Fixed Point and Subfixed Point for Fuzzy Mappings in Generalized Metric Fuzzy Spaces

Abstract

A generalized metric in space of set of fuzzy sets is introduced. We prove some common fixed point for contractive iterate at the point and orbitally contractive at the point fuzzy mappings and subfixed point results for family of mappings satisfying generalized contractive conditions in generalized metric fuzzy spaces.

1. Introduction

Uncertainty regarding some experiments may essentially have two origins. It may arise from randomness due to the natural variability of observation or it may be caused by imprecisions due to partial information, for example, expert opinions or sparse data sets. An incomplete data set delivers an imprecise assessment of the information which should be expressed by a [0,1]-fuzzy set instead of a number. In other words, the system is complemented with extra dimension of uncertainty provided by fuzzy set theory. Fuzzy logic is the principal component of an array of methodologies for dealing with problems in which uncertainty and imprecision play important roles.

Fixed point theory in uncertain systems can be treated in different ways and one of them is by using the fuzzy logic. Depending on which segment of the problem is plagued with some sort of uncertainty the appropriate structure of the basic space in which the problem is considered could be used. If the distance between elements is imprecise, then the fuzziness is included in metric, as it was done in the definition of fuzzy metric spaces by Kaleva and Seikkala [1]. This model has a lot of similarities with probabilistic metric spaces (Menger spaces). Fixed point techniques, methods, and results are closely related in these two structures. Some of fixed point results in fuzzy metric space could be found in [2–7].

If affiliations of elements are imprecise, then the system could be treated as a fuzzy system. Appropriate structure is introduced depending on the related problem. The treatment of the problem involving fixed point theory has to be placed in spaces with adequate topological structure.

We denote by 𝒦(𝑋) the set of compact subsets of 𝑋 and by ℱ(𝑋) the set of fuzzy sets with compact 𝛼-levels defined over 𝑋, where 𝑋 has some metric structure. Mustafa and Sims [8] introduced the definition of a generalized metric space, briefly, 𝐺-metric spaces. In [9] the 𝐺-metric is introduced in 𝒦(𝑋) and, in [10], the similar construction is made to establish the 𝐺-metric in the set ℱ(𝑋). In both cases the structure of the basic 𝐺-metric space [3–5] is used to define the Hausdorff 𝐺-metric by the metric 𝑑_𝐺 derived from 𝐺-metric 𝐺. In our paper the basic space is the metric space (instead of 𝐺-metric space) and, using the same idea as in [9, 10], the Hausdorff 𝐺-metric 𝒢 is introduced, but metric 𝑑 of the original metric space (𝑋,𝑑) is used instead of the derived 𝑑_𝐺 metric. In spite of the fact that the relation of the basic and the derived spaces is simpler than in [10] the structure of generalized metric space (ℱ(𝑋),𝒢) is not reduced. Further, we analyse the existence and uniqueness of a common fixed point for the family {𝑓_𝑖} of self-mappings in the set of fuzzy sets ℱ(𝑋) endowed with generalized metric 𝒢. The different type of generalized contractive condition is considered using a nondecreasing, right continuous function Φ∶[0,∞)→[0,∞). Putting different additional conditions on Φ, we can follow how Φ influences the other conditions, related technique of proving, and the final fixed point result.

For more fixed point results for mappings defined in 𝐺-metric spaces of fuzzy sets, we refer the reader to [10–12].

2. Preliminaries

Definition 1 (see [8]). Let 𝑋 be a nonempty set, and let 𝐺∶𝑋×𝑋×𝑋→ℝ⁺ be a function satisfying the following properties:(𝐺1)𝐺(𝑥,𝑦,𝑧)=0 if 𝑥=𝑦=𝑧;(𝐺2)0<𝐺(𝑥,𝑥,𝑦), for all 𝑥,𝑦∈𝑋, with 𝑥≠𝑦;(𝐺3)𝐺(𝑥,𝑥,𝑦)≤𝐺(𝑥,𝑦,𝑧), for all 𝑥,𝑦,𝑧∈𝑋, with 𝑧≠𝑦;(𝐺4)𝐺(𝑥,𝑦,𝑧)=𝐺(𝑥,𝑧,𝑦)=𝐺(𝑦,𝑧,𝑥)=⋯, (symmetry in all three variables);(𝐺5)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑎,𝑎)+𝐺(𝑎,𝑦,𝑧), for all 𝑥,𝑦,𝑧,𝑎∈𝑋. Then function 𝐺 is called a generalized metric, abbreviated 𝐺-metric, on 𝑋 and the pair (𝑋,𝐺) is called a 𝐺-metric space. If 𝐺(𝑥,𝑦,𝑦)=𝐺(𝑦,𝑥,𝑥) for all 𝑥,𝑦∈𝑋, then (𝑋,𝐺) is symmetric.

Clearly, these properties are satisfied when 𝐺(𝑥,𝑦,𝑧) is the perimeter of the triangle with vertices at 𝑥, 𝑦, and 𝑧∈ℝ²; moreover taking 𝑎 in the interior of the triangle shows that (𝐺5) is the best possible.

Example 2 (see [8]). Let (𝑋,𝑑) be an ordinary metric space; then (𝑋,𝑑) defines 𝐺-metrics on 𝑋 by 𝐺_𝑠(),𝐺𝑥,𝑦,𝑧=𝑑𝑥,𝑦+𝑑𝑦,𝑧+𝑑𝑥,𝑧_𝑚𝑥,𝑦,𝑧=max𝑑𝑥,𝑦,𝑑𝑦,𝑧,𝑑(𝑥,𝑧). (1)

The following useful properties of a 𝐺-metric are readily derived from the axioms.

Proposition 3 (see [8]). Let (𝑋,𝐺) be a 𝐺-metric space; then for any 𝑥,𝑦,𝑧, and 𝑎 from 𝑋 it follows that(1)if  𝐺(𝑥,𝑦,𝑧)=0, then  𝑥=𝑦=𝑧,(2)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑥,𝑦)+𝐺(𝑥,𝑥,𝑧),(3)𝐺(𝑥,𝑦,𝑦)≤2𝐺(𝑦,𝑥,𝑥),(4)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑎,𝑧)+𝐺(𝑎,𝑦,𝑧),(5)𝐺(𝑥,𝑦,𝑧)≤(2/3)(𝐺(𝑥,𝑦,𝑎)+𝐺(𝑥,𝑎,𝑧)+𝐺(𝑎,𝑦,𝑧)),(6)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑎,𝑎)+𝐺(𝑦,𝑎,𝑎)+𝐺(𝑧,𝑎,𝑎).

Definition 4 (see [8]). Let (𝑋,𝐺) be a 𝐺-metric space; and let {𝑥_𝑛} be a sequence of points of 𝑋. A point 𝑥∈𝑋 is said to be the limit of the sequence {𝑥_𝑛} if lim_{𝑛,𝑚→∞}𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)=0, and one says that the sequence {𝑥_𝑛} is 𝐺-convergent to 𝑥.

Proposition 5 (see [8]). Let (𝑋,𝐺) be a 𝐺-metric space, then for a sequence {𝑥_𝑛}⊆𝑋 and a point 𝑥∈𝑋 the following are equivalent: (1){𝑥_𝑛}  is  𝐺-convergent to  𝑥,(2)𝐺(𝑥_𝑛,𝑥_𝑛,𝑥)→0  as  𝑛→∞,(3)𝐺(𝑥_𝑛,𝑥,𝑥)→0  as  𝑛→∞.

Definition 6 (see [8]). Let (𝑋,𝐺) be a 𝐺-metric space; a sequence {𝑥_𝑛} is called 𝐺-Cauchy if, for every 𝜖>0, there is 𝑁∈ℕ such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)<𝜖, for all 𝑛,𝑚,𝑙≥𝑁, that is, if 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)→0 as 𝑛,𝑚,𝑙→∞.

Proposition 7 (see [8]). In a 𝐺-metric space (𝑋,𝐺), the following are equivalent: (1)the sequence  {𝑥_𝑛}  is  𝐺-Cauchy,(2)for every  𝜀>0, there exists an  𝑛₀∈ℕ  such that  𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑚)<𝜀, for all  𝑛,𝑚≥𝑛₀.

A 𝐺-metric space (𝑋,𝐺) is 𝐺-complete (or complete 𝐺-metric) if every 𝐺-Cauchy sequence in (𝑋,𝐺) is 𝐺-convergent in (𝑋,𝐺).

Proposition 8 (see [8]). Every 𝐺-metric space (𝑋,𝐺) defines a metric space (𝑋,𝑑_𝐺) by 𝑑_𝐺𝑥,𝑦=𝐺𝑥,𝑦,𝑦+𝐺𝑦,𝑥,𝑥,∀𝑥,𝑦∈𝑋. (2) Note that if (𝑋,𝐺) is a symmetric 𝐺-metric space, then 𝑑_𝐺𝑥,𝑦=2𝐺𝑥,𝑦,𝑦,∀𝑥,𝑦∈𝑋. (3)

However, if (𝑋,𝐺) is nonsymmetric, then by 𝐺-metric properties 32𝐺𝑥,𝑦,𝑦≤𝑑_𝐺𝑥,𝑦≤3𝐺𝑥,𝑦,𝑦,∀𝑥,𝑦∈𝑋, (4) and in general these inequalities cannot be improved.

Proposition 9 (see [8]). Let (𝑋,𝐺) be a 𝐺-metric space; then the function 𝐺(𝑥,𝑦,𝑧) is jointly continuous in all three of its variables.

In [13] it was shown that if (𝑋,𝐺) is a 𝐺-metric space, putting 𝜚(𝑥,𝑦)=𝐺(𝑥,𝑦,𝑦), (𝑋,𝜚) is a quasi-metric space (generally, 𝜚 is not symmetric). It is well known that any quasi-metric induces different metrics and mostly used are(𝜇)𝜇(𝑥,𝑦)=𝜚(𝑥,𝑦)+𝜚(𝑦,𝑥),(𝜌)𝜌(𝑥,𝑦)=max{𝜚(𝑥,𝑦),𝜚(𝑦,𝑥)}.

The following result is an immediate consequence of the above definitions and relations.

Theorem 10. Let (𝑋,𝐺) be a 𝐺-metric space and let 𝐷∈{𝜇,𝜌}. Then (1){𝑥_𝑛}⊂𝑋 is 𝐺-convergent to 𝑥∈𝑋 if and only if {𝑥_𝑛} is convergent to 𝑥 in (𝑋,𝐷);(2){𝑥_𝑛}⊂𝑋 is 𝐺-Cauchy if and only if {𝑥_𝑛} is Cauchy in (𝑋,𝐷);(3)(𝑋,𝐺) is 𝐺-complete if and only if (𝑋,𝐷) is complete.

Recently, Samet et al. [14] and Jleli and Samet [13] observed that some fixed point theorems in context of 𝐺-metric space can be proved (by simple transformation) using related existing results in the setting of (quasi) metric space. Namely, if the contraction condition of the fixed point theorem on 𝐺-metric space can be reduced to two variables, then one can construct an equivalent fixed point theorem in setting of usual metric space. This idea is not completely new, but it was not successfully used before; see [15]. Karapinar and Agarval in [16] continued to develop Jleli-Samet technique in 𝐺-metric space, but, on the other side, they proved fixed point theorems on the context of 𝐺-metric space for which Jleli-Samet technique is not applicable. So, in some cases, as it is noticed even in Jleli and Samet paper [13], when the contraction condition is of nonlinear type, this strategy cannot be always successfully used. This is exactly the case in our paper where in fixed point results the use of Jleli-Samet technique does not give satisfactory results. If, for instance, the function 𝑓 in (17) is not independent of the variable 𝜇∈ℱ(𝑋), then 𝑛(𝜇), 𝜇∈ℱ(𝑋), the exponent factor in contraction conditions in our theorems, is not the constant function (as it is the case in our paper), implying that conditions which the contractor Φ in related metric space must satisfy become significantly more restrictive if the Jleli-Samet technique is used. But, using directly 𝐺-metric 𝐺, the proofs of theorems in our paper are given. The conclusion is that results from our paper cannot be deduced from the usual one in metric or quasi-metric space and cannot be derived from the results of Samet et al. [14] and Jleli and Samet [13].

For some fixed point results in 𝐺-metric spaces we refer to [17–22].

3. Fuzzy Generalized Metric Space

Let (𝑋,𝑑) be a metric space and let 𝒦(𝑋) be the set of all nonempty compact subsets of 𝑋. For all 𝑥∈𝑋 and all 𝐴,𝐵∈𝒦(𝑋), let 𝑑(𝑥,𝐴)=inf_𝑎∈𝐴𝑑(𝑥,𝑎) and 𝑑(𝐴,𝐵)=inf_{𝑎∈𝐴,𝑏∈𝐵}𝑑(𝑎,𝑏). The Hausdorff metric on 𝒦(𝑋) is defined by 𝐻(𝐴,𝐵)=max{sup_𝑎∈𝐴𝑑(𝑎,𝐵),sup_𝑏∈𝐵𝑑(𝑏,𝐴)}. By ℱ(𝑋) we denote the set of all fuzzy sets 𝜇∶𝑋→[0,1] with compact 𝛼-levels 𝜇_𝛼, 𝛼∈(0,1], where 𝜇_𝛼={𝑥∈𝑋∶𝜇(𝑥)≥𝛼} and bounded support 𝜇₀=∪_𝛼∈(0,1]𝜇_𝛼. In the set ℱ(𝑋) the metric is introduced by 𝛿(𝜇,𝜈)=sup_𝛼∈(0,1]𝐻(𝜇_𝛼,𝜈_𝛼). If 𝜇,𝜈∈ℱ(𝑋), then (𝜇⊆𝜈)⇔(𝜇(𝑥)≤𝜈(𝑥)) for all 𝑥∈𝑋 and (𝜇=𝜈)⇔(𝜇⊆𝜈∧𝜈⊆𝜇).

The function 𝒢∶ℱ(𝑋)×ℱ(𝑋)×ℱ(𝑋)→ℝ⁺ is defined by 𝒢𝜇,𝜈,𝜂=maxℒ𝜇,𝜈,𝜂,ℒ𝜂,𝜇,𝜈,ℒ𝜈,𝜂,𝜇, (5) where ℒ𝜇,𝜈,𝜂=sup_𝛼∈(0,1]sup_{𝑥∈𝜇𝛼}𝑑𝑥,𝜈_𝛼+𝑑𝜈_𝛼,𝜂_𝛼+𝑑𝑥,𝜂_𝛼. (6)

Lemma 11. For all 𝜇,𝜈∈ℱ(𝑋), 𝛿(𝜇,𝜈)<𝒢(𝜇,𝜈,𝜈)≤2𝛿(𝜇,𝜈).

Proof. Since ℒ𝜇,𝜈,𝜈=sup_𝛼∈(0,1]2sup_{𝑥∈𝜇𝛼}𝑑𝑥,𝜈_𝛼,ℒ𝜈,𝜇,𝜈=ℒ𝜈,𝜈,𝜇=sup_𝛼∈(0,1]sup_{𝑦∈𝜈𝛼}𝑑𝑦,𝜇_𝛼+𝑑𝜇_𝛼,𝜈_𝛼, (7) from (5) and (6), we obtain 𝒢𝜇,𝜈,𝜈=maxsup_(]𝛼∈0,12sup_{𝑥∈𝜇𝛼}𝑑𝑥,𝜈_𝛼,sup_𝛼∈(0,1]sup_{𝑦∈𝜈𝛼}𝑑𝑦,𝜇_𝛼+𝑑𝜇_𝛼,𝜈_𝛼=sup_(]𝛼∈0,1max2sup_{𝑥∈𝜇𝛼}𝑑𝑥,𝜈_𝛼,sup_{𝑦∈𝜈𝛼}𝑑𝑦,𝜇_𝛼+𝑑𝜇_𝛼,𝜈_𝛼. (8) The next two inequalities, 𝒢𝜇,𝜈,𝜈=sup_(]𝛼∈0,1max2sup_{𝑥∈𝜇𝛼}𝑑𝑥,𝜈_𝛼,sup_{𝑦∈𝜈𝛼}𝑑𝑦,𝜇_𝛼+𝑑𝜇_𝛼,𝜈_𝛼≤2sup_(]𝛼∈0,1maxsup_{𝑥∈𝜇𝛼}𝑑𝑥,𝜈_𝛼,sup_{𝑦∈𝜈𝛼}𝑑𝑦,𝜇_𝛼=2sup_𝛼∈(0,1]𝐻𝜇_𝛼,𝜈_𝛼,𝒢=2𝛿𝜇,𝜈𝜇,𝜈,𝜈=sup_(]𝛼∈0,1max2sup_{𝑥∈𝜇𝛼}𝑑𝑥,𝜈_𝛼,sup_{𝑦∈𝜈𝛼}𝑑𝑦,𝜇_𝛼+𝑑𝜇_𝛼,𝜈_𝛼>sup_(]𝛼∈0,1maxsup_{𝑥∈𝜇𝛼}𝑑𝑥,𝜈_𝛼,sup_{𝑦∈𝜈𝛼}𝑑𝑦,𝜇_𝛼=sup_(]𝛼∈0,1𝐻𝜇_𝛼,𝜈_𝛼=𝛿𝜇,𝜈, (9) imply the relation 𝛿(𝜇,𝜈)<𝒢(𝜇,𝜈,𝜈)≤2𝛿(𝜇,𝜈), what we had to prove.

Proposition 12. If (𝑋,𝑑) is a complete metric space, then (ℱ(𝑋),𝒢) is a complete 𝐺-metric fuzzy space.

Proof. Properties (𝐺1), (𝐺2), and (𝐺4) from Definition 1 are obvious, so the proof is omitted.
(𝐺3) Since 𝑑(𝑥,𝐴)≤𝑑(𝑥,𝐵)+𝑑(𝐴,𝐵) for all 𝑥∈𝑋 and all 𝐴,𝐵∈𝒦(𝑋), we have ℒ𝜇,𝜈,𝜈=sup_𝛼∈(0,1]sup_{𝑥∈𝜇𝛼}𝑑𝑥,𝜈_𝛼+𝑑𝑥,𝜈_𝛼+𝑑𝜈_𝛼,𝜈_𝛼≤sup_(]𝛼∈0,1sup_{𝑥∈𝜇𝛼}𝑑𝑥,𝜈_𝛼+𝑑𝑥,𝜂_𝛼+𝑑𝜂_𝛼,𝜈_𝛼=ℒ𝜇,𝜈,𝜂. (10) By the same way we show that ℒ𝜈,𝜇,𝜈≤ℒ𝜂,𝜇,𝜈,ℒ𝜈,𝜈,𝜇≤ℒ𝜈,𝜂,𝜇, (11) which implies that 𝒢(𝜇,𝜈,𝜈)≤𝒢(𝜇,𝜈,𝜂).
(𝐺5) To prove that 𝒢(𝜇,𝜈,𝜂)≤𝒢(𝜇,𝜃,𝜃)+𝒢(𝜃,𝜈,𝜂), we consider the related inequality for ℒ: ℒ𝜇,𝜈,𝜂=sup_𝛼∈(0,1]sup_{𝑥∈𝜇𝛼}𝑑𝑥,𝜈_𝛼+𝑑𝑥,𝜂_𝛼+𝑑𝜂_𝛼,𝜈_𝛼≤sup_(]𝛼∈0,1sup_{𝑥∈𝜇𝛼}𝑑𝑥,𝜃_𝛼+𝑑𝜃_𝛼,𝜈_𝛼+𝑑𝑥,𝜃_𝛼+𝑑𝜃_𝛼,𝜂_𝛼+𝑑𝜂_𝛼,𝜈_𝛼≤sup_(]𝛼∈0,1sup_{𝑥∈𝜇𝛼}𝑑𝑥,𝜃_𝛼+sup_{𝑡∈𝜃𝛼}𝑑𝑡,𝜈_𝛼+𝑑𝑥,𝜃_𝛼+sup_{𝑡∈𝜃𝛼}𝑑𝑡,𝜂_𝛼+𝑑𝜂_𝛼,𝜈_𝛼=sup_𝛼∈(0,1]sup_{𝑥∈𝜇𝛼}𝑑𝑥,𝜃_𝛼+𝑑𝑥,𝜃_𝛼+𝑑𝜃_𝛼,𝜃_𝛼+sup_{𝑡∈𝜃𝛼}𝑑𝑡,𝜈_𝛼+𝑑𝑡,𝜂_𝛼+𝑑𝜂_𝛼,𝜈_𝛼=ℒ𝜇,𝜃,𝜃+ℒ𝜃,𝜈,𝜂. (12) Analogously, ℒ(𝜂,𝜇,𝜈)≤ℒ(𝜃,𝜇,𝜃)+ℒ(𝜂,𝜃,𝜈) and ℒ(𝜈,𝜂,𝜇)≤ℒ(𝜃,𝜃,𝜇)+ℒ(𝜈,𝜂,𝜃). All three inequalities together imply that 𝒢(𝜇,𝜈,𝜂)≤𝒢(𝜇,𝜃,𝜃)+𝒢(𝜃,𝜈,𝜂).
The completeness of (ℱ(𝑋),𝒢) is a consequence of the completeness of (ℱ(𝑋),𝛿) and inequality from the previous lemma.

Proposition 13. 𝐺-metric fuzzy space (ℱ(𝑋),𝒢) is not symmetric.

Proof. We prove that 𝒢(𝜇,𝜈,𝜈)≠𝒢(𝜈,𝜇,𝜇). If sup_𝛼∈(0,1]sup_{𝑥∈𝜇𝛼}𝑑𝑥,𝜈_𝛼=𝐴,sup_𝛼∈(0,1]sup_{𝑦∈𝜈𝛼}𝑑𝑦,𝜇_𝛼=𝑏,sup_𝛼∈(0,1]𝑑𝜈_𝛼,𝜇_𝛼=𝐶, (13)
then 𝒢{}{}.𝜇,𝜈,𝜈=max2𝐴,𝐵+𝐶≠𝒢𝜈,𝜇,𝜇=max2𝐵,𝐴+𝐶 (14)

4. Fixed Point for Contractive Iterate at the Point and Orbitally Contractive at the Point Fuzzy Mappings

A generalization of the contraction principle can be obtained using different type of a nondecreasing right continuous function Φ∶[0,∞)→[0,∞). The most usual additional properties imposed on Φ are given using a combination of the next seven conditions:(𝜙₁)Φ(0)=0,(𝜙₂)Φ(𝑥)<𝑥, for all 𝑥>0,(𝜙₃)lim_𝑖→∞Φ^𝑖(𝑥)=0, for all 𝑥>0,(𝜙₄){𝑥_𝑖}⊂[0,∞) is a sequence such that 𝑥_𝑖+1≤Φ(𝑥_𝑖); then lim_𝑖→∞𝑥_𝑖=0.(𝜙₅)for any 𝑥≥0 there exists a 𝑦(𝑥)≥0, 𝑦(𝑥)=sup_𝑦≥0{𝑦≤𝑥+Φ(𝑦)},(𝜙₆)lim_𝑥→∞(𝑥−Φ(𝑥))=∞, (𝜙₇)∑^∞_𝑖=1Φ^𝑖(𝑥)<∞, for all 𝑥>0.

Some of the noted properties of Φ are equivalent, some of them imply other, and some of them are incompatible. The next lemma discusses some of the relations between properties (𝜙₁)--(𝜙₇), especially those which are used in this paper to define a generalized contraction.

Lemma 14. Let Φ∶[0,∞)→[0,∞) be a nondecreasing right continuous function. Then (i)(𝜙₂)⇔(𝜙₃)⇔(𝜙₄), (ii)(𝜙₇)⇒(𝜙_𝑘)⇒(𝜙₁), where 𝑘∈{2,3,4},(iii)(𝜙₅)+(𝜙₂)⇏(𝜙₇) and (𝜙₆)+(𝜙₂)⇏(𝜙₇),(iv)(𝜙₇)⇏(𝜙₅) and (𝜙₇)⇏(𝜙₆).

Proof. (i) It is enough to prove that (𝜙₄)⇒(𝜙₂)⇔(𝜙₃)⇒(𝜙₄).
(𝜙₂)⇒(𝜙₃). We assume that for some 𝑥>0, lim_𝑖→∞Φ^𝑖(𝑥)=𝑎>0. Since {Φ^𝑖(𝑥)} is nonincreasing sequence, by the right continuity of Φ, Φ(𝑎)=Φlim_𝑖→∞Φ^𝑖(𝑥)=lim_𝑖→∞Φ^𝑖+1(𝑥)=𝑎>0; that is, 0<𝑎=Φ(𝑎) which contradicts (𝜙₂).
(𝜙₃)⇒(𝜙₂). If for some 𝑥>0, Φ(𝑥)≥𝑥, then, knowing that Φ is nondecreasing, Φ^𝑖(𝑥)≥Φ^𝑖−1(𝑥)≥⋯≥Φ(𝑥)≥𝑥>0. It means that lim_𝑖→∞Φ^𝑖(𝑥)≠0, which contradicts (𝜙₃).
(𝜙₃)⇒(𝜙₄). Let {𝑥_𝑖}⊂[0,∞) be any sequence such that 𝑥_𝑖+1≤Φ(𝑥_𝑖). Then 𝑥_𝑖≤Φ(𝑥_𝑖−1)≤Φ²(𝑥_𝑖−2)≤⋯≤Φ^𝑖(𝑥₀) and lim_𝑖→∞𝑥_𝑖≤lim_𝑖→∞Φ^𝑖(𝑥₀)=0.
(𝜙₄)⇒(𝜙₂). If (𝜙₂) does not hold, that is, there exists a 𝑦>0, 𝑦≤Φ(𝑦), then putting 𝑥_𝑖=𝑦, for all 𝑖=0,1,…, we have the sequence with 𝑦=𝑥_𝑖+1≤Φ(𝑦)=Φ(𝑥_𝑖), but that sequence does not converge to 0.
(ii) is obvious, so the proof is omitted.
(iii) Function Φ(𝑥)=(𝑛+2)⁻¹,(𝑛+1)⁻¹≤𝑥<𝑛⁻¹3,𝑛∈ℕ,₋₁,1≤𝑥 (15) satisfies (𝜙₅), (𝜙₆) and (𝜙₂) but not (𝜙₇).
(iv) Function Φ(𝑥)=𝑥2²,0≤𝑥<2,𝑥−1,2≤𝑥 (16) satisfies (𝜙₇) but not (𝜙₅) nor (𝜙₆).

Theorem 15. Let (𝑋,𝑑) be a complete metric space, let (ℱ(𝑋),𝒢) be related 𝐺-metric fuzzy space, and let ℬ⊆ℱ(𝑋). Further, let {𝑓_𝑖} be the sequence of self-mappings of ℱ(𝑋) such that, for all 𝑖∈ℕ, 𝑓_𝑖(ℬ)⊆ℬ and for each 𝜇∈ℱ(𝑋) there exists an 𝑛(𝜇)∈ℕ such that 𝒢𝑓_𝑖^𝑛(𝜇)(𝜈),𝑓_𝑗^𝑛(𝜇)𝜇,𝑓_𝑗^𝑛(𝜇)𝜇𝒢,2≤Φmax𝜈,𝜇,𝜇₋₁𝒢𝜈,𝑓_𝑗^𝑛(𝜇)𝜇,𝑓_𝑗^𝑛(𝜇)𝜇+𝒢𝜇,𝑓_𝑗^𝑛(𝜇)𝜇,𝑓_𝑗^𝑛(𝜇)𝜇,2⁻¹𝒢𝜈,𝑓_𝑗^𝑛(𝜇)𝜇,𝑓_𝑗^𝑛(𝜇)𝜇+𝒢𝑓_𝑖^𝑛(𝜇)(𝜈),𝜇,𝜇, (17) for all 𝑖,𝑗∈ℕ and all 𝜈∈ℬ, where Φ satisfies (𝜙₂). If there exists 𝜇^∗∈ℬ such that 𝑓^{𝑛(𝜇∗)}_𝑖(𝜇^∗)=𝜇^∗ for all 𝑖∈ℕ, then 𝜇^∗ is a unique common fixed point for {𝑓_𝑖} in ℬ and, for every 𝜇₁∈ℱ(𝑋), the sequence 𝜇_𝑗+1=𝑓^{𝑛(𝜇∗)}_𝑖(𝜇_𝑗), 𝑖∈ℕ, converges to 𝜇^∗.

Proof. First we prove that 𝜇^∗ is a unique point in ℬ with the property that 𝑓^{𝑛(𝜇∗)}_𝑖(𝜇^∗)=𝜇^∗, 𝑖∈ℕ. If 𝜈∈ℬ, 𝜈≠𝜇^∗, 𝑓^{𝑛(𝜇∗)}_𝑖(𝜈)=𝜈, 𝑖∈ℕ, then 𝒢𝜈,𝜇^∗,𝜇^∗≤𝒢𝑓^{𝑛(𝜇∗)}_𝑖(𝜈),𝑓^{𝑛(𝜇∗)}_𝑗𝜇^∗,𝑓^{𝑛(𝜇∗)}_𝑗𝜇^∗≤Φ𝒢𝜈,𝜇^∗,𝜇^∗. (18) By the property Φ(𝑡)<𝑡, 𝑡>0, since 𝒢(𝜈,𝜇^∗,𝜇^∗)>0, we have the contradiction; that is, the assumption 𝜇^∗≠𝜈 is not correct.
Further, since 𝑓_𝑖𝜇^∗=𝑓_𝑖𝑓^{𝑛(𝜇∗)}_𝑖𝜇^∗=𝑓^𝑛(𝜇∗_𝑖⁾⁺¹𝜇^∗=𝑓^{𝑛(𝜇∗)}_𝑖𝑓_𝑖𝜇^∗, (19) it follows that 𝑓_𝑖(𝜇^∗)=𝜇^∗ for all 𝑖∈ℕ.
Now, for some 𝜇₁∈ℬ, we form the sequence 𝜇_𝑖+1=𝑓^{𝑛(𝜇∗)}_𝑖(𝜇_𝑖).
If 𝜇₁=𝜇^∗, then 𝜇_𝑖=𝑓^{𝑛(𝜇∗)}_𝑖−1(𝑓^{𝑛(𝜇∗)}_𝑖−2⋯(𝑓^{𝑛(𝜇∗)}₁(𝜇^∗))⋯)=𝜇^∗ and the sequence {𝜇_𝑖} converges to 𝜇^∗.
If 𝜇_𝑖≠𝜇^∗, in order to prove that the sequence {𝜇_𝑖} converges to 𝜇^∗, we consider the sequence 𝒢(𝜇_𝑖+1,𝜇^∗,𝜇^∗), 𝑖∈ℕ: 𝒢𝜇_𝑖+1,𝜇^∗,𝜇^∗=𝒢𝑓^{𝑛(𝜇∗)}_𝑖𝜇_𝑖,𝑓^{𝑛(𝜇∗)}_𝑗𝜇^∗,𝑓^{𝑛(𝜇∗)}_𝑗𝜇^∗≤Φmax𝒢𝜇_𝑖,𝜇^∗,𝜇^∗,2⁻¹𝒢𝜇_𝑖,𝜇^∗,𝜇^∗+𝒢𝜇^∗,𝜇^∗,𝜇^∗,2⁻¹𝒢𝜇_𝑖,𝜇^∗,𝜇^∗+𝒢𝜇_𝑖+1,𝜇^∗,𝜇^∗=Φmax𝒢𝜇_𝑖,𝜇^∗,𝜇^∗,2⁻¹𝒢𝜇_𝑖,𝜇^∗,𝜇^∗+𝒢𝜇_𝑖+1,𝜇^∗,𝜇^∗. (20) If we choose the option that max𝒢𝜇_𝑖,𝜇^∗,𝜇^∗,2⁻¹𝒢𝜇_𝑖,𝜇^∗,𝜇^∗+𝒢𝜇_𝑖+1,𝜇^∗,𝜇^∗=2⁻¹𝒢𝜇_𝑖,𝜇^∗,𝜇^∗+𝒢𝜇_𝑖+1,𝜇^∗,𝜇^∗, (21) it implies that 𝒢𝜇_𝑖,𝜇^∗,𝜇^∗≤𝒢𝜇_𝑖+1,𝜇^∗,𝜇^∗. (22) On the other hand, in that case 𝒢𝜇_𝑖+1,𝜇^∗,𝜇^∗≤Φ2⁻¹𝒢𝜇_𝑖,𝜇^∗,𝜇^∗+𝒢𝜇_𝑖+1,𝜇^∗,𝜇^∗<2⁻¹𝒢𝜇_𝑖,𝜇^∗,𝜇^∗+2⁻¹𝒢𝜇_𝑖+1,𝜇^∗,𝜇^∗; (23) that is 𝒢𝜇_𝑖+1,𝜇^∗,𝜇^∗<𝒢𝜇_𝑖,𝜇^∗,𝜇^∗. (24) It is obvious that (22) contradicts (24). So, 𝒢𝜇_𝑖+1,𝜇^∗,𝜇^∗≤Φ𝒢𝜇_𝑖,𝜇^∗,𝜇^∗. (25) Now, applying that procedure 𝑖 times and letting 𝑖→∞, we get 𝒢𝜇_𝑖+1,𝜇^∗,𝜇^∗≤Φ𝒢𝜇_𝑖,𝜇^∗,𝜇^∗≤⋯≤Φ^𝑖𝒢𝜇₁,𝜇^∗,𝜇^∗. (26) Since 𝜇₁≠𝜇^∗, 𝒢(𝜇_𝑖,𝜇^∗,𝜇^∗)>0 and lim_𝑖→∞𝒢(𝜇_𝑖+1,𝜇^∗,𝜇^∗)=0. The last relation proves that the sequence {𝜇_𝑖} converges to 𝜇^∗.

Theorem 16. Let (𝑋,𝑑) be a complete metric space, 𝑓∶ℱ(𝑋)→ℱ(𝑋), where (ℱ(𝑋),𝒢) is related 𝐺-metric fuzzy space, and let Φ∶[0,∞)→[0,∞) be a subadditive mapping satisfying (𝜙₇). If for some 𝜇₀∈ℱ(𝑋) the orbit 𝒪(𝑓;𝜇₀) is complete and for each 𝜇∈𝒪(𝑓;𝜇₀) there exists an 𝑛(𝜇)∈ℕ such that 𝒢𝑓^𝑛(𝜇)(𝜈),𝑓^𝑛(𝜇)𝜇,𝑓^𝑛(𝜇)𝜇≤Φ𝒢𝜈,𝜇,𝜇, (27) for all 𝜈∈𝒪(𝑓;𝜇₀), then the sequence 𝜇_𝑖+1=𝑓^{𝑛(𝜇𝑖)}(𝜇_𝑖), 𝑖∈ℕ₀, converges to some 𝜇^∗∈ℱ(𝑋).
If inequality (27) holds for all 𝜇∈𝒪(𝑓;𝜇₀), then 𝑓^{𝑛(𝜇∗)}(𝜇^∗)=𝜇^∗ and 𝑓^𝑖(𝜇)→𝜇^∗ for all 𝜇∈𝒪(𝑓;𝜇₀). If 𝑓(𝒪(𝑓;𝜇₀))⊆𝒪(𝑓;𝜇₀), then 𝜇^∗ is the fixed point of 𝑓.

Proof. First, we show that {𝜇_𝑖}_𝑖∈ℕ0⊂ℱ(𝑋) is a Cauchy sequence. For sufficiently large 𝑚∈ℕ, there exist 𝑘,𝑟∈ℕ, 1≤𝑟<𝑛(𝜇₀) such that 𝑚=𝑘⋅𝑛(𝜇₀)+𝑟. Using (27), we get 𝒢𝑓^𝑚𝜇₀,𝜇₀,𝜇₀≤𝒢𝑓^{𝑘𝑛(𝜇0)+𝑟}𝜇₀,𝑓^𝑛(𝜇0)𝜇₀,𝑓^𝑛(𝜇0)𝜇₀+𝒢𝑓^𝑛(𝜇0)𝜇₀,𝜇₀,𝜇₀≤Φ𝒢𝑓^{(𝑘−1)𝑛(𝜇0)+𝑟}𝜇₀,𝜇₀,𝜇₀+𝒢𝑓^𝑛(𝜇0)𝜇₀,𝜇₀,𝜇₀≤Φ𝒢𝑓^{(𝑘−1)𝑛(𝜇0)+𝑟}𝜇₀,𝑓^𝑛(𝜇0)𝜇₀,𝑓^𝑛𝜇₀+𝒢𝑓^𝑛(𝜇0)𝜇₀,𝜇₀,𝜇₀+𝒢𝑓^𝑛(𝜇0)𝜇₀,𝜇₀,𝜇₀≤Φ²𝒢𝑓^{(𝑘−2)𝑛(𝜇0)+𝑟}𝜇₀,𝜇₀,𝜇₀+Φ𝒢𝑓^𝑛(𝜇0)𝜇₀,𝜇₀,𝜇₀+𝒢𝑓^𝑛(𝜇0)𝜇₀,𝜇₀,𝜇₀≤⋯≤Φ^𝑘𝒢𝑓^𝑟𝜇₀,𝜇₀,𝜇₀+^𝑘−1_𝑖=1Φ^𝑖𝒢𝑓^𝑛(𝜇0),𝜇₀,𝜇₀. (28) Putting 𝐴=max{𝒢(𝑓^𝑝(𝜇₀),𝜇₀,𝜇₀)∶1≤𝑝≤𝑛(𝜇₀)}, for all 𝑚∈ℕ, the next inequality holds: 𝒢𝑓^𝑚𝜇₀,𝜇₀,𝜇₀≤^𝑘_𝑠=1Φ^𝑠(𝐴)≤^∞_𝑠=1Φ^𝑠(𝐴)=𝐵<∞, (29) and, consequently, 𝒢𝜇_𝑚,𝜇_𝑚,𝜇_𝑚+1=𝒢𝑓^{𝑛(𝜇𝑚−1)}𝜇_𝑚−1,𝑓^{𝑛(𝜇𝑚−1)}𝜇_𝑚−1,𝑓^{𝑛(𝜇𝑚)}𝑓^{𝑛(𝜇𝑚−1)}𝜇_𝑚−1≤Φ𝒢𝜇_𝑚−1,𝜇_𝑚−1,𝑓^{𝑛(𝜇𝑚)}𝜇_𝑚−1≤⋯≤Φ^𝑚𝒢𝜇₀,𝜇₀,𝑓^{𝑛(𝜇𝑚)}𝜇₀≤Φ^𝑚(𝐵), (30) for all 𝑚∈ℕ. Using the last inequality, for every 𝑖,𝑗∈ℕ, 𝑖<𝑗, we have 𝒢𝜇_𝑖,𝜇_𝑖,𝜇_𝑗≤𝒢𝜇_𝑖,𝜇_𝑖,𝜇_𝑖+1+⋯+𝒢𝜇_𝑘−1,𝜇_𝑘−1,𝜇_𝑘≤^𝑗_𝑠=𝑖Φ^𝑠(𝐵), (31) implying that {𝜇_𝑖}_𝑖∈ℕ is a Cauchy sequence. Since (ℱ(𝑋),𝒢) is a complete 𝐺-metric fuzzy space, there exists an 𝜇^∗∈ℱ(𝑋) such that lim_𝑖→∞𝜇_𝑖=𝜇^∗.
In the second part of the theorem, inequality (27) holds for all 𝜇∈𝒪(𝑓;𝜇₀). Then, the elements 𝜇_𝑖 of the sequence {𝜇_𝑖}_𝑖∈ℕ from the previous part of the proof satisfy the next two relations: 𝒢𝑓^{𝑛(𝜇∗)}𝜇^∗,𝑓^{𝑛(𝜇∗)}𝜇^∗,𝑓^{𝑛(𝜇∗)}𝜇_𝑖≤Φ𝒢𝜇^∗,𝜇^∗,𝜇_𝑖,𝒢𝑓^{𝑛(𝜇∗)}𝜇_𝑖,𝜇_𝑖,𝜇_𝑖=𝒢𝑓^{𝑛(𝜇∗)}𝑓^{𝑛(𝜇𝑖−1)}𝜇_𝑖−1,𝑓^{𝑛(𝜇𝑖−1)}𝜇_𝑖−1,𝑓^{𝑛(𝜇𝑖−1)}𝜇_𝑖−1≤Φ𝒢𝑓^{𝑛(𝜇∗)}𝜇_𝑖−1,𝜇_𝑖−1,𝜇_𝑖−1≤Φ^𝑖𝒢𝑓^{𝑛(𝜇∗)}𝜇₀,𝜇₀,𝜇₀. (32)(33) By (32) lim_𝑖→∞𝑓^{𝑛(𝜇∗)}𝜇_𝑖=𝑓^{𝑛(𝜇∗)}𝜇^∗ (34) and by (33) lim_𝑖→∞𝒢𝑓^{𝑛(𝜇∗)}𝜇_𝑖,𝜇_𝑖,𝜇_𝑖=𝒢𝑓^{𝑛(𝜇∗)}𝜇^∗,𝜇^∗,𝜇^∗=0. (35) Hence, 𝑓^{𝑛(𝜇∗)}(𝜇^∗)=𝜇^∗.
To show that 𝜇^∗ is a unique fixed point of 𝑓^{𝑛(𝜇∗)} in 𝒪(𝑓;𝜇₀), we assume that there exists another point 𝑥^∗∗∈𝒪(𝑓;𝜇₀) with the same property. Then 𝒢𝑥^∗∗,𝜇^∗,𝜇^∗=𝒢𝑓^{𝑛(𝜇∗)}𝑥^∗∗,𝑓^{𝑛(𝜇∗)}𝜇^∗,𝑓^{𝑛(𝜇∗)}𝜇^∗≤Φ𝒢𝑥^∗∗,𝜇^∗,𝜇^∗. (36) That is 𝑥^∗∗=𝜇^∗. Further, if 𝑓(𝒪(𝑓;𝜇₀))⊆𝒪(𝑓;𝜇₀), then 𝑓𝜇^∗=𝑓(𝑓^{𝑛(𝜇∗)}𝜇^∗)=𝑓^{𝑛(𝜇∗)}(𝑓𝜇^∗), implying that 𝑓𝜇^∗=𝜇^∗.

5. Subfixed Point for Generalized Contraction Family of Fuzzy Mappings

We say that 𝜇^∗∈ℱ(𝑋) is a subfixed point of the mapping 𝑓∶ℱ(𝑋)→ℱ(𝑋) if and only if 𝜇^∗⊆𝑓(𝜇^∗).

The proof of the next two propositions is the same as in [10]; only, instead of the derived metric 𝑑_𝐺, the metric 𝑑 from original metric space (𝑋,𝑑) is used.

Proposition 17. If 𝜇₁,𝜇₂,𝜈₂∈ℱ(𝑋) and 𝜇₁≤𝜇₂, then there exists a 𝜈₁∈ℱ(𝑋) such that 𝒢(𝜇₁,𝜈₁,𝜈₁)≤𝒢(𝜇₂,𝜈₂,𝜈₂).

Proposition 18. If 𝜇,𝜈,𝜂∈ℱ(𝑋) and 𝜇⊆𝜈, then (i)ℒ(𝜇,𝜂,𝜂)≤ℒ(𝜈,𝜂,𝜂),(ii)ℒ(𝜂,𝜈,𝜈)≤ℒ(𝜂,𝜇,𝜇),(iii)ℒ(𝜇,𝜈,𝜈)=0⇔𝜇⊆𝜈.

In the next two theorems we consider the existence of a fuzzy set 𝜇^∗ which represents a common subfixed point for the family of self-mappings {𝑓_𝑖}, that is, the point such that 𝜇^∗⊆𝑓_𝑖(𝜇^∗) for all 𝑖∈ℕ.

Let (𝑋,𝑑) be a complete metric space, and let (ℱ(𝑋),𝒢) be related 𝐺-metric fuzzy space. Further, let for all 𝜇,𝜈∈ℱ(𝑋) and all 𝑖,𝑗∈ℕ, 𝑖≠𝑗, 𝒢𝑓_𝑖(𝜈),𝑓_𝑗𝜇,𝑓_𝑗𝜇≤Φmax𝒢𝜈,𝜇,𝜇,ℒ𝜈,𝑓_𝑖(𝜈),𝑓_𝑖(𝜈),ℒ𝜇,𝑓_𝑗𝜇,𝑓_𝑗𝜇,ℒ𝜈,𝑓_𝑗𝜇,𝑓_𝑗𝜇,ℒ𝜇,𝑓_𝑖(𝜈),𝑓_𝑖(𝜈), (37) where Φ∶[0,∞)→[0,∞).

If 𝜇₀ is any element from ℱ(𝑋) and a 𝜇₁∈ℱ(𝑋) is chosen such that 𝜇₁⊆𝑓₁(𝜇₀), then, by Proposition 17, there exists a 𝜇₂⊆𝑓₂(𝜇₁), such that 𝒢𝜇₁,𝜇₂,𝜇₂≤𝒢𝑓₁𝜇₀,𝑓₂𝜇₁,𝑓₂𝜇₁. (38) By the same principle as for the first three members, the sequence {𝜇_𝑖} is formed such that 𝜇_𝑖+1⊆𝑓_𝑖+1𝜇_𝑖𝒢𝜇,∀𝑖∈ℕ,_𝑖,𝜇_𝑖+1,𝜇_𝑖+1≤𝒢𝑓_𝑖𝜇_𝑖−1,𝑓_𝑖+1𝜇_𝑖,𝑓_𝑖+1𝜇_𝑖. (39)(40)

Lemma 19. If Φ in (37) satisfies (𝜙₅) together with (𝜙₂) or (𝜙₃) or (𝜙₄), the sequence {𝜇_𝑖} defined in (39) is a Cauchy sequence.

Proof. By (37), for any 𝑖,𝑗∈ℕ, we have 𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗≤𝒢𝑓_𝑖𝜇_𝑖−1,𝑓_𝑗𝜇_𝑗−1,𝑓_𝑗𝜇_𝑗−1≤Φmax𝒢𝜇_𝑖−1,𝜇_𝑗−1,𝜇_𝑗−1,ℒ𝜇_𝑖−1,𝑓_𝑖𝜇_𝑖−1,𝑓_𝑖𝜇_𝑖−1,ℒ𝜇_𝑗−1,𝑓_𝑖𝜇_𝑖−1,𝑓_𝑖𝜇_𝑖−1,ℒ𝜇_𝑗−1,𝑓_𝑗𝜇_𝑗−1,𝑓_𝑗𝜇_𝑗−1,ℒ𝜇_𝑖−1,𝑓_𝑗𝜇_𝑗−1,𝑓_𝑗𝜇_𝑗−1≤Φmax𝒢𝜇_𝑖−1,𝜇_𝑗−1,𝜇_𝑗−1,ℒ𝜇_𝑖−1,𝑓_𝑖𝜇_𝑖−1,𝑓_𝑖𝜇_𝑖−1,ℒ𝜇_𝑗−1,𝑓_𝑖𝜇_𝑖−1,𝑓_𝑖𝜇_𝑖−1,ℒ𝜇_𝑗−1,𝑓_𝑗𝜇_𝑗−1,𝑓_𝑗𝜇_𝑗−1,ℒ𝜇_𝑖−1,𝑓_𝑗𝜇_𝑗−1,𝑓_𝑗𝜇_𝑗−1≤Φmax𝒢𝜇_𝑖−1,𝜇_𝑗−1,𝜇_𝑗−1,ℒ𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖,ℒ𝜇_𝑗−1,𝜇_𝑖,𝜇_𝑖,ℒ𝜇_𝑗−1,𝜇_𝑗,𝜇_𝑗,ℒ𝜇_𝑖−1,𝜇_𝑗,𝜇_𝑗≤Φmax𝒢𝜇_𝑖−1,𝜇_𝑗−1,𝜇_𝑗−1,𝒢𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖,𝒢𝜇_𝑗−1,𝜇_𝑖,𝜇_𝑖,𝒢𝜇_𝑗−1,𝜇_𝑗,𝜇_𝑗,𝒢𝜇_𝑖−1,𝜇_𝑗,𝜇_𝑗. (41) Considering relation (41) for different values 𝑘≤𝑖, 𝑗≤𝑛, we get sup_{𝑘≤𝑖,𝑗≤𝑛}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗≤Φsup_{𝑘−1≤𝑖,𝑗≤𝑛}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗, (42) for all 𝑘,𝑛∈ℕ, 2≤𝑘<𝑛. Knowing that sup_{1≤𝑖,𝑗≤𝑛}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗≤max𝒢𝜇₁,𝜇₂,𝜇₂,𝒢𝜇₂,𝜇₁,𝜇₁+sup_{2≤𝑖,𝑗≤𝑛}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗 (43) and using (42) with 𝑘=2, we obtain sup_{1≤𝑖,𝑗≤𝑛}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗≤max𝒢𝜇₁,𝜇₂,𝜇₂,𝒢𝜇₂,𝜇₁,𝜇₁+Φsup_{1≤𝑖,𝑗≤𝑛}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗. (44) Putting 𝑥=max{𝒢(𝜇₁,𝜇₂,𝜇₂),𝒢(𝜇₂,𝜇₁,𝜇₁)} and applying the property (𝜙₅) in (44), there exists a 𝑦(𝑥)>0 such that sup_{1≤𝑖,𝑗≤𝑛}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗≤𝑦(𝑥). (45) Taking 𝑘=2 in (42) again, using the last relation, we get sup_{2≤𝑖,𝑗≤𝑛}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗≤Φ𝑦(𝑥). (46) Continuing this process, we obtain sup_{𝑘≤𝑖,𝑗≤𝑛}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗≤Φ^𝑘−1𝑦(𝑥), (47) where 𝑘,𝑛∈ℕ, 2≤𝑘<𝑛. Now, letting 𝑘→∞ and using the properties of Φ, finally we show that {𝜇_𝑖} is a Cauchy sequence; that is lim_𝑘→∞sup_{𝑘≤𝑖,𝑗≤𝑛}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗≤lim_𝑘→∞Φ^𝑘−1𝑦(𝑥)=0. (48) Using Lemma 14(ii), property (𝜙₃) can be replaced by (𝜙₂) or (𝜙₄).

Lemma 20. If Φ in (37) satisfies (𝜙₆) together with (𝜙₂) or (𝜙₃) or (𝜙₄), the sequence {𝜇_𝑖} defined in (39) is a Cauchy sequence.

Proof. Using properties of Φ satisfying (𝜙₆) together with (𝜙₄), we prove by contradiction that lim_𝑛→∞sup_{1≤𝑖,𝑗≤𝑛}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗=sup_{1≤𝑖,𝑗}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗<∞. (49) The assumption that (49) is not the case leads to lim_𝑛→∞sup_{1≤𝑖,𝑗≤𝑛}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗=∞. (50) On the other hand, by (𝜙₆) and (44) we obtain ∞=lim_𝑛→∞sup_{1≤𝑖,𝑗≤𝑛}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗−𝜙sup_{1≤𝑖,𝑗≤𝑛}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗≤max𝒢𝜇₁,𝜇₂,𝜇₂,𝒢𝜇₂,𝜇₁,𝜇₁, (51) which is a contradiction implying that (49) is true. Now we define a decreasing sequence {𝑥_𝑘}⊂(0,∞) by 𝑥_𝑘=sup_{𝑘≤𝑖,𝑗}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗 (52) and, using (41), 𝑥_𝑘≤𝜙𝑥_𝑘−1≤⋯≤𝜙^𝑘−1𝑥₁. (53) Letting 𝑘→∞, finally we prove that lim_𝑘→∞sup_{𝑘≤𝑖,𝑗}𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗=0. (54) So, {𝜇_𝑖} is a Cauchy sequence. Using Lemma 14(ii), property (𝜙₄) can be replaced by (𝜙₂) or (𝜙₃).

Theorem 21. If all assumptions from Lemma 19 or from Lemma 20 are satisfied, then there exists a 𝜈^∗∈ℱ(𝑋) such that 𝜈^∗⊆𝑓_𝑖(𝜈^∗) for all 𝑖∈ℕ.

Proof. In Lemma 19 or Lemma 20 it was proved that {𝜇_𝑖} is a Cauchy sequence and, by the completeness of (ℱ(𝑋),𝒢), lim_𝑖→∞𝜇_𝑖=𝜇^∗∈ℱ(𝑋). To prove that 𝜇^∗⊆𝑓_𝑗(𝜇^∗), we proceed as follows: ℒ𝜇^∗,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗≤lim_𝑖→∞ℒ𝜇^∗,𝜇_𝑖,𝜇_𝑖+lim_𝑖→∞ℒ𝜇_𝑖,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗=lim_𝑖→∞𝒢𝑓_𝑖𝜇_𝑖−1,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗≤lim_𝑖→∞Φmax𝒢𝜇_𝑖−1,𝜇^∗,𝜇^∗,ℒ𝜇_𝑖−1,𝑓_𝑖𝜇_𝑖−1,𝑓_𝑖𝜇_𝑖−1,ℒ𝜇^∗,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗,ℒ𝜇_𝑖−1,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗,ℒ𝜇^∗,𝑓_𝑖𝜇_𝑖−1,𝑓_𝑖𝜇_𝑖−1≤lim_𝑖→∞Φmax𝒢𝜇_𝑖−1,𝜇^∗,𝜇^∗,ℒ𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖,ℒ𝜇^∗,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗,ℒ𝜇_𝑖−1,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗,ℒ𝜇^∗,𝜇_𝑖,𝜇_𝑖≤Φmax𝒢𝜇^∗,𝜇^∗,𝜇^∗,ℒ𝜇^∗,𝜇^∗,𝜇^∗,ℒ𝜇^∗,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗,ℒ𝜇^∗,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗,ℒ𝜇^∗,𝜇^∗,𝜇^∗=Φmax0,0,ℒ𝜇^∗,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗,0+ℒ𝜇^∗,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗𝜇,0≤ℒ^∗,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗. (55)
Hence, ℒ(𝜇^∗,𝑓_𝑗(𝜇^∗),𝑓_𝑗(𝜇^∗))=0⇒𝜇^∗⊆𝑓_𝑗(𝜇^∗), for all 𝑗∈ℕ, what we had to prove.

Let 𝜑∶[0,∞)⁵→[0,∞) be nondecreasing continuous from the right function with respect to each of the five variables such that _𝑘∈ℕ𝜑^𝑘(𝑡,𝑡,𝑡,2𝑡,0)<∞, (56) for all 𝑡>0. Obviously, for all 𝑡>0, lim_𝑘𝜑^𝑘()()𝜑()𝑡,𝑡,𝑡,2𝑡,0=0,𝜑𝑡,𝑡,𝑡,2𝑡,0<𝑡,0,0,0,0,0=0. (57) Further, let (𝑋,𝑑) be a complete metric space, let (ℱ(𝑋),𝒢) be related 𝐺-metric fuzzy space, and the family of self-mapings {𝑓_𝑖} satisfy the next inequality: 𝒢𝑓_𝑖(𝜈),𝑓_𝑗𝜇,𝑓_𝑗𝜇≤𝜑𝒢𝜈,𝜇,𝜇,ℒ𝜈,𝑓_𝑖(𝜈),𝑓_𝑖(𝜈),ℒ𝜇,𝑓_𝑗𝜇,𝑓_𝑗𝜇,ℒ𝜈,𝑓_𝑗𝜇,𝑓_𝑗𝜇,ℒ𝜇,𝑓_𝑖(𝜈),𝑓_𝑖(𝜈), (58) for all 𝜇,𝜈∈ℱ(𝑋) and all 𝑖,𝑗∈ℕ.

Using the same arguments as in (39), the sequence {𝜇_𝑖} is formed: 𝜇_𝑖+1⊆𝑓_𝑖+1𝜇_𝑖,∀𝑖∈ℕ, (59) with property 𝒢𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1≤𝒢𝑓_𝑖𝜇_𝑖−1,𝑓_𝑖+1𝜇_𝑖,𝑓_𝑖+1𝜇_𝑖. (60)

Lemma 22. The sequence {𝜇_𝑖} defined in (59) is a Cauchy sequence.

Proof. In order to prove that {𝜇_𝑖} is a Cauchy sequence, we consider the sequence {𝒢(𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1)}. Using relations (58) and (60) and the implication 𝜇_𝑖⊆𝑓_𝑖(𝜇_𝑖−1)⇒ℒ(𝜇_𝑖,𝑓_𝑖(𝜇_𝑖−1),𝑓_𝑖(𝜇_𝑖−1))=0, we get 𝒢𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1≤𝒢𝑓_𝑖𝜇_𝑖−1,𝑓_𝑖+1𝜇_𝑖,𝑓_𝑖+1𝜇_𝑖≤𝜑𝒢𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖,ℒ𝜇_𝑖−1,𝑓_𝑖𝜇_𝑖−1,𝑓_𝑖𝜇_𝑖−1,ℒ𝜇_𝑖,𝑓_𝑖+1𝜇_𝑖,𝑓_𝑖+1𝜇_𝑖,ℒ𝜇_𝑖−1,𝑓_𝑖+1𝜇_𝑖,𝑓_𝑖+1𝜇_𝑖,ℒ𝜇_𝑖,𝑓_𝑖𝜇_𝑖−1,𝑓_𝑖𝜇_𝑖−1≤𝜑𝒢𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖,ℒ𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖,ℒ𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1,ℒ𝜇_𝑖−1,𝜇_𝑖+1,𝜇_𝑖+1𝒢𝜇,0≤𝜑_𝑖−1,𝜇_𝑖,𝜇_𝑖,𝒢𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖,𝒢𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1,𝒢𝜇_𝑖−1,𝜇_𝑖+1,𝜇_𝑖+1,0. (61) Since 𝒢(𝜇_𝑖−1,𝜇_𝑖+1,𝜇_𝑖+1)≤𝒢(𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖)+𝒢(𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1), the next relation holds: 𝒢𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1≤𝜑𝒢𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖,𝒢𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖,𝒢𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1,𝒢𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖+𝒢𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1,0. (62) The assumption 𝒢(𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1)≰𝒢(𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖), which is equivalent to 𝒢(𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1)>𝒢(𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖), leads to inequality 𝒢𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1≤𝜑𝒢𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1,𝒢𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1,𝒢𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1,2𝒢𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1𝜇,0<𝒢_𝑖,𝜇_𝑖+1,𝜇_𝑖+1, (63) which is a contradiction. In the last transformation we used nondecreasingness of 𝜙 and the property 𝜑(𝑡,𝑡,𝑡,2𝑡,0)<𝑡, 𝑡∈ℝ⁺. Hence, 𝒢(𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1)≤𝒢(𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖). With notation 𝒢(𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖)=𝑡_𝑖, we have 𝑡_𝑖+1≤𝜑𝑡_𝑖,𝑡_𝑖,𝑡_𝑖,2𝑡_𝑖,0<𝑡_𝑖≤⋯≤𝜑^𝑖𝑡₁,𝑡₁,𝑡₁,2𝑡₁,0. (64)
If 𝜇₀=𝜇₁, then 𝑡₁=𝒢(𝜇₀,𝜇₁,𝜇₁)=0. Since 𝜑(0,0,0,0,0)=0, from inequality (64) we get (𝜇₀=𝜇₁)⇒(𝜇₁=𝜇₂)⇒⋯⇒(𝜇_𝑖−1=𝜇_𝑖)⇒⋯. Further, 𝜇₀=𝜇_𝑖⊆𝑓_𝑖(𝜇_𝑖−1)=𝑓_𝑖(𝜇₀), which means that 𝜇^∗=𝜇₀ and the proof is completed.
If 𝜇₀≠𝜇₁, we prove that lim_𝑖→∞𝒢(𝜇_𝑖,𝜇_𝑗,𝜇_𝑗)=0 whenever 𝑖<𝑗: 𝒢𝜇_𝑖,𝜇_𝑗,𝜇_𝑗≤𝒢𝜇_𝑖,𝜇_𝑖+1,𝜇_𝑖+1+𝒢𝜇_𝑖+1,𝜇_𝑖+2,𝜇_𝑖+2+⋯+𝒢𝜇_𝑗−1,𝜇_𝑗,𝜇_𝑗=^𝑗−1_𝑘=𝑖𝒢𝜇_𝑘,𝜇_𝑘+1,𝜇_𝑘+1=^𝑗−1_𝑘=𝑖𝑡_𝑘+1≤^𝑗−1_𝑘=𝑖𝜑^𝑘𝑡₁,𝑡₁,𝑡₁,2𝑡₁,0, (65) and, since 𝑡₁=𝒢(𝜇₀,𝜇₁,𝜇₁)>0, lim_𝑖→∞𝒢(𝜇_𝑖,𝜇_𝑗,𝜇_𝑗)=0. Also, 𝒢(𝜇_𝑗,𝜇_𝑖,𝜇_𝑖)≤2𝒢(𝜇_𝑖,𝜇_𝑗,𝜇_𝑗)→0. So, we have proved that {𝜇_𝑖} is a Cauchy sequence and, consequently, there exists a 𝜇^∗∈ℱ(𝑋) such that lim_𝑖→∞𝜇_𝑖=𝜇^∗; that is, lim_𝑖→∞𝒢(𝜇_𝑖,𝜇^∗,𝜇^∗)=lim_𝑖→∞𝒢(𝜇^∗,𝜇_𝑖,𝜇_𝑖)=0.

Theorem 23. If all assumptions from the previous lemma are satisfied, then there exists a 𝜈^∗∈ℱ(𝑋) such that 𝜈^∗⊆𝑓_𝑖(𝜈^∗) for all 𝑖∈ℕ.

Proof. If ℒ(𝜇^∗,𝑓_𝑗(𝜇^∗),𝑓_𝑗(𝜇^∗))=0 for all 𝑗∈ℕ, then 𝜇^∗⊆𝑓_𝑗(𝜇^∗). So, assume that, for some 𝑗∈ℕ, ℒ(𝜇^∗,𝑓_𝑗(𝜇^∗),𝑓_𝑗(𝜇^∗))>0. Then ℒ𝜇^∗,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗≤lim_𝑖→∞ℒ𝜇^∗,𝜇_𝑖,𝜇_𝑖+lim_𝑖→∞ℒ𝜇_𝑖,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗≤lim_𝑖→∞ℒ𝑓_𝑖𝜇_𝑖−1,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗≤lim_𝑖→∞𝒢𝑓_𝑖𝜇_𝑖−1,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗≤lim_𝑖→∞𝜑𝒢𝜇_𝑖−1,𝜇^∗,𝜇^∗,ℒ𝜇_𝑖−1,𝑓_𝑖𝜇_𝑖−1,𝑓_𝑖𝜇_𝑖−1,ℒ𝜇^∗,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗,ℒ𝜇_𝑖−1,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗,ℒ𝜇^∗,𝑓_𝑖𝜇_𝑖−1,𝑓_𝑖𝜇_𝑖−1≤lim_𝑖→∞𝜑𝒢𝜇_𝑖−1,𝜇^∗,𝜇^∗,ℒ𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖,ℒ𝜇^∗,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗,ℒ𝜇_𝑖−1,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗,ℒ𝜇^∗,𝜇_𝑖,𝜇_𝑖. (66) If we put 𝑎=ℒ(𝜇^∗,𝑓_𝑗(𝜇^∗),𝑓_𝑗(𝜇^∗))>0, since lim_𝑖→∞𝜇_𝑖=𝜇^∗, for every 𝜀∈(0,𝑎) there exists an 𝑖₀=max{𝑖₁,𝑖₂,𝑖₃,𝑖₄}, where (i)𝒢(𝜇_𝑖−1,𝜇^∗,𝜇^∗)<𝜀<𝑎 for all 𝑖>𝑖₁,(ii)ℒ(𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖)<𝜀<𝑎 for all 𝑖>𝑖₂,(iii)ℒ(𝜇_𝑖−1,𝑓_𝑗(𝜇^∗),𝑓_𝑗(𝜇^∗))≤ℒ(𝜇_𝑖−1,𝜇_𝑖,𝜇_𝑖)+ℒ(𝜇_𝑖,𝑓_𝑗(𝜇^∗),𝑓_𝑗(𝜇^∗))<(𝜀/2)+𝑎+(𝜀/2)<2𝑎, for all 𝑖>𝑖₃,(iv)ℒ(𝜇^∗,𝜇_𝑖,𝜇_𝑖)<𝜀 for all 𝑖>𝑖₄. Now, relation (66) becomes ℒ𝜇^∗,𝑓_𝑗𝜇^∗,𝑓_𝑗𝜇^∗()()≤𝜑𝜀,𝜀,𝑎,𝜀+𝑎,𝜀≤𝜑𝑎,𝑎,𝑎,2𝑎,𝜀 (67) and, letting 𝜀→0, we get 𝑎≤𝜑(𝑎,𝑎,𝑎,2𝑎,0)<𝑎, where 𝑎=ℒ(𝜇^∗,𝑓_𝑗(𝜇^∗),𝑓_𝑗(𝜇^∗)). Hence, ℒ(𝜇^∗,𝑓_𝑗(𝜇^∗),𝑓_𝑗(𝜇^∗))=0⇒𝜇^∗⊆𝑓_𝑗(𝜇^∗), for all 𝑗∈ℕ, what we had to prove.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was partially supported by Ministarstvo Nauke i Životne Sredine Republike Srbije.