Fixed Point Theory and Applications for Function SpacesView this Special Issue
On Extended Branciari -Distance Spaces and Applications to Fractional Differential Equations
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  defined the notion of -metric space which is further used by Czerwik in [7, 8]. In , 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 . 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 , Kamran et al. introduced the notion of extended -metric space as a generalization of -metric space and proved the following result.
Definition 2 . 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 . 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 , .
Theorem 4 . 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 , Abdeljawad et al. defined the notion of extended Branciari -distance (EBbDS, in short) by combining the extended -metric and Branciari distance.
Definition 5 . 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 . 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 , Samet et al. define the notion of -admissible mappings which is further extended by Sintunavarat  and named as weakly -admissible mapping.
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 , 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, .
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 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
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 , 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, .
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 .
Definition 15. Let