T(M(bj,bj+1,pϑj),T(M(bj+1,bj+2,pϑj+1,M(bj+k1,bj+k,pϑj+k1)).

    By (2.4) follows that

    M(bj,bj+1,p)M(b0,b1,pqn),nN,p>0,

    and for j>k>i0 we have

    M(bj,bj+k,p)T(M(b0,b1,pϑiqi),T(M(b0,b1,pϑi+1qi+1),M(b0,b1,pϑj+k1qj+k1))=Tj+k1s=jM(b0,b1,pϑsqs)T+s=jM(b0,b1,pςs),p>0,

    where ς=qϑ. Since ς(q,1) by (2.2) follows that {bn} is a Cauchy sequence.

    Based on that Θ is κorbitally complete fuzzy metric space and {bn} is a Cauchy sequence it follows that limn+κnb0=uΘ, and using orbital continuity of κ we have that κu=u.

    Example 2. Let Θ=[0,1],k[12,1),κb={kb,b01,b=0,M(b,l,p)=e|bl|p and T=TP.

    Further, condition (2.1), with q=m, will be checked.

    Case 1. Let b,l0. Then

    max{M(κb,κl,p),M(κb,b,p),M(κl,l,p)}+min{1M(b,κl,p),1M(κb,l,p)}max{M(κb,κl,p),M(κb,b,p),M(κl,l,p)}ek|bl|p=M(b,l,pq),p>0.

    Case 2. Let b=0 and l0. Then

    max{M(κb,κl,p),M(κb,b,P),M(κl,l,p)}+min{1M(b,κl,p),1M(κb,l,p)}e(1k)lp=e(1k)|bl|pek|bl|p=M(b,l,pq),p>0.

    Case 3. Let b=l=0. Then

    max{M(κb,κl,p),M(κb,b,p),M(κl,l,p)}+min{1M(b,κl,p),1M(κb,l,p)}=2e1pek|bl|p=M(b,l,pq),p>0.

    So, condition (2.1) is satisfied for all b,lΘ with q=k, but since κ is not orbital continuous then the statement of the Theorem 2.1 does not have to be true, and there is not a fixed point of the mapping κ.

    Example 3. Let Θ=[0,1],k[12,1),κb={kb,brationalb,birrational.,M(b,l,p)=e|bl|p and T=TP.

    Then, condition (2.1) is satisfied for all b,lΘ with q=k,κ is not continuous but it is orbital continuous and by Theorem 2.1 κ has fixed point. Moreover, κ has infinitely many fixed points.

    Remark 1.

    (ⅰ) Using Proposition 1.3 we conclude that the Theorem 2.1 holds if instead of condition (2.2) we use a triangular norm of Hadžić type.

    (ⅱ) Let (bn)nN be a sequence from (0,1) such that:

    a) +n=1(1bn)λ,λ(0,+) convergent. Then limn+(Tλ)+i=nbi=1,{D,AA}.

    b) +n=1(1bn),λ(1,+] convergent. Then limn+(TSWλ)+i=nbn=1.

    Therefore, Theorem 2.1 is valid if instead of an arbitrary triangular norm that satisfies the condition (2.2) one uses the Dombi (TDλ)λ(0,+), Aczˊel-Alsina (TAAλ)λ(0,+) or Sugeno-Weber (TSVλ)λ[1,+) family of triangular norms with additional conditions: +i=1(11ςi)λ for Dombi and Aczˊel-Alsina family of triangular norms, as well as +i=1(11ςi) for the Sugeno-Weber family of triangular norms. Namely, this statement is obvious by using the proposition given in [13].

    For more information on the mentioned families of triangular norms, readers are referred to the book [13].

    Theorem 2.2. Let (Θ,M,T) be a fuzzy metric such that limp+M(b,l,p)=1. Let F:ΘC(Θ) be orbitally upper-semicontinuous, Θ is Forbitally complete and F satisfies the following:

    For every b,lΘ,uFb and 0<δ<1, there exists vFb such that

    max{M(u,v,p),M(u,b,p),M(v,l,p)}+min{1M(u,l,p),1M(v,b,p)}M(b,l,pδq), (2.5)

    for some q(0,1) and every p>max{q1q,δ}. Also, one of the conditions (i) or (ii) is satisfied:

    (i) F is weakly demicompact

    or

    (ii) there exists b0,b1Θ,b1Fb0 and ς(q,1) such that triangular norm T satisfies:

    limn+T+i=nM(b0,b1,pςi)=1.

    Then there exist bΘ such that bFb.

    Proof. Let b0,b1Θ such that b1Fb0. Using (2.5), for u=b1,b=b0,l=b1 and δ=q there exist b2Θ such that b2Fb1 and

    max{M(b1,b2,p),M(b0,b1,p),M(b1,b2,p)}+min{1M(b0,b2,p),1M(b1,b1,p)}M(b0,b1,pqq).

    Then,

    max{M(b1,b2,p),M(b0,b1,p)}M(b0,b1,pqq).

    Suppose that max{M(b1,b2,p),M(b0,b1,p)}=M(b0,b1,p), we get M(b0,b1,p)M(b0,b1,pqq), which means that ppqq. Since this is contradictory with assumption (p>q1q), we conclude that

    M(b1,b2,p)M(b0,b1,pqq). (2.6)

    Further, for δ=q2 there exists b3Fb2 such that by (2.5) we have

    max{M(b2,b3,p),M(b1,b2,p),M(b2,b3,p)}+min{1M(b1,b3,p),1M(b2,b2,p)}M(b1,b2,pq2q).

    which implies that max{M(b2,b3,p),M(b1,b2,p)}=M(b2,b3,p). So,

    M(b2,b3,p)M(b1,b2,pq2q)M(b0,b1,p2q2q2).

    Continuing, for δ=q3,δ=q4, we can construct sequence {bn}nN from Θ such that the following conditions are satisfied:

    (a) bn+1Fbn,

    (b) M(bn,bn+1,p)M(bn1,bn,pqnq),nN.

    Using (b) we have that

    M(bn,bn+1,p)M(b1,b0,pnqnqn),nN.

    Since, limn+M(b1,b0,pqnn)=1 we conclude that

    limn+M(bn,bn+1,p)=1. (2.7)

    If we suppose that F is weakly demicompact (condition (i)), using (2.7) and bn+1Fbn we conclude that there exists a convergent subsequence (bnk)kN of the sequence (bn)nN. It remains to be proved that a sequence (bn)nN is convergent if triangular norm T satisfies condition (ii).

    Let ϑ=qς. As ϑ(q,1) follows that +i=1ϑi is convergent, and there exists m0N such that +i=m0ϑi<1. So, for all m>m0 and sN we have

    p>p+i=m0ϑi>pm+si=mϑi.

    Then,

    M(bm+s+1,bm,p)M(bm+s+1,bm,pm+si=mϑi)T(T(Tstimes(M(bm+s+1,bm+s,pϑm+s),M(bm+s,bm+s1,pϑm+s1)),,M(bm+1,bm,pϑm))T(T(Tstimes(M(b1,b0,pϑm+s(m+s)qm+sqm+s),M(b1,b0,pϑm+s1(m+s1)qm+s1qm+s1),M(b1,b0,pϑmmqmqm))=T(T(Tstimes(M(b1,b0,p(qϑ)m+s(m+s)),M(b1,b0,p(qϑ)m+s1(m+s1)),M(b1,b0,p(qϑ)mm))=Tm+si=mM(b1,b0,pςii).

    Since, ς(q,1), there exist m1(t)>m0 such that pςmm>p2ςm, for every m>m1(p). Now, for all sN we have

    M(bm+s+1,bm,p)Tm+si=mM(b1,b0,p2ςi)T+i=mM(b1,b0,p2ςi).

    Using assumption that limm+T+i=mM(b1,b0,1ςi)=1, we conclude that limm+T+i=mM(b1,b0,p2ςi)=1, for every p>0. So, for every p>0,λ(0,1), there exists m2(p,λ)>m1(p) such that M(bm+s+1,bm,p)>1λ, for all m>m2(p,λ) and every sN. Since, the sequence (bn)nN is a Cauchy and the space Θ is Forbitally complete we have that limn+bn exists.

    So, in both cases (i) and (ii) there exists a subsequence (bnk)kN such that

    u=limk+bnkΘ.

    The upper semi-continuity of F implies that uFu.

    Theorem 2.3. Let (Θ,M,T) be a fuzzy metric such that M is increasing by t and limp+M(b,y,p)=1. Let F:ΘC(Θ) be orbitally upper-semicontinuous. If Θ is Forbitally complete and F satisfies the following:

    There exists q(0,1) such that for every b,lΘ,p>0,

    max{H(Fb,Fl,p),E(Fb,b,p),E(Fl,l,p)}+(min{1E(Fb,l,p),1E(b,Fl,p)})M(b,l,pq). (2.8)

    and if one of the conditions (i) or (ii) from the Theorem 2.2 is satisfied, then there exists bΘ such that bFb.

    Proof. Let a>0 be an arbitrary small real number less then 1 and let t0>0 be arbitrary. Since M(b,l,p) is increasing by p, for every bΘ there exists lFb,(lb, otherwise b is a fixed point) such that:

    M(b,l,p0)E(b,Fb,qap0). (2.9)

    Let κ:ΘΘ be a function such that κb=l,bΘ. We take arbitrary bΘ and consider a orbit of κ defined as κbn1=bn,nN where b0=b. Note that bnFbn1 implies E(bn,Fbn,p)H(Fbn1,Fbn,p) and E(bn,Fbn1,p)=1,p>0. Now, we have using (2.8) for b=bn1 and l=bn,nN:

    max{E(Fbn1,bn1,p),E(Fbn,bn,p)}=max{H(Fbn1,Fbn,p),E(Fbn1,bn1,p),E(Fbn,bn,p)}(min{1E(Fbn1,bn,p),1E(bn1,Fbn,p)})M(bn1,bn,pq),p>0.

    Then, by (2.9), we have

    max{M(bn,bn1,p0),M(bn+1,bn,p0)}max{E(Fbn1,bn1,qap0),E(Fbn,bn,qap0)}M(bn1,bn,p0q1a),

    which implies that

    M(bn+1,bn,p0)M(bn1,bn,p0q1a),nN.

    Finally, we conclude that

    M(bn+1,bn,p)M(bn1,bn,pq1a),nN,p>0.

    Let q1=q1a. Then q1(0,1) and we can use the same technique as in Theorem 2.1 to conclude that the sequence {bn} is a Cauchy sequence and by the assumption that he mapping F is orbitally complete we conclude that there exists bΘ such that bFb.

    Let Φ be collection of all continuous mappings φ:[0,1][0,1] such that φ(p)>p,p(0,1),φ(1)=1. Such type of collection Φ is, together with contraction condition proposed by Ćirić in [5], used in [1] to obtain nonunique fixed point results in bmetric space. In the following theorem collection Φ within slightly modified contraction condition is observed in fuzzy metric spaces and existence of unique fixed point is proved. Open question: is it possible to obtain nonunique result with original Ćirić's condition, using collection Φ, as it is done in [1].

    Theorem 2.4. Let (Θ,M,T) be κorbitally complete fuzzy metric space and κ:ΘΘ be an orbitally continuous mapping on Θ such that limp+M(b,l,p)=1 and limt0+M(bn,bn+1,p)>0,nN. If κ satisfies the following condition

    M(κb,κl,p)+min{1M(κb,b,p),1M(κl,l,p),1M(b,κl,p),1M(κb,l,p)}φ(M(b,l,p)). (2.10)

    φΦ,b,lΘ,p>0, then for each bΘ the sequence {κnb}+n=1 converges to a fixed point of κ.

    Proof. Let b0Θ is arbitrary. Now, we can construct a sequence {bn}nN{0} such that bn+1=bn, for all nN{0}. If bn0=bn0+1, for some n0N{0}, then the proof is finished. So, suppose that bnbn+1, for all nN{0}.

    By (2.10), for b=bn1,l=bn,nN, we have

    M(bn+1,bn,p)φ(M(bn,bn1,p))>M(bn,bn1,p),p>0,

    and conclude that

    M(bn+1,bn,p)>M(bn,bn1,p)>>M(b1,b0,p),nN,p>0.

    Therefore, since the sequence {M(bn,bn+1,p)},nN is monotone increasing we have that there exists a1 such that

    limn+M(bn,bn+1,p)=a,p>0.

    Suppose that a<1, then using (2.10) we have contradiction

    aφ(a)>a,

    and conclude that a=1.

    Further, we need to prove that {bn} is a Cauchy sequence. Suppose that is not true and by Lemma 1.9, we have that there exist ε(0,1),p0>0 and sequences {bmk} and {bnk} such that limk+M(bmk,bnk,p0)=1ε. By (2.10)

    M(bmk,bnk,p0)+min{1M(bmk1,bmk,p0),1M(bnk1,bnk,p0),1M(bmk,bnk1,p0),1M(bnk,bmk1,p0)}φ(M(bmk1,bnk1,p0)),

    and when k+ we get contradiction

    1εφ(1ε)>1ε.

    So, {bn} is a Cauchy sequence.

    Since (Θ,M,T) is complete there exists uΘ such that limn+bn=u. By condition (2.10), with b=u,l=bn, we have

    M(κu,bn+1,p)+min{1M(u,κu,p),1M(bn,bn+1,p),1M(u,bn+1,p),1M(κu,bn,p)}φ(M(u,bn,p)),nN,p>0.

    If we take n+ it follows that u is a fixed point for κ:

    M(κu,u,p)φ(M(u,u,p))=φ(1)=1.

    Moreover, u is the unique fixed point for κ. Suppose that different u and v are fixed points and take (2.10) with b=u,l=v:

    M(κu,κv,p)+min{1M(u,κu,p),1M(v,κv,p),1M(u,κv,p),1M(κu,v,p)}φ(M(u,v,p)),p>0.

    So, we have contradiction

    M(κu,κv,p)φ(M(u,v,p))>M(u,v,p)=M(κu,κv,p).

    Example 4. Let Θ=R,κu=u2,M(b,l,p)=e|bl|p and T=TP and φ(p)=p. Then all conditions of Theorem 2.4 are satisfied and 0 is the unique fixed point.

    Using the countable extension of the triangular norm, we were able to prove theorems about the fixed point for the single-valued and multi-valued cases within fuzzy metric spaces. The mapping is not assumed to be continuous, but orbitally continuous. A fixed point is not necessarily unique as illustrated by an example. Potentially, obtained results could be applied for solving different integral equations and integral operators as it is done, for example, in [15,16,18].

    The first and second authors are supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia (Project No. 451-03-68/2022-14/200134).

    The authors declare that they have no conflicts of interest.



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