Definition 1. Let G be any set. A fuzzy set A in G is a function whose domain is G and the range is [0,1].

Definition 2 (see [35].)A binary operation ∗ : [0,1] × [0,1]⟶[0,1] would be a continuous t-norm if ∗ fulfils the following conditions:

• (i)

  ∗ is associative and commutative

• (ii)

  ∗ is continuous

• (iii)

  1∗α = α,  ∀ α ∈ [0,1]

• (iv)

  α∗β ≤ γ∗δ whenever α ≤ γ and β ≤ δ,  for α, β, γ, δ ∈ [0,1]

Definition 3 (see [9].)Let E be a real Banach space, and P is a subset of E. Then, P is called a cone if

• (i)

  P is closed and nonempty and P ≠ {0}

• (ii)

  If α, β ∈ R, α, β ≥ 0 and g, h ∈ P, then αg + βh ∈ P

• (iii)

  If both g ∈ P and −g ∈ P, then g = 0

A partial ordering on a given cone P ⊂ E is defined by $$. g ⪯ h stands for $$ and g ≠ h, while g ≪ h stands for h − g ∈ int(P). In this paper, all cones have a nonempty interior.

Definition 4 (see [21].)A 3-tuple (G, M_c, ∗) is said to be an FM-space if G is any set, ∗ is continuous t-norm, and M_c is a fuzzy set on G² × (0, ∞) satisfying

• (i)

  M_c(g, h, t) > 0

• (ii)

  M_c(g, h, t) = 0 if and only if g = h

• (iii)

  M_c(g, h, t) = M_c(h, g, t)

• (iv)

  M_c(g, k, t) + M_c(k, h, s) ≤ M_c(g, h, t + s)

• (v)

  M_c(g, h, ⋅) : (0, ∞)⟶[0,1] is continuous, for g, h, k ∈ G and t, s > 0

Definition 5 (see [25].)A 3-tuple (G, M_c, ∗) is said to be an FCM-space if P is a cone of E, G is an arbitrary set, ∗ is continuous t-norm, and M_c is a fuzzy set on G² × int(P) satisfying

• (i)

  M_c(g, h, t) > 0

• (ii)

  M_c(g, h, t) = 0 if and only if g = h

• (iii)

  M_c(g, h, t) = M_c(h, g, t)

• (iv)

  M_c(g, k, t) + M_c(k, h, s) ≤ M_c(g, h, t + s)

• (v)

  M_c(g, h, ⋅) : int(P)⟶[0,1] is continuous, for g, h, k ∈ G, and t, s ≫ 0

Definition 6 (see [25].)Let a 3-tuple (G, M_c, ∗) be an FCM-space, h₁ ∈ G, and a sequence {h_ℓ} in G is

• (i)

  Converging to h₁ if γ ∈ (0,1) and t ≫ 0 and there is ℓ₁ ∈ ℕ such that M_c(h_ℓ, h₁, t) > 1 − γ, for ℓ ≥ ℓ₁. We may write this lim_ℓ⟶∞h_ℓ = h₁ or h_ℓ⟶h₁ as ℓ⟶∞;

• (ii)

  Cauchy sequence if γ ∈ (0,1) and t ≫ 0 and there is ℓ₁ ∈ ℕ such that M_c(h_ℓ, h_j, t) > 1 − γ, for ℓ, j ≥ ℓ₁;

• (iii)

  (G, M_c, ∗) complete if every Cauchy sequence is convergent in G;

• (iv)

  Fuzzy cone contractive if ∃ α ∈ (0,1), satisfying

Lemma 1. (see [25]). Let (G, M_c, ∗) be an FCM-space and let a sequence {h_ℓ} in G converge to a point h₁ ∈ G if M_c(h_ℓ, h₁, t)⟶1 as ℓ⟶∞, for t ≫ 0.

Definition 7. (see [26]). Let (G, M_c, ∗) be an FCM-space. The fuzzy cone metric M_c is triangular if

Definition 8. (see [25]). Let (G, M_c, ∗) be an FCM-space and $$. Then $$ is said to be fuzzy cone contractive if there exists α ∈ (0,1) such that

Definition 9 (see [31].)An element (g, h) ∈ G × G is called coupled fixed point of a mapping $$ if

Now, in the following main results, we shall prove some unique coupled FP theorems depending on another function which is continuous, one-one, and subsequently convergent in FCM-spaces. We present some illustrative examples in support of our results. As a further study, we shall present two Volterra integral equations to ensure the existence of common solution to support our work.

Theorem 1. Let $$ be a mapping in a complete FCM-space (G, M_c, ∗) in which M_c is triangular and $$ is a continuous, one-one, and subsequently convergent self-mapping on G, that is, $$, satisfying

for all g, h, ξ, η ∈ G, t ≫ 0, and α, β, γ ∈ [0,1] with α + 2β + 2γ < 1. Then $$ has a unique coupled FP. Also, if $$ converges sequently, then for every g₀ ∈ G the iterative sequence $$ converges to this coupled FP.

Proof. Consider any g₀, h₀ ∈ G; we define sequences {g_ℓ} and {h_ℓ} in G such that

Now, from (5) for t ≫ 0, we have

After simplification, we get that

where θ = (α + β + γ)/(1 − β − γ) < 1. Similarly, from (5), for t ≫ 0, we have where θ is the same as in (8). Now, from (8) and (9) and by induction, for t ≫ 0, we have that

It is shown that $$ is a fuzzy cone contractive sequence; therefore,

Now, for j > ℓ and for t ≫ 0, we have

Hence, proving that $$ is a Cauchy sequence, we have that

Now, for sequence $$ and from (5), for t ≫ 0, we have

After simplification, we get that

where θ is the same as in (8). Similarly, from (5) for t ≫ 0, we have where θ is the same as in (8). Now, from (15) and (16) and by induction, for t ≫ 0, we have that

It is shown that $$ is a fuzzy cone contractive sequence.

Now, for j > ℓ and for t ≫ 0, we have

Hence, proving that $$ is a Cauchy sequence, we have that

Since G is complete, $$ and $$ are Cauchy sequences in G; therefore, $$ and $$ as ℓ⟶∞; that is, $$ and $$. Since $$ is subsequently convergent, {g_ℓ} has a convergent subsequence. So there exist g ∈ G and {g_ℓ(k)} in G such that lim_k⟶∞g_ℓ(k) = g. Since $$ is continuous, lim_k⟶∞g_ℓ(k) = g, and $$. Now, from (5), for t ≫ 0, we have

After simplification, for t ≫ 0, we have

Hence, we get that $$; this implies that $$. Since $$ is one-one, $$. Next, we have to prove that $$. Then, from (5), for t ≫ 0, we have

After simplification, for t ≫ 0, we have

Hence, we get that $$; this implies that $$. Since $$ is one-one, we get $$.

For uniqueness, suppose that (g₁, h₁) and (h₁, g₁) are coupled fixed-point pairs in G × G such that $$ and $$. Now, from (5), for t ≫ 0, we have

Hence, we get that $$ for t ≫ 0; this implies that g = g₁. Similarly, again from (5), for t ≫ 0, we have

Hence, we get that $$ for t ≫ 0, and this implies that h = h₁.

Corollary 1. Let $$ be a mapping in a complete FCM-space (G, M_c, ∗) in which M_c is triangular and $$ is a continuous, one-one, and subsequently convergent self-mapping on G, that is, $$, satisfying

for all g, h, ξ, η ∈ G, t ≫ 0, and α, β ∈ [0,1] with α + 2β < 1. Then $$ has a unique coupled FP. Also, if $$ converges sequently, then, for every g₀ ∈ G, the iterative sequence $$ converges to this coupled FP.

Corollary 2. Let $$ be a mapping in a complete FCM-space (G, M_c, ∗) in which M_c is triangular and $$ is a continuous, one-one, and subsequently convergent self-mapping on G, that is, $$, satisfying

for all g, h, ξ, η ∈ G, t ≫ 0, and α, γ ∈ [0,1] with α + 2γ < 1. Then $$ has a unique coupled FP. Also, if $$ converges sequently, then, for every g₀ ∈ G, the iterative sequence $$ converges to this coupled FP.

If we use $$ as an identity self-mapping, that is, $$ in Theorem 1, then we get the following corollary.

Corollary 3. Let $$ be a mapping in a complete FCM-space (G, M_c, ∗) in which M_c is triangular, satisfying

for all g, h, ξ, η ∈ G, t ≫ 0, and α, β, γ ∈ [0,1] with α + 2β + 2γ < 1. Then $$ has a unique coupled FP in G.

Example 1. Let G = {0} ∪ {(1/2), (1/3), (1/4), …}, and let a fuzzy metric M_c : G² × (0, ∞)⟶[0,1] be defined by

∀g, h ∈ G and t > 0. Then easily one can verify that M_c is triangular and (G, M_c, ∗) is a complete FCM-space. We define $$ by $$ and $$ for all n, ℓ ≥ 2; and a mapping $$ is defined as $$ and $$. Then, by using (30), for t ≫ 0, we have

Let (1/t)(1/(n + ℓ + 2)^n+ℓ+2) ≤ (1/5t)[(2/nⁿ) − (1/(n + ℓ + 2)^n+ℓ+2)]. Then, from (31), we have

Hence, we get that

Thus, inequality (33) satisfies all the conditions of Theorem 1 with α = (1/5) and β = γ = (1/10) and $$ has a unique coupled fixed point; that is, $$.

Theorem 2. Let $$ be a mapping in a complete FCM-space (G, M_c, ∗) in which M_c is triangular and $$ is a continuous, one-one, and subsequently convergent self-mapping on G, that is, $$, satisfying

for all g, h, ξ, η ∈ G, t ≫ 0, and a, b, c ∈ [0,1] with α + 2β + γ < 1. Then $$ has a unique coupled FP. Also, if $$ converges sequently, then, for each g₀ ∈ G, the iterative sequence $$ converges to this coupled FP.

Proof. Let any g₀, h₀ ∈ G, and we define sequence {g_ℓ} by

Now, from (34), for t ≫ 0, we have

After simplification, we get that

where η = (α + β)/(1 − β − γ) < 1. Similarly, again from (34), for t ≫ 0, we have where η is the same as in (37). Now, from (37) and (38) and by induction, for t ≫ 0, we have

Hence, we get that $$ is a fuzzy cone contractive sequence; therefore,

Now, for j > ℓ and for t ≫ 0, we have

Hence, proving that $$ is a Cauchy sequence, we have that

Now, again from (34), for t ≫ 0, we have

After simplification, we get that

where η value is the same as in (37). Similarly, again from (34), for t ≫ 0, we have where η value is the same as in (37). Now, from (44) and (45) and by induction, for t ≫ 0, we have that

Hence, proving that $$ is a fuzzy cone contractive sequence,

Now, for j > ℓ and for t ≫ 0, we have

Hence, proving that $$ is a Cauchy sequence,

Since G is complete, $$ and $$ are Cauchy sequences in G; therefore $$ and $$ as ℓ⟶∞; that is, $$ and $$. Since $$ is subsequently convergent, {g_ℓ} has a convergent subsequence. So there exist g ∈ G and {g_ℓ(k)} such that lim_k⟶∞g_ℓ(k) = g. Since $$ is continuous, lim_k⟶∞g_ℓ(k) = g and $$. Now, from (34) and (39), for t ≫ 0, we have

After simplification, for t ≫ 0, we have that

Hence, we get that $$, and this implies that $$. Since $$ is one-one, $$. Now, again from (34) and (46), for t ≫ 0, we have

After simplification, for t ≫ 0, we have that

Hence, we get that $$ for t ≫ 0, and this implies that $$. Since $$ is one-one, $$.

For uniqueness, let (g₁, h₁) and (h₁, g₁) be coupled fixed-point pairs in G × G such that $$ and $$. Now, from (34), for t ≫ 0, we have

Hence, we get that $$, and this implies that g = g₁. Similarly, again from (34), for t ≫ 0, we have

Hence, we get that $$ for t ≫ 0, and this implies that h = h₁.

Corollary 4. Let $$ be a mapping in a complete FCM-space (G, M_c, ∗) in which M_c is triangular and $$ is a continuous, one-one, and subsequently convergent self-mapping on G, that is, $$, satisfying

for all g, h, ξ, η ∈ G, t ≫ 0, and α, γ ∈ [0,1] with α + γ < 1. Then $$ has a unique coupled FP. Also, if $$ converges sequently, then, for each g₀ ∈ G, the iterative sequence $$ converges to this coupled FP.

If we use $$ as an identity self-mapping, that is, $$ in Theorem 2, then we get the following corollary.

Corollary 5. Let $$ be a mapping in a complete FCM-space (G, M_c, ∗) in which M_c is triangular, satisfying

for all g, h, ξ, η ∈ G, t ≫ 0, and α, β, γ ∈ [0,1] with α + 2β + γ < 1. Then $$ has a unique coupled FP in G.

Example 2. Let G = {0} ∪ {(1/2), (1/3), (1/4), …}, and let M_c : G² × (0, ∞)⟶[0,1] be defined as

for all g, h ∈ G and t > 0. Then, it could be verified that M_c is triangular and (G, M_c, ∗) is a complete FCM-space. We define $$ by $$ and $$ for all n, ℓ ≥ 2; and a mapping $$ is defined as $$ and $$. Then, by using (58), for t ≫ 0, we have

Let (1/t)(1/(n + ℓ + 1)^n+ℓ+1) ≤ (1/5t)((2/nⁿ)  − (1/(n + ℓ + 1)^n+ℓ+1)). Then, from (??), for t ≫ 0, we have

We have

Therefore,

This implies that

Thus, inequality (63) satisfies all the conditions of Theorem 2 with α = (1/5) and β = γ = (1/10) and $$ has a unique coupled FP; that is, $$ and $$.

Theorem 3. The two VIEs are

where ζ ∈ [0,1] ⊂ R and ℏ₁, ℏ₂ ∈ G. Assume that K₁, K₂ : [0,1]² × R⟶R such that U_{(g, h)}, V_{(ξ, η)} ∈ G for g, η ∈ U, and h, ξ ∈ V and U, V ⊂ G. Therefore, we define

If there exist λ ∈ [0,1] such that

where where $$, then the two VIEs (67) have a unique common solution in G.

Proof. Define operators $$ and $$:

Then,

where $$ and $$. Now, we may have the three following cases:

• (1)

  If $$, then, from (66), (71) and (72), we have

  $$ (73)

•  

  for g, η ∈ U, and h, ξ ∈ V. Hence, operators $$ and $$ satisfy all the conditions of Theorem 1 with λ = α and β = γ = 0 in (5). Thus, the two VIEs in (67) have a unique common solution in G. Then integral equations have a unique common solution.

• (2)

  If $$, then, from (66), (71) and (72), we have

  $$ (74)

•  

  for g, η ∈ U, and h, ξ ∈ V. Hence, operators $$ and $$ satisfy all the conditions of Theorem 1 with λ = β and α = γ = 0 in (5). Thus, the two VIEs in (67) have a unique common solution in G. Then integral equations have a unique common solution.

• (3)

  If $$, then, from (66), (71) and (72), we have

for g, η ∈ U, and h, ξ ∈ V. Hence, operators $$ and $$ satisfy all the conditions of Theorem 1 with λ = γ and α = β = 0 in (5). Thus, the two VIEs in (67) have a unique common solution in G. Then integral equations have a unique common solution.