A note on three different contractions in partially ordered complex valued $G_b$-metric spaces

1.   Introduction

Fixed point theory in metric spaces occupies an extremely important position in modern mathematics, it has been generalized in various aspects. For example, $G$-metric spaces ^[1] were introduced and $G_b$-metric spaces were reported in ^[2], which successfully popularized the general metric and promoted the research of various types of fixed point theorems. These theorems are accompanied with different contractive conditions (see ^{[3,4,5,6,7,8,9,10,11,12,21,22,23,24,25,26,27]}), especially the new Geraghty contraction was given in ^[13] and the JS contraction was given in ^[14].

Recently, Shoaib et al. ^[15] introduced the ordered dislocated quasi $G$-metric spaces, and obtained some new fixed point results for a dominated mapping on a close ball in this space. On the other hand, Ege ^[16] also proposed the complex valued $G_b$-metric spaces as a new notion, the Banach contraction principle and Kannan's fixed point theorem were proved for this space. Moreover, there are also other interesting fixed point theorems in this space (see ^{[17,18,19,20]}).

In this work, we study some problems about the common solutions of the operator equations $F_nx = ux\, \, (u\geq1, n\in\, \mathbb{N}^{*})$ in complete partially ordered complex valued $G_b$-metric spaces, introduce the complex valued $C^{p}$-class function and a type of Geraghty contraction to this space respectively, and we obtain the common solutions in a closed ball. Furthermore, we also introduce a type of JS contraction to this space and investigate a new theorem.

Firstly, we recall some basic concepts, which will be used later. For a real Banach space $E$, a nonempty closed subset $Q\subset\, E$ is called a cone, if

$(a)$ for all $\zeta\in\, Q$ and $\tau\geq\, 0$, $\tau\zeta\in\, Q$;

$(b)$ for all $\zeta_1, \zeta_2\in\, Q$, $\zeta_1+\zeta_2\in\, Q$;

$(c)$ $Q\cap(-Q) = 0$.

For $\xi_1, \xi_2\in\, E$, given a cone $Q$, we define a partial order $\preceq$ on $E$, which is induced by $Q$, i.e., $\xi_1\preceq\xi_2$ iff $\xi_2-\xi_1\in\, Q$. Furthermore, $\xi_1, \xi_2$ are said to be comparable if $\xi_1\preceq\xi_2$ or $\xi_2\preceq\xi_1$.

On the other hand, for all $\xi_1, \xi_2\in\mathbb{C}$, the partial order $\precsim$ on $\mathbb{C}$ is defined as follows:

Therefore, $\xi_1\precsim\xi_2$ if one of the following conditions holds:

$(C_1)$ $Re(\xi_1) = Re(\xi_2)$ and $Im(\xi_1) = Im(\xi_2)$;

$(C_2)$ $Re(\xi_1) = Re(\xi_2)$ and $Im(\xi_1) < Im(\xi_2)$;

$(C_3)$ $Re(\xi_1) < Re(\xi_2)$ and $Im(\xi_1) = Im(\xi_2)$;

$(C_4)$ $Re(\xi_1) < Re(\xi_2)$ and $Im(\xi_1) < Im(\xi_2)$.

Moreover, we denote $\xi_1\prec\xi_2$ if only $(C_4)$ holds. Obviously, $0\precsim\xi_1\precsim\xi_2\Rightarrow |\xi_1|\leq|\xi_2|$, where $|\xi_i|$ is the magnitude of $\xi_i$, $i = 1, 2$. For more details, see ^[25].

Definition 1.1. (^[16]) Let $X$ be a nonempty set, for a real number $s\geq\, 1$, if the mapping $G_b:X\times X\times X \rightarrow \mathbb{C}$ satisfies:

$(CG_b1)$ $G_b(\zeta_1, \zeta_2, \zeta_3) = 0$ if $\zeta_1 = \zeta_2 = \zeta_3$;

$(CG_b2)$ $G_b(\zeta_1, \zeta_1, \zeta_2)\succ\, 0$ for all $\zeta_1, \zeta_2\in\, X$ with $\zeta_1\neq \zeta_2$;

$(CG_b3)$ $G_b(\zeta_1, \zeta_1, \zeta_2)\precsim\, G_b(\zeta_1, \zeta_2, \zeta_3)$ for all $\zeta_1, \zeta_2, \zeta_3\in\, X$ with $\zeta_3\neq\zeta_2$;

$(CG_b4)$ $G_b(\zeta_1, \zeta_2, \zeta_3) = G_b(R\{\zeta_1, \zeta_2, \zeta_3\})$, where $R$ is an arbitrary permutation of $\{\zeta_1, \zeta_2, \zeta_3\}$;

$(CG_b5)$ $G_b(\zeta_1, \zeta_2, \zeta_3)\precsim\, s[G_b(\zeta_1, \upsilon, \upsilon)+G_b(\upsilon, \zeta_2, \zeta_3)]$ for all $\zeta_1, \zeta_2, \zeta_3, \upsilon\in\, X$.

Then the function $G_b$ is called a complex valued $G_b$-metric on $X$, the pair $(X, G_b)$ is called a complex valued $G_b$-metric space.

Proposition 1.1. (^[16]) For a complex valued $G_b$-metric space $(X, G_b)$ and all $\zeta_1, \zeta_2, \zeta_3\in\, X$, we have

(1) $G_b(\zeta_1, \zeta_2, \zeta_3)\precsim\, s[G_b(\zeta_1, \zeta_1, \zeta_2)+G_b(\zeta_1, \zeta_1, \zeta_3)]$;

(2) $G_b(\zeta_1, \zeta_2, \zeta_2)\precsim\, 2s[G_b(\zeta_1, \zeta_1, \zeta_2)]$.

Definition 1.2. (^[16]) Let $\{x_n\}$ be a sequence in a complex valued $G_b$-metric space $(X, G_b)$,

(1) $\{x_n\}$ is called complex valued $G_b$-convergent to $\zeta\in X$, if for any $\epsilon\in\mathbb{C}$ with $\epsilon\succ\, 0$, there exists $\xi\in\mathbb{N}$ such that $G_b(\zeta, x_n, x_m)\prec\epsilon$ for all $n, m\geq\xi$. We write $x_n\rightarrow \zeta$ as $n\rightarrow \infty$, or $\lim\limits_{n\rightarrow \infty}x_n = \zeta$;

(2) $\{x_n\}$ is called complex valued $G_b$-Cauchy, if for any $\epsilon\in\mathbb{C}$ with $\epsilon\succ\, 0$, there exists $\xi\in\mathbb{N}$ such that $G_b(x_n, x_m, x_l)\prec\epsilon$ for all $n, m, l\geq\xi$;

(3) $(X, G_b)$ is said to be complex valued $G_b$-complete, if any complex valued $G_b$-Cauchy sequence $\{x_n\}$ is complex valued $G_b$-convergent.

Theorem 1.1. (^[16]) Let $\{x_n\}$ be a sequence in a complex valued $G_b$-metric space $(X, G_b)$, and $\zeta\in\, X$, the following are equivalent:

(1) $\{x_n\}$ is complex valued $G_b$-convergent to $\zeta$;

(2) $|G_b(x_n, x_m, \zeta)|\rightarrow \, 0$ as $n, m\rightarrow \infty$;

(3) $|G_b(x_n, \zeta, \zeta)|\rightarrow \, 0$ as $n\rightarrow \infty$;

(4) $|G_b(x_n, x_n, \zeta)|\rightarrow \, 0$ as $n\rightarrow \infty$.

Theorem 1.2. (^[16]) A sequence $\{x_n\}$ is complex valued $G_b$-Cauchy sequence is equivalent to $|G_b(x_n, x_m, x_l)|\rightarrow \, 0$ as $n, m, l\rightarrow \infty$.

Definition 1.3. (^[28]) Let $Q\subset\mathbb{R}^{m}$ be a cone, a mapping $S:Q\rightarrow \mathbb{R}^{m}$ is said to be dominated if $Sx\preceq\, x$ for all $x\in\, Q$.

Theorem 1.3. (^[15]) Let $(X, \preceq, G)$ be an ordered complete dislocated quasi $G$-metric space, $S:X\rightarrow \, X$ be a mapping and $x_0$ be an arbitrary point in $X$. Suppose there exists $k\in[0, \frac{1}{2})$ with

for all comparable elements $x, y, z\in\overline{B(x_0, r)}$, and

where $\theta = \frac{k}{1-2k}$. If for nonincreasing sequence $\{x_n\}\rightarrow \, u$ implies that $u\preceq\, x_n$. Then there exists a point $x^{\star}$ in $\overline{B(x_0, r)}$ such that $x^{\star} = Sx^{\star}$ and $G(x^{\star}, x^{\star}, x^{\star}) = 0$. Moreover, if for any three points $x, y, z\in\overline{B(x_0, r)}$, there exists a point $v$ in $\overline{B(x_0, r)}$ such that $v\preceq\, x$ and $v\preceq\, y$, $v\preceq\, z$, where

then the point $x^{\star}$ is unique.

2.   Results and examples

In this section, let $X = \mathbb{R}^{N}$, $\Omega_1 = \{Z_1\in\mathbb{C}:0\precsim\, Z_1\}$, $\Omega_2 = \{Z_2\in\mathbb{C}:0\prec\, Z_2\}$, $\Omega_3 = \{Z_3\in\mathbb{C}:1\prec\, Z_3\}$. $(X, G_b, \preceq)$ is called a partially ordered complex valued $G_b$-metric space, which shows $(X, G_b)$ is a complex valued $G_b$-metric space and $(X, \preceq)$ is a partially ordered set.

Let $(X, G_b)$ be a complex valued $G_b$-metric space, for any $x_0\in\, X$, $r\in\mathbb{C}$ and $r\succ0$, the $G_b$-ball with ball center $x_0$ is $\overline{B(x_0, r)} = \{x\in\, X|\, \, G_b(x_0, x, x)\precsim\, r\}$. Moreover, for all $n\in\, \mathbb{N}^{*}$ and $x_1, x_2, ..., x_n\in\mathbb{C}$, the function $\max\{x_1, x_2, ..., x_n\}\succsim\, x_j$, $j = 1, 2, ..., n$.

Definition 2.1. A continuous mapping $P:\Omega_1^{3}\rightarrow \mathbb{C}$ is called complex valued $C^{p}$-class function, if it satisfies $r\precsim\, P(r, s, t)$ for all $r, s, t\in\, \Omega_1$.

Example 2.1. Some examples of complex valued $C^{p}$-class function are given as follows:

$(1)$ $P(r, s, t) = r+s+t$, where $r, s, t\in\Omega_1;$

$(2)$ $P(r, s, t) = mr$, where $m\in[1, \infty)$ and $r, s, t\in\Omega_1;$

$(3)$ $P(r, s, t) = \eta(r)r$, where $\eta:\Omega_1\rightarrow [1, \infty)$ and $r, s, t\in\Omega_1.$

Theorem 2.1. Let $(X, G_b, \preceq)$ be a complete partially ordered complex valued $G_b$-metric space with $s\geq1$, $Q\subset\, X$ be a cone, $x_0$ be an arbitrary element in $Q$, $\{S_n:X\rightarrow \, X, \, \, n\in\, \mathbb{N}^{*}\}$ be a dominated mapping sequence. If there exist $r\in\Omega_2$, and nonnegative numbers $\alpha, \beta, \gamma$ satisfy $\alpha-2s\gamma\neq0, \frac{\beta}{\alpha-2\gamma}\in[0, \delta], \delta < \frac{1}{s}$, such that

for any comparable elements $x, y, z$ in $\overline{B(x_0, r)}$, where $\overline{B(x_0, r)}\subset\, Q, \, \, i, j\in\mathbb{N}^{*}$, $P$ is a complex valued $C^{p}$-class function, $\psi:\Omega_1\rightarrow \, \Omega_1$ is a nondecreasing function, $\varphi:\Omega_1\rightarrow \mathbb{C}$ is a continuous function. And

Define the operator equations $F_nx = ux$ by $F_n = uS_n$, $u\geq1$. If a nonincreasing sequence $\{x_n\}\rightarrow \, \kappa$ such that $\kappa\preceq\, x_n$, then the operator equations have at least a common solution $x^{*}$ in $\overline{B(x_0, r)}$. Moreover, if there exists an element $v$ in $\overline{B(x_0, r)}$ such that $v\preceq\, x^{*}$, and

then the operator equations have an unique solution.

Proof. By selecting the ball centre $x_0$ in $\overline{B(x_0, r)}$, we construct a sequence $\{x_n\}$, where $x_{n+1} = S_{n+1}x_n\preceq\, x_n, \, n\in\mathbb{N}$. From (2.2), we obtain $x_1\in\overline{B(x_0, r)}$. Using (2.1), we have

Since the function $\psi$ is nondecreasing, we can easily get

Hence, $G_b(x_0, x_2, x_2)\precsim\, s[G_b(x_0, x_1, x_1)+G_b(x_1, x_2, x_2)]\precsim\, s(1+\delta)G_b(x_0, x_1, x_1).$ Using (2.2), we get $G_b(x_0, x_2, x_2)\precsim(1-\delta^2)r\prec\, r$, that is $x_2\in\overline{B(x_0, r)}.$

Now we prove $\{x_n\}\subset\overline{B(x_0, r)}$. Suppose that $x_3, x_4, ..., x_k\in\overline{B(x_0, r)}$, according to (2.1), we have

Thus $G_b(x_k, x_{k+1}, x_{k+1})\precsim\, \frac{\beta}{\alpha-2\gamma}G_b(x_{k-1}, x_k, x_k)\precsim\delta\, G_b(x_{k-1}, x_k, x_k)$, it can easily get that

By using $(CG_b5)$ and (2.4), it follows that

i.e., $x_{k+1}\in\overline{B(x_0, r)}$, therefore, $\{x_n\}\subset\overline{B(x_0, r)}$.

Now we show that $\{x_n\}$ is a complex valued $G_b$-Cauchy sequence, from (2.4), we obtain

thus for all $n, m\in\mathbb{N}^{*}, n < m$, we have

which implies that

Therefore, $\{x_n\}$ is a complex valued $G_b$-Cauchy sequence, and there exists an element $x^{*}$ in $\overline{B(x_0, r)}$ such that $x_n\rightarrow \, x^{*}$.

Next we prove $x^{*}$ is the common solution of the operator equations. For any $j\in\mathbb{N}^{*}$, we have

Furthermore, since $S_jx^{*}\preceq\, x^{*}\preceq\, x_n\preceq\, x_{n-1}$, using (2.1), it can be easily get that

Hence,

That is,

Let $n\rightarrow \infty$ at both sides of the above inequality, we obtain $\lim\limits_{n\rightarrow \infty}G_b(x^{*}, S_jx^{*}, S_jx^{*}) = 0$, i.e. $x^{*} = S_jx^{*}$. According to the arbitrariness of $j$, we get $x^{*}$ is a common solution of the operator equations.

Uniqueness. Assume that $y^{*}$ is another solution of the operator equations, $y^{*}\neq\, x^{*}$ and $y^{*}\in\overline{B(x_0, r)}$.

Case 1. If $x^{*}$ and $y^{*}$ are comparable, using (2.1), it follows that

as a result, $x^{*} = y^{*}$.

Case 2. If $x^{*}$ and $y^{*}$ are not comparable, then there exists an element $v\in\overline{B(x_0, r)}$ such that $v\preceq\, x^{*}$ and $v\preceq\, y^{*}$, for any $j\in\mathbb{N}^{*}$, we will show $\{S^{n}_jx_n\}\subset\overline{B(x_0, r)}$. Owing to (2.1) and (2.3), we have

i.e.,

Hence,

that is $S_jv\in\overline{B(x_0, r)}$. Suppose that $S_j^{2}v, S_j^{3}v, ..., S_j^{k}v\in\overline{B(x_0, r)}$, obviously, $S_j^{k}v\preceq\, S_j^{k-1}v\preceq...\preceq\, S_j^{2}v\preceq\, S_jv\preceq\, v\preceq\, x^{*}\preceq\, x_n\preceq...\preceq\, x_0.$ From (2.1), we can immediately obtain

so we have

as a result,

In addition, using (2.1), (2.3), (2.5) and (2.6), we can also immediately obtain

i.e.,

Thus,

which implies $S_j^{k+1}v\in\overline{B(x_0, r)}$, so $\{S^{n}_jx_n\}\subset\overline{B(x_0, r)}$. From (2.6), we obtain

and

From (2.1), we can easily get

Owing to (2.7), we have

Similarly,

According to (2.7), we also have

Since $G_b(x^{*}, y^{*}, y^{*})\precsim\, s[G_b(x^{*}, S_j^{n}v, S_j^{n}v)+G_b(S_j^{n}v, y^{*}, y^{*})]$, using (2.8) and (2.9), we obtain

Therefore, $x^{*} = y^{*}$, the proof is completed.

Following the proof process of Theorem 2.1, we can obtain the following corollary.

Corollary 2.1. Let $(X, G_b, \preceq)$ be a complete partially ordered complex valued $G_b$-metric space with $s\geq1$, $Q\subset\, X$ be a cone, $\{S_n:X\rightarrow \, Q, \, \, n\in\, \mathbb{N}^{*}\}$ be a dominated mapping sequence. If there exist nonnegative numbers $\alpha, \beta, \gamma$ satisfy $\alpha-2s\gamma\neq0, \frac{\beta}{\alpha-2\gamma}\in[0, \frac{1}{s})$, such that

for any comparable elements $x, y, z$ in $Q$, where $i, j\in\mathbb{N}^{*}$, $\eta:\Omega_1\rightarrow [1, \infty)$, $\psi:\Omega_1\rightarrow \, \Omega_1$ is a nondecreasing function.

Define the operator equations $F_nx = ux$ by $F_n = uS_n$, $u\geq1$. If a nonincreasing sequence $\{x_n\}\rightarrow \, \kappa$ such that $\kappa\preceq\, x_n$, then the operator equations have at least a common solution $x^{*}$ in $Q$. Moreover, if there exists an element $v$ in $Q$ such that $v\preceq\, x^{*}$, then the operator equations have an unique solution.

Example 2.2. Let $X = R$, $Q = [0, \infty)$, $\alpha = 5, \beta = \gamma = 1, \delta = \frac{1}{3}$, $G_b:X\times X\times X \rightarrow \mathbb{C}$ be defined by $G_b(\xi_1, \xi_2, \xi_3) = \max\{|\xi_1-\xi_2|^2, |\xi_2-\xi_3|^2, |\xi_1-\xi_3|^2\}+\max\{|\xi_1-\xi_2|^2, |\xi_2-\xi_3|^2, |\xi_1-\xi_3|^2\}i$ with $s = 2$, and $\psi(r) = \eta(r)r = r$ for any $r$ in $\Omega_1$.

For any $\xi$ in $X$, $0 < \nu^{n}\leq\frac{1}{4}$ and $n\in\mathbb{N}^{*}$, take $S_{n}\xi = \nu^{n}\xi$ and $F_n = uS_n$, where $u\geq1$. The partial order $\preceq$ on $X$ is the usual order $\leq$ of $R$, for any $\xi_1, \xi_2, \xi_3$ in $Q$, we have

and

Hence,

It follows that the operator equations $F_n\xi = u\xi$ have a common solution $\xi^{*} = 0$ in $Q$, and there exists an element $v = 0$ in $Q$ such that $v\leq\xi^{*}$. Therefore, all conditions of Corollary 2.1 are satisfied, the operator equations $F_n\xi = u\xi$ have an unique solution $\xi^{*} = 0$.

Let $\mathscr{B}$ be the set of functions $\beta:\Omega_1\rightarrow [0, \frac{1}{s})$, which satisfies if $\lim\limits_{n\rightarrow \infty}\beta(x_n) = \frac{1}{s}$, then $\lim\limits_{n\rightarrow \infty}x_n = 0.$

Theorem 2.2. Let $(X, G_b, \preceq)$ be a complete partially ordered complex valued $G_b$-metric space with $s\geq1$, $Q\subset\, X$ be a cone, $x_0$ be an arbitrary element in $Q$, $\{S_n:X\rightarrow \, X, n\in\, \mathbb{N}^{*}\}$ be a dominated mapping sequence. Suppose that there exist $\beta\in\mathscr{B}$, $i, j\in\mathbb{N}^{*}$ and $r\in\Omega_2$, such that

for any comparable elements $x, y, z$ in $\overline{B(x_0, r)}$, where $\overline{B(x_0, r)}\subset\, Q$,

and

where $\delta\in(0, \frac{1}{s})$.

Define the operator equations $F_nx = ux$ by $F_n = uS_n$, $u\geq1$. If a nonincreasing sequence $\{x_n\}\rightarrow \kappa$ such that $\kappa\preceq\, x_n$, then the operator equations have at least a common solution $x^{*}$ in $\overline{B(x_0, r)}$.

Proof. By selecting the ball centre $x_0$ in $\overline{B(x_0, r)}$, we construct a sequence $\{x_n\}$, where $x_{n+1} = S_{n+1}x_n\preceq\, x_n, \, n\in\mathbb{N}$. From (2.12), we know $x_1\in\overline{B(x_0, r)}$. Using (2.10), we have

where

Since

and

thus $M(x_0, x_1, x_1)\precsim\max\{G_b(x_0, x_1, x_1), \, \, G_b(x_1, x_2, x_2)\}$.

If $\max\{G_b(x_0, x_1, x_1), \, \, G_b(x_1, x_2, x_2)\} = G_b(x_1, x_2, x_2)$, then we have

which is a contradiction, thus

and

So we have

as a result, $x_2\in\overline{B(x_0, r)}.$

Now we will show $\{x_n\}\subset\overline{B(x_0, r)}$. Assume that $x_3, x_4, ..., x_k\in\overline{B(x_0, r)}$, owing to (2.10), we get

Following the above proof process, we can obtain

Thus,

By using $(CG_b5)$ and (2.15), it follows that

Hence, $x_{k+1}\in\overline{B(x_0, r)}$, so $\{x_n\}\subset\overline{B(x_0, r)}$. As a result, for all $n\in\mathbb{N}^{*}$,

thus we have $G_b(x_n, x_{n+1}, x_{n+1})\prec\frac{1}{s}G_b(x_{n-1}, x_n, x_n)$.

If $s > 1$, then $G_b(x_n, x_{n+1}, x_{n+1})\prec(\frac{1}{s})^{n}G_b(x_0, x_1, x_1)\rightarrow \, 0$ as $n\rightarrow \, \infty$.

If $s = 1$, then $G_b(x_n, x_{n+1}, x_{n+1})\prec\, G_b(x_{n-1}, x_n, x_n)$, which implies that $\{G_b(x_n, x_{n+1}, x_{n+1})\}$ is a decreasing sequence.

Suppose that

owing to (2.14) and (2.16), we obtain

thus $\lim\limits_{n\rightarrow \infty}\beta(M(x_{n-1}, x_n, x_n)) = 1$, which implies $\lim\limits_{n\rightarrow \infty}G_b(x_{n-1}, x_n, x_n) = 0$, contradiction. As a result, $\lim\limits_{n\rightarrow \infty}G_b(x_n, x_{n+1}, x_{n+1}) = 0$.

Now we prove $\{x_n\}$ is a complex valued $G_b$-Cauchy sequence. Suppose that contrary, then there exist $\epsilon\succ0$ and two subsequences $x_{m_k}$ and $x_{n_k}$ of $x_n$, such that

So we have

Let $k\rightarrow \infty$, we get

Furthermore, using (2.10) and (2.14),

thus we have

Therefore, $\lim\limits_{k\rightarrow \infty}\beta(M(x_{m_k}, x_{n_k-1}, x_{n_k-1})) = \frac{1}{s}$, thus $\lim\limits_{k\rightarrow \infty}G_b(x_{m_k}, x_{n_k-1}, x_{n_k-1}) = 0.$ As a result,

which is a contradiction. Therefore, $\{x_n\}$ is a complex valued $G_b$-Cauchy sequence, and there exists an element $x^{*}$ in $\overline{B(x_0, r)}$ such that $x_n\rightarrow \, x^{*}$.

Finally, we show that $x^{*}$ is a common solution of the operator equations. Let $x = x_{i-1}, y = z = x^{*}$ in (2.10), we have

where

It can be easily deduced that $\lim\limits_{i\rightarrow \infty}M(x_{i-1}, x^{*}, x^{*}) = 0$ and $\lim\limits_{i\rightarrow \infty}G_b(S_ix_{i-1}, S_jx^{*}, S_jx^{*}) = 0$, thus

As a result, $x^{*} = S_jx^{*}$, owing to the arbitrariness of $j$, we obtain that $x^{*}$ is a common solution of the operator equations, the proof is completed.

Similarly, following the proof process of Theorem 2.2, the following corollary will be established.

Corollary 2.2. Let $(X, G_b, \preceq)$ be a complete partially ordered complex valued $G_b$-metric space with $s\geq1$, $Q\subset\, X$ be a cone, $\{S_n:X\rightarrow \, Q, n\in\, \mathbb{N}^{*}\}$ be a dominated mapping sequence. Suppose that there exist $i, j\in\mathbb{N}^{*}$ such that

for any comparable elements $x, y, z$ in $Q$, where $\lambda\in[0, \frac{1}{s})$, and

Define the operator equations $F_nx = ux$ by $F_n = uS_n$, $u\geq1$. If a nonincreasing sequence $\{x_n\}\rightarrow \kappa$ such that $\kappa\preceq\, x_n$, then the operator equations have at least a common solution $x^{*}$ in $Q$.

Example 2.3. Let $X = R$, $Q = [0, \infty)$, $G_b:X\times X\times X \rightarrow \mathbb{C}$ be defined by $G_b(\xi_1, \xi_2, \xi_3) = (|\xi_1-\xi_2|+|\xi_2-\xi_3|+|\xi_1-\xi_3|)^2+(|\xi_1-\xi_2|+|\xi_2-\xi_3|+|\xi_1-\xi_3|)^2i$ with $s = 2$, $\delta = \frac{1}{5}$, $x_0 = 1$, $r = 4+4i$. For all $t\in\Omega_1$, take

Obviously, $\frac{1}{3}\leq\beta(t) < \frac{1}{2}$, and

Moreover, for any $\xi$ in $X$, let $S_{n}\xi = \frac{|\xi|}{\sqrt{3}n}, n\in\mathbb{N}^{*}$ and $F_n = uS_n$, where $u\geq1$. The partial order $\preceq$ on $X$ is the usual order $\leq$ of $R$, for any $\xi_1, \xi_2, \xi_3$ in $\overline{B(1, 4+4i)}$, we have

and

It follows that

and

It is clearly that all conditions of Theorem 2.2 are satisfied, as a result, the operator equations $F_n\xi = u\xi$ have a common solution $\xi^{*} = 0$ in $\overline{B(1, 4+4i)}$.

On the other hand, let $\Theta$ be the set of functions $\theta:\Omega_2\rightarrow \, \Omega_3$, which satisfies the following conditions:

$\Theta_1:$ $\theta$ is continuous;

$\Theta_2:$ $\theta$ is nondecreasing, i.e. $\theta(x_1)\succsim\theta(x_2)$ if $x_1\succsim\, x_2$;

$\Theta_3:$ $\lim\limits_{n\rightarrow \infty}\theta(x_n) = 1 \Leftrightarrow \lim\limits_{n\rightarrow \infty}x_n = 0^{+}$, where $\{x_n\}\subset\, \Omega_2$.

Theorem 2.3. Let $(X, G_b, \preceq)$ be a complete partially ordered complex valued $G_b$-metric space with $s\geq1$, $Q\subset\, X$ be a cone, $\{S_n:X\rightarrow \, Q, \, \, n\in\mathbb{N}^{*}\}$ be a dominated mapping sequence. Suppose that there exist $\theta\in\Theta$, $i, j\in\mathbb{N}^{*}, k\in(0, 1), \alpha\geq0$ such that

for any comparable elements $x, y, z$ in $Q$, where $G_b(S_ix, S_jy, S_jz)\neq\, 0$, and

Define the operator equations $F_nx = ux$ by $F_n = uS_n$, $u\geq1$. If a nonincreasing sequence $\{x_n\}\rightarrow \kappa$ such that $\kappa\preceq\, x_n$, then the operator equations have at least a common solution $x^{*}$ in $Q$. Moreover, if there exists an element $v$ in $Q$ such that $v\preceq\, x^{*}$, and

then the operator equations have an unique solution.

Proof. By selecting a point $x_0$ in $Q$, we construct a sequence $\{x_n\}$, where $x_{n+1} = S_{n+1}x_n\preceq\, x_n, \, n\in\mathbb{N}$. Let $x = x_{n-1}, y = z = x_n$ in (2.17), we have

where

thus we get

If $\max\{G_b(x_{n-1}, x_n, x_n), G_b(x_n, x_{n+1}, x_{n+1})\} = G_b(x_n, x_{n+1}, x_{n+1})$, then

hence,

It follows that

and

therefore,

Now we show $\{x_n\}$ is a complex valued $G_b$-Cauchy sequence. If not, then there exist $\epsilon\succ0$ and two subsequences $x_{m_i}$ and $x_{n_i}$ of $x_n$, where $i\leq\, n_i\leq\, m_i$, such that

Using $(CG_b5)$, we have

let $i\rightarrow \infty$ at the above inequality, we get

In addition, owing to (2.17), we obtain

i.e.,

where

Since

obviously, $M(x_{m_{i-1}}, x_{n_i}, x_{n_i}) = G_b(x_{n_i}, x_{n_i}, x_{m_{i-1}})$, it follows that

Using (2.20) and (2.21), we have

which is a contradiction with $k\in(0, 1)$. As a result, $\{x_n\}$ is a complex valued $G_b$-Cauchy sequence, and there exists an element $x^{*}$ in $Q$ such that $x_n\rightarrow \, x^{*}$.

Now we prove that $x^{*}$ is a common solution of the operator equations. For all $i, j\in\mathbb{N}^{*}$, we have

and let $i\rightarrow \infty$ at the above inequality, we get

In addition, since $x^{*}\preceq\, x_{i-1}$, according to (2.17), we obtain

where

If $M(x_{i-1}, x^{*}, x^{*}) = G_b(x^{*}, S_jx^{*}, S_jx^{*})$, using (2.22), it follows that

contradiction, thus we can easily get

or

hence,

From (2.22), we have

As a result, $x^{*} = S_jx^{*}$, owing to the arbitrariness of $j$, we obtain $x^{*}$ is a common solution of the operator equations.

Uniqueness. If $y^{*}$ is another solution of the operator equations, $y^{*}\neq\, x^{*}$, then $G_b(x^{*}, y^{*}, y^{*})\neq\, 0$.

Case 1. $x^{*}$ and $y^{*}$ are comparable, using (2.17), it follows that

Obviously, $M(x^{*}, y^{*}, y^{*}) = G_b(x^{*}, y^{*}, y^{*})$, so we have

which is a contradiction. As a result, $y^{*} = x^{*}$.

Case 2. $x^{*}$ and $y^{*}$ are not comparable, then there exists an element $v\in\, Q$ such that $v\preceq\, x^{*}$ and $v\preceq\, y^{*}$, for any $i, j\in\mathbb{N}^{*}$, we have

and

From (2.17), we get

where

According to (2.19), we obtain $M(S^{n-1}_ix^{*}, S^{n-1}_jv, S^{n-1}_jv) = G_b(S^{n-1}_ix^{*}, S^{n-1}_jv, S^{n-1}_jv)$, and

so that we have

It follows that

hence,

Similarly, using (2.17) and (2.19), we get

where

Therefore,

let $n\rightarrow \infty$, we have

so we obtain

and also

Using (2.23) and (2.24), we also have

Owing to $G_b(x^{*}, y^{*}, y^{*}) = G_b(S^{n}_ix^{*}, S^{n}_jy^{*}, S^{n}_jy^{*})$, as a result, $x^{*} = y^{*}$, the proof is completed.

Example 2.4. Let $X = R$, $Q = [0, \infty)$, $\theta(t) = 1+t$, $\alpha = 0$, $k = \frac{1}{2}$, $G_b:X\times X\times X \rightarrow \mathbb{C}$ be defined by $G_b(\xi_1, \xi_2, \xi_3) = \max\{|\xi_1-\xi_2|^2, |\xi_2-\xi_3|^2, |\xi_1-\xi_3|^2\}+\max\{|\xi_1-\xi_2|^2, |\xi_2-\xi_3|^2, |\xi_1-\xi_3|^2\}i$ with $s = 2$. For any $\xi$ in $X$, take $S_{n}\xi = \frac{|\xi|}{4n}$ and $F_n = uS_n$, where $u\geq1, n\in\mathbb{N}^{*}$, the partial order $\preceq$ on $X$ is the usual order $\leq$ of $R$.

Suppose that $\xi_1\geq\xi_2\geq\xi_3$, if $\xi_1-\xi_3\leq\, 1$ for any $\xi_1, \xi_2, \xi_3$ in $Q$, or $\xi_1, \xi_2, \xi_3\in[0, 1]$, we can easily obtain

and

Hence,

and

Thus we obtain

It follows that the operator equations $F_n\xi = u\xi$ have a common solution $\xi^{*} = 0$ in $Q$ and (2.19) is established with $v = 0$. Therefore, all conditions of Theorem 2.3 are satisfied, the operator equations $F_n\xi = u\xi$ have an unique solution $\xi^{*} = 0$.

The following two corollaries can be easily obtained, if we let $\theta(t) = e^{|t|}+t$ and $\theta(t) = 2-\frac{2}{\pi}\arctan(\frac{1}{|t|^{\gamma}})$ in Theorem 2.3 respectively.

Corollary 2.3. Let $(X, G_b, \preceq)$ be a complete partially ordered complex valued $G_b$-metric space with $s\geq1$, $Q\subset\, X$ be a cone, $\{S_n:X\rightarrow \, Q, \, \, n\in\mathbb{N}^{*}\}$ be a dominated mapping sequence. Suppose that there exist $i, j\in\mathbb{N}^{*}, k\in(0, 1), \alpha\geq0$ such that

for any comparable elements $x, y, z$ in $Q$, where $G_b(S_ix, S_jy, S_jz)\neq\, 0$, and

Define the operator equations $F_nx = ux$ by $F_n = uS_n$, $u\geq1$. If a nonincreasing sequence $\{x_n\}\rightarrow \kappa$ such that $\kappa\preceq\, x_n$, then the operator equations have at least a common solution $x^{*}$ in $Q$. Moreover, if there exists an element $v$ in $Q$ such that $v\preceq\, x^{*}$, and

then the operator equations have an unique solution.

Corollary 2.4. Let $(X, G_b, \preceq)$ be a complete partially ordered complex valued $G_b$-metric space with $s\geq1$, $Q\subset\, X$ be a cone, $\{S_n:X\rightarrow \, Q, \, \, n\in\mathbb{N}^{*}\}$ be a dominated mapping sequence. Suppose that there exist $i, j\in\mathbb{N}^{*}, \gamma, k\in(0, 1), \alpha\geq0$ such that

for any comparable elements $x, y, z$ in $Q$, where $G_b(S_ix, S_jy, S_jz)\neq\, 0$, and

Define the operator equations $F_nx = ux$ by $F_n = uS_n$, $u\geq1$. If a nonincreasing sequence $\{x_n\}\rightarrow \kappa$ such that $\kappa\preceq\, x_n$, then the operator equations have at least a common solution $x^{*}$ in $Q$. Moreover, if there exists an element $v$ in $Q$ such that $v\preceq\, x^{*}$, and

then the operator equations have an unique solution.

3.   Conclusions

In this paper, we have obtained some new theorems for the common solutions of the operator equations $F_nx = ux\, \, (u\geq1, n\in\, \mathbb{N}^{*})$ via complex valued $C^{p}$-class function, a type of Geraghty contraction and a type of JS contraction in complete partially ordered complex valued $G_b$-metric spaces, and some of which are established in a closed ball. These new results generalize many known results in complex valued $G_b$-metric spaces and $G_b$-metric spaces, in addition, it would be interesting and worthwhile to further investigate some similar problems in other types of spaces.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant No. 11771198).

Conflict of interest

The authors declare no conflict of interest.