Fixed Point Results for an Almost Generalized <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2724395pt" id="M1" height="8.14084pt" version="1.1" viewBox="-0.0657574 -7.8684 9.91493 8.14084" width="9.91493pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"/></g></svg>-</span>Admissible <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06579971pt" id="M2" height="11.8958pt" version="1.1" viewBox="-0.0657574 -11.83 12.0508 11.8958" width="12.0508pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-91" d="M669 636L663 650H260C230 650 221 652 208 675H186C177 621 162 548 144 481L173 479C195 534 211 560 226 578C248 604 266 615 373 615H544C460 511 114 112 23 16L31 0H561C577 34 611 135 622 170L594 182C566 125 544 88 515 65C481 38 416 35 319 35C249 35 197 37 152 41C270 184 542 501 669 636Z"/></g></svg>-</span>Contraction in the Setting of Partially Ordered b-Metric Spaces

Teweldemedhin, Solomon Gebregiorgis; Tola, Kidane Koyas

doi:https://doi.org/10.1155/2021/5988007

Abstract and Applied Analysis

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 5988007 | https://doi.org/10.1155/2021/5988007

Fixed Point Results for an Almost Generalized -Admissible -Contraction in the Setting of Partially Ordered b-Metric Spaces

Solomon Gebregiorgis Teweldemedhin¹and Kidane Koyas Tola¹

Academic Editor: A. Ghareeb

Received18 Aug 2021

Revised25 Oct 2021

Accepted04 Nov 2021

Published16 Nov 2021

Abstract

In this paper, we introduce an almost generalized -admissible -contraction with the help of a simulation function and study fixed point results in the setting of partially ordered b-metric spaces. The presented results generalize and unify several related fixed point results in the existing literature. Finally, we verify our results by using two examples. Moreover, one of our fixed point results is applied to guarantee the existence of a solution of an integral equation.

1. Introduction

Metric fixed point theory is a vivid topic, which furnishes useful methods and notions for dealing with various problems. In particular, we refer to the existence of solutions of mathematical problems reducible to equivalent fixed point problems. Thus, we recall that Banach contraction principle [1] is at the foundation of this theory. Due to its usefulness, Banach contraction principle has been extended and generalized in various spaces using different conditions either by modifying the basic contractive condition or by generalizing the ambient spaces or both. For some extensions of Banach contraction principle in metric spaces, see References [2–12].

The existence of fixed point in partially ordered sets has been considered by Turinici in ordered metrizable uniform spaces [11]. The applications of fixed point results in partially ordered metric spaces were studied by Ran and Reurings [13] to solve matrix equations and by Nieto and Rodríguez-López [14] to obtain solutions of certain partial differential equations with periodic boundary conditions. Many researchers have focused on different contractive conditions in complete metric spaces endowed with a partial order and obtained many fixed point results in such spaces. For further works in this direction, see References [3, 15–19]. The concept of metric spaces has been generalized in many directions. The notion of a b-metric space was introduced by Bakhtin in [20] and later extensively used by Czerwik in [21, 22]. Since then, several papers have been published on the fixed point theory of various classes of single-valued and multivalued operators in (ordered) b-metric spaces. For further works in this direction, see References [2, 4, 23–27].

Khojasteh et al. [28] introduced the notion of Z-contraction and studied existence and uniqueness of fixed points for -contraction type operators. This class of -contractions unifies large types of nonlinear contractions existing in the literature. Afterwards, Karapinar [25] originated the concept of -admissible -contraction. For more works in this line of research, see References [3, 4, 9, 12, 25]. Recently, Melliani et al. [9] introduced a new concept of -admissible almost type -contraction and proved the existence of fixed points for admissible almost type -contractions in a complete metric space.

Inspired and motivated by the works of [9, 25], the purpose of this paper is to introduce a new class of mappings, namely, an almost generalized -admissible -contraction, and prove the existence and uniqueness of fixed points for such mappings in the setting of partially ordered b-metric spaces.

2. Preliminaries

In this section, we present some notions, definitions, and theorems used in the sequel.

Throughout this paper, we shall use and to represent the set of real numbers and the set of nonnegative real numbers, respectively.

Definition 1 (see [21]). Given a nonempty set A function is called b-metric if there is a real number such that for all , the following conditions hold: (i)(ii)(iii)The triplet is called a b-metric space.

Definition 2 (see [29]). Let be a b-metric space. Then, a sequence in is said to be (a)b-convergent if there exists such that as In this case, we write (b)b-Cauchy sequence if as (c)b-complete if every b-Cauchy sequence in is b-convergent.

Remark 3 (see [29]). In a b-metric space , the following assertions hold: (1)(R1) A convergent sequence has a unique limit.(2)(R2) Each convergent sequence is a Cauchy sequence.(3)(R3) In general, a b-metric is not continuous.(4)(R4) In general, a b-metric does not induce a topology on

Definition 4. A partially ordered set (poset) is a system , where is nonempty set and is a binary relation of satisfying (i)(reflexivity).(ii), then (antisymmetry).(iii), then (transitivity) for all

Definition 5. Let be a nonempty set. Then, is called partially ordered b-metric spaces if (i) is a b-metric space and(ii) is a partially ordered set.Now, we give an example to show that a b-metric is not necessarily metric.

Example 1 (see [23]). Let be a metric space and , where is a real number. Then, is a b-metric with . However, if is a metric space, then is not necessarily a metric space. For example, if and , then is a b-metric on with but it is not a metric on .

Definition 6 (see [18]). Let be a partially ordered set and is a self-mapping; we say is monotone nondecreasing with respect to if for

Definition 7. Let () be a partially ordered set and ; then, and are said to be comparable elements of if

Theorem 8 (see [30]). Let be a complete metric space and be a map satisfying for all where with Then, has a unique fixed point.

Theorem 9 (see [18]). Let be a complete partially ordered metric space. Let be a continuous and nondecreasing map satisfying for all with and with If there exists such that , then has a fixed point.

Definition 10 (see [5]). Let be a metric space. A map is called an almost contraction or contraction if there exist constants and such that for all

Definition 11 (see [8]). A function is called an altering distance function if (i)is nondecreasing and continuous;(ii)if and only if

Definition 12 (see [28]). A simulation function is a map that satisfies the following conditions:

then
The collection of all simulation functions is denoted by .

Example 2 (see [28]). Let where is defined as follows: (i) for all where (ii) for all where is an upper semicontinuous function such that if and only if and for all (iii) for all where are continuous functions such that if and only if and for all

Definition 13 (see [22]). Let be a metric space, be a map, and . Then, is called -contraction with respect to if the following condition is satisfied: for all

Definition 14 (see [10]). Let be a nonempty set. Let and be the maps. Then, is called -admissible if implies , for each

Definition 15 (see [31]). Let be a nonempty set. Let and be the maps. Then, is said to be -orbital admissible mapping if implies , for each

Definition 16 (see [31]). Let be a nonempty set. Let and be the maps. Then, is said to be a triangular -orbital admissible if (i) is -orbital admissible;(ii) and implies for each

Definition 17 (see [25]). Let be a nonempty set and be a self-map defined on a metric space . If there exist and such that for all , then is called an -admissible -contraction with respect to .

Theorem 18 (see [25]). Let be a complete metric space and be an -admissible -contraction with respect to . Suppose that (i) is triangular -orbital admissible(ii)There exists such that (iii) is continuousThen, there exists such that .

3. Main Result

In this section, we present our main findings.

Definition 19. Let be a partially ordered b-metric space with parameter . Let and be the maps. Assume that there exist a simulation function and a constant such that for all with where Then, is called an almost generalized -admissible -contraction with respect to .

Theorem 20. Let be a complete partially ordered b-metric space and be an almost generalized -admissible -contraction with respect to . Suppose that (i) is nondecreasing and continuous(ii)There exists such that and (iii) is triangular -orbital admissibleThen, has a fixed point.

Proof. Due to condition (ii), there exists such that and
Define an iterative sequence in as follows: for every .

If there exists some nonnegative integer such that, then is a fixed point of .

Now, we shall assume that for all So, we have for all integers Due to (ii) and is -admissible, we have which in turn implies that .

Inductively, we obtain that

Since is nondecreasing map and , we have the following:

Hence, ≼ for .

Using Equation (9), putting and for , we get the following: where

If , then Equation (15) becomes the following:

It follows that

Using Equation (13), we have the following: which is a contradiction.

Hence,

Now Equation (15) becomes .

It follows that

Furthermore, using Equation (15) and since , we have the following:

It follows that

Hence, the sequence is nonincreasing and bounded below. Accordingly, there exists such that

Assuming and using Equation (20), we get the following:

Letting , , and using , we obtain the following: which is a contradiction.

Thus, we have the following:

Now, we need to show that is a Cauchy sequence in .

Suppose is not a Cauchy sequence in , then there exists an such that where are two sequences of positive integerswith for all positive integers Moreover, is chosen as the smallest integer satisfying Equation (25).

Thus, we have the following:

By applying the triangle inequality and using Equations (25) and (26), we get the following:

Now, taking the upper limit as in Equation (27) and using Equation (24), we get the following:

Similarly, applying the triangle inequality twice, we get the following:

Furthermore,

Taking the upper limit as in Equations (29) and (30) and combining, we get the following:

Again,

Furthermore,

Taking the upper limit as in Equations (32) and (33), using Equation (31), and combining, we get the following:

Similarly, we can show that

Now, for simplicity, we denote the following expressions as follows: which give and

Using Equations (24), (34), and (35), and the property of lim sup, we can see that

Since is triangular -orbital admissible, we have the following:

Using Equation (38), we have the following:

Using Equation (9) with and , we obtain the following:

It follows that

Combining Equations (39) and (41), we get the following:

If , then after rearranging, collecting like terms, and applying the triangle inequality, Equation (42) becomes the following:

Taking the upper limit as in Equation (43) and using Equation (31), we get the following: which is a contradiction due to Equation (9).

Hence,

Using Equations (32) and (45), we get the following:

Using Equations (45) and (46) together with Equations (24), (34), and (35) and applying the property of , it follows that

Taking the lower limit in Equation (39) as and using Equation (25), we get the following:

Taking the upper limit in Equation (40) and using Equations (37), (47), and (48), we get the following: which is a contradiction.

Hence, is a Cauchy sequence in a complete b-metric space. So, there exists such that

Since is continuous, we obtain the following:

Thus, is a fixed point of .

Theorem 21. Let be a complete partially ordered b-metric space and is an almost generalized -admissible -contraction with respect to . Suppose that (i) is nondecreasing(ii)There exists such that and (iii) is triangular -orbital admissible(iv)There exists a nondecreasing sequence in such that with and Then, has a fixed point.

Proof. Following the proof of Theorem 20, we know that the sequence defined by for all converges to some
Hence, by (iii), we have the following: for all

By (i), there exists with and since is a nondecreasing map, we have the following:

Thus, is a nondecreasing sequence that converges to .

It follows that

Now, we show that .

Applying Equation (9) with and , we get the following: where

From Equation (54), it follows that

Using Equations (52) and (56), we have the following:

Since , we have the following:

If , then , then we are done. Assume

Letting and and using , we obtain the following:

which is a contradiction. Thus, we have ; that is,

Theorem 22. In addition to the hypotheses of Theorem 20 or Theorem 21, suppose that for every, there exists such that and Then, has a unique fixed point.

Proof. Referring to Theorem 20 or Theorem 21, the sets of fixed points of are nonempty. Now, we shall show the uniqueness of fixed point. To prove the uniqueness of the fixed point, assume that there exist such that and with
Assume that there exists such that and , then as in the proof of Theorem 20, we define the sequence such that for all

Using the hypotheses of Theorem 22, Equation (60), and proceeding inductively, we get the following:

Due to the monotone property of , we have the following:

Since for all , then for some positive integer , and hence, for all . It follows that as .

Using Equation (61) and the fact that as , we have for all

Now, suppose that for all , so for all , then for all

Applying Equation (9) with and , we get the following:

It follows that

Here,

Moreover,

If , then substituting the corresponding values of and in Equation (64) and using the fact that , it becomes the following:

It follows that which is a contradiction. Hence,

Again, substituting the corresponding values of and in Equation (64) and using the fact that , it becomes the following:

Hence, the sequence is nonincreasing and bounded below. Accordingly, there exists such that Assume

Using Equation (69), we get the following:

Letting and , and using , we obtain the following: which is a contradiction.

Thus, we have ; that is, as.

Similarly as.

Due to the uniqueness of the limit, it implies that Thus, has a unique fixed point.

Now, we give corollaries of our main theorem, Theorem 20.

If we take for all in Theorem 20, then we have the following result.

Corollary 23. Let be a complete partially ordered b-metric space with parameter . Let be a map. Suppose there exists a simulation function such that for all with where and are the same as in Theorem 20.
Also, assume that the following conditions hold; (i) is nondecreasing and continuous(ii)There exists such that Then, has a fixed point.
Similarly, we can deduce the following results.

Corollary 24. Let be a complete partially ordered b-metric space with parameter . Let and be the maps. Suppose there exists a simulation function such that for all with where is defined in Theorem 20.
Also, assume that the following conditions hold; (i) is nondecreasing and continuous(ii)There exists such that and (iii) is triangular -orbital admissibleThen, T has a fixed point.

Corollary 25. Let be a complete partially ordered b-metric space with parameter . Let be a map. Suppose there exists a simulation function such that for all with where is defined in Theorem 20.
Also, assume that the following conditions hold; (i) is nondecreasing and continuous(ii)There exists such that Then, has a fixed point.
Now, we provide an example in support of Theorem 20.

Example 3. Let and let be defined by the following: for all .
Hence, is a complete b-metric space with parameter
Define a partial order on as follows: Then, is a partially ordered set.
We define the maps and as follows: where Clearly, is a continuous, nondecreasing, and triangular -orbital admissible map.
Moreover, we choose ; then, and
Now, we verify Equation (9) by choosing and for all where For simplicity, let and
The cases where we have are trivial. So, we consider only the following three cases.
Case (i): When and Hence,
Case (ii): When and Hence,
Case (iii): When and Hence,
Case (iv): When and Hence,
From cases (i)-(iv) considered above, satisfies Equation (9) and hence all the hypotheses of Theorem 20. Thus, has a fixed point which is
The following is an example in support of Corollary 24.

Example 4. Let and let is the same as in Example 3..
Define a partial order on X as follows: Then, is a partially ordered set.
We define the following mappings: Clearly, is a continuous, nondecreasing, and triangular -orbital admissible mapping.
We choose and then and
Now, we verify Equation (9) by choosing the simulation function: for all where

Case 1.
For simplicity, let and
In this case, and Moreover, Hence,

Case 2.
In this case, and Moreover, Hence,

Therefore, satisfies Equation (73) and hence all the assertions of Corollary 24. Thus, has a unique fixed point, namely,

4. An Application to an Integral Equation

In this section, we give an application of our result to an integral equation.

Consider the following integral equation: where are given continuous functions and is a constant such that .

Let be the set of continuous real functions defined on . Define the b-metric by the following: for all Consider Then, is a complete partially ordered b-metric space with the usual order.

Now, define a mapping by the following:

We will use the following assumptions to prove the existence of a solution of Equation (90): (a)(b)(c)For all , (d)There exists a function such that for and for all with

The existence of a solution to Equation (90) is equivalent to the existence of a fixed point of Now, we prove the following result.

Theorem 26. Under the assumptions (a)-(d), Equation (90) has a unique solution in .

Proof. where and are defined as in Theorem 20. Hence, satisfies all the conditions of Corollary 23. Therefore, has a fixed point; that is, Equation (90) has a solution in .

5. Conclusion

In this paper, we introduce a new class of maps, namely, an almost generalized -admissible -contraction, and establish fixed point theorems. Moreover, we prove the existence and uniqueness of fixed points in the setting of partially ordered b-metric spaces with the help of simulation function. Our results unify several related results in the existing literature. The given results not only unify several existing results but also extend and improve them. Finally, we verify the established theorems by some examples and provide an application of our result to an integral equation.

Data Availability

The authors confirm that the data supporting the findings of this study are available within this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors would like to thank the College of Natural Sciences, Jimma University, for funding this research work.

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Copyright © 2021 Solomon Gebregiorgis Teweldemedhin and Kidane Koyas Tola. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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