# Set-Valued Fixed-Point Theorems for Generalized Contractive Mappings on Fuzzy Metric Spaces

Authors: S. K. Elagan; Dumitru Baleanu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The purpose of this paper is to introduce new types of asymptotically (g,φ)-contractions which generalize the Binayak
S. Choudhury type contraction on fuzzy metric spaces and prove
some fixed-point theorems for single- and multivalued mappings
on fuzzy metric spaces. Hence, our results can be viewed as a
generalization and improvement of many recent results.


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## Body

## 1. Introduction and Preliminaries

The concept of fuzzy metric space was introduced in different ways by some authors (see, i.e., [1, 2]) and further,the fixed-point theory in this kind of spaces has been intensively studied (see [3–5]). Contraction mappings in probabilistic and fuzzy metric spaces have considered by many authors. Singh and Chauhan were the first to introduce contraction mapping principle in probabilistic metric space [6]. The result has been known as Seghal contraction. The structures of these spaces allow to extend the contraction mapping principle to these spaces in more than one inequivalent ways. One such concept is C-contraction which was originally introduced by O. Hadžic in [7] and subsequently studied and generalized in several works like [2, 8, 9]. In [7] O. Hadžic introduced the notion of a C-contraction in probabilistic metric space. In [10], Goleţ introduced the concept of g-contraction where g is a bijective function to generalize the Hicks-type contraction. Some other works may be noted in [4, 11–20].In this paper we establish two coincidence point results for three mappings. For this purpose we consider fuzzy(g,φ)-contraction and fuzzy asymptotically (g,φ)-contraction, and prove several important fixed-point theorems for single- and multivalued mappings.Definition 1.1 (see [18]).
A binary operation*:[0,1]×[0,1]→[0,1] is said to be a continuous t-norm if it is satisfies the following conditions: (a)
* is associative and commutative;(b)
* is continuous;(c)
a*1=a for all a∈[0,1];(d)
a*b≤c*d whenever a≤c and b≤d for each a,b,c,d∈[0,1].In [8], Kramosil and Michálek gave the following definition of fuzzy metric space.Definition 1.2.
The 3-tuple(X,M,*) is said to be a fuzzy metric space if X is an arbitrary set, * is a continuous t-norm, and M is a fuzzy set on X2×[0,∞), satisfying the following conditions. For all x,y,z∈X and t,s>0,(M1)
M(x,y,0)=0;(M2)
M(x,y,t)=1 if and only if x=y;(M3)
M(x,y,t)=M(y,x,t);(M4)
M(x,z,t+s)≥M(x,y,t)*M(y,z,s);(M5)
M(x,y,·):(0,∞)→[0,1] is left-continuous.Example 1.3 (see [21]).
Let(X,d) be a metric space. Define a*b=a·b (or a*b=min⁡{a,b}) and for all x,y∈X and ϵ>0,
(1.1)M(x,y,ϵ)=ϵϵ+d(x,y).
Then (X,M,*) is a fuzzy metric space. We call this fuzzy metric M induced by the metric d the standard fuzzy metric. On the other hand, note that there exists no metric on X satisfying (1.1).Lemma 1.4.
Let* be a continuous t-norm according to Definition 1.1. Then the condition
(1.2)a*a≥aforeverya∈[0,1]
holds if and only if
(1.3)a*b=min⁡⁡{a,b}foreverya,b∈[0,1].Proof.
Suppose (1.2) holds. Let a,b∈[0,1] such that a≤b. Then
(1.4)min⁡⁡{a,b}=a=a*1≥a*b≥a*a≥a=min⁡⁡{a,b}.
Therefore, for all a,b∈[0,1], we have a*b=min⁡⁡{a,b}. Suppose (1.3) holds. Then clearly
(1.5)a*a=min⁡⁡{a,a}=a
and so (1.2) holds.Definition 1.5 (see [8]).
Let(X,M,*) be a fuzzy metric space: a sequence {xn} in X is said to be convergent to a point x∈X (denoted xn→x) if
(1.6)lim⁡n→∞M(xn,x,t)=1foreacht>0.Definition 1.6 (see [21]).
Let(X,M,*) be a fuzzy metric space: a sequence {xn} in X is called a Cauchy sequence if and only if for any ϵ>0,t>0, there exists n0∈ℕ such that
(1.7)M(xn,xm,t)>1-ϵ
for all n,m≥n0.Definition 1.7 (see [8]).
Let(X,M,*) be a fuzzy metric space: a sequence {xn} in X is called a Cauchy sequence if
(1.8)lim⁡n→∞M(xn+p,xn,t)=1foreacht>0,p>0.Definition 1.8.
Let(X,M1,*) and (Y,M2,*) be two fuzzy metric spaces. A mapping f:X→Y is said to be (uniformly) continuous if for each ϵ>0, s>0, there exist δ>0, t>0 such that
(1.9)M1(x,y,s)>1-δ⟹M2(f(x),f(y),t)>1-ϵ,
for each x,y∈X.Definition 1.9.
Let(X,M,*) be a fuzzy metric space and let T:X→X be a self-mapping on X. The mapping T is called asymptotically regular at x∈X if
(1.10)lim⁡n→∞M(Tnx,Tn+1x,t)=1.


## 2. Main Results

LetΦ be the class of all mappings φ:[0,∞)→[0,∞) with the following properties:(i)
φ is strictly increasing;(ii)
φ is right-continuous;(iii)
φ(t)<t for all t>0.Lemma 2.1.
For allt>0,lim⁡n→∞⁡φn(t)=0, where φn is the n-iteration of φ.Proof.
Suppose thatlim⁡n→∞⁡φn(t0)=l<0 for some t0∈(0,∞). By the monotonicity and right continuity of φ, we have
(2.1)l=lim⁡n→∞φn+1(t0)=φ(lim⁡n→∞φn(t0))=φ(l)<l,
which is a contradiction.Definition 2.2.
Let(X,M,*) be a fuzzy metric space and φ∈Φ. We say that the mapping f:X→X is a fuzzy (g,φ)-contraction if there exists a bijective function g:X→X such that for every x,y∈X, t>0 the following implication holds:
(2.2)M(g(x),g(y),t)>1-t⟹M(f(x),f(y),φ(t))>1-φ(t).Note that, ifφ(t)=kt for k∈(0,1),t>0, then (2.2) is a g-contraction in the sense of Goleţ [10]. On the other hand, if g is an identity function, then (2.2) is called g-H-contraction due to Miheţ [9]. Thus, our definition (g,φ)-contraction is a generalization of the Goleţ and Miheţ's type contraction principle in the fuzzy settings.Definition 2.3.
Let(X,M,*) be a fuzzy metric space and let φ∈Φ. Let f,g, and T are three mappings defined on (X,M,*) with values into itself and let one take T as asymptotically regular at x∈X. Then f is called fuzzy asymptotically (g,φ)-contraction with respect to T if
(2.3)M(gTnx,gTn+1x,t)>1-t⟹M(fTnx,fTn+1x,φ(t))>1-φ(t).
Note that, if φ(t)=kt for k∈(0,1),t>0, then (2.3) is asymptotically g-contraction with respect to T in the sense of Binayak S. Choudhury.Lemma 2.4.
Letf satisfy the condition given in Definition 2.2. Then g-1f is a continuous mapping on M(gx,gy,t) with values into itself.Proof.
Let{xn} be a sequence in X such that xn→x in X under the fuzzy metric (X,Mg,*); this implies that M(g(xn),g(x),t)=Mg(xn,x,t)→1 as n→∞ for all t>0. By definition, it follows that
(2.4)M(f(xn),f(x),φ(t))⟶1,
as n→∞ for all t>0, which implies that
(2.5)M(gg-1f(xn),gg-1f(x),φ(t))⟶1,
as n→∞ for all t>0, which implies that,
(2.6)Mg(g-1f(xn),g-1f(x),φ(t))⟶1,
as n→∞ for all t>0. This shows that g-1f is continuous mapping on (X,Mg,*) with values into itself.Theorem 2.5.
Letf,g,T be three mappings defined on complete fuzzy metric space (X,M,*) with values into itself where g is bijective, T is asymptotically regular at x∈X, and f is fuzzy asymptotically (g,φ)-contraction with respect to T with fT=Tf,gT=Tg at x∈X, and g-1fT is continuous in X. Then fTx=gx.Proof.
Sincef is fuzzy asymptotically (g,φ)-contraction with respect to T, we get for t>0,
(2.7)M(gTnx,gTn+1x,t)>1-t⟹M(fTnx,fTn+1x,φ(t))>1-φ(t),
where t>0, this implies that
(2.8)Mg(g-1fTTn-1x,g-1fTTnx,φ(t))>1-φ(t)
which implies that
(2.9)Mg(αTn-1x,αTnx,φ(t))>1-φ(t),
where α=g-1fT; this implies that
(2.10)M(fαTn-1x,fαTnx,φ2(t))>1-φ2(t),
which implies that
(2.11)M(fg-1fTTTn-2x,fg-1fTTTn-1x,φ2(t))>1-φ2(t),
which implies that
(2.12)M(gg-1fTg-1fTTn-2x,gg-1fTg-1fTTn-1x,φ2(t))>1-φ2(t),
which implies that
(2.13)Mg(α2Tn-2x,α2Tn-1x,φ2(t))>1-φ2(t).
Continuing this process, we get
(2.14)Mg(αnx,αnTx,φn(t))>1-φn(t).
For every ϵ>0 and λ∈(0,1) there exists n0∈ℕ such that φn(t)≤min⁡⁡(ϵ,λ) whenever n≥n0. Thus, we have
(2.15)Mg(αnx,αnTx,ϵ)≥Mg(αnx,αnTx,φn(t))>1-φn(t)>1-λ,
which implies that Mg(αnx,αnTx,ϵ)>1-λ. Let {xn} be the sequence defined as xn+1=αxn. Now, taking Tx=xm-n and x=x0, then we get from the above inequality
(2.16)Mg(xn,xm,ϵ)>1-λ,
for every n,m≥n0. This implies that {xn} is a fuzzy Cauchy sequence in (X,M,*). Since (X,M,*) is complete fuzzy metric space, then also (X,Mg,*) is complete fuzzy metric space, there exists z∈X such that xn→z (n→∞) under Mg. Again, α is continuous on (X,Mg,*), it follows that αx=x, which implies that fTx=gx.



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*Source: 832807-2012-05-22.xml*