Coupled Fixed Point Theorems for a Pair of Weakly Compatible Maps along with CLRg Property in Fuzzy Metric Spaces

Definition 2.1 (see [1].)A fuzzy set A in X is a function with domain X and values in [0,1].

Definition 2.2 (see [10].)A binary operation * : [0,1]×[0,1]→[0,1] is continuous t-norm, if ([0,1], *) is a topological abelian monoid with unit 1 such that a*b ≤ c*d whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0,1].

Some examples are below:

• (i)

  *(a, b) = ab,

• (ii)

  *(a, b) = min (a, b).

Definition 2.3 (see [11].)Let sup _t∈(0,1)Δ(t, t) = 1. A t-norm Δ is said to be of H-type if the family of functions $$ is equicontinuous at t = 1, where

A t-norm Δ is an H-type t-norm if and only if for any λ ∈ (0,1), there exists δ(λ)∈(0,1) such that Δ^m(t)>(1 − λ) for all m ∈ ℕ, when t > (1 − δ).

The t-norm Δ_M = min  is an example of t-norm, of H-type.

Definition 2.4 (see [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:

• (FM-1)

  M(x, y, 0) > 0 for all x, y ∈ X,

• (FM-2)

  M(x, y, t) = 1 if and only if x = y, for all x, y ∈ X and t > 0,

• (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 continuous for all x, y ∈ X.

Definition 2.5 (see [2].)Let (X, M, *) be a fuzzy metric space. A sequence {x_n} ∈ X is said to be:

• (i)

  convergent to a point x ∈ X, if for all t > 0,

  $$ ()

• (ii)

  a Cauchy sequence, if for all t > 0 and p > 0,

  $$ ()

Lemma 2.6 (see [12].)Let x_n → x and y_n → y, then for all t > 0:

• (i)

  lim _n→∞M(x_n, y_n, t) ≥ M(x, y, t),

• (ii)

  lim _n→∞M(x_n, y_n, t) = M(x, y, t) if M(x, y, t) is continuous.

Definition 2.7 (see [7].)Define Φ = {ϕ : ℝ⁺ → ℝ⁺}, and each ϕ ∈ Φ satisfies the following conditions:

•  

  ϕ ϕ is nondecreasing;

•  

  ϕ ϕ is upper semicontinuous from the right;

•  

  ϕ $$ for all t > 0, where ϕⁿ⁺¹(t) = ϕ(ϕⁿ(t)), n ∈ ℕ.

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

Definition 2.8 (see [4].)An element (x, y) ∈ X × X is called:

• (i)

  a coupled fixed point of the mapping f : X × X → X if f(x, y) = x, f(y, x) = y,

• (ii)

  a coupled coincidence point of the mappings f : X × X → X and g : X → X if f(x, y) = g(x),  f(y, x) = g(y),

• (iii)

  a common coupled fixed point of the mappings f : X × X → X and g : X → X if x = f(x, y) = g(x),  y = f(y, x) = g(y).

Definition 2.9 (see [6].)An element x ∈ X is called a common fixed point of the mappings f : X × X → X and g : X → X if x = f(x, x) = g(x).

Definition 2.10 (see [6].)The mappings f : X × X → X and g : X → X are called:

• (i)

  commutative if gf(x, y) = f(gx, gy) for all x, y ∈ X,

• (ii)

  compatible if

  $$ ()
  for all t > 0 whenever {x_n} and {y_n} are sequences in X, such that  lim _n→∞f(x_n, y_n) = lim _n→∞g(x_n) = x, and lim _n→∞f(y_n, x_n) = lim _n→∞g(y_n) = y, for some x, y ∈ X.

Definition 2.11 (see [13].)The maps f : X × X → X and g : X → X are called w-compatible if gf(x, y) = f(gx, gy) whenever f(x, y) = g(x), f(y, x) = g(y).

We note that the maps f : X × X → X and g : X → X are called weakly compatible if

implies gf(x, y) = f(gx, gy),  gf(y, x) = f(gy, gx), for all x, y ∈ X.

Example 2.12. Let (X, M, *) be a fuzzy metric space, * being a continuous norm with X = [0,1). Define M(x, y, t) = t/(t+|x − y|) for all t > 0,  x, y ∈ X. Also define the maps f : X × X → X and g : X → X by f(x, y) = (x²/2) + (y²/2) and g(x) = x/2, respectively. Note that (0, 0) is the coupled coincidence point of f and g in X. It is clear that the pair (f, g) is weakly compatible on X.

We next show that the pair (f, g) is not compatible.

Consider the sequences {x_n} = {(1/2) + (1/n)} and {y_n} = {(1/2) − (1/n)}, n ≥ 3, then

but which is not convergent to 1 as n → ∞.

Hence the pair (f,  g) is not compatible.

Example 2.13. Let (X, M, *) be a fuzzy metric space, * being a continuous norm with X = [2,20]. Define M(x, y, t) = t/(t+|x − y|) for all t > 0,  x, y ∈ X. Define the maps f : X × X → X and g : X → X by

The only coupled coincidence point of the pair (f, g) is (2,2). The mappings f and g are noncompatible, since for the sequences {x_n} = {y_n} = {5 + (1/n)}, n ≥ 1 we have f(x_n, y_n) = 2, g(x_n) → 2, f(y_n, x_n) = 2, g(y_n) → 2, M(f(gx_n, gy_n), g(f(x_n, y_n)), t) = t/(t + 4)↛1 as n → ∞. But they are weakly compatible since they commute at their coupled coincidence point (2,2).

Definition 2.14. Let (X, d) be a metric space. Two mappings f : X × X → X and g : X → X are said to satisfy E.A. property if there exists sequences {x_n}, {y_n} ∈ X such that

for some x, y ∈ X.

Example 2.15. Let (−∞, ∞) be a usual metric space. Define mappings f : X × X → X and g : X → X by f(x, y) = x² + y² and g(x) = 2x for all x, y ∈ X. Consider the sequences {x_n} = {1/n} and {y_n} = {−1/n}. Since

therefore, f and g satisfy E.A. property, since 0 ∈ X.

Remark 2.16. It is to be noted that property E.A. need not imply compatibility, since in Example 2.12, the maps f and g defined are not compatible, but satisfy property E.A., since for the sequences {x_n} = {(1/2) + (1/n)} and {x_n} = {(1/2) − (1/n)} we have

since 1/4 ∈ X.

Definition 2.17. Let (X, d) be a metric space. Two mappings f : X → X and g : X → X are said to satisfy (CLRg) property if there exists a sequence {x_n} ∈ X  such that   lim _n→∞f(x_n) = lim _n→∞g(x_n) = g(p)  for some  p ∈ X.

Definition 2.18. Let (X, d) be a metric space. Two mappings f : X × X → X and g : X → X are said to satisfy (CLRg) property if there exists sequences {x_n}, {y_n} ∈ X such that

for some p, q ∈ X.

Example 2.19. Let X = [0, ∞) be a metric space under usual metric. Define mappings f : X × X → X and g : X → X by f(x, y) = x + y + 2 and g(x) = 2(1 + x) for all x, y ∈ X. We consider the sequences {x_n} = {1 + (1/n)} and {x_n} = {1 − (1/n)}. Since

therefore, the maps f and g satisfy (CLRg) property.

Example 2.20. Let X = [0, ∞) be a metric space under usual metric. Define mappings f : X × X → X and g : X → X by

We consider the sequences {x_n} = {1/n} and {y_n} = {1 + (1/n)}. Since therefore, the maps f and g  satisfy (CLRg) property but the maps are not continuous.

Example 2.21. Let (X, M, *) be a fuzzy metric space, * being a continuous norm,  X = [0,1/2), and M(x, y, t) = t/(t+|x − y|) for all x, y ∈ X and t > 0.

Define the maps f : X × X → X and g : X → X by f(x, y) = (x² + y²)/2 and g(x) = x/3, respectively.

Consider the sequences {x_n} = {(1/3) + (1/n)} and {y_n} = {(1/3) − (1/n)}, n > 7. Then

Further there exists the point 1/3 in X such that g(1/3) = 1/9, so that the pair (f, g) satisfies (CLRg) property. But, does not converge to 1 as n → ∞.

Hence, the pair (f, g) is not compatible.

Theorem 3.1. Let (X, M, *) be a complete fuzzy metric space where * is a continuous t-norm of H-type. Let f : X × X → X and g : X → X be two mappings and there exists ϕ ∈ Φ such that

• (2)
  $$ ()
  for all x, y, u, v ∈ X and t > 0. Suppose that f(X × X)⊆g(X), g is continuous, f and g are compatible maps. Then there exists a unique point x ∈ X such that x = g(x) = f(x, x), that is, f and g have a unique common fixed point in X.

Theorem 3.2. Let (X, M, *) be a Fuzzy Metric Space, * being continuous t-norm of H-type. Let f : X × X → X and g : X → X be two mappings and there exists ϕ ∈ Φ satisfying (2) with the following conditions:

• (3)

  the pair (f, g) is weakly compatible,

• (4)

  the pair (f, g) satisfy (CLRg) property.

Then f and g have a coupled coincidence point in X. Moreover, there exists a unique point x ∈ X such that x = f(x, x) = g(x).

Proof . Since f and g satisfy (CLRg) property, there exists sequences {x_n} and {y_n} in X such that

for some p, q ∈ X.

Step  1. To show that f and g have a coupled coincidence point. From (2),

Taking limit n → ∞, we get M(g(p), f(p, q), t) = 1, that is, f(p, q) = g(p) = x.

Similarly, f(q, p) = g(q) = y.

Since f and g are weakly compatible, so that f(p, q) = g(p) = x(say) and f(q, p) = g(q) = y(say) implies gf(p, q) = f(g(p), g(q)) and gf(q, p) = f(g(q), g(p)), that is, g(x) = f(x, y) and g(y) = f(y, x). Hence f and g have a coupled coincidence point.

Step  2. To show that g(x) = x, and   g(y) = y. Since * is a t-norm of H-type, for any ϵ > 0, there exists δ > 0 such that

for all p ∈ ℕ.

Since lim _t→∞M(x, y, t) = 1 for all x, y ∈ X, there exists t₀ > 0 such that

Also since ϕ ∈ Φ using condition (ϕ − 3), we have $$.

Then for any t > 0, there exists n₀ ∈ ℕ such that $$. From (2), we have

Similarly, we can also get Continuing in the same way, we can get for all n ∈ ℕ, Then, we have So, for any ϵ > 0, we have M(gx, x, t)≥(1 − ϵ) for all t > 0.

This implies g(x) = x. Similarly, g(y) = y.

Step  3. Next we shall show that x = y. Since * is a t-norm of H-type, for any ϵ > 0 there exists δ > 0 such that

for all p ∈ ℕ.

Since lim _t→∞M(x, y, t) = 1 for all x, y ∈ X, there exists t₀ > 0 such that M(x, y, t₀)≥(1 − δ).

Also since ϕ ∈ Φ, using condition (ϕ-3), we have $$. Then for any t > 0, there exists n₀ ∈ ℕ such that

Using condition (2), we have Continuing in the same way, we can get for all n₀ ∈ ℕ, Then we have which implies that x = y. Thus, we have proved that f and g have a common fixed point x ∈ X.

Step  4. We now prove the uniqueness of x. Let z be any point in X such that z ≠ x with g(z) = z = f(z, z). Since * is a t-norm of H-type, for any ϵ > 0, there exists δ > 0 such that

for all p ∈ ℕ. Since lim _t→∞M(x, y, t) = 1 for all x, y ∈ X, there exists t₀ > 0 such that M(x, z, t₀)≥(1 − δ). Also since ϕ ∈ Φ and using condition (ϕ-3), we have $$. Then for any t > 0, there exists n₀ ∈ ℕ such that Using condition (2), we have Continuing in the same way, we can get for all n ∈ ℕ, Then we have which implies that x = z.

Hence f and g have a unique common fixed point in X.

Remark 3.3. We still get a unique common fixed point if weakly compatible notion is replaced by w-compatible notion.

Corollary 3.4. Let (X, M, *) be a fuzzy metric space where * is a continuous t-norm of H-type. Let f : X × X → X and g : X → X be two mappings and there exists ϕ ∈ Φ satisfying (2) and (3) with the following condition:

• (5)

  the pair (f, g) satisfy E.A. property.

If g(X) is a closed subspace of X, then f and g have a unique common fixed point in X.

Proof. Since f and g satisfy E.A. property, there exists sequences {x_n} and {y_n} in X such that

for some x, y ∈ X.

It follows from g(X) being a closed subspace of X that x = g(p), y = g(q) for some p, q ∈ X and then f and g satisfy the (CLRg) property. By Theorem 3.2, we get that f and g have a unique common fixed point in X.

Corollary 3.5. Let (X, M, *) be a fuzzy metric space where * is a continuous t-norm of H-type. Let f : X × X → X and g : X → X be two mappings and there exists ϕ ∈ Φ satisfying (2), (3), and (5).

Suppose that f(X × X)⊆g(X), if range of one of the maps f or g is a closed subspace of X, then f and g have a unique common fixed point in X.

Proof. It follows immediately from Corollary 3.5.

Corollary 3.6. Let (X, M, *) be a fuzzy metric space where * is a continuous t-norm of H-type. Let f : X × X → X and g : X → X be two mappings and there exists ϕ ∈ Φ satisfying the following conditions, for all x, y, u, v ∈ X and t > 0:

• (6)

  M(f(x, y), f(u, v), ϕ(t)) ≥ M(x, u, t)*M(y, v, t),

• (7)

  there exists sequences {x_n} and {y_n} in X such that

  $$ ()
  for some x, y ∈ X.