Abstract
The objective of this manuscript is to present new tripled fixed point results for mixed-monotone mappings by a pivotal lemma in the setting of partially ordered complete metric spaces. Our outcomes sum up, enrich, and generalize several results in the current writing. Moreover, some examples have been discussed to strengthen and support our theoretical results. Finally, the theoretical results are applied to study the existence and uniqueness of the solution to an integro-differential equation.
1. Brief Introduction
The fixed point (FP) technique is considered a fundamental pillar and a powerful tool in nonlinear analysis because of its many vital applications in many disciplines such as computer science, engineering, economics, biology, chemistry, and physics. In mathematics, this technique plays a prominent role in the study of statistical models, dynamical systems, game-theoretic models, differential equations, and many others. More clearly, for example, this method is mainly applied in finding the analytical solution to some differential and integral equations, fractional equations, integro-differential equations (IDEs), and functional analysis which facilitates the way to find numerical solutions to such problems. These problems were addressed by Fredholm [1], Rus [2], Hammad and De La Sen [3, 4], Ameer et al. [5], Hussain et al. [6, 7] and Younis et al. [8–10]
In [11], the concepts of the coupled FP and a mixed-monotone mapping were initiated, and some exciting work in partially ordered metric spaces (POMSs) have been discussed by the same authors. This idea was investigated by many authors such as Berinde [12], Choudhury and Maity [13], and Aydi et al. [14]. Moreover, in abstract spaces, this concept has many applications in integral and functional equations; see the papers of Cirić et al. [15], Ding et al. [16], Hammad et al. [17, 18], Luong and Thuan [19], Choudhury and Kundu [20], Agarwal et al. [21], Radenović [22], and Hammad et al. [23].
In 2011, coupled FP notions are generalized to tripled fixed points (TFPs) concepts by Berinde and Borcut [24] in the setting of POMSs. Via the mentioned spaces, Borcut and Berinde [25, 26], Karapnar et al. [27] presented pivotal results about TFP theorems and the applications in this direction introduced by Mustafa et al. [28] and Hammad and De la Sen [29, 30].
Definition 1 [25]. We say that a trio (where ) is a TFP of a self-mapping if , , and .
Definition 2 [26]. A trio on a nonempty set is called a tripled coincidence point of the two self-mappings and if , , and .
Definition 3 [26]. Assume that is a set. A trio is called a tripled common FP of and , if , , and .
Definition 4 [24]. Assume that is a partially ordered set on the product space defined as follows:
Under this partial order, we state the following definitions.
Definition 5 [24]. A mapping on a partially ordered set has a mixed-monotone property, if for any , we have
Definition 6 [14]. A mapping on a partially ordered set has a mixed -monotone property, where , if for any , we have
Definition 7 [27]. Assume that is a nonempty set. We say that the mappings and are commutative if , for all .
The first contribution of the TFP for a mixed-monotone mapping in a partially ordered set was presented as follows:
Theorem 8 [24]. Let be a complete partially ordered metric space (CPOMS). Assume that , so that (i) has a mixed-monotone property(ii)Either is continuous or has the following properties: (a), if the nondecreasing sequence (b), if the nonincreasing sequence , for all (iii)There is with such thatfor any , for which , , and . If there are so that , , and Then has a TFP.
In this manuscript, we utilize a pivotal lemma to obtain new TFP results for mixed-monotone mappings in CPOMSs. Our results unify, extend, and generalize the papers [19, 31, 32]. Also, some examples and a corollary are given. Later on, we apply the theoretical results to obtain the solution of a system of IDEs as an application.
2. Main Results
We begin this part with the pivotal lemma below.
Lemma 9. Assume that is a partially ordered set and and are two mappings. Suppose that the following assumptions hold:
() There is , so that for , , and , () There is with , so that for , Then, there is , so that , , and .
Proof. We split the proof into three steps: (Step 1)Since , then there is a nonnegative integer , such that for , we get , i.e., . Take , , and By Stipulation () and (5), we have(Step 2)Since , then for each , we obtain , i.e., Apply Stipulation () and (6), we get(Step 3)Similar to Step 1, since , then we can write . By Stipulation () and (7), we haveThis completes the proof.
Remark 10. The results (5)-(7) of Lemma 9 still hold if we reserve the symbols “” and “,” that is, so that , , and
Theorem 11. Let be a CPOMS and be a zero element in . Assume that is mixed -monotone mapping, is self-mapping, and , so that for any . Suppose that the hypotheses below hold:
(i)(ii) and are continuous and commute(iii) verifies stipulations () and () of Lemma 9(iv)For any with , , , and we haveThen, the following conclusions are fulfilled:
() For a triplet , construct three sequences , , and in verifying
for all . Then, , , and , as
() and have a tripled coincidence point . Moreover, assume that , , and are comparable, and for each , is comparable to
() and have a unique common fixed point , that is,
Proof. We shall prove Conclusion () By Condition (iii), there are , , and , so that , , and , where Since , by Condition (i), this yields that there exists , so that , , and Generally, we can build the three sequences , , and in , so that
Since is a mixed -monotone, then we get
By induction for , one can write
It follows from (12) and (13) that for , we have
If , then by (16), we find directly that
this is a contradiction. So, we should take . It is clear that for each and , we have
Since then by (16), we obtain that
Set . Using (12)–(17), one can obtain
thus, we have
According to the proof of Lemma 9 and Condition (ii) (the continuity), there is , , so that
that is,
Therefore, . By the triangle inequality and (21), one can write for any positive integers with , so we have
It follows that . Similarly, we can show that and . This illustrates that , , and are Cauchy sequences. The completeness of leads to the conclusion that there are so that
Next, we shall show Conclusion Since is continuous, then by (25), we get
Also, by the commutativity of and , we have
By (26) and (27), we deduce that
Therefore, the trio is a tripled coincidence point of and .
Finally, to prove Conclusion , assume that , , , and , for . Since and by Definition 4, we have
which yields
It follows from (13) and (30) and mixed -monotonicity of that
This leads to
By induction, for , we have
Applying (12), we observe that
In the same manner, we have
From another direction, since , and the mixed -monotonicity of , then , i.e., ; similarly, one can obtain and By the proof of Conclusion we have
Taking the limit as , we get
This implies together with (25) that
Hence, by (34), (35), (37), and (38), we have
With the help of (39) and the triangle inequality, we can write
Thus, . According to (39), we obtain which proves that is a common FP of and .
To discuss the uniqueness, suppose that is another common FP of and , thus . By the above results and Definition 4, we get
That is, for , , and , where , we have
This leads to
Hence, , which leads to . Therefore, is a unique common FP of and . This finishes the proof.
Examples below verify the assumptions of Theorem 11.
Example 1. Let be endowed with
Define the order relation by
It is obvious that is a CPOMS. Define the mappings and by respectively. It is clear that , and are continuous, and have a mixed -monotone property.
Now, let us verify Condition (12) of Theorem 11 for all with , , and . Consider that the function is given by
Now, we consider the cases below:
() If and , then , and we have
() If and , then , and we have
() If and , then , and we have
() If and , then we have
The four cases indicate that the requirements of Theorem 11 are fulfilled and is a unique TFP.
Example 2. Assume that the first requirements of Example 1 hold with the usual order “.” Define the mappings and by respectively. It is obvious that is a CPOMS, , and are continuous, and have a mixed -monotone property.
Now, let us verify Condition (12) of Theorem 11 for all with , , and . Let be a function defined by
Now, consider
Hence, all conditions of Theorem 11 are satisfied and and have a unique common TFP in for all .
If we set (the identity mapping on ) in Theorem 11, we deduce the result below:
Corollary 12. Let be a CPOMS and be a zero element in . Assume that is a mixed-monotone mapping and is so that for any Suppose that the assumptions below are satisfied:
(i) is continuous(ii) verifies stipulations () and () of Lemma 9(iii)For any with , , and , and we haveThen, the following conclusions are fulfilled:
() For a triplet , construct three sequences , , and in verifying
for . Then, , , and , as
() has a TFP Moreover, assume that , , and are comparable and for each , is comparable to
() has a unique FP , that is
3. An Application to Integro-Differential Equation
In fact, this part is a fundamental pillar of our paper, where the theoretical results presented in the above section are involved in order to obtain the existence of the solution to an IDE of the following form: for , where , , and (for ) are given continuous functions.
Assume that , , , , , and are nonnegative real continuous functions which are differentiable on , where , , and are the first derivative of , , and with respect to , respectively.
Also, suppose that is continuously differentiable with respect to its first variable, where .
In order to find the existence solution of Problem (57), we shall derive the hypotheses below:
() and (for ) are continuous functions so that for any fixed
() Define the mappings and by such that for any fixed
() For any fixed , there is so that
() there is , so that for any fixed , ,
Now, our main theorem of this section is stated as follows:
Theorem 13. Under hypotheses ()-(), System (57) has a unique solution .
Proof. The proof is splitting into the following steps:
(St1) Construct a CPOMS. Assume that is the set of all nonnegative real continuous functions on , . Define a metric on by
where . Define the partial ordered by
Then, a trio is a CPOMS if , , and , whenever , , and , for all
(St2) Construct the mappings and . For this purpose, we involve a derivative with respect to on both sides of (59) and (60), respectively, for , , we get
(St3) Show that is a mixed -monotone and and are commuting. If , then by (68), we have
since , and , then, we get . Moreover by hypothesis (), for any fixed , If
then, we have
Similarly, if
and if
then this implies that is a mixed -monotone. By the definition of and , we can write
(St4) Fulfill Condition (12) of Theorem 11. It follows from (60), (63), and (68) that
From (59), (60), (67)–(76) and by definition of we conclude that
that is
Set for It is clear that for . Through Inequality (78), Hypothesis () of Theorem 11 is verified. Suppose that , , and , then by ((65)), we have
that is
This means , , and . Thus, all requirements of Theorem 11 are fulfilled. So, there is a unique common FP for the mappings and in the form of , so that , which is the unique solution to Problem (57).
The example below supports our application.
Example 3. Consider the following problem:
Problem (80) is a special form of Problem (57), with the following constraints:
, for all and , .
Here, we considered , , and Thus, , , and , respectively. It is clear that the function is continuously differentiable with respect to its first variable, where . Moreover, , , and are nonnegative real continuous functions on Also, the two mappings take the following form:
Now, we are going to satisfy the hypotheses of Theorem 13. (1)For , , and , the functions and (for ) are continuous functions, so thatThis obviously leads to Similarly, one can write (2)From (82) and (83), and assuming that , we haveSimilarly, one can write (3)For any fixed , there is , so thatIn the same manner, we have
Also, we have
Similarly, with , one can write (4)There is , so that for any fixed , , , and , we have
Hence, we find that all hypotheses of Theorem 13 are fulfilled. So, the Problem (80) has a unique solution.
4. Conclusion
Due to the multiple applications of fixed point theory, it has become widespread in many scientific disciplines, especially in nonlinear analysis. It contributes significantly to the study of the existence and uniqueness of the solution to many differential and integral equations, as well as integro-differential equations. So, the main objectives of this paper have been to present some new tripled fixed point results for mixed monotone mappings in the framework of partially ordered metric spaces, and these new results have extended to a lot of papers in the literature. Furthermore, to support the proposed results, some illustrative examples have been given and the existence and uniqueness of the solution to the integro-differential equation have been obtained.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly in writing this article.
Acknowledgments
The authors are grateful to the Spanish Government and the European Commission for Grant IT1207-19.