 Research
 Open access
 Published:
Some applications of fixed point results for monotone multivalued and integral type contractive mappings
Fixed Point Theory and Algorithms for Sciences and Engineering volume 2023, Article number: 11 (2023)
Abstract
The motivation of the present paper is to introduce and establish some new fixed point results for monotone multivalued functions in partially ordered complete \(D^{*}\)metric spaces, where the partial ordered set \((X,\leq )\) is obtained via a pair of functions \(( \Upsilon,\Omega )\). Moreover, several existence and uniqueness coupled fixed point theorems of mappings satisfying contractive conditions have been investigated and verified in the setting of partially ordered complete \(D^{*}\)metric spaces by using the concept of integral type contractions with respect to partially ordered \(D^{*}\)metric space. Furthermore, we present appropriate examples as an application for our main results. Our results generalize the work of Ghasab, Majani, and Rad on the study of integral type contraction and coupled fixed point theorems in the ordered Gmetric spaces.
1 Introduction and preliminaries
The fixed point theorems in partially ordered metric spaces play a major role in verifying the existence and uniqueness of solutions for some differential and integral equations. The theory of multivalued mappings is a branch of mathematics that has received great attention in the last decades and has various applications in convex optimization, optimal control theory, and differential inclusions. Let \((X, d)\) be a complete metric space. A mapping \(T: X \rightarrow X\) is a contraction mapping if there exists a constant \({q} \in (0,1)\) such that \({d}({T}({x}),{T}({y})) \leq {qd}({x}, {y})\) for all \({x}, {y} \in {X}\). Then the Banach fixed point theorem states that T always has a unique fixed point in X. After witnessing the application of Banach fixed point theorem in giving the existence and uniqueness solutions for many integral and differential equations, various generalizations or extensions of Banach fixed point theorem were carried out. By considering subsequences in the sequence of iterates, Edelstein [1] weakened the condition in the Banach fixed point theorem, and later people knew this as Edelstein’s fixed point theorem. Meanwhile, Boyd and Wong [2] introduced a continuous function \(\Upsilon:[0, \infty ) \rightarrow [0, \infty )\) to replace \(q{d}({x}, {y})\) with \(\psi ({d}({x}, {y}))\) and hence generalized Banach fixed point theorem. In 1976, Caristi [3] proved some general fixed point theorems using the characterization of weakly inward mappings. An analogue version of Banach fixed point theorem in the setting of partially ordered sets was proved in [4] and is also known as the Ran–Reurings fixed point theorem.
However, there is also a failed attempt on generalizing Banach fixed point theorem by Dhage et al. [5] who introduced the Dmetric space topology, see [6–8] for details. As a correction to the topology introduced by Dhage et al., in 2007, Sedghi et al. [9] established the meaning of \({D}^{*}\)metric spaces, which is a probable modification of Dmetric spaces proved by the author in [5]. Afterwards, many authors [10, 11] proved several fixed point theorems in these spaces.
Some coupled fixed point results for a mixed monotone mapping in ordered metric spaces have been established (see [12, 13]); for more details on coupled fixed point and n tuples fixed point theorems, we refer the reader to paper [14] and the references therein. Fixed point problems have further considered the concept of partially ordered complete generalized \({D}^{*}\)metric spaces. Al. Jumaili in [15] used the meaning of \({D}^{*}\)metric spaces and presented some coincidence fixed point theorems for functions satisfying contractive conditions concerning nondecreasing φmappings in partially ordered complete generalized \({D}^{*}\)metric spaces.
Recently, Ghasab et al. [16] applied the idea of integral kind contractions and proved some coupled fixed point theorems for such contractions in ordered Gmetric spaces. Our objective in this article is to introduce and investigate several new fixed point theorems for monotone multivalued functions in partially ordered complete \({D}^{*}\)metric spaces, wherever the partial ordered set \(({X}, \leq )\) is obtained via a pair of functions \((\Upsilon, \Omega )\). Furthermore, we study and establish some existence and uniqueness coupled fixed point theorems for mappings satisfying contractive conditions in the setting of partially ordered complete \({D}^{*}\)metric spaces by using the concept of integral type contractions. In addition, we provide suitable examples as an application for our main results. In our research, in the beginning, we explain several definitions and fundamental conclusion under the concept of \(D^{*}\)metric spaces, because we think these explanations give readers the opportunity to understand more easily in subsequent parts.
Definition 1.1
([9, Definition 1.1])
Let X be a nonempty set. Let \({D}^{*}: X \times X \times X \rightarrow [ 0, \infty ) \) be a function that satisfies the following conditions for all \(x, y, z, b \in X\):

(i)
\(D^{*}(x, y, z) \geq 0\);

(ii)
\(D^{*}(x, y, z)=0\) if and only if \(x=y=z\);

(iii)
\(D^{*}(x, y, z)=D^{*}({P}\{x, y, z\})\), where P is a permutation function;

(iv)
\(D^{*}(x, y, z) \leq D^{*}(x, y, b)+D^{*}(b, z, z)\).
Then the function \(D^{*}\) is called a \(D^{*}\)metric on X, and the pair \((X, D^{*} )\) is called a \(D^{*}\)metric space.
It was remarked in [9, Remark 1.2] that the equality \(D^{*}(x, x, y)=D^{*}(x, y, y)\) holds true for all \(x, y \in X\). Also, some examples of \(D^{*}\) metric have been presented in the same reference. We need the following definition to study the \(D^{*}\)metric spaces.
Definition 1.2
([9, Definition 1.4])
Suppose that \((X, D^{*} )\) is a \(D^{*}\)metric space. Then we say that:

(i)
A sequence \(\{x_{s} \}\) in X converges to a point \(x \in X\) if and only if \(D^{*} (x_{s}, x_{s}, x )=\) \(D^{*} (x, x, x_{\mathrm{s}} ) \rightarrow 0\) as \(\mathrm{s} \rightarrow \infty \). That is, for each \(\varepsilon >0\), there exists a positive integer \(\mathrm{s}_{\mathrm{o}}\) such that for all \(\mathrm{s} \geq \mathrm{s}_{\mathrm{o}} \Rightarrow D^{*} (x, x, x_{\mathrm{s}} )<\varepsilon \). This is equivalent to: for each \(\varepsilon >0\), there exists a positive integer \(\mathrm{s}_{o}\) such that \(D^{*} (x, x_{\mathrm{s}}, x_{r} )<\varepsilon \) for all \(\mathrm{s}, r \geq \mathrm{s}_{o}\).

(ii)
A sequence \(\{x_{s} \}\) in X is a Cauchy sequence if for given \(\varepsilon >0\) there exists a positive integer \(\mathrm{s}_{\mathrm{o}}\) such that, for each \(\mathrm{s}, r \geq \mathrm{s}_{o}, D^{*} (x_{\mathrm{s}}, x_{ \mathrm{s}}, x_{r} )<\varepsilon \).

(iii)
A space \((X, D^{*})\) is a complete \(D^{*}\)metric if every Cauchy sequence in \((X, D^{*})\) is convergent in \((X, D^{*})\).
Lemma 1.1
([17, Lemma 1.9])
Let \((X, D^{*} )\) be a \(D^{*}\)metric space. If a sequence \(\{x_{s} \}\) is convergent to \(x \in X\), then it is a Cauchy sequence.
Lemma 1.2
([9, Lemma 1.7])
Let \((X, D^{*} )\) be a \(D^{*}\)metric space. Then \(D^{*}\) is a continuous function on \(X^{3}\), that is,
whenever a sequence \(\{ (x_{s}, y_{s}, z_{s} ) \}\) in \(X^{3}\) converges to a point \((x, y, z)\in X^{3}\).
Definition 1.3
([3])
Let \((X,d)\) be a metric space and \(\Omega: X \rightarrow [ 0, \infty ) \) be a functional. Define the relation ≤ on X by \(x \leq y\) if and only if \(d(x, y) \leq \Omega (x)\Omega (y)\). Then ≤ is a partial order relation on X induced by Ω and \((X, \leq )\) is called an ordered metric space introduced by Ω.
Definition 1.4
([3])
The following two classes of mappings Ψ and Φ are defined as \(\Psi =(\Upsilon  \Upsilon:[0,\infty ) \rightarrow [0,\infty ) \text{ is continuous and nondecreasing with }\Upsilon (\tau )=0\text{ iff }\tau =0\}\) and \(\Phi =\{\Omega  \Omega:[0,\infty ) \rightarrow [0,\infty )\text{ is lower semicontinuous, }\Omega (\tau )>0\text{ for all }\tau >0\text{ and }\Omega (0)=0\}\).
Definition 1.5
([18, Definition 1.2])
An element \((a, b) \in X^{2}\) is said to be a coupled fixed point of the mapping \(F: X^{2} \rightarrow X\) if \(F(a, b)=a\) and \(F(b, a)=b\).
Definition 1.6
([18, Definition 1.1])
Suppose that \((X, \leq )\) is an ordered partial metric space. If relation ⊑ is defined on \(X^{2}\) by \((a, b) \sqsubseteq (u, v)\) iff \(a \leq u\) and \(b \leq y\), then \((X^{2}, \sqsubseteq )\) is an ordered partial metric space.
Definition 1.7
([18, Definition 1.1])
Suppose that \((X, \leq )\) is a partially ordered set. The mapping \({F}: X^{2} \rightarrow X\) is said to have the mixed monotone property if F is monotone nondecreasing in its first argument and is monotone nonincreasing in its second argument; i.e., for all \(x_{1}, x_{2} \in X, x_{1} \leq x_{2}\) \(\Rightarrow {F} (x_{1}, y ) \leq {F} (x_{2}, y )\) for all \(y \in x\), and for all \(y_{1}, y_{2} \in X, y_{1} \leq y_{2} \Rightarrow {F} (x, y_{2} ) \leq {F} (x, y_{1} )\) for all \(x \in X\).
Definition 1.8
([19, definition 2.1])
A function \(f: X \rightarrow \mathbb{R}\) is called lower semicontinuous if for any \(\{x_{n} \} \subset X\) and \(x \in X\)
.
2 Multivalued functions and \(D^{*}\)metric spaces
This section is devoted to introducing and studying several new fixed point theorems for monotone multivalued functions in partially ordered complete \({D}^{*}\)metric spaces by using the setting of \({D}^{*}\)metric spaces. Throughout this article, we assume that the function \(\Upsilon:[0,\infty ) \longrightarrow [0,\infty )\) has the following properties:

(i)
ϒ is nondecreasing and continuous;

(ii)
\(\Upsilon ^{1}(\{0\})=\{0\}\);

(iii)
\(\Upsilon (a+b) \leq \Upsilon (a)+\Upsilon (b)\) for all \(a,b \in [0,+\infty )\).
Definition 2.1
Let \((X, D^{*},\leq )\) be a \(D^{*}\)metric space, and let \(\Upsilon:[0,\infty ) \longrightarrow [0,\infty )\) be a realvalued mapping, and \(\Omega:X\rightarrow [0,\infty )\) be a functional. We define a relation ≤ as follows: \(x \leq y\) if and only if
for all \(x,y \in X\).
Proposition 2.1
Let \((X, D^{*})\) be a \(D^{*}\)metric space, then ≤ is a partial order on X and \((X, \leq )\) is a partially ordered set.
Proof
First, we prove that ≤ is reflexive. Since \(\Upsilon (D^{*}(x, x, x) )= \Omega (x)\Omega (x) \) for all \(x \in X\), it follows that ≤ is reflexive. Secondly, we prove ≤ is antisymmetric. If \(x, y \in X\) such that \(x \leq y\) and \(y \leq x\), then \(\Upsilon (D^{*}(x, x, y) ) \leq \Omega (x)\Omega (y)\) and \(\Upsilon (D^{*}(y, y, x) ) \leq \Omega (y)\Omega (x)\). Hence \(\Upsilon (D^{*}(x, x, y) )+\Upsilon (D^{*}(y, y, x) )=0\). Thus, \(\Upsilon (D^{*}(x, x, y) )=\Upsilon (D^{*}(y, y, x) )=0 \). So \(\Upsilon (D^{*}(x, x, y) )=0\), and so \(x=y\), which shows that ≤ is antisymmetric. In addition, ≤ is transitive because if \(x, y, z \in X\) such that \(x \leq y\) and \(y \leq z\), hence
and
Hence, \(\Upsilon (D^{*}(x, x, y))+\Upsilon (D^{*}(y, y, z)) \leq \Omega (x) \Omega (z)\). Utilizing part (iii) of the definition of \(D^{*}\)metric space and property (iii) of the function ϒ, we obtain
Thus, we get \(x \leq z\). □
Remark 2.1
From the proof of Proposition 2.1, we see that \((X, \Upsilon \circ D^{*})\) is again a \(D^{*}\)metric space.
In the next work we suppose that \((X, D^{*}, \leq )\) is an ordered \(D^{*}\)metric space induced via \((\Upsilon, \Omega )\).
Definition 2.2
Suppose that \((X, D^{*}, \leq )\) is an ordered \(D^{*}\)metric space induced via \((\Upsilon, \Omega )\), the ordered intervals in X are defined as follows:

(i)
\([x, y]=\{z \in X: x \leq z \leq y\}\);

(ii)
\([x,\infty )=\{z \in X: x \leq z\}\);

(iii)
\((\infty, x]=\{z \in X: z \leq x\}\).
Definition 2.3
([19, Definition 2.3])
Suppose that \({F}: X \rightarrow 2^{X}\) is a multivalued mapping. We say that F is upper semicontinuous if whenever \(\{x_{s} \}\in X\) and \(\{y_{s} \} \in {F} (x_{s} )\) with \(x_{s} \rightarrow p \in X\) and \(y_{s} \rightarrow \omega \in X\), then \(\omega \in {F}(p)\).
Definition 2.4
([19, Definition 2.4])
An element \(x \in X\) is said to be a fixed point of a multivalued mapping \({F}: X \rightarrow 2^{X}\) if \(x \in {F}(x)\).
We illustrate our first result in the following theorem.
Theorem 2.1
Assume that \((X, D^{*}, \leq )\) is a partially ordered complete \(D^{*}\)metric space induced via \((\Upsilon, \Omega )\), where \(\Omega: X \rightarrow [0,\infty ) \) is a mapping which is bounded below. Let \({F}:X \rightarrow 2^{X}\) be a multivalued mapping and \({M}=\{x \in X: {F}(x) \cap [x,\infty ) \neq \emptyset \}\). Assume that

(i)
F is upper semicontinuous;

(ii)
If \(x \in {M}\), then \(y \in {M}\) for all \(y \in {F}(x)\cap [ x,\infty )\);

(iii)
\({F}(p)\cap [ p,\infty )\neq \emptyset \) for some \(p\in X \).
Then there exists a sequence \(\{x_{s} \}\) such that \(x_{s1} \leq x_{s} \in {F} (x_{s1} )\) for all \(s \in \mathbb{N} \), and F has a fixed point \(\mathrm{x}_{o}\) such that \(x_{\mathrm{s}} \rightarrow x_{o} \). In addition, if Ω is lower semicontinuous, then \(x_{s} \leq x_{0}\) for all s.
Proof
By(iii), there exists \(p\in X \) such that \(p \in {M}\). Thus choose \(q \in {F}(p) \cap [ p,+\infty )\), and we have \(p \leq q\). By(ii), we have \(q \in {M}\). Choose \(\tau \in {F}(q) \cap [ q,+\infty )\), and we have \(q\leq \tau \). Continuing in this manner, we obtain a sequence \(\{x_{s} \}\) in X such that \(x_{s1} \leq x_{s} \in {F} (x_{s1} )\) for all \(s \in \mathbb{N}\).
Now, since \((X, D^{*}, \leq )\) is a partially ordered \(D^{*}\)metric space induced via \((\Upsilon, \Omega )\), we obtain that
Since ϒ is a nonnegative mapping, we have that \(\Omega (x_{s1})\Omega (x_{s}) \geq 0\) for all \(s \in \mathbb{N}\). Therefore, \(\Omega (x_{s1}) \geq \Omega (x_{s})\) for all \(s \in \mathbb{N}\). Since Ω is a mapping that is bounded below, we get \({\Omega (x_{s})}\) is a decreasing sequence that is bounded below. So, via the completeness property of R, we obtain \(\lim_{s \rightarrow +\infty} \Omega (x_{s})=\inf \{x_{s}: s \in \mathbb{N} \}\). Thus,
Therefore, \(\lim_{s, r \rightarrow +\infty} \Upsilon (D^{*}(x_{s}, x_{s}, x_{r}))=0\).
Now, utilizing the continuity of the mapping ϒ and the property that \(\Upsilon ^{1}(\{0\})=\{0\}\), we obtain that \(\lim_{s, r \rightarrow +\infty} D^{*} (x_{s}, x_{s}, x_{r} )=0 \). Therefore, \(\{x_{s} \}\) is a Cauchy sequence in x. Since X is complete, there exists \(x_{0} \in X\) such that \(\{x_{s} \}\) is \(D^{*}\)convergent to \(x_{o}\). Since \(x_{s1} \in X\), \(x_{s} \in {F} (x_{s1} ), x_{s1} \rightarrow x_{o}\), and \(x_{s} \rightarrow x_{o}\), via the definition of upper semicontinuity of F, we have \(\mathrm{x}_{o} \in {F} (\mathrm{x}_{o} )\).
Now, assume that Ω is lower semicontinuous, then for all \(s \in \mathbb{N}\), we have
Thus, \(x_{s} \leq x_{o} \) for all \(s \in \mathbb{N}\). □
The next result shows that assumption (ii) can be replaced with a new assumption on F.
Corollary 2.1
Suppose that \((X, D^{*}, \leq )\) is a partially ordered complete \(D^{*}\)metric space induced via \((\Upsilon, \Omega )\), where \(\Omega: X \rightarrow [0,\infty )\) is a mapping that is bounded below, and let \({F}: X \rightarrow 2^{X}\) be a multivalued mapping where

(i)
F is upper semicontinuous;

(ii)
F satisfies the condition of monotonic sequence: for all \(x, y \in X\) with \(x \leq y\) and every \(\alpha \in {F}(x)\), there exists \(\beta \in {F}(y)\) such that \(\alpha \leq \beta \);

(iii)
There exists \(p \in X\) such that \({F}(p) \cap [p,+\infty ) \neq \emptyset \).
Then there exists a sequence \(\{x_{s} \}\) in X with \(x_{s1} \leq x_{s} \in {F} (x_{s1} )\) for all \(s \in \mathbb{N}\), and F has a fixed point \(x_{o}\) such that \(x_{s} \rightarrow x_{0}\). Furthermore, if Ω is lower semicontinuous, then \(x_{s} \leq x_{0}\) for all s.
Proof
By(ii), we have \(p\in {M}\). Let \(y \in{F}(p)\cap [ p,+\infty )\). Then the—condition of F implies that there exists \(z \in F(y)\) such that \(y\leq z\). In other words, \(z\in {F}(y)\cap [ y,+\infty )\neq \emptyset \). Hence \(y \in {M}\) and the proof is complete by applying Theorem 2.1. □
Corollary 2.2
Suppose that \((X, D^{*}, \leq )\) is a partially ordered complete \(D^{*}\)metric space induced via \((\Upsilon, \Omega )\) such that \(\Omega:X \rightarrow [0,\infty )\) is a mapping which is bounded below, and let \({f}:X \rightarrow X\) be a mapping. Assume the following:

(i)
f is continuous;

(ii)
f satisfies the condition of monotonic increasing sequence: for any \(\alpha \in f(x) \), there exists \(\beta \in f(y)\) such that \(\alpha \leq \beta \);

(iii)
There exists \(p \in X\) such that \(p \leq {f}( p )\).
Then there exists a sequence \(\{x_{s} \}\) in X with \(x_{s1} \leq x_{s} \in {f} (x_{s1} )\) for all \(s \in \mathbb{N}\), and f has a fixed point \(x_{o}\) such that \(x_{s} \rightarrow x_{o} \). As well, if Ω is lower semicontinuous, then \(x_{s} \leq x_{o}\) for all s.
Proof
Define a multivalued mapping, \({F}: X \rightarrow 2^{X}\) via \({F}(x)=\{{f}(x)\}\) for all \(x \in X\), then F and X satisfy all the assumption of Theorem 2.1. So, we obtain the result from Theorem 2.1. □
The following results are an analogous version of the previous results by replacing the condition of bounded below with the condition of bounded above. The proofs are similar and hence omitted.
Theorem 2.2
Let \((X, D^{*}, \leq )\) be a partially ordered complete \(D^{*}\)metric space induced via \((\Upsilon, \Omega )\), where \(\Omega: X \rightarrow R\) is a mapping that is bounded above. Suppose that \({F}: X \rightarrow 2^{X}\) is a multivalued mapping and \({M}=\{x \in X: {F}(x) \cap (\infty, x] \neq \emptyset \}\). Assume that

(i)
F is upper semicontinuous;

(ii)
For all \(x \in {M}, {F}(x) \cap {M} \cap (\infty, x] \neq \emptyset \).
Then there exists a sequence \(\{x_{s} \}\) such that \(x_{s1} \geq x_{s} \in {F} (x_{s1} ) \) for all \(s \in \mathbb{N}\), and F has a fixed point \(x_{o}\) such that \(x_{s} \rightarrow x_{o}\). Also, if Ω is lower semicontinuous, then \(x_{s} \geq x_{o}\) for all s.
Corollary 2.3
Suppose that \((X, D^{*}, \leq )\) is a partially ordered complete \(D^{*}\)metric space induced via \((\Upsilon, \Omega )\), where \(\Omega:X \rightarrow R\) is a mapping that is bounded above, and let \({F}: X \rightarrow 2^{X}\) be a multivalued mapping, where

(i)
F is upper semicontinuous;

(ii)
F satisfies the condition of monotonic sequence: for all \(x, y \in X\) with \(x \geq y\) and every \(\alpha \in {F}(x)\), there exists \(\beta \in {F}(y)\) such that \(\alpha \geq \beta \);

(iii)
There exists \(p \in X\) such that \({F}(p) \cap (\infty, p] \neq \emptyset \).
Then there exists a sequence \(\{x_{s} \}\) in X with \(x_{s1} \geq x_{s} \in {F} (x_{s1} )\) for all \(s \in \mathbb{N}\), and F has a fixed point \(x_{o}\) such that \(x_{s} \rightarrow x_{o}\). Furthermore, if Ω is lower semicontinuous, then \(x_{s} \geq x_{o} \) for all s.
Corollary 2.4
Suppose that \((X, D^{*}, \leq )\) is a partially ordered complete \(D^{*}\)metric space induced via \((\Upsilon, \Omega )\) such that \(\Omega: X \rightarrow [0,+\infty )\) is a mapping that is bounded below, and let f: \(X \rightarrow X\) be a mapping where

(i)
f is continuous;

(ii)
f satisfies the condition of monotonic increasing sequence. \(\alpha \in {f}(x)\), there exists \(\beta \in {f}(y)\) such that \(\alpha \leq \beta \);

(iii)
There exists \(p \in X\) such that \(p \geq {f}( p )\).
Then there exists a sequence \(\{x_{s} \}\) in X with \(x_{s1} \geq x_{s} \in {f} (x_{s1} )\) for all \(s \in \mathbb{N}\), and f has a fixed point \(x_{\circ}\) such that \(x_{s} \rightarrow x_{o} \). As well, if Ω is lower semicontinuous, then \(x_{s} \geq x_{o}\) for all s.
3 Coupled fixed point theorems in \(D^{*}\)metric spaces
This section is devoted to introducing and studying several results of coupled fixed point satisfying contractive conditions in the setting of partially ordered complete \(D^{*}\)metric spaces by using the idea of integral type contractions. For convenience, we explain the following properties of mappings \(\Phi, \Psi:[0,\infty ) \rightarrow [ 0,\infty )\):

(i)
Ψ is nondecreasing on \([ 0,\infty )\);

(ii)
\(\Psi (t) \leq t\) for all \(t > 0\);

(iii)
Ψ is an additive mapping;

(iv)
\(\sum_{s=1}^{\infty} s \Psi ^{s}(t)<\infty \) for all \(t > 0\).
Also,

(i)
Φ is nonincreasing on \([0,\infty )\);

(ii)
Φ is Lebesgue integrable;

(iii)
For every \(\epsilon >0, \int _{0}^{\epsilon} \Phi (t) \mathrm{d}t >0\);

(iv)
Φ is a continuous mapping.
Now, we introduce the main theorem in our manuscript.
Theorem 3.1
Suppose that \((X, D^{*}, \leq )\) is a partially ordered complete \(D^{*}\)metric space, and let \({F}: X^{2} \rightarrow X\) be a continuous mapping with the mixed monotone property on X such that
where \(x, y, z, p, q, d\) ∈X and \(\Phi:[0,\infty ) \rightarrow [0,\infty )\) is a Lebesgueintegrable mapping that is summable (i.e., with finite integral) with \(d \leq p \leq x\) and \(y \leq q\leq z\), where either \(p \neq d\) or \(q\neq z\). If there exist \(y_{0}, x_{0} \in X\) such that \(x_{0} \leq {F}(x_{0}, y_{0})\) and \({F}(y_{0}, x_{0}) \leq y_{0}\), then F has a coupled fixed point in X.
Proof
By assumption, there exist \(x_{o}, y_{o} \in X \) such that \(x_{o} \leq {F}(x_{o}, y_{o})\) and \({F}(x_{o}, y_{o})\) \(\leq y_{o}\). Define \(x_{1}, y_{1} \in X\) as follows: \(x_{1}=x_{\circ}\leq {F}( x _{o}, y _{ o } )\) and \(y_{1}={F}( y _{ o }, x _{ o } ) \leq y_{o}\). Suppose that \(x_{2}={F}( x _{ 1 }, y _{ 1 } )\) and \(y _{ 2 }={F}( y _{ 1 }, x _{ 1 } ) \). Therefore we get
By utilizing the property of mixed monotonicity for the mapping F, we obtain:
If we continue the above proceedings for each \(s \geq 0\), we obtain the following:
such that
Notice that if \((x_{s+1}, y_{s+1} )= (x_{s}, y_{s} )\), then F has a coupled fixed point. Now, suppose that \((x_{s+1}, y_{s+1} ) \neq (x_{s}, y_{s} )\) for all \(s \geq 0\), that is, let either \((x_{s+1}={F}(x_{s},y_{s})\neq x_{s}\) or \(y_{s+1}={F} (y_{s}, x_{s} ) \neq y_{s}\). By (3.1), we obtain
In the same way we get
Since the mapping Φ is nonincreasing, for every \(a,b \geq 0\), we get
Also, since Ψ is linear and nondecreasing, it follows from (3.1), (3.2), and (3.4) that for all \(s\geq 0\)
In a similar manner, we get
Assume that \(r,s \in \mathbb{N}\) such that \(r>s\). It then follows from Definition 1.1 that
Consequently, it follows from (3.5) that
Since \(\sum_{i=s}^{\infty} i \Psi ^{i}(t)<\infty \) for all \(t \in [0,+\infty )\), it follows that \(\lim_{s, r \rightarrow \infty} D^{*} (x_{s}, x_{s}, x_{r} )=0\) and the sequence \(\{x_{s} \}\) is a Cauchy sequence in X. Similarly, we can deduce that \(\{y_{s} \}\) is a Cauchy sequence in X. Since X is a complete \(D^{*}\)metric space, there exist \(x, y \in X\) such that \(\lim_{s \rightarrow \infty} x_{s}=x\) and \(\lim_{s \rightarrow \infty} y_{s}=y\). Since F is continuous, it follows that \({F}(x, y)=x\) and \({F}(y, x)=y \); that is, \((x, y)\) is a coupled fixed point of the mapping F. □
Theorem 3.2
Assume that \((X, D^{*}, \leq )\) is a partially ordered complete \(D^{*}\)metric space satisfying the following conditions:

(i)
If \(\{x_{s} \}\) is a nondecreasing convergent sequence to \(x \in X\), then \(x_{s} \leq x \) for all s;

(ii)
If \(\{y_{s} \}\) is a nonincreasing convergent sequence to \(y \in X\), then \(y_{s} \geq y \) for all s, and let \({F}: X^{2} \rightarrow X\) be a mapping having the mixed monotone property on X such that
where \(x, y, z, p, q, d \in X\) and \(\Phi:[0,\infty ) \rightarrow [0,\infty )\) is a Lebesgueintegrable mapping that is summable (i.e., with finite integral) with \(d \leq p \leq x\) and \(y \leq q \leq z\), where either \(p \neq d\) or \(q \neq z\); if there exist \(x_{o}, y_{o} \in X\) such that \(x_{o} \leq {F} (x_{o}, y_{o} )\) and \({F} (y_{o}, x_{o} ) \leq y_{o}\), then F has a coupled fixed point in X.
Proof
By using a similar approach as that in proving Theorem 3.1, we obtain two Cauchy sequences \(\{x_{s} \}\) and \(\{y_{s} \}\) ∈X. It then follows from conditions (i) and (ii) that there exist x and \(y \in X\) such that \(x_{s}\leq x\) and \(y_{s}\leq y\) for all \(s\geq 0\). If \(x_{\mathrm{s}}=x \) and \(y_{\mathrm{s}}=y\) for some s, then, as shown in the proof of Theorem 3.1, we obtain \(x_{s+1}=x\) and \(y_{s+1}=y\); that is, \((x, y)\) is a coupled fixed point. So, without loss of generality, we assume that either \(x_{s} \neq x\) or \(y_{s} \neq y\). Therefore, by using 3.1, we obtain
Hence, via (3.7) with as \(s \rightarrow \infty \), we get \(D^{*}({F}(x, y), {F}(x, y), x)=0\), which gives that \({F}(x, y)=x \). In the same way, we can illustrate that \(D^{*}({F}(y, x), {F}(y, x), y)=0\) and thus \({F}(y, x)=y\). Therefore, the proof of the theorem is completed. □
We illustrate in the following theorem that the coupled fixed point of F can be unique.
Theorem 3.3
Let \((X, D^{*}, \leq )\) be a partially ordered complete \(D^{*}\)metric space satisfying the following conditions:

(i)
If \(\{x_{s} \}\) ia a nondecreasing convergent sequence to \(x \in X\), then \(x_{s} \leq x \) for all s;

(ii)
If \(\{y_{s} \}\) is a nonincreasing convergent sequence to \(y \in X\), then \(y_{s} \geq y \) for all s;

(iii)
For all \((x, y), (x_{1}, y_{1} ) \in X^{2}\), there exists \((z_{1}, z_{2} ) \in X^{2}\) such that is comparable with \((x,y)\) and \((x_{1}, y_{1} )\).
And let \({F}: X^{2} \rightarrow X\) be a continuous mapping having the mixed monotone property on X such that
where \(x, y, z, p, q, d \in X\) and \(\Phi:[0,\infty ) \rightarrow [0,\infty )\) is a Lebesgueintegrable mapping that is summable (i.e., with finite integral) with \(d \leq p \leq x\) and \(y \leq q \leq z\), where either \(p \neq d\) or \(q \neq z\), if there exist \(x_{o}, y_{o} \in X\) such that \(x_{\circ} \leqslant {F} (x_{o}, y_{o} )\) and \({F} (y_{o}, x_{o} ) \leq y_{o}\), then F has a unique coupled fixed point in \((X,D^{*} )\).
Proof
Suppose that \((x_{1}, y_{1} )\) is another fixed point of F. Now we discuss the following cases. Case one. Assume that \((x, y)\) and \((x_{1}, y_{1} )\) are comparable with respect to the partial ordering ⊑ in \(X^{2}\) as introduced in Definition 1.6. We assume that \(x \leq x_{1}\) and \(y \leq y_{1}\) without restriction of generality. Now, by using the conditions of Theorem 3.1, we obtain
Suppose that \(s \rightarrow \infty \), therefore via (3.8) we get \(x=x_{1}\). By using same method, we can verify that \(y=y_{1}\). Case two. Let \((x, y)\) be not comparable with \((x_{1}, y_{1} )\). So, by condition (iii) there exists \((z_{1}, z_{2} ) \in X^{2}\), which is comparable to \((x, y)\) and \((x_{1}, y_{1} )\). We can assume that \(z_{1} \leq x, z_{2} \leq y, z_{1} \leq x_{1}\), and \(z_{2} \leq y_{1}\) without restriction of generality. By using the conditions of Theorem 3.1, we get
Assume that \(s \rightarrow \infty \), then by (3.9) we obtain \(D^{*} ({F}^{s}(x, y), {F}^{s} (z_{1}, z_{2} ), {F}^{s} (z_{1}, z_{2} ) )=0\). Thus, \(\lim_{s \rightarrow \infty} {F}^{s}(x, y)=\lim_{s \rightarrow \infty} {F}^{s} (z_{1}, z_{2} )=x\). By means of above, we get \(\lim_{s \rightarrow \infty} {F}^{s} (x_{1}, y_{1} )= \lim_{s \rightarrow \infty} {F}^{s} (z_{1}, z_{2} )=x_{1}\). Thus, \(x=x_{1}\). In the same way, we obtain \(y=y_{1}\). Then, in all cases, we have \((x, y)= (x_{1}, y_{1} )\), that is, a mapping F has a unique coupled fixed point. □
Theorem 3.4
Let \((X, D^{*}, \leq )\) be a partially ordered complete \(D^{*}\)metric space satisfying the following conditions:

(i)
If \(\{x_{s} \}\) is a nondecreasing convergent sequence to \(x \in X\), then \(x_{s} \leq x \) for all s;

(ii)
If \(\{y_{s} \}\) is a nonincreasing convergent sequence to \(y \in X\), then \(y_{s} \geq y \) for all s;

(iii)
Every pair of element of X has an upper and a lower bound in X.
Also, let \({F}: X^{2} \rightarrow X\) be a continuous mapping having the mixed monotone property on X such that:
where \(x, y, z, p, q, d \in X\) and \(\Phi:[0,\infty ) \rightarrow [0,\infty )\) is a Lebesgueintegrable mapping that is summable (i.e., with finite integral) with \(d \leq p \leq x\) and \(y \leq q \leq z\), where either \(p \neq d\) or \(q \neq z\). If there exist \(x_{o}, y_{o} \in X\) such that \(x_{\circ} \leq F( x _{ o }, y _{ o } )\) and \({F} (y_{o}, x_{o} ) \leq y_{o}\), then \(x=y\).
Proof
We first assume that x and y are comparable with respect to the partial ordering ⊏ in X. We can assume that \(x \leq y\) and \(y \leq y\) without restriction of generality. So, by the same method of Theorem 3.3, we obtain \(x=y\). Now, suppose that x and y are not comparable. Then there exists an upper bound or a lower bound of x and y; that is, there exists \(z \in X\) comparable with x and y. This means that we can assume that \(x \leq z\) and \(y \leq z\). By applying Theorem 3.3, we obtain \((x, y)=(z,z)\). Thus, we have \(x=y\). □
Example 3.1
Let \(X=[0,1]\) and \(D^{*}: X \times X \times X \rightarrow \mathbb{R}^{+}\) be a mapping defined by \(D^{*}(a, b, c)=ab+ac+bc \) for all \(a, b, c \in X\). Therefore \((X, D^{*} )\) is a complete \(D^{*}\)metric space. Now, assume that \(\Psi (t)=\frac{1}{2} t \) for all \(t \in [0,\infty )\), and let \({F}:X\times X \rightarrow X\) be a mapping defined by \({F}(a, b)=\frac{a+b}{16}\). Since \(a+b(m+n)\leq am+bn\) holds for all \(a, b, m, n \in X\), it follows that the conditions of Theorem 3.1 hold. In fact, we have
where \(a, b, c, m, n, \ell \in X\). It is obvious that the mapping F satisfies all the conditions of Theorem 3.1. Hence, F has a coupled fixed point.
4 Conclusion
The fixed point results in partially ordered metric spaces play an essential role in constructing methods in mathematics to solve several problems in the pure and applied mathematical sciences. On the other hand, the study of partially ordered metric spaces plays the most important role in many fields both in pure and applied science such as biology, medicine, physics, and computer science (see [20, 21]). Therefore, we introduced and investigated some of new fixed point theorems for monotone multivalued functions in partially ordered complete generalized \({D}^{*}\)metric spaces. In addition, we presented and verified some of coupled fixed point theorems for mappings satisfying integral type conditions in partially ordered \({D}^{*}\)metric spaces. We hope that our results will be useful for the future studies on generalized metric spaces to carry out general framework for the practical applications and to solve the complicated problems containing uncertainties in environment, medical, engineering, economics, and in dynamical systems of various types.
Availability of data and materials
No data were used to support this study.
References
Edelstein, M.: On fixed and periodic points under contractive mappings. J. Lond. Math. Soc. 37, 74–79 (1962)
Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)
Caristi, J.: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 215, 241–251 (1976)
Ran, A., Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132(5), 1435–1443 (2004)
Dhage, B.C.: Generalised metric spaces and mappings with fixed point. Bull. Calcutta Math. Soc. 84(4), 329–336 (1992)
Naidu, S.V.R., Rao, K.P.R., Srinivasa Rao, N.: On the topology of Dmetric spaces and generation of Dmetric spaces from metric spaces. Int. J. Math. Math. Sci. 49–52, 2719–2740 (2004)
Naidu, S.V.R., Rao, K.P.R., Srinivasa Rao, N.: On the concepts of balls in a Dmetric space. Int. J. Math. Math. Sci. 1, 133–141 (2005)
Naidu, S.V.R., Rao, K.P.R., Srinivasa Rao, N.: On convergent sequences and fixed point theorems in Dmetric spaces. Int. J. Math. Math. Sci. 12, 1969–1988 (2005)
Sedghi, S., Shobe, N., Zhou, H.: A common fixed point theorem in \(D^{\ast}\)metric spaces. Fixed Point Theory Appl. 2007, Article ID 27906 (2007)
Luong, N.V., Thuan, N.X.: Common fixed point theorem in compact d*metric spaces. Int. Math. Forum 6(13), 605–612 (2011)
Veerapandi, T., Pillai, A.M.: Some common fixed point theorems in \({D}^{*}\)—metric spaces. Afr. J. Math. Comput. Sci. Res. 4(12), 357–367 (2011)
Agarwal, R.P., Kadelburg, Z., Radenović, S.: On coupled fixed point results in asymmetric Gmetric spaces. J. Inequal. Appl. 2013, Article ID 528 (2013)
Radenović, S.: Coupled fixed point theorems for monotone mappings in partially ordered metric spaces. Kragujev. J. Math. 38(2), 249–257 (2014)
Soleimani Rad, G., Shukla, S., Rahimi, H.: Some relations between ntuple fixed point and fixed point results. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 109(2), 471–481 (2015)
Jumaili, A.M.F.A.: Some coincidence and fixed point results in partially ordered complete generalized \(D^{*}\)metric spaces. Eur. J. Pure Appl. Math. 10(5), 1023–1034 (2017)
Ghasab, E.L., Majani, H., Soleimani Rad, G.: Integral type contraction and coupled fixed point theorems in ordered Gmetric spaces. J. Linear Topol. Algebra 9(2), 113–120 (2020)
Aage, C.T., Salunke, J.N.: Some fixed points theorems in generalized \(D^{*}\)metric spaces. Appl. Sci. 12, 1–13 (2010)
Gnana Bhaskar, T., Lakshmikantham, V.: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65(7), 1379–1393 (2006)
Feng, Y., Liu, S.: Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings. J. Math. Anal. Appl. 317, 103–112 (2006)
Secelean, N.A.: Iterated function systems consisting of Fcontractions. Fixed Point Theory Appl. 2013, Article ID 277 (2013)
Khan, S.U., Arshad, M., Hussain, A., Nazam, M.: Two new types of fixed point theorems for Fcontraction. J. Adv. Stud. Topol. 7, 251–260 (2016)
Acknowledgements
The authors would like to thank the reviewers and the editor for their valuable comments.
Funding
No funds have been taken for this project.
Author information
Authors and Affiliations
Contributions
Author contribution; The first author prepare the manuscript based on the research idea proposed by the second author. The third author check through and made correction and the fourth author made final review and approved the final copy.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare no competing interests.
Additional information
Abbreviations
Not applicable.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Majid, N.A., Jumaili, A.A., Ng, Z.C. et al. Some applications of fixed point results for monotone multivalued and integral type contractive mappings. Fixed Point Theory Algorithms Sci Eng 2023, 11 (2023). https://doi.org/10.1186/s13663023007489
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13663023007489