On strict common fixed points of hybrid mappings in 2-metric spaces

A sequence {x_n} in a 2-metric space (X, d) is said to be convergent to a point x ∈ X (denoted by lim_n→∞x_n = x) if lim_n→∞d(x_n, x, z) = 0 for all z ∈ X.

A sequence {x_n} in a 2-metric space (X, d) is said to be Cauchy sequence if lim_n,m→∞d(x_n, x_m, z) = 0 for all z ∈ X.

A 2-metric space (X, d) is said to be complete if every Cauchy sequence in X is convergent.

Remark 2.1 Naidu and Prasad, 1986

In general, a convergent sequence in a 2-metric space (X, d) need not be Cauchy, but every convergent sequence is a Cauchy sequence whenever 2-metric d is continuous on X.

Definition 2.4 Murthy et al., 1992

A pair of self mappings (S, T) of a 2-metric space (X, d) is said to be compatible if lim_n→∞ d(STx_n, TSx_n, z) = 0 for all z ∈ X, whenever {x_n} is a sequence in X such that lim_n→∞ Sx_n = lim_n→∞Tx_n = t for some t ∈ X.

A pair of self mappings (S, T) of a nonempty set X is said to be weakly compatible if Sx = Tx (for some x ∈ X) implies STx = TSx.

A sequence {A_n} of subsets of a 2-metric space (X, d) is said to be convergent to a subset A of X if:

1. given a ∈ A, there exists {a_n} in X such that a_n ∈ A_n for n = 1, 2, 3, … and lim_n→∞d(a_n, a, z) = 0 for each z ∈ X, and

2. given ∊ > 0, there exists a positive integer N such that A_n ⊂ A_∊ for n > N where A_∊ is the union of all open balls with centers in A and radius ∊.

The mappings I : X → X and F : X → B(X) are said to be weakly commuting at x if IFx ∈ B(X) and

If F is a single-valued mapping, then the set IFx becomes singleton. Therefore, δ(IFx, IFx, z) = 0 and condition (2.1) reduces to the condition given by Khan (1984), that is D(FIx, IFx, z) ⩽ D(Ix, Fx, z).

The mappings I : X → X and F : X → B(X) are said to be compatible if lim_n→∞D(FIx_n, IFx_n, z) = 0 for all z ∈ X, whenever {x_n} is a sequence in X such that lim_n→∞Ix_n = t ∈ A = lim_n→∞ Fx_n for some t ∈ X and A ∈ B(X).

The mappings I : X → X and F : X → B(X) are said to be δ-compatible if lim_n→∞δ(FIx_n, IFx_n, z) = 0 for all z ∈ X, whenever {x_n} is a sequence in X such that IFx_n ∈ B(X), Fx_n → {t} and Ix_n → t for some t ∈ X.

Let I : X → X and F : X → B(X). A point x ∈ X is said to be a fixed point (strict fixed point) of F if x ∈ Fx (Fx = {x}). Also, a point x ∈ X is said to be a coincidence point (strict coincidence point) of (I, F) if Ix ∈ Fx (Fx = {Ix}).

Definition 2.11 Jungck and Rhoades, 1998

The mappings I : X → X and F : X → B(X) are said to be weakly compatible if they commute at all strict coincidence points, i.e., for each x in X such that Fx = {Ix}, we have FIx = IFx.

Remark 2.3 Jungck and Rhoades, 1998

Any δ-compatible pair (I, F) is weakly compatible but not conversely.

The mappings I : X → X and F : X → B(X) are said to be strict occasionally weakly compatible if the pair commutes at some of it’s strict coincidence points.

If I, J : X → X and F, G : X → B(X) are mappings which satisfy

1. ∪G(X) ⊆ I(X) and ∪F(X) ⊆ J(X),

2. $$\delta (\mathit{Fx}\text{,}\mathit{Gy}\text{,}C)⩽\alpha \max \left\{\delta (\mathit{Ix}\text{,}\mathit{Jy}\text{,}C)\text{,}\delta (\mathit{Ix}\text{,}\mathit{Fx}\text{,}C)\text{,}\delta (\mathit{Jy}\text{,}\mathit{Gy}\text{,}C)\right\}+(1-\alpha )[\mathit{aD}(\mathit{Ix}\text{,}\mathit{Gy}\text{,}C)+\mathit{bD}(\mathit{Jy}\text{,}\mathit{Fx}\text{,}C)]$$ for all x, y ∈ X and C ∈ B(X), where 0 ⩽ α < 1, a + b < 1, a, b ⩾ 0 and α∣a − b∣ < 1 − (a + b),

3. I(X) (or J(X)) is complete subspace of (X, d),

4. both the pairs (F, I) and (G, J) are weakly compatible, then F, G, I and J have a unique common fixed point in X.

Let Φ be the set of all continuous functions $\varphi :{\mathfrak{R}}_{+}^6\to \mathfrak{R}$ satisfying the following conditions:

• (ϕ₁)

  ϕ is nondecreasing in variable t₁ and nonincreasing in variables t₂ … , t₆.

• (ϕ₂)

  there exists h, k > 0 with hk < 1 such that for u, v ⩾ 0

  • (ϕ_a):

    ϕ(u, v, v, u, u + v,0) ⩽ 0 implies u ⩽ hv,

  • (ϕ_b):

    ϕ(u, v, u, v,0, u + v) ⩽ 0 implies u ⩽ kv.

• (ϕ₃)

  ϕ(t, t,0,0, t, t) > 0 ∀t > 0.

Define $\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6):{\mathfrak{R}}_{+}^6\to \mathfrak{R}$ as $$\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6)=t_1-\alpha \max \left\{t_2\text{,}{t}_3\text{,}{t}_4\text{,}{\displaystyle \frac{1}{2}}(t_5+t_6)\right\}\text{,}\text{where}\alpha \in (0\text{,}1)\text{.}$$ Setting h = k = α < 1, the requirements of Definition 3.1 are met out.

Define $\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6):{\mathfrak{R}}_{+}^6\to \mathfrak{R}$ as $$\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6)=t_1^2-c_1\max \{t_2^2\text{,}{t}_3^2\text{,}{t}_4^2\}-c_2\max \{t_3{t}_5\text{,}{t}_4{t}_6\}-c_3{t}_5{t}_6\text{,}$$ where c₁ > 0, c₂, c₃ ⩾ 0, c₁ + 2c₂ < 1 and c₁ + c₃ < 1.

Choosing $h=k=\sqrt{c_1+2c_2}<1$ , one can easily verify the requirements of Definition 3.1.

Define $\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6):{\mathfrak{R}}_{+}^6\to \mathfrak{R}$ as $$\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6)=t_1-\alpha t_2-\beta \min \{t_3\text{,}{t}_5\}-\eta \min \{t_4\text{,}{t}_6\}\text{,}$$ where α,β,η > 0, α + β < 1, α + η < 1 and (α + β)(α + η) < 1.

Setting h = α + β < 1, k = α + η < 1 with hk < 1, one can easily check the requirements of Definition 3.1.

Define $\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6):{\mathfrak{R}}_{+}^6\to \mathfrak{R}$ as $$\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6)=t_1-\alpha \max \{t_2\text{,}{t}_3\text{,}{t}_4\}-(1-\alpha )(\beta t_5+\eta t_6)\text{,}$$ where 0 ⩽ α < 1, β + η < 1, β,η ⩾ 0 and α∣β − η∣ < 1 − (β + η). Choosing $h=\max \left\{\frac{\alpha +(1-\alpha )\beta }{1-(1-\alpha )\beta }\text{,}\frac{\beta }{1-\beta }\right\}$ , $k=\max \left\{\frac{\alpha +(1-\alpha )\eta }{1-(1-\alpha )\eta }\text{,}\frac{\eta }{1-\eta }\right\}$ with hk < 1 (see Abd El-Monsef et al., 2009, p. 1438), one can easily verify the requirements of Definition 3.1.

Define $\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6):{\mathfrak{R}}_{+}^6\to \mathfrak{R}$ as $$\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6)=t_1-\psi \left(\max \left\{t_2\text{,}{t}_3\text{,}{t}_4\text{,}\frac{t_5+t_6}{2}\right\}\right)$$ where $\psi :{\mathfrak{R}}^{+}\to {\mathfrak{R}}^{+}$ is an upper semi-continuous function such that ψ(t) < t for all t > 0.

Define $\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6):{\mathfrak{R}}_{+}^6\to \mathfrak{R}$ as $$\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6)=t_1-\psi (t_2\text{,}{t}_3\text{,}{t}_4\text{,}{t}_5\text{,}{t}_6)$$ where $\psi :{\mathfrak{R}}_{+}^5\to {\mathfrak{R}}_{+}$ is an upper semi-continuous and increasing function in t₂, … , t₆ such that ψ(t, t, t,α t,βt) < t for all t > 0 and α,β ⩾ 0 with α + β = 2.

Define $\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6):{\mathfrak{R}}_{+}^6\to \mathfrak{R}$ as $$\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6)={\int }_0^{t_1}\psi (t)\mathrm{d}t-\alpha {\int }_0^{\max \{t_2\text{,}{t}_3\text{,}{t}_4\text{,}\frac{t_5+t_6}{2}\}}\psi (t)\mathrm{d}t$$ where α ∈ (0,1) and $\psi :{\mathfrak{R}}^{+}\to {\mathfrak{R}}^{+}$ is a Lebesgue integrable function which is summable and ${\int }_0^{∊}\psi (t)\mathrm{d}t>0$ for all ∊ > 0.

Define $\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6):{\mathfrak{R}}_{+}^6\to \mathfrak{R}$ as $$\varphi (t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_6)={\int }_0^{t_1}\psi (t)\mathrm{d}t-\alpha \max \left\{{\int }_0^{t_2}\psi (t)\mathrm{d}t\text{,}{\int }_0^{t_3}\psi (t)\mathrm{d}t\text{,}{\int }_0^{t_4}\psi (t)\mathrm{d}t\text{,}\frac{1}{2}\left({\int }_0^{t_5}+{\int }_0^{t_6}\right)\psi (t)\mathrm{d}t\right\}$$ where α ∈ (0,1) and $\psi :{\mathfrak{R}}^{+}\to {\mathfrak{R}}^{+}$ is a Lebesgue integrable function which is summable and satisfies ${\int }_0^{∊}\psi (t)\mathrm{d}t>0$ for all ∊ > 0.

Let (X, d) be a 2-metric space wherein the mappings I : X → X and F:X → B(X) are strict OWC pair. If I and F have a unique point of strict coincidence {z} = {Ix} = Fx, then z is the unique common fixed point of I and F which also remains a strict fixed point of F.

Since the mappings I and F are strict OWC, there exists a point x ∈ X with {z} = {Ix} = Fx implies that FIx = IFx. Therefore {Iz} = {IIx} = IFx = FIx = Fz = {u} which shows that u is a point of strict coincidence of I and F. Now, in view of the uniqueness of point of coincidence, one infers z = u and henceforth {z} = {Iz} = Fz which shows that z is a common fixed point of I and F. Suppose that v ≠ z is another common fixed point of I and F which is also a strict fixed point for F, then {v} = {Iv} = Fv implies that v is a point of strict coincidence of I and F. Now, due to the uniqueness of point of strict coincidence one gets v = z. This concludes the proof. □

Let (X, d) be a 2-metric space wherein I, J : X → X and F, G : X → B(X) are the mappings which satisfy the inequality

Firstly, we show that Ix = Jy. Let on contrary that Ix ≠ Jy, then using (4.1) and (ϕ₁), we obtain $$\varphi (\delta (\mathit{Ix}\text{,}\mathit{Jy}\text{,}C)\text{,}\delta (\mathit{Ix}\text{,}\mathit{Jy}\text{,}C)\text{,}0\text{,}0\text{,}\delta (\mathit{Ix}\text{,}\mathit{Jy}\text{,}C)\text{,}\delta (\mathit{Ix}\text{,}\mathit{Jy}\text{,}C))⩽0$$ a contradiction to (ϕ₃). Hence Ix = Jy. Thus u = {Ix} = Fx = {Jy} = Gy. Suppose that there is some z ∈ X, z ≠  x with {w} = {Iz} = Fz. Then using (4.1) and (ϕ₁), we obtain $$\varphi (\delta (\mathit{Iz}\text{,}\mathit{Jy}\text{,}C)\text{,}\delta (\mathit{Iz}\text{,}\mathit{Jy}\text{,}C)\text{,}0\text{,}0\text{,}\delta (\mathit{Iz}\text{,}\mathit{Jy}\text{,}C)\text{,}\delta (\mathit{Iz}\text{,}\mathit{Jy}\text{,}C))⩽0$$ a contradiction to (ϕ₃) provided δ(Iz, Jy, C) = 0. Hence {w} = {Iz} = Fz = {Jy} = Gy, u = {Ix} = Fx, and u is the unique point of strict coincidence of I and F. Similarly, one can show that v is the unique point of strict coincidence of J and G. This completes the proof. □

If I, J : X → X and F, G : X → B(X) are mappings which satisfy (4.1) and (4.2), then (for every n ∈ N), δ(Y_n, Y_n+1, Y_n+2) = 0.

By using (4.1) and (ϕ₁), we can have $$\begin{array}{cc}\hfill & \varphi (\delta ({\mathit{Fx}}_{2n+2}\text{,}{\mathit{Gx}}_{2n+1}\text{,}{Y}_{2n})\text{,}\delta ({\mathit{Ix}}_{2n+2}\text{,}{\mathit{Jx}}_{2n+1}\text{,}{Y}_{2n})\text{,}\delta ({\mathit{Ix}}_{2n+2}\text{,}{\mathit{Fx}}_{2n+2}\text{,}{Y}_{2n})\text{,}\hfill \\ \hfill & \delta ({\mathit{Jx}}_{2n+1}\text{,}{\mathit{Gx}}_{2n+1}\text{,}{Y}_{2n})\text{,}D({\mathit{Ix}}_{2n+2}\text{,}{\mathit{Gx}}_{2n+1}\text{,}{Y}_{2n})\text{,}D({\mathit{Fx}}_{2n+2}\text{,}{\mathit{Jx}}_{2n+1}\text{,}{Y}_{2n}))⩽0\hfill \\ \hfill & \mathit{or}\varphi (\delta (Y_{2n+2}\text{,}{Y}_{2n+1}\text{,}{Y}_{2n})\text{,}\delta (Y_{2n+1}\text{,}{Y}_{2n}\text{,}{Y}_{2n})\text{,}\delta (Y_{2n+1}\text{,}{Y}_{2n+2}\text{,}{Y}_{2n})\text{,}\hfill \\ \hfill & \delta (Y_{2n}\text{,}{Y}_{2n+1}\text{,}{Y}_{2n})\text{,}D(Y_{2n+1}\text{,}{Y}_{2n+1}\text{,}{Y}_{2n})\text{,}D(Y_{2n}\text{,}{Y}_{2n+2}\text{,}{Y}_{2n}))⩽0\hfill \\ \hfill & \mathit{or}\varphi (\delta (Y_{2n+2}\text{,}{Y}_{2n+1}\text{,}{Y}_{2n})\text{,}0\text{,}\delta (Y_{2n+1}\text{,}{Y}_{2n+2}\text{,}{Y}_{2n})\text{,}0\text{,}0\text{,}D(Y_{2n}\text{,}{Y}_{2n+2}\text{,}{Y}_{2n}))⩽0\hfill \\ \hfill & \mathit{or}\varphi (\delta (Y_{2n+2}\text{,}{Y}_{2n+1}\text{,}{Y}_{2n})\text{,}0\text{,}\delta (Y_{2n+1}\text{,}{Y}_{2n+2}\text{,}{Y}_{2n})\text{,}0\text{,}0\text{,}\delta (Y_{2n+1}\text{,}{Y}_{2n+2}\text{,}{Y}_{2n}))⩽0\hfill \end{array}$$ which implies (due to (ϕ_b)) δ(Y_2n, Y_2n+1, Y_2n+2) = 0. Similarly, using (ϕ_a), we can also show that δ(Y_2n+1, Y_2n+2, Y_2n+3) = 0. Thus, in all, δ(Y_n, Y_n+1, Y_n+2) = 0. □

Lemma 4.2 Abd El-Monsef et al., 2007

If {A_n} and {B_n} are sequences in B(X) converging to A and B respectively, then δ(A_n, B_n, C) converges to δ(A, B, C) for every C ∈ B(X).

Let I, J : X → X and F, G : X → B(X) be the mappings such that (4.1) and (4.2) hold (for all x, y ∈ X and for all C ∈  B(X)). If I(X) (or J(X)) is a complete subspace of X, then

1. I and F have a strict coincidence point,

2. J and G have a strict coincidence point.

Owing to (4.1), (4.2), (4.3) and (ϕ₁), we can write $$\begin{array}{cc}\hfill & \varphi (\delta ({\mathit{Fx}}_{2n}\text{,}{\mathit{Gx}}_{2n+1}\text{,}C)\text{,}\delta ({\mathit{Ix}}_{2n}\text{,}{\mathit{Jx}}_{2n+1}\text{,}C)\text{,}\delta ({\mathit{Ix}}_{2n}\text{,}{\mathit{Fx}}_{2n}\text{,}C)\text{,}\hfill \\ \hfill & \delta ({\mathit{Jx}}_{2n+1}\text{,}{\mathit{Gx}}_{2n+1}\text{,}C)\text{,}D({\mathit{Ix}}_{2n}\text{,}{\mathit{Gx}}_{2n+1}\text{,}C)\text{,}D({\mathit{Fx}}_{2n}\text{,}{\mathit{Jx}}_{2n+1}\text{,}C))⩽0\hfill \\ \hfill & \text{or}\varphi \left(\delta (Y_{2n}\text{,}{Y}_{2n+1}\text{,}C)\text{,}\delta (Y_{2n-1}\text{,}{Y}_{2n}\text{,}C)\text{,}\delta (Y_{2n-1}\text{,}{Y}_{2n}\text{,}C)\text{,}\right.\hfill \\ \hfill & \left.\delta (Y_{2n}\text{,}{Y}_{2n+1}\text{,}C)\text{,}\delta (Y_{2n-1}\text{,}{Y}_{2n+1}\text{,}C)\text{,}\delta (Y_{2n}\text{,}{Y}_{2n}\text{,}C)\right)⩽0\text{.}\hfill \end{array}$$ Since δ(Y_2n−1, Y_2n+1, C) ⩽ δ(Y_2n−1, Y_2n, C) + δ(Y_2n, Y_2n+1, C) + δ(Y_2n−1, Y_2n+1, Y_2n) and δ(Y_2n−1, Y_2n+1, Y_2n) = 0 (due to Lemma 4.1), therefore $$\begin{array}{cc}\hfill & \varphi (\delta (Y_{2n}\text{,}{Y}_{2n+1}\text{,}C)\text{,}\delta (Y_{2n-1}\text{,}{Y}_{2n}\text{,}C)\text{,}\delta (Y_{2n-1}\text{,}{Y}_{2n}\text{,}C)\text{,}\hfill \\ \hfill & \delta (Y_{2n}\text{,}{Y}_{2n+1}\text{,}C)\text{,}\delta (Y_{2n-1}\text{,}{Y}_{2n}\text{,}C)+\delta (Y_{2n}\text{,}{Y}_{2n+1}\text{,}C)\text{,}0)⩽0\text{.}\hfill \end{array}$$ (due to (ϕ_a)) gives rise

For all C ∈ B(X) and m > n, we have (by Lemma 4.1) $$\delta (Y_n\text{,}{Y}_m\text{,}C)⩽\delta (Y_n\text{,}{Y}_{n+1}\text{,}{Y}_{n+2})+\delta (Y_{n+1}\text{,}{Y}_{n+2}\text{,}{Y}_{n+3})+\cdots +\delta (Y_{m-2}\text{,}{Y}_{m-1}\text{,}{Y}_m)+\delta (Y_{m-1}\text{,}{Y}_m\text{,}C)\text{,}$$ which on letting n, m → ∞ gives rise that limδ(Y_n, Y_m, C) = 0.

Suppose that J(X) is complete and Jx_2n+1 ∈ Fx_2n = Y_2n, for n = 0,1, 2, … , we can have $$d({\mathit{Jx}}_{2m+1}\text{,}{\mathit{Jx}}_{2n+1}\text{,}C)⩽\delta (Y_{2m}\text{,}{Y}_{2n}\text{,}C)$$ which implies that limd(Jx_2m+1, Jx_2n+1, C) = 0. Hence {Jx_2n+1} is a Cauchy sequence and is also convergent to a limit p ∈ J(X), hence p = Jv for some v ∈ X. But Ix_2n ∈ Gx_2n−1 = Y_2n−1, so that we obtain $$\lim \delta ({\mathit{Ix}}_{2n}\text{,}{\mathit{Jx}}_{2n+1}\text{,}C)⩽\lim \delta (Y_{2n-1}\text{,}{Y}_{2n}\text{,}C)=0\text{.}$$ Consequently, lim Ix_2n = p. Moreover, we obtain $$\delta ({\mathit{Fx}}_{2n}\text{,}p\text{,}C)⩽\delta ({\mathit{Fx}}_{2n}\text{,}{\mathit{Ix}}_{2n}\text{,}C)+\delta ({\mathit{Ix}}_{2n}\text{,}p\text{,}C)+\delta ({\mathit{Fx}}_{2n}\text{,}p\text{,}{\mathit{Ix}}_{2n})\text{.}$$ Since δ(Fx_2n, Ix_2n, C) ⩽ δ(Y_2n, Y_2n−1, C) implies limδ(Fx_2n, Ix_2n, C) = 0, therefore limδ(Fx_2n, p, C) = 0. Similarly, we can have limδ(Gx_2n−1, p, C) = 0. Using the inequality (4.1), we obtain $$\begin{array}{cc}\hfill & \varphi (\delta ({\mathit{Fx}}_{2n}\text{,}\mathit{Gv}\text{,}C)\text{,}\delta ({\mathit{Ix}}_{2n}\text{,}\mathit{Jv}\text{,}C)\text{,}\delta ({\mathit{Ix}}_{2n}\text{,}{\mathit{Fx}}_{2n}\text{,}C)\text{,}\delta (\mathit{Jv}\text{,}\mathit{Gv}\text{,}C)\text{,}\hfill \\ \hfill & D({\mathit{Ix}}_{2n}\text{,}\mathit{Gv}\text{,}C)\text{,}D(\mathit{Jv}\text{,}{\mathit{Fx}}_{2n}\text{,}C))⩽0\text{.}\hfill \end{array}$$ Since δ(Jx_2n+1, Gv, C) ⩽ δ(Fx_2n, Gv, C), then by (ϕ₁), we have $$\begin{array}{cc}\hfill & \varphi (\delta ({\mathit{Jx}}_{2n+1}\text{,}\mathit{Gv}\text{,}C)\text{,}\delta ({\mathit{Ix}}_{2n}\text{,}\mathit{Jv}\text{,}C)\text{,}\delta ({\mathit{Ix}}_{2n}\text{,}{\mathit{Fx}}_{2n}\text{,}C)\text{,}\delta (\mathit{Jv}\text{,}\mathit{Gv}\text{,}C)\text{,}\hfill \\ \hfill & \delta ({\mathit{Ix}}_{2n}\text{,}\mathit{Gv}\text{,}C)\text{,}\delta (\mathit{Jv}\text{,}{\mathit{Fx}}_{2n}\text{,}C))⩽0\text{.}\hfill \end{array}$$ Letting n → ∞, we obtain $$\varphi (\delta (p\text{,}\mathit{Gv}\text{,}C)\text{,}0\text{,}0\text{,}\delta (p\text{,}\mathit{Gv}\text{,}C)\text{,}\delta (p\text{,}\mathit{Gv}\text{,}C)\text{,}0)⩽0$$ which implies by (ϕ_a) that δ(p, Gv, C) = 0, i.e., Gv = {p}. Therefore, Gv = {Jv} = {p} and v is a strict coincidence point of J and G.

Since G(X) ⊂ I(X), there exists u ∈ X such that {Iu} = Gv = {Jv}. By (4.1) and (ϕ₁) we obtain $$\begin{array}{cc}\hfill & \varphi (\delta (\mathit{Fu}\text{,}\mathit{Gv}\text{,}C)\text{,}\delta (\mathit{Iu}\text{,}\mathit{Jv}\text{,}C)\text{,}\delta (\mathit{Iu}\text{,}\mathit{Fu}\text{,}C)\text{,}\delta (\mathit{Jv}\text{,}\mathit{Gv}\text{,}C)\text{,}D(\mathit{Iu}\text{,}\mathit{Gv}\text{,}C)\text{,}D(\mathit{Fu}\text{,}\mathit{Jv}\text{,}C))⩽0\hfill \\ \hfill & \varphi (\delta (\mathit{Fu}\text{,}p\text{,}C)\text{,}0\text{,}\delta (p\text{,}\mathit{Fu}\text{,}C)\text{,}0\text{,}0\text{,}\delta (\mathit{Fu}\text{,}p\text{,}C))⩽0\text{.}\hfill \end{array}$$ By (ϕ_b), we obtain (Fu, p, C) = 0 which implies Fu = {p}. Hence u is a strict coincidence point of I and F. Therefore, {p} = {Iu} = Fu = {Jv} = Gv.

In view of Theorem 4.2, {p} = {Iu} = Fu is the unique point of strict coincidence of I and F. Similarly, {p} = {Jv} = Gv is the unique point of strict coincidence of J and G. Since (I, F) and (J, G) are strict OWC and p is a unique point of coincidence, then by Theorem 4.1, p is the unique common fixed point of I, J, F and G which is a strict common fixed point for F and G. In case I(X) is complete, the proof is similar. This completes the proof. □

The conclusions of Theorem 4.3 remain valid if inequality (4.1) is replaced by any one of the following contraction conditions: $$(a_1)\delta (\mathit{Fx}\text{,}\mathit{Gy}\text{,}C)⩽\alpha \max \left\{\delta (\mathit{Ix}\text{,}\mathit{Jy}\text{,}C)\text{,}\delta (\mathit{Ix}\text{,}\mathit{Fx}\text{,}C)\text{,}\delta (\mathit{Jy}\text{,}\mathit{Gy}\text{,}C)\text{,}\frac{1}{2}[D(\mathit{Ix}\text{,}\mathit{Gy}\text{,}C)+D(\mathit{Jy}\text{,}\mathit{Fx}\text{,}C)]\right\}\text{where}\alpha \in (0\text{,}1)\text{.}$$ $$(a_2){\delta }^2(\mathit{Fx}\text{,}\mathit{Gy}\text{,}C)⩽c_1\max \left\{{\delta }^2(\mathit{Ix}\text{,}\mathit{Jy}\text{,}C)\text{,}{\delta }^2(\mathit{Ix}\text{,}\mathit{Fx}\text{,}C)\text{,}{\delta }^2(\mathit{Jy}\text{,}\mathit{Gy}\text{,}C)\right\}+c_2\max \left\{\delta (\mathit{Ix}\text{,}\mathit{Fx}\text{,}C)D(\mathit{Ix}\text{,}\mathit{Gy}\text{,}C)\text{,}\delta (\mathit{Jy}\text{,}\mathit{Gy}\text{,}C)D(\mathit{Jy}\text{,}\mathit{Fx}\text{,}C)\right\}+c_{3}D(\mathit{Ix}\text{,}\mathit{Gy}\text{,}C)D(\mathit{Jy}\text{,}\mathit{Fx}\text{,}C)$$ where c₁ > 0, c₂, c₃ ⩾ 0, c₁ + 2c₂ < 1 and c₁ + c₃ < 1. $$(a_3)\delta (\mathit{Fx}\text{,}\mathit{Gy}\text{,}C)⩽\alpha \delta (\mathit{Ix}\text{,}\mathit{Jy}\text{,}C)+\beta \min \{\delta (\mathit{Ix}\text{,}\mathit{Fx}\text{,}C)\text{,}D(\mathit{Ix}\text{,}\mathit{Gy}\text{,}C)\}+\eta \min \{\delta (\mathit{Jy}\text{,}\mathit{Gy}\text{,}C)\text{,}D(\mathit{Jy}\text{,}\mathit{Fx}\text{,}C)\}$$ where α,β,η > 0,α + β < 1,α + η > 1 and (α + β)(α + η) < 1. $$(a_4)\delta (\mathit{Fx}\text{,}\mathit{Gy}\text{,}C)⩽\alpha \max \left\{\delta (\mathit{Ix}\text{,}\mathit{Jy}\text{,}C)\text{,}\delta (\mathit{Ix}\text{,}\mathit{Fx}\text{,}C)\text{,}\delta (\mathit{Jy}\text{,}\mathit{Gy}\text{,}C)\right\}+(1-\alpha )(\beta D(\mathit{Ix}\text{,}\mathit{Gy}\text{,}C)+\eta D(\mathit{Jy}\text{,}\mathit{Fx}\text{,}C))$$ where 0 ⩽ α < 1,β,η ⩾ 0,β + η < 1 and α∣β − η∣ < 1 − (β + η). $$(a_5)\delta (\mathit{Fx}\text{,}\mathit{Gy}\text{,}C)⩽\psi \left(\max \left\{\delta (\mathit{Ix}\text{,}\mathit{Jy}\text{,}C)\text{,}\delta (\mathit{Ix}\text{,}\mathit{Fx}\text{,}C)\text{,}\delta (\mathit{Jy}\text{,}\mathit{Gy}\text{,}C)\text{,}{\displaystyle \frac{1}{2}}[D(\mathit{Ix}\text{,}\mathit{Gy}\text{,}C)+D(\mathit{Jy}\text{,}\mathit{Fx}\text{,}C)]\right\}\right)$$ where $\psi :{\mathfrak{R}}^{+}\to {\mathfrak{R}}^{+}$ is an upper semi-continuous function such that ψ(t) < t for all t > 0. $$(a_6)\delta (\mathit{Fx}\text{,}\mathit{Gy}\text{,}C)⩽\psi \left(\delta (\mathit{Ix}\text{,}\mathit{Jy}\text{,}C)\text{,}\delta (\mathit{Ix}\text{,}\mathit{Fx}\text{,}C)\text{,}\delta (\mathit{Jy}\text{,}\mathit{Gy}\text{,}C)\text{,}D(\mathit{Ix}\text{,}\mathit{Gy}\text{,}C)\text{,}D(\mathit{Jy}\text{,}\mathit{Fx}\text{,}C)\right)$$ where $\psi :{\mathfrak{R}}_{+}^5\to {\mathfrak{R}}_{+}$ is an upper semi-continuous function such that ψ(t, t, t,αt,βt) < t for all t > 0 and α,β ⩾ 0 with α + β = 2. $$(a_7){\int }_0^{\delta (\mathit{Fx}\text{,}\mathit{Gy}\text{,}C)}\psi (t)\mathrm{d}t⩽\alpha {\int }_0^{\max \{\delta (\mathit{Ix}\text{,}\mathit{Jy}\text{,}C)\text{,}\delta (\mathit{Ix}\text{,}\mathit{Fx}\text{,}C)\text{,}\delta (\mathit{Jy}\text{,}\mathit{Gy}\text{,}C)\text{,}\frac{1}{2}[D(\mathit{Ix}\text{,}\mathit{Gy}\text{,}C)+D(\mathit{Jy}\text{,}\mathit{Fx}\text{,}C)]\}}\psi (t)\mathrm{d}t$$ where α ∈ (0,1) and $\psi :{\mathfrak{R}}^{+}\to {\mathfrak{R}}^{+}$ is a Lebesgue integrable function which is summable and satisfies ${\int }_0^{∊}\psi (t)\mathrm{d}t>0$ for all ∊ > 0. $$(a_8){\int }_0^{\delta (\mathit{Fx}\text{,}\mathit{Gy}\text{,}C)}\psi (t)\mathrm{d}t⩽\alpha \max \left\{{\int }_0^{\delta (\mathit{Ix}\text{,}\mathit{Jy}\text{,}C)}\psi (t)\mathrm{d}t\text{,}{\int }_0^{\delta (\mathit{Ix}\text{,}\mathit{Fx}\text{,}C)}\psi (t)\mathrm{d}t\text{,}{\int }_0^{\delta (\mathit{Jy}\text{,}\mathit{Gy}\text{,}C)}\psi (t)\mathrm{d}t\text{,}\frac{1}{2}\left[{\int }_0^{D(\mathit{Ix}\text{,}\mathit{Gy}\text{,}C)}\psi (t)\mathrm{d}t+{\int }_0^{D(\mathit{Jy}\text{,}\mathit{Fx}\text{,}C)}\psi (t)\mathrm{d}t\right]\right\}$$ where α ∈ (0,1) and $\psi :{\mathfrak{R}}^{+}\to {\mathfrak{R}}^{+}$ is a Lebesgue integrable function which is summable and satisfies ${\int }_0^{∊}\psi (t)\mathrm{d}t>0$ for all ∊ > 0.

In view of Theorem 4.3 with inequality (a₄), we obtain a generalization of Theorem 2.1 besides some relevant results contained in Abd El-Monsef et al. (2007). Using inequalities (a₁–a₃) and (a₅–a₈) together with Theorem 4.3, we obtain generalization and extension of relevant results from Jungck and Rhoades (1998), Khan (1984), Naidu and Prasad (1986), Popa et al. (2010), Sessa et al. (1986) and also obtain some new results.