Fixed Point Results for Perov–Ćirić–Prešić-Type Θ-Contractions with Applications

Abstract

The aim of this paper is to introduce the notion of Perov–Ćirić–Prešić-type $\Theta$-contractions and to obtain some generalized fixed point theorems in the setting of vector-valued metric spaces. We derive some fixed point results as consequences of our main results. A nontrivial example is also provided to support the validity of our established results. As an application, we investigate the solution of a semilinear operator system in Banach space.

1. Introduction

In fixed point theory, the Banach contraction principle [1] plays a key and essential role that ensures the existence and uniqueness of a fixed point in the background of a complete metric space (CMS). Due to its actuality and convenience, various researchers have developed many interesting accessories and augmentations of this principle. Ćirić [2] gave a generalized contractive condition on a complete metric space and established a fixed point theorem as a generalization of the Banach contraction principle. In 1965, Prešić [3] generalized the prominent principle and applied the derived results to confirm the convergence of a particular type of sequence. Later on, Ćirić and Prešić [4] presented a new result and generalized the main result of Prešić [3]. For more details, we refer the reader to the literature (see [5,6,7,8,9,10,11,12,13,14]).

On the other hand, Perov [15] introduced the notion of a vector-valued metric space (VVMS) in 1964 and generalized the Banach contraction principle by replacing the contractive factor with a matrix convergent to zero. Then, numerous studies with these features were established (see [16,17,18,19,20,21,22]).

In 2012, Samet and Jleli [23] introduced a new contraction named a $\Theta$-contraction and generalized the celebrated Banach contraction principle. Recently, Altun et al. [24] defined the notion of a Perov-type $\Theta$-contraction and proved some fixed point theorems to generalize the main result of Perov [15], as well as the result of Jleli and Samet [23].

In this paper, we define the notion of the Perov–Ćirić–Prešić-type $\Theta$-contraction in the framework of vector-valued metric spaces to obtain some generalized fixed point results. To support the authenticity of our main result, we provide a nontrivial example and the solution of a semilinear operator system in Banach space, as an application.

2. Preliminaries

In fixed point theory, Banach’s contraction principle (BCP) [1] is one of the important results. It states that if $(\mathcal{S},\sigma )$ is a complete metric space, $\mathcal{L}:(\mathcal{S},\sigma )\to (\mathcal{S},\sigma ),$ and there exists $\lambda \in [0,1)$ such that

$$\sigma (\mathcal{L}\varrho ,\mathcal{L}\omega )\le \lambda \sigma (\varrho ,\omega )$$

(1)

for all $\varrho ,\omega \in \mathcal{S}$, then there exists a unique point ${\varrho }^{*}\in \mathcal{S}$ such that ${\varrho }^{*}=$$\mathcal{L}{\varrho }^{*}$.

In 1965, Prešić [3] gave the following result as a generalization of this famous principle.

Theorem 1

(see [3]). Let $(\mathcal{S},\sigma )$ be a CMS and k a positive integer. If $\mathcal{L}:{\mathcal{S}}^k\to \mathcal{S}$ is a mapping such that

$$\begin{array}{ccc}& & \sigma (\mathcal{L}({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_k),\mathcal{L}({\varrho }_2,\dots ,{\varrho }_k,{\varrho }_{k+1}))\hfill \\ & \le & {\lambda }_1\sigma ({\varrho }_1,{\varrho }_2)+{\lambda }_2\sigma ({\varrho }_2,{\varrho }_3)+\dots +{\lambda }_k\sigma ({\varrho }_k,{\varrho }_{k+1})\hfill \end{array}$$

(2)

for all ${\varrho }_1,\dots ,{\varrho }_{k+1}\in \mathcal{S}$, where ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_k$ are non-negative constants such that ${\lambda }_1+{\lambda }_2+\dots +{\lambda }_k<1$, then there exists a unique point ${\varrho }^{*}\in$ $\mathcal{S}$ such that $\mathcal{L}({\varrho }^{*},\dots ,{\varrho }^{*})={\varrho }^{*}$. Moreover if ${\varrho }_1,\dots ,{\varrho }_k$ are arbitrary points in $\mathcal{S}$ and for $n\in \mathbb{N},$

$${\varrho }_{n+k}=\mathcal{L}({\varrho }_n,{\varrho }_{n+1},\dots ,{\varrho }_{n+k-1})$$

(3)

then the sequence $\left\{{\varrho }_n\right\}$ is convergent and ${\varrho }^{*}=lim{\varrho }_n=\mathcal{L}(lim{\varrho }_n,lim{\varrho }_n,...,lim{\varrho }_n).$

A mapping $\mathcal{L}:{\mathcal{S}}^k\to \mathcal{S}$ satisfying the inequality (2) is said to be a Prešić operator. A point ${\varrho }^{*}\in \mathcal{S}$ is called a fixed point of $\mathcal{L}$ if ${\varrho }^{*}=\mathcal{L}({\varrho }^{*},\dots ,{\varrho }^{*})$. If $k=1$ in Theorem 1, then we obtain the BCP as a specific case. From now onward, k is a positive integer.

Ćirić and Prešić [4] established the following theorem and generalized the above result.

Theorem 2

(see [4]). Let $(\mathcal{S},\sigma )$ be a complete metric space. If $\mathcal{L}:{\mathcal{S}}^k\to \mathcal{S}$ satisfies

$$\begin{array}{ccc}& & \sigma (\mathcal{L}({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_k),\mathcal{L}({\varrho }_2,\dots ,{\varrho }_k,{\varrho }_{k+1}))\hfill \\ & \le & \lambda max\{\sigma ({\varrho }_1,{\varrho }_2),\sigma ({\varrho }_2,{\varrho }_3),\dots ,\sigma ({\varrho }_k,{\varrho }_{k+1})\}\hfill \end{array}$$

(4)

for any ${\varrho }_1,\dots ,{\varrho }_{k+1}\in \mathcal{S}$, where $0<\lambda <1$, then there exists ${\varrho }^{*}\in \mathcal{S}$ such that $\mathcal{L}({\varrho }^{*},\dots ,{\varrho }^{*})={\varrho }^{*}$. Moreover, for any arbitrary points ${\varrho }_1,\dots ,{\varrho }_k\in \mathcal{S}$, the sequence given in (3) is convergent and

$$\underset{n\to \infty }{lim}{\varrho }_n=\mathcal{L}(\underset{n\to \infty }{lim}{\varrho }_n,\dots ,\underset{n\to \infty }{lim}{\varrho }_n).$$

If, in addition,

$$\sigma (\mathcal{L}({\varrho }^{*},\dots ,{\varrho }^{*}),\mathcal{L}({\varrho }^{/},\dots ,{\varrho }^{/}))<\sigma ({\varrho }^{*},{\varrho }^{/})$$

holds for all ${\varrho }^{*},{\varrho }^{/}\in \mathcal{S}$, with ${\varrho }^{*}\ne {\varrho }^{/}$, then the fixed point in $\mathcal{S}$ is unique.

On the other hand, we use the following notions from [25]:

(i)

$\mathbf{0}$ as the zero matrix of order $m\times 1$;

(ii)

${\mathbb{R}}_{+}^m$, the set of $m\times 1$ real matrices with positive components in $[0,\infty );$

(iii)

${\mathcal{M}}_m^m\left({\mathbb{R}}_{+}\right)$, the set of all $m\times m$ matrices with positive components in $[0,\infty );$

(iv)

⊝ as the zero matrix of order $m\times m$;

(v)

I as the identity matrix of order $m\times m$;

(vi)

$\mathbb{C}$ as the set of complex numbers.

If $\Xi \in$${\mathcal{M}}_m^m\left({\mathbb{R}}_{+}\right)$, then ${\Xi }^T$ denotes the transpose of the matrix $\Xi$. Let $\wp ={\left({\wp }_i\right)}_{i=1}^m,\hslash ={\left({\hslash }_i\right)}_{i=1}^m\in {\mathbb{R}}^m,$ then

$$\wp ⪯\hslash \iff {\wp }_i\le {\hslash }_i,\mathrm{for}\mathrm{all}i\in \{1,2,\cdots ,m\}.$$

In the whole paper, $\wp ⪯\hslash$ and $\hslash ⪰\wp$ will have same meaning. In addition, $\wp \prec \hslash$ implies ${\wp }_i<{\hslash }_i$ for each $i\in \{1,2,\cdots ,m\}$, where ⪯ is the coordinate-wise ordering on ${\mathbb{R}}^m$ (see [24,25,26,27]).

Definition 1

(see [15]). Let $\mathcal{S}\ne \varnothing$ and $\sigma :\mathcal{S}\times \mathcal{S}\to {\mathbb{R}}^m$ be a function satisfying

(i) 

$\sigma (\varrho ,\omega )=\mathbf{0}$ if and only if $\varrho =\omega ;$

(ii) 

$\sigma (\varrho ,\omega )=\sigma (\omega ,\varrho );$

(iii) 

$\sigma (\varrho ,\omega )⪯\sigma (\varrho ,z)+\sigma (z,\omega )$

for all $\varrho ,\omega ,z\in \mathcal{S},$ then ($\mathcal{S},\sigma )$ is said to be vector-valued metric space (VVMS).

Let $\Xi \in {\mathcal{M}}_m^m\left({\mathbb{R}}_{+}\right),$ then Ξ$⟶0$⟺${\Xi }^n\to ⊝$ as $n\to \infty$ (See [25]).

Theorem 3

(see [25]). Let $\Xi \in {\mathcal{M}}_m^m\left({\mathbb{R}}_{+}\right)$. Then, these statements are equivalent:

1. 

Ξ$⟶0$ as $n⟶\infty$;

2. 

for $\left|\lambda \right|<1$ for every $\lambda \in \mathbb{C}$ with $det(\Xi -\lambda I)=0$;

3. 

$det(I-\Xi )\ne 0$ and

$${(I-\Xi )}^{-1}=I+\Xi +\cdots +{\Xi }^n+\cdots .$$

Perov [15] presented the following result in vector-valued metric spaces in this way.

Theorem 4

(see [15]). Let $(\mathcal{S},\sigma )$ be a complete VVMS and $\mathcal{L}:\mathcal{S}\to \mathcal{S}$ be a Perov contraction, that is, a mapping with the property that there exists $\Xi \in {\mathcal{M}}_m^m\left({\mathbb{R}}_{+}\right)$, which converges to zero such that

$$\sigma (\mathcal{L}\varrho ,\mathcal{L}\omega )⪯\Xi \sigma (\varrho ,\omega ).$$

Then, there exists a unique point $\varrho \in \mathcal{S}$ such that $\mathcal{L}\varrho =\varrho$. Moreover, the sequence $\left\{{\varrho }_n\right\}$ defined by ${\varrho }_n={\mathcal{L}}^n{\varrho }_0$ converges to ϱ.

In 2012, Jleli and Samet [23] gave the notion of $\Theta$-contractions as follows.

Definition 2.

We designate by Ω the class of all mappings $\Theta :{\mathbb{R}}^{+}\to (1,\infty )$ satisfying these properties:

(${\Theta }_1$) 

$0<{\varrho }_1<{\varrho }_2$$⟹\Theta \left({\varrho }_1\right)\le \Theta \left({\varrho }_2\right);$

(${\Theta }_2$) 

for $\left\{{\varrho }_n\right\}\subseteq {\mathbb{R}}^{+}$, ${lim}_{n\to \infty }\Theta \left({\varrho }_n\right)=1$ if and only if ${lim}_{n\to \infty }\left({\varrho }_n\right)=0;$

(${\Theta }_3$) 

there exists $h\in (0,1)$ and $\mathfrak{q}\in (0,\infty ]$ such that ${lim}_{\varrho \to 0^{+}}\frac{\Theta \left(\varrho \right)-1}{{\varrho }^h}=\mathfrak{q}.$

Recently, Altun et al. [24] considered the technique of Jleli and Samet [23] in vector-valued metric spaces and defined the notion of a Perov-type $\Theta$-contraction as follows.

Definition 3

(Altun et al. [24]). Let $\Theta :{\mathbb{R}}_{+0}^m\to {\mathbb{R}}_{+1}^m$, where ${\mathbb{R}}_{+j}^m$ is the set of $m\times 1$ real matrices with every element greater than j. Suppose that

(${\Theta }_1^{*}$) 

for all $\wp ={\left({\wp }_i\right)}_{i=1}^m,\hslash ={\left({\hslash }_i\right)}_{i=1}^m\in {\mathbb{R}}_{+0}^m$ such that $\wp ⪯\hslash ,$ then $\Theta (\wp )⪯\Theta (\hslash ),$

(${\Theta }_2^{*}$) 

for $\left\{{\wp }_n\right\}=({\wp }_n^{\left(1\right)},{\wp }_n^{\left(2\right)},\cdots ,{\wp }_n^{\left(m\right)})$ of ${\mathbb{R}}_{+0}^m$

$$\underset{n\to \infty }{lim}{\wp }_n^{\left(i\right)}=0^{+}⟺\underset{n\to \infty }{lim}{\hslash }_n^{\left(i\right)}=1$$

for $i\in \{1,2,\cdots m\}$, where

$$\Theta \left(({\wp }_n^{\left(1\right)},{\wp }_n^{\left(2\right)},\cdots ,{\wp }_n^{\left(m\right)})\right)=({\hslash }_n^{\left(1\right)},{\hslash }_n^{\left(2\right)},\cdots ,{\hslash }_n^{\left(m\right)}),$$

(${\Theta }_3^{*}$) 

there exist $h\in (0,1)$ and $\mathfrak{q}\in (0,\infty ]$ such that ${lim}_{{\wp }_i\to 0^{+}}\frac{{\hslash }_i-1}{{\wp }_i^h}=\mathfrak{q},$ for $i\in \{1,2,\cdots m\},$ where

$$\Theta \left(({\wp }_1,{\wp }_2,\cdots ,{\wp }_m)\right)=({\hslash }_1,{\hslash }_2,\cdots ,{\hslash }_m).$$

We represent by ${\Xi }^m$ the family of all mappings $\Theta$ fulfilling (${\Theta }_1^{*}$)–(${\Theta }_3^{*}$).

Example 1.

Define $\Theta :{\mathbb{R}}_{+0}^m\to {\mathbb{R}}_{+1}^m$ by

$$\Theta \left(({\wp }_1,{\wp }_2,\cdots ,{\wp }_m)\right)=(exp\sqrt{{\wp }_1},exp\sqrt{{\wp }_2},\cdots ,exp\sqrt{{\wp }_m}),$$

then $\Theta \in {\Xi }^m$.

Let us use the notation ${\Lambda }^{\left[\lambda \right]}:={\left({\Lambda }_i^{{\lambda }_i}\right)}_{i=1}^m$ for $\Lambda ={\left({\Lambda }_i\right)}_{i=1}^m\in {\mathbb{R}}_{+}^m$ and $\lambda ={\left({\lambda }_i\right)}_{i=1}^m\in {\mathbb{R}}_{+}^m$.

By considering the class ${\Xi }^m$, Altun et al. [24] introduced the notion of the Perov-type $\Theta$-contraction in this way.

Definition 4

(Altun et al. [24]). Let $(\mathcal{S},\sigma )$ be a VVMS and $\mathcal{L}:\mathcal{S}\to \mathcal{S}$. If there exist $\Theta \in {\Xi }^m$ and $\lambda ={\left({\lambda }_i\right)}_{i=1}^m\in {\mathbb{R}}_{+}^m$ with ${\lambda }_i<1,$ ∀$i\in \{1,2,\cdots ,m\}$ such that

$$\Theta \left(\sigma (\mathcal{L}\varrho ,\mathcal{L}\omega )\right)⪯{\left[\Theta \left(\sigma \right(\varrho ,\omega \left)\right)\right]}^{\left[\lambda \right]},$$

(5)

for all $\varrho ,\omega \in \mathcal{S}$ with $\sigma (\mathcal{L}\varrho ,\mathcal{L}\omega )\succ \mathbf{0},$ then $\mathcal{L}$ is said to be a Perov-type Θ-contraction.

Altun et al. [24] proved the following result.

Theorem 5

(Altun et al. [24]). Let $(\mathcal{S},\sigma )$ be a complete VVMS and $\mathcal{L}:\mathcal{S}\to \mathcal{S}$ be a Perov-type Θ-contraction, then $\mathcal{L}$ has a unique fixed point.

3. Main Result

We define a Perov–Ćirić–Prešić-type $\Theta$-contraction in vector-valued metric spaces as follows:

Definition 5.

Let $(\mathcal{S},\sigma )$ be a vector-valued metric space and let $\mathcal{L}:{\mathcal{S}}^k\to \mathcal{S}$ (k is a positive integer). If there exist $\Theta \in {\Xi }^m$ and $\lambda ={\left({\lambda }_i\right)}_{i=1}^m\in {\mathbb{R}}_{+}^m$ with ${\lambda }_i<1,$ for $i\in \{1,2,\cdots ,m\}$ such that

$$\Theta \left(\sigma (\mathcal{L}({\varrho }_1,{\varrho }_2,...,{\varrho }_k),\mathcal{L}({\varrho }_2,...,{\varrho }_{k+1}))\right)⪯{\left[\Theta \left(sup\left\{\sigma ({\varrho }_1,{\varrho }_2),\sigma ({\varrho }_2,{\varrho }_3),...,\sigma ({\varrho }_k,{\varrho }_{k+1})\right\}\right)\right]}^{\left[\lambda \right]}$$

(6)

for all ${\varrho }_1,{\varrho }_2,...,{\varrho }_{k+1}\in \mathcal{S}$ with $\sigma (\mathcal{L}({\varrho }_1,{\varrho }_2,...,{\varrho }_k),\mathcal{L}({\varrho }_2,...,{\varrho }_{k+1}))\succ \mathbf{0},$ then $\mathcal{L}$ is called a Perov–Ćirić–Prešić-type Θ-contraction.

Theorem 6.

Let $(\mathcal{S},\sigma )$ be a complete VVMS and $\mathcal{L}:{\mathcal{S}}^k\to \mathcal{S}$ be a Perov–Ćirić–Prešić-type Θ-contraction, then $\mathcal{L}$ has a unique fixed point. Furthermore, suppose there exists a sequence $\left\{{\varrho }_n\right\}$ in $\mathcal{S}$ such that ${\varrho }_{n+k}=\mathcal{L}({\varrho }_n,{\varrho }_{n+1}...,{\varrho }_{n+k+1})$ and $\sigma ({\varrho }_{n+k},{\varrho }_{n+k+1})\succ \mathbf{0},$ for all $n\in \mathbb{N}$. In addition, if ${\varrho }_n\to \varrho ,$ with $\sigma ({\varrho }_n,\varrho )\succ \mathbf{0},$ for all $n\in \mathbb{N},$ then the sequence $\left\{{\varrho }_n\right\}$ converges to a fixed point of $\mathcal{L}$. Furthermore, if

$$\sigma \left(\mathcal{L}\right(\varrho ,\varrho ,...,\varrho ),\mathcal{L}(\omega ,...,\omega \left)\right)<\sigma (\varrho ,\omega )$$

for all $\varrho ,\omega \in \mathcal{S}$ with $\varrho \ne \omega ,$ then the fixed point of $\mathcal{L}$ is unique.

Proof. 

For any $n\in \mathbb{N},$ we obtain

$$\begin{array}{ccc}\hfill \Theta \left(\sigma ({\varrho }_{n+k},{\varrho }_{n+k+1})\right)& =& \Theta \left(\sigma (\mathcal{L}({\varrho }_n,...,{\varrho }_{n+k-1}),\mathcal{L}({\varrho }_{n+1},...,{\varrho }_{n+k}))\right)\hfill \\ & \le & {\left[\Theta \left(sup\left\{\sigma ({\varrho }_n,{\varrho }_{n+1}),\sigma ({\varrho }_{n+1},{\varrho }_{n+2}),...,\sigma ({\varrho }_{n+k-1},{\varrho }_{n+k})\right\}\right)\right]}^{\left[\lambda \right]}.\hfill \end{array}$$

(7)

Therefore,

$$\begin{array}{ccc}\hfill \Theta \left(\sigma ({\varrho }_{k+1},{\varrho }_{k+2})\right)& =& \Theta \left(\sigma (\mathcal{L}({\varrho }_1,...,{\varrho }_k),\mathcal{L}({\varrho }_2,...,{\varrho }_{k+1}))\right)\hfill \\ & \le & {\left[\Theta \left(sup\left\{\sigma ({\varrho }_1,{\varrho }_2),\sigma ({\varrho }_2,{\varrho }_3),...,\sigma ({\varrho }_k,{\varrho }_{k+1})\right\}\right)\right]}^{\left[\lambda \right]}\hfill \\ & =& {\left[\Theta \left(M\right)\right]}^{\left[\lambda \right]}\hfill \end{array}$$

(8)

where $M=sup\left\{\sigma ({\varrho }_1,{\varrho }_2),\sigma ({\varrho }_2,{\varrho }_3),...,\sigma ({\varrho }_k,{\varrho }_{k+1})\right\}.$ Now,

$$\begin{array}{ccc}\hfill \Theta \left(\sigma ({\varrho }_{k+2},{\varrho }_{k+3})\right)& =& \Theta \left(\sigma (\mathcal{L}({\varrho }_2,...,{\varrho }_{k+1}),\mathcal{L}({\varrho }_3,...,{\varrho }_{k+2}))\right)\hfill \\ & \le & {\left[\Theta \left(sup\left\{\sigma ({\varrho }_2,{\varrho }_3),\sigma ({\varrho }_3,{\varrho }_4),...,\sigma ({\varrho }_{k+1},{\varrho }_{k+2})\right\}\right)\right]}^{\left[\lambda \right]}\hfill \\ & \le & {\left[\Theta \left(max\left\{M,{\Theta }^{-1}{\left[\Theta \left(M\right)\right]}^{\left[\lambda \right]}\right\}\right)\right]}^{\left[\lambda \right]}\hfill \\ & =& {\left[\Theta \left(M\right)\right]}^{\left[\lambda \right]}.\hfill \end{array}$$

(9)

Continuing this approach, we have

$$\begin{array}{ccc}\hfill \Theta \left(\sigma ({\varrho }_{2k},{\varrho }_{2k+1})\right)& =& \Theta \left(\sigma (\mathcal{L}({\varrho }_k,...,{\varrho }_{2k-1}),\mathcal{L}({\varrho }_{k+1},...,{\varrho }_{2k}))\right)\hfill \\ & \le & {\left[\Theta \left(sup\left\{\sigma ({\varrho }_k,{\varrho }_{k+1}),\sigma ({\varrho }_{k+1},{\varrho }_{k+2}),...,\sigma ({\varrho }_{2k-1},{\varrho }_{2k})\right\}\right)\right]}^{\left[\lambda \right]}\hfill \\ & \le & {\left[\Theta \left(max\left\{M,{\Theta }^{-1}{\left[\Theta \left(M\right)\right]}^{\left[\lambda \right]}\right\}\right)\right]}^{\left[\lambda \right]}\hfill \\ & =& {\left[\Theta \left(M\right)\right]}^{\left[\lambda \right]}\hfill \end{array}$$

(10)

and

$$\begin{array}{ccc}\hfill \Theta \left(\sigma ({\varrho }_{2k+1},{\varrho }_{2k+2})\right)& =& \Theta \left(\sigma (\mathcal{L}({\varrho }_{k+1},...,{\varrho }_{2k}),\mathcal{L}({\varrho }_{k+2},...,{\varrho }_{2k+1}))\right)\hfill \\ & \le & {\left[\Theta \left(sup\left\{\sigma ({\varrho }_{k+1},{\varrho }_{k+2}),\sigma ({\varrho }_{k+2},{\varrho }_{k+3}),...,\sigma ({\varrho }_{2k},{\varrho }_{2k+1})\right\}\right)\right]}^{\left[\lambda \right]}\hfill \\ & \le & {\left[\Theta \left({\Theta }^{-1}{\left[\Theta \left(M\right)\right]}^{\left[\lambda \right]}\right)\right]}^{\left[\lambda \right]}\hfill \\ & =& {\left[\Theta \left(M\right)\right]}^{\left[{\lambda }^2\right]}\hfill \end{array}$$

(11)

⋯

$$\begin{array}{ccc}\hfill \Theta \left(\sigma ({\varrho }_{3k},{\varrho }_{3k+1})\right)& =& \Theta \left(\sigma (\mathcal{L}({\varrho }_{2k},...,{\varrho }_{3k-1}),\mathcal{L}({\varrho }_{2k+1},...,{\varrho }_{3k}))\right)\hfill \\ & \le & {\left[\Theta \left(sup\left\{\sigma ({\varrho }_{2k},{\varrho }_{2k+1}),\sigma ({\varrho }_{2k+1},{\varrho }_{2k+2}),...,\sigma ({\varrho }_{3k-1},{\varrho }_{3k})\right\}\right)\right]}^{\left[\lambda \right]}\hfill \\ & \le & {\left[\Theta \left(max\left\{{\Theta }^{-1}{\left[\Theta \left(M\right)\right]}^{\left[\lambda \right]},{\Theta }^{-1}{\left[\Theta \left(M\right)\right]}^{\left[{\lambda }^2\right]}\right\}\right)\right]}^{\left[\lambda \right]}\hfill \\ & =& {\left[\Theta \left(M\right)\right]}^{\left[{\lambda }^2\right]}\hfill \end{array}$$

(12)

and

$$\begin{array}{ccc}\hfill \Theta \left(\sigma ({\varrho }_{3k+1},{\varrho }_{3k+2})\right)& =& \Theta \left(\sigma (\mathcal{L}({\varrho }_{2k+1},...,{\varrho }_{3k}),\mathcal{L}({\varrho }_{2k+2},...,{\varrho }_{3k+1}))\right)\hfill \\ & \le & {\left[\Theta \left(sup\left\{\sigma ({\varrho }_{2k+1},{\varrho }_{2k+2}),\sigma ({\varrho }_{2k+2},{\varrho }_{2k+3}),...,\sigma ({\varrho }_{3k},{\varrho }_{3k+1})\right\}\right)\right]}^{\left[\lambda \right]}\hfill \\ & \le & {\left[\Theta \left({\Theta }^{-1}{\left[\Theta \left(M\right)\right]}^{\left[{\lambda }^2\right]}\right)\right]}^{\left[\lambda \right]}\hfill \\ & =& {\left[\Theta \left(M\right)\right]}^{\left[{\lambda }^3\right]}.\hfill \end{array}$$

(13)

Continuing in this way, we have

$$\Theta \left(\sigma ({\varrho }_{pk+i},{\varrho }_{pk+i+1})\right)⪯{\left[\Theta \left(M\right)\right]}^{\left[{\lambda }^p\right]}$$

(14)

for all $p\in \mathbb{N}$ and $i=\left\{1,2,...k\right\}.$ Now, taking $\sigma ({\varrho }_{n+1},{\varrho }_n)=({\wp }_n^{\left(1\right)},{\wp }_n^{\left(2\right)},\cdots ,{\wp }_n^{\left(m\right)})$ and $\Theta \left(({\wp }_n^{\left(1\right)},{\wp }_n^{\left(2\right)},\cdots ,{\wp }_n^{\left(m\right)})\right)=({\hslash }_n^{\left(1\right)},{\hslash }_n^{\left(2\right)},\cdots ,{\hslash }_n^{\left(m\right)}),$ we obtain

$$\begin{array}{ccc}\hfill ({\hslash }_{pk+i}^{\left(1\right)},{\hslash }_{pk+i}^{\left(2\right)},\cdots ,{\hslash }_{pk+i}^{\left(m\right)})& =& \Theta \left(({\wp }_{pk+i}^{\left(1\right)},{\wp }_{pk+i}^{\left(2\right)},\cdots ,{\wp }_{pk+i}^{\left(m\right)})\right)\hfill \\ & \le & {\left[\Theta \left(M\right)\right]}^{\left[{\lambda }^p\right]}\hfill \\ & =& ({\left(h_1\right)}^{\left[{\lambda }_1^p\right]},{\left(h_2\right)}^{\left[{\lambda }_2^p\right]},\cdots ,{\left(h_m\right)}^{\left[{\lambda }_m^p\right]})\hfill \end{array}$$

(15)

where $M={\left(M_i\right)}_{i=1}^m$ and $\left[\Theta \left({\left(M_i\right)}_{i=1}^m\right)\right]={\left(h_i\right)}_{i=1}^m.$ Thus,

$${\hslash }_{pk+i}^{\left(j\right)}\le {\left(h_j\right)}^{\left[{\lambda }_j^p\right]}$$

(16)

for all $j\in \{1,2,...,m\}.$ In the limit, we have ${lim}_{p\to \infty }{\hslash }_{pk+i}^{\left(j\right)}=1.$ Therefore, ${lim}_{p\to \infty }{\wp }_{pk+i}^{\left(j\right)}=0^{+},$ for all $j\in \{1,2,...,m\}.$ Hence, ${lim}_{p\to \infty }\sigma ({\varrho }_{pk+i},{\varrho }_{pk+i+1})=\theta .$ By (${\Theta }_3$), there exist $h\in (0,1)$ and $\mathfrak{q}\in (0,\infty ]$ such that

$$\underset{n\to \infty }{lim}\frac{{\hslash }_{pk+i}^{\left(j\right)}-1}{{\left[{\wp }_{pk+i}^{\left(j\right)}\right]}^h}=\mathfrak{q}$$

for all $j\in \{1,2,\cdots ,m\}$.

Assume that ${\eta }_1<\infty .$ In this case, let ${\eta }_2=\frac{{\eta }_1}{2}>0.$ By the definition, there exist $n_0\in \mathbb{N}$ such that, for all $n\ge n_0,$ we have

$$\left|\frac{{\hslash }_{pk+i}^{\left(j\right)}-1}{{\left[{\wp }_{pk+i}^{\left(j\right)}\right]}^h}-{\eta }_1\right|\le {\eta }_2$$

for all $j\in \{1,2,\cdots ,m\}$. This yields

$$\frac{{\hslash }_{pk+i}^{\left(j\right)}-1}{{\left[{\wp }_{pk+i}^{\left(j\right)}\right]}^h}\ge {\eta }_1-{\eta }_2={\eta }_2$$

for all $j\in \{1,2,\cdots ,m\}$. Then,

$${\eta }_{2}p{\left[{\wp }_{pk+i}^{\left(j\right)}\right]}^h\le p\left[{\hslash }_{pk+i}^{\left(j\right)}-1\right].$$

(17)

Assume that ${\eta }_1=\infty .$ Let ${\eta }_2>0$. By the definition, there exists $n_0\in \mathbb{N}$ such that $n\ge n_0$, and we have

$$\frac{{\hslash }_{pk+i}^{\left(j\right)}-1}{{\left[{\wp }_{pk+i}^{\left(j\right)}\right]}^h}\ge {\eta }_2$$

for all $j\in \{1,2,\cdots ,m\}$. This yields

$${\eta }_{2}p{\left[{\wp }_{pk+i}^{\left(j\right)}\right]}^h\le p\left[{\hslash }_{pk+i}^{\left(j\right)}-1\right].$$

From these cases and (16), we obtain

$${\eta }_{2}p{\left[{\wp }_{pk+i}^{\left(j\right)}\right]}^h\le p\left[{\left(h_j\right)}^{\left[{\lambda }_j^p\right]}-1\right]$$

(18)

for all $j\in \{1,2,\cdots ,m\}$ and for some ${\eta }_2>0$. Taking $p\to \infty$ in (18),

$$\underset{p\to \infty }{lim}p{\left[{\wp }_{pk+i}^{\left(j\right)}\right]}^h=0$$

for all $j\in \{1,2,\cdots ,m\}.$ Hence for $j\in \{1,2,\cdots ,m\},$ there exists $p_j\in \mathbb{N}$ such that $p{\left[{\wp }_{pk+i}^{\left(j\right)}\right]}^h\le 1,$ for all $p\ge p_j.$ Hence ${\wp }_{pk+i}^{\left(j\right)}\le \frac{1}{p^{1/h}},$ for all $p\ge p_j.$ Setting $p_0=max\{p_j:1\le j\le m\},$ we obtain

$${\wp }_{pk+i}^{\left(j\right)}\le \frac{1}{p^{1/h}}$$

(19)

for all $p\ge p_0$ and all $i\in \{1,2,...,k\}.$ We prove that $\left\{{\varrho }_n\right\}$ is a Cauchy sequence. Now, let $m,n\in \mathbb{N}$ such that $p_{0}k\le n<m.$ Then, there exist $p,q\in \mathbb{N}$ and $i,j\in \{1,2,...,k\}$ such that $p_0\le p\le q,n=pk+i$ and $m=qk+j.$ Now, by (19) and the triangle inequality of VVMS, we obtain

$$\begin{array}{ccc}\hfill \sigma ({\varrho }_n,{\varrho }_m)& ⪯& \sigma ({\varrho }_{pk+i},{\varrho }_{qk+j})⪯\sum _{\tau =p}^q\sum _{l=\mathfrak{q}}^k\sigma ({\varrho }_{\tau k+\mathfrak{l}},{\varrho }_{\tau k+\mathfrak{l}+1})\hfill \\ & ⪯& \sum _{\tau =p}^{q}k\left(\frac{1}{{\tau }^{\frac{1}{h}}},...,\frac{1}{{\tau }^{\frac{1}{h}}}\right)\hfill \\ & ⪯& k\left(\sum _{\tau =p}^q\frac{1}{{\tau }^{\frac{1}{h}}},...,\sum _{\tau =p}^q\frac{1}{{\tau }^{\frac{1}{h}}}\right).\hfill \end{array}$$

(20)

As $n,m\to \infty ,$ we have $p,q\to \infty .$ Thus, the last term in (20) converges to $\theta$. This implies that $\left\{{\varrho }_n\right\}$ is a Cauchy sequence in $\left(\mathcal{S},\sigma \right)$. As $\left(\mathcal{S},\sigma \right)$ is a complete VVMS, so $\left\{{\varrho }_n\right\}$ converges to some point ${\varrho }^{*}\in \mathcal{S}$, i.e, ${lim}_{n\to \infty }{\varrho }_n={\varrho }^{*}.$ Now, we shall prove that ${\varrho }^{*}$ is a fixed point of $\mathcal{L}$. To see this, we have

$$\begin{array}{ccc}\hfill \Theta \left(\sigma ({\varrho }_{n+k},\mathcal{L}({\varrho }^{*},...,{\varrho }^{*}))\right)& =& \Theta \left(\sigma (\mathcal{L}({\varrho }_n,{\varrho }_{n+1}...,{\varrho }_{n+k-1}),\mathcal{L}({\varrho }^{*},...,{\varrho }^{*}))\right)\hfill \\ & ⪯& \Theta \left(\begin{array}{c}\sigma (\mathcal{L}({\varrho }_n,{\varrho }_{n+1}...,{\varrho }_{n+k-1}),\mathcal{L}({\varrho }_{n+1},{\varrho }_{n+2}...,{\varrho }_{n+k-1},{\varrho }^{*}))\\ +\sigma (\mathcal{L}({\varrho }_{n+1},{\varrho }_{n+2}...,{\varrho }_{n+k-1},{\varrho }^{*}),\mathcal{L}({\varrho }_{n+2},{\varrho }_{n+3}...,{\varrho }_{n+k-1},{\varrho }^{*},{\varrho }^{*}))\\ +...+\sigma (\mathcal{L}{\varrho }_{n+k-1},{\varrho }^{*}...,{\varrho }^{*}),\mathcal{L}({\varrho }^{*},...,{\varrho }^{*}))\end{array}\right)\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}& & \sigma ({\varrho }_{n+k},\mathcal{L}({\varrho }^{*},...,{\varrho }^{*}))\hfill \\ & ⪯& {\Theta }^{-1}{\left[\Theta \left(max\left(\sigma ({\varrho }_n,{\varrho }_{n+1}),\sigma ({\varrho }_{n+1},{\varrho }_{n+2}),...,\sigma ({\varrho }_{n+k-1},{\varrho }^{*})\right)\right)\right]}^{\lambda }\hfill \\ & & +{\Theta }^{-1}{\left[\Theta \left(max\left(\sigma ({\varrho }_{n+1},{\varrho }_{n+2}),\sigma ({\varrho }_{n+2},{\varrho }_{n+3}),...,\sigma ({\varrho }_{n+k-1},{\varrho }^{*})\right)\right)\right]}^{\lambda }\hfill \end{array}$$

$$+...+{\Theta }^{-1}{\left[\Theta \left(\left(\sigma ({\varrho }_{n+k-1},{\varrho }^{*})\right)\right)\right]}^{\lambda }\to \theta$$

as $n\to \infty .$ Thus,

$$\sigma ({\varrho }^{*},\mathcal{L}({\varrho }^{*},...,{\varrho }^{*}))=\underset{n\to \infty }{lim}\sigma ({\varrho }_{n+k},\mathcal{L}({\varrho }^{*},...,{\varrho }^{*}))=\theta .$$

Therefore, ${\varrho }^{*}=\mathcal{L}({\varrho }^{*},...,{\varrho }^{*}).$ Suppose ${\varrho }^{*}$, $\overline{\varrho }$ are two distinct fixed points of $\mathcal{L}$. Now, by hypothesis, we have

$$\sigma ({\varrho }^{*},\overline{\varrho })=\sigma (\mathcal{L}({\varrho }^{*},{\varrho }^{*}...,{\varrho }^{*}),\mathcal{L}(\overline{\varrho },\overline{\varrho }...,\overline{\varrho }))<\sigma ({\varrho }^{*},\overline{\varrho })$$

which is a contradiction. Thus, the fixed point of $\mathcal{L}$ is unique. □

Remark 1.

Taking $\Theta :{\mathbb{R}}_{+0}^m\to {\mathbb{R}}_{+1}^m$, by

$$\Theta \left(({\wp }_1,{\wp }_2,\cdots ,{\wp }_m)\right)=(exp\sqrt{{\wp }_1},exp\sqrt{{\wp }_2},\cdots ,exp\sqrt{{\wp }_m}),$$

in Theorem 6, we obtain Ćirić- and Prešić-type [4] results in the setting of a vector-valued metric space.

Theorem 7.

Let $(\mathcal{S},\sigma )$ be a complete VVMS and $\mathcal{L}:{\mathcal{S}}^k\to \mathcal{S}$. Assume that there exist $\lambda ={\left({\lambda }_i\right)}_{i=1}^m\in {\mathbb{R}}_{+}^m$ with ${\lambda }_i<1,$ for $i\in \{1,2,\cdots ,m\}$ such that

$$\sigma (\mathcal{L}({\varrho }_1,{\varrho }_2,...,{\varrho }_k),\mathcal{L}({\varrho }_2,...,{\varrho }_{k+1}))⪯Asup\left\{\sigma ({\varrho }_1,{\varrho }_2),\sigma ({\varrho }_2,{\varrho }_3),...,\sigma ({\varrho }_k,{\varrho }_{k+1})\right\}$$

for all ${\varrho }_1,{\varrho }_2,...,{\varrho }_{k+1}\in \mathcal{S},$ where

$$A={\left(\begin{array}{cccc}{\lambda }_1^2& 0& \cdots & 0\\ 0& {\lambda }_2^2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & {\lambda }_m^2\end{array}\right)}_{m\times m}$$

and a sequence $\left\{{\varrho }_n\right\}$ in $\mathcal{S}$ such that ${\varrho }_{n+k}=\mathcal{L}({\varrho }_n,{\varrho }_{n+1}...,{\varrho }_{n+k+1})$ and $\sigma ({\varrho }_{n+k},{\varrho }_{n+k+1})\succ \mathbf{0}$ for all $n\in \mathbb{N}$. In addition, if ${\varrho }_n\to \varrho ,$ with $\sigma ({\varrho }_n,\varrho )\succ \mathbf{0},$ for all $n\in \mathbb{N},$ then the sequence $\left\{{\varrho }_n\right\}$ converges to a fixed point of $\mathcal{L}$. Moreover, if for all $\varrho ,\omega \in \mathcal{S}$ with $\varrho \ne \omega ,$

$$\sigma \left(\mathcal{L}\right(\varrho ,\varrho ,...,\varrho ),\mathcal{L}(\omega ,...,\omega \left)\right)<\sigma (\varrho ,\omega )$$

then the fixed point of $\mathcal{L}$ is unique.

Remark 2.

In above result, as $max\{{\lambda }_i:i\in \{1,2,\cdots ,m\}\}<1$, so the matrix A is convergent to zero.

Example 2.

Let $\mathcal{S}=\left\{0,1,2,...\right\}$ and σ:$\mathcal{S}\times \mathcal{S}\to {\mathbb{R}}^2$. By

$$\sigma (\varrho ,\omega )=\left\{\begin{array}{c}\left(0,0\right),\mathit{if}\varrho =\omega \\ \left(\varrho +\omega ,\varrho +\omega \right),\mathit{if}\varrho \ne \omega \end{array}\right.$$

then ($\mathcal{S}$,σ) is a complete vector-valued metric space. Define $\mathcal{L}:\mathcal{S}\times \mathcal{S}\to \mathcal{S}$ by

$$\mathcal{L}\left(\varrho ,\omega \right)=\left\{\begin{array}{c}0,\mathit{if}\varrho ,\omega \in \{0,1\}\\ max\{\varrho ,\omega \}-1,\mathit{if}\varrho ,\omega \ge 2\end{array}\right.$$

and

$$\Theta ({\wp }_1,{\wp }_2)=(exp\sqrt{{\wp }_{1}exp{\wp }_1},exp\sqrt{{\wp }_{2}exp{\wp }_2}).$$

By simple calculation, we have $\Theta \in {\Xi }^2.$ Now, we assert that $\mathcal{L}$ is a Perov–Ciric–Presic-type Θ-contraction with constant $\lambda =(exp(-\frac{1}{2}),exp(-\frac{1}{2})),$ that is

$$\Theta \left(\sigma (\mathcal{L}({\varrho }_1,{\varrho }_1),\mathcal{L}({\varrho }_1,{\varrho }_2))\right)⪯{\left[\Theta \left(sup\left\{\sigma ({\varrho }_1,{\varrho }_1),\sigma ({\varrho }_1,{\varrho }_2)\right\}\right)\right]}^{\left[\lambda \right]}$$

for all ${\varrho }_1,{\varrho }_2\in \mathcal{S}$ with $\sigma (\mathcal{L}({\varrho }_1,{\varrho }_1),\mathcal{L}({\varrho }_1,{\varrho }_2))\succ \mathbf{0}.$ For this, it is sufficient to show

$$\frac{\sigma (\mathcal{L}({\varrho }_1,{\varrho }_1),\mathcal{L}({\varrho }_1,{\varrho }_2))}{sup\left\{\sigma ({\varrho }_1,{\varrho }_1),\sigma ({\varrho }_1,{\varrho }_2)\right\}}{e}^{\sigma (\mathcal{L}({\varrho }_1,{\varrho }_1),\mathcal{L}({\varrho }_1,{\varrho }_2))-sup\left\{\sigma ({\varrho }_1,{\varrho }_1),\sigma ({\varrho }_1,{\varrho }_2)\right\}}\le e^{-1}$$

$$\frac{\sigma (\mathcal{L}({\varrho }_1,{\varrho }_1),\mathcal{L}({\varrho }_1,{\varrho }_2))}{sup\{2{\varrho }_1,{\varrho }_1+{\varrho }_2\}}{e}^{\sigma (\mathcal{L}({\varrho }_1,{\varrho }_1),\mathcal{L}({\varrho }_1,{\varrho }_2))-sup\left\{\sigma ({\varrho }_1,{\varrho }_1),\sigma ({\varrho }_1,{\varrho }_2)\right\}}\le e^{-1}.$$

(21)

First, observe that $\sigma (\mathcal{L}({\varrho }_1,{\varrho }_1),\mathcal{L}({\varrho }_1,{\varrho }_2))\succ 0$⟺ the set $\left\{{\varrho }_1,{\varrho }_2\right\}\cap \{0,1\}$ is a singleton or empty. As Equation (21) is symmetric with regard to ${\varrho }_1$ and ${\varrho }_2$, we may suppose that ${\varrho }_1$ > ${\varrho }_2$ in these two cases.

Case 1. Let $\left\{{\varrho }_1,{\varrho }_2\right\}\cap \{0,1\}$ be a singleton. Then, $\mathcal{L}({\varrho }_1,{\varrho }_1)+\mathcal{L}({\varrho }_1,{\varrho }_2)={\varrho }_1-1$ and $sup\{2{\varrho }_1,{\varrho }_1+{\varrho }_2\}=2{\varrho }_1.$ Thus, we have

$$\begin{array}{ccc}& & \frac{\sigma (\mathcal{L}({\varrho }_1,{\varrho }_1),\mathcal{L}({\varrho }_1,{\varrho }_2))}{sup\{2{\varrho }_1,{\varrho }_1+{\varrho }_2\}}{e}^{\sigma (\mathcal{L}({\varrho }_1,{\varrho }_1),\mathcal{L}({\varrho }_1,{\varrho }_2))-sup\left\{\sigma ({\varrho }_1,{\varrho }_1),\sigma ({\varrho }_1,{\varrho }_2)\right\}}\hfill \\ & =& \frac{{\varrho }_1-1}{2{\varrho }_1}{e}^{-0-{\varrho }_1}\le e^{-1}.\hfill \end{array}$$

Case 2. Let $\left\{{\varrho }_1,{\varrho }_2\right\}\cap \{0,1\}$ be empty. Then, $\mathcal{L}({\varrho }_1,{\varrho }_1)+\mathcal{L}({\varrho }_1,{\varrho }_2)=2{\varrho }_1-2$ and $sup\{2{\varrho }_1,{\varrho }_1+{\varrho }_2\}=2{\varrho }_1.$ Thus, we have

$$\begin{array}{ccc}& & \frac{\sigma (\mathcal{L}({\varrho }_1,{\varrho }_1),\mathcal{L}({\varrho }_1,{\varrho }_2))}{sup\{2{\varrho }_1,{\varrho }_1+{\varrho }_2\}}{e}^{\sigma (\mathcal{L}({\varrho }_1,{\varrho }_1),\mathcal{L}({\varrho }_1,{\varrho }_2))-sup\left\{\sigma ({\varrho }_1,{\varrho }_1),\sigma ({\varrho }_1,{\varrho }_2)\right\}}\hfill \\ & =& \frac{2{\varrho }_1-2}{2{\varrho }_1}{e}^{-2}\le e^{-1}.\hfill \end{array}$$

Thus, by Theorem 6, $\mathcal{L}$ has a fixed point that is $\mathcal{L}(0,0)=0$

4. Application

Let $\left(E,{∥·∥}_E\right)$ be a Banach space and $A_1,\cdots ,A_k:E^k\to E$ be k nonlinear operators. In this section, we discuss the solution of the following semilinear operator system:

$$\begin{array}{ccc}\hfill A_1\left(l_1,l_2,...,l_k\right)& =& l_1\hfill \\ & & ·\hfill \\ & & ·\hfill \\ & & ·\hfill \\ \hfill A_k\left(l_1,l_2,...,l_k\right)& =& l_k.\hfill \end{array}$$

(22)

Let $\Delta =E^k$. Define $\sigma :\Delta \times \Delta \to {\mathbb{R}}^k$ by

$$\sigma (u,v)=\left({∥l_1-e_1∥}_E,...,{∥l_k-e_k∥}_E\right).$$

for $u=(l_1,l_2,...,l_k),$$v=(e_1,e_2,...,e_k)\in \Delta .$ Clearly, ($\Delta ,\sigma$) is a complete VVMS. If we define $\mathcal{L}:{\Delta }^k\to \Delta$ by

$$\mathcal{L}(u,u,\dots ,u)=\left(A_1(l_1,l_2,...,l_k),...,A_k(l_1,l_2,...,l_k)\right)$$

(23)

then (22) can be written as a fixed point problem such as

$$\mathcal{L}(u,u,\dots ,u)=(l_1,l_2,...,l_k)=u$$

(24)

in the space $\Delta$. Thus, applying Theorem 6, we discuss the existence of a solution of problem (24).

Theorem 8.

Assume that ∃${\lambda }_i(i=1,2,...,k)<1$ such that

$${∥A_i(l_1,l_2,...,l_k)-A_i(e_1,e_2,...,e_k)∥}_E\le {\lambda }_i{∥l_i-e_i∥}_E$$

(25)

for all $u=(l_1,l_2,...,l_k),v=(e_1,e_2,...,e_k)\in E^k$ with $l_i\ne e_i.$ Then, (22) has a unique solution in $E^k$.

Proof. 

By the inequality (25), we have

$$e^{\sqrt{{∥A_i(l_1,l_2,...,l_k)-A_i(e_1,e_2,...,e_k)∥}_E}}\le e^{\sqrt{{\lambda }_i}\sqrt{{∥l_i-e_i∥}_E}}={\left[e^{\sqrt{{∥l_i-e_i∥}_E}}\right]}^{\sqrt{{\lambda }_i}}$$

for all $i=1,\cdots ,k$. Thus, we have

$$\left(e^{\sqrt{{∥A_1\left(u_1\right)-A_1\left(u_2\right)∥}_E}},...,e^{\sqrt{{∥A_k\left(u_1\right)-A_k\left(u_2\right)∥}_E}}\right)⪯\left({\left[e^{\sqrt{{∥l_1-e_1∥}_E}}\right]}^{\sqrt{{\lambda }_1}},...,{\left[e^{\sqrt{{∥l_1-e_1∥}_E}}\right]}^{\sqrt{{\lambda }_k}}\right).$$

Taking the function $\Theta \in {\Xi }^k$ as $\Theta ({\wp }_1,{\wp }_2,...,{\wp }_k)=(exp\left\{\sqrt{{\wp }_1}\right\},exp\{\sqrt{{\wp }_{2}exp\left\{{\wp }_2\right\}},...,$$\Theta exp\left\{\sqrt{{\wp }_k}\right\}\left\}\right)$, the above inequality can be written as

$$\begin{array}{ccc}\hfill \Theta \left({∥A_i(l_1,l_2,...,l_k)-A_i(e_1,e_2,...,e_k)∥}_E\right)& \le & {\left(\Theta \left({∥l_1-e_1∥}_E,...,{∥l_k-e_k∥}_E\right)\right)}^{\lambda }\hfill \\ & =& {\left(\Theta \left(\sigma (u_1,u_2),...,\sigma (u_k,u_{k+1})\right)\right)}^{\lambda }\hfill \\ & \le & {\left(\Theta \left(sup\left\{\sigma (u_1,u_2),...,\sigma (u_k,u_{k+1})\right\}\right)\right)}^{\lambda }\hfill \end{array}$$

where $\lambda =(\sqrt{{\lambda }_1},\sqrt{{\lambda }_2},...,\sqrt{{\lambda }_k})$, or equivalently

$$\Theta \left(\sigma \left(\mathcal{L}(u_1,u_2,\dots ,u_k),\mathcal{L}(u_2,u_3,\dots ,u_{k+1})\right)\right)\le {\left(\Theta \left(sup\left\{\sigma (u_1,u_2),...,\sigma (u_k,u_{k+1})\right\}\right)\right)}^{\lambda }.$$

Thus, applying Theorem 6, $\mathcal{L}$ possesses a unique fixed point in $\Delta$$=E^k$, or equivalently, the semilinear operator system (25) has a unique solution in $E^k$. □

5. Conclusions

Ćirić and Prešić extended the concept of a Prešić contraction to a Ćirić–Prešić-type contraction in the setting of a complete metric space. Recently, Altun et al. [24] introduced the Perov-type $\Theta$-contraction in a vector-valued metric space. In this paper, we introduced the notion of the Perov–Ćirić–Prešić-type $\Theta$-contraction, which is a generalization of the Perov–Prešić-type $\Theta$-contraction in the setting of a vector-valued metric space. Since a Perov–Prešić-type $\Theta$-contraction is an extension of the Perov-type $\Theta$-contraction introduced by Altun et al. [24], our main theorem, Theorem 6, is a generalization of the main result of Altun et al. [24]. Moreover, we derived Theorem 7 from our main theorem, Theorem 6, and this is a generalization of the main Ćirić–Prešić result [4] in the sense of a vector-valued metric space. As an application of our main results, we investigated the solution of a semilinear operator system.

In this area, our future work will focus on studying the fixed points for the Perov–Ćirić–Prešić-type $\Theta$-contraction in the context of vector-valued metric spaces endowed with a graph of the solution of the matrix difference equation, as an application.