Abstract

This paper aims to prove the existence and uniqueness of some fixed point for nonlinear contractive mappings in the setting of metric spaces and partially ordered metrics spaces satisfying a contraction condition of rational type. These contributions extend the existing literature on metric spaces and fixed point theory. Through illustrative examples, we showcase the practical applicability of our proposed notions and results, demonstrating their effectiveness in real-world scenarios.

1. Introduction

Metric fixed point theory is widely recognized to have been originated in the work of Stefan Banach in 1922 called the classical Banach contraction principle (BCP) ¹ which is one of the most notable results which has played a vital role in the development of a metric fixed point theory. The principle has been generalized by numerous authors in the different directions by improving the underlying contraction conditions, enhancing the number of involved mappings, weakening the involved metrical notions, and enlarging the class of ambient spaces ^{2, 3, 4, 5, 6, 7}.

In 2004, Ran and Reurings ⁸ obtained a new variant of the classical Banach contraction principle to a complete metric space endowed with partial order relation, which was slightly modified by Nieto and Rodriguez-Lopez ⁹ in 2005 and established fixed point results. Other generalizations of Banach contraction principle can be found in ^{10, 11, 12, 13 14, 15, 16 17, 18, 19, 20, 21 28, 29, 30}.

Recently, Raji et. al. ²² obtained the existence and uniqueness of fixed points for rational type contraction mappings in a metric space that is equipped with a partial order.

Based on the above insight, we prove the existence and uniqueness of some fixed point for nonlinear contractive mappings in the setting of metric spaces and partially ordered metrics spaces satisfying a contraction condition of rational type. These contributions extend the existing literature on metric spaces and fixed point theory. To bolster our findings, we showcase the practical applicability of our proposed notions and results.

2. Preliminaries

The following are some of the definitions that are relevant in our study.

Definition 2.1 ²³ The triple is called partially ordered metric spaces, if is a partial ordered set and is a metric space.

Definition 2.2 ^{24, 25} If is a complete metric space, then the triple is called complete partially ordered metric space.

Definition 2.3 ^{26, 27} Let be a partial ordered set and let be a mapping. Then

(1). elements are comparable, if or holds;

(2). a non empty set is called well ordered set, if every two elements of it are comparable.

Definition 2.4 ⁷ A partially ordered metric space is called ordered complete if for each convergent sequence , if is a non-decreasing sequence in such that implies , for all that is, .

3. Main Results

Definition 3.1 Suppose is a partially ordered set and is said to be monotone non-decreasing if for all

implies (3.1)

Theorem 3.2 Let be a complete partially ordered metric space. Suppose that is a continuous self-mapping on is monotone non-decreasing mapping satisfying

(3.2)

for all and for some with If there exists with , then has a fixed point.

Proof If then we have the result. Suppose that Since is a monotone non-decreasing mapping, we obtain by induction that

(3.3)

Inductively, we can construct a sequence a sequence in such that for every Since is monotone non-decreasing mapping, we obtain

If there exists such that then from , is a fixed point and the proof is finish. Suppose that , for all

Since for all from (3.2) we have

By triangular inequality , we have

(3.4)

which implies that

(3.5)

Using mathematical induction, we have

(3.6)

where We shall now prove that is a Cauchy sequence. For we have

(3.7)

which implies that as Thus, is a Cauchy sequence in a complete metric space Therefore, there exists such that By the continuity of we have

Hence is a fixed point of

The prove of Theorem 3.2 is still valid for not necessarily continuous, assuming the following hypothesis in

If is a non-decreasing sequence in such that then

Theorem 3.3 Let be a complete partially ordered metric space. Suppose that is a continuous self-mapping on is monotone non-decreasing mapping satisfying

(3.8)

for all and for some with

Assume that is a non-decreasing sequence in such that then If there exists with then has a fixed point.

Proof From the proof of Theorem 3.2, is a Cauchy sequence. Since is a non-decreasing sequence in such that then Particularly, for all

Since is monotone non-decreasing mapping for all or, equivalently, for all Moreover, as and we have

Now, we construct as for all Since . Similarly, we have is a non-decreasing sequence and for certain so we have Since for all using (3.8), we have

(3.9)

Letting we have As we have Particularly, and consequence,

Hence is a fixed point of

We will now present example to illustrate where it can be appreciated that hypotheses in Theorem 3.2 and 3.3 do not guarantee uniqueness of fixed point.

Example 3.4. Let , and define the usual order

(3.10)

Consider , a partially ordered set, whose different elements are not comparable. Beside, is a complete metric space considering , the Euclidean distance. The identity map is trivially continuous and nondecreasing and condition (3.2) holds for any , since elements in are only comparable to themselves. Moreover, . In this case, there are two fixed points in , the hypotheses in Theorem 3.2 holds. Theorem 3.3 is also applicable since is a monotone nondecreasing sequence converging to .

We now present a sufficient condition for the uniqueness of the fixed point in Theorem 3.2 and 3.3. The condition that guarantee uniqueness of fixed point can be found in ¹⁰, such that

for every pair has a lower bound or upper bound. (3.11)

Furthermore, ⁹ proved the condition (3.11) is equivalent,

for every , there exists which is comparable to and (3.12)

Consequently,

Theorem 3.5. Adding condition (3.12) to the hypotheses of Theorem 3.2 (or Theorem 3.3), then has a unique fixed point.

Proof Suppose that for every there exists that is comparable to and From Theorem 3.2 (or Theorem 3.3), the set of fixed points of is non-empty. Suppose that are two fixed points of

We have the following two cases:

Case 1. Suppose and are comparable and , then using (3.2) we get

which implies that as . Hence

Case 2. If is not comparable to there exists that is comparable to and By monotonicity is comparable to and for . If there exists such that , then as is a fixed point, the sequence is constant, and, consequently, On the other hand, if for using the contractive condition, we have, for

By triangular inequality , we have

which implies that

Using mathematical induction, we have , for , and as , we have

Similarly, we can prove that Now, the uniqueness of the limit implies

Hence has a unique fixed point.

Example 3.6 Suppose and let be a partial order given by . The elements in are only comparable to themselves with , a complete metric space where is the Euclidean distance. Let be defined as

(3.13)

Observe that is trivially continuous and nondecreasing, and assumption in (3.2) of Theorem 3.2 satisfied. Since elements in are only comparable to themselves. Observe also, and by Theorem 3.2, has a fixed point (1.1).

We present additional example to illustrate or support our result.

Example 3.7 Consider , where represents the usual order relation in hypotheses (3.8) is valid. Indeed, if

(3.14)

then and are respectively, upper and lower bound of and . If is a monotone nondecreasing sequence in , converging to , then and are monotone nondecreasing sequences which converge, respectively, to and in , then and for all , and is an upper bound of all terms in the sequence

4. Application

The Some application of the main results to a self mapping involving an integral type contraction.

Let us consider the set of all functions defined on satisfying the following conditions:

1. Each is Lebesque integrable mapping on each compact subset of

2. For any , we have .

Corollary 4.1 Let be a complete partially ordered metric space. Suppose that is a continuous self-mapping on is monotone non-decreasing mapping satisfying

(4.1)

for all and for some with If there exists with then has a fixed point.

Corollary 4.2 Let be a complete partially ordered metric space. Suppose that is a continuous self-mapping on is monotone non-decreasing mapping satisfying

(4.2)

for all and for some with

Assume that is a non-decreasing sequence in such that then If there exists with , then has a fixed point.

5. Conclusion

The main findings of this study demonstrate the existence and uniqueness of some fixed point for nonlinear contractive mappings in the setting of metric spaces and partially ordered metrics spaces satisfying a contraction condition of rational type. These contributions extend the existing literature on metric spaces and fixed point theory. Through illustrative examples, we showcased the practical applicability of our proposed notions and results. This study provides significant advancements in the understanding of metric spaces, with potential applications in differential equations.

Acknowledgements

We would want to thank everyone who has assisted us in finishing this task from the bottom of our hearts.

References