1 Introduction

In 1958, Luxemburg [1, p. 541] expanded the notion of metric spaces by introducing the concept of a generalized complete metric space (or simply GCMS) as follows:

Definition 1.1

Let W be a set, and \(\varrho : W \times W \rightarrow [0, \infty ]\) be a function. Then, the pair \((W, \varrho )\) is termed a GCMS if and only if for \(u, v, w \in W\):

  1. 1.

    \(\varrho (u, v) = 0\) if and only if \(u=v\),

  2. 2.

    \(\varrho (u, v) = \varrho (v, u)\),

  3. 3.

    \(\varrho (u, v) \le \varrho (u, w) + \varrho (w, v)\),

  4. 4.

    every \(\varrho\)-Cauchy sequence in W is \(\varrho\)-convergent, that is \(\lim \nolimits _{n, m\rightarrow \infty }\varrho (u_{n}, u_{m})=0\) for a sequence \((u_{n}) \in W\) for \(n\in \mathbb {N},\) implies the existence of an element \(u\in W\) with \(\lim \nolimits _{n\rightarrow \infty } \varrho (u, u_{n})=0.\)

This concept diverges from the concept of a complete metric due to the fact that not every pair of points in W necessarily has finite distance. He also established convergence results for the sequences of functions obtained by Picard’s method of successive approximations to find a unique solution of a first order differential equation. Using this notion, Diaz and Margolis [2] proved a “theorem of alternative” for contraction mappings in the GCMS. The result of Daiz and Margolis [2] provides an interesting generalization of the well-known Banach contraction principle (BCP) in the sense that every complete metric space is a GCMS. This result was further extended by Covitz and Nadler [3] to multivalued mappings. For further interesting generalizations of the BCP, readers are referred to [4,5,6].

Another important generalization of the BCP was introduced by Matkowski [7], who extended the BCP to a system of mappings on finite products of metric spaces. These types of results are fruitful to study the existence of solutions of a system of functional equations (see [8, 9]). Reddy and Subrahmanyam [9] generalized Krasnoselski’s fixed point theorem [6] to Matkowski’s operators and used it to find convex solutions for the system of functional equations. Khantwal and Gairola [8] extended Matkowski’s result to establish the existence of bounded solutions of the system of functional equations. For more comprehensive study of this topic, we refer to [8, 10,11,12,13,14,15,16,17,18,19] and references therein.

In 2014, Jleli and Samet [5] introduced a new type of contractive mapping called the \(\theta\)-contraction (or the JS-contraction) and established some fixed point theorems which extend some existing results. For more details in the direction one can refer to [20,21,22,23,24].

Motivated by the work of the aforementioned authors, we establish some theorems of the alternative for a system of mappings in the GCMSs. Our results extend and generalize certain results of Daiz and Margolis [2], Jleli and Samet [5], Matkowski [25], and many others. Additionally, we present some illustrative examples to validate our results.

2 Preliminaries

The result of Diaz and Margolis [2] is given below:

Theorem 2.1

Let \((W, \varrho )\) be a GCMS and that the function \(T:W\rightarrow W\) satisfies the condition: there exists a constant \(0< r <1\) such that whenever \(\varrho (u, v)< \infty\) one has

$$\begin{aligned} \varrho (Tu, Tv) \le r \varrho (u, v). \end{aligned}$$

Let \(u_0 \in W\), and consider the sequence of successive approximations: \(u_0, Tu_0, T^2u_0, \dots .\) Then the following alternative holds: either

  1. (A)

    for every integer \(n=0, 1, 2, \dots\), one has

    $$\begin{aligned}\varrho (T^{n}u_0, T^{n+1}u_0) =\infty , ~~\text { or }\end{aligned}$$
  2. (B)

    the sequence of successive approximations \(u_0, Tu_0, T^{2}u_0, \dots\) is convergent to a fixed point of T.

Jleli and Samet [5] introduced the notion of \(\theta\)-contraction mappings as follows.

Definition 2.2

A mapping \(\theta : (0, \infty ) \rightarrow (1, \infty )\) is said to be \(\theta\)-contraction if the following properties hold:

(\(\theta _1\))::

\(\theta\) is non-decreasing,

(\(\theta _2\))::

for each \((\xi _n) \subset (0, \infty )\), \(\lim _{n\rightarrow \infty } \theta (\xi _n)=1\) and \(\lim _{n\rightarrow \infty } \xi _n =0^{+}\),

(\(\theta _3\))::

there exists \(r\in (0, 1)\) and \(l\in (0, \infty ]\) such that \(\lim _{\xi \rightarrow 0^{+}} \dfrac{\theta (\xi )-1}{\xi ^r} =l\).

For convenience we denote by \(\Theta\), the set of all functions satisfying (\(\theta _1\))-(\(\theta _3\)). Some examples of functions belonging to \(\Theta\) are defined as:

Example 2.3

Let \(\theta :(0, \infty ) \rightarrow (1, \infty )\) be defined by

  1. (i)

    \(\theta (\xi ):= e^{\sqrt{\xi }}\),

  2. (ii)

    \(\theta (\xi ):= e^{\sqrt{\xi e^{\xi }}}\),

  3. (iii)

    \(\theta (\xi ):= \dfrac{2}{\pi } \arctan \left( \frac{1}{\xi ^{a}}\right) , ~ 0< a <1, ~ \xi >0.\)

Then every \(\theta (\xi )\) satisfies all the properties of \(\Theta\).

Jleli and Samet [5] obtained the following result by considering the class \(\Theta\).

Theorem 2.4

Let \((W,\varrho )\) be a complete metric space, and \(T: W \rightarrow W\) be a mapping. Suppose that there exist \(\theta \in \Theta\) and \(\lambda \in (0, 1)\) such that for all \(u, v \in W\), \(\varrho (Tu, Tv) >0\) implies that

$$\begin{aligned} \theta (\varrho (Tu, Tv)) \le \left[ \theta (\varrho (u, v))\right] ^\lambda . \end{aligned}$$

Then T has a unique fixed point in W.

We observe that BCP follows easily from Theorem 2.4 by taking \(\theta (\xi ) =e^{\sqrt{\xi }}\).

Now onwards, in this paper we adopt the following notations and results.

Let \(\Lambda = \{1, 2, \dots , n\},~ (\Omega _{i},\rho _{i}), ~i\in \Lambda\) be metric spaces, and \(\Omega :=\Omega _{1}\times \dots \times \Omega _{n}\). Assume that \(\Im _{i}: \Omega \rightarrow \Omega _{i},~ i\in \Lambda\) are mappings, \(\mathbb {N}\) denotes the set of natural numbers, \(\mathbb {R}\) denotes the set of real numbers and \(\left( \omega ^{m}\right) = \left( \omega _{1}^{m},\ldots ,\omega _{n}^{m} \right)\), \(m \in \mathbb {N}\) be a sequence in \(\Omega\).

Consider a square matrix \((a_{ik})\) of size \(n\times n\) with non-negative entries. We define the elements \(c_{ik}^{(0)}\) according to the following rule:

$$\begin{aligned} c_{ik}^{(0)}=\left\{ \begin{array}{cc} a_{ik}, & i\ne k \\ 1-a_{ik}, & i=k \end{array} \right. ~~ i,k \in \Lambda , \end{aligned}$$
(1)

for all \(i,k \in \Lambda\). Then, recursively, we determine \(c_{ik}^{(j)}\) as follows:

$$\begin{aligned} c_{ik}^{(j+1)}&=\left\{ \begin{array}{cc} { c_{11}^{(j)}}{c_{i+1,k+1}^{(j)}}+{c_{i+1,1}^{(j)}}{c_{1,k+1}^{(j)}}, & i \ne k\\ \\ {c_{11}^{(j)}}{c_{i+1,k+1}^{(j)}}- {c_{i+1,1}^{(j)}}{c_{1,k+1}^{(j)}}, & i=k \end{array} \right. \end{aligned}$$
(2)
$$\begin{aligned} j =&1, \dots , n-1;~~~~~~~~~~~i, k=1,\dots , n-j. \end{aligned}$$

where \(j = 1, \dots , n-1\) and \(i, k=1,\dots , n-j\).

The following lemma is due to Matkowski [7, 25].

Lemma 2.5

Let \(c_{ik}^{(0)}\) for \(i,k \in \Lambda ,~n\ge 2.\) The system of inequalities

$$\begin{aligned} \sum \limits _{k=1, k\ne i}^{n}c_{ik}^{(0)}q_{k}<c_{ii}^{(0)}q_{i}, ~~ i\in \Lambda , \end{aligned}$$

has a positive solution \(q_{1},\dots , q_{n}\) if and only if

$$\begin{aligned} c_{ii}^{(j)}>0, j=1,\dots ,n-1, i=1,\dots , n-j . \end{aligned}$$
(3)

Additionally, there exists a positive number \(h<1\) such that

$$\begin{aligned} \sum \limits _{k=1}^{n}a_{ik}q_{k}\le hq_{i},~~ i\in \Lambda , \end{aligned}$$
(4)

for some positive number \(q_{1},\dots , q_{n}\). Moreover, h can be chosen as

$$\begin{aligned}h=\max _{i}\left\{ q_{i}^{-1}\sum \limits _{k=1}^{n}a_{ik}q_{k} \right\} .\end{aligned}$$

Remark 2.6

It’s worth noting that the condition (3) holds if the spectral radius of the matrix \((a_{ik})\) is less than one (see [25]).

The result of Matkowski in [7, 25] can be stated as follows:

Theorem 2.7

Let \((\Omega _{i}, \rho _{i}), ~i\in \Lambda ,\) be complete metric spaces and \(\Im _{i}:\Omega \rightarrow \Omega _{i}, ~ i\in \Lambda ,\) be mappings. If there exist non-negative numbers \(a_{ik},~ i, k \in \Lambda ,\) such that

$$\begin{aligned} \rho _{i}(\Im _{i}\omega , \Im _{i}\bar{\omega })\le \sum \limits _{k=1}^{n}a_{ik}\rho _{k}(\omega _{k},\bar{\omega _{k}}) \end{aligned}$$
(5)

for every \(\omega = (\omega _{1}, \dots , \omega _{n}),~ \bar{\omega } = ( \bar{\omega }, \dots , \bar{\omega }),~ i \in \Lambda\) and the numbers defined by (1)–(2) satisfy inequalities (3), then the system of equations

$$\begin{aligned} \omega _{i}=\Im _{i}(\omega _{1}, \dots , \omega _{n}),~ i\in \Lambda \end{aligned}$$
(6)

has exactly one solution \(z =(z_{1},\dots , z_{n}) \in \Omega\) such that \(z_{i}\in \Omega _{i}, ~ i\in \Lambda\). Moreover, for arbitrary \(\omega ^{1}\in \Omega\), the sequence of successive approximations

$$\begin{aligned} \omega _{i}^{m+1} =\Im _{i} ( \omega _{1}^{m}, \dots , \omega _{n}^{m}) \text { for } i\in \Lambda , ~ m \in \mathbb {N} \end{aligned}$$
(7)

converges and

$$\begin{aligned}z_{i}=\lim \limits _{m \rightarrow \infty }\omega _{i}^{m},~~ i \in \Lambda .\end{aligned}$$

3 Main results

In this section, we consider \(\theta : (0, \infty ]\rightarrow (1, \infty ]\) satisfies conditions \((\theta _{1})-(\theta _{3})\) with \(\theta (\alpha ) =\infty\) if and only if \(\alpha =\infty\). Further, we denote the class of such mappings by \(\Theta ^{\prime }\). Now, we present the following result for a system of mappings in GCMSs (generalizing many other results).

Theorem 3.1

Assume that \((\Omega _{i}, \rho _{i}), ~i\in \Lambda ,\) are GCMSs and \(\Im _{i}:\Omega \rightarrow \Omega _{i}, ~ i\in \Lambda ,\) are mappings. Suppose there exist \(\theta \in \Theta ^{\prime }\) and non-negative real numbers \(a_{ik}\), where \(i,k=1,\ldots ,n\), defined in (1) and (2), are such that (3) and the following condition holds:

$$\begin{aligned} \rho _{i}(\Im _{i}\omega , \Im _{i}\bar{\omega }) \ne 0 \Longrightarrow \theta \left( \rho _{i}(\Im _{i}\omega , \Im _{i}\bar{\omega })\right) \le \prod _{k=1}^{n} \left[ \theta \left( \rho _{k}(\omega _{k},\bar{\omega }_{k})\right) \right] ^{a_{ik}}, \end{aligned}$$
(8)

whenever \(\rho _{i}(\omega _{k}, \bar{\omega }_{k}) < \infty\) for all \(\omega _{k}, \bar{\omega }_{k} \in \Omega _{k}\) and \(i, k \in \Lambda\). Then either

  1. (a)

    for every integer \(j \in \mathbb {N}\), there exists an \(i\in \Lambda\) such that

    $$\begin{aligned} \rho _{i} (\omega _{i}^{j}, \Im _{i}(\omega _{1}^{j}, \dots , \omega _{n}^{j})) = \infty , ~~~ \text { or } \end{aligned}$$
    (9)
  2. (b)

    the sequence of successive approximations \(( \omega ^{m} ) = (\omega _{1}^{m}, \dots , \omega _{n}^{m})\) given by (6) converges to a unique solution \(z= (z_{1}, \dots , z_{n})\in \Omega\) of the system (7).

Proof

Pick \(\omega _{i}^{1}\in \Omega _{i}\) for each \(i \in \Lambda\) and define

$$\begin{aligned} \omega _{i}^{m+1}= \Im _{i}\omega ^{m}~ \text { for } i\in \Lambda , ~m\in \mathbb {N}. \end{aligned}$$

Now, two mutually exclusive possibilities arise for the sequence successive approximation: either

  1. (i)

    for any \(j \in \mathbb {N}\), there exists an \(i\in \Lambda\) such that

    $$\begin{aligned}\rho _{i} (\omega _{i}^{j}, \Im _{i}(\omega _{1}^{j}, \dots , \omega _{n}^{j})) = \infty ,\end{aligned}$$

    which is precisely the alternative (a) of the conclusion of the Theorem 3.1 or

  2. (ii)

    there exists an \(s \in \mathbb {N}\) such that for every \(i\in \Lambda\)

    $$\begin{aligned}\rho _{i}(\omega _{i}^{s}, \omega _{i}^{s+1})< \infty .\end{aligned}$$

To complete the proof, we need to show that \(\text {(ii)}\) implies (b) of the conclusion of the Theorem 3.1.

In case (ii) holds, let \(N = N(\omega ^{1})\) denote a particular one such that

$$\begin{aligned}\rho _{i}(\omega _{i}^{N}, \omega _{i}^{N+1})< \infty \Leftrightarrow \ln \left( \theta (\rho _{i} (\omega _{i}^{N}, \omega _{i}^{N+1}))\right) < \infty .\end{aligned}$$

Then, by (8) it follows that

$$\begin{aligned}\ln \theta (\rho _{i}(\omega _{i}^{N+1}, \omega _{i}^{N+2})) \le \ln \theta (\rho _{i}(\Im _{i}\omega ^{N}, \Im _{i}\omega ^{N+1})) \le \sum \limits _{k=1}^{n} {a_{ik}} \ln \theta (\rho _{k}(\omega _{k}^{N}, \omega _{k}^{N+1})) < \infty , \end{aligned}$$

and consequently by induction

$$\begin{aligned}\ln \left[ \theta \left( \rho _{i}(\omega _{i}^{N + \ell }, \omega _{i}^{N + \ell +1})\right) \right]< \infty \Leftrightarrow \rho _{i}(\omega _{i}^{N + \ell }, \omega _{i}^{N + \ell +1}) < \infty \text { for } \ell \in \mathbb {N}\cup \{0\}, ~ i\in \Lambda .\end{aligned}$$

From (3) and Lemma 2.5, it follows that there exist positive numbers \(q_{i},~ i \in \Lambda\) and \(h\in (0, 1)\) such that the inequalities (4) hold. Note that if \(q_{1}, \dots , q_{n}\) fulfil the inequalities (4), then so do \(aq_{1}, \dots , aq_{n}\) for every \(a>0\).

Now, without restriction of generality, we may assume that

$$\begin{aligned}0<\ln \left[ \theta \left( \rho _{i}(\omega _{i}^{1}, \omega _{i}^{2}) \right) \right] \le q_{i},~ i\in \Lambda .\end{aligned}$$

Then for all \(m \ge N\), we have

$$\begin{aligned} \ln \left[ \theta \left( \rho _{i} (\omega _{i}^{m}, \omega _{i}^{m+1}) \right) \right]&\le \sum \limits _{k=1}^{n}a_{ik}\ln \left[ \theta \left( \rho _{k} ( \omega _{k}^{m-1}, \omega _{k}^{m} ) \right) \right] \nonumber \\&\le \sum \limits _{k=1}^{n}a_{ik} \left( \sum \limits _{j=1}^{n}a_{ij} \ln \left[ \theta \left( \rho _{j} ( \omega _{j}^{m-2}, \omega _{j}^{m-1} ) \right) \right] \right) \nonumber \\&\vdots \nonumber \\&\le h^{m-2} q_{i} \end{aligned}$$
(10)

or

$$\begin{aligned} \theta \left( \rho _{i} (\omega _{i}^{m}, \omega _{i}^{m+1}) \right) \le e^{h^{m-2} q_{i}} \text { for } m \ge N \in \mathbb {N}. \end{aligned}$$
(11)

Making \(m\rightarrow \infty\) and by virtue of \((\theta _{2})\), we have

$$\begin{aligned} \lim \limits _{m \rightarrow \infty } \theta (\rho _{i}(\omega _{i}^{m}, \omega _{i}^{m+1})) =1 \Leftrightarrow \lim \limits _{m \rightarrow \infty } \rho _{i}(\omega _{i}^{m}, \omega _{i}^{m+1}) = 0 \text { for } i\in \Lambda . \end{aligned}$$

Let \(\rho _{i}(\omega _{i}^{m}, \omega _{i}^{m+1})=\xi _{i_m}\) for \(i \in \Lambda\) and \(m\in \mathbb {N}\). To prove \((\omega _{i}^{m})\) is a Cauchy sequence, note that in view of condition \((\theta _{3})\), there exist \(r\in (0, 1)\) and \(\lambda \in (0, \infty ]\) such that

$$\begin{aligned} \lim \limits _{m\rightarrow +\infty } \dfrac{\theta (\xi _{i_m})-1}{(\xi _{i_m})^{r}}= \lambda , ~ i\in \Lambda . \end{aligned}$$
(12)

Take \(\delta \in (0, \lambda )\). By the definition of limit, there exists \(m_{1} \in \mathbb {N}\) such that

$$\begin{aligned}(\xi _{i_m})^{r} \le \delta ^{-1}\left[ \theta (\xi _{i_m}) -1\right] \text { for } m > m_{1}.\end{aligned}$$

From above inequality, we deduce that

$$\begin{aligned}m(\xi _{i_m})^{r} \le m\delta ^{-1}\left[ \theta \left( e^{h^{m-2}q_{i}}\right) -1\right] \text { for } m > m_{1}.\end{aligned}$$

This implies that

$$\begin{aligned} \lim \limits _{m\rightarrow \infty } m(\xi _{i_m})^{r} =\lim \limits _{m\rightarrow \infty } m(\rho _{k}(\omega _{k}^{m}, \omega _{k}^{m+1}) )^{r} =0.\end{aligned}$$

Hence there exists \(m_{2} \in \mathbb {N}\) such that

$$\begin{aligned} \rho _{k}(\omega _{k}^{m}, \omega _{k}^{m+1}) \le \dfrac{1}{m^{1/r}} \text { for } m\ge m_{2} . \end{aligned}$$
(13)

Let \(p> m > m_2\). Then, using the triangle inequality and (13), we have

$$\begin{aligned}\rho _{i}(\omega _{i}^{m}, \omega _{i}^{p}) \le \sum \limits _{k=m}^{p-1}\rho _{i}(\omega _{i}^{k}, \omega _{i}^{k+1}) \le \sum \limits _{k=m}^{p-1}\dfrac{1}{k^{1/r}} \le \sum \limits _{k=m}^{\infty }\dfrac{1}{k^{1/r}}.\end{aligned}$$

Since \(0< r <1\), the series \(\sum \nolimits _{k=m}^{\infty }\dfrac{1}{k^{1/r}}\) converges and hence \(\rho _{i}(\omega _{i}^{k}, \omega _{i}^{k+1}) \rightarrow 0\) as \(m \rightarrow \infty\) for each \(i \in \Lambda\). This proves that \((\omega _{i}^{m} )\) is a Cauchy sequence in \(\Omega _{i}\). From the completeness of \((\Omega _{i}, \rho _{i})\), there exists \(z_{i} \in \Omega _{i}\) such that \(\omega _{i}^{m} \rightarrow z_{i}\) as \(m\rightarrow \infty\). By continuity of \(\Im _{i}\), \(\omega _{i}^{m+1} = \Im _{i}\omega ^{m} =\Im _{i}z.\) The uniqueness of limit yields that \(z_{i}= \Im _{i}z.\)

We will prove that the solution just obtained is unique. Suppose that \(z_{1}, \dots z_{n}\) and \(v_{1}, \dots , v_{n}\) are solutions of system (6). Without loss of generality, we can assume that

$$\begin{aligned} \ln \left[ \theta \left( \rho _{i}(z_{i}, v_{i})\right) \right] \le q_{i}, ~ i \in \Lambda . \end{aligned}$$
(14)

Now, from the equations

$$\begin{aligned}z_{i}= \Im _{i}(z_{1}, \dots , z_{n}) ~ \text { and } ~ v_{i}= \Im _{i} (v_{1}, \dots , v_{n}), ~ i \in \Lambda ,\end{aligned}$$

and from (8) and (4), we obtain by induction

$$\begin{aligned}\ln \left[ \theta \left( \rho _{i}(z_{i}, v_{i})\right) \right] \le h^{m}q_{i}, ~ i\in \Lambda , ~ m\in \mathbb {N}.\end{aligned}$$

Making \(m\rightarrow \infty\), we get

$$\begin{aligned}\rho _{i}(z_{i}, v_{i})=0 \text { or } z_{i}=v_{i}~ \text { for all } i \in \Lambda ,\end{aligned}$$

which completes the proof of the theorem. \(\square\)

Remark 3.2

For different choices of n, \(\theta\), and \(\Omega _{i}\), we can extend the following results:

  1. 1.

    When \(n=1\) and \(\theta (\xi ) = e^{\sqrt{\xi }}\), we get the Diaz and Margolis’s [2] result for a contraction mapping in the GCMS.

  2. 2.

    Choosing \(\Omega _{i}, ~ i\in \Lambda\) as a complete metric space and \(\theta (\xi ) = e^{\xi }\), we obtain Theorem 2.7.

  3. 3.

    For \(n=1\), \(\theta (\xi ) = e^{\sqrt{\xi }}\), and considering \(\Omega _{1}\) as a complete metric space, we obtain the BCP [26].

  4. 4.

    The system of equations (7) may possess more than one fixed points in \(\Omega\) (refer to [1]).

Now we present an existence result for a solution of the system of equations (7) in GCMSs, which includes the result of Luxemburg [1, p. 95], and Diaz and Margolis [2, p. 308] as a special case.

Theorem 3.3

Let \((\Omega _{i}, \rho _{i})\) for \(i\in \Lambda\) be GCMSs and \(\Im _{i}:\Omega \rightarrow \Omega _{i}\) for \(i\in \Lambda\) be mappings. Assume that there exist \(\theta \in \Theta ^{\prime }\) and non-negative real numbers \(a_{ik}\), where \(i,k=1,\ldots ,n\), defined in (1) and (2), such that (3) holds, and the following condition is satisfied:

$$\begin{aligned} \rho _{i}(\Im _{i}\omega , \Im _{i}\bar{\omega }) \ne 0 \Longrightarrow \theta \left( \rho _{i}(\Im _{i}\omega , \Im _{i}\bar{\omega })\right) \le \prod _{k=1}^{n} \left[ \theta \left( \rho _{k}(\omega _{k},\bar{\omega }_{k})\right) \right] ^{a_{ik}}, \end{aligned}$$
(15)

whenever \(\theta \left( \rho _{k}(\omega _{k}, \bar{\omega }_{k})\right) < C\) for all \(\omega _{k}, \bar{\omega }_{k} \in \Omega _{k}\) and \(i, k \in \Lambda\). Then the following alternative holds: either

  1. (c)

    for every integer \(s \in \mathbb {N}\) there exists an \(i\in \Lambda\) such that

    $$\begin{aligned} \theta \left( \rho _{i} (\omega _{i}^{s}, \Im _{i}(\omega _{1}^{s}, \dots , \omega _{n}^{s}))\right) \ge C, ~~~ \text { or } \end{aligned}$$
  2. (d)

    the sequence of successive approximations \(( \omega ^{m}) = (\omega _{1}^{m}, \dots , \omega _{n}^{m})\) given by (6) converges to a unique solution \(z= (z_{1}, \dots , z_{n})\in \Omega\) of the system of equations (7).

Proof

From (3) and Lemma 2.5, it follows that there exist positive numbers \(q_{i},~ i\in \Lambda\), and \(h\in (0, 1)\) such that the inequalities (4) hold. The homogeneity of (4) implies that numbers \(aq_{i},~ i \in \Lambda\) for every \(a > 0\), also fulfil the inequalities (4). We may assume that

$$\begin{aligned}C > \max \{ e^{q_{i}}, i\in \Lambda \}. \end{aligned}$$

Pick \(\omega _{i}^{1}\in \Omega _{i}, ~i \in \Lambda\) and define

$$\begin{aligned} \omega _{i}^{m+1}= \Im _{i}\omega ^{m}~ \text { for } i\in \Lambda , ~m\in \mathbb {N}. \end{aligned}$$

Now, two mutually exclusive possibilities arise for the sequence successive approximation: either

  1. (i)

    for any \(j \in \mathbb {N}\), there exists an \(i\in \Lambda\) such that

    $$\begin{aligned}\theta \left( \rho _{i} (\omega _{i}^{j}, \Im _{i}(\omega _{1}^{j}, \dots , \omega _{n}^{j}))\right) \ge C,\end{aligned}$$

    which corresponds precisely to alternative (c) of the conclusion of Theorem 3.3, or

  2. (ii)

    there exists an integer \(s \in \mathbb {N}\) such that for every \(i\in \Lambda\)

    $$\begin{aligned}\theta \left( \rho _{i}(\omega _{i}^{s}, \omega _{i}^{s+1})\right) < C.\end{aligned}$$

To complete the proof, it only remains to prove that \(\text {(ii)}\) implies alternative (d) of the Theorem 3.3.

In case (ii) holds, let \(N = N(\omega ^{1})\) be a particular one such that

$$\begin{aligned}\theta \left( \rho _{i}(\omega _{i}^{N}, \omega _{i}^{N+1})\right) < C, ~ i\in \Lambda .\end{aligned}$$

Now, without loss of generality, we may assume that

$$\begin{aligned}0< \ln \left[ \theta \left( \rho _{i}(\omega _{i}^{N}, \omega _{i}^{N+1}) \right) \right] \le q_{i},~ i\in \Lambda .\end{aligned}$$

Then, by (15) it follows that

$$\begin{aligned} \ln \theta (\rho _{i}(\omega _{i}^{N+1}, \omega _{i}^{N+2}))&\le \ln \theta (\rho _{i}(\Im _{i}\omega ^{N}, \Im _{i}\omega ^{N+1})) \\&\le \sum \limits _{k=1}^{n} {a_{ik}} \ln \theta (\rho _{k}(\omega _{k}^{N}, \omega _{k}^{N+1})) \\&\le \sum \limits _{k=1}^{n} {a_{ik}} q_{k} \le hq_{i} < \ln (C). \end{aligned}$$

Since \(0\le h<1\), hence, by induction

$$\begin{aligned} \theta \left( \rho _{i}(\omega _{i}^{N + \ell }, \omega _{i}^{N + \ell +1})\right) < C, ~~ \ell \in \mathbb {N}\cup \{0\}, ~ i\in \Lambda .\end{aligned}$$

Now, following the proof of Theorem 3.1, we conclude alternative (d) of the Theorem 3.3. \(\square\)

Theorem 3.4

Assume that \((\Omega _{i}, \rho _{i})\) for \(i\in \Lambda\) are complete metric spaces, and \(\Im _{i}:\Omega \rightarrow \Omega _{i}\) for \(i\in \Lambda\) are mappings. Suppose there exist \(\theta \in \Theta\) and non-negative real numbers \(a_{ik}\), where \(i,k=1,\ldots ,n\), defined in (1) and (2), such that (3) holds, and the following condition is satisfied:

$$\begin{aligned} \rho _{i}(\Im _{i}\omega , \Im _{i}\bar{\omega }) \ne 0 \Longrightarrow \theta \left( \rho _{i}(\Im _{i}\omega , \Im _{i}\bar{\omega })\right) \le \prod _{k=1}^{n} \left[ \theta \left( \rho _{k}(\omega _{k},\bar{\omega }_{k})\right) \right] ^{a_{ik}}, \end{aligned}$$
(16)

for all \(\omega _{k}, \bar{\omega }_{k} \in \Omega _{k}\) and \(i, k \in \Lambda\). Then, the sequence of successive approximations \((\omega ^{m}) = (\omega _{1}^{m}, \dots , \omega _{n}^{m})\) given by (6) converges to a unique solution \(z= (z_{1}, \dots , z_{n})\in \Omega\) of the system (7).

Proof

The proof of this theorem can be seen as follows: if \(\Omega _{k},~ k\in \Lambda\) are complete metric spaces then \(\rho _{k}(\omega _{k}, \bar{\omega _{k}}) < \infty\) for every \(\omega _{k}, \bar{\omega _{k}} \in \Omega _{k}, ~ k\in \Lambda\) and the conclusion of alternative (a) is excluded from Theorem 3.1. \(\square\)

Remark 3.5

If \(n=1\) and \(\theta (\xi ) = e^{\sqrt{\xi }}\), we get the result presented by Dugundji and Granas [6, p. 10–11].

The following example illustrates the utility of our result.

Example 3.6

Let \(\Omega _{1}= \left\{ \omega _{m}=\frac{m(m+1)}{2}:~ m \in \mathbb {N} \right\}\), \(\Omega _{2}= \left\{ \omega _{m} =\frac{m(m+1)(m+2)}{6}:~ m \in \mathbb {N} \right\}\) and \((\Omega _{i}, \rho _{i}), ~ i=1, 2\) are usual metric spaces. Define \(\Im _{1}: \Omega _{1} \times \Omega _{2}\rightarrow \Omega _{1}\) by

$$\begin{aligned} \Im _{1}(\omega _{m}, \bar{\omega }_{m})= \left\{ \begin{array}{cl} 1, & \text { when } \omega _{m}= \omega _{1}, \\ \omega _{m-1}, & \text { otherwise,} \end{array} \right. \end{aligned}$$

and \(\Im _{2}: \Omega _{1} \times \Omega _{2}\rightarrow \Omega _{2}\) by

$$\begin{aligned} \Im _{2}(\omega _{m}, \bar{\omega }_{m})= \left\{ \begin{array}{cl} 1, & \text { when } \bar{\omega }_{m}= \bar{\omega }_{1}, \\ \bar{\omega }_{m-1}, & \text { otherwise} \end{array} \right. \end{aligned}$$

for all \((\omega _{m}, \bar{\omega }_{m}) \in \Omega _{1} \times \Omega _{2}, ~ m\in \mathbb {N}\). Then, it is easy to see that \((\Omega _{i}, \rho _{i}),~ i=1, 2\) are complete metric spaces and \(\Im _{i}, ~i=1, 2\) are continuous mappings. Also, for \(\omega =(1, 1)\) and \(\bar{\omega }=(\bar{\omega }_{100}, \bar{\omega }_{100}) \in \Omega _{1}\times \Omega _{2}\), we have

$$\begin{aligned}\rho _{i}(\Im _{i}\omega , \Im _{i}\bar{\omega }) \ge \sum \limits _{k=1}^{2} a_{ik} \rho _{k}(\omega _{k}, \bar{\omega }_{k})~ \text { for } i=1, 2.\end{aligned}$$

for any choices of \((a_{ik})\) as defined in Theorem 3.4. This implies that the system \(\{\Im _{1}, \Im _{2}\}\) does not satisfy Matkowski’s contraction (5). However, if we take

$$\begin{aligned}a_{11} = a_{22}=0.9,~ a_{21} = a_{12} = 0 ~ \text { and } ~\theta (\xi )= e^{\sqrt{\xi e^{\xi }}},\end{aligned}$$

then, it is not difficult to prove that the system \(\{\Im _{1}, \Im _{1}\}\) satisfies contraction (16). Let

$$\begin{aligned} e^{\sqrt{\rho _{i}(\Im _{i}\omega , \Im _{i}\bar{\omega })e^{\rho _{i}(\Im _{i}\omega , \Im _{i}\bar{\omega })}}}&\le e^{\sqrt{ {a_{ii}} \rho _{i}(\omega _{i}, \bar{\omega }_{i})e^{\rho _{i}(\omega _{i}, \bar{\omega }_{i})}}} \\ \rho _{i}(\Im _{i}\omega , \Im _{i}\bar{\omega })e^{\rho _{i}(\Im _{i}\omega , \Im _{i}\bar{\omega })}&\le {a_{ii}}^{2} \rho _{i}(\omega _{i}, \bar{\omega }_{i})e^{\rho _{i}(\omega _{i}, \bar{\omega }_{i})}\\ \dfrac{\rho _{i}(\Im _{i}\omega , \Im _{i}\bar{\omega })e^{\rho _{i}(\Im _{i}\omega , \Im _{i}\bar{\omega })-\rho _{i}(\omega _{i}, \bar{\omega }_{i})}}{\rho _{i}(\omega _{i}, \bar{\omega }_{i})}&\le {a_{ii}}^{2}. \end{aligned}$$
Case I::

Let \(i=1\). We consider two subcases. In first subcase, let \(\omega =(1,~ *)\) and \(\bar{\omega }= (\bar{\omega }_{m}, ~*)\) for \(m > 2\), where \(*\) indicates arbitrary point in \(\Omega _2\). Then

$$\begin{aligned} \dfrac{\rho _{1}(\Im _{1}\omega , \Im _{1}\bar{\omega })e^{\rho _{1}(\Im _{1}\omega , \Im _{1}\bar{\omega })-\rho _{1}(\omega _{1}, \bar{\omega }_{1})}}{\rho _{1}(\omega _{1}, \bar{\omega }_{1})} = \dfrac{m^2-m-2}{m^2+m-2}e^{-m} \le e^{-1}. \end{aligned}$$

In the second subcase, let \(\omega =(\omega _{p},~ *)\) and \(\bar{\omega }= (\bar{\omega }_{m}, ~*)\) for \(m>p>1\). Then

$$\begin{aligned} \dfrac{\rho _{1}(\Im _{1}\omega , \Im _{1}\bar{\omega })e^{\rho _{1}(\Im _{1}\omega , \Im _{1}\bar{\omega })-\rho _{1}(\omega _{1}, \bar{\omega }_{1})}}{\rho _{1}(\omega _{1}, \bar{\omega }_{1})} = \dfrac{m+p-1}{m + p +1}e^{p-m} \le e^{-1}. \end{aligned}$$
Case II::

Let \(i=2\). We consider two subcases. In first subcase, let \(\omega =(*, ~1)\) and \(\bar{\omega }= (*, ~ \bar{\omega }_{m})\) for \(m > 2\), where \(*\) indicates arbitrary point in \(\Omega _1\). Then

$$\begin{aligned} \dfrac{\rho _{2}(\Im _{2}\omega , \Im _{2}\bar{\omega })e^{\rho _{2}(\Im _{2}\omega , \Im _{2}\bar{\omega })-\rho _{2}(\omega _{2}, \bar{\omega }_{2})}}{\rho _{2}(\omega _{2}, \bar{\omega }_{2})} = \dfrac{m^3-m-6}{m^3 +3m^2 +2m-6}e^{-3m(m+1)} \le e^{-1}. \end{aligned}$$

In the second subcase, let \(\omega =(*, ~\omega _{p})\) and \(\bar{\omega }= (*, ~\bar{\omega }_{m})\) for \(m>p>1\). Then

$$\begin{aligned} \dfrac{\rho _{2}(\Im _{2}\omega , \Im _{2}\bar{\omega })e^{\rho _{2}(\Im _{2}\omega , \Im _{2}\bar{\omega })-\rho _{2}(\omega _{2}, \bar{\omega }_{2})}}{\rho _{2}(\omega _{2}, \bar{\omega }_{2})} = \dfrac{m^2 + p^2 +mp-1}{m^2 + p^2 +mp+ 3m +3p +2}e^{-3(m+p+1)} \le e^{-1}. \end{aligned}$$

Hence, all the assumptions of Theorem 3.4 are verified and the system \(\{\Im _{1}, \Im _{2}\}\) has a unique solution at (1, 1).