Several Fixed-Point Theorems for <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06580067pt" id="M1" height="11.4652pt" version="1.1" viewBox="-0.0657574 -11.3994 10.5867 11.4652" width="10.5867pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-71" d="M584 650H137L131 622C214 614 217 612 200 521L125 127C109 41 101 35 23 28L17 0H288L294 28C201 35 193 42 209 128L242 309H348C440 309 442 300 443 226H471L510 422H482C452 354 449 348 357 348H251L295 575C302 609 304 615 338 615H426C502 615 517 604 526 581C534 560 536 524 537 492L565 494C574 554 583 631 584 650Z"/></g></svg>-</span>Contractions in Complete Branciari <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M2" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 8.20973 12.533" width="8.20973pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g></svg>-</span>Metric Spaces and Applications

Chen, Lili; Huang, Shuai; Li, Chaobo; Zhao, Yanfeng

doi:https://doi.org/10.1155/2020/7963242

Journal of Function Spaces

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

Research Article | Open Access

Volume 2020 | Article ID 7963242 | https://doi.org/10.1155/2020/7963242

Several Fixed-Point Theorems for -Contractions in Complete Branciari -Metric Spaces and Applications

Lili Chen,^1,2Shuai Huang,¹Chaobo Li,¹and Yanfeng Zhao²

Academic Editor: Erdal Karapinar

Received19 Apr 2020

Revised28 May 2020

Accepted09 Sept 2020

Published25 Nov 2020

Abstract

In this paper, we prove the existence and uniqueness of fixed points for -contractions in complete Branciari -metric spaces. Furthermore, an example for supporting the related result is shown. We also present the concept of the weak well-posedness of the fixed-point problem of the mapping and discuss the weak well-posedness of the fixed-point problem of an -contraction in complete Branciari -metric spaces. Besides, we investigate the problem of common fixed points for -contractions in above spaces. As an application, we apply our main results to solving the existence and uniqueness of solutions for a class of the integral equation and the dynamic programming problem, respectively.

1. Introduction

In the last decades, one can observe a huge amount of interest for the development of fixed-point theory, since it has a lot of applications, especially in metric spaces, see [1–6]. In 1993, Czerwik [7] introduced the concepts of -metric spaces by weakening the coefficient of the triangle inequality and generalized Banach’s Contraction Principle to these spaces. Subsequently, Boriceanu [8] obtained some concrete examples of -metric spaces and studied the fixed-point properties of set-valued operators in -metric spaces. In 2000, Branciari [9] suggested a new distance function to improve the notion of metric via substituting the triangle inequality with the quadrilateral inequality and extended Banach’s Contraction Principle to Branciari metric spaces. In 2015, George et al. [10] combined Branciari metric spaces with -metric spaces to introduce a generalization of metric spaces-rectangular -metric spaces (or Branciari -metric spaces). They also proved Banach’s Contraction Principle and presented the generalized form of Kannan’s fixed-point theorem in above spaces. In 2018, Karapinar [11] revisited Kannan’s fixed-point theorem under the aspect of interpolation and proposed a new Kannan-type contraction to maximize the rate of convergence. For more details about fixed-point theorems in metric spaces, please see [12–17].

In fact, fixed-point properties of Branciari -metric spaces have aroused great interest of many researchers who have extended the results in Branciari metric spaces and -metric spaces to Branciari -metric spaces. However, sometimes it is difficult to achieve the desired results in Branciari -spaces; there are still a large number of problems to be solved. Based on the above-mentioned discussions, we discuss the problem of the existence and uniqueness of fixed points for -contractions in complete Branciari -metric spaces which extend the results in [18]. What is more, an example for supporting the result is presented. The concept of the weak well-posedness of the fixed-point problem of the mapping is also introduced, and the weak well-posedness of the fixed-point problem of an -contraction satisfying some conditions in complete Branciari -metric spaces is investigated. Besides, the problem of common fixed points for two -contractions is studied in above spaces. As an application, by using main results, the existence and uniqueness of solutions for a class of the integral equation and the dynamic programming problem are solved, respectively.

2. Preliminaries

Firstly, we recall some basic notations and facts that we are going to use later.

Definition 1 (see [19]). Let be a nonempty set and be a real number. A function is called a -metric if the following conditions are satisfied, for every
(A1) if and only if
(A2)
(A3)

In this case, the pair is called a -metric space with constant .

Definition 2 (see [10]). Let be a nonempty set and be a real number. A function is called a Branciari -metric if the following conditions are satisfied, for every
(B1) if and only if
(B2)
(B3) for all distinct points

In this case, the pair is called a Branciari -metric space with constant .

Definition 3 (see [18]). A function belongs to if it satisfies the following conditions:
(F1) is strictly increasing
(F2) There exists such that

In [18], they omitted Wardowski’s (F2) condition from the above definition. Explicitly, they do not require that (WF2) if is a sequence of positive real numbers, then if and only if

The reason for this is the following lemma.

Lemma 4 (see [18]). If is an increasing function and is a decreasing sequence such that , then .

We can also see some properties concerning and .

Definition 5 (see [19]). Let and . We say that belongs to if it also satisfies if inf and are such that and , then

In [20], the authors introduced the following condition (F4) in Definition3.1:

if is a sequence such that , for all and for some , then , for all .

Proposition 6 (see [18]). If is increasing, then is equivalent to .

Definition 7 (see [18]). Let be a -metric space with constant and be an operator. If there exist and , such that for all , the inequality implies then, is called an -contraction.

3. Main Results

In this section, we mainly focus on the existence and uniqueness of fixed points for -contractions, the weak well-posedness of the fixed-point problem of an -contraction, and criteria for common fixed points of two -contractions in complete Branciari -metric space. Moreover, the existence of fixed points for Hardy-Rogers-type set-valued -contractions in above spaces is shown. In the sequel, we denote by the set of real numbers and by the set of positive integer numbers.

3.1. Existence and Uniqueness of Fixed Points for -Contractions

Theorem 8. Let be a complete Branciari -metric space with constant and be an -contraction for . Then, has a unique fixed point . Furthermore, for any , the sequence satisfying is convergent.

Proof. Firstly, we show that has at most one fixed point. Suppose that and are two different fixed points of . That is, . It follows that Therefore, we can apply to get which implies , a contradiction. Hence, has at most one fixed point.
Next, we prove the existence of fixed points of . For any , set , , and with and . Now, we consider the following two cases: (i)If there exists such that , then we have . This shows , a fixed point of . Therefore, the proof is finished(ii)If , for any , then we have , for each . Hence, implies , for every By Proposition 6, we obtain Furthermore, for any , we have Since , then the inequality (6) implies On the other hand, from (5) and (F1), it follows that the sequence is decreasing. Furthermore, we can apply Lemma 4 to get According to (F2), there exists such that Multiplying (6) by results which implies . It follows that there exists such that , for all . Thus, which shows that the series is convergent.

Now, we prove that is a Cauchy sequence. For all , we drive the proof into two cases. (a)If is odd, we can obtain

Hence, we deduce that (b)If is even, we can get

Furthermore, we conclude that

Since , we can assume that . By a similar method, replacing with in (11), there exists such that which implies and . Together (a) with (b), for every , letting , we deduce that

Thus, is a Cauchy sequence. Since is complete, there exists such that . On the other hand, we notice that holds for all with . Since is increasing, then

It follows that

Hence, we get , which completes the proof.

Next, we will give an example to support Theorem 8.

Example 1. Let , where and . For any , we define by

Then, it is not difficult to see that is a Branciari -metric space with constant .

Let be a mapping satisfying

Now, we verify that is an -contraction satisfying the condition . We take , , and . Then, and

Let be a mapping defined by , then it is easy to see that . In addition, we can get

Hence, satisfies the conditions of Theorem 8 and is the unique fixed point of .

3.2. Weak Well-Posedness for the Fixed-Point Problem of an -Contraction

The notion of well-posedness of a fixed-point problem has evoked much interest to several researchers [21–23]. Now, we introduce the concept of weak well-posedness for the fixed-point problem of the mapping .

Definition 9. Let be a metric space and be a mapping. The fixed-point problem of is said to be weakly well-posed if (1) has a unique fixed point in (2)For any sequence in with and, we have

Theorem 10. Let be a complete Branciari -metric space with constant and be an -contraction for with the continuous differentiable function and . Then, the fixed-point problem of is weakly well-posed.

Proof. From Theorem 8, it follows that satisfies (5). Let be the unique fixed point of . For any sequence in with and , in general, the sequence can be decomposed into two subsequences , where and for any . Consequently, we can prove the following two assertions:
(W1) if is such that for all , then
(W2) if is such that for all , then

In fact, (W1) is trivial, since as . Now, we prove (W2). Because of the uniqueness of the fixed point of , we can assume that there exists such that for . If for any , then we can apply to get

Since is strictly increasing, we have . Then,

Then there exists a constant between and such that which implies

Actually, we can assume that there exists a distinct subsequence of {}. Otherwise, there exists and such that for . Since , we get . If , then because of the uniqueness of the fixed point of . For , we obtain . Since is an -contraction, together with (19), we have which is a contradiction. Hence, there exist () such that . Then,

Combining with (28), we deduce that

Since

Then,

Therefore, we get which implies since is continuous and . Hence, we deduce that . From (30), it follows that .

3.3. The Problem of Common Fixed Points for Two -Contractions

Now, we continue to solve the problem of common fixed points for two -contraction mappings in a complete Branciari -metric space.

Theorem 11. Let be a complete Branciari -metric space with constant . If there exist and , such that are two -contraction mappings on and satisfy for any with , , and for any . Then, there exists a unique common fixed point of the pair mappings .

Proof. Let , we define by Combining with (34) and (35), we have Let , it is not difficult to see since . Thanks to the strictly monotone increasing property of , we deduce Similarly, Hence, For any , we obtain Notice that Since is strictly monotone increasing, we have By induction, we get

Next, we will show that is a Cauchy sequence in . We consider the following two cases. (i)Let , if is odd and , we have(ii)Let , if is even and , we have

Letting , we obtain that for all . Hence, is a Cauchy sequence in . From the completeness of , it follows that there exists such that

Next, we verify that is a fixed point of the mapping . Indeed, if , then

By using the strictly monotone increasing property of , we get We can also see that

It follows that

Hence, we deduce that which contradicts the fact . Therefore, we obtain , i.e., . Similarly, we can get . Therefore, we have

Next, we will show that and have a unique common fixed point. Suppose that , are two distinct common fixed points of and . Then,

In view of the strictly monotone increasing property of , we obtain , which is a contradiction. It means .

Next, we discuss the existence of fixed points for Hardy-Rogers-type set-valued -contractions in a complete Branciari -metric space.

Lemma 12 (see [24]). If is right continuous and satisfies (F1), then for all with .

Definition 13 (see [24]). Let be a Branciari -metric space and be the family of all nonempty closed and bounded subsets of . Let be a function defined by for all , where . Then, defines a metric on called the Hausdorff metric induced by .

Theorem 14. Let be a complete Branciari -metric space with constant . Assume that there exist and such that satisfies for all and with and . If is right continuous, then has a fixed point in .

Proof. Let be an arbitrary point of and choose . If , then is a fixed point of and the proof is finished. Assume that , then . Since is right continuous and is closed, it follows from Lemma 12 that Noticing that , then there exists such that Consequently, we get which implies Since is strictly increasing, we deduce and hence, Since , we obtain Combining with (58), we get By induction, we can get a sequence satisfying is decreasing, , , and for all .
Let and . Then, we have for all . Similarly to the proof of Theorem 8 (while ), we deduce that is a Cauchy sequence. Since is complete, there exists such that as . Now, we prove that is a fixed point of . Since we deduce that It follows that Noting that we obtain Combining with (66) and (68), we get Since , then which implies . Hence, is a fixed point of since is closed.

4. Applications

4.1. Existence of Solutions of Integral Equations

Firstly, we will apply Theorem 8 to solving the existence and uniqueness of the solution of the following integral equation: where is a function such that , , and are two continuous functions.

Let be the set of real continuous functions defined on and the operator be defined by

We will prove the following result.

Theorem 15. Let and be given by for all .

Suppose there exists such that and for and . Then, the integral equation (70) has a unique solution in .

Proof. Clearly, each fixed point of in (71) is a solution of (70). Meanwhile, it is not difficult to see that is a complete Branciari -metric space with . Then, we obtain Consequently, for each , we have which implies for each . From Definition 7, the mapping is an -contraction with . By applying Theorem 8, the operator has a unique fixed point, that is, the integral equation (70) has a unique solution.

4.2. Existence of the Common Solution of Functional Equations

Finally, we will apply Theorem 11 to solving the existence and uniqueness of the solution of the dynamic programming problem. In particular, the problem of dynamic programming related to multistage process reduces to solving the existence and uniqueness of the solution of the following functional equations: where , , and . We assume that and are Banach spaces, is a state space, and is a decision space. For details, see also [25].

In this section, we will show that the equations (76) and (77) have one common solution. Let be the set of all bounded real-valued functions defined on a nonempty set . We define the norm by for all , it is not difficult to see that is a Banach space. Moreover, we define for all . Then, is a complete Branciari -metric space with . Consider the operators given by

Now, we prove the following theorem.

Theorem 16. Let be defined as in (80) and (81), respectively. Suppose that the following hypotheses hold: (i) and are bounded functions(ii)There exists such that, for all , , and (iii)There exists such that, for all , , and where for all with , , and

Then, there exists a unique common fixed point of the pair mappings .

Proof. Indeed, let , then is bounded, there exists such that for all . By hypothesis (i), there exist , and such that for all and . Furthermore, by (ii), for all and , we have Consequently, for all , , and , we get Similarly, which show that and are bounded. Let , , , there exist such that Then by (89) and (92), we obtain Similarly, by (90) and (91), we deduce Besides, we also find Since is arbitrary, we conclude Hence, we deduce Similarly, we can get Then, which show that and are two -contraction mappings on with . By Theorem 11, there exists a unique common fixed point of the pair mappings which is the solution of the equations (76) and (77).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

Acknowledgments

This research was funded by the University Nursing Program for Young Scholar with Creative Talents in Heilongjiang Province under Grant UNPYSCT-2017078, the Postdoctoral Science Foundation of Heilongjiang Province under Grant LBH-Q18067, and the Introduction and Cultivation Project of Young and Innovative Talents in Universities of Shandong Province.

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Copyright © 2020 Lili Chen et al. 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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