Abstract
The aim of the paper is to establish some theorems of the alternative for a system of mappings in generalized complete metric spaces. Our results extend and generalize the well-known results of Daiz and Margolis (Bull Am Math Soc 74:305–309, 1968), Matkowski (Bull Acad Polon Sci Sér Sci Math Astron Phys 21:323–324, 1973), Jleli and Samet (J Inequal Appl 38:8, 2014), and many others. We also present some illustrative examples to validate our results.
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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.
\(\varrho (u, v) = 0\) if and only if \(u=v\),
-
2.
\(\varrho (u, v) = \varrho (v, u)\),
-
3.
\(\varrho (u, v) \le \varrho (u, w) + \varrho (w, v)\),
-
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
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
-
(A)
for every integer \(n=0, 1, 2, \dots\), one has
$$\begin{aligned}\varrho (T^{n}u_0, T^{n+1}u_0) =\infty , ~~\text { or }\end{aligned}$$ -
(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
-
(i)
\(\theta (\xi ):= e^{\sqrt{\xi }}\),
-
(ii)
\(\theta (\xi ):= e^{\sqrt{\xi e^{\xi }}}\),
-
(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
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:
for all \(i,k \in \Lambda\). Then, recursively, we determine \(c_{ik}^{(j)}\) as follows:
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
has a positive solution \(q_{1},\dots , q_{n}\) if and only if
Additionally, there exists a positive number \(h<1\) such that
for some positive number \(q_{1},\dots , q_{n}\). Moreover, h can be chosen as
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
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
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
converges and
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:
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
-
(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) -
(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
Now, two mutually exclusive possibilities arise for the sequence successive approximation: either
-
(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
-
(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
Then, by (8) it follows that
and consequently by induction
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
Then for all \(m \ge N\), we have
or
Making \(m\rightarrow \infty\) and by virtue of \((\theta _{2})\), we have
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
Take \(\delta \in (0, \lambda )\). By the definition of limit, there exists \(m_{1} \in \mathbb {N}\) such that
From above inequality, we deduce that
This implies that
Hence there exists \(m_{2} \in \mathbb {N}\) such that
Let \(p> m > m_2\). Then, using the triangle inequality and (13), we have
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
Now, from the equations
and from (8) and (4), we obtain by induction
Making \(m\rightarrow \infty\), we get
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.
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.
Choosing \(\Omega _{i}, ~ i\in \Lambda\) as a complete metric space and \(\theta (\xi ) = e^{\xi }\), we obtain Theorem 2.7.
-
3.
For \(n=1\), \(\theta (\xi ) = e^{\sqrt{\xi }}\), and considering \(\Omega _{1}\) as a complete metric space, we obtain the BCP [26].
-
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:
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
-
(c)
for every integer \(s \in \mathbb {N}\) there exists an \(i\in \Lambda\) such that
$$\begin{aligned} \theta \left( \rho _{i} (\omega _{i}^{s}, \Im _{i}(\omega _{1}^{s}, \dots , \omega _{n}^{s}))\right) \ge C, ~~~ \text { or } \end{aligned}$$ -
(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
Pick \(\omega _{i}^{1}\in \Omega _{i}, ~i \in \Lambda\) and define
Now, two mutually exclusive possibilities arise for the sequence successive approximation: either
-
(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
-
(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
Now, without loss of generality, we may assume that
Then, by (15) it follows that
Since \(0\le h<1\), hence, by induction
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:
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
and \(\Im _{2}: \Omega _{1} \times \Omega _{2}\rightarrow \Omega _{2}\) by
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
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
then, it is not difficult to prove that the system \(\{\Im _{1}, \Im _{1}\}\) satisfies contraction (16). Let
- 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).
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Acknowledgements
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Khantwal, D. Theorem of the alternative for a system of mappings in generalized complete metric spaces. J Anal (2024). https://doi.org/10.1007/s41478-024-00832-2
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DOI: https://doi.org/10.1007/s41478-024-00832-2