Definition 1 (see [3].)Let Φ be a nonempty set and let G : Φ × Φ × Φ⟶ℝ₊ be a function satisfying

(G₁) G(r, s, t) = 0 if r = s = t

(G₂) 0 < G(r, r, s) for all r, s ∈ Φ with r ≠ s

(G₃) G(r, r, s) ≤ G(r, s, t), for all r, s, t ∈ Φ with t ≠ s

(G₄) G(r, s, t) = G(r, t, s) = G(s, r, t) = ⋯ (symmetry in all variables)

(G₅) G(r, s, t) ≤ G(r, u, u) + G(u, s, t), for all r, s, t, u ∈ Φ (rectangle inequality)

Example 2 (see [4].)Let (Φ, d) be a usual metric space; then, (Φ, G_k) and (Φ, G_m) are G-metric spaces, where

Definition 3 (see [4].)Let (Φ, G) be a G-metric space and let {r_n} be a sequence of points of Φ. Then, {r_n} is said to be G-convergent to r if lim_n,m⟶∞G(r, r_n, r_m) = 0; that is, for any ε > 0, there exists n₀ ∈ ℕ such that G(r, r_n, r_m) < ε, ∀n, m ≥ n₀. We refer to r as the limit of the sequence {r_n}.

Proposition 4 (see [4].)Let (Φ, G) be a G-metric space. Then, the following are equivalent:

• (i)

  {r_n} is G-convergent to r

• (ii)

  G(r, r_n, r_m)⟶0, as n⟶∞

• (iii)

  G(r_n, r, r)⟶0, as n⟶∞

• (iv)

  G(r_n, r_n, r)⟶0, as n⟶∞

Definition 5 (see [4].)Let (Φ, G) be a G-metric space. A sequence {r_n} is called G-Cauchy if for any ε > 0, we can find n₀ ∈ ℕ such that G(r_n, r_m, r_l) < ε, ∀n, m, l ≥ n₀, that is, G(r_n, r_m, r_l)⟶0, as n, m, l⟶∞.

Proposition 6 (see [4].)If (Φ, G) is a G-metric space, the following statements are equivalent:

• (i)

  The sequence {r_n} is G-Cauchy

• (ii)

  For every ε > 0, there exists n₀ ∈ ℕ such that G(r_n, r_m, r_m) < ε, ∀n, m ≥ n₀

Definition 7 (see [4].)Let (Φ, G) and (Φ^′, G^′) be G-metric spaces and f : (Φ, G)⟶(Φ^′, G^′) be a function. Then, f is G-continuous at a point u ∈ Φ if and only if for any ε > 0, there exists δ > 0 such that r, s ∈ Φ; and G(u, r, s) < δ implies G^′(f(u), f(r), f(s)) < ε. A function f is G-continuous on Φ if and only if it is G-continuous at all u ∈ Φ.

Proposition 8 (see [4].)Let (Φ, G) and (Φ^′, G^′) be G-metric spaces. Then, a function f : (Φ, G)⟶(Φ^′, G^′) is said to be G-continuous at a point r ∈ Φ if and only if it is G-sequentially continuous at r; that is, whenever {r_n} is G-convergent to r, {fr_n} is G-convergent to fr.

Definition 9 (see [4].)A G-metric space (Φ, G) is called symmetric G-metric space if

Proposition 10 (see [4].)Let (Φ, G) be a G-metric space. Then, the function G(r, s, t) is jointly continuous in all variables.

Proposition 11 (see [4].)Every G-metric space (Φ, G) defines a metric space (Φ, d_G) by

Note that for a symmetric G-metric space (Φ, G),

On the other hand, if (Φ, G) is not symmetric, then by the G-metric properties,

and that in general, these inequalities are sharp.

Definition 12 (see [4].)A G-metric space (Φ, G) is referred to as G-complete (or complete G-metric) if every G-Cauchy sequence in (Φ, G) is G-convergent in (Φ, G).

Proposition 13 (see [4].)A G-metric space (Φ, G) is G-complete if and only if (Φ, d_G) is a complete metric space.

Mustafa [2] proved the following result in the framework of G-metric space.

Theorem 14 (see [2].)Let (Φ, G) be a complete G-metric space, and let Γ : Φ⟶Φ be a mapping satisfying the following condition:

for all r, s, t ∈ Φ where 0 ≤ k < 1; then, Γ has a unique fixed point (say u, i.e., Γu = u), and Γ is G-continuous at u.

Definition 15 (see [9].)Let Ψ be the set of all functions ϕ : ℝ₊⟶ℝ₊ satisfying

• (i)

  ϕ is monotone increasing, that is, t₁ ≤ t₂ implies ϕ(t₁) ≤ ϕ(t₂)

• (ii)

  the series $$ is convergent for all t > 0

Then, ϕ is called a (c)-comparison function.

Remark 16. If ϕ ∈ Ψ, then ϕ(t) < t for any t > 0, ϕ(0) = 0, and ϕ is continuous at 0.

Definition 17 (see [1].)Let (Φ, d) be a complete metric space. A self-mapping Γ : Φ⟶Φ is called a Jaggi-type hybrid contraction; if there exists ϕ ∈ Φ such that

λ₁, λ₂ ≥ 0 with λ₁ + λ₂ = 1 and Fix(Γ) = {r ∈ Φ : Γr = r}.

Definition 18. Let (Φ, G) be a G-metric space. A self-mapping Γ : Φ⟶Φ is called a Jaggi-type hybrid (G − ϕ)-contraction, if there exists ϕ ∈ Φ such that

λ₁, λ₂ ≥ 0 with λ₁ + λ₂ = 1 and Fix(Γ) = {r ∈ Φ : Γr = r}.

Theorem 19. Let (Φ, G) be a complete G-metric space and let Γ : Φ⟶Φ be a continuous Jaggi-type hybrid (G − ϕ)-contraction on (Φ, G). Then, Γ has a fixed point in Φ (say c), and for any c₀ ∈ Φ, the sequence $$ converges to c.

Proof. Let r₀ ∈ Φ be an arbitrary point and define a sequence $$ in Φ by r_n = Γⁿr₀. If there exists some n ∈ ℕ such that Γr_n = r_n+1 = r_n, then r_n is a fixed point of Γ, and so the proof is complete. Assume now that r_n ≠ r_n−1 for any n ∈ ℕ. Since Γ is a Jaggi-type hybrid (G − ϕ)-contraction, then we have from (9) that

Case 1. For q > 0, we have

Case 2. For q = 0, we have

Theorem 20. Let (Φ, G) be a complete G-metric space and let Γ : Φ⟶Φ be a Jaggi-type hybrid (G − ϕ)-contraction. If for some integer i > 2, we have that Γⁱ is continuous, then Γ has a unique fixed point in Φ.

Proof. In Theorem 19, we have established that there exists a G-Cauchy sequence $$ in (Φ, G) with r_n = Γr_n−1 such that r_n⟶c for some c in Φ. Let $$ be a subsequence of $$ where n_l = l · i for all l ∈ ℕ, i > 2 fixed. Notice that Γ⁰ is an identity self-mapping on Φ so that $$. Hence, by the continuity of Γⁱ, we have

that is, c is a fixed point of Γⁱ.

Example 21. Let Φ = [−1, 1] and let Γ : Φ⟶Φ be a self-mapping on Φ defined by

for all r ∈ Φ. Define G : Φ × Φ × Φ⟶ℝ₊ by

Then, (Φ, G) is a complete G-metric space and Γ is continuous for all r ∈ Φ. Define ϕ ∈ Ψ by ϕ(x) = x/2 for all x ≥ 0.

To see that Γ is a Jaggi-type hybrid (G − ϕ)-contraction, notice that G(Γr, Γs, Γ²s) = 0 for all r, s ∈ (−1, 1). Hence, inequality (9) holds for all r, s ∈ (−1, 1).

Corollary 22 (see Theorem 14.)Let (Φ, G) be a complete G-metric space, and let Γ : Φ⟶Φ be a mapping satisfying the following condition:

for all r, s, t ∈ Φ where 0 ≤ k < 1; then, Γ has a unique fixed point (say u) and Γ is G-continuous at u.

Proof. Consider Definition (18) and let Γs = t, λ₁ = 0, λ₂ = 1, q > 0, and ϕ(p) = kp for all p ≥ 0 and k ∈ [0, 1). Clearly, ϕ ∈ Ψ and Γ is a Jaggi-type hybrid (G − ϕ)-contraction. Hence, (9) coincides with (6) of Theorem 14 due to Mustafa [2]. Therefore, it is easy to see that we can find a unique point u in Φ such that Γu = u and Γ is G-continuous at u.

Corollary 23 (see [10], Theorem 3.1.)Let (Φ, G) be a complete G-metric space. Suppose the mapping Γ : Φ⟶Φ satisfies

for all r, s, t ∈ Φ. Then, Γ has a unique fixed point (say u) and Γ is G-continuous at u.

Proof. Consider Definition 18 and let Γs = t, λ₁ = 0, λ₂ = 1 and q > 0. Then,

for all r, s, t ∈ Φ. Hence, inequality (9) becomes

for all r, s, t ∈ Φ and ϕ ∈ Ψ. This coincides with Theorem 3.1 due to Shatanawi [10] and so the proof follows in a similar manner.

Corollary 24. Let (Φ, G) be a complete G-metric space. If there exists μ ∈ (0, 1) such that for all r, s ∈ Φ, the mapping Γ : Φ⟶Φ satisfies

then Γ has a fixed point in Φ.

Theorem 25. Assume that the following conditions are satisfied:

(C₁) $$ and f : [a, b] × ℝ⟶ℝ are continuous

(C₂) For all r, s ∈ Φ, x ∈ [a, b], we have |f(x, r(x)) − f(x, s(x))| ≤ |r(x) − s(x)|

(C₃) $$ for some μ < 1

Proof. Observe that for any r, s ∈ Φ, using (55) and the above hypotheses, we obtain

Remark 26.

• (i)

  We can deduce a number of corollaries by particularizing some of the parameters in Definition 18

• (ii)

  None of the results presented in this work can be expressed in the form G(r, s, s) or G(r, r, s). Hence, they cannot be obtained from their corresponding versions in metric space