A tripled coincidence point technique for solving integral equations via an upper class of type II

1.   Background and preliminaries

The fixed point theory is one of the most powerful and productive tools from the nonlinear analysis and it can be considered to be the kernel of the nonlinear analysis since 1960. It has been classified into two major areas: metric fixed point theory and topological fixed point theory. The fixed point theory finds its roots with the method of successive approximations to prove the existence of solutions of differential equations introduced independently by Liouville ^[1] in 1837 and Picard ^[2] in 1890. While, officially, it was launched at the start of the 20th century as an important part of functional analysis. The best known result from the fixed point theory is Banach Contraction Principle ^[3] (1922), which can be considered to be the beginning of this theory. In a metric space, setting it can be briefly stated as follows: Every contraction on a complete metric space admits a unique fixed point. Furthermore, it provides a fixed point approximation algorithm as the limit of an iterated sequence. This result has been extended and generalized last years in various directions. Among these extensions, we cite the main result of Jleli and Samet ^[4], where they initiated a new generalized metric, called as $JS$-metric. It is given as follows:

Definition 1.1. ^[4] Let $\xi :\mho \times \mho \rightarrow \lbrack 0, +\infty)$ be a mapping justifying, for each $\vartheta, \varpi \in \mho$,

$(i)$ if $\xi (\vartheta, \varpi) = 0$, then $\vartheta = \varpi;$

$(ii)$ $\xi (\vartheta, \varpi) = \xi (\varpi, \vartheta);$

$(iii)$ there is $Z > 0$ so that

Then $\xi$ is a $JS$-metric on $\mho$ and the pair $\left(\mho, \xi \right)$ is called a $JS$-metric space.

Definition 1.2. Let $\left(\mho, \xi \right)$ be a $JS$-metric space and $\{x_n\}$ be a sequence in $\left(\mho, \xi \right)$.

$(i)$ $\{x_n\}$ is said to be $\xi$-convergent to $x$ in $\mho$ if $\lim_{n\rightarrow \infty} \xi(x_n, x) = 0$.

$(ii)$ $\{x_n\}$ is said to be $\xi$-Cauchy sequence if $\lim_{n, m\rightarrow \infty} \xi(x_n, x_m) = 0$.

$(iii)$ $\left(\mho, \xi \right)$ is said to be $\xi$-complete if every $\xi$-Cauchy sequence in $\mho$ is $\xi$-convergent to some point $x\in \mho$.

After the work of Jleli and Samet ^[4], there is an immense literature in fixed point theory and its applications in this setting. For more details, see ^[5,6,7].

In paper ^[8], the notions of mixed-monotone functions and coupled fixed points were initiated and studied. Under partially ordered metric spaces (POMSs) and abstract spaces, some main results in this direction have been driven, for broadening, see ^{[4,8,11,12,13,14]}.

In 2011, Berinde and Borcut ^[15], introduced the definition of mixed monotone property and the definition of tripled fixed point for a mapping $T : \mho ^{3} = :\mho\times \mho\times\mho \rightarrow \mho$ and established tripled fixed point theorems for contractive type mappings having that property in partially ordered metric spaces. Later, Borcut ^[16] and Berinde and Borcut ^[17] have introduced the notion of a tripled coincidence point for a pair of nonlinear contractive mappings $T : \mho ^{3} \rightarrow \mho$ and $f: \mho \rightarrow \mho$. Subsequently, Aydi et al. ^[18] have proved some new tripled fixed point theorems in abstract metric spaces. Further works dealing in this direction have been appeared, see ^[19,20,21].

Combining the notion of triangular $\alpha -$admissible property and the concept of upper class of type $II$, the aim of this paper is to establish some tripled coincidence point results for a pair of generalized contraction type mappings $\Xi :\mho ^{3}\rightarrow \mho$ and $r:\mho \rightarrow \mho$ in the context of $JS$-metric spaces equipped with a partial order. The obtained results are supported by some concrete examples. At the end, we ensure the existence of a solution for a system of non-homogeneous and homogeneous integral equations.

2.   Main results

We start this section with the following concepts.

Consider a nonempty set $\mho$. Assume that $\Xi :\mho ^{3}\rightarrow \mho$ and $r:\mho \rightarrow \mho$ are two mappings. We say that $r$ commutes with $\Xi$ if

According to ^[4], for a partial order $\preceq,$ define $\nabla _{\preceq } = \{\left(\vartheta, \varpi \right) \in \mho ^{2}:\vartheta \preceq \varpi \}.$ Then $\Xi$ has the $\preceq -r$ monotone property, if for each $\vartheta, \varpi, \varkappa \in \mho,$

and

After that, we generalize the notion of triangular $\alpha -$admissible property as follows:

Definition 2.1. Assume that $\Xi :\mho ^{3}\rightarrow \mho$, $r:\mho \rightarrow \mho$ and $\alpha :\mho ^{3}\times \mho ^{3}\rightarrow \lbrack 0, +\infty]$ so that the postulates below hold:

$(i)$ If $\alpha \left(\left(r\vartheta, r\varpi, r\varkappa \right), \left(r\widetilde{\vartheta }, r\widetilde{\varpi }, r \widetilde{\varkappa }\right) \right) \geq 1,$ then

$(ii)$ If ${\alpha}\left(\left(r\vartheta, r\varpi, r\varkappa \right), \left(r\widetilde{\vartheta }, r\widetilde{\varpi }, r\widetilde{ \varkappa }\right) \right) \geq 1$ and

then

Here, we say that $\Xi$ and $r$ is a generalized triangular $\alpha -$ admissible $(\alpha _{A}^{GT},$ for short).

Next, we extend the concept of upper class of type $II$ ^[11] as follows.

Definition 2.2. Let $\wp :[0, +\infty]^{3}\rightarrow \mathbb{R} \cup \{+\infty \}$ and $\ell :[0, +\infty]^{2}\rightarrow \mathbb{R} \cup \{+\infty \}$ be two given mappings. We say that the pair $(\wp, \ell)$ is an upper class of type II $(\Omega _{II},$ for short) if for all $\vartheta, \varpi, \varkappa, \widetilde{\vartheta }, \widetilde{\varpi }, \widetilde{\varkappa }\in \lbrack 0, +\infty],$ the postulates below hold:

$(i)$ $\wp (1, 1, \varpi)\leq \wp (\vartheta, \varpi, \varkappa),$ whenever $1\leq \varpi, \varkappa;$

$(ii)$ $\ell (\widetilde{\vartheta }, \widetilde{\varpi })\leq \ell (1, \widetilde{\varpi }),$ whenever $\widetilde{\vartheta }\leq 1;$

$(iii)$ $\varkappa \leq \widetilde{\vartheta } \widetilde{\varpi },$ whenever $\wp (1, 1, \varkappa)\leq \ell (\widetilde{ \vartheta }, \widetilde{\varpi }).$

Example 2.3. Let $\wp, \ell :[0, +\infty]^{2}\rightarrow \mathbb{R} \cup \{+\infty \}$ be a function described as follows:

$(1)$ $\wp \left(\vartheta, \varpi, \varkappa \right) = \left\{ \begin{array}{cc} (\varkappa +z)^{\vartheta \varpi }, & \text{if }\vartheta, \varpi, \varkappa \in \lbrack 0, +\infty), \\ +\infty, & \text{otherwise, } \end{array} \right.$ $\ \ell \left(\widetilde{\vartheta }, \widetilde{\varpi }\right) = \left\{ \begin{array}{cc} \widetilde{\vartheta }\widetilde{\varpi }+z, & \text{if }\widetilde{ \vartheta }, \widetilde{\varpi }\in \lbrack 0, +\infty), \\ +\infty, & \text{otherwise}. \end{array} \right.$

$(2)$ $\wp \left(\vartheta, \varpi, \varkappa \right) = \left\{ \begin{array}{cc} (\vartheta +z)^{\varkappa \varpi }, & \text{if }\vartheta, \varpi, \varkappa \in \lbrack 0, +\infty), \\ +\infty, & \text{otherwise, } \end{array} \right.$ $\ \ell \left(\widetilde{\vartheta }, \widetilde{\varpi }\right) = \left\{ \begin{array}{cc} \left(1+z\right) ^{\widetilde{\vartheta }\widetilde{\varpi }}, & \text{if } \widetilde{\vartheta }, \widetilde{\varpi }\in \lbrack 0, +\infty), \\ +\infty, & \text{otherwise}. \end{array} \right.$

$(3)$ $\wp \left(\vartheta, \varpi, \varkappa \right) = \left\{ \begin{array}{cc} \varkappa, & \text{if }\vartheta, \varpi, \varkappa \in \lbrack 0, +\infty), \\ +\infty, & \text{otherwise, } \end{array} \right.$ $\ \ell \left(\widetilde{\vartheta }, \widetilde{\varpi }\right) = \left\{ \begin{array}{cc} \widetilde{\vartheta }\widetilde{\varpi }, & \text{if }\widetilde{\vartheta }, \widetilde{\varpi }, \widetilde{\varkappa }\in \lbrack 0, +\infty), \\ +\infty, & \text{otherwise}. \end{array} \right.$

$(4)$ $\wp \left(\vartheta, \varpi, \varkappa \right) = \left\{ \begin{array}{cc} \vartheta ^{i}\varpi ^{j}\varkappa ^{k} & \text{if }\vartheta, \varpi, \varkappa \in \lbrack 0, +\infty), \\ +\infty, & \text{otherwise, } \end{array} \right.$ $\ \ell \left(\widetilde{\vartheta }, \widetilde{\varpi }\right) = \left\{ \begin{array}{cc} \left(\widetilde{\vartheta }\widetilde{\varpi }\right) ^{k}, & \text{if } \widetilde{\vartheta }, \widetilde{\varpi }\in \lbrack 0, +\infty), \\ +\infty, & \text{otherwise}, \end{array} \right.$

for $z > 1$ and $i, j, k\in \mathbb{N}.$ Then the pair $(\wp, \ell)$ is $\Omega _{II}.$

Here, the triplet $(\mho, \xi, \preceq)$ is a complete POJSM-space and $\widetilde{\Phi }$ consists of all $\phi :[0, +\infty]^{3}\rightarrow \lbrack 0, 1)$ satisfying:

$(\phi _{1})$ For any $\vartheta, \varpi \in \lbrack 0, +\infty],$ $\phi (\vartheta, \varpi) = \phi (\varpi, \vartheta).$

$(\phi _{2})$ For $\{\vartheta _{i}\},$ $\{\varpi _{i}\}\subseteq \lbrack 0, +\infty],$ $\lim\limits_{i\rightarrow \infty }\phi (\vartheta _{i}, \varpi _{i}, \varkappa _{i}) = 1\Longrightarrow \lim\limits_{i\rightarrow \infty }\vartheta _{i} = \lim\limits_{i\rightarrow \infty }\varpi _{i} = \lim\limits_{i\rightarrow \infty }\varkappa _{i} = 0.$

Also, we define the mappings $\Xi :\mho ^{3}\rightarrow \mho$ and $r:\mho \rightarrow \mho$ so that the following properties hold:

(a) $\Xi \left(\mho ^{3}\right) \subseteq r(\mho);$

(b) $\Xi$ is $\preceq -r$ monotone;

(c) $r$ commutes with $\Xi$ and is $\xi -$continuous.

Now, our first theorem becomes valid for presentation, which generalizes the results of ^[7].

Theorem 2.4. Assume that the postulates below are fulfilled:

($\triangle _{i}$) For any $\Omega _{II}$ of the pair $(\wp, \ell),$ there exist $\alpha :\mho ^{3}\times \mho ^{3}\rightarrow \lbrack 0, +\infty]$ and $\phi \in \Phi$ fulfilling, for any $\left(r\vartheta, r\widetilde{ \vartheta }\right) \in \nabla _{\preceq },$ $\left(r\varpi, r\widetilde{ \varpi }\right) \in \nabla _{\preceq }$ and $\left(r\varkappa, r\widetilde{ \varkappa }\right) \in \nabla _{\preceq },$ the below inequality is verified:

where

($\triangle _{ii}$) $\Xi$ and $r$ is $\alpha _{A}^{GT},$ and there exist $\vartheta _{0}, \varpi _{0}, \varkappa _{0}\in \mho,$

so that

and

($\triangle _{iii}$) If $\lim\limits_{i\rightarrow \infty }\xi \left(r\vartheta _{i}, r\vartheta _{i+1}\right) = 0,$ $\lim\limits_{i\rightarrow \infty }\xi \left(r\varpi _{i}, r\varpi _{i+1}\right) = 0$ and $\lim\limits_{i\rightarrow \infty }\xi \left(r\varkappa _{i}, r\varkappa _{i+1}\right) = 0,$ then

where $\{\vartheta _{i}\},$ $\{\varpi _{i}\}$ and $\{\varkappa _{i}\}$ are sequences in $\mho;$ ($\triangle _{iv}$) $\Xi$ is $\xi -$continuous.

Then $\Xi$ and $r$ have a tripled coincidence point (TCP) in $\mho.$

Proof. Assume that $\vartheta _{0}, \varpi _{0}, \varkappa _{0}\in \mho$ justifying assertion $(\triangle _{ii}).$ Because $\Xi \left(\mho ^{3}\right) \subseteq r(\mho),$ we can select $\vartheta _{1}, \varpi _{1}, \varkappa _{1}\in \mho$ so that $r\vartheta _{1} = \Xi \left(\vartheta _{0}, \varpi _{0}, \varkappa _{0}\right),$ $r\varpi _{1} = \Xi \left(\varpi _{0}, \varkappa _{0}, \vartheta _{0}\right)$ and $r\varkappa _{1} = \Xi \left(\varkappa _{0}, \vartheta _{0}, \varpi _{0}\right).$ Analogously, $r\vartheta _{2} = \Xi \left(\vartheta _{1}, \varpi _{1}, \varkappa _{1}\right),$ $r\varpi _{2} = \Xi \left(\varpi _{1}, \varkappa _{1}, \vartheta _{1}\right)$ and $r\varkappa _{2} = \Xi \left(\varkappa _{1}, \vartheta _{1}, \varpi _{1}\right).$ In the same scenario, $\{\vartheta _{i}\},$ $\{\varpi _{i}\}$ and $\{\varkappa _{i}\}$ are obtained with

For some natural number $i_{0},$ if $r\vartheta _{i_{0}+1} = r\vartheta _{i_{0}},$ $r\varpi _{i_{0}+1} = r\varpi _{i_{0}}$ and $r\varkappa _{i_{0}+1} = r\varkappa _{i_{0}},$ then $\Xi$ and $r$ have a TCP. So, for some positive integer $i,$ assume that

Based on assumption $(\triangle _{ii}),$ we have

Because $\Xi$ is $\preceq -r$ monotone,

that is,

Repeating the same approach, we get

From the transitivity of $\preceq,$ one can obtain

Again, applying the postulate $(\triangle _{ii}),$ we have

Since $\Xi$ and $r$ are $\alpha _{A}^{GT},$

By induction, one can deduce that

Analogously, we can obtain

and

Since $\Xi$ and $r$ are $\alpha _{A}^{GT},$

Next, we want to prove $\lim\limits_{i\rightarrow \infty }\xi \left(r\vartheta _{i}, r\vartheta _{i+1}\right) = 0,$ $\lim\limits_{i\rightarrow \infty }\xi \left(r\varpi _{i}, r\varpi _{i+1}\right) = 0$ and $\lim\limits_{i\rightarrow \infty }\xi \left(r\varkappa _{i}, r\varkappa _{i+1}\right) = 0.$ For this, we use the opposite technique, suppose that either $\lim\limits_{i\rightarrow \infty }\xi \left(r\vartheta _{i}, r\vartheta _{i+1}\right) \neq 0$, or $\lim\limits_{i\rightarrow \infty }\xi \left(r\varpi _{i}, r\varpi _{i+1}\right) \neq 0$ or $\lim\limits_{i\rightarrow \infty }\xi \left(r\varkappa _{i}, r\varkappa _{i+1}\right) \neq 0.$ Then there is $\epsilon > 0$ for which we have a subsequence $\{i_{s}\}$ so that $s\leq i_{s}$ and

Consider

Again,

and

The inequalities (2.1)–(2.3) lead to

and

Because $\phi (\vartheta, \varpi, \varkappa)\in \lbrack 0, 1)$ for any $\vartheta, \varpi, \varkappa \in \lbrack 0, +\infty],$ we get

It follows from (2.4)–(2.7) that $\max \left\{ \xi \left(r\vartheta _{i_{s}}, r\vartheta _{i_{s}+1}\right), \xi \left(r\varpi _{i_{s}}, r\varpi _{i_{s}+1}\right), \xi \left(r\varkappa _{i_{s}}, r\varkappa _{i_{s}+1}\right) \right\}$

Using this concept, we can write

Select $v_{s}$ so that

Consider

If $\nabla < 1,$ then

This contradicts the hypothesis. If $\nabla = 1,$ for suitability, we assume that

Because $\phi \in \widetilde{\Phi },$

That is, there is an $s_{0}\in \mathbb{N}$ so that

Thus, we get

a contradiction, hence, we have

Now, we claim that $\{r\vartheta _{i}\},$ $\{r\varpi _{i}\}$ and $\{r\varkappa _{i}\}$ are $\xi -$Cauchy sequences. Suppose that $\{r\vartheta _{i}\},$ $\{r\varpi _{i}\}$ and $\{r\varkappa _{i}\}$ are not $\xi -$Cauchy sequences, so for each $s\in \mathbb{N}$ with subsequences $i_{s}, j_{s}\geq s$ such that

Consider

and

Based on (2.9)–(2.11) and the above properties, one can obtain

and

respectively. Using (2.8), we get

From (2.12)–(2.15), we have

Thus,

Select $v_{s}$ so that

Define

If $\nabla ^{\ast } < 1,$ then

This is impossible to happen because of our hypothesis.

If $\nabla ^{\ast } = 1,$ for convenience, consider

Because $\phi \in \widetilde{\Phi },$

That is, there is an $s_{0}\in \mathbb{N}$ so that

Thus, we get

This is a contradiction, therefore, $\{r\vartheta _{i}\},$ $\{r\varpi _{i}\}$ and $\{r\varkappa _{i}\}$ are $\xi -$Cauchy sequences. The completeness of $(\mho, \xi)$ implies that for some $u, u^{\prime }, u^{\prime \prime }\in \mho,$

Since $r$ is continuous, we have

Also, the continuity of $\Xi$ leads to

By alteration between $\Xi$ and $r,$ we have that $\Xi \left(u, u^{\prime }, u^{\prime \prime }\right) = ru,$ $\Xi \left(u^{\prime }, u^{\prime \prime }, u\right) = ru^{\prime }$ and $\Xi \left(u^{\prime \prime }, u, u^{\prime }\right) = ru^{\prime \prime },$ therefore $\Xi$ and $r$ have a TCP.

Corollary 2.5. The results of Theorem 2.4 are still true if we replace the stipulation $(\triangle _{i})$ with one of the hypotheses below: (note $\aleph \left(\left(r\vartheta, r\widetilde{\vartheta }\right), \left(r\varpi, r\widetilde{\varpi }\right), \left(r\varkappa, r\widetilde{ \varkappa }\right) \right)$ is defined in the above theorem):

$(\dagger _{1})$ There exist $\phi \in \Phi$ and $z > 1,$ fulfilling, for any $\left(r\vartheta, r\widetilde{\vartheta }\right) \in \nabla _{\preceq },$ $\left(r\varpi, r\widetilde{\varpi }\right) \in \nabla _{\preceq }$ and $\left(r\varkappa, r\widetilde{\varkappa }\right) \in \nabla _{\preceq },$ the following inequality is satisfied:

$(\dagger _{2})$ There exist $\phi \in \Phi$ and $z > 1,$ fulfilling, for any $\left(r\vartheta, r\widetilde{\vartheta }\right) \in \nabla _{\preceq },$ $\left(r\varpi, r\widetilde{\varpi }\right) \in \nabla _{\preceq }$ and $\left(r\varkappa, r\widetilde{\varkappa }\right) \in \nabla _{\preceq },$ the following inequality is satisfied:

$(\dagger _{3})$ There exists $\phi \in \Phi,$ justifying, for any $\left(r\vartheta, r\widetilde{\vartheta }\right) \in \nabla _{\preceq },$ $\left(r\varpi, r\widetilde{\varpi }\right) \in \nabla _{\preceq }$ and $\left(r\varkappa, r\widetilde{\varkappa }\right) \in \nabla _{\preceq },$ the following inequality holds:

$(\dagger _{4})$ There exists $\phi \in \Phi,$ satisfying, for any $\left(r\vartheta, r\widetilde{\vartheta }\right) \in \nabla _{\preceq },$ $\left(r\varpi, r\widetilde{\varpi }\right) \in \nabla _{\preceq }$ and $\left(r\varkappa, r\widetilde{\varkappa }\right) \in \nabla _{\preceq },$ the inequality below is obtained:

for all positive integers $j$ and $k.$

Proof. The proof follows immediately from Example 2.3 by applying the following in Theorem 2.4.

$(\dagger _{1})$ Use the value of $\wp$ and $\ell$ from Example 2.3(1).

$(\dagger _{2})$ Use the value of $\wp$ and $\ell$ from Example 2.3(2).

$(\dagger _{3})$ Use the value of $\wp$ and $\ell$ from Example 2.3(3).

$(\dagger _{4})$ Use the value of $\wp$ and $\ell$ from Example 2.3(4).

Remark 2.6 ● In a $b-$metric and a $b-$metric-like space, Corollary 2.5 holds.

● The stipulation $(\dagger _{3})$ of Corollary 2.5 generalizes Theorem 3.2 of ^[13].

● The stipulation $(\dagger _{4})$ of Corollary 2.5 extends and unifies Theorem 3.1 of ^[7] when $j = k = 1.$

Example 2.7. Consider $\mho = [0, +\infty].$ Define $\xi \left(\vartheta, \varpi \right) = \max \{\vartheta, \varpi \},$ $\vartheta, \varpi \in \mho.$ Describe the mappings $\Xi :\mho ^{3}\rightarrow \mho,$ $r:\mho \rightarrow \mho$ and $\alpha :\mho ^{3}\times \mho ^{3}\rightarrow \lbrack 0, +\infty]$ by

and

respectively. Assume that $\vartheta \leq \widetilde{\vartheta },$ $\varpi \leq \widetilde{\varpi }$ and $\varkappa \leq \widetilde{\varkappa }.$ Then

Hence, Stipulation $(\dagger _{3})$ of Corollary 2.5 holds for $\phi (m, z) = \frac{1}{3},$ where $m, z\in \lbrack 0, +\infty].$ In an easy way, it can be shown that all the hypotheses of Theorem 2.4 are fulfilled, and then there is a TCP for the mappings $\Xi$ and $r.$ Notice that, here $\xi$ is not a metric on $\mho$ because, if we take $\vartheta \leq \varpi \leq \varkappa \leq \widetilde{\varkappa }$, and $\vartheta \leq \widetilde{\varpi }\leq \widetilde{\vartheta }\leq \widetilde{ \varkappa },$ we have

Therefore, in the sense of TCPs, Theorem 3.1 in ^[13] and one of the hypotheses of Theorems 3.1 and 3.2 in ^[7] fails in studying the existence of a TCP for $r$ and $\Xi.$

Let $\Phi ^{\ast }$ represent the class of mappings $\phi ^{\ast }:[0, +\infty]\rightarrow \lbrack 0, 1)$ so that

The second part of this section is neglected the continuity condition for the mapping $\Xi$ as follows:

Theorem 2.8. Let the following postulates be satisfied:

$(h_{i})$ For any $\Omega _{II}$ of the pair $(\wp, \ell),$ there exist $\alpha :\mho ^{3}\times \mho ^{3}\rightarrow \lbrack 0, +\infty]$ and $\phi ^{\ast }\in \Phi ^{\ast }$ fulfilling, for any $\left(r\vartheta, r \widetilde{\vartheta }\right) \in \nabla _{\preceq },$ $\left(r\varpi, r \widetilde{\varpi }\right) \in \nabla _{\preceq }$ and $\left(r\varkappa, r \widetilde{\varkappa }\right) \in \nabla _{\preceq },$ the below inequality is justified:

where

$(h_{ii})$ $\Xi$ and $r$ is $\alpha _{A}^{GT}$ and there exist $\vartheta _{0}, \varpi _{0}, \varkappa _{0}\in \mho,$

so that

and

$(h_{iii})$ If $\lim\limits_{i\rightarrow \infty }\xi \left(r\vartheta _{i}, r\vartheta _{i+1}\right) = 0,$ $\lim\limits_{i\rightarrow \infty }\xi \left(r\varpi _{i}, r\varpi _{i+1}\right) = 0$ and $\lim\limits_{i\rightarrow \infty }\xi \left(r\varkappa _{i}, r\varkappa _{i+1}\right) = 0,$ then

where $\{\vartheta _{i}\},$ $\{\varpi _{i}\}$ and $\{\varkappa _{i}\}$ are sequences in $\mho;$ $(h_{iv})$ If $\{\vartheta _{i}\},$ $\{\varpi _{i}\}$ and $\{\varkappa _{i}\}$ are sequences in $\mho$ with $\left(r\vartheta _{i}, r\vartheta _{i+1}\right),$ $\left(r\varpi _{i}, r\varpi _{i+1}\right)$, $\left(r\varkappa _{i}, r\varkappa _{i+1}\right) \in \nabla _{\preceq },$

and $\lim\limits_{i\rightarrow \infty }\xi \left(r\vartheta _{i}, u\right) = 0,$ $\lim\limits_{i\rightarrow \infty }\xi \left(r\varpi _{i}, u^{\prime }\right) = 0$, $\lim\limits_{i\rightarrow \infty }\xi \left(r\varkappa _{i}, u^{\prime \prime }\right) = 0,$ for each $u, u^{\prime }, u^{\prime \prime }\in \mho,$ then $\left(r\vartheta _{i}, ru\right),$ $\left(r\varpi _{i}, ru^{\prime }\right)$, $\left(r\varkappa _{i}, u^{\prime \prime }\right) \in \nabla _{\preceq },$

$(h_{v})$ There is $\Re \in (0, 1]$ so that

Then $\Xi$ and $r$ have a TCP.

Proof. Based on Theorem 2.4, the sequences $\{r\vartheta _{i}\},$ $\{r\varpi _{i}\}$ and $\{r\varkappa _{i}\}$ are obtained. Furthermore, replacing $\phi \left(\xi \left(r\vartheta, r\widetilde{\vartheta }\right), \xi \left(r\varpi, r\widetilde{\varpi }\right), \xi \left(r\varkappa, r \widetilde{\varkappa }\right) \right)$ with $\phi ^{\ast }\left(\xi \left(r\vartheta, r\widetilde{\vartheta }\right), \xi \left(r\varpi, r\widetilde{ \varpi }\right), \xi \left(r\varkappa, r\widetilde{\varkappa }\right) \right)$ in Theorem 2.4, where $\vartheta, \widetilde{ \vartheta }, \varpi, \widetilde{\varpi }, \varkappa, \widetilde{\varkappa }\in \mho,$ we conclude that the three sequences are Cauchy in $(\mho, \xi).$ The completness of this space leads to for some $u, u^{\prime }, u^{\prime \prime }\in \mho,$

Since $r$ is continuous,

By postulates $(\triangle _{i})$ and $(\triangle _{ii}),$ we get

and

where

Accordingly,

and

Assume that $ru\neq \Xi \left(u, u^{\prime }, u^{\prime \prime }\right)$ or $ru^{\prime }\neq \Xi \left(u^{\prime }, u^{\prime \prime }, u\right)$ or $ru^{^{\prime \prime }}\neq \Xi \left(u^{\prime \prime }, u, u^{\prime }\right),$ that is,

Applying hypothesis $(h_{v}),$ there is $\Re \in (0, 1]$ so that

and

Hence

Because $1\leq \frac{1}{\Re },$ we have

this implies that

Then there exists a subsequence $\max \left\{ \begin{array}{c} \xi \left(\Xi \left(r\vartheta _{i_{s}}, r\varpi _{i_{s}}, r\varkappa _{i_{s}}\right), \Xi \left(u, u^{\prime }, u^{\prime \prime }\right) \right), \\ \xi \left(\Xi \left(r\varpi _{i_{s}}, r\varkappa _{i_{s}}, r\vartheta _{i_{s}}\right), \Xi \left(u^{\prime }, u^{\prime \prime }, u\right) \right), \\ \xi \left(\Xi \left(r\varkappa _{i_{s}}, r\vartheta _{i_{s}}, r\varpi _{i_{s}}\right), \Xi \left(u^{\prime \prime }, u, u^{\prime }\right) \right) \end{array} \right\}$ so that

Passing $i\rightarrow \infty$ in (2.16), we obtain that

Using (2.17)–(2.19), one can write

Putting $s\rightarrow \infty$ on both sides, we get

Thus, $\lim_{s\rightarrow \infty }Q\left(\left(rr\vartheta _{i_{s}}, ru\right), \left(rr\varpi _{i_{s}}, ru^{\prime }\right), \left(rr\varkappa _{i_{s}}, ru^{\prime \prime }\right) \right) = 0.$ This contradicts (2.20). Hence $ru = \Xi \left(u, u^{\prime }, u^{\prime \prime }\right)$ and $ru^{\prime } = \Xi \left(u^{\prime }, u^{\prime \prime }, u\right)$ and $ru^{^{\prime \prime }} = \Xi \left(u^{\prime \prime }, u, u^{\prime }\right).$ Therefore, the element$\left(u, u^{\prime }, u^{\prime \prime }\right)$ is a TCP of $r$ and $\Xi.$

Corollary 2.9. The statements of Theorem 2.8 are still valid if we replace the stipulation $(h_{i})$ with one of the assumptions below: (note $Q\left(\left(r\vartheta, r\widetilde{\vartheta }\right), \left(r\varpi, r \widetilde{\varpi }\right), \left(r\varkappa, r\widetilde{\varkappa } \right) \right)$ is defined in Theorem 2.8):

$(1)$ There exist $\phi ^{\ast }\in \Phi ^{\ast }$ and $z > 1,$ fulfilling, for any $\left(r\vartheta, r\widetilde{\vartheta }\right) \in \nabla _{\preceq },$ $\left(r\varpi, r\widetilde{\varpi }\right) \in \nabla _{\preceq }$ and $\left(r\varkappa, r\widetilde{\varkappa }\right) \in \nabla _{\preceq },$ the following inequality is obtained:

$(2)$ There exist $\phi ^{\ast }\in \Phi ^{\ast }$ and $z > 1,$ fulfilling, for any $\left(r\vartheta, r\widetilde{\vartheta }\right) \in \nabla _{\preceq },$ $\left(r\varpi, r\widetilde{\varpi }\right) \in \nabla _{\preceq }$ and $\left(r\varkappa, r\widetilde{\varkappa }\right) \in \nabla _{\preceq },$ the following inequality is verified:

$(3)$ There is $\phi ^{\ast }\in \Phi ^{\ast },$ fulfilling, for any $\left(r\vartheta, r\widetilde{\vartheta }\right) \in \nabla _{\preceq },$ $\left(r\varpi, r\widetilde{\varpi }\right) \in \nabla _{\preceq }$ and $\left(r\varkappa, r\widetilde{\varkappa }\right) \in \nabla _{\preceq }$ so that

$(4)$ There is $\phi ^{\ast }\in \Phi ^{\ast },$ satisfying, for any $\left(r\vartheta, r\widetilde{\vartheta }\right) \in \nabla _{\preceq },$ $\left(r\varpi, r\widetilde{\varpi }\right) \in \nabla _{\preceq }$ and $\left(r\varkappa, r\widetilde{\varkappa }\right) \in \nabla _{\preceq },$ the inequality below is obtained:

for all positive integers $j$ and $k.$

Proof. The proof follows immediately from Example 2.3 by applying the following in Theorem 2.8:

$(1)$ Use the value of $\wp$ and $\ell$ from Example 2.3(1).

$(2)$ Use the value of $\wp$ and $\ell$ from Example 2.3(2).

$(3)$ Use the value of $\wp$ and $\ell$ from Example 2.3(3).

$(4)$ Use the value of $\wp$ and $\ell$ from Example 2.3(4).

Example 2.10. Consider $\mho = [0, +\infty].$ Define $\xi \left(\vartheta, \varpi \right) = \left\vert \vartheta \right\vert +\left\vert \varpi \right\vert,$ for all $\vartheta, \varpi \in \mho.$ Define $\Xi :\mho ^{3}\rightarrow \mho,$ $r:\mho \rightarrow \mho$ and $\alpha :\mho ^{3}\times \mho ^{3}\rightarrow \lbrack 0, +\infty]$ by

and

respectively. Suppose that $\vartheta \leq \widetilde{\vartheta },$ $\varpi \leq \widetilde{\varpi }$ and $\varkappa \leq \widetilde{\varkappa }.$ When $z > 1$, we have

So, the condition $(2)$ of Corollary 2.9 is fulfilled for $\phi ^{\ast }(m) = \frac{3}{4},$ where $m\in \lbrack 0, +\infty].$ In an easy way, it can be shown that all assumptions of Theorem 2.8 are satisfied, therefore $\Xi$ and $r$ have a TCP.

3.   Solve a system of integral equations

This section serves as the basis and core of our paper in which the existence of a solution to a system of integral equations is discussed. This system takes the following form:

where $b\in \lbrack 0, 1].$ Consider $\mho = C[0, 1]$ equipped with $\xi \left(\vartheta, \varpi \right) = \sup_{b\in \lbrack 0, 1]}\left\vert \vartheta (b)-\varpi (b)\right\vert,$ for each $\vartheta, \varpi \in \mho.$ Clearly, $(\mho, \xi, \preceq)$ is a complete POJSM-space. Based on the theoretical results presented in the upper section of the paper, we are able to present theorems related to the existence of a solution to system (3.1) as follows:

Theorem 3.1. Consider the integral equations (3.1) via the following postulates:

$(\heartsuit _{i})$ The functions $\Lambda :[0, 1]\rightarrow \mathbb{R},$ $\Im _{s}:[0, 1]\times \lbrack 0, 1]\rightarrow \mathbb{R} ^{+}$ and $\Game _{s}:[0, 1]\times \mathbb{R} \rightarrow \mathbb{R} ^{+}$ are continuous, where $s = 1, 2, 3;$

$(\heartsuit _{ii})$ For $\vartheta _{1}, \vartheta _{2}\in \mho,$ if $\vartheta _{1}\leq \vartheta _{2},$ then $\Game _{1}(b, \vartheta _{1}(b))\leq \Game _{1}(b, \vartheta _{2}(b)),$ $\Game _{2}(b, \vartheta _{1}(b))\leq \Game _{2}(b, \vartheta _{2}(b))$ and $\Game _{3}(b, \vartheta _{1}(b))\leq \Game _{3}(b, \vartheta _{2}(b));$

$(\heartsuit _{iii})$ There exist $\widehat{A},$ $\widehat{B}$ and $\widehat{C}$ in $\mathbb{R}^{+}$ so that $3\widehat{A}\widehat{B}\widehat{C} < 1,$ $\int\limits_{0}^{1} \Im _{s}\left(b, c\right) dc\leq \widehat{A},$ $\int\limits_{0}^{1}\Im _{s}\left(b, c\right) \Game _{s}(c, \vartheta (c))dc\leq \widehat{C}$ and

where $\vartheta, \widetilde{\vartheta }\in \mho,$ $b\in \lbrack 0, 1]$ and $s = 1, 2, 3;$

$(\heartsuit _{iv})$ There exist $\vartheta _{0}, \varpi _{0}, \varkappa _{0}\in \mho$ such that $\vartheta _{0}\leq \Xi \left(\vartheta _{0}, \varpi _{0}, \varkappa _{0}\right),$ $\varpi _{0}\leq \Xi \left(\varpi _{0}, \varkappa _{0}, \vartheta _{0}\right)$ and $\varkappa _{0}\leq \Xi \left(\varkappa _{0}, \vartheta _{0}, \varpi _{0}\right);$

$(\heartsuit _{v})$ If $\{\vartheta _{i}\}, \{\varpi _{i}\}, \{\varkappa _{i}\}\in \mho$ so that $\lim\limits_{i\rightarrow \infty }\left\vert \vartheta _{i}-\vartheta _{i+1}\right\vert = 0,$ $\lim\limits_{i\rightarrow \infty }\left\vert \varpi _{i}-\varpi _{i+1}\right\vert = 0$ and $\lim\limits_{i\rightarrow \infty }\left\vert \varkappa _{i}-\varkappa _{i+1}\right\vert = 0,$ then

where $\vartheta _{0}, \varpi _{0}, \varkappa _{0}\in \mho.$

Then Problem (3.1) possesses a solution.

Proof. Define the mappings $\Xi :\mho ^{3}\rightarrow \mho$ and $r:\mho \rightarrow \mho$ by

and $r\vartheta = \vartheta$ for $\vartheta, \varpi, \varkappa \in \mho.$ Clearly $\Xi$ is $\xi -$continuous $\Xi \left(\mho ^{3}\right) \subseteq r(\mho)$ and $r$ is $\xi -$continuous and commutes with $\Xi.$ After that, we claim that $\Xi$ is $\preceq -r$ monotone. Suppose that $\vartheta _{1}, \vartheta _{2}, \varpi, \varkappa \in \mho,$ $\vartheta _{1}\leq \vartheta _{2}.$ Based on postulate $(\heartsuit _{ii}),$ we get

Similarly, one can show that $\Xi \left(\varkappa, \vartheta _{1}, \varpi \right) (b)\leq \Xi \left(\varkappa, \vartheta _{2}, \varpi \right) (b).$ Hence, $\Xi$ has the $\preceq -r$ monotone property.

Now, for $\left(\vartheta, \varpi, \varkappa \right), \left(\widetilde{ \vartheta }, \widetilde{\varpi }, \widetilde{\varkappa }\right) \in \mho ^{3},$ describe $\alpha :\mho ^{3}\times \mho ^{3}\rightarrow \lbrack 0, +\infty]$ by $\alpha \left(\left(\vartheta, \varpi, \varkappa \right), \left(\widetilde{\vartheta }, \widetilde{\varpi }, \widetilde{\varkappa }\right) \right) = 1.$ Obviously, $\Xi$ and $r$ is $\alpha _{A}^{GT}.$ Also, assumptions $(\triangle _{ii})$ and $(\triangle _{iii})$ of Theorem 2.4 follow immediately from postulates $(\heartsuit _{iv})$ and $(\heartsuit _{v})$, respectively. In order to be able to validate Theorem 2.4, it remains to fulfill condition $(\triangle _{i})$, which can be replaced by condition $(\dagger _{3})$ of Corollary 2.5. Therefore, let $\vartheta, \varpi, \varkappa, \widetilde{\vartheta }, \widetilde{\varpi }, \widetilde{\varkappa }\in \mho.$ If $\vartheta \preceq \widetilde{\vartheta },$ $\varpi \preceq \widetilde{\varpi }$ and $\varkappa \preceq \widetilde{\varkappa },$ then using postulate $(\heartsuit _{iii})$, we have

Consider $\phi = 3\widehat{A}\widehat{B}\widehat{C} < 1,$ then $\phi \in \widetilde{\Phi }.$ Hence the condition $(\dagger _{3})$ of Corollary 2.5 is fulfilled. Therefore, a tripled fixed point of $\Xi$ exists. Thus, there is a solution to the problem (3.1).

It should be noted that in the above theorem, $JS$-metric $\xi$ is a usual metric, therefore, the results seem easy to obtain, for this reason, we define $JS$-metric $\xi :\mho ^{2}\rightarrow \lbrack 0, \infty]$ by

Clearly, under this distance, $JS$-metric $\xi$ is not a metric. As a result, $(\mho, \xi, \preceq)$ is complete partially ordered. Moreover, we can guarantee the existence of the solution to system (3.1) only if it is homogeneous.

Theorem 3.2. From the integral equations (3.1), let $\Xi :\mho ^{3}\rightarrow \mho$ be described as

where $b\in \lbrack 0, 1].$ Assume also,

$(\clubsuit _{1})$ the functions $\Im _{s}:[0, 1]\times \lbrack 0, 1]\rightarrow \mathbb{R} ^{+}$ and $\Game _{s}:[0, 1]\times \mathbb{R} \rightarrow \mathbb{R} ^{+}$ are continuous, where $s = 1, 2, 3;$

$(\clubsuit _{2})$ for $\vartheta _{1}, \vartheta _{2}\in \mho,$ if $\vartheta _{1}\leq \vartheta _{2},$ then $\Game _{1}(b, \vartheta _{1}(b))\leq \Game _{1}(b, \vartheta _{2}(b)),$ $\Game _{2}(b, \vartheta _{1}(b))\leq \Game _{2}(b, \vartheta _{2}(b))$ and $\Game _{3}(b, \vartheta _{1}(b))\leq \Game _{3}(b, \vartheta _{2}(b));$

$(\clubsuit _{3})$ there exist $\widehat{A},$ $\widehat{B}$ and $\widehat{C}$ in $\mathbb{R}^{+}$ so that $3\widehat{A}\widehat{B}\widehat{C} < 1,$ $\int\limits_{0}^{1}\Im _{s}\left(b, c\right) dc\leq \widehat{A},$ $\int\limits_{0}^{1}\Im _{s}\left(b, c\right) \Game _{s}(c, \vartheta (c))dc\leq \widehat{C}$ and

where $\vartheta, \widetilde{\vartheta }\in \mho,$ $b\in \lbrack 0, 1]$ and $s = 1, 2, 3;$

$(\clubsuit _{4})$ there exist $\vartheta _{0}, \varpi _{0}, \varkappa _{0}\in \mho$ so that $\vartheta _{0}\leq \Xi \left(\vartheta _{0}, \varpi _{0}, \varkappa _{0}\right),$ $\varpi _{0}\leq \Xi \left(\varpi _{0}, \varkappa _{0}, \vartheta _{0}\right)$ and $\varkappa _{0}\leq \Xi \left(\varkappa _{0}, \vartheta _{0}, \varpi _{0}\right).$

Then Problem (3.1) has a solution provided that the integral equations are homogeneous.

Proof. Assume that $\alpha \left(\left(\vartheta, \varpi, \varkappa \right), \left(\widetilde{\vartheta }, \widetilde{\varpi }, \widetilde{\varkappa } \right) \right) = 1$ for any $\left(\vartheta, \varpi, \varkappa \right), \left(\widetilde{\vartheta }, \widetilde{\varpi }, \widetilde{\varkappa } \right) \in \mho ^{3}.$ Repeating the same arguments used in Theorem 3.1, under this $JS$-{metric }$\xi$ and considering $r$ is the identity mapping, we have the $\xi -$continuities of $r$ and $\Xi$ and all assumption except condition $(\triangle _{i})$ of Theorem 2.4 are valid.

So, our task is to prove the condition $(\triangle _{i})$ of Theorem 2.4. Again, this condition can be replaced by assumption $(\dagger _{3})$ of Corollary 2.5. Let $\vartheta, \varpi, \varkappa, \widetilde{\vartheta }, \widetilde{\varpi }, \widetilde{\varkappa }\in \mho.$ If $\vartheta \preceq \widetilde{\vartheta },$ $\varpi \preceq \widetilde{ \varpi }$ and $\varkappa \preceq \widetilde{\varkappa },$ then by $(\clubsuit _{3})$, we get

Analogously, if we take $\phi = 3\widehat{A}\widehat{B}\widehat{C} < 1,$ then $\phi \in \widetilde{\Phi }.$ Hence, the assumption $(\dagger _{3})$ of Corollary 2.5 is satisfied. Hence, $\Xi$ possesses a tripled fixed point, which is a unique solution for the problem (3.1) under the condition of homogeneity for these equations.

The following example supports Theorem 3.2:

Example 3.3. Consider the problem below:

where $b\in \lbrack 0, 1].$ By comparing this system with system (3.1) in the homogeneous case, we can write

for $b, c\in \lbrack 0, 1].$ It is easy to see that $\Im _{s}$ and $\Game _{s}$ are continuous, and $\Game _{s}(c, \vartheta)\geq 0$ for $s = 1, 2, 3.$ Furthermore, $\Game _{s}(b, \vartheta (b))\leq \Game _{s}(b, \varpi (b))$ whenever $\vartheta \leq \varpi$ for all $s = 1, 2, 3.$ Moreover, for non-positive-valued $\vartheta _{0}, \varpi _{0}, \varkappa _{0},$ the assumption $(\clubsuit _{4})$ of Theorem 3.2 holds.

Now, consider

and

so, assume that $\widehat{B} = \frac{1}{2}.$ Also, one can write

Putting $\widehat{C} = \frac{1}{3}\ln (2),$ we have $\int\limits_{0}^{1}\Im _{s}\left(b, c\right) \Game _{s}(c, \vartheta (c))dc\leq \widehat{C}$ for $s = 1, 2, 3.$ Moreover,

select $\widehat{A} = \frac{1}{3},$ then we have $\int\limits_{0}^{1}\Im _{s}\left(b, c\right) dc\leq \widehat{A}.$ Consequently, $3\widehat{A} \widehat{B}\widehat{C} = \frac{1}{6}\ln (2) < 1.$ Therefore, all hypotheses of Theorem 3.2 are fulfilled. This implies that Problem (3.2) owns a solution in $C[0, 1]$.

4.   Conclusions and future works

The existence of a tripled coincidence point of a generalized contraction type mapping, which is considered in $JS$-metric spaces endowed with a partial order, was investigated in this article. Furthermore, the theoretical results have been supported by illustrative examples. Ultimately, as an application, the existence of a solution for a system of non-homogeneous and homogeneous integral equations is provided. Moreover, a numerical example of the system is derived. Along with the works presented in ^[4,24,25] as future works, we launch the following two inquiries: What would the results look like if the $JS$-metric space was replaced by a Modular space? What if we used the variational principle?

Acknowledgments

The authors would like to acknowledge the support of Prince Sultan University for paying the Article Processing Charges (APC) of this publication through the Theoretical and Applied Sciences Lab.

Conflict of interest

The authors declare that they have no competing interests concerning the publication of this article.