On Extended Branciari -Distance Spaces and Applications to Fractional Differential Equations

Abstract

In this work, we define new -rational contractive conditions and establish fixed-points results based on aforesaid contractive conditions for a mapping in extended Branciari -distance spaces. We furnish two examples to justify the work. Further, we discuss results on weak well-posed property, weak limit shadowing property, and generalized -Ulam-Hyers stability in the underlying space. Finally, as an application of our main result, we obtain sufficient conditions for the existence of solutions of a nonlinear fractional differential equation with integral boundary conditions.

1. Introduction and Preliminaries

The distance notion in the metric fixed-point theory is introduced and generalized in different ways by many authors [1–5]. Bakhtin [6] defined the notion of -metric space which is further used by Czerwik in [7, 8]. In [9], Branciari extended the metric space and introduced the notion of the Branciari distance by changing the property of triangle inequality with quadrilateral one.

Definition 1 [9]. Let be a set and let such that, for all and all (bd1) if and only if (self-distance/indistancy)(bd2) (symmetry)(bd3) (quadrilateral inequality).

The symbol the denotes Branciari distance space and abbreviated as “BDS.”

In [10], Kamran et al. introduced the notion of extended -metric space as a generalization of -metric space and proved the following result.

Definition 2 [10]. Let be a set and . We say that a function is an extended -metric (-metric, in short) if it satisfies (eb1) if and only if (eb2) (symmetry)(eb3),for all . The symbol denotes a -metric space.

Theorem 3 [10]. Let be a complete extended -metric space such that is a continuous functional. Let satisfy for all where such that for each , , here , . Then has precisely one fixed-point . Moreover, for each , .

In [3], Mitrović et al. extended Theorem 3 and proved the following:

Theorem 4 [3]. Let be a complete extended -metric space such that is a continuous functional. Let satisfy for all where are nonnegative real numbers with . Then, has a unique fixed-point . Moreover, there exists a sequence in which converges to such that for every .

In [11], Abdeljawad et al. defined the notion of extended Branciari -distance (EBbDS, in short) by combining the extended -metric and Branciari distance.

Definition 5 [11]. Let be a set and . We say that a function is an extended Branciari -metric (-metric, in short) if it satisfies (ebb1) if and only if (ebb2)(ebb3),for all , all distinct . The symbol denotes the extended Branciari -distance space. For , will be called a Branciari -distance space (BbDS, in short).

Example 1. Let and define by with . Note that for all , and if and only if . Also, . Hence, it is clear that is an EBbDS, but it is neither an BDS nor metric space.

Definition 6 [11]. Let be a set endowed with extended Branciari -distance . (a)A sequence in converges to if for every there exists such that for all . For this particular case, we write (b)A sequence in is called Cauchy if for every there exists such that for all (c)An -metric space is complete if every Cauchy sequence in is convergent.On the other hand, in [12], Samet et al. define the notion of -admissible mappings which is further extended by Sintunavarat [13] and named as weakly -admissible mapping.

Definition 7. For a set, let and be two mappings. Then is called (1)[12] -admissible if (2)[13] weakly -admissible if

For a set and a mapping , we use

It is noted that

The notion of well-posedness of a fixed-point problem (fpp) has evoked much interest of several mathematicians, for example, Popa [14, 15] and others. In the paper [16], authors defined a weak well-posed (wwp) property in BbDS and in the papers [17, 18]; the authors have discussed limit shadowing property of fixed-point problems.

The aim of this work is to introduce -rational contraction in an EBbDS and prove the existence of fixed points of such rational contraction in an EBbDS. We also discuss the weak well-posedness, limit shadowing property, and generalized weak-Ulam-Hyers stability of fixed-point problems in a EBbDS. As an application of our main result, we obtain sufficient conditions for the existence of solutions of a nonlinear fractional differential equation with integral boundary conditions. By doing these work, we generalize Theorems 3 and 4 in the sense that we use a more general contractive condition which depends on the variable (Lipschitz constants), function on the left-side of contractive condition, and proved results on the weakly -admissible mapping on more general space structures. It is justifies the usefulness of these terms through illustrations, and the results are real generalization as the considered distances are neither metric space not Branciari distance space.

2. Main Results

2.1. -Rational Contractive Mapping and Fixed Points

We start with introducing the notion of -rational contraction in a EBbDS as follows.

Definition 8. Let be an EBbDS and and . A mapping is said to be an -rational contraction, if there exist with which implies

We denote by the collection of all -rational contractive mappings on .

The set of all fixed points of a self-mapping on a set will be denoted by .

We are now in a position to state and prove the result.

Theorem 9. Let be a complete EBbDS and . Let be a mapping satisfying the following: (A1)(A2)There exists such that (A3) is continuous.

Then, . Furthermore, for any , the sequence satisfying is convergent.

Proof. By virtue of condition (A2), there exists such that . Define the sequence by . If there exists such that , then , and we are complete. Therefore, we assume that for all
It follows that It follows from and that Continuing this process, we obtain

Step 1. First, we prove that It follows from that If for some , then from (13), we have , which is a contradiction since and . Thus, for all , and the sequence is a decreasing sequence of real numbers. Therefore, there exists such that Again applying the limit in (13), we get which leads to as . Thus, we get

Step 2. At this step, we will prove that is a Cauchy sequence, that is, for , we prove Using (ebb3), we have Applying and using (12), we get Hence, is a Cauchy sequence. Since is a complete , then there exists a point such that as , that is Next, we prove that . Indeed, we write Since is continuous, on letting , we obtain , that is, , and hence, is a fixed point of .
To prove the uniqueness of fixed-point , we impose an additional requirement. (A4)For every pair and of fixed points of , .

Theorem 10. In addition of condition (A4) in Theorem 9, is a singleton set.

Proof. Following Theorem 9, . To prove is a singleton set, assume that there exist with , and by (A4), we have . It follows from that which implies that a contradiction, and hence, .

2.2. Illustrations

Example 2. Let . Define so that for all , and , otherwise. Then is a EBbDS with but neither a BDS nor a metric space . For instance but Consider the self-mapping on , and and for all .

It is easy to see that . We will check that satisfies (8) for with and . We demonstrate by three nontrivial possible cases. Here, .

Case 1. , (or vice versa if , change places). Then, , , and Therefore, (8) implies that , and (8) holds true.

Case 2. , (or vice versa if , change places). Then, , , and and it is easily seen that (8) is fulfilled.

Thus, all the conditions are fulfilled, and has a unique fixed point (which is ).

Note that in this example the use of weakly -admissibility and was crucial because, e.g., if we take , , we get , and and no contractive condition for any can be chosen which would holds for these points.

Example 3. Consider and define by . Then, is a EBbDS with but neither a BDS not a metric space . For instance but for all .
Consider the self-mapping on given by . Taking and such that for all , and for , it is obvious to see . Here, .
Then equation (8) for would be of the form holds whenever and .

For example, we demonstrate (34) is true for two cases:

Case 1. , (or vice versa if , change places). Then, (34) will be which is true.

Case 2. , (or vice versa if , change places). Then, (34) will be which holds true.

Similarly, it can be verified for any with . Thus, all the conditions are fulfilled, and the is a singleton set.

2.3. Weak Well-Posedness, Weak Limit Shadowing, and Generalized -Ulam-Hyers Stability

The notion of well-posedness of an fpp has evoked much interest of several mathematicians, for example, Popa [14, 15] and others. In the paper [16], the authors defined a weak well-posed (wwp) property in BbDS. In what follows, we extend this notion to EBbDS.

Definition 11. Let be a complete EBbDS and be a mapping. The fpp of is said to be weak well-posed if it satisfies the following: (1) is a singleton set in (2)For any sequence in with and

Theorem 12. Let be a complete EBbDS and be a mapping satisfying all the conditions of Theorem 9 and a sequence in such that , , and . Then, the fpp of is wwp.

Proof. Let be a sequence in such that and , for ; we obtain from (ebb3) that Taking limit WLOG, we can assume that there exists a distinct subsequence of . Otherwise, there exists and such that for . Since , we get . If , then due to uniqueness of the fixed point of . For , we obtain . So, we have For and , we have Therefore, since , we get So , i.e., , a contradiction. Hence, there exist such that . Then which as . On replacing the value in (39), we get Again, since and , we have which implies On placing in (39), we get Therefore, .

The limit shadowing property of fpps has been discussed in the papers [17, 18]. We define weak limit shadowing property (wlsp) in EBbDS.

Definition 13. Let be a complete EBbDS and be a mapping. The fpp of is said to have wlsp in if assuming that in satisfies as and , it follows that there exists such that as .

Theorem 14. Let be a complete EBbDS and be an -contractive mapping for and with in such that , and . Then, has the wlsp.

Proof. Since is a fixed point of , we have , and let in such that , ; then, by virtue of Theorem 12, we have , and therefore, we can write .

In the following, we define the generalized -Ulam-Hyers stability (G-UHS) of fixed-point problem (fpp) in EBbDS as an extension of -metric space case discussed in [19, 20] (see also [21]).

Definition 15. Let be a complete EBbDS and be a mapping. The fixed-point equation (FPE) is called the generalized weak-Ulam-Hyers stable (G-UHS in short) in the setting of EBbDS if there exists an increasing function , continuous at , with , such that for each and an -solution , that is there exists a solution of (48) such that

If for all , where , then FPE (48) is said to be -UHS in the setting of EBbDS.

Theorem 16. Let be a complete EBbDS and be an -contractive mapping for and and also that the function is strictly increasing and onto. Then the FPE (48) is G-UHS.

Proof. Following Theorem 14, we have , that is, is a solution of the FPE (48) with . Let and be an -solution of FPE (48), that is Since , and are -solutions. Since we have , so

Let us discuss the two possible cases.

Case 1. If , then we get that is which implies that

Case 2. If , then (12) gives It shows that the inequality (50) is true for all cases, and thus the FPE (48) is G-UHS.

3. Application

In this section, we discuss the existence of solutions of a nonlinear fractional differential equation (FDE) [22] as an application of Theorem 9. Some other FDE-related work can be seen in [23–25].

The Caputo fractional derivative of order is defined as where is a continuous function, denotes the integer part of the positive real number , and is the gamma function.

Consider the nonlinear FDE with the integral boundary conditions where , , and are a continuous function.

Let be endowed with the EBbDS function and .

Theorem 17. Let be the operator defined by for , . Also, let be a given function. Assume the following: (F1) is a continuous function, non-decreasing in the second variable(F2)There exists such that for all (F3) and for all imply that for all (F4)There exists such that for with , and , we have where and . Then, the problems (58) and (59) have at least one solution .

Proof. Define a function by where . It is obvious to check that the assumption (F2) implies the condition (A2) of Theorem 9. Assumption (F3) clearly implies that .
Let be , i.e., for all . For each , by the definition (61) of operator , we have (using Cauchy-Schwartz inequality) that is Applying (F4) and small calculations, we get This implies that for all with where Thus, . Therefore, all the requirements of Theorem 9 are fulfilled, and we conclude that there is a fixed-point of the operator . It is well known (see, e.g., [22], Theorem 17) that in this case is also a solution of the integral equation (61) and the FDE (58) with the condition (59).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

The authors are thankful to the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University, Al-Kharj, Kingdom of Saudi Arabia, for supporting this research. The authors are thankful to the learned reviewer for his valuable comments.