A Common Fixed Point Theorem in Fuzzy Metric Spaces with Nonlinear Contractive Type Condition Defined Using Φ-Function

Definition 1 (see [5].)A binary operation * : [0,1]×[0,1]→[0,1] is a continuous t-norm if ([0,1], *) is a topological monoid with unit 1 such that a*b ≤ c*d whenever a ≤ c and b ≤ d are satisfied, for all a, b, c, d ∈ [0,1].

Example 2. a*b = ab,  a*b = min{a, b}.

Definition 3 (see [1].)A fuzzy set A in X is a function with domain in X and values in [0,1].

Definition 4 (see [3].)The 3-tuple (X, M, *) is said to be a fuzzy metric space in the sense of Kramosil and Michálek, where X is an arbitrary set, * is a continuous t-norm, and M is a fuzzy set defined on set X² × [0, ∞) satisfying the following conditions:

• (FM-1)

  M(x, y, 0) = 0,

• (FM-2)

  M(x, y, t) = 1, for every t > 0 if and only if x = y,

• (FM-3)

  M(x, y, t) = M(y, x, t), for all x, y ∈ X and t > 0,

• (FM-4)

  M(x, y, t)*M(y, z, s) ≤ M(x, z, t + s), for all x, y, z ∈ X and t, s > 0,

• (FM-5)

  M(x, y, ·):(0, ∞)  →  [0,1] is a left continuous function, for all x, y ∈ X.

Remark 5 (see [4].)Let (X, d) be a metric space and a*b = ab and

for k, m, n ∈ ℝ⁺. Then, (X, M, *) is a fuzzy metric space. The 3-tuple (X, M, *) is a fuzzy metric space, also, if we observe a t-norm a*b = min {a, b} instead of a*b = ab. Taking that k = m = n = 1 in (1), we get

A fuzzy metric given by (2) is called the standard fuzzy metric, and it is induced by a metric d.

Lemma 6 (see [6].)Let (X, M, *) be a fuzzy metric space. Then, M(x, y, ·) is a nondecreasing function, for all x, y ∈ X.

Definition 7 (see [4].)A sequence {x_n} in a fuzzy metric space (X, M, *) is said to converge to x ∈ X if and only if for every ε > 0 and t > 0, there exists n₀ ∈ N such that M(x_n, x, t) > 1 − ε, for every n > n₀, that  is, lim _n→∞M(x_n, x, t) = 1, for every t > 0. In that case, write one lim _n→∞x_n = x.

Definition 8 (see [4].)A sequence {x_n} in a fuzzy metric space (X, M, *) is said to be a Cauchy sequence if for every ε > 0 and t > 0, there exists n₀ ∈ ℕ such that M(x_n, x_m, t) > 1 − ε, for all n, m > n₀. A fuzzy metric space is said to be complete if every Cauchy sequence is convergent.

Lemma 9. Let (X, M, *) be a fuzzy metric space and lim _n→∞x_n = x and lim _n→∞y_n = y. Then, for every fixed t > 0, it holds

Definition 10 (see [4].)Let (X, M, *) be a fuzzy metric space. Open ball B(x, r, t) with center in x ∈ X, and radius r ∈ (0,1), for t > 0 is defined as follows:

Topology in fuzzy metric spaces is defined as follows:

Theorem 11 (see [4].)Every open ball is an open set.

Theorem 12 (see [4].)Every fuzzy metric space is Hausdorff.

Definition 13. Let (X, M, *) be a fuzzy metric space and A⊆X. The closure of the set A is the smallest closed set containing A, denoted by $$.

Remark 14. $$ if and only if there exists a sequence {x_n} in A such that x_n → x.

Definition 15 (see [7].)Let (X, M, *) be a fuzzy metric space. A collection {F_n} _n∈ℕ is said to have fuzzy diameter zero if for each r ∈ (0,1) and each t > 0, there exists n₀ ∈ ℕ such that M(x, y, t) > 1 − r, for all $$.

Theorem 16 (see [7].)A fuzzy metric space (X, M, *) is complete if and only if every nested sequence {F_n} _n∈ℕ of nonempty closed sets with fuzzy diameter zero has nonempty intersection.

Remark 17 (see [7].)The element x ∈ ⋂_n∈ℕ F_n is unique.

Definition 18. Let (X, M, *) be a fuzzy metric space. Let the mappings δ_A(t):(0, ∞)→[0,1] be defined as

The constant δ_A  =  sup _t>0δ_A(t) will be called fuzzy diameter of set A. If δ_A = 1 the set A will be called F-strongly bounded.

Lemma 19. Let (X, M, *) be a fuzzy metric space. A set A⊆X is an F-strongly bounded if and only if for each r ∈ (0,1), there exists t > 0 such that M(x, y, t) > 1 − r, for all x, y ∈ A.

Proof. The proof follows from the definitions of sup A and inf  A of nonempty sets.

Example 20. Let (X, M, *) be a fuzzy metric space induced by a metric d on X given in Remark 5. A⊆X is metrically bounded if and only if it is an F-strongly bounded.

Proof. Let A⊆X be metrically bounded, that is, d(x, y) < k, for some k ∈ ℝ and all x, y ∈ A. Let r ∈ (0,1) be arbitrary. We will prove that there exists t > 0 such that M(x, y, t) > 1 − r, for all x, y ∈ A. Let us take t such that t > k(1 − r)/r. For this t, it follows that tr > k(1 − r) from which it follows that 0 > k(1 − r) − tr. Finally, from the last inequality, we get that t > k(1 − r) + t − tr, that  is, t/(t + k) > 1 − r. Since d(x, y) < k, for all x, y ∈ A, it follows that M(x, y, t) = t/(t + d(x, y)) > t/(t + k) > 1 − r, for all x, y ∈ A. Applying Lemma 19, we get that A is an F-strongly bounded.

Conversely, if A is an F-strongly bounded set, then for arbitrary r ∈ (0,1), there exists t > 0 such that M(x, y, t) = t/(t + d(x, y)) > 1 − r, for all x, y ∈ A. From this inequality, it follows that d(x, y) < rt/(1 − r), for all x, y ∈ A, that is, the set A is metrically bounded. This completes the proof.

Definition 21 (see [13].)A function h : [0, ∞)→[0, ∞) is an altering distance function if

• (i)

  h is monotone increasing and continuous,

• (ii)

  h(t) = 0 if and only if t = 0.

Definition 22 (see [14].)A function ϕ : [0, ∞)  →  [0, ∞) is said to be a Φ-function if the following conditions hold

• (i)

  ϕ(t) = 0 if and only if t = 0,

• (ii)

  ϕ is strictly increasing and ϕ(t) → ∞ as t → ∞,

• (iii)

  ϕ is left continuous in (0, ∞),

• (iv)

  ϕ is continuous at 0.

The class of all Φ-functions will be denoted by Φ.

Theorem 23 (see [14].)Let (X, ℱ, T_M) be a complete Menger PM-space, with continuous t-norm T_M given by T_M(a, b) = min {a, b}, and let f be a continuous self-mapping on X such that for every x, y ∈ X, and all t > 0 holds

where ϕ is a ϕ-function and 0 < c < 1. Then, f has a unique fixed point.

Theorem 24 (see [12].)Let (X, ℱ, T) be a complete Menger PM-space with continuous t-norm T, and let f be a self-mapping on X such that for every x, y ∈ X, and all t > 0 holds

where ϕ is a Φ-function and 0 < c < 1. If there exists x ∈ X such that the orbit of f in x, 𝒪(f, x) = {f^mx, m ∈ ℕ∖{0}} is probabilistic bounded, then f has a unique fixed point.

Definition 25. Let (X, M, *) be a fuzzy metric space, and let f and g be self-mappings of X. The mappings f and g will be called R-weakly commutings, if there exists some positive real number R such that

for all t > 0 and each x ∈ X.

Lemma 26. Let (X, M, *) be a fuzzy metric space. Let ϕ be a Φ-function and 0 < c < 1. If for x, y ∈ X, it holds that

for all t > 0, then x = y.

Proof. Let us suppose that M(x, y, ϕ(t)) ≥ M(x, y, ϕ(t/c)) and x ≠ y. From this condition, we have M(x, y, ϕ(t)) ≤ M(x, y, ϕ(ct)) and, by induction, we have M(x, y, ϕ(t)) ≤ M(x, y, ϕ(cⁿt)). Taking limit as n →  ∞, we get that M(x, y, ϕ(t)) = 0, for all t > 0, which is a contradiction, that is, x = y.

Theorem 27. Let (X, M, *) be a complete fuzzy metric space. Let f and g be R-weakly commuting self-mappings on X, and let f be continuous such that g(X)⊆f(X). Let, for all x, y ∈ X and every t > 0,

hold, where ϕ is a Φ-function and 0 < c < 1. If there exists a point u₀ ∈ X and n₀ ∈ ℕ such that the set where f(u_i) = g(u_i−1) (i ∈ ℕ) is an F-strongly bounded set, then the mappings g and f have a unique common fixed point in X.

Proof. First, we will prove that the mapping g is continuous. We will prove that for sequence y_n → y, it follows that g(y_n) → g(y) as n → ∞. Let t > 0 be arbitrary. For arbitrary t > 0, there exists s > 0 such that t > ϕ(s). Since f is continuous, it follows that f(y_n) → f(y), that is, for all t > 0, it holds that M(f(y_n), f(y), (t)) → 1 as n → ∞. Since the function M(x, y, ·) is nondecreasing for all x, y ∈ X, applying (11), we have

as n → ∞, that  is, M(g(y_n), g(y), t) → 1 as n → ∞, for all t > 0, that is, g(y_n) → g(y) as n → ∞.

For u₀ ∈ X from g(X)⊆f(X), it follows that there exists a point u₁ ∈ X such that g(u₀) = f(u₁). By induction, a sequence {u_n} can be chosen such that g(u_n−1) = f(u_n), and the set $$ is an F-strongly bounded.

Let us consider nested sequence of nonempty closed sets defined by

We will prove that the family $$ has fuzzy diameter zero.

Let r ∈ (0,1) and t > 0 be arbitrary. We will prove that there exists n^* ∈ ℕ such that M(x, y, t) > 1 − r, for all $$. For arbitrary k ≥ n₀,   k ∈ ℕ, from $$, it follows that F_k is an F-strongly bounded set, that is, there exists t₀ > 0 such that

Since ϕ is a Φ-function, it follows that there exists l ∈ ℕ such that ϕ(t/c^l) > t₀. Let n^* = l + k and $$ be arbitrary. There exist sequences {g(u_n(i)−1)},  {g(u_n(j)−1)} in $$ such that lim _n(i)→∞ g(u_n(i)−1) = x and lim _n(j)→∞g(u_n(j)−1) = y.

Since for arbitrary t > 0, there exists s > 0 such that t > ϕ(s), from (11), we have

Thus, by induction, we get

Since ϕ(t/c^l) > t₀ and because M(x, y, ·) is a nondecreasing function, from previous inequalities, it follows that

As {g(u_n(i)−l−1)} and {g(u_n(j)−l−1)} are sequences in F_k from (15) and (18), it follows that, for all i, j ∈ ℕ, it holds

Taking liminf  as n(i),  n(j) → ∞ and applying Lemma 9, we get that M(x, y, t) > 1 − r, for all $$, that is, the family $$ has fuzzy diameter zero.

Applying Theorem 16, we conclude that this family has nonempty intersection, which consists of exactly one point z. Since the family $$ has fuzzy diameter zero and z ∈ F_n, for all n ≥ n₀, then for each r ∈ (0,1) and each t > 0 there exists k₀ ≥ n₀ such that for all n ≥ k₀ holds

From the last, it follows that for each r ∈ (0,1) and each t > 0 holds

Taking that r → 0, we get that for each t > 0, it holds that

that is, lim _n→∞g(u_n−1) = z. From the definition of {f(u_n)}, we have lim _n→∞f(u_n) = z.

Since f and g are R-weakly commutings, we have that for each t > 0 it holds that

Taking liminf  as n → ∞ from previous inequality, for each t > 0, it holds that

that is, it holds that

Let us prove that f(z) = z. From (11), it follows that for every t > 0, it holds that

Taking liminf  as n → ∞, we have that M(g(z), z, ϕ(t)) ≥ M(f(z), z, ϕ(t/c)) holds, for every t > 0. Since f(z) = g(z), it follows that

Applying Lemma 26, we get that f(z) = z, that  is, z is a fixed point of f. From (25), it follows that g(z) = z, that  is, z is a common fixed point of f and g.

Let us prove that z is a unique common fixed point. For this purpose, let us suppose that there exists another common fixed point, denoted by u. From the starting condition, follow

for every t > 0. Therefore, we get that for every t > 0. Finally, applying Lemma 26, it follows that z = u. This completes the proof.

Theorem 28. Let (X, M, *) be a complete fuzzy metric space. Let g be a self-mapping on X such that for all x, y ∈ X and every t > 0, it holds

where ϕ is a Φ-function and 0 < c < 1. If there exists a point u₀ ∈ X and n₀ ∈ ℕ such that the set where u_i = g(u_i−1) (i ∈ ℕ) is an F-strongly bounded set, then the mapping g has a unique common fixed point in X.

Example 29. Let (X, M, *) be a complete fuzzy metric space induced by the metric d(x, y) = |x − y| on X = [0, +∞) ⊂ ℝ given in Remark 5. Let f(x) = 4x,  g(x) = 4x/(4 + x),  g(X) = [0,4) ⊂ X = f(X),  ϕ(t) = tm and c = 1/4.

We will prove that all the conditions of Theorem 27 are satisfied. Because

we conclude that f(x) and g(x) are not commuting mappings. On the other hand, we have that

Since 12x²/(4 + x)(1 + x) ≤ (4x² + 12x)/(1 + x), for every x ≥ 0, we have

that is, for R = 1, the condition (9) is satisfied, that is, we conclude that f(x) and g(x) are R-weakly commutings, for R = 1.

We will prove that the condition (11) is satisfied too. Since 16 | x − y|/(4 + x)(4 + y)≤|x − y|, for all x, y ≥ 0, then we have

Since all the conditions of Theorem 27 are satisfied, we have that f(x) and g(x) have a unique common fixed point. It is easy to see that this point is x = 0.