A different approach to complex valued $G_{b}$-metric spaces

Abstract

In this work, we introduce a generalized fixed point theorem using a complex C-class function as a new tool in complex valued $G_{b}$-metric spaces. Moreover, we define $\alpha -(F,\psi ,\varphi )$-contractive type and α-admissible mapping. Then we prove a fixed point theorem using these notions and the complex C-class function. The obtained results generalize some facts in the literature.

1 Introduction

Fixed point theory has great importance in science and mathematics. Since this area has been developed very fast over the past two decades due to huge applications in various fields such as nonlinear analysis, topology and engineering problems, it has attracted considerable attention from researchers.

In 1989, Bakhtin [1] presented b-metric spaces. Since then, researchers have performed significant studies such as [2–6] in this type metric space. After the complex valued metric space was defined as a new concept, this idea has been used many times. For example, the complex valued b-metric spaces are given in [7]. G-metric spaces [8] have been defined and then researchers have obtained important results (see [9–15]).

After introducing $G_{b}$-metric spaces in [16], the Banach and Kannan fixed point theorems [17] were proved for $G_{b}$-metric spaces. There are also other significant studies [18–21] on $G_{b}$-metric spaces.

In recent times, Ansari [22] has investigated the notion of C-class function. He has presented new fixed point results using this function. For some of them, see [23–30].

This paper starts with Sect. 2 which consists of the required background. Then a common fixed point theorem has been proved and a corollary with an illustrating example is presented. After introducing the $\alpha -(F,\psi ,\varphi )$-contractive type and α-admissible mapping and complex C-class function, we give the proof of a new fixed point theorem.

2 Preliminaries

In this part, some useful notions and facts will be given. A partial order ≾ on $\mathbb{C}$, which is the set of complex numbers, can be defined as follows:

$$ \tau _{1}\precsim \tau _{2} :\quad \Leftrightarrow \quad \Im (\tau _{1}) \leq \Im (\tau _{2}) \quad \text{and} \quad \Re (\tau _{1})\leq \Re (\tau _{2}). $$

We write $\tau _{1}\precsim \tau _{2}$ if one of the following holds:

($C_{1}$):

$\Im (\tau _{1})=\Im (\tau _{2})$ and $\Re (\tau _{1})=\Re (z_{2})$,

($C_{2}$):

$\Im (\tau _{1})=\Im (\tau _{2})$ and $\Re (\tau _{1})<\Re (z_{2})$,

($C_{3}$):

$\Im (\tau _{1})<\Im (\tau _{2})$ and $\Re (\tau _{1})=\Re (z_{2})$,

($C_{4}$):

$\Im (\tau _{1})<\Im (\tau _{2})$ and $\Re (\tau _{1})<\Re (z_{2})$.

We use $\tau _{1}\precnsim \tau _{2}$ if $\tau _{1}\neq \tau _{2}$ and one of $(C_{2})$, $(C_{3})$ and $(C_{4})$ holds and we denote $\tau _{1}\prec \tau _{2}$ if only $(C_{4})$ holds.

• (1) If $u,v\in \mathbb{R}$ with $u\leq v$, then $u\tau \prec v\tau$ for each $\tau \in \mathbb{C}$.

• (2) If $0\precsim \tau _{1}\precnsim \tau _{2}$, then $\vert \tau _{1} \vert < \vert \tau _{2} \vert$.

• (3) If $\tau _{1}\precsim \tau _{2}$ and $\tau _{2}\prec \tau _{3}$, then $\tau _{1}\prec \tau _{3}$.

Definition 2.1

([17])

For a nonempty set X and a real number $s\geq 1$, if for a map $G:X\times X\times X\rightarrow \mathbb{C}$ holds the following:

($\mathit{CG}_{b}1$):

$G(\xi _{1},\xi _{2},\xi _{3})=0$ if $\xi _{1}=\xi _{2}=\xi _{3}$,

($\mathit{CG}_{b}2$):

$0\prec G(\xi _{1},\xi _{1},\xi _{2})$ for all $\xi _{1},\xi _{2}\in X$ with $\xi _{1}\neq \xi _{2}$,

($\mathit{CG}_{b}3$):

$G(\xi _{1},\xi _{1},\xi _{2})\precsim G(\xi _{1},\xi _{2},\xi _{3})$ for all $\xi _{1},\xi _{2},\xi _{3}\in X$ with $\xi _{2}\neq \xi _{3}$,

($\mathit{CG}_{b}4$):

$G(\xi _{1},\xi _{2},\xi _{3})=G(\rho \{\xi _{1},\xi _{2},\xi _{3}\})$, where ρ is a permutation of $\xi _{1}$, $\xi _{2}$, $\xi _{3}$,

($\mathit{CG}_{b}5$):

$G(\xi _{1},\xi _{2},\xi _{3})\precsim s(G(\xi _{1},\kappa ,\kappa )+G( \kappa ,\xi _{2},\xi _{3}))$ for all $\xi _{1},\xi _{2},\xi _{3},\kappa \in X$,

we say that G is a complex valued $G_{b}$-metric and the pair $(X,G)$ is a complex valued $G_{b}$-metric space.

Definition 2.2

([17])

Let $\{x_{n}\}$ be a sequence in a complex valued $G_{b}$-metric space $(X,G)$.

• (1) $\{x_{n}\}$ is complex valued $G_{b}$-convergent to ξ if, for every $\kappa \in \mathbb{C}$ with $0\prec \kappa$, there is a natural number ω such that $G(\xi ,x_{n},x_{m})\prec \kappa$ for all $n,m\geq \omega$.

• (2) $\{x_{n}\}$ is said to be complex valued $G_{b}$-Cauchy if, for every $\kappa \in \mathbb{C}$ with $0\prec \kappa$, there exists $\omega \in \mathbb{N}$ such that $G(x_{n},x_{m},x_{l})\prec \kappa$ for all $n,m,l\geq \omega$.

• (3) $(X,G)$ is called complex valued $G_{b}$-complete if every complex valued $G_{b}$-Cauchy sequence is complex valued $G_{b}$-convergent.

Ege [17] proves that a sequence $\{x_{n}\}$ in a complex valued $G_{b}$-metric space is complex valued $G_{b}$-convergent to ξ iff $\vert G(\xi ,x_{n},x_{m}) \vert \rightarrow 0$ as $n,m\rightarrow \infty$.

Theorem 2.3

([17])

For a sequence$\{x_{n}\}$in a complex valued$G_{b}$-metric space$(X,G)$, the following statements are equivalent:

• (1) $\{x_{n}\}$is complex valued$G_{b}$-convergent to a point ξ.

• (2) $\vert G(x_{n},x_{n},\xi ) \vert \rightarrow 0$as$n\rightarrow \infty$.

• (3) $\vert G(x_{n},\xi ,\xi ) \vert \rightarrow 0$as$n\rightarrow \infty$.

• (4) $\vert G(x_{m},x_{n},\xi ) \vert \rightarrow 0$as$m,n\rightarrow \infty$.

Theorem 2.4

([17])

A sequence$\{x_{n}\}$is a complex valued$G_{b}$-Cauchy sequence if and only if$\vert G(x_{n},x_{m},x_{l}) \vert \rightarrow 0$as$n,m,l\rightarrow \infty$.

The notion of a C-class function was presented in [22]. For any $\mu ,t\in {}[ 0,\infty )$, for a continuous function $F:[0,\infty )^{2}\rightarrow \mathbb{R}$ holds the following:

• (i) $F(\mu ,\chi )\leq \mu$;

• (ii) $F(\mu ,\chi )=\mu$ implies that either $\mu =0$ or $\chi =0$.

Then F is said to be C-class function. $\mathcal{C}$ denotes the class of all C-functions.

Example 2.5

([22])

The following are examples of C-class functions:

• (i) $F(\mu ,\chi )=\mu -\chi$.

• (ii) $F(\mu ,\chi )=m\mu$, for some $m\in (0,1)$.

• (iii) $F(\mu ,\chi )=\frac{\mu }{(1+\chi )^{r}}$, for a positive real number r.

• (iv) $F(\mu ,\chi )=(\mu +l)^{(1/(1+\chi )^{r})}-l$, where $l>1$ for $r\in (0,\infty )$.

• (v) $F(\mu ,\chi )=\mu \log _{\chi +u}u$ for $u>1$.

Let $\varPhi _{u}$ be the class of the continuous functions $\varphi :[0,\infty )\rightarrow [0,\infty )$ satisfying $\varphi (\chi )>0$ for $\chi >0$ and $\varphi (0)\geq 0$.

Definition 2.6

([25])

For any $\mu ,t\in S= \{ z\in \mathbb{C}:0\precsim z \}$, if a continuous function $F:S^{2}\rightarrow \mathbb{C}$ satisfies the following:

• (i) $F(\mu ,\chi )\precsim \mu$,

• (ii) if $F(\mu ,\chi )=\mu$, then either $\mu =0$ or $\chi =0$,

then it is called a complex C-class function. We denote the class of all complex C-class functions by the same symbol $\mathcal{C}$.

As an example, we can give the following: Let $S= \{ z\in \mathbb{C}:0\precsim z \}$.

• (1) $F(\mu ,\chi )=\phi (\mu )$ where $\phi :S\rightarrow S$ is continuous, $\phi (0)=0$ and $\phi (\chi )\succ 0$ if $\chi \succ 0$.

• (2) $F(\mu ,\chi )=\mu \beta (\mu )$, where $\beta :[0,\infty )\rightarrow {}[ 0,1)$ is continuous and $\mu \in S$.

Let Ψ denote the class of continuous functions $\psi :S\rightarrow S$ satisfying $\psi (\chi )\succ 0$ iff $\chi \succ 0$ and $\varphi (0)=0$.

$\varPhi _{u}$ will denote the class of continuous functions $\varphi :S\rightarrow S$ satisfying $\varphi (\chi )\succ 0$ iff $\chi \succ 0$ and $\varphi (0)\succeq 0$.

Our aim is to give some different generalizations of the following theorems from the literature using C-class functions.

Theorem 2.7

([18])

Let$\{T_{n}\}$be a sequence of self-mappings of a complete complex valued$G_{b}$-metric space$(X,G)$such that

$$ G\bigl(T_{i}(x),T_{j}(y),T_{j}(z)\bigr)\precsim \beta _{i,j}\bigl[G\bigl(x,T_{i}(x),T_{i}(x) \bigr)+G\bigl(y,T_{j}(y),T_{j}(z)\bigr)\bigr]+ \gamma _{i,j} G(x,y,z) $$

for$x,y,z\in X$with$x\neq y$, $0\leq \beta _{i,j},\gamma _{i,j}<1$, $i,j=1,2,\ldots$ .

If$\sum_{i=1}^{\infty }( \frac{\beta _{i,i+1}+\gamma _{i,i+1}}{1-\beta _{i,i+1}})$is anα-series, then$\{T_{n}\}$has a unique common fixed point in X.

Theorem 2.8

([18])

Let$(X,G)$be a complete complex valued$G_{b}$-metric space and$T:X\rightarrow X$be an$\alpha -\psi$contractive mapping of typeAsatisfying the following conditions:

• (i) Tisα-admissible,

• (ii) There exists$x_{0}\in X$such that$\alpha (x_{0},Tx_{0},Tx_{0})\geq 1$,

• (iii) If$\{x_{n}\}$is a sequence inXsuch that$\alpha (x_{n},x_{n+1},x_{n+1})\geq 1$for allnand$x_{n}\rightarrow x\in X$as$n\rightarrow \infty$, then$\alpha (x_{n},x,x_{n+1})\geq 1$for all n.

Consider an element$z\in X$such that$\alpha (x,z,z)\geq 1$and$\alpha (y,z,z)\geq 1$for all$x,y\in X$. ThenThas a unique fixed point.

3 Main results

Theorem 3.1

Let$\{T_{n}\}$be a sequence of self-mappings of a complete complex valued$G_{b}$-metric space$(X,G)$such that

$$\begin{aligned}& \psi \bigl(G\bigl(T_{i}(x),T_{j}(y),T_{j}(z) \bigr)\bigr) \\& \quad \precsim F \biggl(\psi \biggl( \frac{1}{\alpha _{i,j}+\beta _{i,j}+\delta _{i,j}}\bigl[\alpha _{i,j}G\bigl(x,T_{i}(x),T_{i}(x)\bigr)+ \beta _{i,j}G\bigl(y,T_{j}(y),T_{j}(z)\bigr) \\& \quad\quad {}+\delta _{i,j}G(x,y,z)\bigr]\biggr), \\& \quad\quad {}\varphi \biggl(\frac{1}{\alpha _{i,j}+\beta _{i,j}+\delta _{i,j}}\bigl[ \alpha _{i,j}G \bigl(x,T_{i}(x),T_{i}(x)\bigr)+\beta _{i,j}G \bigl(y,T_{j}(y),T_{j}(z)\bigr) \\& \quad\quad {} +\delta _{i,j}G(x,y,z)\bigr]\biggr) \biggr) \end{aligned}$$

(3.1)

for$x,y,z\in X$with$x\neq y$, $0\leq \alpha _{i,j}, \beta _{i,j}, \delta _{i,j}$and$\alpha _{i,j}+\beta _{i,j}+\delta _{i,j}>0$, $i,j=1,2,\ldots$ , where$\psi \in \varPsi$, $F\in C$and$\varphi \in \varPhi _{u}$. $\{T_{n}\}$has a unique common fixed point in X.

Proof

Consider a sequence as $x_{n}=T_{n}(x_{n-1})$ for an element $x_{0}\in X$ where $n=1,2,\ldots$ . If we use (3.1), we obtain

$$\begin{aligned}& \psi \bigl(G(x_{1},x_{2},x_{2}) \bigr) \\& \quad = \psi \bigl(G\bigl(T_{1}(x_{0}),T_{2}(x_{1}),T_{2}(x_{1}) \bigr)\bigr) \\& \quad \precsim F \biggl(\psi \biggl( \frac{1}{\alpha _{1,2}+\beta _{1,2}+\delta _{1,2}}\bigl[\alpha _{1,2}G\bigl(x_{0},T_{1}(x_{0}),T_{1}(x_{0}) \bigr)+ \beta _{1,2}G\bigl(x_{1},T_{2}(x_{1}),T_{2}(x_{1}) \bigr) \\& \quad \quad {} +\delta _{1,2}G(x_{0},x_{1},x_{1}) \bigr]\biggr), \\& \quad\quad {}\varphi \biggl(\frac{1}{\alpha _{1,2}+\beta _{1,2}+\delta _{1,2}}\bigl[\alpha _{1,2}G \bigl(x_{0},T_{1}(x_{0}),T_{1}(x_{0}) \bigr)+ \beta _{1,2}G\bigl(x_{1},T_{2}(x_{1}),T_{2}(x_{1}) \bigr) \\& \quad \quad {} +\delta _{1,2}G(x_{0},x_{1},x_{1}) \bigr]\biggr) \biggr) \\& \quad = F \biggl(\psi \biggl(\frac{1}{\alpha _{1,2}+\beta _{1,2}+\delta _{1,2}}\bigl[ \alpha _{1,2}G(x_{0},x_{1},x_{1})+ \beta _{1,2}G(x_{1},x_{2},x_{2})+ \delta _{1,2}G(x_{0},x_{1},x_{1})\bigr] \biggr), \\& \quad \quad {}\varphi \biggl(\frac{1}{\alpha _{1,2}+\beta _{1,2}+\delta _{1,2}}\bigl[\alpha _{1,2}G(x_{0},x_{1},x_{1})+ \beta _{1,2}G(x_{1},x_{2},x_{2})+\delta _{1,2}G(x_{0},x_{1},x_{1})\bigr]\biggr) \biggr) \\& \quad \precsim \psi \biggl( \frac{\alpha _{1,2}+\delta _{1,2}}{\alpha _{1,2}+\beta _{1,2}+\delta _{1,2}}G(x_{0},x_{1},x_{1})+ \frac{\beta _{1,2}}{\alpha _{1,2}+\beta _{1,2}+\delta _{1,2}}G(x_{1},x_{2},x_{2})\biggr). \end{aligned}$$

From the property of F, ψ and monotonocity increasing of ψ, we get

$$ G(x_{1},x_{2},x_{2})\precsim G(x_{0},x_{1},x_{1}). $$

Moreover, by the following inequalities:

$$\begin{aligned}& \psi \bigl(G(x_{2},x_{3},x_{3}) \bigr) \\& \quad = \psi \bigl(G\bigl(T_{2}(x_{1}),T_{3}(x_{2}),T_{3}(x_{2}) \bigr)\bigr) \\& \quad \precsim F \biggl(\psi \biggl( \frac{1}{\alpha _{2,3}+\beta _{2,3}+\delta _{2,3}}\bigl[\alpha _{2,3}G \bigl(x_{1},T_{2}(x_{1}),T_{2}(x_{1}) \bigr)+ \beta _{2,3}G\bigl(x_{2},T_{3}(x_{2}),T_{3}(x_{2}) \bigr) \\& \quad \quad {} +\delta _{2,3}G(x_{1},x_{2},x_{2}) \bigr]\biggr), \\& \quad \quad {}\varphi \biggl(\frac{1}{\alpha _{2,3}+\beta _{2,3}+\delta _{2,3}}\bigl[\alpha _{2,3}G \bigl(x_{1},T_{2}(x_{1}),T_{2}(x_{1}) \bigr)+ \beta _{2,3}G\bigl(x_{2},T_{3}(x_{2}),T_{3}(x_{2}) \bigr) \\& \quad \quad {} +\delta _{2,3}G(x_{1},x_{2},x_{2}) \bigr]\biggr) \biggr) \\& \quad = F \biggl(\psi \biggl(\frac{1}{\alpha _{2,3}+\beta _{2,3}+\delta _{2,3}}\bigl[ \alpha _{2,3}G(x_{1},x_{2},x_{2})+ \beta _{2,3}G(x_{2},x_{3},x_{3})+ \delta _{2,3}G(x_{1},x_{2},x_{2})\bigr]\biggr), \\& \quad \quad {}\varphi \biggl(\frac{1}{\alpha _{2,3}+\beta _{2,3}+\delta _{2,3}}\bigl[\alpha _{2,3}G(x_{1},x_{2},x_{2})+ \beta _{2,3}G(x_{2},x_{3},x_{3})+\delta _{2,3}G(x_{1},x_{2},x_{2})\bigr]\biggr) \biggr) \\& \quad \precsim \psi \biggl( \frac{\alpha _{2,3}+\delta _{2,3}}{\alpha _{2,3}+\beta _{2,3}+\delta _{2,3}}G(x_{1},x_{2},x_{2})+ \frac{\beta _{2,3}}{\alpha _{2,3}+\beta _{2,3}+\delta _{2,3}}G(x_{2},x_{3},x_{3})\biggr), \end{aligned}$$

we obtain

$$ G(x_{2},x_{3},x_{3})\precsim G(x_{1},x_{2},x_{2}). $$

If the same procedure is applied repeatedly

$$\begin{aligned} \begin{aligned}[b] \psi \bigl(G(x_{n},x_{n+1},x_{n+1}) \bigr)={}&\psi \bigl(G\bigl(T_{n}(x_{n-1}),T_{n+1}(x_{n}),T_{n+1}(x_{n}) \bigr)\bigr) \\ \precsim {}&F \biggl(\psi \biggl( \frac{1}{\alpha _{n,n+1}+\beta _{n,n+1}+\delta _{n,n+1}}\bigl[\alpha _{n,n+1}G\bigl(x_{n-1},T_{n}(x_{n-1}),T_{n}(x_{n-1}) \bigr) \\ &{} +\beta _{n,n+1}G\bigl(x_{n},T_{n+1}(x_{n}),T_{n+1}(x_{n}) \bigr)+\delta _{n,n+1}G(x_{n-1},x_{n},x_{n}) \bigr]\biggr), \\ &{}\varphi \biggl(\frac{1}{\alpha _{n,n+1}+\beta _{n,n+1}+\delta _{n,n+1}}\bigl[G\bigl(x_{n-1},T_{n}(x_{n-1}),T_{n}(x_{n-1}) \bigr) \\ &{} +\beta _{n,n+1}G\bigl(x_{n},T_{n+1}(x_{n}),T_{n+1}(x_{n}) \bigr)+\delta _{n,n+1}G(x_{n-1},x_{n},x_{n}) \bigr]\biggr) \biggr) \\ ={}&F \biggl(\psi \biggl( \frac{1}{\alpha _{n,n+1}+\beta _{n,n+1}+\delta _{n,n+1}}\bigl[\alpha _{n,n+1}G(x_{n-1},x_{n},x_{n}) \\ &{} +\beta _{n,n+1}G(x_{n},x_{n+1},x_{n+1})+ \delta _{n,n+1}G(x_{n-1},x_{n},x_{n})\bigr] \biggr), \\ &{}\varphi \biggl(\frac{1}{\alpha _{n,n+1}+\beta _{n,n+1}+\delta _{n,n+1}}\bigl[ \alpha _{n,n+1}G(x_{n-1},x_{n},x_{n}) \\ &{} +\beta _{n,n+1}G(x_{n},x_{n+1},x_{n+1})+ \delta _{n,n+1}G(x_{n-1},x_{n},x_{n})\bigr] \biggr) \biggr), \end{aligned} \end{aligned}$$

(3.2)

we get $G(x_{n},x_{n+1},x_{n+1})\precsim G(x_{n-1},x_{n},x_{n})$. Thus $\{G(x_{n},x_{n+1},x_{n+1})\}$ is a decreasing sequence in $\mathbb{C}$. So we say that it is $G_{b}$-convergent to $0\preceq \chi \in \mathbb{C}$. We assert that $\chi =0$. To show this, assume that $\chi \succ 0$. If we take the limit of (3.2), we get

$$ \begin{aligned} \psi (\chi )\precsim {}&F \biggl(\psi \biggl( \frac{1}{\alpha _{n,n+1}+\beta _{n,n+1}+\delta _{n,n+1}}[\alpha _{n,n+1} \chi +\beta _{n,n+1}\chi +\delta _{n,n+1}\chi ]\biggr), \\ &{} \varphi \biggl(\frac{1}{\alpha _{n,n+1}+\beta _{n,n+1}+\delta _{n,n+1}}[ \alpha _{n,n+1}\chi +\beta _{n,n+1}\chi +\delta _{n,n+1}\chi ]\biggr) \biggr) \\ ={}&F \bigl(\psi (\chi ), \varphi (\chi ) \bigr) \end{aligned} $$

which implies $\psi (\chi )=0$ or $\varphi (\chi )=0$, namely $\chi =0$. But this is a contradiction. So $\chi =0$. i.e.,

$$ \lim_{n\rightarrow \infty }G(x_{n},x_{n+1},x_{n+1})=0. $$

(3.3)

We will show that the sequence $\{x_{n}\}$ is a $G_{b}$-Cauchy by assuming the contrary. If we use (3.1), we obtain

$$ \begin{aligned} \psi \bigl(G(x_{n},x_{m},x_{m}) \bigr)&=\psi \bigl(G\bigl(T_{n}(x_{n-1}),T_{m}(x_{m-1}),T_{m}(x_{m-1}) \bigr)\bigr) \\ &\precsim F \biggl(\psi \biggl( \frac{1}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}\bigl[\alpha _{n,m}G \bigl(x_{n-1},T_{n}(x_{n-1}),T_{n}(x_{n-1}) \bigr) \\ & \quad {}+\beta _{n,m}G\bigl(x_{m-1},T_{m}(x_{m-1}),T_{m}(x_{m-1}) \bigr)+\delta _{n,m}G(x_{n-1},x_{m-1},x_{m-1}) \bigr]\biggr), \\ & \quad {}\varphi \biggl(\frac{1}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}\bigl[ \alpha _{n,m}G \bigl(x_{n-1},T_{n}(x_{n-1}),T_{n}(x_{n-1}) \bigr) \\ & \quad {}+\beta _{n,m}G\bigl(x_{m-1},T_{m}(x_{m-1}),T_{m}(x_{m-1}) \bigr)+G(x_{n-1},x_{m-1},x_{m-1})\bigr]\biggr) \biggr) \\ &=F \biggl(\psi \biggl(\frac{1}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}\bigl[ \alpha _{n,m}G(x_{n-1},x_{n},x_{n}) \\ & \quad{} +\beta _{n,m}G(x_{m-1},x_{m},x_{m})+ \delta _{n,m}G(x_{n-1},x_{m-1},x_{m-1})\bigr] \biggr), \\ & \quad{} \varphi \biggl(\frac{1}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}\bigl[ \alpha _{n,m}G(x_{n-1},x_{n},x_{n}) \\ & \quad{} +\beta _{n,m}G(x_{m-1},x_{m},x_{m})+G(x_{n-1},x_{m-1},x_{m-1}) \bigr]\biggr) \biggr). \end{aligned} $$

Using the same procedure, we get

$$ \begin{aligned} \psi (\varepsilon )\precsim {}& F \biggl(\psi \biggl( \frac{1}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}[\alpha _{n,m} \varepsilon +\beta _{n,m} \varepsilon +\delta _{n,m}\varepsilon ]\biggr), \\ &{}\varphi \biggl(\frac{1}{\alpha _{n,m}+\beta _{n,m}+ \delta _{n,m}}[\alpha _{n,m}\varepsilon +\beta _{n,m}\varepsilon + \delta _{n,m}\varepsilon ]\biggr) \biggr) \\ ={}&F \bigl(\psi (\varepsilon ),\varphi (\varepsilon ) \bigr), \end{aligned} $$

which implies $\psi (\varepsilon )=0$ or $\varphi (\varepsilon )=0$. Namely, $\varepsilon =0$ but this is a contradiction. So $\{x_{n}\}$ is a complex valued $G_{b}$-Cauchy sequence. By the $G_{b}$-completeness of X, $\{x_{n}\}$ converges to an element v in X. From (3.1), we have

$$ \begin{aligned} \psi \bigl(G\bigl(x_{n},T_{m}(v),T_{m}(v) \bigr)\bigr)={}&\psi \bigl(G\bigl(T_{n}(x_{n-1}),T_{m}(v),T_{m}(v) \bigr)\bigr) \\ \precsim {}& F \biggl(\psi \biggl( \frac{1}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}\bigl[\alpha _{n,m}G\bigl(x_{n-1},T_{n}(x_{n-1}),T_{n}(x_{n-1}) \bigr) \\ &{} +\beta _{n,m}G\bigl(v,T_{m}(v),T_{m}(v) \bigr)+\delta _{n,m}G(x_{n-1},v,v)\bigr]\biggr), \\ &{}\varphi \biggl(\frac{1}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}\bigl[\alpha _{n,m}G \bigl(x_{n-1},T_{n}(x_{n-1}),T_{n}(x_{n-1}) \bigr) \\ &{} +\beta _{n,m}G\bigl(v,T_{m}(v),T_{m}(v) \bigr)+\delta _{n,m}G(x_{n-1},v,v)\bigr]\biggr) \biggr) \\ ={}&F \biggl(\psi \biggl(\frac{1}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}\bigl[ \alpha _{n,m}G(x_{n-1},x_{n},x_{n}) \\ &{} +\beta _{n,m}G\bigl(v,T_{m}(v),T_{m}(v) \bigr)+\delta _{n,m}G(x_{n-1},v,v)\bigr]\biggr), \\ &{}\varphi \biggl(\frac{1}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}\bigl[\alpha _{n,m}G(x_{n-1},x_{n},x_{n}) \\ &{} +\beta _{n,m}G\bigl(v,T_{m}(v),T_{m}(v) \bigr)+\delta _{n,m}G(x_{n-1},v,v)\bigr]\biggr) \biggr) \end{aligned} $$

for every positive integer m. If we take the limit as $n\rightarrow \infty$ and use $(\mathit{CG}_{b}1)$, we have

$$ \begin{aligned} \psi \bigl(G\bigl(v,T_{m}(v),T_{m}(v) \bigr)\bigr)\precsim {}&F \biggl(\psi \biggl( \frac{1}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}\bigl[\alpha _{n,m}G(v,v,v) \\ &{} +\beta _{n,m}G\bigl(v,T_{m}(v),T_{m}(v) \bigr)+\delta _{n,m}G(v,v,v)\bigr]\biggr), \\ &{}\varphi \biggl(\frac{1}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}\bigl[\alpha _{n,m}G(v,v,v) \\ &{} +\beta _{n,m}G\bigl(v,T_{m}(v),T_{m}(v) \bigr)+\delta _{n,m}G(v,v,v)\bigr]\biggr) \biggr) \\ ={}&F \biggl(\psi \biggl( \frac{\beta _{n,m}}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}G\bigl(v,T_{m}(v),T_{m}(v) \bigr)\biggr), \\ &{}\varphi \biggl( \frac{\beta _{n,m}}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}G\bigl(v,T_{m}(v),T_{m}(v) \bigr)\biggr) \biggr), \end{aligned} $$

which implies

$$\begin{aligned}& \psi \biggl(\frac{\beta _{n,m}}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}G\bigl(v,T_{m}(v),T_{m}(v) \bigr)\biggr)=0 \quad \text{or} \\& \varphi \biggl( \frac{\beta _{n,m}}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}G \bigl(v,T_{m}(v),T_{m}(v)\bigr)\biggr)=0. \end{aligned}$$

That is, $\frac{\beta _{n,m}}{\alpha _{n,m}+\beta _{n,m}+\delta _{n,m}}G(v,T_{m}(v),T_{m}(v))=0$, we deduce that $T_{m}(v)=v$. Therefore, v is a common fixed point of $\{T_{m}\}$.

We now prove the uniqueness. Assume that u is a different common fixed point of $\{T_{m}\}$ where $u\neq v$. Then (3.1) gives the following result:

$$ \begin{aligned} \psi \bigl(G(v,u,u)\bigr)&=\psi \bigl(G \bigl(T_{m}(v),T_{m}(u),T_{m}(u)\bigr)\bigr) \\ &\precsim F \biggl(\psi \biggl( \frac{1}{\alpha _{m,m}+\beta _{m,m}+\delta _{m,m}} \\ &\quad{}\times \bigl[\alpha _{m,m}G \bigl(v,T_{m}(v),T_{m}(v)\bigr)+ \beta _{m,m}G \bigl(u,T_{m}(u),T_{m}(u)\bigr) \\ & \quad {}+\delta _{m,m}G(v,u,u)\bigr]\biggr), \\ & \quad {}\varphi \biggl(\frac{1}{\alpha _{m,m}+\beta _{m,m}+\delta _{m,m}} \\ &\quad{}\times\bigl[ \alpha _{m,m}G \bigl(v,T_{m}(v),T_{m}(v)\bigr)+\beta _{m,m}G \bigl(u,T_{m}(u),T_{m}(u)\bigr) \\ & \quad {}+\delta _{m,m}G(v,u,u)\bigr]\biggr) \biggr). \end{aligned} $$

By the limit as $m\rightarrow \infty$, we obtain

$$ \begin{aligned} & \psi \bigl(G(v,u,u)\bigr) \\ &\quad \precsim F \biggl(\psi \biggl( \frac{1}{\alpha _{m,m}+\beta _{m,m}+\delta _{m,m}}\bigl[\alpha _{m,m}G(v,v,v)+ \beta _{m,m}G(u,u,u)+\delta _{m,m}G(v,u,u)\bigr]\biggr), \\ &\quad\quad {}\varphi \biggl(\frac{1}{\alpha _{m,m}+\beta _{m,m}+\delta _{m,m}}\bigl[\alpha _{m,m}G(v,v,v)+ \beta _{m,m}G(u,u,u)+\delta _{m,m}G(v,u,u)\bigr]\biggr) \biggr) \\ &\quad =F \biggl(\psi \biggl( \frac{\beta _{m,m}}{\alpha _{m,m}+\beta _{m,m}+\delta _{m,m}}G(v,u,u)\biggr), \varphi \biggl( \frac{\beta _{m,m}}{\alpha _{m,m}+\beta _{m,m}+\delta _{m,m}}G(v,u,u)\biggr) \biggr), \end{aligned} $$

which implies $\psi (\frac{\beta _{m,m}}{\alpha _{m,m}+\beta _{m,m}+\delta _{m,m}}G(v,u,u))=0$ or $\varphi ( \frac{\beta {m,m}}{\alpha _{m,m}+\beta _{m,m}+\delta _{m,m}}G(v,u,u))=0$. As a result, we have $v=u$. This completes the proof. □

Taking $F(\mu ,\chi )=\mu \eta (\mu )$, where $\eta :[0,\infty )\rightarrow {}[ 0,1)$ is continuous function and $\mu \in S= \{ z\in \mathbb{C}:0\precsim z \}$ in Theorem 3.1, we have the following.

Corollary 3.2

Let$\{T_{n}\}$be a sequence of self-mappings of complex valued$G_{b}$-complete metric space$(X,G)$such that

$$ \begin{aligned}[b] & \psi \bigl(G\bigl(T_{i}(x),T_{j}(y),T_{j}(z) \bigr)\bigr) \\ &\quad \precsim \psi \biggl( \frac{1}{\alpha _{i,j}+\beta _{i,j}+\delta _{i,j}}\bigl[\alpha _{i,j}G\bigl(x,T_{i}(x),T_{i}(x)\bigr)+ \beta _{i,j}G\bigl(y,T_{j}(y),T_{j}(z)\bigr) \\ &\quad\quad {} +\delta _{i,j}G(x,y,z)\bigr]\biggr)\eta \biggl(\psi \biggl( \frac{1}{\alpha _{i,j}+\beta _{i,j}+\delta _{i,j}}\bigl[\alpha _{i,j}G\bigl(x,T_{i}(x),T_{i}(x) \bigr) \\ &\quad\quad {}+\beta _{i,j}G\bigl(y,T_{j}(y),T_{j}(z)\bigr)+ \delta _{i,j}G(x,y,z)\bigr]\biggr)\biggr) \end{aligned} $$

(3.4)

for$x,y,z\in X$with$x\neq y$, $0\leq \alpha _{i,j}, \beta _{i,j}, \delta _{i,j}$and$\alpha _{i,j}+\beta _{i,j}+\delta _{i,j}>0$, $i,j=1,2,\ldots$ , where$\eta :[0,\infty )\rightarrow {}[ 0,1)$is continuous and$\psi \in \varPsi$. Then$\{T_{n}\}$has a unique common fixed point in X.

Example 3.3

Consider the set $X=[-1,1]$. $(X,G)$ is a complex valued $G_{b}$-metric space [17] where $G:X\times X\times X\rightarrow \mathbb{C}$ is defined for all $u,v,w\in X$ as follows:

$$ G(u,v,w)=\bigl( \vert u-v \vert + \vert v-w \vert + \vert w-u \vert \bigr)^{2}. $$

Let $S=\{z\in \mathbb{C}:0\precsim z\}$. Define the following maps:

• $F:X\times X\rightarrow X$ with $F(\mu ,\chi )=\frac{\mu }{2}i$, where $\mu \in S$.

• $T_{n}(u)=u$ for all $n\in \mathbb{N}$ and $u\in X$.

• $\psi :S\rightarrow S$ with $\psi (\mu )=\mu$.

Then $\{T_{n}\}$ satisfies (3.1) for $u,v,w\in X$ with $u\neq v$, $0\leq \alpha _{i,j}, \beta _{i,j}, \delta _{i,j}$ and $\alpha _{i,j}+\beta _{i,j}+\delta _{i,j}>0$, where $i,j=1,2,\ldots$ . 0 is the unique common fixed point of $\{T_{n}\}$.

Let us define the $\alpha -(F,\psi ,\varphi )$-contractive self-mapping as a new concept in complex valued $G_{b}$-metric space.

Definition 3.4

Let $(X,G)$ be a complex valued $G_{b}$-metric space. A mapping $T:X\rightarrow X$ is called $\alpha -(F,\psi ,\varphi )$-contractive mapping of type A if there exist functions $\alpha :X\times X\times X\rightarrow {}[ 0,\infty )$, $F\in C$, $\psi \in \varPsi$ (which has a property such that $\lim_{n\rightarrow \infty }\psi ^{n}(t)=0$ for $t\in \mathbb{C}$) and $\varphi \in \varPhi _{u}$ such that

$$ \begin{aligned}[b] & \alpha (x,y,Tx)\psi \bigl(G\bigl(Tx,Ty,T^{2}x\bigr)\bigr) \\ &\quad \precsim F \bigl(\psi \bigl(G(x,y,Tx)\bigr), \varphi \bigl(G(x,y,Tx)\bigr)\bigr) \quad \text{for all } x,y,z\in X. \end{aligned} $$

(3.5)

Definition 3.5

([18])

Let $(X,G)$ be a complex valued $G_{b}$-metric space and $\alpha :X\times X\times X\rightarrow [0,\infty )$ be a given mapping. A mapping $T:X\rightarrow X$ is said to be α-admissible if $x,y\in X$, $\alpha (x,y,z)\geq 1$ implies $\alpha (Tx,Ty,Tz)\geq 1$.

Theorem 3.6

Suppose that$(X,G)$is a complex valued$G_{b}$-complete metric space. Let$T:X\rightarrow X$be an$\alpha -(F,\psi ,\varphi )$-contractive mapping of typeAand satisfy the following:

• (i) Tisα-admissible,

• (ii) there is an element$x_{0}$inXsuch that$\alpha (x_{0},Tx_{0},Tx_{0})\geq 1$,

• (iii) if a sequence$\{x_{n}\}$inXsatisfies$\alpha (x_{n},x_{n+1},x_{n+1})\geq 1$for allnand$x_{n}\rightarrow x\in X$as$n\rightarrow \infty$, then$\alpha (x_{n},x,x_{n+1})\geq 1$for all n.

T has a unique fixed point if there is an element$z\in X$such that$\alpha (x,z,z)\geq 1$and$\alpha (y,z,z)\geq 1$for all$x,y\in X$.

Proof

Assume that there is an element $x_{0}\in X$ such that $\alpha (x_{0},Tx_{0},Tx_{0})\geq 1$. Consider a sequence $\{x_{n}\}$ in X defined as $x_{n+1}=Tx_{n}$. If $x_{n}=x_{n+1}$ for some $n\in \mathbb{N}$, since $x_{n}$ is a fixed point for T, we assume that $x_{n}\neq x_{n+1}$ for all $n\in \mathbb{N}$.

Using $(i)$, we obtain $\alpha (x_{0},x_{1},x_{1})=\alpha (x_{0},Tx_{0},Tx_{0})\geq 1$ implies that $\alpha (Tx_{0},Tx_{1},Tx_{1})=\alpha (x_{1},x_{2},x_{2})\geq 1$. From induction

$$ \alpha (x_{n},x_{n+1},x_{n+1})\geq 1 \quad \text{for all } n\in \mathbb{N}. $$

(3.6)

Since

$$ G(x_{n},x_{n+1},x_{n+1})=G(Tx_{n-1},Tx_{n},Tx_{n})=G \bigl(Tx_{n-1},Tx_{n},T^{2}x_{n-1}\bigr) $$

and by Definition 3.4, we get

$$\begin{aligned} \psi \bigl(G(x_{n},x_{n+1},x_{n+1}) \bigr)&\precsim \alpha (x_{n-1},x_{n},x_{n}) \psi \bigl(G(x_{n},x_{n+1},x_{n+1})\bigr) \\ &=\alpha (x_{n-1},x_{n},x_{n})\psi \bigl(G \bigl(Tx_{n-1},Tx_{n},T^{2}x_{n-1}\bigr) \bigr) \\ &\precsim F\bigl(\psi \bigl(G(x_{n-1},x_{n},x_{n}) \bigr), \varphi \bigl(G(x_{n-1},x_{n},x_{n})\bigr) \bigr) \\ &\precsim \psi \bigl(G(x_{n-1},x_{n},x_{n})\bigr). \end{aligned}$$

Therefore we get $\psi (G(x_{n},x_{n+1},x_{n+1}))\precsim \psi (G(x_{n-1},x_{n},x_{n}))$ as $\alpha (x_{n-1},x_{n},x_{n})\geq 1$. Since ψ is non-decreasing, we conclude

$$ G(x_{n},x_{n+1},x_{n+1})\precsim G(x_{n-1},x_{n},x_{n}) \quad \text{for all } n\geq 1. $$

(3.7)

Hence $\{G(x_{n},x_{n+1},x_{n+1})\}$ is the decreasing sequence in $\mathbb{C}$ and so it is $G_{b}$-convergent to $0\preceq \chi \in \mathbb{C}$.

We will show that $\chi =0$. Suppose, to the contrary, that $\chi \succ 0$. The limit case in (3.2) shows that

$$ \psi (\chi )\precsim F \bigl(\psi (\chi ),\varphi (\chi ) \bigr), $$

which implies $\psi (\chi )=0$ or $\varphi (\chi )=0$. Namely, $\chi =0$. But it is a contradiction. Thus, $\chi =0$. i.e.,

$$ \lim_{n\rightarrow \infty }G(x_{n},x_{n+1},x_{n+1})=0. $$

(3.8)

By $(\mathit{CG}_{b}5)$ and (3.7), we get

$$ \begin{aligned} G(x_{n},x_{p},x_{p})& \precsim s\bigl[G(x_{n},x_{n+1},x_{n+1}) \bigr]+s^{2}\bigl[G(x_{n+1},x_{n+2},x_{n+2}) \bigr]+ \cdots +s^{p-n}\bigl[G(x_{p-1},x_{p},x_{p}) \bigr] \\ &\precsim \sum_{k=n}^{p-1}s^{k-n+1}G(x_{k},x_{k+1},x_{k+1}) \\ & \quad {}\vdots \\ &\precsim \sum_{k=n}^{p-1}s^{k-n+1}G(x_{0},x_{1},x_{1}) \end{aligned} $$

and consequently from (3.8)

$$ \lim_{n,p\rightarrow \infty } \bigl\vert G(x_{n},x_{p},x_{p}) \bigr\vert =0. $$

The sequence $\{x_{n}\}$ is a complex valued $G_{b}$-Cauchy. Since $(X,G)$ is a complex valued $G_{b}$-complete, there is an element $\upsilon ^{\ast }\in X$ such that $x_{n}\rightarrow \upsilon ^{\ast }$ as $n\rightarrow \infty$. Considering (3.6) and (iii), then we obtain

$$ \alpha \bigl(x_{n},\upsilon ^{\ast },\upsilon ^{\ast } \bigr)\geq 1 \quad \text{for all } n\geq 0. $$

(3.9)

By (3.5) and (3.9), we find

$$\begin{aligned} G\bigl(x_{n+1},T\upsilon ^{*},x_{n+2}\bigr)&=G\bigl(Tx_{n},T\upsilon ^{*},T^{2}x_{n}\bigr) \\ &\precsim \alpha \bigl(x_{n},\upsilon ^{*},x_{n+1} \bigr)G\bigl(Tx_{n},T\upsilon ^{*},T^{2}x_{n} \bigr) \\ &\precsim F\bigl(\psi \bigl(G\bigl(x_{n},\upsilon ^{*},x_{n+1} \bigr)\bigr),\varphi \bigl(G\bigl(x_{n}, \upsilon ^{*},x_{n+1} \bigr)\bigr)\bigr) \\ &\precsim \psi \bigl(G\bigl(x_{n},\upsilon ^{*},x_{n+1} \bigr)\bigr). \end{aligned}$$

Taking the limit,

$$ G\bigl(\upsilon ^{*},T\upsilon ^{*},\upsilon ^{*}\bigr)\precsim \psi \bigl(G\bigl( \upsilon ^{*}, \upsilon ^{*},\upsilon ^{*}\bigr)\bigr). $$

From Theorem 2.3 and $(\mathit{CG}_{b}1)$, we have

$$ \lim_{n\rightarrow \infty } \bigl\vert G\bigl(\upsilon ^{\ast },T \upsilon ^{ \ast },\upsilon ^{\ast }\bigr) \bigr\vert =0 $$

as ψ is continuous at $\chi =0$. As a result, $\upsilon ^{\ast }=T\upsilon ^{\ast }$.

To complete the proof, we show the uniqueness. Suppose that $\vartheta ^{*}\neq \upsilon ^{*}$ is another fixed point of T. Then there is a point $z\in X$ such that $\alpha (\upsilon ^{*},\upsilon ^{*},z)\geq 1$ and $\alpha (\vartheta ^{*},\vartheta ^{*},z)\geq 1$. By induction and (i), we have

$$ \alpha \bigl(\upsilon ^{*},\upsilon ^{*},T^{n}z\bigr)\geq 1 \quad \text{and} \quad \alpha \bigl( \vartheta ^{*},\vartheta ^{*},T^{n}z\bigr)\geq 1 $$

(3.10)

for all $n=1,2,\ldots$ . Equations (3.5) and (3.10) give the following result:

$$ \begin{aligned} G\bigl(\upsilon ^{*},T^{n}z, \upsilon ^{*}\bigr)&=G\bigl(T\upsilon ^{*},T \bigl(T^{n-1}z\bigr),T^{2} \upsilon ^{*}\bigr) \\ &\precsim \alpha \bigl(\upsilon ^{*},T^{n-1}z,T\upsilon ^{*}\bigr)G\bigl(T\upsilon ^{*},T\bigl(T^{n-1}z \bigr),T^{2} \upsilon ^{*}\bigr) \\ &\precsim F\bigl(\psi \bigl(G\bigl(\upsilon ^{*},T^{n-1}z,T \upsilon ^{*}\bigr)\bigr),\varphi \bigl(G\bigl( \upsilon ^{*},T^{n-1}z,T\upsilon ^{*}\bigr)\bigr)\bigr) \\ &\precsim \psi \bigl(G\bigl(\upsilon ^{*},T^{n-1}z,T\upsilon ^{*}\bigr)\bigr) \\ &=\psi \bigl(G\bigl(\upsilon ^{*},T^{n-1}z,\upsilon ^{*}\bigr)\bigr). \end{aligned} $$

Using induction, we get

$$ G\bigl(\upsilon ^{*},T^{n}z,\upsilon ^{*}\bigr) \precsim \psi ^{n}\bigl(G\bigl(\upsilon ^{*},z, \upsilon ^{*}\bigr)\bigr) $$

for all natural numbers n. From $(\mathit{CG}_{b}4)$, we obtain $G(\upsilon ^{*},\upsilon ^{*},T^{n}z)\precsim \psi ^{n}(G(\upsilon ^{*}, \upsilon ^{*},z))$. Taking the limit, we observe

$$ \bigl\vert G\bigl(\upsilon ^{*},\upsilon ^{*},T^{n}z \bigr) \bigr\vert =0. $$

So $\{T^{n}z\}$ is $G_{b}$-convergent to $\upsilon ^{*}$. It can be observed that $\{T^{n}z\}$ is $G_{b}$-convergent to $\vartheta ^{*}$. The uniqueness of the limit gives $\upsilon ^{*}=\vartheta ^{*}$. As a result, T has a unique fixed point. □

4 Conclusions

We have introduced a generalized fixed point theorem using the complex C-class function as a new tool in complex valued $G_{b}$-metric spaces. Moreover, we have defined an $\alpha -(F,\psi ,\varphi )$-contractive type and α-admissible mapping. Then we have proved a fixed point theorem using these notions and the complex C-class function. The obtained results generalize some facts in the literature.