A Coupled Fixed Point Theorem in Fuzzy Metric Space Satisfying 𝜙-Contractive Condition

Abstract

The intent of this paper is to prove a coupled fixed point theorem for two pairs of compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous) mappings, satisfying 𝜙-contractive conditions in a fuzzy metric space. We also furnish some illustrative examples to support our results.

1. Introduction

The evolution of fuzzy mathematics commenced with the introduction of the notion of fuzzy sets by Zadeh [1], where the concept of uncertainty was introduced in the theory of sets, in a nonprobabilistic manner. Fuzzy set theory has applications in applied sciences such as mathematical programming, model theory, engineering sciences, image processing, and control theory. In 1975, Kramosil and Michalek [2] introduced the concept of fuzzy metric space as a generalization of the statistical (probabilistic) metric space. Afterwards, Grabiec [3] defined the completeness of the fuzzy metric space and extended the Banach contraction principle to fuzzy metric spaces. Since then, many authors contributed to the development of this theory, also in relation to fixed point theory (e.g., [4–9]).

Mishra et al. [10] extended the notion of compatible mappings (introduced by Jungck [11] in metric spaces) to fuzzy metric spaces and proved common fixed point theorems in presence of continuity of at least one of the mappings, completeness of the underlying space, and containment of the ranges amongst involved mappings. Further, Singh and Jain [12] weakened the notion of compatibility by using the notion of weakly compatible, mappings in fuzzy metric spaces and showed that every pair of compatible mappings is weakly compatible but converse is not true. Inspired by Bouhadjera and Godet-Thobie [13, 14], Gopal and Imdad [15] extended the notions of subcompatibility and subsequential continuity to fuzzy metric spaces and proved fixed point theorems using these notions together due to Imdad et al. [16]. In recent past, several authors proved various fixed point theorems employing more general contractive conditions (e.g., [17–26]).

On the other hand, Bhaskar and Lakshmikantham [27] and Lakshmikantham and Ćirić [28] gave some coupled fixed point theorems in partially ordered metric spaces (see also [29–31]). In 2010, Sedghi et al. [32] proved common coupled fixed point theorems in fuzzy metric spaces for commuting mappings. Motivated by the results of [33], Hu [34] proved a coupled fixed point theorem for compatible mappings satisfying 𝜙-contractive conditions in fuzzy metric spaces with continuous t-norm of H-type and generalized the result of Sedghi et al. [32]. In an interesting note, Zhu and Xiao [35] showed that the results contained in Sedghi et al. [32] are not true in their present form.

Inspired by the work of Zhu and Xiao [35], we prove coupled common fixed point theorems for two pairs of mappings satisfying a general contractive condition in fuzzy metric spaces, by using the notions of compatibility and subsequential continuity (alternately subcompatibility and reciprocal continuity). Our results improve many known common coupled fixed point theorems available in the existing literature. We support our results with two illustrative examples.

2. Preliminaries

In this section, we collect some basic notions and results. In the sequel ℝ⁺ will denote the set of all positive real numbers while ℕ will denote the set of natural numbers.

Definition 1 (see [1]). Let 𝑋 be any set. A fuzzy set 𝐴 in 𝑋 is a function with domain 𝑋 and values in [0,1].

Definition 2 (see [36]). A binary operation ∗∶[0,1]×[0,1]→[0,1] is a continuous 𝑡-norm if ∗ satisfies the following conditions: (a)∗ is commutative and associative; (b)∗ is continuous; (c)𝑎∗1=𝑎 for all 𝑎∈[0,1]; (d)𝑎∗𝑏≤𝑐∗𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑 for all 𝑎,𝑏,𝑐,𝑑∈[0,1].

Definition 3 (see [37]). One says that a 𝑡-norm ∗ is of H-type if the family {∗^𝑛} of its iterates is equicontinuous at 𝑥=1; that is, for any 𝜆∈(0,1), there exists 𝛿(𝜆)∈(0,1) such that 𝑥>1−𝛿 implies ∗^𝑛(𝑥)>1−𝜆, for all 𝑛∈ℕ.

The 𝑡-norm ∗_𝑚=min{𝑎,𝑏} for all 𝑎,𝑏∈[0,1] is an example of 𝑡-norm of H-type, but there are some other 𝑡-norms ∗ of H-type (see [37]).

Definition 4 (see [2]). A 3-tuple (𝑋,𝑀,∗) is said to be a fuzzy metric space if 𝑋 is an arbitrary nonempty set, ∗ is a continuous 𝑡-norm, and 𝑀 is a fuzzy set in 𝑋²×(0,+∞) satisfying the following conditions, for each 𝑥,𝑦,𝑧∈𝑋 and 𝑡,𝑠>0: (a)𝑀(𝑥,𝑦,𝑡)>0; (b)𝑀(𝑥,𝑦,𝑡)=1 for all 𝑡>0 if and only if 𝑥=𝑦; (c)𝑀(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡); (d)𝑀(𝑥,𝑦,𝑡)∗𝑀(𝑦,𝑧,𝑠)≤𝑀(𝑥,𝑧,𝑡+𝑠); (e)𝑀(𝑥,𝑦,⋅)∶(0,∞)→(0,1] is continuous.

Example 5 (see [7]). Let (𝑋,𝑑) be a metric space. Define the 𝑡-norm 𝑎∗𝑏=𝑎𝑏 for all 𝑎,𝑏∈[0,1], and, for all 𝑥,𝑦∈𝑋 and 𝑡>0, 𝑀=𝑡𝑥,𝑦,𝑡.𝑡+𝑑𝑥,𝑦 (1) Then (𝑋,𝑀,∗) is a fuzzy metric space, and the fuzzy metric 𝑀 induced by the metric 𝑑 is often referred, as the standard fuzzy metric.

Example 6 (see [32]). Let (𝑋,𝑑) be a metric space and 𝜓 be an increasing and continuous function from ℝ⁺ into (0,1) such that lim_𝑠→∞𝜓(𝑠)=1. Four typical examples of these functions are 𝜓(𝑠)=𝑠/(𝑠+1), 𝜓(𝑠)=sin(𝜋𝑠/(2𝑠+1)), 𝜓(𝑠)=1−𝑒^−𝑠, and 𝜓(𝑠)=𝑒^−1/𝑠. Let 𝑎∗𝑏=𝑎𝑏 for all 𝑎,𝑏∈[0,1], and, for each 𝑥,𝑦∈𝑋 and 𝑡>0, define 𝑀𝑥,𝑦,𝑡=𝜓(𝑡)^{𝑑(𝑥,𝑦)}. (2) It is easy to see that (𝑋,𝑀,∗) is a fuzzy metric space.

Definition 7 (see [34]). Define Φ={𝜙∶ℝ⁺→ℝ⁺} such that 𝜙∈Φ satisfies the following conditions: (𝜙-1)𝜙 is nondecreasing; (𝜙-2)𝜙 is upper semicontinuous from the right; (𝜙-3)∑^∞_𝑛=0𝜙^𝑛(𝑠)<+∞ for all 𝑠>0, where 𝜙^𝑛+1(𝑠)=𝜙(𝜙^𝑛(𝑠)), 𝑛∈ℕ.

Clearly if 𝜙∈Φ, then 𝜙(𝑠)<𝑠 for all 𝑠>0.

Definition 8 (see [27]). An element (𝑥,𝑦)∈𝑋×𝑋 is called (a)a coupled fixed point of the mapping 𝑓∶𝑋×𝑋→𝑋 if 𝑓𝑥,𝑦=𝑥,𝑓𝑦,𝑥=𝑦; (3)(b)a coupled coincidence point of the mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 if 𝑓𝑥,𝑦=𝑔(𝑥),𝑓𝑦,𝑥=𝑔𝑦; (4)(c)a common coupled fixed point of the mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 if (𝑥),𝑦.𝑥=𝑓𝑥,𝑦=𝑔𝑦=𝑓𝑦,𝑥=𝑔 (5)

Definition 9 (see [27]). An element 𝑥∈𝑋 is called a common fixed point of the mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 if 𝑥=𝑔(𝑥)=𝑓(𝑥,𝑥). (6)

Definition 10 (see [34]). The mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are called compatible if lim_𝑛→∞𝑀𝑔𝑓𝑥_𝑛,𝑦_𝑛,𝑓𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑡=1,lim_𝑛→∞𝑀𝑔𝑓𝑦_𝑛,𝑥_𝑛,𝑓𝑔𝑦_𝑛,𝑔𝑥_𝑛,𝑡=1, (7) for all 𝑡>0, whenever {𝑥_𝑛} and {𝑦_𝑛} are sequences in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝛼,lim_𝑛→∞𝑓𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=𝛽, (8) for some 𝛼,𝛽∈𝑋.

Now we introduce the following notions.

Definition 11. The mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are said to be reciprocally continuous if, for sequences {𝑥_𝑛}, {𝑦_𝑛} in 𝑋, one has lim_𝑛→∞𝑓𝑔𝑥_𝑛,𝑔𝑦_𝑛=𝑓𝛼,𝛽,lim_𝑛→∞𝑔𝑓𝑥_𝑛,𝑦_𝑛=𝑔𝛼,lim_𝑛→∞𝑓𝑔𝑦_𝑛,𝑔𝑥_𝑛=𝑓𝛽,𝛼,lim_𝑛→∞𝑔𝑓𝑦_𝑛,𝑥_𝑛=𝑔𝛽, (9) whenever lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝛼,lim_𝑛→∞𝑓𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=𝛽, (10) for some 𝛼,𝛽∈𝑋.

If two self-mappings are continuous, then they are obviously reciprocally continuous, but the converse is not true. Moreover, in the setting of common fixed point theorems for compatible pairs of self mappings satisfying contractive conditions, continuity of one of the mappings implies their reciprocal continuity but not conversely (see [38]).

Definition 12. The mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are said to be subsequentially continuous if and only if there exist sequences {𝑥_𝑛}, {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝛼,lim_𝑛→∞𝑓𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=𝛽, (11) for some 𝛼,𝛽∈𝑋, and lim_𝑛→∞𝑓𝑔𝑥_𝑛,𝑔𝑦_𝑛=𝑓𝛼,𝛽,lim_𝑛→∞𝑔𝑓𝑥_𝑛,𝑦_𝑛=𝑔𝛼,lim_𝑛→∞𝑓𝑔𝑦_𝑛,𝑔𝑥_𝑛=𝑓𝛽,𝛼,lim_𝑛→∞𝑔𝑓𝑦_𝑛,𝑥_𝑛=𝑔𝛽. (12)

One can easily check that if two self mappings 𝑓 and 𝑔 are both continuous, hence also reciprocally continuous mappings but 𝑓 and 𝑔 are not sub-sequentially continuous (see [38, Example 1]).

Definition 13. The mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are said to be subcompatible if and only if there exist sequences {𝑥_𝑛}, {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝛼,lim_𝑛→∞𝑓𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=𝛽, (13) for some 𝛼,𝛽∈𝑋, and lim_𝑛→∞𝑀𝑓𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑓𝑥_𝑛,𝑦_𝑛,𝑡=1,lim_𝑛→∞𝑀𝑓𝑔𝑦_𝑛,𝑔𝑥_𝑛,𝑔𝑓𝑦_𝑛,𝑥_𝑛,𝑡=1, (14) for all 𝑡>0.

3. Results

In this section, we state and prove our fixed point results.

Theorem 14. Let (𝑋,𝑀,∗) be a fuzzy metric space, where ∗ is a continuous t-norm of H-type such that 𝑀(𝑥,𝑦,𝑡)→1 as 𝑡→∞, for all 𝑥,𝑦∈𝑋. Let 𝐴,𝐵∶𝑋×𝑋→𝑋 and 𝑆,𝑇∶𝑋→𝑋 be four mappings such that(a)the pairs (𝐴,𝑆) and (𝐵,𝑇) are compatible and subsequentially continuous; (b)there exists 𝜙∈Φ such that 𝑀𝐴()(𝑡)(),𝑥,𝑦,𝐵𝑢,𝑣,𝜙≥𝑀𝑆𝑥,𝑇𝑢,𝑡∗𝑀𝑆𝑦,𝑇𝑣,𝑡 (15) for all 𝑥,𝑦,𝑢,𝑣∈𝑋 and 𝑡>0. Then there exists a unique point 𝛼 in 𝑋 such that 𝛼=𝑆𝛼=𝑇𝛼=𝐴(𝛼,𝛼)=𝐵(𝛼,𝛼).

Proof. Since the mappings 𝐴 and 𝑆 are subsequentially continuous and compatible, there exist sequences {𝑥_𝑛}, {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝛼,lim_𝑛→∞𝐴𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝑆𝑦_𝑛=𝛽, (16) for all 𝛼,𝛽∈𝑋, and lim_𝑛→∞𝑀𝐴𝑆𝑥_𝑛,𝑆𝑦_𝑛,𝑆𝐴𝑥_𝑛,𝑦_𝑛,𝑡=1,lim_𝑛→∞𝑀𝐴𝑆𝑦_𝑛,𝑆𝑥_𝑛,𝑆𝐴𝑦_𝑛,𝑥_𝑛,𝑡=1, (17) that is 𝐴(𝛼,𝛽)=𝑆𝛼 and 𝐴(𝛽,𝛼)=𝑆𝛽. Similarly, with respect to the pair (𝐵,𝑇), there exist sequences {𝑥^′_𝑛}, {𝑦^′_𝑛} in 𝑋 such that lim_𝑛→∞𝐵𝑥^′_𝑛,𝑦^′_𝑛=lim_𝑛→∞𝑇𝑥^′_𝑛=𝛼^′,lim_𝑛→∞𝐵𝑦^′_𝑛,𝑥^′_𝑛=lim_𝑛→∞𝑇𝑦^′_𝑛=𝛽^′, (18) for all 𝛼^′,𝛽^′∈𝑋, and lim_𝑛→∞𝑀𝐵𝑇𝑥^′_𝑛,𝑇𝑦^′_𝑛,𝑇𝐵𝑥^′_𝑛,𝑦^′_𝑛,𝑡=1,lim_𝑛→∞𝑀𝐵𝑇𝑦^′_𝑛,𝑇𝑥^′_𝑛,𝑇𝐵𝑦^′_𝑛,𝑥^′_𝑛,𝑡=1, (19) that is 𝐵(𝛼^′,𝛽^′)=𝑇𝛼^′ and 𝐵(𝛽^′,𝛼^′)=𝑇𝛽^′. Hence (𝛼,𝛽)∈𝑋×𝑋 is a coupled coincidence point of the pair (𝐴,𝑆), whereas (𝛼^′,𝛽^′)∈𝑋×𝑋 is a coupled coincidence point of the pair (𝐵,𝑇).
Now we assert that (𝛼,𝛽)=(𝛼^′,𝛽^′), that is, 𝛼=𝛼^′ and 𝛽=𝛽^′. Since ∗ is a 𝑡-norm of H-type, for any 𝜆>0, there exists an 𝜇>0 such that (1−𝜇)∗(1−𝜇)∗⋯∗(1−𝜇)_𝑝≥1−𝜆, (20) for all 𝑝∈ℕ.
Since 𝑀(𝑥,𝑦,⋅) is continuous and lim_𝑡→∞𝑀(𝑥,𝑦,𝑡)=1 for all 𝑥,𝑦∈𝑋, there exists 𝑡₀>0 such that 𝑀(𝛼,𝛼^′,𝑡₀)≥1−𝜇 and 𝑀(𝛽,𝛽^′,𝑡₀)≥1−𝜇.
On the other hand, since 𝜙∈Φ, by condition (𝜙-3), we have ∑^∞_𝑛=1𝜙^𝑛(𝑡₀)<∞. Then for any 𝑡>0, there exists 𝑛₀∈ℕ such that 𝑡>∑^∞_𝑝=𝑛0𝜙^𝑝(𝑡₀). On using inequality (15) with 𝑥=𝑥_𝑛,  𝑦=𝑦_𝑛,  𝑢=𝑥^′_𝑛,  and 𝑣=𝑦^′_𝑛, we have 𝑀𝐴𝑥_𝑛,𝑦_𝑛,𝐵𝑥^′_𝑛,𝑦^′_𝑛,𝜙𝑡₀≥𝑀𝑆𝑥_𝑛,𝑇𝑥^′_𝑛,𝑡₀∗𝑀𝑆𝑦_𝑛,𝑇𝑦^′_𝑛,𝑡₀. (21) Letting 𝑛→∞, we get 𝑀𝛼,𝛼^′,𝜙𝑡₀≥𝑀𝛼,𝛼^′,𝑡₀∗𝑀𝛽,𝛽^′,𝑡₀. (22) Again using inequality (15) with 𝑥=𝑦_𝑛, 𝑦=𝑥_𝑛, 𝑢=𝑦^′_𝑛, and 𝑣=𝑥^′_𝑛, we have 𝑀𝐴𝑦_𝑛,𝑥_𝑛,𝐵𝑦^′_𝑛,𝑥^′_𝑛,𝜙𝑡₀≥𝑀𝑆𝑦_𝑛,𝑇𝑦^′_𝑛,𝑡₀∗𝑀𝑆𝑥_𝑛,𝑇𝑥^′_𝑛,𝑡₀. (23) Letting 𝑛→∞, we get 𝑀𝛽,𝛽^′,𝜙𝑡₀≥𝑀𝛽,𝛽^′,𝑡₀∗𝑀𝛼,𝛼^′,𝑡₀. (24) From (22) and (24), we obtain 𝑀𝛼,𝛼^′,𝜙𝑡₀∗𝑀𝛽,𝛽^′,𝜙𝑡₀≥𝑀𝛼,𝛼^′,𝑡₀²∗𝑀𝛽,𝛽^′,𝑡₀². (25) In general, for all 𝑛∈ℕ, we have 𝑀𝛼,𝛼^′,𝜙^𝑛𝑡₀∗𝑀𝛽,𝛽^′,𝜙^𝑛𝑡₀≥𝑀𝛼,𝛼^′,𝜙^𝑛−1𝑡₀²∗𝑀𝛽,𝛽^′,𝜙^𝑛−1𝑡₀²≥𝑀𝛼,𝛼^′,𝑡₀^2𝑛∗𝑀𝛽,𝛽^′,𝑡₀^2𝑛. (26) Then, we have 𝑀𝛼,𝛼^′,𝑡∗𝑀𝛽,𝛽^′≥𝑀,𝑡𝛼,𝛼^′,^∞_𝑝=𝑛0𝜙^𝑝𝑡₀∗𝑀𝛽,𝛽^′,^∞_𝑝=𝑛0𝜙^𝑝𝑡₀≥𝑀𝛼,𝛼^′,𝜙^𝑛0𝑡₀∗𝑀𝛽,𝛽^′,𝜙^𝑛0𝑡₀≥𝑀𝛼,𝛼^′,𝑡₀^2𝑛0∗𝑀𝛽,𝛽^′,𝑡₀^2𝑛0≥(1−𝜇)∗(1−𝜇)∗⋯∗(1−𝜇)_202𝑛≥1−𝜆. (27) So for any 𝜆>0, we have 𝑀𝛼,𝛼^′,𝑡∗𝑀𝛽,𝛽^′,𝑡≥1−𝜆, (28) for all 𝑡>0, and so 𝛼=𝛼^′ and 𝛽=𝛽^′. Therefore we have 𝐴𝐵𝛼,𝛽=𝑆𝛼,𝐴𝛽,𝛼=𝑆𝛽,𝛼,𝛽=𝑇𝛼,𝐵𝛽,𝛼=𝑇𝛽. (29) Next, we show that 𝑆𝛼=𝑇𝛼 and 𝑆𝛽=𝑇𝛽. Since ∗ is a 𝑡-norm of H-type, for any 𝜆>0, there exists an 𝜇>0 such that 1−𝜇∗1−𝜇∗⋯∗(1−𝜇)_𝑝≥1−𝜆, (30) for all 𝑝∈ℕ.
Since 𝑀(𝑥,𝑦,⋅) is continuous and lim_𝑡→+∞𝑀(𝑥,𝑦,𝑡)=1 for all 𝑥,𝑦∈𝑋, there exists 𝑡₀>0 such that 𝑀(𝑆𝛼,𝑇𝛼,𝑡₀)≥1−𝜇 and 𝑀(𝑆𝛽,𝑇𝛽,𝑡₀)≥1−𝜇.
Since 𝜙∈Φ, by condition (𝜙-3), we have ∑^∞_𝑛=1𝜙^𝑛(𝑡₀)<∞. Then for any 𝑡>0, there exists 𝑛₀∈ℕ such that 𝑡>∑^∞_𝑝=𝑛0𝜙^𝑝(𝑡₀). On using inequality (15) with 𝑥=𝑢=𝛼, 𝑦=𝑣=𝛽, we have 𝑀𝐴𝛼,𝛽,𝐵𝛼,𝛽,𝜙𝑡₀≥𝑀𝑆𝛼,𝑇𝛼,𝑡₀∗𝑀𝑆𝛽,𝑇𝛽,𝑡₀, (31) and so 𝑀𝑆𝛼,𝑇𝛼,𝜙𝑡₀≥𝑀𝑆𝛼,𝑇𝛼,𝑡₀∗𝑀𝑆𝛽,𝑇𝛽,𝑡₀. (32) Similarly, we can obtain 𝑀𝑆𝛽,𝑇𝛽,𝜙𝑡₀≥𝑀𝑆𝛽,𝑇𝛽,𝑡₀∗𝑀𝑆𝛼,𝑇𝛼,𝑡₀. (33) From (32) and (33), we have 𝑀𝑆𝛼,𝑇𝛼,𝜙𝑡₀∗𝑀𝑆𝛽,𝑇𝛽,𝜙𝑡₀≥𝑀𝑆𝛼,𝑇𝛼,𝑡₀²∗𝑀𝑆𝛽,𝑇𝛽,𝑡₀². (34) In general, for all 𝑛∈ℕ, we get 𝑀𝑆𝛼,𝑇𝛼,𝜙^𝑛𝑡₀∗𝑀𝑆𝛽,𝑇𝛽,𝜙^𝑛𝑡₀≥𝑀𝑆𝛼,𝑇𝛼,𝜙^𝑛−1𝑡₀²∗𝑀𝑆𝛽,𝑇𝛽,𝜙^𝑛−1𝑡₀²≥𝑀𝑆𝛼,𝑇𝛼,𝑡₀^2𝑛∗𝑀𝑆𝛽,𝑇𝛽,𝑡₀^2𝑛. (35) Then, we have 𝑀()≥𝑀𝑆𝛼,𝑇𝛼,𝑡∗𝑀𝑆𝛽,𝑇𝛽,𝑡𝑆𝛼,𝑇𝛼,^∞_𝑝=𝑛0𝜙^𝑝𝑡₀∗𝑀𝑆𝛽,𝑇𝛽,^∞_𝑝=𝑛0𝜙^𝑝𝑡₀≥𝑀𝑆𝛼,𝑇𝛼,𝜙^𝑛0𝑡₀∗𝑀𝑆𝛽,𝑇𝛽,𝜙^𝑛0𝑡₀≥𝑀𝑆𝛼,𝑇𝛼,𝑡₀^2𝑛0∗𝑀𝑆𝛽,𝑇𝛽,𝑡₀^2𝑛0≥1−𝜇∗1−𝜇∗⋯∗1−𝜇_202𝑛≥1−𝜆. (36) So for any 𝜆>0, we obtain 𝑀(𝑆𝛼,𝑇𝛼,𝑡)∗𝑀𝑆𝛽,𝑇𝛽,𝑡≥1−𝜆, (37) for all 𝑡>0, and hence 𝑆𝛼=𝑇𝛼 and 𝑆𝛽=𝑇𝛽. Therefore ,.𝑆𝛼=𝑇𝛼=𝐴𝛼,𝛽=𝐵𝛼,𝛽𝑆𝛽=𝑇𝛽=𝐴𝛽,𝛼=𝐵𝛽,𝛼 (38) Now we show that 𝑆𝛼=𝛼 and 𝑆𝛽=𝛽. Since ∗ is a 𝑡-norm of H-type, for any 𝜆>0, there exists an 𝜇>0 such that 1−𝜇∗1−𝜇∗⋯∗(1−𝜇)_𝑝≥1−𝜆, (39) for all 𝑝∈ℕ.
Since 𝑀(𝑥,𝑦,⋅) is continuous and lim_𝑡→+∞𝑀(𝑥,𝑦,𝑡)=1 for all 𝑥,𝑦∈𝑋, there exists 𝑡₀>0 such that 𝑀(𝑆𝛼,𝛼,𝑡₀)≥1−𝜇 and 𝑀(𝑆𝛽,𝛽,𝑡₀)≥1−𝜇.
On the other hand, since 𝜙∈Φ, by condition (𝜙-3) we have ∑^∞_𝑛=1𝜙^𝑛(𝑡₀)<∞. Then for any 𝑡>0, there exists 𝑛₀∈ℕ such that 𝑡>∑^∞_𝑝=𝑛0𝜙^𝑝(𝑡₀). On using inequality (15) with 𝑥=𝛼, 𝑦=𝛽, 𝑢=𝑥^′_𝑛, 𝑣=𝑦^′_𝑛, we have 𝑀𝐴𝛼,𝛽,𝐵𝑥^′_𝑛,𝑦^′_𝑛,𝜙𝑡₀≥𝑀𝑆𝛼,𝑇𝑥^′_𝑛,𝑡₀∗𝑀𝑆𝛽,𝑇𝑦^′_𝑛,𝑡₀. (40) Letting 𝑛→∞, we obtain 𝑀𝑆𝛼,𝛼,𝜙𝑡₀≥𝑀𝑆𝛼,𝛼,𝑡₀∗𝑀𝑆𝛽,𝛽,𝑡₀. (41) Similarly, we can get 𝑀𝑆𝛽,𝛽,𝜙𝑡₀≥𝑀𝑆𝛽,𝛽,𝑡₀∗𝑀𝑆𝛼,𝛼,𝑡₀. (42) Consequently, from (41) and (42), we have 𝑀𝑆𝛼,𝛼,𝜙𝑡₀∗𝑀𝑆𝛽,𝛽,𝜙𝑡₀≥𝑀𝑆𝛼,𝛼,𝑡₀²∗𝑀𝑆𝛽,𝛽,𝑡₀². (43) In general, for all 𝑛∈ℕ, we get 𝑀𝑆𝛼,𝛼,𝜙^𝑛𝑡₀∗𝑀𝑆𝛽,𝛽,𝜙^𝑛𝑡₀≥𝑀𝑆𝛼,𝛼,𝜙^𝑛−1𝑡₀²∗𝑀𝑆𝛽,𝛽,𝜙^𝑛−1𝑡₀²≥𝑀𝑆𝛼,𝛼,𝑡₀^2𝑛∗𝑀𝑆𝛽,𝛽,𝑡₀^2𝑛. (44) Then, we have 𝑀()≥𝑀𝑆𝛼,𝛼,𝑡∗𝑀𝑆𝛽,𝛽,𝑡𝑆𝛼,𝛼,^∞_𝑝=𝑛0𝜙^𝑝𝑡₀∗𝑀𝑆𝛽,𝛽,^∞_𝑝=𝑛0𝜙^𝑝𝑡₀≥𝑀𝑆𝛼,𝛼,𝜙^𝑛0𝑡₀∗𝑀𝑆𝛽,𝛽,𝜙^𝑛0𝑡₀≥𝑀𝑆𝛼,𝛼,𝑡₀^2𝑛0∗𝑀𝑆𝛽,𝛽,𝑡₀^2𝑛0≥1−𝜇∗1−𝜇∗⋯∗1−𝜇_202𝑛≥1−𝜆. (45) Therefore for any 𝜆>0, we obtain 𝑀(𝑆𝛼,𝛼,𝑡)∗𝑀𝑆𝛽,𝛽,𝑡≥1−𝜆, (46) for all 𝑡>0 and so 𝑆𝛼=𝛼 and 𝑆𝛽=𝛽. Thus ,.𝛼=𝑆𝛼=𝑇𝛼=𝐴𝛼,𝛽=𝐵𝛼,𝛽𝛽=𝑆𝛽=𝑇𝛽=𝐴𝛽,𝛼=𝐵𝛽,𝛼 (47) Finally, we assert that 𝛼=𝛽. Since ∗ is a 𝑡-norm of H-type, for any 𝜆>0, there exists an 𝜇>0 such that 1−𝜇∗1−𝜇∗⋯∗(1−𝜇)_𝑝≥1−𝜆, (48) for all 𝑝∈ℕ.
Since 𝑀(𝑥,𝑦,⋅) is continuous and lim_𝑡→+∞𝑀(𝑥,𝑦,𝑡)=1 for all 𝑥,𝑦∈𝑋, there exists 𝑡₀>0 such that 𝑀(𝛼,𝛽,𝑡₀)≥1−𝜇.
Also, since 𝜙∈Φ, by condition (𝜙-3), we have ∑^∞_𝑛=1𝜙^𝑛(𝑡₀)<∞. Then for any 𝑡>0, there exists 𝑛₀∈ℕ such that  𝑡>∑^∞_𝑝=𝑛0𝜙^𝑝(𝑡₀). On using inequality (15) with 𝑥=𝑣=𝛼, 𝑦=𝑢=𝛽, we have 𝑀𝐴𝛼,𝛽,𝐵𝛽,𝛼,𝜙𝑡₀≥𝑀𝑆𝛼,𝑇𝛽,𝑡₀∗𝑀𝑆𝛽,𝑇𝛼,𝑡₀, (49) and so 𝑀𝛼,𝛽,𝜙𝑡₀≥𝑀𝛼,𝛽,𝑡₀∗𝑀𝛽,𝛼,𝑡₀. (50) Thus we have 𝑀𝛼,𝛽,𝑡≥𝑀𝛼,𝛽,^∞_𝑝=𝑛0𝜙^𝑝𝑡₀≥𝑀𝛼,𝛽,𝜙^𝑛0𝑡₀≥𝑀𝛼,𝛽,𝑡₀^2𝑛0∗𝑀𝛽,𝛼,𝑡₀^2𝑛0≥(1−𝜇)∗(1−𝜇)∗⋯∗(1−𝜇)_202𝑛≥1−𝜆, (51) which implies that 𝛼=𝛽. Therefore, we proved that there exists 𝛼 in 𝑋 such that 𝛼=𝑆𝛼=𝑇𝛼=𝐴(𝛼,𝛼)=𝐵(𝛼,𝛼). (52) The uniqueness of such a point follows immediately from inequality (15) and so we omit the details.

Remark 15. The conclusion of Theorem 14 remains true if we substitute condition (a) with the following condition: (a′)the pairs (𝐴,𝑆) and (𝐵,𝑇) are subcompatible and reciprocally continuous.

From Theorem 14, taking 𝐴=𝐵 and 𝑆=𝑇, we deduce the following natural result.

Corollary 16. Let (𝑋,𝑀,∗) be a fuzzy metric space, where ∗ is a continuous t-norm of H-type such that 𝑀(𝑥,𝑦,𝑡)→1 as 𝑡→∞,  for all 𝑥,𝑦∈𝑋. Let 𝐴∶𝑋×𝑋→𝑋 and 𝑆∶𝑋→𝑋 be compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous) mappings such that 𝑀𝐴()(𝑡)(),𝑥,𝑦,𝐴𝑢,𝑣,𝜙≥𝑀𝑆𝑥,𝑆𝑢,𝑡∗𝑀𝑆𝑦,𝑆𝑣,𝑡 (53) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, 𝜙∈Φ and 𝑡>0. Then there exists a unique point 𝛼 in 𝑋 such that 𝛼=𝑆𝛼=𝐴(𝛼,𝛼).

Next, we illustrate our results providing the following examples.

Example 17. Let 𝑋=[0,+∞),  𝑎∗𝑏=𝑎𝑏 for all 𝑎,𝑏∈[0,1] and 𝜓(𝑠)=𝑠/(𝑠+1) for all 𝑠∈ℝ⁺. Then (𝑋,𝑀,∗) is a fuzzy metric space, where 𝑀𝑥,𝑦,𝑡=𝜓(𝑡)^{|𝑥−𝑦|}, (54) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Let 𝜙(𝑠)=𝑠/2, and let the mappings 𝐴∶𝑋×𝑋→𝑋, 𝑆∶𝑋→𝑋 be defined as 𝐴𝑥,𝑦=3𝑥+3𝑦−5,if𝑥,𝑦∈(1,∞),𝑥+𝑦6,otherwise,𝑆(𝑥)=3𝑥−2,if(),𝑥𝑥∈1,∞6,if𝑥∈[0,1]. (55) In view of Definition 10, to prove compatibility, we have only to consider sequences {𝑥_𝑛} and {𝑦_𝑛} converging to zero from the right. In such case we have lim_𝑛→∞𝐴𝑥_𝑛,𝑦_𝑛=0=lim_𝑛→∞𝑆𝑥_𝑛,lim_𝑛→∞𝐴𝑦_𝑛,𝑥_𝑛=0=lim_𝑛→∞𝑆𝑦_𝑛. (56) Next, we get lim_𝑛→∞𝐴𝑆𝑥_𝑛,𝑆𝑦_𝑛=0=𝐴(0,0),lim_𝑛→∞𝑆𝐴𝑥_𝑛,𝑦_𝑛=0=𝑆(0),lim_𝑛→∞𝐴𝑆𝑦_𝑛,𝑆𝑥_𝑛=0=𝐴(0,0),lim_𝑛→∞𝑆𝐴𝑦_𝑛,𝑥_𝑛=0=𝑆(0). (57)
Consequently lim_𝑛→∞𝑀𝐴𝑆𝑥_𝑛𝑆𝑦_𝑛,𝑆𝐴𝑥_𝑛,𝑦_𝑛,𝑡=1,lim_𝑛→∞𝑀𝐴𝑆𝑦_𝑛𝑆𝑥_𝑛,𝑆𝐴𝑦_𝑛,𝑥_𝑛,𝑡=1, (58) for all 𝑡>0.
On the other hand, to prove subsequential continuity, in view of Definition 12, we have only to consider sequences {𝑥_𝑛} and {𝑦_𝑛} converging to one from the right. In such case we have lim_𝑛→∞𝐴𝑥_𝑛,𝑦_𝑛=1=lim_𝑛→∞𝑆𝑥_𝑛,lim_𝑛→∞𝐴𝑦_𝑛,𝑥_𝑛=1=lim_𝑛→∞𝑆𝑦_𝑛. (59) Also, note that, for the same sequences, we get lim_𝑛→∞𝐴𝑆𝑥_𝑛,𝑆𝑦_𝑛=1≠𝐴(1,1),lim_𝑛→∞𝑆𝐴𝑥_𝑛,𝑦_𝑛=1≠𝑆(1),lim_𝑛→∞𝐴𝑆𝑦_𝑛,𝑆𝑥_𝑛=1≠𝐴(1,1),lim_𝑛→∞𝑆𝐴𝑦_𝑛,𝑥_𝑛=1≠𝑆(1), (60) but lim_𝑛→∞𝑀𝐴𝑆𝑥_𝑛,𝑆𝑦_𝑛,𝑆𝐴𝑥_𝑛,𝑦_𝑛,𝑡=1,lim_𝑛→∞𝑀𝐴𝑆𝑦_𝑛,𝑆𝑥_𝑛,𝑆𝐴𝑦_𝑛,𝑥_𝑛,𝑡=1. (61) Thus, the mappings 𝐴 and 𝑆 are compatible as well as subsequentially continuous but not reciprocally continuous. Next, by a routine calculation, one can verify that condition (53) holds true. For instance, for all 𝑡>0 and 𝑥,𝑦,𝑢,𝑣∈[0,1], we have 𝑀𝐴()(𝑡)𝐴(),𝑡𝑥,𝑦,𝐴𝑢,𝑣,𝜙=𝑀𝑥,𝑦,𝐴𝑢,𝑣2=𝜓𝑡2^{|𝑥−𝑢+𝑦−𝑣|/6}≥𝜓𝑡2^{|𝑥−𝑢|/6}⋅𝜓𝑡2^{|𝑦−𝑣|/6}=𝑡𝑡+2^{|𝑥−𝑢|/6}⋅𝑡𝑡+2^{|𝑦−𝑣|/6}≥𝑡𝑡+1^{|𝑥−𝑢|/6}⋅𝑡𝑡+1^{|𝑦−𝑣|/6}=𝜓(𝑡)^{|𝑥−𝑢|/6}⋅𝜓(𝑡)^{|𝑦−𝑣|/6}()().=𝑀𝑆𝑥,𝑆𝑢,𝑡⋅𝑀𝑆𝑦,𝑆𝑣,𝑡=𝑀𝑆𝑥,𝑆𝑢,𝑡∗𝑀𝑆𝑦,𝑆𝑣,𝑡 (62)

Therefore, all the conditions of Corollary 16 are satisfied and (0,0) is the unique common fixed point of the pair (𝐴,𝑆). It is noted that this example cannot be covered by those fixed point theorems which involve compatibility and reciprocal continuity both.

Example 18. In the setting of Example 17 (besides retaining the rest), let 𝑋=(−∞,∞), and let the mappings 𝐴∶𝑋×𝑋→𝑋, 𝑆∶𝑋→𝑋 be defined as 𝐴𝑥,𝑦=𝑥+𝑦4,if𝑥,𝑦∈(−∞,1),3𝑥+3𝑦−5,if𝑥,𝑦∈[1,∞),𝑥−𝑦4,if()[),𝑆(𝑥)=𝑥∈−∞,1,𝑦∈1,∞𝑥+1,if𝑥∈(−∞,1),3𝑥−2,if𝑥∈[1,∞). (63) In view of Definitions 11 and 13, to prove reciprocal continuity and subcompatibility, we have only to consider sequences {𝑥_𝑛} and {𝑦_𝑛} converging to one from the right. For such sequences, we get lim_𝑛→∞𝐴𝑥_𝑛,𝑦_𝑛=1=lim_𝑛→∞𝑆𝑥_𝑛,lim_𝑛→∞𝐴𝑦_𝑛,𝑥_𝑛=1=lim_𝑛→∞𝑆𝑦_𝑛. (64) Also, we deduce that lim_𝑛→∞𝐴𝑆𝑥_𝑛,𝑆𝑦_𝑛=1=𝐴(1,1),lim_𝑛→∞𝑆𝐴𝑥_𝑛,𝑦_𝑛=1=𝑆(1),lim_𝑛→∞𝐴𝑆𝑦_𝑛,𝑆𝑥_𝑛=1=𝐴(1,1),lim_𝑛→∞𝑆𝐴𝑦_𝑛,𝑥_𝑛=1=𝑆(1). (65) Therefore, we have lim_𝑛→∞𝑀𝐴𝑆𝑥_𝑛,𝑆𝑦_𝑛,𝑆𝐴𝑥_𝑛,𝑦_𝑛,𝑡=1,lim_𝑛→∞𝑀𝐴𝑆𝑦_𝑛,𝑆𝑥_𝑛,𝑆𝐴𝑦_𝑛,𝑥_𝑛,𝑡=1, (66) for all 𝑡>0. Finally, to show that the mappings 𝐴 and 𝑆 are not compatible, it suffices to consider the particular sequences {𝑥_𝑛}={1/𝑛−2}_𝑛∈ℕ and {𝑦_𝑛}={1/3𝑛−2}_𝑛∈ℕ in 𝑋. In fact, in such case, we have lim_𝑛→∞𝐴𝑥_𝑛,𝑦_𝑛=−1=lim_𝑛→∞𝑆𝑥_𝑛,lim_𝑛→∞𝐴𝑦_𝑛,𝑥_𝑛=−1=lim_𝑛→∞𝑆𝑦_𝑛. (67) Next, we deduce that lim_𝑛→∞𝐴𝑆𝑥_𝑛,𝑆𝑦_𝑛1=−2=𝐴(−1,−1),lim_𝑛→∞𝑆𝐴𝑥_𝑛,𝑦_𝑛=0=𝑆(−1),lim_𝑛→∞𝐴𝑆𝑦_𝑛,𝑆𝑥_𝑛1=−2=𝐴(−1,−1),lim_𝑛→∞𝑆𝐴𝑦_𝑛,𝑥_𝑛=0=𝑆(−1). (68) Consequently, we obtain lim_𝑛→∞𝑀𝐴𝑆𝑥_𝑛,𝑆𝑦_𝑛,𝑆𝐴𝑥_𝑛,𝑦_𝑛,𝑡≠1,lim_𝑛→∞𝑀𝐴𝑆𝑦_𝑛,𝑆𝑥_𝑛,𝑆𝐴𝑦_𝑛,𝑥_𝑛,𝑡≠1, (69) for all 𝑡>0. Thus, the mappings 𝐴 and 𝑆 are reciprocally continuous as well as subcompatible but not compatible. Next, by a routine calculation, one can verify that condition (53) holds true. For instance, for all 𝑡>0 and 𝑥,𝑦,𝑢,𝑣∈[1,∞), we have 𝑀𝐴()(𝑡)𝐴(),𝑡𝑥,𝑦,𝐴𝑢,𝑣,𝜙=𝑀𝑥,𝑦,𝐴𝑢,𝑣2=𝜓𝑡2^{|3𝑥−3𝑢+3𝑦−3𝑣|}≥𝜓𝑡2^{3|𝑥−𝑢|}⋅𝜓𝑡2^{3|𝑦−𝑣|}=𝑡𝑡+2^{3|𝑥−𝑢|}⋅𝑡𝑡+2^{3|𝑦−𝑣|}≥𝑡𝑡+1^{3|𝑥−𝑢|}.𝑡𝑡+1^{3|𝑦−𝑣|}=𝜓(𝑡)^{3|𝑥−𝑢|}⋅𝜓(𝑡)^{3|𝑦−𝑣|}()().=𝑀𝑆𝑥,𝑆𝑢,𝑡⋅𝑀𝑆𝑦,𝑆𝑣,𝑡=𝑀𝑆𝑥,𝑆𝑢,𝑡∗𝑀𝑆𝑦,𝑆𝑣,𝑡 (70)
Therefore, all the conditions of Corollary 16 are satisfied, and (1, 1) is the unique common fixed point of the pair (𝐴,𝑆). It is also noted that this example cannot be covered by those fixed point theorems which involve compatibility and reciprocal continuity both.

Remark 19. The conclusions of Theorem 14 and Corollary 16 remain true if we assume 𝜙(𝑠)=𝑘𝑠, where 𝑘∈(0,1).

4. Conclusion

Theorem 14 is proved for two pairs of compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous) mappings in fuzzy metric spaces, wherein conditions on completeness (or closedness) of the underlying space (or subspaces) together with conditions on continuity in respect to anyone of the involved mappings are relaxed. Theorem 14 improves the results of Jain et al. [39, Theorem 3.2, Corollary 3.2, Theorem 3.3, Theorem 3.4, Theorem 4.1] and Hu [34, Theorem 1]. A natural result is also obtained for a pair of mappings (see Corollary 16). Finally, Examples 17 and 18 are furnished to demonstrate the usefulness of Corollary 16. In view of Remark 19, Theorem 14 and Corollary 16 improve the results of Sedghi et al. [32, Theorem 2.5, Corollary 2.6] and Jain et al. [39, Corollary 3.1].

Acknowledgments

The authors would like to express their sincere thanks to the editor and reviewer(s) for their valuable suggestions.