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

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.

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], Golet 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, 𝑀=𝜖𝑥,𝑦,𝜖.𝜖+𝑑𝑥,𝑦 (1.1) 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 𝑎∗𝑎≥𝑎forevery𝑎∈[0,1] (1.2) holds if and only if 𝑎∗𝑏=min{𝑎,𝑏}forevery𝑎,𝑏∈[0,1]. (1.3)

Proof. Suppose (1.2) holds. Let 𝑎,𝑏∈[0,1] such that 𝑎≤𝑏. Then min{𝑎,𝑏}=𝑎=𝑎∗1≥𝑎∗𝑏≥𝑎∗𝑎≥𝑎=min{𝑎,𝑏}. (1.4) Therefore, for all 𝑎,𝑏∈[0,1], we have 𝑎∗𝑏=min{𝑎,𝑏}. Suppose (1.3) holds. Then clearly 𝑎∗𝑎=min{𝑎,𝑎}=𝑎 (1.5) and so (1.2) holds.

Definition 1.5 (see [8]). Let (𝑋,𝑀,∗) be a fuzzy metric space: a sequence {𝑥_𝑛} in 𝑋 is said to be convergent to a point 𝑥∈𝑋 (denoted 𝑥_𝑛→𝑥) if lim_𝑛→∞𝑀𝑥_𝑛,𝑥,𝑡=1foreach𝑡>0. (1.6)

Definition 1.6 (see [21]). Let (𝑋,𝑀,∗) be a fuzzy metric space: a sequence {𝑥_𝑛} in 𝑋 is called a Cauchy sequence if and only if for any 𝜖>0,𝑡>0, there exists 𝑛₀∈ℕ such that 𝑀𝑥_𝑛,𝑥_𝑚,𝑡>1−𝜖 (1.7) for all 𝑛,𝑚≥𝑛₀.

Definition 1.7 (see [8]). Let (𝑋,𝑀,∗) be a fuzzy metric space: a sequence {𝑥_𝑛} in 𝑋 is called a Cauchy sequence if lim_𝑛→∞𝑀𝑥_𝑛+𝑝,𝑥_𝑛,𝑡=1foreach𝑡>0,𝑝>0. (1.8)

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 𝑀₁𝑥,𝑦,𝑠>1−𝛿⟹𝑀₂𝑓(𝑥),𝑓𝑦,𝑡>1−𝜖, (1.9) 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 lim_𝑛→∞𝑀𝑇^𝑛𝑥,𝑇^𝑛+1𝑥,𝑡=1. (1.10)

2. Main Results

Let Φ 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 all 𝑡>0,lim_𝑛→∞𝜑^𝑛(𝑡)=0, where 𝜑^𝑛 is the 𝑛-iteration of 𝜑.

Proof. Suppose that lim_𝑛→∞𝜑^𝑛(𝑡₀)=𝑙<0 for some 𝑡₀∈(0,∞). By the monotonicity and right continuity of 𝜑, we have 𝑙=lim_𝑛→∞𝜑^𝑛+1𝑡₀=𝜑lim_𝑛→∞𝜑^𝑛𝑡₀=𝜑(𝑙)<𝑙, (2.1) 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: 𝑀𝑔(𝑥),𝑔𝑦,𝑡>1−𝑡⟹𝑀𝑓(𝑥),𝑓𝑦,𝜑(𝑡)>1−𝜑(𝑡). (2.2)

Note that, if 𝜑(𝑡)=𝑘𝑡 for 𝑘∈(0,1),𝑡>0, then (2.2) is a 𝑔-contraction in the sense of Golet [10]. On the other hand, if 𝑔 is an identity function, then (2.2) is called 𝑔-𝐻-contraction due to Mihet [9]. Thus, our definition (𝑔,𝜑)-contraction is a generalization of the Golet and Mihet'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 𝑀𝑔𝑇^𝑛𝑥,𝑔𝑇^𝑛+1𝑥,𝑡>1−𝑡⟹𝑀𝑓𝑇^𝑛𝑥,𝑓𝑇^𝑛+1𝑥,𝜑(𝑡)>1−𝜑(𝑡). (2.3) 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. Let 𝑓 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 {𝑥_𝑛} be a sequence in 𝑋 such that 𝑥_𝑛→𝑥 in 𝑋 under the fuzzy metric (𝑋,𝑀^𝑔,∗); this implies that 𝑀(𝑔(𝑥_𝑛),g(𝑥),𝑡)=𝑀^𝑔(𝑥_𝑛,𝑥,𝑡)→1 as 𝑛→∞ for all 𝑡>0. By definition, it follows that 𝑀𝑓𝑥_𝑛,𝑓(𝑥),𝜑(𝑡)⟶1, (2.4) as 𝑛→∞ for all 𝑡>0, which implies that 𝑀𝑔𝑔⁻¹𝑓𝑥_𝑛,𝑔𝑔⁻¹𝑓(𝑥),𝜑(𝑡)⟶1, (2.5) as 𝑛→∞ for all 𝑡>0, which implies that, 𝑀^𝑔𝑔⁻¹𝑓𝑥_𝑛,𝑔⁻¹𝑓(𝑥),𝜑(𝑡)⟶1, (2.6) as 𝑛→∞ for all 𝑡>0. This shows that 𝑔⁻¹𝑓 is continuous mapping on (𝑋,𝑀^𝑔,∗) with values into itself.

Theorem 2.5. Let 𝑓,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. Since 𝑓 is fuzzy asymptotically (𝑔,𝜑)-contraction with respect to 𝑇, we get for 𝑡>0, 𝑀𝑔𝑇^𝑛𝑥,𝑔𝑇^𝑛+1𝑥,𝑡>1−𝑡⟹𝑀𝑓𝑇^𝑛𝑥,𝑓𝑇^𝑛+1𝑥,𝜑(𝑡)>1−𝜑(𝑡), (2.7) where 𝑡>0, this implies that 𝑀^𝑔𝑔⁻¹𝑓𝑇𝑇^𝑛−1𝑥,𝑔⁻¹𝑓𝑇𝑇^𝑛𝑥,𝜑(𝑡)>1−𝜑(𝑡) (2.8) which implies that 𝑀^𝑔𝛼𝑇^𝑛−1𝑥,𝛼𝑇^𝑛𝑥,𝜑(𝑡)>1−𝜑(𝑡), (2.9) where 𝛼=𝑔⁻¹𝑓𝑇; this implies that 𝑀𝑓𝛼𝑇^𝑛−1𝑥,𝑓𝛼𝑇^𝑛𝑥,𝜑²(𝑡)>1−𝜑²(𝑡), (2.10) which implies that 𝑀𝑓𝑔⁻¹𝑓𝑇𝑇𝑇^𝑛−2𝑥,𝑓𝑔⁻¹𝑓𝑇𝑇𝑇^𝑛−1𝑥,𝜑²(𝑡)>1−𝜑²(𝑡), (2.11) which implies that 𝑀𝑔𝑔⁻¹𝑓𝑇𝑔⁻¹𝑓𝑇𝑇^𝑛−2𝑥,𝑔𝑔⁻¹𝑓𝑇𝑔⁻¹𝑓𝑇𝑇^𝑛−1𝑥,𝜑²(𝑡)>1−𝜑²(𝑡), (2.12) which implies that 𝑀^𝑔𝛼²𝑇^𝑛−2𝑥,𝛼²𝑇^𝑛−1𝑥,𝜑²(𝑡)>1−𝜑²(𝑡). (2.13) Continuing this process, we get 𝑀^𝑔𝛼^𝑛𝑥,𝛼^𝑛𝑇𝑥,𝜑^𝑛(𝑡)>1−𝜑^𝑛(𝑡). (2.14) For every 𝜖>0 and 𝜆∈(0,1) there exists 𝑛₀∈ℕ such that 𝜑^𝑛(𝑡)≤min(𝜖,𝜆) whenever 𝑛≥𝑛₀. Thus, we have 𝑀^𝑔𝛼^𝑛𝑥,𝛼^𝑛𝑇𝑥,𝜖≥𝑀^𝑔𝛼^𝑛𝑥,𝛼^𝑛𝑇𝑥,𝜑^𝑛(𝑡)>1−𝜑^𝑛(𝑡)>1−𝜆, (2.15) which implies that 𝑀^𝑔(𝛼^𝑛𝑥,𝛼^𝑛𝑇𝑥,𝜖)>1−𝜆. Let {𝑥_𝑛} be the sequence defined as 𝑥_𝑛+1=𝛼𝑥_𝑛. Now, taking 𝑇𝑥=𝑥_𝑚−𝑛 and 𝑥=𝑥₀, then we get from the above inequality 𝑀^𝑔𝑥_𝑛,𝑥_𝑚,𝜖>1−𝜆, (2.16) for every 𝑛,𝑚≥𝑛₀. This implies that {𝑥_𝑛} is a fuzzy Cauchy sequence in (𝑋,𝑀,∗). Since (𝑋,𝑀,∗) is complete fuzzy metric space, then also (𝑋,𝑀^𝑔,∗) is complete fuzzy metric space, there exists 𝑧∈𝑋 such that 𝑥_𝑛→𝑧 (𝑛→∞) under 𝑀^𝑔. Again, 𝛼 is continuous on (𝑋,𝑀^𝑔,∗), it follows that 𝛼𝑥=𝑥, which implies that 𝑓𝑇𝑥=𝑔𝑥.