Abstract
This research focuses on proving the results of tripled fixed point and coincidence point in generalized metric spaces endowed with vector-valued metrics and matrix equations. The results from this study are illustrated by two applications.
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1 Introduction
Perov described the Banach contraction principle for contraction mappings on spaces equipped with vector-valued metrics in [11]. Later, by a different method, the results of Perov in [5] were generalized and their fixed point property of a self-mapping over generalized metric space \((X,d)\) was studied.
In this article \(M_{m,m}(\mathbb{R^{+}})\) represents the set of all \(m\times m\) matrices with components in \(\mathbb{R^{+}}\), Θ represents the matrix zero and I represents the identity matrix and \(\mathbb{N}_{0}=\mathbb{N}\cup \{0\}\).
Let \(A\in M_{m,m}(\mathbb{R^{+}})\), then A is called convergent to zero, if and only if \(A^{n}\rightarrow 0\) as \(n\rightarrow \infty \). We refer to [14, 15] for more details.
Let \(\alpha, \beta \in \mathbb{R}^{m}\), where \(\alpha =(\alpha _{1}, \ldots, \alpha _{m})\), \(\beta =(\beta _{1},\ldots, \beta _{m})\) and \(c\in \mathbb{R}\). Note that \(\alpha _{i}\leq \beta _{i}\) (resp. \(\alpha _{i}<\beta _{i}\)) for each \(1\leq i\leq m\) and also \(\alpha _{i}\leq c\) (resp. \(\alpha _{i}< c\)) for \(1\leq i\leq m\), respectively. We define
and
These are addition and multiplication on \(\mathbb{R}^{m}\) (see [5, 7, 8]).
Definition 1.1
([5])
Let X be a non-empty set. A mapping \(d:X^{2}\longrightarrow \mathbb{R}^{m}\) is called a vector-valued metric on X, if the following properties hold:
-
(1)
\(d(x^{1}, x^{2})\geq 0\) for each \(x^{1}, x^{2}\in X\), if \(d(x^{1}, x^{2})=0\), if and only if \(x^{1}=x^{2}\);
-
(2)
\(d(x^{1},x^{2})=d(x^{2},x^{1})\) for each \(x^{1},x^{2}\in X\);
-
(3)
\(d(x^{1},x^{2})\leq d(x^{1}, x^{3})+ d(x^{3}, x^{2})\) for each \(x^{1}, x^{2}, x^{3}\in X\).
If \(x^{1}, x^{2} \in \mathbb{R}^{m}, x^{1} = (x^{1}_{1}, \ldots, x^{1}_{m})\) and \(x^{2} = (x^{2}_{1}, \ldots, x^{2}_{m})\), then \(x^{1} \leq x^{2}\) if and only if \(x^{1}_{i} \leq x^{2}_{i}\) for \(1 \leq i \leq m\). A set X is called a generalized metric space, equipped with a vector-valued metric d and denoted by \((X, d)\).
Now, we need the following equivalent propositions. Their proofs are classic results in matrix analysis (see for more details [1, 12, 13]).
-
(1)
\(A\rightarrow 0\);
-
(2)
\(A^{n}\rightarrow 0\) as \(n\rightarrow \infty \);
-
(3)
\(|\lambda |<1\), for each \(\lambda \in \mathbb{C}\) with \(\det (A-\lambda I)=0\);
-
(4)
the matrix \(I-A\) is nonsingular and
$$ (I-A)^{-1}=I+A+\cdots +A^{n}+\cdots; $$ -
(5)
\(A^{n}q\longrightarrow 0\) and \(qA^{n}\longrightarrow 0\) as \(n\longrightarrow \infty \), for each \(q\in \mathbb{R}^{m}\).
Denote the set of all matrices \(A\in M_{m,m}(\mathbb{R^{+}})\) where \(A^{n}\longrightarrow 0\) by \(\mathcal{Z}M\). For the sake of simplicity, we identify the row and column vectors in \(\mathbb{R}^{m}\).
Definition 1.2
([3])
An element \((x^{1}, x^{2})\in X^{2}\) is called a coupled fixed point of the mapping \(F:X^{2}\longrightarrow X\) if \(F(x^{1}, x^{2})=x^{1}, F(x^{2}, x^{1})=x^{2}\).
Definition 1.3
([10])
Suppose that \(F:X^{2}\longrightarrow X\) and \(g:X\longrightarrow X\) are given. An element \((x^{1}, x^{2})\in X^{2}\) is called a coupled coincidence point of the mappings F and g if \(F(x^{1}, x^{2})=gx^{1}\) and \(F(x^{2}, x^{1}) = gx^{2}\). Then \((gx^{1}, gx^{2})\) is called a coupled coincidence point.
Definition 1.4
([15])
Let \((X, d,\preceq )\) be a partially ordered complete metric space. We consider partially ordered set X. We define on \(X^{3}\) the following order, for \((x^{1}, x^{2}, x^{3}), (u^{1}, u^{2}, u^{3}) \in X^{3}\),
Definition 1.5
([2])
.Let \((X,\preceq )\) be a partially ordered set and \(F:X^{3}\rightarrow X\). We say that F has the mixed monotone property if for any \(x^{1}, x^{2}, x^{3} \in X\)
that is, \(F(x^{1}, x^{2}, x^{3})\) is monotone non-decreasing in \(x^{1}\) and \(x^{3}\) and is monotone non-increasing in \(x^{2}\).
Now, we present a triple fixed point of the second kind that used for mixed monotone mappings (see [9]).
Definition 1.6
([9])
An element \((x^{1}, x^{2}, x^{3})\in X^{3}\) is called a triple fixed point of the mapping \(F:X^{3}\longrightarrow X\) if
Definition 1.7
([2])
Let \((X, d)\) be the complete generalized metric space. The mapping \(\overline{d}: X^{3}\rightarrow \mathbb{R}^{m}\) with
defines a metric on \(X^{3} \), which, for convenience, we denote by d, too.
Definition 1.8
([4])
Let \((X,\preceq )\) be a partially ordered set, \(F:X^{3}\longrightarrow X\) and \(g:X\longrightarrow X\) be given. We say F has the g-mixed monotone property if for any \(x^{1}, x^{2}, x^{3} \in X\),
that is, \(F(x^{1}, x^{2}, x^{3})\) is monotone non-decreasing in \(x^{1}\) and \(x^{3}\), and monotone non-increasing in \(x^{2}\).
Definition 1.9
([15])
Let \(F:X^{3}\longrightarrow X\) and \(g:X\longrightarrow X\) be given. F and g are called compatible if
where \(U_{123}=F(x^{1}_{n}, x^{2}_{n},x^{3}_{n})\) and \(V_{123}=(gx^{1}_{n}, gx^{2}_{n}, gx^{3}_{n})\),
where \(U_{212}=F(x^{2}_{n}, x^{1}_{n},x^{2}_{n})\) and \(V_{212}=(gx^{2}_{n}, gx^{1}_{n}, gx^{2}_{n})\),
where \(U_{321}=F(x^{3}_{n}, x^{2}_{n}, x^{1}_{n})\) and \(V_{321}=(gx^{3}_{n}, gx^{2}_{n}, gx^{1}_{n})\),
whenever \(\{x^{1}_{n}\}, \{x^{2}_{n}\}\), and \(\{x^{3}_{n}\}\) are sequences in X, such that
for some \(x^{1}, x^{2}, x^{3} \in X\).
Definition 1.10
([15])
Let \(F:X^{3}\longrightarrow X\) and \(g:X\longrightarrow X\). The mappings F and g are called weakly reciprocally continuous if
whenever \(\{x^{1}_{n}\}, \{x^{2}_{n}\}\), and \(\{x^{3}_{n}\}\) are sequences in X, such that
for some \(x^{1}, x^{2}, x^{3} \in X\).
Definition 1.11
([15])
Let \(F:X^{3}\longrightarrow X\) and \(g:X\longrightarrow X\). The mappings F and g are called reciprocally continuous if
whenever \(\{x^{1}_{n}\}, \{x^{2}_{n}\}\), and \(\{x^{3}_{n}\}\) are sequences in X, such that
for some \(x^{1}, x^{2}, x^{3} \in X\).
Definition 1.12
([15])
Let \((X, d,\preceq )\) be a partially ordered metric space. We say that X is regular if the following properties hold:
-
(i)
if a non-decreasing sequence \(x^{1}_{n}\rightarrow x^{1}\), then \(x^{1}_{n} \preceq x^{1}\) for all \(n \geq 0\),
-
(ii)
if a non-increasing sequence \(x^{2}_{n}\rightarrow x^{2}\), then \(x^{2} \preceq x^{2}_{n}\) for all \(n \geq 0\).
For the main result of this article, we study existence and uniqueness of triple common fixed point for a sequence of mappings \(T_{n}: X^{3} \rightarrow X\) and \(g:X\rightarrow X\), where \((X, d)\) is a complete generalized metric space.
First, we have the following two definitions from [6, 15].
Definition 1.13
([15])
Let \((X, d)\) be a metric space and let \(T_{n}:X^{3}\rightarrow X\) and \(g:X\longrightarrow X\) are given. The sequence \(\{T_{n}\}_{n\in \mathbb{N}_{0}}\) and the mapping g are said to be compatible if
where \(U'_{123}=T_{n}(x^{1}_{n}, x^{2}_{n},x^{3}_{n})\) and \(V'_{123}=(gx^{1}_{n}, gx^{2}_{n}, gx^{3}_{n})\)
where \(U'_{212}=T_{n}(x^{2}_{n}, x^{1}_{n},x^{2}_{n})\) and \(V'_{212}=(gx^{2}_{n}, gx^{1}_{n}, gx^{2}_{n})\)
where \(U'_{321}=T_{n}(x^{3}_{n}, x^{2}_{n}, x^{1}_{n})\) and \(V'_{321}=(gx^{3}_{n}, gx^{2}_{n}, gx^{1}_{n})\), whenever \(\{x^{1}_{n}\}, \{x^{2}_{n}\}\), and \(\{x^{3}_{n}\}\) are sequences in X, such that
for some \(x^{1}, x^{2}, x^{3} \in X\).
Definition 1.14
([15])
Let \((X, d)\) be a metric space and let \(T_{n}:X^{3}\rightarrow X\) and \(g:X\longrightarrow X\) are given. \(\{T_{n}\}_{n\in \mathbb{N}_{0}}\) and g are called weakly reciprocally continuous if
whenever \(\{x^{1}_{n}\}, \{x^{2}_{n}\}\), and \(\{x^{3}_{n}\}\) are sequences in X, such that
for some \(x^{1}, x^{2}, x^{3} \in X\).
2 Main results
We start with the following statement, which we will use in the main theorem. Inspired by Definition 1.8 we have the following definition.
Definition 2.1
Let \((X,\preceq )\) be a partially ordered set, \(T_{n}:X^{3}\rightarrow X, n\in \mathbb{N}_{0}\), and \(g:X\rightarrow X\). We say that \(\{T_{n}\}_{n\in \mathbb{N}_{0}}\) has the g-mixed monotone property if for any \(x^{1}, x^{2}, x^{3}, x^{\prime 1}, x^{\prime 2}, x^{\prime 3}\in X\),
imply that
Definition 2.2
Suppose that \(T_{i}:X^{3}\rightarrow X\) and \(g:X\rightarrow X\) are given. We say \(\{T_{i}\}_{i\in \mathbb{N}_{0}}\) and g satisfy the \(( K ) \) property if
for \(x^{1}, x^{2}, x^{3}, u^{1}, u^{2}, u^{3} \in X\) with \(gx^{1} \succeq gu^{1}, gu^{2} \succeq gx^{2}, gx^{3} \succeq gu^{3}\) or \(gx^{1} \preceq gu^{1}, gu^{2} \preceq gx^{2}, gx^{3} \preceq gu^{3}\), \(I\neq A = (a_{ij}), I\neq B = (b_{ij}) \in M_{m, m}(\mathbb{R}^{+})\), \((A + B)(I-A)^{-1} \in \mathcal{Z}M\).
Definition 2.3
If \(T_{0}\) and g have a non-decreasing transcendence point in \(x^{1}_{0}, x^{3}_{0}\) and a non-increasing transcendence point in \(x^{2}_{0}\), then we say \(T_{0}\) and g have a mixed triple transcendence point, if there exist \(x^{1}_{0}, x^{2}_{0}, x^{3}_{0} \in X\) such that
Lemma 2.4
Let\((X, d, \preceq )\)be a partially ordered complete generalized metric space. Letgbe a self-mapping onXand\(\{T_{i}\}_{i\in \mathbb{N}_{0}}\)be a sequence of mappings from\(X^{3}\)intoXand having ag-mixed monotone property with\(T_{i}(X^{3}) \subseteq g(X)\). If\(T_{0}\)andghave a mixed triple transcendence point, then
-
(a)
there are sequences\(\{x^{1}_{n}\}, \{x^{2}_{n}\}\)and\(\{x^{3}_{n}\}\)inXsuch that
$$\begin{aligned} &gx^{1}_{n}= T_{n-1} \bigl(x^{1}_{n-1}, x^{2}_{n-1}, x^{3}_{n-1}\bigr),\qquad gx^{2}_{n} = T_{n-1} \bigl(x^{2}_{n-1}, x^{1}_{n-1}, x^{2}_{n-1}\bigr)\quad\textit{and} \\ & gx^{3}_{n} = T_{n-1} \bigl(x^{3}_{n-1}, x^{1}_{n-1}, x^{2}_{n-1}\bigr); \end{aligned}$$ -
(b)
\(\{gx^{1}_{n}\}, \{gx^{3}_{n}\}\)are non-decreasing sequences and\(\{gx^{2}_{n}\}\)is a non-increasing sequence.
Proof
(a) By hypothesis, let for \(x^{1}_{0}, x^{2}_{0}, x^{3}_{0} \in X\) the condition (2.4) hold. Since \(T_{0}(X^{3}) \subseteq g(X)\), we can define \(x^{1}_{1}, x^{2}_{1}, x^{3}_{1}\in X\) such that
Again since \(T_{0}(X^{3}) \subseteq g(X)\), there exist \(x^{1}_{2}, x^{2}_{2}, x^{3}_{2} \in X\) such that
Continuing this technique, we get
(b) Now, by mathematical induction, we show that
for all \(n \geq 0\). To this end, since (2.4) holds, in the light of
we have
that is, (2.6) holds for \(n = 0\). We assume that (2.6) holds for some \(n > 0\). Now, by (2.5) and (2.6), the result is achieved. Thus, we are done. □
Before expressing the main theorems, we first give the following examples.
Example 2.5
-
1.
and are matrices in \(\mathcal{Z}M\). It is easy to see that \((A+B)(I-A)^{-1}\in \mathcal{Z}M\).
-
2.
and are matrices in \(\mathcal{Z}M\). It is easy to see that \((A+B)(I-A)^{-1}\in \mathcal{Z}M\).
-
3.
Let \(A=\alpha I\) and \(B=((I-\alpha )^{3}-\alpha )I\) be matrices in \(\mathcal{Z}M\). Then for \(\alpha = \frac{1}{4}, \frac{1}{5}, \frac{1}{7}, \frac{1}{8}\) it is clear that \((A+B)(I-A)^{-1}\in \mathcal{Z}M\).
Theorem 2.6
In addition to the conditions of Lemma 2.4, let\(g(X)\subseteq X\)be complete, \(\{T_{i}\}_{i\in \mathbb{N}_{0}}\)andgbe compatible, weakly reciprocally continuous, wheregis monotonic non-decreasing, continuous, and satisfies the condition (K). If\(g(X)\)is regular and\(A, B\)are nonzero matrices in\(\mathcal{Z}M\), then\(\{T_{i}\}_{i\in \mathbb{N}_{0}}\)andghave a triple coincidence point.
Proof
Let \(\{x^{1}_{n}\}, \{x^{2}_{n}\}\textit{ and }\{x^{3}_{n}\}\) be the same sequences which are constructed in Lemma 2.4. By (2.3), we get
It follows that
and similarly
and
We set \(C = (A + B)(I-A)^{-1}\), for all \(n \in \mathbb{N}\), then
Moreover, with repeated use of the triangle inequality and for \(p > \varTheta \), we get
We have
Now, taking the limit as \(n\rightarrow +\infty \), we conclude
This implies that
Thus, \(\{gx^{1}_{n}\}, \{gx^{2}_{n}\}\) and \(\{gx^{3}_{n}\}\) are Cauchy sequences in X. Since \(g(X)\) is complete, there exists \((x^{\prime 1}, x^{\prime 2}, x^{\prime 3}) \in X ^{3}\), with
By construction, we have
and
Since \(\{T_{i}\}_{i\in \mathbb{N}_{0}}\) and g are weakly reciprocally continuous and compatible, we have
and
Since \(\{gx^{1}_{n}\}\) and \(\{gx^{3}_{n}\}\) are non-decreasing and \(\{gx^{2}_{n}\}\) is non-increasing, using the regularity of X, we have \(gx^{1}_{n} \preceq x^{1}, x^{2} \preceq gx^{2}_{n}\) and \(gx^{3}_{n} \preceq x^{3}\) for all \(n \geq 0\). So by (2.3), we get
Taking the limit as \(n\rightarrow +\infty \), we obtain \(gx^{1}=T_{i}(x^{1}, x^{2}, x^{3}) \). Similarly, it can be proved that \(gx^{2}=T_{i}(x^{2}, x^{1}, x^{2})\) and \(gx^{3} =T_{i}(x^{3}, x^{2}, x^{1})\). Thus, \((x^{1}, x^{2}, x^{3})\) is a triple coincidence point of \(\{T_{i}\}_{i \in \mathbb{N}}\) and g. □
If in Theorem 2.6g is the identity mapping, then we have the following corollary.
Corollary 2.7
Let\((X, d, \preceq )\)be a partially ordered complete generalized metric space. Let\(\{T_{i}\}_{i\in \mathbb{N}\cup \{0\}}\)be a mixed monotone sequence of mappings from\(X^{3}\)intoX, where\(\{T_{m}\}\)and\(\mathrm{Id}: X \rightarrow X\)satisfy the\((K)\)property. Also\(T_{0}\)andIdhave a mixed transcendence point. If\(g(X)\)is regular, then there exists\((x^{1}, x^{2}, x^{3}) \in X^{3}\), such that\(x^{1} = T_{i}(x^{1}, x^{2}, x^{3}), x^{2}= T_{i}(x^{2}, x^{1}, x^{2})\), and\(x^{3}= T_{i}(x^{3}, x^{2}, x^{1})\)for\(i \in \mathbb{N}_{0}\).
Definition 2.8
We say that \((x^{1}, x^{2}, x^{3})\) is a triple comparable with \((u^{1}, u^{2}, u^{3})\) if and only if
If in the above definition we replace \((x^{1}, x^{2}, x^{3} )\) and \((u^{1}, u^{2}, u^{3})\) with \(( gx^{1}, gx^{2}, gx^{3})\) and \((gu^{1}, gu^{2}, gu^{3})\), we call \((x^{1}, x^{2}, x^{3}) \) a triple comparable with \((u^{1}, u^{2}, u^{3})\) with respect to g.
Theorem 2.9
Let\((X, d, \preceq )\)be a partially ordered complete generalized metric space. Letgbe a self-mapping onXand\(\{T_{i}\}_{i\in \mathbb{N}_{0}}\)be a sequence of mappings from\(X^{3}\)intoX. Let\(\{T_{i}\}_{i\in \mathbb{N}_{0}}\)andgsatisfy the condition (K) and have triple coincidence points comparable with respect tog, then\(\{T_{i}\}_{i \in \mathbb{N}_{0}}\)andghave a unique triple common fixed point.
Proof
According to Theorem 2.6, the set of tripled coincidence points is non-empty. First, we show that, if \((x^{1}, x^{2}, x^{3})\) and \((x^{\prime 1}, x^{\prime 2}, x^{\prime 3})\) are triple coincidence points, that is, if
then \(gx^{1} = gx^{\prime 1}, gx^{2} = gx^{\prime 2}\) and \(gx^{3}=gx^{\prime 3}\). Since the set of triple coincidence points is a triple comparable, applying condition (2.3) implies
Therefore, as \(I\neq B\in \mathcal{Z}M\), \(d(gx^{1}, gx^{\prime 1}) = \varTheta \), that is, \(gx^{1} = gx^{\prime 1}\). Similarly, it can be proved that \(gx^{2} = gx^{\prime 2}\) and \(gx^{3} = gx^{\prime 3}\). So \(gx^{1} =gx^{2} = gx^{3} =gx^{\prime 1}= gx^{\prime 2}=gx^{\prime 3}\).
Therefore, \(\{T_{i}\}_{i \in \mathbb{N}}\) and g have a unique triple coincidence point \((gx^{1}, gx^{1})\). Since two compatible mappings commute at their coincidence points, thus, clearly, \(\{T_{i}\}_{i \in \mathbb{N}}\) and g have a unique tripled common fixed point whenever \(\{T_{i}\}_{i \in \mathbb{N}}\) and g are weakly compatible. □
Example 2.10
Let \(X = [0, 1]\). Define
Then \((X, d)\) is a partially ordered complete generalized metric space. Define
Because A and B are nonzero matrices in \(\mathcal{Z}M\) and considering the mapping \(T_{i}: X^{3} \rightarrow X\) and \(g:X\rightarrow X\) with
it can be easily verified by mathematical induction that the inequality (2.3) holds for all \(x^{1}, x^{2}, x^{3} \in X\), that is, we see that the greatest value of the first side happens when \(i=1, j\rightarrow \infty \), in this case for \(i=1\) we have
Now for \(j=j+1\) we have
So
Thus all the hypotheses of Theorem 2.6 are satisfied and \((0,0, 0)\) is the triple coincident point of g and \(\{T_{i}\}_{i \in \mathbb{N}_{0}}\). Moreover, using the same \(\{T_{i}\}_{i \in \mathbb{N}_{0}}\) and g in Theorem 2.9, \((0,0, 0)\) is the unique triple common fixed point of g and \(\{T_{i}\}_{i \in \mathbb{N}_{0}}\).
Before explaining the application, it is necessary to provide the following definition, which we will use in Theorem 3.1.
Definition 2.11
Let \(A=( a_{ij})\) and \(B=( b_{ij}) \) be two matrices in \(\mathcal{Z}M\). Then
Clearly if \(A\leq B\) then \(\max \lbrace A,B \rbrace =B\).
3 Application 1
In this part, we will use the results of Sect. 2 to extract some results for the existence and uniqueness of solutions of the integral equations system. Consider the following integral equations system:
for all \(t, s \in [0, T]\), for some \(T > 0\).
Let \(X=C([0, T], \mathbb{R})\) be continuous real functions, defined on the interval \([0, T]\), endowed with a metric
We define the partial order “⪯” on X as follows:
for \(x^{1}, x^{2}\in X, x^{1} \preceq x^{2} \Leftrightarrow x^{1}(t) \preceq x^{2}(t)\) for any \(t\in [0, T]\).
Thus, \((X, d, \preceq )\) is a partially ordered complete generalized metric space. For (3.1) we consider the following hypotheses:
-
(i)
\(f, g, h \in [0, T]\times [0, T] \times \mathbb{R}\longrightarrow \mathbb{R}^{2}\) are continuous;
-
(ii)
\(v:[0, T]\longrightarrow \mathbb{R}\) is continuous;
-
(iii)
there exists \(\rho:[0, T]\longrightarrow M_{2\times 2}([0, T])\), such that, for all \(x^{1}, x^{2}\in X\),
$$\begin{aligned} \begin{aligned} &0\leq \bigl\vert f \bigl(t, s, x^{1}(s) \bigr) - f \bigl(t, s, x^{2}(s) \bigr) \bigr\vert \leq \rho _{1}(t)d\bigl(x^{1}, x^{2}\bigr), \\ &0\leq \bigl\vert g\bigl(t, s, x^{2}(s)\bigr) - g\bigl(t, s, x^{1}(s) \bigr) \bigr\vert \leq \rho _{2}(t)d \bigl(x^{1}, x^{2}\bigr), \\ &0\leq \bigl\vert h\bigl(t, s, x^{1}(s) \bigr) - h \bigl(t, s, x^{2}(s) \bigr) \bigr\vert \leq \rho _{3}(t)d \bigl(x^{1}, x^{2}\bigr), \end{aligned} \end{aligned}$$(3.2)for all \(s, t\in [0, T] \) with and . Because A and B are nonzero matrices in \(\mathcal{Z}M\);
-
(iv)
we suppose that \(\rho _{1}(t)+\rho _{2}(t)+\rho _{3}(t)<1\) and
$$ \rho (t) = \max \bigl\{ \rho _{1}(t), \rho _{2}(t), \rho _{3}(t)\bigr\} ; $$ -
(v)
there are functions \(\alpha, \beta, \gamma: [0, T] \longrightarrow \mathbb{R}\) which are continuous, such that
$$\begin{aligned} &\alpha \leq \int ^{T}_{0} (f \bigl(t, s, \alpha (s)\bigr) + g \bigl(t, s, \beta (s)\bigr)+ h \bigl(t, s, \gamma (s)\bigr)\, ds+v(t), \\ &\beta \geq \int ^{T}_{0} (f \bigl(t, s, \beta (s)\bigr) + g \bigl(t, s, \alpha (s)\bigr)+ h \bigl(t, s, \beta (s)\bigr)\, ds+v(t), \\ &\gamma \leq \int ^{T}_{0} (f \bigl(t, s, \gamma (s)\bigr) + g \bigl(t, s, \beta (s)\bigr)+ h \bigl(t, s, \alpha (s)\bigr)\, ds+v(t). \end{aligned}$$
Theorem 3.1
Under hypotheses (i)–(v), (3.1) has a unique solution inX.
Proof
We consider the operator defined by \(T_{i}: X ^{3}\longrightarrow X\), with
for any \(x^{1}, x^{2}, x^{3}\in X\) and \(t, s \in [0, T]\).
We prove that the operator \(\{T_{i}\}_{i \in \mathbb{N}}\) fulfills the conditions of Corollary 2.7. First, we show that \(\{T_{i}\}_{i \in \mathbb{N}}\) has the mixed monotone property. Let \(x^{1}, u^{1}\in X\) with \(x^{1}\leq u^{1}\) and \(t, s \in [0, T]\), then we have
Given that \(x^{1}(t)\leq u^{1}(t)\) for all \(t\in [0, T]\) and based on our assumption (3.2), we have
that is, \(T_{i}(u^{1}, x^{2}, x^{3} )(t)\geq T_{i}(x^{1}, x^{2}, x^{3} )(t) \). For \(x^{2}, u^{2}\in X\) with \(x^{2}\leq u^{2}\) and \(t, s \in [0, T]\), then we have
Given that \(x^{2}(t)\leq u^{2}(t)\) for all \(t\in [0, T]\) and based on our assumption (3.2), we have
that is, \(T_{i}(x^{1}, x^{2}, x^{3} )(t)\geq T_{i}(x^{1}, u^{2}, x^{3} )(t)\). Similarly, we have
that is, \(T_{i}(x^{1}, x^{2}, x^{3} )(t)\leq T_{i}(x^{1}, x^{2}, u^{3} )(t) \). So, \(\{T_{i}\}_{i \in \mathbb{N}}\) has the mixed monotone property. Now, we estimate \(d(T_{i}(x^{1}, x^{2}, x^{3} ), T_{j}(u^{1}, u^{2}, u^{3} ))\) for \(x^{1} \preceq u^{1}, u^{2} \preceq x^{2}, x^{3} \preceq u^{3}\) or \(x^{1} \succeq u^{1}, u^{2} \succeq x^{2}, x^{3} \succeq u^{3}\) and with \(\{T_{i}\}_{i \in \mathbb{N}}\) having the mixed monotone property, we get
Now, for all \(t\in [0, T]\) by using (3.2), we have
Consequently,
Let \(\alpha, \beta, \gamma \) be the same as (v); then we have
If \(x^{1}_{0}= \alpha, x^{2}_{0}=\beta, x^{3}_{0}=\gamma \), then all assumptions of Corollary 2.7 are fulfilled. So, there exists a triple fixed point \((x^{1}, x^{2}, x^{3})\) for the operator \(\{T_{i}\}_{i \in \mathbb{N}}\); that is, \(T_{i}(x^{1}, x^{2}, x^{3})=x^{1},T_{i}(x^{2}, x^{1}, x^{2}) = x^{2}\), and \(T_{i}(x^{3}, x^{2}, x^{1}) =x^{3}\) for \(i \in \mathbb{N}\). □
4 Application 2
Now if we consider the sequence of the integral equations system below, in which
for all \(t, s \in [0, T]\), for some \(T > 0\), then, similar to Theorem 3.1, this sequence of the integral equations system with the conditions given below will have a simultaneous solution.
Let \(X=C([0, T], \mathbb{R})\) be equipped with metric defined in Sect. 3 and “⪯” be the partial order on X. Thus, \((X, d, \preceq )\) is a partially ordered complete generalized metric space. For (4.1) we consider the following hypotheses:
-
(i)
\(f_{i}, g_{i}, h_{i} \in [0, T]\times [0, T] \times \mathbb{R} \longrightarrow \mathbb{R}^{2}\) are continuous;
-
(ii)
\(v:[0, T]\longrightarrow \mathbb{R}\) is continuous;
-
(iii)
there exists \(\rho:[0, T]\longrightarrow M_{2\times 2}([0, T])\), such that, for all \(x^{1}, x^{2}\in X\), we have
$$\begin{aligned} \begin{aligned} &0\leq \bigl\vert f_{i} \bigl(t, s, x^{1}(s) \bigr) - f_{i} \bigl(t, s, x^{2}(s) \bigr) \bigr\vert \leq \rho _{1}(t)d\bigl(x^{1}, x^{2}\bigr), \\ &0\leq \bigl\vert g_{i}\bigl(t, s, x^{2}(s) \bigr) - g_{i}\bigl(t, s, x^{1}(s) \bigr) \bigr\vert \leq \rho _{2}(t)d\bigl(x^{1}, x^{2}\bigr), \\ &0\leq \bigl\vert h_{i}\bigl(t, s, x^{1}(s) \bigr) - h_{i} \bigl(t, s, x^{2}(s) \bigr) \bigr\vert \leq \rho _{3}(t)d\bigl(x^{1}, x^{2}\bigr), \end{aligned} \end{aligned}$$(4.2)for all \(s, t\in [0, T] \) with and ;
-
(iv)
we suppose that \(\rho _{1}(t)+\rho _{2}(t)+\rho _{3}(t)<1\) and
$$ \rho (t) = \max \bigl\{ \rho _{1}(t), \rho _{2}(t), \rho _{3}(t)\bigr\} ; $$ -
(v)
there are functions \(\alpha, \beta, \gamma: [0, T] \longrightarrow \mathbb{R}\) which are continuous, such that
$$\begin{aligned} &\alpha \leq \int ^{T}_{0} (f_{i} \bigl(t, s, \alpha (s)\bigr) + g_{i} \bigl(t, s, \beta (s)\bigr)+ h_{i} \bigl(t, s, \gamma (s)\bigr) \,ds+v(t), \\ &\beta \geq \int ^{T}_{0} (f_{i} \bigl(t, s, \beta (s)\bigr) + g_{i} \bigl(t, s, \alpha (s)\bigr)+ h_{i} \bigl(t, s, \beta (s)\bigr)\, ds+v(t), \\ &\gamma \leq \int ^{T}_{0} (f_{i} \bigl(t, s, \gamma (s)\bigr) + g_{i} \bigl(t, s, \beta (s)\bigr)+ h_{i} \bigl(t, s, \alpha (s)\bigr)\, ds+v(t). \end{aligned}$$
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Hadi Bonab, S., Abazari, R., Bagheri Vakilabad, A. et al. Generalized metric spaces endowed with vector-valued metrics and matrix equations by tripled fixed point theorems. J Inequal Appl 2020, 204 (2020). https://doi.org/10.1186/s13660-020-02472-z
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DOI: https://doi.org/10.1186/s13660-020-02472-z