**2. Preliminaries**

**Definition 1.** *[6] Let X be a nonempty set. Assume that, for all x*, *y*, ∈ *X and distinct u*1, ··· , *uv* ∈ *X* − {*x*, *y*}*, dv* : *X* × *X* → *R satisfies :*

*Axioms* **2020**, *9*, 129


*Then,* (*X*, *dv*) *is a bv*(*s*)*-metric space.*

**Definition 2.** *[6] In the bv*(*s*)*-metric space* (*X*, *dv*)*, the sequence* < *un* >


Clearly, *b*1(1)-metric space is the usual metric space, whereas *b*1(*s*), *b*2(1), *b*2(*s*), and *bv*(1)-metric spaces are, respectively, the *b*-metric space ([1]), rectangular metric space ([2]), rectangular b-metric space ([3]), and *v*-generalized metric space ([2]).

**Lemma 1.** *[6] If* (*X*, *dv*) *is a bv*(*s*)*-metric space, then* (*X*, *dv*) *is a b*2*v*(*s*2)*-metric space.*

**Definition 3.** *An element* (*u*, *v*) ∈ *X* × *X is called a coupled coincidence point of S* : *X* × *X* → *X and g* : *X* → *X if g*(*u*) = *S*(*u*, *v*) *and g*(*v*) = *S*(*v*, *u*)*. In this case, we also say that* (*g*(*u*), *g*(*v*)) *is the point of coupled coincidence of S and g. If u* = *g*(*u*) = *S*(*u*, *v*) *and v* = *g*(*v*) = *S*(*v*, *u*)*, then we say that* (*u*, *v*) *is a common coupled fixed point of S and g.*

We will denote by *COCP*{*S*, *g*} and *CCOFP*{*S*, *g*} respectively the set of all coupled coincidence points and the set of all common coupled fixed points of *S* and *g*.

**Definition 4.** *S* : *X* × *X* → *X and g* : *X* → *X are said to be weakly compatible if and only if S*(*g*(*u*), *g*(*v*)) = *g*(*S*(*u*, *v*)) *for all* (*u*, *v*) ∈ *COCP*{*S*, *g*}*.*

### **3. Main Results**

We will start this section by proving the following lemma which is an extension of Lemma 1.12 of [6] to two sequences:

**Lemma 2.** *Let* (*X*, *dv*) *be a bv*(*s*)*-metric space and let* < *un* > *and* < *vn* > *be two sequences in X such that un* = *un*+1, *vn* = *vn*+<sup>1</sup> (*n* ≥ 0)*. Suppose that λ* ∈ [0, 1) *and c*1, *c*<sup>2</sup> *are real nonnegative numbers such that*

$$K\_{m,n} \le \lambda K\_{m-1,n-1} + c\_1 \lambda^m + c\_2 \lambda^n, \text{ for all } m, n \in \mathbb{N},\tag{1}$$

*where Km*,*<sup>n</sup>* = max{*dv*(*um*, *un*), *dv*(*vm*, *vn*)} *or Km*,*<sup>n</sup>* = *dv*(*um*, *un*) + *dv*(*vm*, *vn*)*. Then,* < *un* > *and* < *vn* > *are Cauchy sequences.*

**Proof.** From (1), we have

$$\begin{array}{rcl} \mathbb{K}\_{n,n+1} & \leq & \lambda \mathbb{K}\_{n-1,n} + c\_1 \lambda^n + c\_2 \lambda^{n+1} \\\\ & \leq \cdots \\ & \leq \lambda^n \mathbb{K}\_{0,1} + c\_1 n \lambda^n + c\_2 n \lambda^{n+1} \\\\ & \leq \lambda^n \mathbb{K}\_{0,1} + \mathbb{C}\_0 n \lambda^n. \end{array} \tag{2}$$

For *m*, *n*, *k* ∈ *N*, by (1), we have

$$K\_{m+k,n+k} \le \begin{array}{c} \lambda \max\{K\_{m+k-1,n+k-1}c\_1\lambda^{m+k-1} + c\_2\lambda^{n+k-1}\} \\ \le \lambda K\_{m+k-1,n+k-1} + c\_1\lambda^{m+k} + c\_2\lambda^{n+k} \\ \dots \\ \le \lambda^k K\_{m,n} + kC\_1\lambda^k(\lambda^m + \lambda^n). \end{array} \tag{3}$$

Since 0 < *λ* < 1, we can find a positive integer *qk* such that 0 < *λqk* < <sup>1</sup> *<sup>s</sup>* . Now, suppose *v* ≥ 2. Then, by using condition 3. of a *bv*(*s*)-metric and inequalities (2) and (3), we have

$$\begin{split} \mathcal{K}\_{m,n} &\leq \begin{aligned} &\leq \left[\mathcal{K}\_{m,m+1} + \mathcal{K}\_{m+1,m+2} + \dots + \mathcal{K}\_{m+\upsilon-3,m+\upsilon-2} + \mathcal{K}\_{m+\upsilon-2,m+q\_{k}} + \mathcal{K}\_{m+q\_{k},n+q\_{k}} + \mathcal{K}\_{m+q\_{k},n}\right] \\ &\leq \left[\mathcal{A}^{m} + \lambda^{m+1} + \dots + \lambda^{m+\upsilon-3}\right] \mathcal{K}\_{0} + \mathfrak{s}\mathcal{C}\_{0}[m\lambda^{m} + (m+1)\lambda^{m+1} + \dots + (m+\upsilon-3)\lambda^{m+\upsilon-2}] \\ &+ \mathfrak{s}[\lambda^{\mathfrak{m}}\mathcal{K}\_{\upsilon-2,q\_{k}} + m\lambda^{\mathfrak{m}}(\lambda^{\upsilon-2} + \lambda^{q\_{k}})\mathcal{K}\_{0}] \\ &+ \mathfrak{s}[\lambda^{\mathfrak{q}}\mathcal{K}\_{\mathfrak{m},\mathfrak{p}} + q\_{k}\lambda^{\mathfrak{q}\_{k}}(\lambda^{\mathfrak{m}} + \lambda^{\mathfrak{n}})\mathcal{K}\_{0}] + \mathfrak{s}[\lambda^{\mathfrak{n}}\mathcal{K}\_{\mathfrak{q},\mathfrak{p}} + n\lambda^{\mathfrak{n}}(\lambda^{\mathfrak{q}\_{k}} + 1)\mathcal{K}\_{0}]. \end{aligned} \end{cases}$$

Then,

$$\begin{array}{rcl} K\_{m,\mathbb{I}} & \leq & \frac{s\lambda^{m}}{(1-s\lambda^{q\_{k}})(1-\lambda)} K\_{0,1} + \frac{s(m+\upsilon-3)\lambda^{m}}{(1-\lambda)(1-s\lambda^{q\_{k}})} \\ & + \frac{s}{1-s\lambda^{q\_{k}}} [\lambda^{m} K\_{\upsilon-2,q\_{k}} + m\lambda^{m}(\lambda^{\upsilon-2} + \lambda^{q\_{k}}) K\_{0,1}] \\ & + \frac{s}{1-s\lambda^{q\_{k}}} [q\_{k}\lambda^{q\_{k}}(\lambda^{m} + \lambda^{n}) K\_{0,1}] + \frac{s}{1-s\lambda^{q\_{k}}} [\lambda^{n} K\_{q\_{k},0} + n\lambda^{n}(\lambda^{q\_{k}} + 1) K\_{0,1}]. \end{array}$$

Thus, from the definition of *Km*,*n*, we see that, as *m*, *n* → +∞, *dv*(*um*, *un*) → 0 and *dv*(*vm*, *vn*) → 0 and thus < *un* > and < *vn* > are Cauchy sequences.

#### *3.1. Coupled Fixed Point Theorems*

We now present our main theorems as follows:

**Theorem 1.** *Let* (*X*, *dv*) *be a bv*(*s*)*-metric space , T* : *X* → *X be a one to one mapping, S*: *X* × *X* → *X and g* : *X* → *X be mappings such that S*(*X* × *X*) ⊂ *g*(*X*)*, Tg*(*X*) *is complete. If there exist real numbers λ*, *μ*, *ν with* <sup>0</sup> <sup>≤</sup> *<sup>λ</sup>* <sup>&</sup>lt; <sup>1</sup>*,* <sup>0</sup> <sup>≤</sup> *<sup>μ</sup>*, *<sup>ν</sup>* <sup>≤</sup> <sup>1</sup>*,* min{*λμ*, *λν*} <sup>&</sup>lt; <sup>1</sup> *<sup>s</sup> such that, for all u*, *v*, *w*, *z* ∈ *X*

$$d\_{\mathbb{P}}(TS(u,v),TS(w,z)) \quad \leq \quad \lambda \max\{d\_{\mathbb{P}}(T\mathbb{g}u, T\mathbb{g}w), d\_{\mathbb{P}}(T\mathbb{g}v, T\mathbb{g}z), \mu d\_{\mathbb{P}}(T\mathbb{g}u, TS(u,v)), \mu d\_{\mathbb{P}}(T\mathbb{g}v, TS(v,u), \mu d\_{\mathbb{P}}(T\mathbb{g}v, T\mathbb{g}z))\}\tag{4}$$
 
$$\nu d\_{\mathbb{P}}(T\mathbb{g}w, TS(w,z)), \nu d\_{\mathbb{P}}(T\mathbb{g}z, TS(z,w))\}$$

*then the following holds :*


**Proof.** 1. We shall start the proof by showing that the sequences < *Tgun* > and < *Tgvn* > are Cauchy sequences, where < *gun* > and < *gvn* > are as mentioned in the hypothesis.

By (4), we have

*dv*(*Tgun*, *Tgun*+1) = *dv*(*TS*(*un*−1, *vn*−1), *TS*(*un*, *vn*)) ≤ *λ* max{*dv*(*Tgun*−1, *Tgun*), *dv*(*Tgvn*−1, *Tgvn*), *μdv*(*Tgun*−1, *TS*(*un*−1, *vn*−1)), *μdv*(*Tgvn*−1, *TS*(*vn*−1, *un*−1)), *νdv*(*Tgun*, *TS*(*un*, *vn*)), *νdv*(*Tgvn*, *TS*(*vn*, *un*))} ≤ *λ* max{*dv*(*Tgun*−1, *Tgun*), *dv*(*Tgvn*−1, *Tgvn*), *dv*(*Tgun*−1, *Tgun*), *dv*(*Tgvn*−1, *Tgvn*), *dv*(*Tgun*, *Tgun*+1), *dv*(*Tgvn*, *Tgvn*+1)}. (5)

Similarly, we get

$$\begin{split} d\_{\upsilon}(T\mathcal{g}\upsilon\_{n}, T\mathcal{g}\upsilon\_{n+1}) &\leq \ \lambda \max\{d\_{\upsilon}(T\mathcal{g}\upsilon\_{n-1}, T\mathcal{g}\upsilon\_{n}), d\_{\upsilon}(T\mathcal{g}u\_{n-1}, T\mathcal{g}u\_{n}), d\_{\upsilon}(T\mathcal{g}\upsilon\_{n-1}, T\mathcal{g}\upsilon\_{n}), \\ d\_{\upsilon}(T\mathcal{g}u\_{n-1}, T\mathcal{g}u\_{n}), d\_{\upsilon}(T\mathcal{g}\upsilon\_{n}, T\mathcal{g}\upsilon\_{n+1}), d\_{\upsilon}(T\mathcal{g}u\_{n}, T\mathcal{g}u\_{n+1})\}. \end{split} \tag{6}$$

Let *Kn* = max{*dv*(*Tgun*, *Tgun*+1), *dv*(*Tgvn*, *Tgvn*+1)}. By (5) and (6), we get

$$\mathcal{K}\_{\mathbb{R}} \le \lambda \max \{ d\_{\mathbb{P}}(T\_{\mathcal{S}}\mathbb{v}\_{n-1}, T\_{\mathcal{S}}\mathbb{v}\_{\mathbb{N}}), d\_{\mathbb{P}}(T\_{\mathcal{S}}\mathbb{v}\_{n-1}, T\_{\mathcal{S}}\mathbb{v}\_{\mathbb{N}}), d\_{\mathbb{P}}(T\_{\mathcal{S}}\mathbb{v}\_{\mathbb{N}}, T\_{\mathcal{S}}\mathbb{v}\_{n+1}), d\_{\mathbb{P}}(T\_{\mathcal{S}}\mathbb{v}\_{\mathbb{N}}, T\_{\mathcal{S}}\mathbb{v}\_{n+1}) \}. \tag{7}$$

If

$$\begin{aligned} \max \{ d\_v(T\mathcal{g}v\_{n-1}, T\mathcal{g}v\_n), d\_v(T\mathcal{g}u\_{n-1}, T\mathcal{g}u\_n), d\_v(T\mathcal{g}v\_n, T\mathcal{g}v\_{n+1}), d\_v(T\mathcal{g}u\_n, T\mathcal{g}u\_{n+1}) \} \\ = d\_v(T\mathcal{g}v\_n, T\mathcal{g}v\_{n+1}) \text{ or } d\_v(T\mathcal{g}u\_n, T\mathcal{g}u\_{n+1}), \end{aligned}$$

then (7) will yield a contradiction. Thus, we have

$$\begin{aligned} \max \{ d\_v(T\mathcal{g}v\_{n-1}, T\mathcal{g}v\_n), d\_v(T\mathcal{g}u\_{n-1}, T\mathcal{g}u\_n), d\_v(T\mathcal{g}v\_n, T\mathcal{g}v\_{n+1}), d\_v(T\mathcal{g}u\_n, T\mathcal{g}u\_{n+1}) \} \\ = \max \{ d\_v(T\mathcal{g}v\_{n-1}, T\mathcal{g}v\_n), d\_v(T\mathcal{g}u\_{n-1}, T\mathcal{g}u\_n) \}, \end{aligned}$$

and then (7) gives

$$\lambda \mathbf{K}\_n \le \lambda \max \{ d\_v(T \mathbf{g} v\_{n-1}, T \mathbf{g} v\_n), d\_v(T \mathbf{g} u\_{n-1}, T \mathbf{g} u\_n) \} = \lambda \mathbf{K}\_{n-1} \preceq \lambda^2 \mathbf{K}\_{n-2} \preceq \dots \preceq \lambda^n \mathbf{K}\_0. \tag{8}$$

For any *m*, *n* ∈ *N*, we have

*dv*(*Tgum*, *Tgun*) = *dv*(*TS*(*um*−1, *vm*−1), *TS*(*un*−1, *vn*−1) ≤ *λ* max{*dv*(*Tgum*−1, *Tgun*−1), *dv*(*Tgvm*−1, *Tgvn*−1), *μdv*(*Tgum*−1, *TS*(*um*−1, *vm*−1)), *μdv*(*Tgvm*−1, *TS*(*vm*−1, *um*−1)), *νdv*(*Tgun*−1, *TS*(*un*−1, *vn*−1)), *νdv*(*Tgvn*−1, *TS*(*vn*−1, *un*−1))} ≤ *λ* max{*dv*(*Tgum*−1, *Tgun*−1), *dv*(*Tgvm*−1, *Tgvn*−1), *dv*(*Tgum*−1, *Tgum*), *dv*(*Tgvm*−1, *Tgvm*), *dv*(*Tgun*−1, *Tgun*), *dv*(*Tgvn*−1, *Tgvn*)}.

Then, by using (8), we get

$$\begin{split} d\_v(T\mathcal{g}\mu\_m, T\mathcal{g}\mu\_n) &\quad \leq \quad \lambda \max\{d\_v(T\mathcal{g}\mu\_{m-1}, T\mathcal{g}\mu\_{n-1}), d\_v(T\mathcal{g}\upsilon\_{m-1}, T\mathcal{g}\upsilon\_{n-1})\} \\ &\quad + (\lambda^m + \lambda^n)K\_0\}. \end{split} \tag{9}$$

Similarly, we have

$$\begin{split} d\_{\mathbb{P}}(T\_{\mathcal{G}}\upsilon\_{m}, T\_{\mathcal{G}}\upsilon\_{n}) &\quad \leq \quad \lambda \max\{d\_{\mathbb{P}}(T\_{\mathcal{G}}\mu\_{m-1}, T\_{\mathcal{G}}\mu\_{n-1}), d\_{\mathbb{P}}(T\_{\mathcal{G}}\upsilon\_{m-1}, T\_{\mathcal{G}}\upsilon\_{n-1})\} \\ &\quad + (\lambda^{m} + \lambda^{n})K\_{0}). \end{split} \tag{10}$$

Let *Km*,*<sup>n</sup>* = max{*dv*(*Tgum*, *Tgun*), *dv*(*Tgvm*, *Tgvn*)}. By (9) and (10), we get

$$\mathcal{K}\_{m,n} \quad \le \quad \lambda \mathcal{K}\_{m-1,n-1} + (\lambda^m + \lambda^n)\mathcal{K}\_0.$$

Thus, we see that inequality (1) is satisfied with *c*<sup>1</sup> = *c*<sup>2</sup> = *K*0. Hence, by Lemma 2, < *Tgun* > and < *Tgvn* > are Cauchy sequences. For *v* = 1, the same follows from Lemma 1. Since (*Tg*(*X*), *d*) is complete, we can find *wx*<sup>0</sup> , *wy*<sup>0</sup> ∈ *X* such that

$$\lim\_{n \to \infty} T\mathcal{g}u\_n = T\mathcal{g}w\_{\mathcal{X}\_0} \\
\text{and} \\
\lim\_{n \to \infty} T\mathcal{g}v\_n = T\mathcal{g}w\_{\mathcal{Y}\_0}.$$

2. Now,

*dv*(*TS*(*wx*<sup>0</sup> , *wy*<sup>0</sup> ), *Tgwx*<sup>0</sup> ) ≤ *s*[*dv*(*TS*(*wx*<sup>0</sup> , *wy*<sup>0</sup> ), *TS*(*un*, *vn*) + *dv*(*TS*(*un*, *vn*), *TS*(*un*+1, *vn*+1)) + ··· + *dv*(*TS*(*un*+*v*−2, *vn*+*v*−2), *TS*(*un*+*v*−1, *vn*+*v*−1) + *dv*(*TS*(*un*+*v*−1, *vn*+*v*−1), *Tgwx*<sup>0</sup> ) ≤ *s*[*λmax*{*dv*(*Tgwx*<sup>0</sup> , *Tgun*), *dv*(*Tgwy*<sup>0</sup> , *Tgvn*), *μdv*(*Tgwx*<sup>0</sup> , *TS*(*wx*<sup>0</sup> , *wy*<sup>0</sup> )), *μdv*(*Tgwy*<sup>0</sup> , *TS*(*wy*<sup>0</sup> , *wx*<sup>0</sup> ), *νdv*(*Tgun*, *TS*(*un*, *vn*)), *νdv*(*Tgvn*, *TS*(*vn*, *un*))} +*dv*(*Tgun*+1, *Tgun*+2) + ··· + *dv*(*Tgun*+*v*−1, *Tgun*+*v*) + *dv*(*Tgun*+*v*, *Tgwx*<sup>0</sup> ) ≤ *s*[*λmax*{*dv*(*Tgwx*<sup>0</sup> , *Tgun*), *dv*(*Tgwy*<sup>0</sup> , *Tgvn*), *μdv*(*Tgwx*<sup>0</sup> , *TS*(*wx*<sup>0</sup> , *wy*<sup>0</sup> )), *μdv*(*Tgwy*<sup>0</sup> , *TS*(*wy*<sup>0</sup> , *wx*<sup>0</sup> ), *νdv*(*Tgun*, *Tgun*+1), *νdv*(*Tgvn*, *Tgvn*+1)} (11)

$$+d\_{\mathbb{P}}(Tg u\_{n+1}, Tg u\_{n+2}) + \dots + d\_{\mathbb{P}}(Tg u\_{n+\nu-1}, Tg u\_{n+\nu} + d\_{\mathbb{P}}(Tg u\_{n+\nu}, Tg u\_{\mathbb{X}\_0}))$$

Note that, since < *Tgun* > and < *Tgvn* > are Cauchy sequences, by definition, *dv*(*Tgun*, *Tgun*+1) → 0, *dv*(*Tgvn*, *Tgvn*+1) → 0 as *n* → ∞. Thus, from (11), as *n* → ∞, we get

$$d\_{\upsilon}(TS(w\_{\mathfrak{x}0}, w\_{\mathfrak{y}0}), T\mathcal{g}w\_{\mathfrak{x}0}) \quad \leq \quad s\lambda \max\{\mu d\_{\upsilon}(Tg w\_{\mathfrak{x}0}, TS(w\_{\mathfrak{x}0}, w\_{\mathfrak{y}0})), \mu d\_{\upsilon}(Tg w\_{\mathfrak{y}0}, TS(w\_{\mathfrak{y}0}, w\_{\mathfrak{x}0}))\}.$$

Similarly, we get

$$\operatorname{ad}\_{\upsilon}(TS(w\_{\mathcal{Y}0}, w\_{\mathcal{X}0}), T\mathcal{g}w\_{\mathcal{Y}0}) \quad \leq \quad s\lambda \max\{\mu\mathrm{d}\_{\upsilon}(T\mathcal{g}w\_{\mathcal{X}0}, TS(w\_{\mathcal{X}0}, w\_{\mathcal{Y}0})), \mu\mathrm{d}\_{\upsilon}(T\mathcal{g}w\_{\mathcal{Y}0}, TS(w\_{\mathcal{Y}0}, w\_{\mathcal{X}0}))\}$$

Thus, we have

$$\begin{split} & \max \{ d\_{\upsilon}(TS(w\_{\overline{x}0}, w\_{\overline{y}0}), T\_{\mathcal{G}}w\_{\overline{x}0}), d\_{\upsilon}(TS(w\_{\overline{y}0}, w\_{\overline{x}0}), T\_{\mathcal{G}}w\_{\overline{y}0}) \} \\ & \leq \quad s\lambda\mu \max \{ d\_{\upsilon}(Tgw\_{\overline{x}0}, TS(w\_{\overline{x}0}, w\_{\overline{y}0})), d\_{\upsilon}(Tgw\_{\overline{y}0}, TS(w\_{\overline{y}0}, w\_{\overline{x}0})) \}. \end{split} \tag{12}$$

Proceeding along the same lines as above, we also have

$$\begin{split} & \max \{ d\_v(T\mathcal{g}w\_{\overline{y}0}, TS(w\_{\overline{x}0}, w\_{\overline{y}0})), d\_v(T\mathcal{g}w\_{\overline{y}0}, TS(w\_{\overline{y}0}, w\_{\overline{x}0})) \} \\ & \leq \quad s\lambda\upsilon \max \{ d\_v(T\mathcal{g}w\_{\overline{x}0}, TS(w\_{\overline{x}0}, w\_{\overline{y}0})), d\_v(T\mathcal{g}w\_{\overline{y}0}, TS(w\_{\overline{y}0}, w\_{\overline{x}0})) \}. \end{split} \tag{13}$$

Using (12) and (13) along with the condition min{*λμ*, *λν*} <sup>&</sup>lt; <sup>1</sup> *<sup>s</sup>* , we get *TS*(*wx*<sup>0</sup> , *wy*<sup>0</sup> ) = *Tgwx*<sup>0</sup> and *TS*(*wy*<sup>0</sup> , *wx*<sup>0</sup> ) = *Tgwy*<sup>0</sup> . As *T* is one to one, we have *S*(*wx*<sup>0</sup> , *wy*<sup>0</sup> ) = *gwx*<sup>0</sup> and *S*(*wy*<sup>0</sup> , *wx*<sup>0</sup> ) = *gwy*<sup>0</sup> . Therefore, (*wx*<sup>0</sup> , *wy*<sup>0</sup> ) ∈ *COCP*{*S*, *g*} .

3. Suppose *S* and *g* are weakly compatible. First, we will show that, if (*w*∗ *<sup>x</sup>*<sup>0</sup> , *w*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> ) ∈ *COCP*{*S*, *g*}, then *gw*∗ *<sup>x</sup>*<sup>0</sup> = *gwx*<sup>0</sup> and *gw*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> = *gwy*<sup>0</sup> , or in other words the point of coupled coincidence of *S* and *g* is unique. By (5), we have

*dv*(*Tgw*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> , *Tgwx*<sup>0</sup> ) = *dv*(*TS*(*w*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> , *w*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> ), *TS*(*wx*<sup>0</sup> , *wy*<sup>0</sup> )) ≤ *λmax*{*dv*(*Tgw*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> , *Tgwx*<sup>0</sup> ), *dv*(*Tgw*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> , *Tgwy*<sup>0</sup> ), *μdv*(*Tgw*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> , *TS*(*w*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> , *w*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> )), *μdv*(*Tgw*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> , *TS*(*w*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> , *w*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> ), *νdv*(*Tgwx*<sup>0</sup> , *TS*(*wx*<sup>0</sup> , *wy*<sup>0</sup> )), *νdv*(*Tgwy*<sup>0</sup> , *TS*(*wy*<sup>0</sup> , *wx*<sup>0</sup> ))} ≤ *λmax*{*dv*(*Tgw*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> , *Tgwx*<sup>0</sup> ), *dv*(*Tgw*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> , *Tgwy*<sup>0</sup> )}.

Similarly, we have

$$d\_{\mathcal{V}}(T\mathcal{g}w^\*\_{\mathcal{Y}0'}T\mathcal{g}w\_{\mathcal{Y}0}) \quad \leq \quad \lambda \max\{d\_{\mathcal{V}}(T\mathcal{g}w^\*\_{\mathcal{X}0'}T\mathcal{g}w\_{\mathcal{X}0}), d\_{\mathcal{V}}(T\mathcal{g}w^\*\_{\mathcal{Y}0'}T\mathcal{g}w\_{\mathcal{Y}0})\}.$$

Thus, from the above two inequalities, we get

$$\max \{ d\_{\upsilon} (T \mathcal{g} w\_{\mathbf{x}\_0}^\*, T \mathcal{g} w\_{\mathbf{x}\_0}), d\_{\upsilon} (T \mathcal{g} w\_{y\_0}^\*, T \mathcal{g} w\_{y\_0}) \} \quad \le \quad \lambda \max \{ d\_{\upsilon} (T \mathcal{g} w\_{\mathbf{x}\_0}^\*, T \mathcal{g} w\_{\mathbf{x}\_0}), d\_{\upsilon} (T \mathcal{g} w\_{y\_0}^\*, T \mathcal{g} w\_{y\_0}) \}$$

which implies that *Tgw*∗ *<sup>x</sup>*<sup>0</sup> = *Tgwx*<sup>0</sup> and *Tgw*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> = *Tgwy*<sup>0</sup> . Since *T* is one to one, we get *gw*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> = *gwx*<sup>0</sup> and *gw*∗ *<sup>y</sup>*<sup>0</sup> = *gwy*<sup>0</sup> , which is the point of coupled coincidence of *S* and *g* is unique. Since *S* and *g* are weakly compatible and, since (*wx*<sup>0</sup> , *wy*<sup>0</sup> ) ∈ *COCP*{*S*, *g*}, we have

$$\operatorname{ggw}\_{x\_0} = \operatorname{gS}(w\_{x\_0}, w\_{y\_0}) = \operatorname{S}(\operatorname{gw}\_{x\_0}, \operatorname{gw}\_{y\_0})$$

and

$$\operatorname{ggw}\_{\mathfrak{y}0} = \operatorname{gS}(w\_{\mathfrak{y}0\prime} w\_{\mathfrak{x}0}) = \operatorname{S}(\operatorname{gw}\_{\mathfrak{y}0\prime} \operatorname{gw}\_{\mathfrak{x}0})$$

which shows that (*gwx*<sup>0</sup> , *gwy*<sup>0</sup> ) ∈ *COCP*{*S*, *g*}. By the uniqueness of the point of coupled coincidence, we get *ggwx*<sup>0</sup> = *gwx*<sup>0</sup> and *ggwy*<sup>0</sup> = *gwy*<sup>0</sup> and thus (*gwx*<sup>0</sup> , *gwy*<sup>0</sup> ) ∈ *CCOFP*{*S*, *g*}. Uniqueness of the coupled fixed point follows easily from (4).

Our next result is a generalized version of Theorem 2.1 of Gu [10].

**Theorem 2.** *Let* (*X*, *dv*)*, T, S and g be as in Theorem 1 and suppose there exist β*1, *β*2, *β*<sup>3</sup> *in the interval [0,1), such that <sup>β</sup>*<sup>1</sup> <sup>+</sup> *<sup>β</sup>*<sup>2</sup> <sup>+</sup> *<sup>β</sup>*<sup>3</sup> <sup>&</sup>lt; <sup>1</sup>*, minimum*{*β*2, *<sup>β</sup>*3} <sup>&</sup>lt; <sup>1</sup> *<sup>s</sup> and for all u*, *v*, *w*, *z* ∈ *X*

$$d\_v(TS(u,v),TS(w,z) + d\_v(TS(v,u),TS(z,w) \le \beta\_1(d\_v(T\_\mathcal{S}u, T\_\mathcal{S}w) + d\_v(T\_\mathcal{S}v, T\_\mathcal{S}z)) + 1))$$

$$\beta\_2(d\_v(T\_\mathcal{S}u, TS(u,v)) + d\_v(T\_\mathcal{S}v, TS(v,u)) + \beta\_3(d\_v(T\_\mathcal{S}w, TS(w,z)) + d\_v(T\_\mathcal{S}z, TS(z,w))).\tag{14}$$

*Then, conclusions 1, 2, and 3 of Theorem 1 are true.*

**Proof.** Let *K <sup>n</sup>* = *dv*(*Tgun*, *Tgun*+1) + *dv*(*Tgvn*, *Tgvn*+1) and *K <sup>m</sup>*,*<sup>n</sup>* = *dv*(*Tgum*, *Tgun*) + *dv*(*Tgvm*, *Tgvn*). From condition (14), we obtain

*dv*(*Tgun*, *Tgun*+1) + *dv*(*Tgvn*, *Tgvn*+1) = *dv*(*TS*(*un*−1, *vn*−1), *TS*(*un*, *vn*)) + *dv*(*TS*(*vn*−1, *un*−1), *TS*(*vn*, *un*)) ≤ *β*1[*dv*(*Tgun*−1, *Tgun*) + *dv*(*Tgvn*−1, *Tgvn*)] + *β*2[*dv*(*Tgun*−1, *TS*(*un*−1, *vn*−1)) +*dv*(*Tgvn*−1, *TS*(*vn*−1, *un*−1))] + *β*3[*dv*(*Tgun*, *TS*(*un*, *vn*)) + *dv*(*Tgvn*, *TS*(*vn*, *un*))] ≤ (*β*<sup>1</sup> + *β*2)[*dv*(*Tgun*−1, *Tgun*) + *dv*(*Tgvn*−1, *Tgvn*)] +*β*3[*dv*(*Tgun*, *Tgun*+1) + *dv*(*Tgvn*, *Tgvn*+1)].

Therefore,

$$d\_{\mathbb{D}}(T\mathbb{g}u\_{\mathbb{H}}, T\mathbb{g}u\_{\mathbb{H}+1}) + d\_{\mathbb{D}}(T\mathbb{g}v\_{\mathbb{H}}, T\mathbb{g}v\_{\mathbb{H}+1}) \le \lambda^{'}[d\_{\mathbb{D}}(T\mathbb{g}u\_{\mathbb{H}-1}, T\mathbb{g}u\_{\mathbb{H}}) + d\_{\mathbb{D}}(T\mathbb{g}v\_{\mathbb{H}-1}, T\mathbb{g}v\_{\mathbb{H}})],$$

where *λ* <sup>=</sup> *<sup>β</sup>*<sup>1</sup> <sup>+</sup> *<sup>β</sup>*<sup>2</sup> 1 − *β*<sup>3</sup> < 1. Thus, we get

$$\mathbf{K}\_n^{\prime} \le \boldsymbol{\lambda}^{\prime} \mathbf{K}\_{n-1}^{\prime} \le \cdots \le \boldsymbol{\lambda}^{\prime n} \mathbf{K}\_0^{\prime}.\tag{15}$$

For any *m*, *n* ∈ N, we have

*dv*(*Tgum*, *Tgun*) + *dv*(*Tgvm*, *Tgvn*) = *dv*(*TS*(*um*−1, *vm*−1), *TS*(*un*−1, *vn*−1) + *dv*(*TS*(*vm*−1, *um*−1), *TS*(*vn*−1, *un*−1) ≤ *β*1[*dv*(*Tgum*−1, *Tgun*−1) + *dv*(*Tgvm*−1, *Tgvn*−1)] +*β*2[*dv*(*Tgum*−1, *TS*(*um*−1, *vm*−1)) + *dv*(*Tgvm*−1, *TS*(*vm*−1, *um*−1))] +*β*3[*dv*(*Tgun*−1, *TS*(*un*−1, *vn*−1)) + *dv*(*Tgvn*−1, *TS*(*vn*−1, *un*−1))] ≤ *β*[*dv*(*Tgum*−1, *Tgun*−1) + *dv*(*Tgvm*−1, *Tgvn*−1)] + *β*2[*dv*(*Tgum*−1, *Tgum*) +*dv*(*Tgvm*−1, *Tgvm*)] + *β*3[*dv*(*Tgun*−1, *Tgun*) + *dv*(*Tgvn*−1, *Tgvn*)].

Then, by using (15), we get

$$\begin{split} d\_{\boldsymbol{\upsilon}}(T\_{\mathcal{G}}\boldsymbol{u}\_{m}, T\_{\mathcal{G}}\boldsymbol{u}\_{n}) + d\_{\boldsymbol{\upsilon}}(T\_{\mathcal{G}}\boldsymbol{\upsilon}\_{m}, T\_{\mathcal{G}}\boldsymbol{\upsilon}\_{n}) &\leq \ \ \ \boldsymbol{\beta}\_{1}[\boldsymbol{d}\_{\boldsymbol{\upsilon}}(T\_{\mathcal{G}}\boldsymbol{u}\_{m-1}, T\_{\mathcal{G}}\boldsymbol{u}\_{n-1}) + \boldsymbol{d}\_{\boldsymbol{\upsilon}}(T\_{\mathcal{G}}\boldsymbol{\upsilon}\_{m-1}, T\_{\mathcal{G}}\boldsymbol{\upsilon}\_{n-1})] \\ &+ (\boldsymbol{\beta}\_{2}\boldsymbol{\lambda}^{\prime\prime m} + \boldsymbol{\beta}\_{3}\boldsymbol{\lambda}^{\prime\prime})\boldsymbol{k}\_{0}^{\prime}). \end{split}$$

That is,

$$\boldsymbol{\kappa}'\_{m,n} \quad \leq \quad \boldsymbol{\lambda} \boldsymbol{\kappa}'\_{m-1,n-1} + (\boldsymbol{\lambda}^m + \boldsymbol{\lambda}^n) \boldsymbol{\kappa}'\_0$$

where *λ* = *β*<sup>1</sup> + *β*<sup>2</sup> + *β*<sup>3</sup> < 1. Now for *m*, *n*,*r* ∈ *N*. Thus, we see that inequality (1) is satisfied with *c*<sup>1</sup> = *c*<sup>2</sup> = *K*0. Hence, by Lemma 2, < *Tgun* > and < *Tgvn* > are Cauchy sequences. For *v* = 1, the same follows from Lemma 1.

Since (*Tg*(*X*), *d*) is complete, we can find *wx*<sup>0</sup> , *wy*<sup>0</sup> ∈ *X* such that

$$\lim\_{n \to \infty} T\mathcal{g}u\_n = T\mathcal{g}w\_{\mathcal{X}\_0} \\
\text{and} \\
\lim\_{n \to \infty} T\mathcal{g}v\_n = T\mathcal{g}w\_{\mathcal{Y}\_0}.$$

Again, from condition 3 in Definition 1, we have

$$\begin{array}{rcl}d\_{\mathbb{D}}(TS(\boldsymbol{w}\_{\boldsymbol{x}\_{0}},\boldsymbol{w}\_{\boldsymbol{y}\_{0}}),\boldsymbol{T}\boldsymbol{g}\boldsymbol{w}\_{\boldsymbol{x}\_{0}})) & \leq & s[d\_{\mathbb{D}}(TS(\boldsymbol{w}\_{\boldsymbol{x}\_{0}},\boldsymbol{w}\_{\boldsymbol{y}\_{0}}),\boldsymbol{TS}(\boldsymbol{u}\_{\boldsymbol{n}},\boldsymbol{p}\_{\boldsymbol{t}}))+d\_{\mathbb{D}}(TS(\boldsymbol{u}\_{\boldsymbol{n}},\boldsymbol{p}\_{\boldsymbol{t}}),\boldsymbol{TS}(\boldsymbol{u}\_{\boldsymbol{n}+1},\boldsymbol{v}\_{\boldsymbol{n}+1}))+\cdots+d\_{\mathbb{D}}(TS(\boldsymbol{u}\_{\boldsymbol{n}+\boldsymbol{v}-1},\boldsymbol{v}\_{\boldsymbol{x}\_{0}+\boldsymbol{v}}))+d\_{\mathbb{D}}(TS(\boldsymbol{u}\_{\boldsymbol{n}+\boldsymbol{v}-1},\boldsymbol{v}\_{\boldsymbol{n}+\boldsymbol{v}}))) \\ & & +d\_{\mathbb{D}}(TS(\boldsymbol{u}\_{\boldsymbol{n}+\boldsymbol{v}-2},\boldsymbol{v}\_{\boldsymbol{n}+\boldsymbol{v}-2}),\boldsymbol{TS}(\boldsymbol{u}\_{\boldsymbol{n}+\boldsymbol{v}-1},\boldsymbol{v}\_{\boldsymbol{n}+\boldsymbol{v}-1}))+ \\ & d\_{\mathbb{D}}(TS(\boldsymbol{u}\_{\boldsymbol{n}+\boldsymbol{v}-1},\boldsymbol{v}\_{\boldsymbol{n}+\boldsymbol{v}-1}),\boldsymbol{Tg}\boldsymbol{w}\_{\boldsymbol{x}}))] \end{array}$$

and

$$\begin{array}{rcl}d\_{\mathbb{D}}(TS(\boldsymbol{w}\_{y\_{0}},\boldsymbol{w}\_{\boldsymbol{u}\_{0}}),\boldsymbol{T}\boldsymbol{g}\boldsymbol{w}\_{y\_{0}})) & \leq & s[d\_{\mathbb{D}}(TS(\boldsymbol{w}\_{y\_{0}},\boldsymbol{w}\_{\boldsymbol{u}\_{0}}),\boldsymbol{TS}(\boldsymbol{v}\_{\boldsymbol{u}},\boldsymbol{u}\_{\boldsymbol{u}}))+d\_{\mathbb{D}}(TS(\boldsymbol{v}\_{\boldsymbol{u}},\boldsymbol{u}\_{\boldsymbol{u}}),\boldsymbol{TS}(\boldsymbol{v}\_{\boldsymbol{u}+1},\boldsymbol{u}\_{\boldsymbol{u}+1}))+\cdots+d\_{\mathbb{D}}(TS(\boldsymbol{v}\_{\boldsymbol{u}+\boldsymbol{v}-1},\boldsymbol{u}\_{\boldsymbol{u}+\boldsymbol{v}}))+d\_{\mathbb{D}}(TS(\boldsymbol{v}\_{\boldsymbol{u}+\boldsymbol{v}-1},\boldsymbol{u}\_{\boldsymbol{u}+\boldsymbol{v}-1}))+d\_{\mathbb{D}}(TS(\boldsymbol{v}\_{\boldsymbol{u}+\boldsymbol{v}-1},\boldsymbol{u}\_{\boldsymbol{u}+\boldsymbol{v}-1}))+d\_{\mathbb{D}}(TS(\boldsymbol{v}\_{\boldsymbol{u}+\boldsymbol{v}-1},\boldsymbol{u}\_{\boldsymbol{u}+\boldsymbol{v}-1})) \\ & d\_{\mathbb{D}}(TS(\boldsymbol{v}\_{\boldsymbol{u}+\boldsymbol{v}-1},\boldsymbol{u}\_{\boldsymbol{u}+\boldsymbol{v}-1}),\boldsymbol{Tg}\boldsymbol{w}\_{\boldsymbol{u}}))]. \end{array}$$

### Therefore,

*dv*(*TS*(*wx*<sup>0</sup> , *wy*<sup>0</sup> ), *Tgwx*<sup>0</sup> ) + *dv*(*TS*(*wy*<sup>0</sup> , *wx*<sup>0</sup> ), *Tgwy*<sup>0</sup> ) ≤ *s*[*dv*(*TS*(*wx*<sup>0</sup> , *wy*<sup>0</sup> ), *TS*(*un*, *vn*) +*dv*(*TS*(*wy*<sup>0</sup> , *wx*<sup>0</sup> ), *TS*(*vn*, *un*) +*dv*(*TS*(*un*, *vn*), *TS*(*un*+1, *vn*+1)) + ··· + *dv*(*TS*(*un*+*v*−2, *vn*+*v*−2), *TS*(*un*+*v*−1, *vn*+*v*−1)) +*dv*(*TS*(*vn*, *un*), *TS*(*vn*+1, *un*+1)) + ··· + *dv*(*TS*(*vn*+*v*−2, *un*+*v*−2), *TS*(*vn*+*v*−1, *un*+*v*−1)) +*dv*(*TS*(*un*+*v*−1, *vn*+*v*−1), *Tgwx*<sup>0</sup> ) + *dv*(*TS*(*vn*+*v*−1, *un*+*v*−1), *Tgwy*<sup>0</sup> )] ≤ *s*[*β*1(*dv*(*Tgwx*<sup>0</sup> , *Tgun*) + *dv*(*Tgwy*<sup>0</sup> , *Tgvn*)) + *β*2(*dv*(*Tgwx*<sup>0</sup> , *TS*(*wx*<sup>0</sup> , *wy*<sup>0</sup> )) + *dv*(*Tgwy*<sup>0</sup> , *TS*(*wy*<sup>0</sup> , *wx*<sup>0</sup> )) + *β*3(*dv*(*Tgun*, *TS*(*un*, *vn*)) + *dv*(*Tgvn*, *TS*(*vn*, *un*)))} +*dv*(*Tgun*, *Tgun*+1) + ··· + *dv*(*Tgun*−1, *Tgun*)++*dv*(*Tgvn*, *Tgvn*+1) + ··· + *dv*(*Tgvn*−1, *Tgvn*) +*dv*(*Tgun*+*v*−1, *Tgwx*<sup>0</sup> ) + *dv*(*Tgvn*+*v*−1, *Tgwy*<sup>0</sup> )].

As *n* → ∞, we get

$$\begin{split} &d\_{\mathbb{P}}(TS(w\_{\mathbb{X}\_{0}}, w\_{\mathbb{Y}\_{0}}), Tgw\_{\mathbb{X}\_{0}}) + d\_{\mathbb{P}}(TS(w\_{\mathbb{Y}\_{0}}, w\_{\mathbb{X}\_{0}}), Tgw\_{\mathbb{Y}\_{0}}) \\ &\leq \quad s\beta\_{2}[d\_{\mathbb{P}}(Tgw\_{\mathbb{X}\_{0}}, TS(w\_{\mathbb{X}\_{0}}, w\_{\mathbb{Y}\_{0}})) + d\_{\mathbb{P}}(Tgw\_{\mathbb{Y}\_{0}}, TS(w\_{\mathbb{Y}\_{0}}, w\_{\mathbb{X}\_{0}}))]. \end{split} \tag{16}$$

Similarly, we can show that

$$\begin{aligned} &d\_{\upsilon}(T\mathcal{g}w\_{\mathbf{x}0}, TS(w\_{\mathbf{x}0}, w\_{\mathbf{y}0})) + d\_{\upsilon}(T\mathcal{g}w\_{\mathbf{y}0}, TS(w\_{\mathbf{y}0}, w\_{\mathbf{x}0})) \\ &\leq \quad s\mathcal{g}\_{3}[d\_{\upsilon}(T\mathcal{g}w\_{\mathbf{x}0}, TS(w\_{\mathbf{x}0}, w\_{\mathbf{y}0})) + d\_{\upsilon}(T\mathcal{g}w\_{\mathbf{y}0}, TS(w\_{\mathbf{y}0}, w\_{\mathbf{x}0}))] \end{aligned} \tag{17}$$

Using (16) and (17) along with the condition min{*β*2, *<sup>β</sup>*3} <sup>&</sup>lt; <sup>1</sup> *<sup>s</sup>* , we get *dv*(*Tgwx*<sup>0</sup> , *TS*(*wx*<sup>0</sup> , *wy*<sup>0</sup> )) + *dv*(*Tgwy*<sup>0</sup> , *TS*(*wy*<sup>0</sup> , *wx*<sup>0</sup> )) = 0, i.e., *TS*(*wx*<sup>0</sup> , *wy*<sup>0</sup> ) = *Tgwx*<sup>0</sup> and *TS*(*wy*<sup>0</sup> , *wx*<sup>0</sup> ) = *Tgwy*<sup>0</sup> . As *T* is one to one, we have *S*(*wx*<sup>0</sup> , *wy*<sup>0</sup> ) = *gwx*<sup>0</sup> and *S*(*wy*<sup>0</sup> , *wx*<sup>0</sup> ) = *gwy*<sup>0</sup> . Therefore, (*wx*<sup>0</sup> , *wy*<sup>0</sup> ) ∈ *COCP*{*S*, *g*} .

If (*w*∗ *<sup>x</sup>*<sup>0</sup> , *w*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> ) ∈ *COCP*{*S*, *g*}, then, by (14), we have

*dv*(*Tgw*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> , *Tgwx*<sup>0</sup> ) + *dv*(*Tgw*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> , *Tgwy*<sup>0</sup> ) = *dv*(*TS*(*w*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> , *w*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> ), *TS*(*wx*<sup>0</sup> , *wy*<sup>0</sup> )) + *dv*(*TS*(*w*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> , *w*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> ), *TS*(*wy*<sup>0</sup> , *wx*<sup>0</sup> )) ≤ *β*1[*dv*(*Tgw*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> , *Tgwx*<sup>0</sup> ) + *dv*(*Tgw*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> , *Tgwy*<sup>0</sup> )] + *β*2[*dv*(*Tgw*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> , *TS*(*w*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> , *w*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> )) +*dv*(*Tgw*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> , *TS*(*w*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> , *w*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> )] + *β*3[*dv*(*Tgwx*<sup>0</sup> , *TS*(*wx*<sup>0</sup> , *wy*<sup>0</sup> )) + *dv*(*Tgwy*<sup>0</sup> , *TS*(*wy*<sup>0</sup> , *wx*<sup>0</sup> ))] ≤ *β*1[*dv*(*Tgw*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> , *Tgwx*<sup>0</sup> ) + *dv*(*Tgw*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> , *Tgwy*<sup>0</sup> )].

Thus, *dv*(*Tgw*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> , *Tgwx*<sup>0</sup> ) + *dv*(*Tgw*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> , *Tgwy*<sup>0</sup> ) = 0, which implies that *Tgw*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> = *Tgwx*<sup>0</sup> and *Tgw*∗ *<sup>y</sup>*<sup>0</sup> = *Tgwy*<sup>0</sup> . Since *T* is one to one, we get *gw*<sup>∗</sup> *<sup>x</sup>*<sup>0</sup> = *gwx*<sup>0</sup> and *gw*<sup>∗</sup> *<sup>y</sup>*<sup>0</sup> = *gwy*<sup>0</sup> , which is the point of coupled coincidence of *S*, and *g* is unique. The remaining part of the proof is the same as in the proof of Theorem 1.

The next results can be proved as in Theorems 1 and 2 and so we will not give the proof.

**Theorem 3.** *Theorem 1 holds if we replace condition (4) with the following condition: There exist <sup>β</sup><sup>i</sup>* <sup>∈</sup> [0, 1), *<sup>i</sup>* ∈ {1, ... , 6} *such that* <sup>∑</sup><sup>6</sup> *<sup>i</sup>*=<sup>1</sup> *<sup>β</sup><sup>i</sup>* <sup>&</sup>lt; <sup>1</sup>*,* min{*β*<sup>3</sup> <sup>+</sup> *<sup>β</sup>*4, *<sup>β</sup>*<sup>5</sup> <sup>+</sup> *<sup>β</sup>*6} <sup>&</sup>lt; <sup>1</sup> *<sup>s</sup> and for all u*, *v*, *w*, *z* ∈ *X,*

$$d\_v(TS(u,v),TS(w,z)) \le \beta\_1 d\_v(Tg\iota, Tgw) + \beta\_2 d\_v(Tg\upsilon, Tgz) + \beta\_3 d\_v(Tg\iota, TS(u,v))$$

$$+ \beta\_4 d\_v(Tg\upsilon, TS(v,u) + \beta\_5 d\_v(Tg\upsilon, TS(w,z))) + \beta\_6 d\_v(Tgz, TS(z,w)).\tag{18}$$

Taking *T* to be the identity mapping in Theorems 1–3, we have the following:

**Corollary 1.** *Let* (*X*, *dv*)*, S, g, λ*, *μ and ν be as in Theorem 1 such that, for all u*, *v*, *w*, *z* ∈ *X, the following holds :*

$$\begin{split} d\_{\mathbb{F}}(S(\mathfrak{u},\boldsymbol{v}), S(\boldsymbol{w},\boldsymbol{z}) &\leq \ \lambda \max\{d\_{\mathbb{F}}(\mathfrak{g}\boldsymbol{u}, \operatorname{\mathcal{g}}\boldsymbol{w}), d\_{\mathbb{F}}(\mathfrak{g}\boldsymbol{v}, \operatorname{\mathcal{g}}\boldsymbol{z}), \operatorname{\mu}d\_{\mathbb{F}}(\mathfrak{g}\boldsymbol{u}, S(\boldsymbol{u}, \boldsymbol{v})), \operatorname{\mu}d\_{\mathbb{F}}(\mathfrak{g}\boldsymbol{v}, S(\boldsymbol{v}, \boldsymbol{u}), \operatorname{\mu}d\_{\mathbb{F}}(\mathfrak{g}\boldsymbol{v}, S(\boldsymbol{w}, \boldsymbol{u})))\} \\ &\quad \operatorname{\nu}d\_{\mathbb{F}}(\mathfrak{g}\boldsymbol{w}, S(\boldsymbol{w}, \boldsymbol{z})), \operatorname{\nu}d\_{\mathbb{F}}(\mathfrak{g}\boldsymbol{z}, S(\boldsymbol{z}, \boldsymbol{w})) \}. \end{split} \tag{19}$$

*Then, COCP*{*S*, *g*} = *φ. Furthermore, if S and g are weakly compatible, then S and g has a unique common coupled fixed point. Moreover, for some arbitrary* (*u*0, *v*0) ∈ *X* × *X, the iterative sequences* (< *gun* > , < *gvn* >) *defined by gun* = *S*(*un*−1, *vn*−1) *and gvn* = *S*(*vn*−1, *un*−1) *converge to the unique common coupled fixed point of S and g.*

**Corollary 2.** *Corollary 1 holds if the condition (19) is replaced with the following condition: There exist <sup>β</sup>*1, *<sup>β</sup>*2, *<sup>β</sup>*<sup>3</sup> *in the interval [0,1), such that <sup>β</sup>*<sup>1</sup> <sup>+</sup> *<sup>β</sup>*<sup>2</sup> <sup>+</sup> *<sup>β</sup>*<sup>3</sup> <sup>&</sup>lt; <sup>1</sup>*,* min{*β*2, *<sup>β</sup>*3} <sup>&</sup>lt; <sup>1</sup> *<sup>s</sup> and for all*

*u*, *v*, *w*, *z* ∈ *X*

$$d\_{\upsilon}(S(\mu, \upsilon), S(w, z) + d\_{\upsilon}(S(\upsilon, \mu), S(z, w) \le \beta\_1(d\_{\upsilon}(\mathcal{g}\mu, \mathcal{g}w) + d\_{\upsilon}(\mathcal{g}\upsilon, \mathcal{g}z)) + \\\beta\_2(d\_{\upsilon}(\mathcal{g}\upsilon, S(\upsilon, \mu)) + \beta\_3(d\_{\upsilon}(\mathcal{g}w, S(w, z)) + d\_{\upsilon}(\mathcal{g}z, S(z, w))).\tag{20}$$

**Corollary 3.** *Corollary 1 holds if the condition (19) is replaced with the following condition: There exist <sup>β</sup><sup>i</sup>* <sup>∈</sup> [0, 1), *<sup>i</sup>* ∈ {1, ... <sup>6</sup>} *such that* <sup>∑</sup><sup>6</sup> *<sup>i</sup>*=<sup>1</sup> *<sup>β</sup><sup>i</sup>* <sup>&</sup>lt; <sup>1</sup>*,* min{*β*<sup>3</sup> <sup>+</sup> *<sup>β</sup>*4, *<sup>β</sup>*<sup>5</sup> <sup>+</sup> *<sup>β</sup>*6} <sup>&</sup>lt; <sup>1</sup> *<sup>s</sup> and, for all u*, *v*, *w*, *z* ∈ *X,*

$$d\_{\mathbb{U}}(S(\mathfrak{u}, \mathfrak{v}), S(\mathfrak{w}, \mathfrak{z})) \le \beta\_1 d\_{\mathbb{U}}(\mathfrak{g}\mathfrak{u}, \mathfrak{g}\mathfrak{w}) + \beta\_2 d\_{\mathbb{U}}(\mathfrak{g}\mathfrak{v}, \mathfrak{g}\mathfrak{z}) + $$

$$\beta\_3 d\_{\mathbb{U}}(\mathfrak{g}\mathfrak{u}, S(\mathfrak{u}, \mathfrak{v})) + \beta\_4 d\_{\mathbb{U}}(\mathfrak{g}\mathfrak{v}, S(\mathfrak{v}, \mathfrak{u})) + \beta\_5 d\_{\mathbb{U}}(\mathfrak{g}\mathfrak{w}, S(\mathfrak{w}, \mathfrak{z})) + \beta\_6 d\_{\mathbb{U}}(\mathfrak{g}\mathfrak{z}, S(\mathfrak{z}, \mathfrak{w})). \tag{21}$$

**Remark 1.** *Since every b-metric space is a b*1(*s*) *metric space, we note that Theorem 1 is a substantial generalization of Theorem 2.2 of Ramesh and Pitchamani [11]. In fact, we do not require continuity and sub sequential convergence of the function T.*

**Remark 2.** *Note that condition (2.1) of Gu [10] implies (20) and hence Corollary 2 gives an improved version of Theorem 2.1 of Gu [10].*

**Remark 3.** *Condition (3.1) of Hussain et al. [12] implies (18) and hence Theorem 3 is an extended and generalized version of Theorem 3.1 of [12].*

### *3.2. Application to a System of Integral Equations*

In this section, we give an application of Theorem 1 to study the existence and uniqueness of solution of a system of nonlinear integral equations.

Let *X* = *C*[0, *A*] be the space of all continuous real valued functions defined on [0, *A*], *A* > 0. Our problem is to find (*u*(*t*), *v*(*t*)) ∈ *X* × *X*, *t* ∈ [0, *A*] such that, for *f* : [0, *A*] × *R* × *R* → *R* and *G* : [0, *A*] × [0, *A*] → *R* and *K* ∈ *C*([0, *A*], the following holds:

$$u(t) = \int\_0^A G(t, r) f(t, u(r), v(r)) dr + K(t)$$

$$v(t) = \int\_0^A G(t, r) f(t, v(r), u(r)) dr + K(t). \tag{22}$$

Now, suppose *F* : *X* × *X* → *X* is given by

$$F(u(t),v(t)) = \int\_0^A G(t,r)f(t,u(r),v(r))dr + K(t).$$

$$F(v(t), u(t)) = \int\_0^A G(t, r) f(t, v(r), u(r)) dr + K(t).$$

Then, (22) is equivalent to the coupled fixed point problem *F*(*u*(*t*), *v*(*t*)) = *u*(*t*), *F*(*v*(*t*), *u*(*t*)) = *v*(*t*).

**Theorem 4.** *The system of Equation (22) has a unique solution provided the following holds:*


(*iii* − *a*) :| *f*(*t*, *u*(*r*), *v*(*r*))) − *f*(*t*, *x*(*r*), *y*(*r*))) | *<sup>p</sup>* <sup>≤</sup> *<sup>λ</sup>pmax*{| *<sup>g</sup>*(*u*(*r*)) <sup>−</sup> *<sup>g</sup>*(*x*(*r*)) <sup>|</sup> *<sup>p</sup>*, <sup>|</sup> *<sup>g</sup>*(*v*(*r*)) <sup>−</sup> *<sup>g</sup>*(*y*(*r*)) <sup>|</sup> *p*, *μ* | *g*(*u*(*r*)) − *F*(*u*(*r*), *v*(*r*)) | *<sup>p</sup>*, *<sup>μ</sup>* <sup>|</sup> *<sup>g</sup>*(*v*(*r*)) <sup>−</sup> *<sup>F</sup>*(*v*(*r*), *<sup>u</sup>*(*r*)) <sup>|</sup> *p*, *ν* | *g*(*x*(*r*)) − *F*(*x*(*r*), *y*(*r*)) | *<sup>p</sup>*, *<sup>ν</sup>* <sup>|</sup> *<sup>g</sup>*(*y*(*r*)) <sup>−</sup> *<sup>F</sup>*(*y*(*r*), *<sup>x</sup>*(*r*)) <sup>|</sup> *p*}.

*(iii-b) F*(*g*(*u*(*t*)), *g*(*v*(*t*))) = *g*(*F*(*u*(*t*), *v*(*t*))) *. (iv) supt*∈[0,*A*] *<sup>A</sup>* <sup>0</sup> | *G*(*t*,*r*) | *<sup>p</sup> dr* <sup>≤</sup> <sup>1</sup> *<sup>λ</sup>p*−<sup>1</sup> *.*

*Moreover, for some arbitrary u*0(*t*), *v*0(*t*) *in X, the sequence* (< *gun*(*t*) >, < *gvn*(*t*) >) *defined by*

$$\mathcal{g}u\_{\mathbb{H}}(t) = \int\_{0}^{A} \mathcal{G}(t, r) f(t, u\_{n-1}(r), v\_{n-1}(r)) dr + \mathcal{K}(t)$$

$$\mathcal{g}v\_{\mathbb{H}}(t) = \int\_{0}^{A} \mathcal{G}(t, r) f(t, v\_{n-1}(r), u\_{n-1}(r)) dr + \mathcal{K}(t) \tag{23}$$

*converges to the unique solution.*

**Proof.** Define *dv* : *X* × *X* → *R* such that for all *u*, *v* ∈ *X*,

$$d\_{\upsilon}(\mu, \upsilon) = \sup\_{t \in [0, A]} |\, |\mu(t) - \upsilon(t)|\, |^{s} \,. \tag{24}$$

Clearly, *dv* is a *bv*((*v* + 1)*s*−1)-metric space. For some *r* ∈ [0, *A*], we have

$$\begin{split} \|F(u(t),v(t)) - F(\mathbf{x}(t),y(t))\| ^p \\ &= |\int\_0^A \mathcal{G}(t,r)f(t,u(r),v(r))dr + \mathcal{g}(t) - \int\_0^A \mathcal{G}(t,r)f(t,\mathbf{x}(r),y(r))dr + \mathcal{g}(t)|\,^p \\ &\le \int\_0^A |\mathcal{G}(t,r)|^p |f(t,u(r),v(r)) - f(t,\mathbf{x}(r),y(r))|^p \, ^p \, d\mathbf{r} \\ &\le \left(\int\_0^A |\mathcal{G}(t,r)|^p \, d\mathbf{r}\right) \lambda^p [\max\{|\,\mathcal{g}(u(r)) - \mathcal{g}(\mathbf{x}(r))|^p, |\,\mathcal{g}(v(r)) - \mathcal{g}(y(r))|^p\} \\ &\le \|\mathcal{g}(u(r)) - F(u(r),v(r))\| ^p \, , \mu \mid \mathcal{g}(v(r)) - F(v(r),u(r)) \, ^p \, \|\mathcal{G}(r)\| \\ &\le \|\mathcal{g}(x(r)) - F(x(r),y(r))\| ^p \, , \nu \mid \mathcal{g}(y(r)) - F(y(r),x(r)) \, \|\,^p \, \|\mathcal{G}(r)\| \\ &\le \left(\int\_0^A |\mathcal{G}(t,r)|^p \, d\mathbf{r}\right) \lambda^p \|\max\{d\_\mathcal{G}(y(\rho),y(\rho)), d\_\mathcal{G}(y(\rho),y(\rho)), \mu d\_\mathcal{G}(y(\rho),F(u,v)), \mu d\_\mathcal{G}(y(\rho),F(v,u)), \mu d\_\mathcal{G}(y(\rho),y(\rho))\}. \end{split}$$

Thus, using condition (iv), we have

$$\begin{split} d\_{\upsilon}(F(\mathsf{u},\upsilon),F(\mathsf{x},\mathsf{y})) &= \sup\_{\mathsf{x}\in[0,A]} |\, |F(\mathsf{u}(t),\upsilon(t)) - F(\mathsf{x}(t),\mathsf{y}(t))| \, |^{p} \\ &\leq \, \lambda [\max\{d\_{\upsilon}(\mathsf{g}(\mathsf{u}),\mathsf{g}(\mathsf{x})),d\_{\upsilon}(\mathsf{g}(\mathsf{v}),\mathsf{g}(\mathsf{y})),\mathsf{pdf}\_{\upsilon}(\mathsf{g}(\mathsf{u}),F(\mathsf{u},\mathsf{v})),\mathsf{pdf}\_{\upsilon}(\mathsf{g}(\mathsf{v}),F(\mathsf{v},\mathsf{u})), \\ &\nu d\_{\upsilon}(\mathsf{g}(\mathsf{x}),F(\mathsf{x},\mathsf{y})),\nu d\_{\upsilon}(\mathsf{g}(\mathsf{y}),F(\mathsf{y},\mathsf{x})) \}. \end{split}$$

Thus, all the conditions of Corollary 1 are satisfied and so *F* has a unique coupled fixed point (*u* , *v* ) ∈ *C*([0, *A*] × *C*([0, *A*], which is the unique solution of (22) and the sequence (< *gun*(*t*) >, < *gvn*(*t*) >) defined by (23) converges to the unique solution of (22).

**Example 1.** *Let X* = *C*[0, 1] *be the space of all continuous real valued functions defined on* [0, 1] *and define d*<sup>3</sup> : *X* × *X* → *R such that, for all u*, *v* ∈ *X,*

$$d\_{\mathcal{G}}(\mathfrak{u}, \upsilon) = \sup\_{t \in [0, 1]} |\mathfrak{u}(t) - \upsilon(t)|^2. \tag{25}$$

*Clearly, d*<sup>3</sup> *is a b*2(3)*-metric. Now, consider the functions f* : [0, 1] × *R* × *R* → *R given by f*(*t*, *u*, *v*) = *t* <sup>2</sup> + <sup>9</sup> <sup>20</sup>*<sup>u</sup>* <sup>+</sup> <sup>8</sup> <sup>20</sup> *v, G* : [0, 1] × [0, 1] → *R given by G*(*t*,*r*) = <sup>√</sup>45(*t*+*r*) <sup>10</sup> *, K* ∈ *C*([0, 1] *given by K*(*t*) = *t. Then, Equation (22) becomes*

$$u(t) = t + \int\_0^1 \frac{\sqrt{45}(t+r)}{10} (t^2 + \frac{9}{20}u(r) + \frac{8}{20}v(r)) dr$$

$$v(t) = t + \int\_0^1 \frac{\sqrt{45}(t+r)}{10} (t^2 + \frac{9}{20}v(r) + \frac{8}{20}u(r)) dr. \tag{26}$$

*Then,*

$$\begin{aligned} \left| \left| f(t, u, v) - f(t, x, y) \right| \right|^2 &= \left| \left| \frac{9}{20} (u - x) + \frac{8}{20} (v - y) \right| \right|^2 \\ &\leq \left| \left| \operatorname{Max} \{ \frac{9}{10} (u - x), \frac{8}{10} (v - y) \} \right| \right|^2 \\ &\leq \frac{81}{100} \operatorname{Max} \{ \left| \left| u - x \right|^2, \left| v - y \right| \right|^2 \}. \end{aligned}$$

*In addition,*

$$\sup\_{t \in [0,1]} \int\_0^1 | \ G(t, r) \mid^2 dr = \int\_0^1 \frac{45}{100} (t + r)^2 dr = 1.05.$$

*We see that all the conditions of Theorem 4 are satisfied, with λ* = <sup>9</sup> <sup>10</sup> , *μ* = 0, *ν* = 0, *p* = 2 *and g* = *IX(Identity mapping). Hence, Theorem 4 ensures a unique solution of (26). Now, for u*0(*t*) = 1 *and v*0(*t*) = 0*, we construct the sequence* (< *un*(*t*) >, < *vn*(*t*) >} *given by*

$$u\_n(t) = t + \int\_0^1 \frac{\sqrt{45}(t+r)}{10} (t^2 + \frac{9}{20} u\_{n-1}(r) + \frac{8}{20} v\_{n-1}(r)) dr$$

$$v\_n(t) = t + \int\_0^1 \frac{\sqrt{45}(t+r)}{10} (t^2 + \frac{9}{20} v\_{n-1}(r) + \frac{8}{20} u\_{n-1}(r)) dr. \tag{27}$$

*Using MATLAB, we see that above sequence converges to* {0.6708*t* <sup>3</sup> + 0.3354*t* <sup>2</sup> + 2.2339*t* + 0.7677, 0.6708*t* <sup>3</sup> + 0.3354*t* <sup>2</sup> <sup>+</sup> 2.2339*<sup>t</sup>* <sup>+</sup> 0.7677}*, and this is the unique solution of the system of nonlinear integral Equation (26). The convergence table is given in Table 1 below.*


**Table 1.** Convergence of sequences < *un*(*t*) > and < *vn*(*t*) >.

**Remark 4.** *Condition (iv) of Theorem 4 above is weaker than the corresponding conditions used in similar theorems of [11,13,14].*

**Remark 5.** *In example 1 above, we see that supt*∈[0,1] 1 | *G*(*t*,*r*) | *dr* = <sup>1</sup> (*<sup>t</sup>* + *<sup>r</sup>*)2*dr* = 1.05 > <sup>1</sup> *and thus condition (v) of Theorem 3.1 of [11], condition (30) of Theorem 3.1 of [13] and condition (iii) of Theorem 3.1 of [14] are not satisfied.*

**Author Contributions:** Investigation, R.G., Z.D.M., and S.R.; Methodology, R.G.; Software, Z.D.M.; Supervision, R.G., Z.D.M., and S.R. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** 1. The authors are thankful to the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University, Al-Kharj, Kingdom of Saudi Arabia, for supporting this research. 2. The authors are thankful to the learned reviewers for their valuable comments which helped in improving this paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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