Coupled coincidences for multi-valued contractions in partially ordered metric spaces

Abstract

In this article, we study the existence of coupled coincidence points for multi-valued nonlinear contractions in partially ordered metric spaces. We do it from two different approaches, the first is Δ-symmetric property recently studied in Samet and Vetro (Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces, Nonlinear Anal. 74, 4260-4268 (2011)) and second one is mixed g-monotone property studied by Lakshmikantham and Ćirić (Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70, 4341-4349 (2009)).

The theorems presented extend certain results due to Ćirić (Multi-valued nonlinear contraction mappings, Nonlinear Anal. 71, 2716-2723 (2009)), Samet and Vetro (Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces, Nonlinear Anal. 74, 4260-4268 (2011)) and many others. We support the results by establishing an illustrative example.

2000 MSC: primary 06F30; 46B20; 47E10.

1. Introduction and preliminaries

Let (X, d) be a metric space. We denote by CB(X) the collection of non-empty closed bounded subsets of X. For A, B ∈ CB(X) and x ∈ X, suppose that

$$D\left(x,A\right)=\underset{a\in A}{inf}d\left(x,a\right)\mathsf{\text{and}}H\left(A,B,\right)=max\left\{\underset{a\in A}{sup}D\left(a,B\right),\underset{b\in B}{sup}D\left(b,A\right)\right\}.$$

Such a mapping H is called a Hausdorff metric on CB(X) induced by d.

Definition 1.1. An element x ∈ X is said to be a fixed point of a multi-valued mapping T: X → CB(X) if and only if x ∈ Tx.

In 1969, Nadler [1] extended the famous Banach Contraction Principle from single-valued mapping to multi-valued mapping and proved the following fixed point theorem for the multi-valued contraction.

Theorem 1.1. Let (X, d) be a complete metric space and let T be a mapping from X into CB(X). Assume that there exists c ∈ [0,1) such that H(Tx, Ty) ≤ cd(x, y) for all x,y ∈ X. Then, T has a fixed point.

The existence of fixed points for various multi-valued contractive mappings has been studied by many authors under different conditions. In 1989, Mizoguchi and Takahashi [2] proved the following interesting fixed point theorem for a weak contraction.

Theorem 1.2. Let (X,d) be a complete metric space and let T be a mapping from X into CB(X). Assume that H (Tx, Ty) ≤ α(d(x,y)) d(x,y) for all x,y ∈ X, where α is a function from [0,∞) into [0,1) satisfying the condition$lim{sup}_{s\to t^{+}}\alpha \left(s\right)<1$for all t ∈ [0, ∞). Then, T has a fixed point.

Let $CL\left(X\right):=\left\{A\subset X|A\ne \Phi ,Ā=A\right\}$, where $Ā$ denotes the closure of A in the metric space (X, d). In this context, Ćirić [3] proved the following interesting theorem.

Theorem 1.3. (See[3]) Let (X,d) be a complete metric space and let T be a mapping from X into CL(X). Let f: X → ℝ be the function defined by f(x) = d(x, Tx) for all x ∈ X. Suppose that f is lower semi-continuous and that there exists a function ϕ: [0, +∞) → [a, 1), 0 < a < 1, satisfying

$$\underset{r\to t^{+}}{limsup}\varphi \left(r\right)<1foreacht\in \left[0,+\infty \right).$$

(1.1)

Assume that for any x ∈ X there is y ∈ Tx satisfying the following two conditions:

$$\sqrt{\varphi \left(f\left(x\right)\right)}d\left(x,y\right)\le f\left(x\right)$$

(1.2)

such that

$$f\left(y\right)\le \varphi \left(f\left(x\right)\right)d\left(x,y\right).$$

(1.3)

Then, there exists z ∈ X such that z ∈ Tz.

Definition 1.2. [4]Let X be a non-empty set and F: X × X → X be a given mapping. An element (x, y) ∈ X × X is said to be a coupled fixed point of the mapping F if F (x, y) = x and F(y, x) = y.

Definition 1.3. [5]Let (x,y) ∈ X × X, F: X × X → X and g: X → X. We say that (x,y) is a coupled coincidence point of F and g if F(x,y) = gx and F(y, x) = gy for x,y ∈ X.

Definition 1.4. A function f: X × X → ℝ is called lower semi-continuous if and only if for any sequence {x _n } ⊂ X, {y _n } ⊂ X and (x,y) ∈ X × X, we have

$$\underset{n\to \infty }{lim}\left(x_n,y_n\right)=\left(x,y\right)\Rightarrow f\left(x,y\right)\le \underset{n\to \infty }{liminf}f\left(x_n,y_n\right).$$

Let (X, d) be a metric space endowed with a partial order and G: X → X be a given mapping. We define the set Δ ⊂ X × X by

$$\Delta :=\left\{\left(x,y\right)\in X\times X|G\left(x\right)\preccurlyeq G\left(y\right)\right\}.$$

In [6], Samet and Vetro introduced the binary relation R on CL(X) defined by

$$ARB\iff A\times B\subseteq \Delta ,$$

where A, B ∈ CL(X).

Definition 1.5. Let F: X × X → CL(X) be a given mapping. We say that F is a Δ-symmetric mapping if and only if (x,y) ∈ Δ ⇒ F(x,y)RF(y,x).

Example 1.1. Suppose that X = [0,1], endowed with the usual order ≤. Let G: [0,1] → [0,1] be the mapping defined by G(x) = M for all x ∈ [0,1], where M is a constant in [0,1]. Then, Δ = [0,1] × [0,1] and F is a Δ-symmetric mapping.

Definition 1.6. [6]Let F: X × X → CL(X) be a given mapping. We say that (x,y) ∈ X × X is a coupled fixed point of F if and only if x ∈ F(x,y) and y ∈ F(y,x).

Definition 1.7. Let F: X × X → CL(X) be a given mapping and let g: X → X. We say that (x,y) ∈ X × X is a coupled coincidence point of F and g if and only if gx ∈ F(x,y) and gy ∈ F(y,x).

In [6], Samet and Vetro proved the following coupled fixed point version of Theorem 1.3.

Theorem 1.4. Let (X, d) be a complete metric space endowed with a partial order ≼. We assume that$\Delta \ne \varnothing$, i.e., there exists (x₀,y₀) ∈ Δ. Let F: X × X → CL(X) be a Δ-symmetric mapping. Suppose that the function f: X × X → [0,+∞) defined by

$$f\left(x,y\right):=D\left(x,F\left(x,y\right)\right)+D\left(y,F\left(y,x\right)\right)forallx,y\in X$$

is lower semi-continuous and that there exists a function ϕ: [0, ∞) → [a, 1), 0 < a < 1, satisfying

$$\underset{r\to t^{+}}{limsup}\varphi \left(r\right)<1foreacht\in \left[0,+\infty \right).$$

Assume that for any (x,y) ∈ Δ there exist u ∈ F(x,y) and v ∈ F(y,x) satisfying

$$\sqrt{\varphi \left(f\left(x,y\right)\right)}\left[d\left(x,u\right)+d\left(y,v\right)\right]\le f\left(x,y\right)$$

such that

$$f\left(u,v\right)\le \varphi \left(f\left(x,y\right)\right)\left[d\left(x,u\right)+d\left(y,v\right)\right].$$

Then, F admits a coupled fixed point, i.e., there exists z = (z₁, z₂) ∈ X × X such that z₁ ∈ F(z₁, z₂) and z₂ ∈ F(z₂, z₁).

In 2006, Bhaskar and Lakshmikantham [4] introduced the notion of a coupled fixed point and established some coupled fixed point theorems in partially ordered metric spaces. They have discussed the existence and uniqueness of a solution for a periodic boundary value problem. Lakshmikantham and Ćirić [5] proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces using mixed g-monotone property. For more details on coupled fixed point theory, we refer the reader to [7–12] and the references therein. Here we study the existence of coupled coincidences for multi-valued nonlinear contractions using two different approaches, first is based on Δ-symmetric property recently studied in [6] and second one is based on mixed g-monotone property studied by Lakshmikantham and Ćirić [5]. The theorems presented extend certain results due to Ćirić [3], Samet and Vetro [6] and many others. We support the results by establishing an illustrative example.

2. Coupled coincidences by Δ-symmetric property

Following is the main result of this section which generalizes the above mentioned results of Ćirić, and Samet and Vetro.

Theorem 2.1. Let (X,d) be a metric space endowed with a partial order ≼ and$\Delta \ne \varnothing$. Suppose that F: X × X → CL(X) is a Δ-symmetric mapping, g: X → X is continuous, gX is complete, the function f: g(X) × g(X) → [0, +∞) defined by

$$f\left(gx,gy\right):=D\left(gx,F\left(x,y\right)\right)+D\left(gy,F\left(y,x\right)\right)forallx,y\in X$$

is lower semi-continuous and that there exists a function ϕ: [0, ∞) → [a, 1), 0 < a < 1, satisfying

$$\underset{r\to t^{+}}{limsup}\varphi \left(r\right)<1foreacht\in \left[0,+\infty \right).$$

(2.1)

Assume that for any (x,y) ∈ Δ there exist gu ∈ F(x,y) and gv ∈ F(y,x) satisfying

$$\sqrt{\varphi \left(f\left(gx,gy\right)\right)}\left[d\left(gx,gu\right)+d\left(gy,gv\right)\right]\le f\left(gx,gy\right)$$

(2.2)

such that

$$f\left(gu,gv\right)\le \varphi \left(f\left(gx,gy\right)\right)\left[d\left(gx,gu\right)+d\left(gy,gv\right)\right].$$

(2.3)

Then, F and g have a coupled coincidence point, i.e., there exists gz = (gz₁, gz₂) ∈ X × X such that gz₁ ∈ F(z₁, z₂) and gz₂ ∈ F(z₂, z₁).

Proof. Since by the definition of ϕ we have ϕ(f(x,y)) < 1 for each (x,y) ∈ X × X, it follows that for any (x,y) ∈ X × X there exist gu ∈ F(x,y) and gv ∈ F(y,x) such that

$$\sqrt{\varphi \left(f\left(gx,gy\right)\right)}d\left(gx,gu\right)\le D\left(gx,F\left(x,y\right)\right)$$

and

$$\sqrt{\varphi \left(f\left(gx,gy\right)\right)}d\left(gy,gv\right)\le D\left(gy,F\left(y,x\right)\right).$$

Hence, for each (x,y) ∈ X × X, there exist gu ∈ F(x,y) and gv ∈ F(y,x) satisfying (2.2).

Let (x₀, y₀) ∈ Δ be arbitrary and fixed. By (2.2) and (2.3), we can choose gx₁ ∈ F(x₀, y₀) and gy₁ ∈ F(y₀, x₀) such that

$$\sqrt{\varphi \left(f\left(g{x}_0,g{y}_0\right)\right)}\left[d\left(g{x}_0,g{x}_1\right)+d\left(g{y}_0,g{y}_1\right)\right]\le f\left(g{x}_0,g{y}_0\right)$$

(2.4)

and

$$f\left(g{x}_1,g{y}_1\right)\le \varphi \left(f\left(g{x}_0,g{y}_0\right)\right)\left[d\left(g{x}_0,g{x}_1\right)+d\left(g{y}_0,g{y}_1\right)\right].$$

(2.5)

From (2.4) and (2.5), we can get

$$\begin{array}{ll}\hfill f\left(g{x}_1,g{y}_1\right)& \le \varphi \left(f\left(g{x}_0,g{y}_0\right)\right)\left[d\left(g{x}_0,g{x}_1\right)+d\left(g{y}_0,g{y}_1\right)\right]\\ =\sqrt{\varphi \left(f\left(g{x}_0,g{y}_0\right)\right)}\left\{\sqrt{\varphi \left(f\left(g{x}_0,g{y}_0\right)\right)}\left[d\left(g{x}_0,g{x}_1\right)+d\left(g{y}_0,g{y}_1\right)\right]\right\}\\ \le \sqrt{\varphi \left(f\left(g{x}_0,g{y}_0\right)\right)}f\left(g{x}_0,g{y}_0\right).\end{array}$$

Thus,

$$f\left(g{x}_1,g{y}_1\right)\le \sqrt{\varphi \left(f\left(g{x}_0,g{y}_0\right)\right)}f\left(g{x}_0,g{y}_0\right).$$

(2.6)

Now, since F is a Δ-symmetric mapping and (x₀, y₀) ∈ Δ, we have

$$F\left(x_0,y_0\right)RF\left(y_0,x_0\right)\Rightarrow \left(x_1,y_1\right)\in \Delta .$$

Also, by (2.2) and (2.3), we can choose gx₂ ∈ F(x₁, y₁) and gy₂ ∈ F(y₁, x₁) such that

$$\sqrt{\varphi \left(f\left(g{x}_1,g{y}_1\right)\right)}\left[d\left(g{x}_1,g{x}_2\right)+d\left(g{y}_1,g{y}_2\right)\right]\le f\left(g{x}_1,g{y}_1\right)$$

and

$$f\left(g{x}_2,g{y}_2\right)\le \varphi \left(f\left(g{x}_1,g{y}_1\right)\right)\left[d\left(g{x}_1,g{x}_2\right)+d\left(g{y}_1,g{y}_2\right)\right].$$

Hence, we get

$$f\left(g{x}_{2}g{y}_2\right)\le \sqrt{\varphi \left(f\left(g{x}_1,g{y}_1\right)\right)}f\left(g{x}_1,g{y}_1\right),$$

with (x₂, y₂) ∈ Δ.

Continuing this process we can choose {gx _n } ⊂ X and {gy _n } ⊂ X such that for all n ∈ ℕ, we have

$$\left(x_n,y_n\right)\in \Delta ,g{x}_{n+1}\in F\left(x_n,y_n\right),g{y}_{n+1}\in F\left(y_n,x_n\right),$$

(2.7)

$$\sqrt{\varphi \left(f\left(g{x}_n,g{y}_n\right)\right)}\left[d\left(g{x}_n,g{x}_{x+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\right]\le f\left(g{x}_n,g{y}_n\right),$$

(2.8)

and

$$f\left(g{x}_{n+1},g{y}_{n+1}\right)\le \sqrt{\varphi \left(f\left(g{x}_n,g{y}_n\right)\right)}f\left(g{x}_n,g{y}_n\right).$$

(2.9)

Now, we shall show that f(gx _n , gy _n ) → 0 as n → ∞. We shall assume that f(gx _n , gy _n ) > 0 for all n ∈ ℕ, since if f(gx _n , gy _n ) = 0 for some n ∈ ℕ, then we get D(gx _n , F(x _n , y _n )) = 0 which implies that $g{x}_n\in \overline{F\left(x_n,y_n\right)}=F\left(x_n,y_n\right)$ and D (gy _n , F(y _n , x _n )) = 0 which implies that gy _n ∈ F(y _n , x _n ). Hence, in this case, (x _n , y _n ) is a coupled coincidence point of F and g and the assertion of the theorem is proved.

From (2.9) and ϕ(t) < 1, we deduce that {f(gx _n , gy _n )} is a strictly decreasing sequence of positive real numbers. Therefore, there is some δ ≥ 0 such that

$$\underset{n\to \infty }{lim}f\left(g{x}_n,g{y}_n\right)=\delta .$$

Now, we will prove that δ = 0. Suppose that this is not the case; taking the limit on both sides of (2.9) and having in mind the assumption (2.1), we have

$$\delta \le \underset{f\left(g{x}_n,g{y}_n\right)\to {\delta }^{+}}{limsup}\sqrt{\varphi \left(f\left(g{x}_n,g{y}_n\right)\right)}\delta <\delta ,$$

a contradiction. Thus, δ = 0, that is,

$$\underset{n\to \infty }{lim}f\left(g{x}_n,g{y}_n\right)=0.$$

(2.10)

Now, let us prove that {gx _n } and {gy _n } are Cauchy sequences in (X, d). Suppose that

$$\alpha =\underset{f\left(g{x}_n,g{y}_n\right)\to 0^{+}}{limsup}\sqrt{\varphi \left(f\left(g{x}_n,g{y}_n\right)\right)}.$$

Then, by assumption (2.1), we have α < 1. Let q be such that α < q < 1. Then, there is some n₀ ∈ ℕ such that

$$\sqrt{\varphi \left(f\left(g{x}_n,g{y}_n\right)\right)}<q\mathsf{\text{for}}\mathsf{\text{each}}n\ge n_0.$$

Thus, from (2.9), we get

$$f\left(g{x}_{n+1},g{y}_{n+1}\right)\le qf\left(g{x}_n,g{y}_n\right)\mathsf{\text{for}}\mathsf{\text{each}}n\ge n_0.$$

Hence, by induction,

$$f\left(g{x}_{n+1},g{y}_{n+1}\right)\le q^{n+1-n_0}f\left(g{x}_{n0},g{y}_{n_0}\right)\mathsf{\text{for}}\mathsf{\text{each}}n\ge n_0.$$

(2.11)

Since ϕ (t) ≥ a > 0 for all t ≥ 0, from (2.8) and (2.11), we obtain

$$d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\le \frac{1}{\sqrt{a}}{q}^{n-n_0}f\left(g{x}_{n_0},g{y}_{n_0}\right)\mathsf{\text{for}}\mathsf{\text{each}}n\ge n_0.$$

(2.12)

From (2.12) and since q < 1, we conclude that {gx _n } and {gy _n } are Cauchy sequences in (X,d).

Now, since gX is complete, there is a w = (w₁, w₂) ∈ gX × gX such that

$$\underset{n\to \infty }{lim}g{x}_n=w_1=g{z}_1\mathsf{\text{and}}\underset{n\to \infty }{lim}g{y}_n=w_2=g{z}_2$$

(2.13)

for some z₁, z₂ in X. We now show that z = (z₁, z₂) is a coupled coincidence point of F and g. Since by assumption f is lower semi-continuous so from (2.10), we get

$$0\le f\left(g{z}_1,g{z}_2\right)=D\left(g{z}_1,F\left(z_1,z_2\right)\right)+D\left(g{z}_2,F\left(z_2,z_1\right)\right)\le \underset{n\to \infty }{liminf}f\left(g{x}_n,g{y}_n\right)=0.$$

Hence,

$$D\left(g{z}_1,F\left(z_1,z_2\right)\right)=D\left(g{z}_2,F\left(z_2,z_1\right)\right)=0,$$

which implies that gz₁ ∈ F(z₁, z₂) and gz₂ ∈ F(z₂, z₁), i.e., z = (z₁, z₂) is a coupled coincidence point of F and g. This completes the proof.

Now, we prove the following theorem.

Theorem 2.2. Let (X, d) be a metric space endowed with a partial order ≼ and$\Delta \ne \varnothing$. Suppose that F: X × X → CL(X) is a Δ-symmetric mapping, g: X → X is continuous and gX is complete. Suppose that the function f: gX × gX → [0,+∞) defined in Theorem 2.1 is lower semi-continuous and that there exists a function ϕ: [0, +∞) → [a, 1), 0 < a < 1, satisfying

$$\underset{r\to t+}{limsup}\varphi \left(r\right)<1foreacht\in \left[0,\infty \right).$$

(2.14)

Assume that for any (x,y) ∈ Δ, there exist gu ∈ F(x,y) and gv ∈ F(y,x) satisfying

$$\sqrt{\varphi \left(d\left(gx,gu\right)+d\left(gy,gv\right)\right)}\left[d\left(gx,gu\right)+d\left(gy,gv\right)\right]\le D\left(gx,F\left(x,y\right)\right)+D\left(gy,F\left(y,x\right)\right)$$

(2.15)

such that

$$D\left(gu,F\left(u,v\right)\right)+D\left(gv,F\left(v,u\right)\right)\le \varphi \left(d\left(gx,gu\right)+d\left(gy,gv\right)\right)\left[d\left(gx,gu\right)+d\left(gy,gv\right)\right].$$

(2.16)

Then, F and g have a coupled coincidence point, i.e., there exists z = (z₁, z₂) ∈ X × X such that gz₁ ∈ F(z₁, z₂) and gz₂ ∈ F(z₂, z₁).

Proof. Replacing ϕ (f(x,y)) with ϕ (d(gx, gu) + d (gy, gv)) and following the lines in the proof of Theorem 2.1, one can construct iterative sequences {x _n } ⊂ X and {y _n } ⊂ X such that for all n ∈ ℕ, we have

$$\left(x_n,y_n\right)\in \Delta ,g{x}_{n+1}\in F\left(x_n,y_n\right),g{y}_{n+1}\in F\left(y_n,x_n\right),$$

(2.17)

$$\begin{array}{c}\sqrt{\varphi \left(d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\right)}\left[d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\right]\\ \le D\left(g{x}_n,F\left(x_n,y_n\right)\right)+D\left(g{y}_n,F\left(y_n,x_n\right)\right)\end{array}$$

(2.18)

and

$$\begin{array}{c}D\left(g{x}_{n+1},F\left(x_{n+1},y_{n+1}\right)\right)+D\left(g{y}_{n+1},F\left(y_{n+1},x_{n+1}\right)\right)\\ \le \sqrt{\phi \left(d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\right)}\left[D\left(g{x}_n,F\left(x_n,y_n\right)\right)+D\left(g{y}_n,F\left(y_n,x_n\right)\right)\right]\end{array}$$

(2.19)

for all n ≥ 0. Again, following the lines of the proof of Theorem 2.1, we conclude that {D (gx _n , F(x _n , y _n )) + D(gy _n , F(y _n , x _n ))} is a strictly decreasing sequence of positive real numbers. Therefore, there is some δ ≥ 0 such that

$$\underset{n\to +\infty }{\lim }\{D(g{x}_n,F(x_n,y_n))+D(g{y}_n,F(y_n,x_n))=\delta .$$

(2.20)

Since in our assumptions there appears ϕ (d(gx _n , gx_{n+ 1}) + d(gy _n , gy_n+1)), we need to prove that {d(gx _n , gx_n+1) + d (gy _n , gy_n+1)} admits a subsequence converging to a certain η⁺ for some η ≥ 0. Since φ (t) ≥ a > 0 for all t ≥ 0, from (2.18) we obtain

$$d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\le \frac{1}{\sqrt{a}}\left[D\left(g{x}_n,F\left(x_n,y_n\right)\right)+D\left(g{y}_n,F\left(y_n,x_n\right)\right)\right].$$

(2.21)

From (2.20) and (2.21), we conclude that the sequence {d(gx _n , gx_n+1)+d(gy _n , gy_n+1)} is bounded. Therefore, there is some θ ≥ 0 such that

$$\underset{n\to +\infty }{liminf}\left\{d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\right\}=\theta .$$

(2.22)

Since gx_n+1∈ F(x _n , y _n ) and gy_n+1∈ F(y _n , x _n ), it follows that

$$d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\ge D\left(g{x}_n,F\left(x_n,y_n\right)\right)+D\left(g{y}_n,F\left(y_n,x_n\right)\right)$$

for each n ≥ 0. This implies that θ ≥ δ. Now, we shall show that θ = δ. If we assume that δ = 0, then from (2.20) and (2.21) we have

$$\underset{n\to +\infty }{lim}\left\{d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\right\}=0.$$

Thus, if δ = 0, then θ = δ. Suppose now that δ > 0 and suppose, to the contrary, that θ > δ. Then, θ - δ > 0 and so from (2.20) and (2.22) there is a positive integer n₀ such that

$$D\left(g{x}_n,F\left(x_n,y_n\right)\right)+D\left(g{y}_n,F\left(y_n,x_n\right)\right)<\delta +\frac{\theta -\delta }{4}$$

(2.23)

and

$$\theta -\frac{\theta -\delta }{4}<d\left(x_n,x_{n+1}\right)+d\left(y_n,y_{n+1}\right)$$

(2.24)

for all n ≥ n₀. Then, combining (2.18), (2.23) and (2.24) we get

$$\begin{array}{c}\sqrt{\phi \left(d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\right)}\left(\theta -\frac{\theta -\delta }{4}\right)\\ <\sqrt{\phi \left(d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\right)}\left[d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\right]\\ \le D\left(g{x}_n,F\left(x_n,y_n\right)\right)+D\left(g{y}_n,F\left(y_n,x_n\right)\right)\\ <\delta +\frac{\theta -\delta }{4}\end{array}$$

for all n ≥ n₀. Hence, we get

$$\sqrt{\phi \left(d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\right)}\le \frac{\theta +3\delta }{3\theta +\delta }$$

(2.25)

for all n ≥ n₀. Set $h=\frac{\theta +3\delta }{3\theta +\delta }<1$. Now, from (2.19) and (2.25), it follows that

$$D\left(g{x}_{n+1},F\left(x_{n+1},y_{n+1}\right)\right)+D\left(g{y}_{n+1},F\left(y_{n+1},x_{n+1}\right)\right)\le h\left[D\left(g{x}_n,F\left(x_n,y_n\right)\right)+D\left(g{y}_n,F\left(y_n,x_n\right)\right)\right]$$

for all n ≥ n₀. Finally, since we assume that δ > 0 and as h < 1, proceeding by induction and combining the above inequalities, it follows that

$$\begin{array}{ll}\hfill \delta & \le D\left(g{x}_{n_0+k_0},F\left(x_{n_0+k_0},y_{n_0+k_0}\right)\right)+D\left(g{y}_{n_0+k_0},F\left(y_{n_0+k_0},x_{n_0+k_0}\right)\right)\\ \le h^{k_0}D\left(g{x}_{n_0},F\left(x_{n_0},y_{n_0}\right)\right)+D\left(g{y}_{n_0},F\left(y_{n_0},x_{n_0}\right)\right)<\delta \end{array}$$

for a positive integer k₀, which is a contradiction to the assumption θ > δ and so we must have θ = δ. Now, we shall show that θ = 0. Since

$$\theta =\delta \le D\left(g{x}_n,F\left(x_n,y_n\right)\right)+D\left(g{y}_n,F\left(y_n,x_n\right)\right)\le d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right),$$

so we can read (2.22) as

$$\underset{n\to +\infty }{liminf}\left\{d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\right\}={\theta }^{+}.$$

Thus, there exists a subsequence $\left\{d\left(g{x}_{n_k},g{x}_{n_k+1}\right)+d\left(g{y}_{n_k},g{y}_{n_k+1}\right)\right\}$ such that

$$\underset{k\to +\infty }{lim}\left\{d\left(g{x}_{n_k},g{x}_{n_k+1}\right)+d\left(g{y}_{n_k},g{y}_{n_k+1}\right)\right\}={\theta }^{+}.$$

Now, by (2.14), we have

$$\underset{\left(d\left(g{x}_{n_k},g{x}_{n_k+1}\right)+d\left(g{y}_{n_k},g{y}_{n_k+1}\right)\right)\to {\theta }^{+}}{limsup}\sqrt{\phi \left(d\left(g{x}_{n_k},g{x}_{n_k+1}\right)+d\left(g{y}_{n_k},g{y}_{n_k+1}\right)\right)}<1.$$

(2.26)

From (2.19),

$$\begin{array}{c}D\left(g{x}_{n_k+1},F\left(x_{n_k+1},y_{n_k+1}\right)\right)+D\left(g{y}_{n_k+1},F\left(y_{n_k+1},x_{n_k+1}\right)\right)\\ \le \sqrt{\phi \left(d\left(g{x}_{n_k},g{x}_{n_k+1}\right)+d\left(g{y}_{n_k},g{y}_{n_k+1}\right)\right)}\left[D\left(g{x}_{n_k},F\left(x_{n_k},y_{n_k}\right)\right)+D\left(g{y}_{n_k},F\left(y_{n_k},x_{n_k}\right)\right)\right].\end{array}$$

Taking the limit as k → +∞ and using (2.20), we get

$$\begin{array}{ll}\hfill \delta & =\underset{k\to +\infty }{limsup}\left\{D\left(g{x}_{n_k+1},F\left(x_{n_k+1},y_{n_k+1}\right)\right)+D\left(g{y}_{n_k+1},F\left(y_{n_k+1},x_{n_k+1}\right)\right)\right\}\\ \le \left(\underset{k\to +\infty }{limsup}\sqrt{\phi \left(d\left(g{x}_{n_k},g{x}_{n_k+1}\right)+d\left(g{y}_{n_k},g{y}_{n_k+1}\right)\right)}\right)\\ \left(\underset{k\to +\infty }{limsup}\left\{D\left(g{x}_{n_k},F\left(x_{n_k},y_{n_k}\right)\right)+D\left(g{y}_{n_k},F\left(y_{n_k},x_{n_k}\right)\right)\right\}\right)\\ =\left(\underset{\left(d\left(g{x}_{n_k},g{x}_{n_k+1}\right)+d\left(g{y}_{n_k},g{y}_{n_k+1}\right)\right)\to {\theta }^{+}}{limsup}\sqrt{\phi \left(d\left(g{x}_{n_k},g{x}_{n_k+1}\right)+d\left(g{y}_{n_k},g{y}_{n_k+1}\right)\right)}\right)\delta .\end{array}$$

From the last inequality, if we suppose that δ > 0, we get

$$1\le \underset{\left(d\left(g{x}_{n_k},g{x}_{n_k+1}\right)+d\left(g{y}_{n_k},g{y}_{n_k+1}\right)\right)\to \theta +}{limsup}\sqrt{\phi \left(d\left(g{x}_{n_k},g{x}_{n_k+1}\right)+d\left(g{y}_{n_k},g{y}_{n_k+1}\right)\right)},$$

a contradiction with (2.26). Thus, δ = 0. Then, from (2.20) and (2.21) we have

$$\alpha =\underset{\left(d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\right)\to 0+}{limsup}\sqrt{\phi \left(d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{y}_n,g{y}_{n+1}\right)\right)}<1.$$

Once again, proceeding as in the proof of Theorem 2.1, one can prove that {gx _n } and {gy _n } are Cauchy sequences in gX and that z = (z₁, z₂) ∈ X × X is a coupled coincidence point of F, g, i.e.

$$g{z}_1\in F\left(z_1,z_2\right)\mathsf{\text{and}}g{z}_2\in F\left(z_2,z_1\right).$$

Example 2.3. Suppose that X = [0,1], equipped with the usual metric d: X × X → [0, + ∞), and G: [0,1] → [0,1] is the mapping defined by

$$G\left(x\right)=M\mathsf{\text{for all }}x\in \left[0,1\right],$$

where M is a constant in [0,1]. Let F: X × X → CL(X) be defined as

$$F\left(x,y\right)=\left\{\begin{array}{cc}\hfill \frac{x^2}{4}\hfill & \hfill \mathsf{\text{if }}y\in \left[0,\frac{15}{32}\right)\cup \left(\frac{15}{32},1\right],\hfill \\ \hfill \left\{\frac{15}{96},\frac{1}{5}\right\}\hfill & \hfill \mathsf{\text{if }}y=\frac{15}{32}.\hfill \end{array}\right.$$

Then, Δ = [0,1] × [0,1] and F is a Δ-symmetric mapping. Define now φ: [0, +∞) → [0,1) by

$$\phi \left(t\right)=\left\{\begin{array}{cc}\hfill \frac{11}{12}t\hfill & \hfill \mathsf{\text{if }}t\in \left[0,\frac{2}{3}\right],\hfill \\ \hfill \frac{11}{18}\hfill & \hfill \mathsf{\text{if }}t\in \left(\frac{2}{3},+\infty \right).\hfill \end{array}\right.$$

Let g: [0,1] → [0,1] be defined as gx = x². Now, we shall show that F(x, y) satisfies all the assumptions of Theorem 2.2. Let

$$f(x,y)=\{\begin{array}{ll}\sqrt{x}+\sqrt{y}-\frac{1}{4}(x+y)\hfill & \text{if }x,y\in [0,\frac{15}{32})\cup (\frac{15}{32},1],\hfill \\ \sqrt{x}-\frac{1}{4}x+\frac{43}{160}\hfill & \text{if }x\in [0,\frac{15}{32})\cup (\frac{15}{32},1]\text{ and }y=\frac{15}{32},\hfill \\ \sqrt{y}-\frac{1}{4}y+\frac{43}{160}\hfill & \text{if }y\in \text{[0,}\frac{15}{32})\cup (\frac{15}{32},1]\text{ and }x=\frac{15}{32},\hfill \\ \frac{43}{80}\hfill & \text{if }x=y=\frac{15}{32}.\hfill \end{array}$$

It is easy to see that the function

$$f\left(gx,gy\right)=\left\{\begin{array}{cc}\hfill x+y-\frac{1}{4}\left(x^2+y^2\right)\hfill & \hfill \mathsf{\text{if }}x,y\in \left[0,\frac{15}{32}\right)\cup \left(\frac{15}{32},1\right],\hfill \\ \hfill x-\frac{1}{4}{x}^2+\frac{43}{160}\hfill & \hfill \mathsf{\text{if }}x\in \left[0,\frac{15}{32}\right)\cup \left(\frac{15}{32},1\right]\mathsf{\text{ and }}y=\frac{15}{32},\hfill \\ \hfill y-\frac{1}{4}{y}^2+\frac{43}{160}\hfill & \hfill \mathsf{\text{if }}y\in \left[0,\frac{15}{32}\right)\cup \left(\frac{15}{32},1\right]\mathsf{\text{ and }}x=\frac{15}{32},\hfill \\ \hfill \frac{43}{80}\hfill & \hfill \mathsf{\text{if }}x=y=\frac{15}{32}\hfill \end{array}\right.$$

is lower semi-continuous. Therefore, for all x, y ∈ [0,1] with $x,y\ne \frac{15}{32}$, there exist $gu\in F\left(x,y\right)=\left\{\frac{x^2}{4}\right\}$ and $gv\in F\left(y,x\right)=\left\{\frac{y^2}{4}\right\}$ such that

$$\begin{array}{ll}\hfill D\left(gu,F\left(u,v\right)\right)+D\left(gv,F\left(v,u\right)\right)& =\frac{x^2}{4}-\frac{x^4}{64}+\frac{y^2}{4}-\frac{y^4}{64}\\ =\frac{1}{4}\left[\left(x+\frac{x^2}{4}\right)\left(x-\frac{x^2}{4}\right)+\left(y+\frac{y^2}{4}\right)\left(y-\frac{y^2}{4}\right)\right]\\ \le \frac{1}{4}\left[\left(x+\frac{x^2}{4}\right)d\left(gx,gu\right)+\left(y+\frac{y^2}{4}\right)d\left(gy,gv\right)\right]\\ \le \frac{1}{2}max\left\{x+\frac{x^2}{4},y+\frac{y^2}{4}\right\}\left[d\left(gx,gu\right)+d\left(gy,gv\right)\right]\\ <\frac{11}{12}max\left\{\left(x-\frac{x^2}{4}\right),\left(y-\frac{y^2}{4}\right)\right\}\left[d\left(gx,gu\right)+d\left(gy,gv\right)\right]\\ \le \phi \left(d\left(gx,gu\right)+d\left(gy,gv\right)\right)\left[d\left(gx,gu\right)+d\left(gy,gv\right)\right].\end{array}$$

Thus, for x, y ∈ [0,1] with $x,y\ne \frac{15}{32}$, the conditions (2.15) and (2.16) are satisfied. Following similar arguments, one can easily show that conditions (2.15) and (2.16) are also satisfied for $x\in \left[0,\frac{15}{32}\right)\cup \left(\frac{15}{32},1\right]$ and $y=\frac{15}{32}$. Finally, for $x=y=\frac{15}{32}$, if we assume that $gu=gv=\frac{15}{96}$, it follows that $d\left(gx,gu\right)+d\left(gy,gv\right)=\frac{15}{24}$.

Consequently, we get

$$\begin{array}{ll}\hfill \sqrt{\phi \left(d\left(gx,gu\right)+d\left(gy,gv\right)\right)}\left[d\left(gx,gu\right)+d\left(gy,gv\right)\right]& =\sqrt{\frac{11}{24}\cdot \frac{15}{24}}\cdot \frac{15}{24}\\ <\frac{43}{80}=D\left(gx,F\left(x,y\right)\right)+D\left(gy,F\left(y,x\right)\right)\end{array}$$

and

$$\begin{array}{ll}\hfill D\left(gu,F\left(u,v\right)\right)+D\left(gv,F\left(v,u\right)\right)& =2\left|\frac{15}{96}-\frac{1}{4}{\left(\frac{15}{96}\right)}^2\right|\\ <\frac{11}{12}\cdot \frac{15}{24}\cdot \frac{15}{24}\\ =\phi \left(d\left(gx,gu\right)+d\left(gy,gv\right)\right)\left[d\left(gx,gu\right)+d\left(gy,gv\right)\right].\\ \hfill \text{(4) }\end{array}$$

Thus, we conclude that all the conditions of Theorem 2.2 are satisfied, and F, g admits a coupled coincidence point z = (0, 0).

3. Coupled coincidences by mixed g-monotone property

Recently, there have been exciting developments in the field of existence of fixed points in partially ordered metric spaces (cf. [13–24]). Using the concept of commuting maps and mixed g-monotone property, Lakshmikantham and Ćirić in [5] established the existence of coupled coincidence point results to generalize the results of Bhaskar and Lakshmikantham [4]. Choudhury and Kundu generalized these results to compatible maps. In this section, we shall extend the concepts of commuting, compatible maps and mixed g-monotone property to the case when F is multi-valued map and prove the extension of the above mentioned results.

Analogous with mixed monotone property, Lakshmikantham and Ćirić [5] introduced the following concept of a mixed g-monotone property.

Definition 3.1. Let (X, ≼) be a partially ordered set and F: X × X → X and g: X → X. We say F has the mixed g-monotone property if F is monotone g-non-decreasing in its first argument and is monotone g-non-increasing in its second argument, that is, for any x,y ∈ X,

$$x_1,x_2\in X,g\left(x_1\right)\preccurlyeq g\left(x_2\right)impliesF\left(x_1,y\right)\preccurlyeq F\left(x_2,y\right)$$

(3.1)

and

$$y_1,y_2\in X,g\left(y_1\right)\preccurlyeq g\left(y_2\right)impliesF\left(x,y_1\right)\succcurlyeq F\left(x,y_2\right).$$

(3.2)

Definition 3.2. Let (X, ≼) be a partially ordered set, F: X × X → CL(X) and let g: X → X be a mapping. We say that the mapping F has the mixed g-monotone property if, for all x₁, x₂, y₁, y₂ ∈ X with gx₁ ≼ gx₂and gy₁ ≽ gy₂, we get for all gu₁ ∈ F(x₁, y₁) there exists gu₂ ∈ F(x₂, y₂) such that gu₁ ≼ gu₂and for all gv₁ ∈ F(y₁,x₁) there exists gv₂ ∈ F(y₂, x₂) such that gv₁ ≽ z gv₂.

Definition 3.3. The mapping F: X × X → CB(X) and g: X → X are said to be compatible if

$$\underset{n\to \infty }{lim}H\left(g\left(F\left(x_n,y_n\right)\right),F\left(g{x}_n,g{y}_n\right)\right)=0$$

and

$$\underset{n\to \infty }{lim}H\left(g\left(F\left(y_n,x_n\right)\right),F\left(g{y}_n,g{x}_n\right)\right)=0,$$

whenever {x _n } and {y _n } are sequences in X, such that x = lim_n→∞gx _n ∈ lim_n→∞F(x _n , y _n ) and y = lim_n→∞gy _n ∈ lim_n→∞F(y _n , x _n ), for all x, y ∈ X are satisfied.

Definition 3.4. The mapping F: X × X → CB(X) and g: X → X are said to be commuting if gF(x, y) ⊆ F(gx, gy) for all x, y ∈ X.

Lemma 3.1. [1]If A,B ∈ CB (X) with H (A, B) < ϵ, then for each a ∈ A there exists an element b ∈ B such that d(a, b) < ϵ.

Lemma 3.2. [1]Let {A _n } be a sequence in CB(X) and lim_n→∞H (A _n , A) = 0 for A ∈ CB (X). If x _n ∈ A _n and lim_n→∞d(x _n , x) = 0, then x ∈ A.

Let (X, ≼) be a partially ordered set and d be a metric on X such that (X, d) is a complete metric space. We define the partial order on the product space X × X as:

for (u,v),(x,y) ∈ X × X, (u, v) ≼ (x, y) if and only if u ≼ x, v ≽ y.

The product metric on X × X is defined as

$$d\left(\left(x_1,y_1\right),\left(x_2,y_2\right)\right):=d\left(x_1,x_2\right)+d\left(y_1,y_2\right)\mathsf{\text{for all }}{x}_i,y_i\in X\left(i=1,2\right).$$

For notational convenience, we use the same symbol d for the product metric as well as for the metric on X.

We begin with the following result that gives the existence of a coupled coincidence point for compatible maps F and g in partially ordered metric spaces, where F is the multi-valued mappings.

Theorem 3.1. Let F: X × X → CB(X), g: X → X be such that:

• (1) there exists κ ∈ (0,1) with

  $$H\left(F\left(x,y\right),F\left(u,v\right)\right)\le \frac{k}{2}d\left(\left(gx,gy\right),\left(gu,gv\right)\right)forall\left(gx,gy\right)\succcurlyeq \left(gu,gv\right);$$

• (2) if gx ₁ ≼ gx ₂, gy ₂ ≼ gy ₁, x _i , y _i ∈ X(i = 1,2), then for all gu ₁ ∈ F(x ₁, y ₁) there exists gu ₂ ∈ F(x ₂, y ₂) with gu ₁ ≼ gu ₂ and for all gv ₁ ∈ F(y ₁, x ₁) there exists gv ₂ ∈ F(y ₂, x ₂) with gv ₂ ≼ gv ₁ provided d((gu ₁, gv ₁), (gu ₂, gv ₂)) < 1; i.e. F has the mixed g-monotone property, provided d((gu ₁, gv ₁), (gu ₂, gv ₂)) < 1;

• (3) there exists x ₀, y ₀ ∈ X, and some gx ₁ ∈ F(x ₀, y ₀), gy ₁ ∈ F(y ₀, x ₀) with gx ₀ ≼ gx ₁, gy ₀ ≽ gy ₁ such that d((gx ₀, gy ₀), (gx ₁, gy ₁)) < 1 - κ, where κ ∈ (0,1);

• (4) if a non-decreasing sequence {x _n } → x, then x _n ≤ x for all n and if a non-increasing sequence {y _n } → y, then y ≤ y _n for all n and gX is complete.

Then, F and g have a coupled coincidence point.

Proof. Let x₀, y₀ ∈ X then by (3) there exists gx₁ ∈ F(x₀, y₀), gy₁ ∈ F(y₀, x₀) with gx₀ ≼ gx₁, gy₀ ≽ gy₁ such that

$$d\left(\left(g{x}_0,g{y}_0\right),\left(g{x}_1,g{y}_1\right)\right)<1-\kappa .$$

(3.3)

Since (gx₀, gy₀) ≼ (gx₁, gy₁) using (1) and (3.3), we have

$$H\left(F\left(x_0,y_0\right),F\left(x_1,y_1\right)\right)\le \frac{\kappa }{2}d\left(\left(g{x}_0,g{y}_0\right),\left(g{x}_1,g{y}_1\right)\right)<\frac{\kappa }{2}\left(1-\kappa \right)$$

and similarly

$$H\left(F\left(y_0,x_0\right),F\left(y_1,x_1\right)\right)\le \frac{\kappa }{2}\left(1-\kappa \right).$$

Using (2) and Lemma 3.1, we have the existence of gx₂ ∈ F(x₁, y₁), gy₂ ∈ F (y₁, x₁) with x₁ ≼ x₂ and y₁ ≽ y₂ such that

$$d\left(g{x}_1,g{x}_2\right)\le \frac{\kappa }{2}\left(1-\kappa \right)$$

(3.4)

and

$$d\left(g{y}_1,g{y}_2\right)\le \frac{\kappa }{2}\left(1-\kappa \right).$$

(3.5)

From (3.4) and (3.5),

$$d\left(\left(g{x}_1,g{y}_1\right),\left(g{x}_2,g{y}_2\right)\right)\le \kappa \left(1-\kappa \right).$$

(3.6)

Again by (1) and (3.6), we have

$$H\left(F\left(x_1,y_1\right),F\left(x_2,y_2\right)\right)\le \frac{{\kappa }^2}{2}\left(1-\kappa \right)$$

and

$$D\left(F\left(y_1,x_1\right),F\left(y_2,x_2\right)\right)\le \frac{{\kappa }^2}{2}\left(1-\kappa \right).$$

From Lemma 3.1 and (2), we have the existence of gx₃ ∈ F(x₂, y₂), gy₃ ∈ F (y₂, x₂) with gx₂ ≼ gx₃, gy₂ ≽ gy₃ such that

$$d\left(g{x}_2,g{x}_3\right)\le \frac{{\kappa }^2}{2}\left(1-\kappa \right)$$

and

$$d\left(g{y}_2,g{y}_3\right)\le \frac{{\kappa }^2}{2}\left(1-\kappa \right).$$

It follows that

$$d\left(\left(g{x}_2,g{y}_2\right),\left(g{x}_3,g{y}_3\right)\right)\le {\kappa }^2\left(1-\kappa \right).$$

Continuing in this way we obtain gx_n+1∈ F (x _n , y _n ), gy_n+1∈ F (y _n , x _n ) with $g{x}_n\preccurlyeq g{x}_{n+1},g{y}_n\succcurlyeq g{y}_{n_1}$ such that

$$d\left(g{x}_n,g{x}_{n+1}\right)\le \frac{{\kappa }^n}{2}\left(1-\kappa \right)$$

and

$$d\left(g{y}_n,g{y}_{n+1}\right)\le \frac{{\kappa }^n}{2}\left(1-\kappa \right).$$

Thus,

$$d\left(\left(g{x}_n,g{y}_n\right),\left(g{x}_{n+1},g{y}_{n+1}\right)\right)\le {\kappa }^n\left(1-\kappa \right).$$

(3.7)

Next, we will show that {gx _n } is a Cauchy sequence in X. Let m > n. Then,

$$\begin{array}{ll}\hfill d\left(g{x}_n,g{x}_m\right)& \le d\left(g{x}_n,g{x}_{n+1}\right)+d\left(g{x}_{n+1},g{x}_{n+2}\right)+d\left(g{x}_{n+2},g{x}_{n+3}\right)+\cdots +d\left(g{x}_{m-1},g{x}_m\right)\\ \le \left[{\kappa }^n+{\kappa }^{n+1}+{\kappa }^{n+2}+\cdots +{\kappa }^{m-1}\right]\frac{\left(1-\kappa \right)}{2}\\ ={\kappa }^n\left[1+\kappa +{\kappa }^2+\cdots +{\kappa }^{m-n-1}\right]\frac{\left(1-\kappa \right)}{2}\\ ={\kappa }^n\left[\frac{1-{\kappa }^{m-n}}{1-\kappa }\right]\frac{\left(1-\kappa \right)}{2}\\ =\frac{{\kappa }^n}{2}\left(1-{\kappa }^{m-n}\right)<\frac{{\kappa }^n}{2},\\ \hfill \text{(6) }\end{array}$$

because κ ∈ (0,1), 1 - κ^m-n< 1. Therefore, d(gx _n , gx _m ) → 0 as n → ∞ implies that {gx _n } is a Cauchy sequence. Similarly, we can show that {gy _n } is also a Cauchy sequence in X. Since gX is complete, there exists x, y ∈ X such that gx _n → gx and gy _n → gy as n → ∞. Finally, we have to show that gx ∈ F(x, y) and gy ∈ F(y, x).

Since {gx _n } is a non-decreasing sequence and {gy _n } is a non-increasing sequence in X such that gx _n → x and gy _n → y as n → ∞, therefore we have gx _n ≼ x and gy _n ≽ y for all n. As n → ∞, (1) implies that

$$H\left(F\left(x_n,y_n\right),F\left(x,y\right)\right)\le \frac{\kappa }{2}d\left(\left(g{x}_n,g{y}_n\right),\left(gx,gy\right)\right)\to 0.$$

Since gx_n+1∈ F(x _n , y _n ) and lim_n→∞d(gx_n+1, gx) = 0, it follows using Lemma 3.2 that gx ∈ F(x, y). Again by (1),

$$H\left(F\left(y_n,x_n\right),F\left(y,x\right)\right)\le \frac{\kappa }{2}d\left(\left(g{y}_n,g{x}_n\right),\left(gy,gx\right)\right)\to 0.$$

Since gy_n+1∈ F(y _n , x _n ) and lim_n→∞d(gy_n+1, gy) = 0, it follows using Lemma 3.2 that gy ∈ F(y, x).

Theorem 3.2. Let F: X × X → CB(X), g: X → X be such that conditions (1)-(3) of Theorem 3.1 hold. Let X be complete, F and g be continuous and compatible. Then, F and g have a coupled coincidence point.

Proof. As in the proof of Theorem 3.1, we obtain the Cauchy sequences {gx _n } and {gy _n } in X. Since X is complete, there exists x, y ∈ X such that gx _n → x and gy _n → y as n → ∞. Finally, we have to show that gx ∈ F(x, y) and gy ∈ F(y, x). Since the mapping F: X × X → CB (X) and g: X → X are compatible, we have

$$\underset{n\to \infty }{lim}H\left(g\left(F\left(x_n,y_n\right)\right),F\left(g{x}_n,g{y}_n\right)\right)=0,$$

because {x _n } is a sequence in X, such that x = lim_n→∞gx_n+1∈ lim_n→∞F(x _n , y _n ) is satisfied. For all n ≥ 0, we have

$$D\left(gx,F\left(g{x}_n,g{y}_n\right)\right)\le D\left(gx,gF\left(x_n,y_n\right)\right)+H\left(gF\left(x_n,y_n\right),F\left(g{x}_n,g{y}_n\right)\right).$$

Taking the limit as n → ∞, and using the fact that g and F are continuous, we get, D (gx, F(x, y)) = 0, which implies that gx ∈ F (x, y).

Similarly, since the mapping F and g are compatible, we have

$$\underset{n\to \infty }{lim}H\left(g\left(F\left(y_n,x_n\right)\right),F\left(g{y}_n,g{x}_n\right)\right)=0,$$

because {y _n } is a sequence in X, such that y = lim_n→∞gy_n+1∈ lim_n→∞F(y _n , x _n ) is satisfied. For all n ≥ 0, we have

$$D\left(gy,F\left(g{y}_n,g{x}_n\right)\right)\le D\left(gy,gF\left(y_n,x_n\right)\right)+H\left(gF\left(y_n,x_n\right),F\left(g{y}_n,g{x}_n\right)\right).$$

Taking the limit as n → ∞, and using the fact that g and F are continuous, we get D (gy, F(y, x)) = 0, which implies that gy ∈ F(y, x).

As commuting maps are compatible, we obtain the following;

Theorem 3.3. Let F: X × X → CB(X), g: X → X be such that conditions (1)-(3) of Theorem 3.1 hold. Let X be complete, F and g be continuous and commuting. Then, F and g have a coupled coincidence point.