An Implicit Relation Approach in Metric Spaces under -Distance and Application to Fractional Differential Equation

Abstract

The purpose of this work is to introduce a new class of implicit relation and implicit type contractive condition in metric spaces under -distance functional. Further, we derive fixed point results under a new class of contractive condition followed by three suitable examples. Next, we discuss results about weak well-posed property, weak limit shadowing property, and generalized -Ulam-Hyers stability of the mappings of a given type. Finally, we obtain sufficient conditions for the existence of solutions for fractional differential equations as an application of the main result.

1. Introduction and Preliminaries

In , Kada et al. [1] introduced the concept of a -distance on a metric space and proved a generalized Caristi fixed point theorem, Ekeland’s -variational principle, and the nonconvex minimization theorem according to Mizoguchi and Takahashi [2].

Definition 1 (see [1]). Let be a metric space. A function is called a -distance on if it satisfies the following properties:
(W1) for any
(W2) is lower semicontinuous in its second variable, i.e., if and , then
(W3) For each , there exists a such that and imply

The following examples show that a -distance is not necessarily a metric.

Example 1. (1)Let be a metric space and . Define and for all . Then, and satisfy (W1)–(W3). Obviously, is not a metric since for any (2)Consider as a metric space with the usual metric. Define

Then, is a -distance on which is not a metric (since it is not symmetric). Note that is a convergent sequence in but for all .

To prove the main theorem, we need the following lemma, proved by Kada et al. [1].

Lemma 2. Let be a metric space and let be a -distance on . Suppose that and are sequences in , and are sequences in converging to , and let . Then, the following assertions hold. (i)If and for all , then . In particular, if , then (ii)If and for all , then converges to (iii)If for all with , then is a Cauchy sequence(iv)If for all , then is a Cauchy sequence

Lemma 3 (see [1, 3]). Let be a -distance on a metric space and be a sequence in such that for each there exists such that implies , i.e., . Then, is a Cauchy sequence.

Recall that the set is called the orbit of the self-map at the point .

Definition 4. Let be metric spaces and be a mapping. Then, (1) (the set of fixed points of )(2)a mapping is called a Picard operator if there exists such that and converges to , for all (3)[4] a metric space is said to be -orbitally complete if every Cauchy sequence contained in (for some in ) converges in (4)a mapping is said to be orbitally -continuous if, for some , the following condition holds: for any and a strictly increasing sequence of positive integersand for any imply that (5) is called orbitally continuous if, for any and a strictly increasing sequence of positive integers, as implies that as

In [5], Samet et al. defined the notion of -admissible mapping which was further sharpened by Karapinar et al. [6] and extended in [7].

Definition 5. For a set , let and be two mappings. Then, is said to be (i)[5] -admissible if(ii)[6] triangular -admissible if is -admissible and

Lemma 6 (see [6]). Let be a triangular -admissible mapping. Assume that there exists such that . Define a sequence by for . Then, for all with .

Similarly, we can state and prove the following lemma.

Lemma 7. Let be a triangular -admissible mapping. Assume that there exists such that . Define a sequence by for . Then, for all with .

To the best of our knowledge, there is no fixed point result in the literature which has been derived by implicit type contractive relation in a metric space under -distance. Also, -distance is not necessarily a metric (examples are given above). Motivated by this fact, there is a need for introducing such type of contractive conditions. With this in mind, in Section 2, we introduce the notion of a new implicit relation and -implicit contractive mapping in the respective structure. Then, we establish unique fixed point results under aforesaid implicit contractive condition for -admissible and orbitally continuous mappings on orbitally complete spaces. We demonstrate the results by three illustrative examples. We note that the symmetry condition and full completeness of the underlying space are not required. In Section 4, some new results on weak well-posed property, weak limit shadowing property, and generalized -Ulam-Hyers stability of mappings of the mentioned type are discussed. In Section 4, a sufficient condition for the existence of solutions for fractional differential equations as an application of the main result is given.

2. Implicit Relation for -Distance on Metric Spaces

In this section, we introduce a modified version of implicit relation and examples discussed in [8, 9].

Let be the set of all continuous functions satisfying the following conditions:

(₁) is nonincreasing in the fifth and sixth variables

(₂) There exists such that for all ,

(_2a) implies that

(_2b) implies that

(₃) and for

Example 2. Let , and .
(_2a) For , we have which implies that . Then, . Hence, , for
(_2b) Similarly as (_2a), if , then for
(₃) For all and for ,

Example 3. Let and .

Similar to Example 2.

Example 4. Let and .
(_2a) For , we have which implies that , that is, , for
(_2b) Similarly as (), if , then
(₃) for all and for

Example 5. Let .
(_2a) For and .
Then, , that is, . Hence, , for .
(_2b) Similarly, if , then
(₃) for all for

Example 6. Let and .
(_2a) For , we have which implies that , that is, , for
(_2b) Similarly as (), if , then
(₃) for all for

Now, we define -implicit contractive mapping in the metric space under -distance using the above introduced implicit relation.

Definition 8. Let be a metric space with -distance , be a given mapping, and be a functional. We say that is an -implicit contractive mapping, if there exists a function such that for all .

If (9) is satisfied for (for some ), we say that is an orbitally -implicit contractive mapping (at ).

Now, we are equipped to state and prove our first main result as follows.

Theorem 9. Let be a metric space with -distance and . Suppose that the following conditions hold: (i)There exists such that and (ii) is a triangular -admissible mapping(iii) is an orbitally -implicit contractive mapping(iv) is -orbitally complete at (v) is orbitally continuousThen, there exists a point . In addition, provided holds.

Proof. Let be the point described in (i). Define a sequence by for . If for some , then obviously . Hence, we suppose that for all . First, we show that Using (ii) and Lemma 6, we have for all . Then, for all , using (9) for , , Denoting for all and applying in the fifth variable, we have It follows from that there is such that and so, that is, the sequence is a nonincresing sequence of real numbers. Therefore, there exists such that Applying the limit in (12), by the continuity of , we get a contradiction, and therefore, .
For , using condition (ii) and Lemma 7, we get for all . Using similar arguments as above, we can prove Next, we show that is a Cauchy sequence in . For this, we show that On the contrary, suppose that condition (18) does not hold. Then, we can find a and increasing sequences of positive integers with such that By (10), there exists a , such that implies that In view of the two last inequalities, we observe that . We may assume that is the minimal index such that (19) holds, so that Now, making use of (19), we get Thus, Using the triangle inequality, we have Taking the limit on both sides and making use of (10), (17), and (23), we obtain Again, using the triangle inequality, we have Taking the limit on both sides and making use of (10), (17), and (23), we obtain Combining (25) and (27), we have From Lemma 6, we have . Therefore, on applying condition (9), we get Now applying in the fifth and sixth variables, we have Applying the limit and using continuity of , we get a contradiction to . Hence, must be a Cauchy sequence in .
Since is -orbitally complete, there exists a point such that . We shall show that is a fixed point of .
Using the orbital continuity of (due to condition (v)), we have . Owing to the uniqueness of the limit, we obtain .
Finally, assume that . Then, by (9) for , we have or It follows from (for ) that there is such that a contradiction. Therefore, .☐

Next, we have the following result.

Theorem 10. The conclusion of Theorem 9 remains true if condition (v) is replaced by the following one:
() For every with ,

Proof. Following the proof of Theorem 9, we observe that the sequence is a Cauchy sequence, and so, there exists a point in such that . Since , for each , there exists an such that implies . Since and is lower semicontinuous, Therefore, . Set , so that Assume that . Then, by the hypothesis (), we have which contradicts our assumption. Therefore, .☐

The last conclusion is derived as in the proof of Theorem 9.

In what follows, we give a sufficient condition for the uniqueness of the fixed point in Theorems 9 and 10.

Theorem 11. In addition to the hypotheses of Theorem 9 (or Theorem 10), if for all fixed points and such that , , then has a unique fixed point.

Proof. Suppose that and are two fixed points of such that . Then, using (9) for , i.e., a contradiction to , and thus, . Also, we have . So, by using Lemma 2, we infer that , i.e., the fixed point of is unique.☐

By choosing from Examples 2–6, we have the following consequences.

Corollary 12. Let all the conditions of Theorems 9 and 10 be satisfied, except that the assumption of orbitally -implicit contractive mapping for is replaced by either of the form where , , , or where , , , or where , , , or where , , .

Then, is a singleton.

3. Illustrations

Example 7. Consider the set with the usual metric . Define a -distance by for all .
Consider the self-mapping on given by . Take . It is simple to show that and that is -orbitally complete at .
Define functional as follows: At in , and . Also, is a triangular -admissible mapping in .
Considering Example 4, we can define as

Here, . One can easily check that for , so that and that belongs to the set . We will show that is an orbitally -implicit contractive mapping.

Take , and so, . Consider two cases.

Case 1. If and , , then (9) reduces to and is fulfilled for , . If and , or , then (9) holds trivially.

Case 2. Let . Then, (9) reduces to and is fulfilled for , .

Thus, is orbitally -implicit contractive mapping. Therefore, all the conditions of Theorem 9 are satisfied, and is the unique fixed point of in .

Example 8. Consider the set with the usual metric . Define a -distance by for all .
Consider the self-mapping on given by

Take . It is simple to show that and that is -orbitally complete at .

Define a function as follows:

At in , and . Also, is a triangular -admissible mapping in .

Considering Example 3, we can define as

Here, . One can easily check that for , so that and that belongs to the set . We will show that is an orbitally -implicit contractive mapping.

Take , and so, . Consider two cases.

Case 1. If and , , then (9) reduces to that is, and is fulfilled, for so that . If and , , or , then (9) holds trivially.

Case 2. Let . Then, inequality (9) has the form that is, that is, and is fulfilled, for so that .
Thus, is an orbitally -implicit contractive mapping.
If , we have so that

Thus, all the conditions of Theorem 10 are satisfied and is the unique fixed point of in .

Example 9. Consider the set with the usual metric . Define a -distance by for all .
Consider the self-mapping on given by Take . It is simple to show that and that is -orbitally complete at .
Define functional as follows: At in , and . Also, is a triangular -admissible mapping in .
Considering Example 6, we can define as and .

Take , and so, . Consider two cases.

Case 1. If and (or and ), , then (9) reduces to and is fulfilled, for , . If and , then (9) holds trivially.

Case 2. Let . Consider and . Then, the inequality (9) has the form which is equivalent to which is obviously true for such that .
It can be noted that for , we have

Thus, all the conditions of Theorem 10 are satisfied except (). Clearly, has no fixed points in .

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

The notion of well-posedness of a fixed point problem (fpp) has evoked much interest of several mathematicians, for example, Popa [10, 11]. In the paper [12], the authors defined a weak well-posed (wwp) property in metric space. In the following, we extend this notion to a -distance in metric space.

Definition 13. Let be a metric space and be a -distance in . Let be a mapping having a unique fixed point such that . The fpp of is said to be wwp with respect to if for any sequence in with and , one has .
To guarantee the wwp of a mapping , we add the following additional condition for functions and call the respective set :
(₄) for all , , implies that there exists such that

Examples 2–4 and 6 satisfy the condition (₄).

Theorem 14. Let be a metric space and be a -distance on . Suppose that all the hypotheses of Theorem 9 hold for . Then, the ffp for is wwp.

Proof. Let be a sequence in such that and , for . We obtain that Taking the limit as , we get 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 . From (9), we have i.e., i.e., It follows from (_2a) that there is such that which on applying gives . Also, we have . So, by Lemma 2, we get , a contradiction. Hence, there exist such that . Then, which as . On replacing the value in (68), we get From (9), we have i.e., which on applying limit as gives It follows from (₄) that there is such that ☐

Combining (74) and (78), we get as .

The limit shadowing property of fixed point problems has been discussed in the papers [13, 14]. We define weak limit shadowing property (wlsp) in metric spaces under -distance.

Definition 15. Let be a complete metric space and be a -distance in . Let 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 16. In addition to the hypotheses of Theorem 9 (or Theorem 10) if in is 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 9, we have , and therefore, we can write .☐

Next, we define generalized--Ulam-Hyers stability (G--UHS) of fixed point problem (fpp) in metric spaces under -distance.

Definition 17. Let be a metric space and be a -distance on . Let be a mapping. The the fixed point equation (FPE) is said to be G--UHS in the setting of metric spaces under -distance, if there exists an increasing function , continuous at , with , such that for each and an -solution , that is, there exists a solution of (79) such that

If for all , where , then FPE (79) is said to be -UHS in the framework of metric spaces under -distance.

Remark 18. If , then Definition 17 reduces to the notion of GUHS in metric spaces. Also, if for all , where , then it reduces to the notion of UHS in metric spaces. Finally, if , then it reduces to the classical UHS.

Theorem 19. Let be a metric space and be a -distance on . Suppose that all the hypotheses of Theorem 9 hold, using the contraction condition in the form (42). Then, the FPE (79) is G--UHS.

Proof. Following Theorem 9, we have ; that is, is a solution of the FPE (79) with . Let and be an -solution of (79), that is, Since , and are -solutions. Now, From the contractive condition (42) for , we get Therefore, from (83), we obtain which implies that i.e., ☐

Thus, the inequality (81) holds, and therefore, the FPE (79) is G--UHS in the metric spaces under -distance.

5. Application

Fractional differential/integral equations (FDE/DIE) have been extensively studied as an application of fixed point theory. In fact, to get the unique solution of FDE, one has to apply the Banach fixed point theorem or its variants. There are different types of FDEs in the literature, but the FDEs in the Caputo sense are the easiest to apply. The main advantage of Caputo derivative is that the derivative of the constant function is while most of the other fractional derivatives do not have such an important property. This property helps in initial value problems to apply fixed point theorems. In paper [15], the existence of solutions for some Atangana-Baleanu fractional differential equations in the Caputo sense have been discussed. Some other FDE-related work can be seen in [16, 17].

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

In this section, we discuss the existence of solutions of following nonlinear fractional differential equation (FDE) [18] as an application of Theorem 9.

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

Let be endowed with the usual distance and -distance by

Theorem 20. Let be the operator defined by for , . Also, let be a given function. Assume that
(F1) is a continuous function, nondecreasing in the second variable
(F2) there exists such that , for all
(F3) for , and for all implies that for all
(F4) there exist with such that for and we have where and . Then, the problem (89)–(90) has at least one solution .

Proof. Define a function by where . It is easy to check that the assumption (F2)-(F3) implies the condition (i)-(ii) of Theorem 9, respectively.
Let , then for each , by the definition (92) of operator , we have that is, After easy calculations, we get This implies that for all , where If we consider where so that , then and the condition (iii) of Theorem 9 is satisfied. 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., [18], Theorem 3.1) that in this case is also a solution of the integral equation (92) and the FDE (89) with condition (90).☐

6. Further Work

On the line of our work, the following two types of FDE can also be discussed: with the two-point boundary conditions

where is a continuous function. with the two-point boundary conditions where is a continuous function.

7. Conclusion

In this work, a new class of implicit relation and implicit type contractive condition under aforesaid implicit relation in the metric spaces under -distance functional have been introduced. Next, some basic fixed point results under respective contractive conditions followed by three suitable examples have been discussed. Further, we have discussed weak well-posed property, weak limit shadowing property, and generalized -Ulam-Hyers stability in the underlying spaces. Finally, sufficient conditions for the existence of solutions for the fractional differential equation as an application of the established result have been discussed.

Data Availability

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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.