Fixed Point, Data Dependence, and Well-Posed Problems for Multivalued Nonlinear Contractions

Iqbal, Iram; Hussain, Nawab; Al-Sulami, Hamed H.; Hassan, Shanza

doi:https://doi.org/10.1155/2021/2200903

Journal of Function Spaces

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

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Unique and Non-Unique Fixed Points and their Applications

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Research Article | Open Access

Volume 2021 | Article ID 2200903 | https://doi.org/10.1155/2021/2200903

Fixed Point, Data Dependence, and Well-Posed Problems for Multivalued Nonlinear Contractions

Iram Iqbal,^1,2Nawab Hussain,³Hamed H. Al-Sulami,³and Shanza Hassan¹

Academic Editor: Santosh Kumar

Received10 May 2021

Revised10 Jul 2021

Accepted07 Aug 2021

Published24 Aug 2021

Abstract

The aim of the paper is to discuss data dependence, existence of fixed points, strict fixed points, and well posedness of some multivalued generalized contractions in the setting of complete metric spaces. Using auxiliary functions, we introduce Wardowski type multivalued nonlinear operators that satisfy a novel class of contractive requirements. Furthermore, the existence and data dependence findings for these multivalued operators are obtained. A nontrivial example is also provided to support the results. The results generalize, improve, and extend existing results in the literature.

1. Introduction and Preliminaries

Let be a metric space (in short ). The set of all nonempty subsets of is denoted by , the set of all nonempty closed subsets of is denoted by , the set of all nonempty closed and bounded subsets of is denoted by , and the set of all nonempty compact subsets of is denoted by . It is obvious that includes . For , define by

where . Such a function is called the Pompei-Hausdorff metric induced by , for more details, see, e.g., [1].

Lemma 1 [2]. Let be a and with . Then, for each and for each , there exists such that

If is a multivalued operator, then an element is called a fixed point for if . The symbol fix denotes the fixed point set of . On the other hand, a strict fixed point for is an element with the property . The set of all strict fixed points of is denoted by SFix .

Banach’s contraction principle [3] is the most fundamental result in metric fixed point theory. Since then, many authors have extended and generalized Banach’s contraction principle in many ways. Extensions of Banach’s contraction principle have spawned a wealth of literature. (see [13, 29]). One of an attractive and important generalization is given by Wardowski in [10]. He introduced a new type of contraction called -contraction and proved a new fixed point theorem concerning -contraction.

Definition 2 [10]. Let be a . A mapping is said to be -contraction if there exists such that for all , where is a function satisfying
() is strictly increasing
() For all sequence , , if and only if
() There exists such that

We denote by the collection of all functions satisfying (), (), and (). Also, define where

() for all with .

Theorem 3 [10]. Let be a complete and be a -contraction. Then, has a unique fixed point and for every , a picard sequence converges to .

Further, Turinici [11] is replaced () by the following condition: ()

Note that, in general, is not continuous. However, by and the properties of the monotone functions, we have the following proposition.

Proposition 4 [11]. Let be a function satisfying and , and then there exists a countable subset such that

Lemma 5 [11]. Let be a function satisfying and . Then, for each sequence in ,

After this, many authors generalized the -contraction in several ways (see [12–22] and references therein). In 2015, Klim and Wardowski [23] extended the concept of -contractive mappings to the case of nonlinear -contractions and proved fixed point theorems via the dynamic processes. In 2017, Wardowski [24] omitted one of the conditions of -contraction and introduced nonlinear -contraction).

Definition 6 [24]. A mapping is said to be a -contraction (or nonlinear -contraction), if there exists and a function satisfying
(H1)
(H2)

Theorem 7 [24]. Let be a complete and let be a -contraction. Then, has a unique fixed point in .

Very recently, Iqbal and Rizwan [25] considered a rich class of functions and generalized Definition 6 to obtain some new fixed point theorems for nonlinear -contractions involving generalized distance. On unifying the concept of Wardowski [10], Nadler [9] and Altun et al. [26] gave the concept of multivalued -contraction as follows.

Definition 8 [26]. Let be a complete and be a mapping. Then, is a multivalued -contraction, if there exists and such that for all ,

Theorem 9 [26]. Let be a complete and be a multivalued -contraction, and then has a fixed point in .

Afterwards, Olgun et al. [27] proved the nonlinear case of Theorem 9 as follows.

Theorem 10 [27]. Let be a complete and , if there exists and satisfying Then, has a fixed point in .

For more directions for nonlinear -contractions, consult [28, 29] and references there in. Next, we denote by the set of all continuous mappings satisfying the following conditions:

() ;

() is subhomogeneous; that is, for all and , we have

() is nondecreasing function; that is, for , , , we have

If , , , then and .

Also, define

Note that .

Example 1. Define by where . Then, . Since , so . Also, note that , so [30].

Example 2. Define by where . Then, .

Example 3. Define by where . Then, .

Now, we prove the following Lemma.

Lemma 11. If and are such that then .

Proof. Without loss of generality, we can suppose that . If , then a contradiction. Thus, .

Now, consider following examples.

Example 4. Let be a function defined by Then, satisfies , , and that is continuous but does not satisfies .

Example 5. Let be a function defined by Then, satisfies and but is not continuous.

Example 6. [31] Let be a function defined by where denotes the integral part of and , . Then, satisfies , , and but is not continuous.

Examples 4–6 clearly show that there exist some functions which does not satisfy the condition of continuity, , , and at a time. By getting inspiration from this, in this paper, we prove fixed point results for contractive conditions involving functions , not necessarily continuous and belongs to by taking support of a continuous function from . Our results generalize many results appearing recently in the literature including Altun et al. [32], Olgun et al., [27] Sgroi and Vetro [33], Vetro [34], Wardowski [24], and Wardowski and Dung [35].

For convenience, we set , the collection of all functions satisfying

Theorem 12. Let be a complete and be a multivalued mapping. Assume that there exists , a nondecreasing real valued function on and a real valued function on satisfying condition and such that the following conditions hold:
(N1) for all
(N2) For all and , implies Then, fix is nonempty.

Proof. Let be an arbitrary point and . Assume that ; otherwise, is a fixed point of , and the proof is complete. Then, and consequently . Compactness of ensures the existence of , such that . From (N1) and (N2), we get

Since is an nondecreasing function, (19) with implies that

By using Lemma 11, (20) implies

Next, arguing as previous, we get , such that with . Also, by using Lemma 11, from (N1) and (N2), we obtain

Continuing in the same manner, we get a sequence such that satisfying with and

for all . (23) implies that is a decreasing sequence of positive real numbers. Hence, from (N1) and (N2), we get

Thus, for all ,

Since , there exists and such that , for all . From (25), we obtain

Taking in (26), we get and by , we have

which further implies that

Now from , there exists such that

Then, from (26), for all , we have

Taking limit , in (30) and using (27) and (29), we have

Observe that from (31), there exist s such that for all . Thus, for all , we have

which further implies that

Now, in order to show that is Cauchy sequence, consider such that . From (33), we get

As a result of the above and the series’ convergence, , we receive that is Cauchy sequence. Since is a complete space, so there exists such that

Now,

Since is nondecreasing function, we obtain for all .

We claim that is fixed point of . On contrary, suppose that and by equation (37), we have

Passing to limit as in the above inequality, we obtain

which implies by Lemma 1 that

which is a contradiction. Hence, . Since is closed, therefore, .

Remark 13. By defining and in Theorem 12, we get back Theorem 2.3 of [27].

Example 7. Consider , and then is complete with metric . Define functions by and for all , respectively. Then is nondecreasing, satisfy the conditions and and for all .

Next, define , and by and for all , respectively. Then and (see Example 1). Observe that

Assume that , and then and . From Figure 1, it is clear that

and . Which further implies that

All hypothesis of Theorem 12 are satisfied and fix .

Observe the following in Example 7: (i) is not continuous at (ii)(iii)(iv).

Corollary 14. Let be a complete and be a multivalued mapping. Assume that there exists , a nondecreasing real valued function on and a real valued function on satisfying condition and such that (N1) and the following condition holds:
implies for all , where Then, fix is nonempty.

Proof. Define by Then, and result follow from Theorem 12.

Remark 15. Corollary 14 generalizes and improves Theorem 2.4 of [35]. In fact, by taking and by defining for all and for all in Corollary 14, then we find Theorem 2.4 of [35]. Corollary 14 shows that condition can be replaced by and the strictness of the monotonicity of is not necessary.

Corollary 16. Let be a complete and be a multivalued mapping. Assume that there exist , a non decreasing real valued function on and a real valued function on satisfying condition and such that (N1) and the following condition holds:
implies for all , where and . Then Fix is nonempty.

Proof. Define by where and . Then and result follows from Theorem 12.

Now, in the next Theorem, we replace the condition of by continuity of in hypothesis of Theorem 12 and obtain another fixed point result.

Theorem 17. Let be a complete and be a multivalued mapping. Assume that there exists , a continuous, nondecreasing real-valued function on and a real valued function on satisfying condition such that following conditions hold:
(N1) for all
(N2) implies for all and .
Then, fix is nonempty.

Proof. Let be an arbitrary point and . Then, as in proof of Theorem 12, we get a sequence such that satisfying with , Taking in (52), we get and by , we have which further implies that Next, we claim that If (55) is not true, then there exists such that for all there exists Also, there exists such that Consider two subsequences and of satisfying Observe that where is chosen as minimal index for which (59) is satisfied. Also, note that because of (58) and (59), the case is impossible. Thus, for all . It implies Using triangle inequality and by (58) and (59), we have Letting limit in (61) and using (53), we get Now, by using (53) and (62), we obtain Then, from , , and monotonicity of , we get Since is continuous, so by passing the limit , using equations (62) and (63), we have Now, since , we have ; so, (65) implies which is a contradiction to (17). Hence, (55) holds, which implies that is a Cauchy sequence. Completeness of ensures the existence of such that By following the same steps as in the proof of Theorem 12, we get . This completes the proof.

Corollary 18. Let be a complete and be a multivalued mapping. Assume that there exists , a continuous, nondecreasing real-valued function on and a real valued function on satisfying condition such that (N1) and the following condition holds: where , and . Then, fix is nonempty.

Proof. Define by where , and . Then, and result follow from Theorem 17.

Remark 19. Corollary 18 improves Theorem 1 of [35]. In fact, by taking and by defining for all in Corollary 18, then we are back to Theorem 1 of [34]. In Corollary 18, condition is weakened to the condition .

Next, we consider that are closed subsets of instead of compact subsets for all and obtain the following theorems.

Theorem 20. Let be a complete and be a multivalued mapping. Assume that there exists , and a real-valued function on such that following holds:
(G1) for all
(G2) implies for all and .
Then, fix () is nonempty.

Proof. Let be an arbitrary point and . Assume that ; otherwise, is a fixed point of , and the proof is complete. Then, since is closed, and consequently, . Due to , we obtain

Then, (71) with (G1) and (G2) gives

Thus, there exists such that

Since is an nondecreasing function, (73) with implies that

By using Lemma 11, (74) implies

Next, arguing as previous, we get with . Also, by using Lemma 11, from (G1) and (G2), we obtain

Continuing in the same manner, we get a sequence such that with and for all . (77) implies that is a decreasing sequence of positive real numbers. Hence, from , (G1), and (G2), we get

Thus, for all ,

Thus, from (79), there exists such that

Since , there exists and such that , for all . From (80), we obtain

Taking in (81), we get and by , we have

Now, from , there exists such that

Then, from (81), for all , we have

Taking limit , in (84) and using (82) and (83), we have

Observe that from (85), there exists such that for all . Thus, for all , we have

Now, in order to show that is Cauchy sequence, consider such that . From (86), we get

As a result of the above and the series’ convergence, , we receive that is Cauchy sequence. Since is a complete space, so there exists such that

Now,

Since is nondecreasing function, we obtain

We claim that is a fixed point of . On contrary, suppose that and by equation (90), we have

Passing to limit as in the above inequality, we obtain which implies by Lemma 1 that which is a contradiction. Hence, . Since is closed, therefore, .

Corollary 21. Let be a complete and be a multivalued mapping. Assume that there exists , and a real-valued function on such that (G1) and the following condition holds: where and . Then, fix is nonempty.

Proof. Define by where and . Then. and result follow from Theorem 20.

Corollary 22. Let be a complete and be a multivalued mapping. Assume that there exists , and a real-valued function on such that (G1) and the following condition holds: where . Then, fix is nonempty.

Proof. Define by where . Then, and result follow from Theorem 20.

Remark 23. Corollary 21 generalizes Theorem 24 of [33]. Indeed, by considering and by defining for all , where in Corollary 21, we obtain Theorem 32 of [33]. Also, by considering and by defining for all , where in Corollary 22, we get back the Theorem 24 of [32].

Theorem 24. Let be a complete and be a multivalued mapping. Assume that there exists , a nondecreasing and continuous real-valued function satisfying condition and a real-valued function on such that following holds:
(G1) for all
(G2) implies for all and .
Then, fix is nonempty.

Proof. Let be an arbitrary point and . Then, as in proof of Theorem 20, we get a sequence such that with , Taking in (100), we get and by , we have Next, we claim that If (102) is not true, then there exists such that for all , there exists Also, there exists such that Consider two subsequences and of ; then, as is proof of Theorem 17, we get Then, from (G1), (G2), and monotonicity of , we get Since is continuous, so by passing the limit , using equations (105) and (106), we have Now, since , we have ; so, (107) implies which is a contradiction to (17). Hence, (102) holds, which implies that is Cauchy sequence. Completeness of ensures the existence of such that By following the same steps as in the proof of Theorem 20, we get . This completes the proof.

Corollary 25. Let be a complete and be a multivalued mapping. Assume that there exists , a nondecreasing and continuous real-valued function satisfying condition and a real-valued function on such that (G1) and the following condition hold: where , , and . Then, fix is nonempty.

Proof. Define by where , , and . Then, and result follow from Theorem 24.

If we restrict in Corollary 22, then defined in the proof of Corollary 22 also satisfies and hence . Consequently, from Theorem 24, we get

Corollary 26. Let be a complete and be a multivalued mapping. Assume that there exists , a nondecreasing and continuous real-valued function satisfying condition and a real-valued function on such that (G1) and the following condition hold: where . Then, fix is nonempty.

2. Data Dependence

Let , be two nonempty sets and . Denote by , the graph of the multivalued operator is . A multivalued operator is said to be closed if is a closed set in . A selection for is a singlevalued operator such that , for each . Mo and Petruel in [36] discussed some basic problems including data dependence of the fixed point theory for a new type contractive multivalued operator. In [37], Rus et al. gave an important abstract notion as follows:

Definition 27. Let be a and a multivalued operator. Then, is a multivalued weakly Picard operator (briefly MWP operator) if for all and , there exists a sequence such that (i), (ii), for all (iii)The sequence is convergent, and its limit is a fixed point of

A sequence satisfying the conditions (i) and (ii) in Definition 27 is also called a sequence of successive approximations of starting from . Now, we present the main result of this section.

Theorem 28. Let be a and be two multivalued operators. Assume that there exists , a nondecreasing real-valued function on and a real valued function on satisfying condition and such that satisfies (N1) and (N2) for all : (i)There exists such that , for all (ii)Then(iii)Fix and (iv) and are MWP operators and

Proof. (a) By Theorem 12, we have that fix , for . Next, we prove that the fixed point set of multivalued operators is closed for . For this, let be a sequence in fix such that as . Then, Since is nondecreasing function, we obtain for all Suppose that , then we have Passing to limit as in the above inequality, we obtain which implies by Lemma 1 that which is a contradiction. Hence, . Since is closed, so .
(b) From the proof of Theorem 12, we immediately get that operators are MWP operators for . Now, we will show that . For this purpose, Let , , be arbitrary. Then, there exists such that and . Next, for , there exists such that and . Then, by using (3.1), we get and Inductively, we will obtain a sequence of successive approximations for starting from , satisfying the following: which further implies for each , Letting , we get that is Cauchy sequence in , and so it converges to an element . As in the proof of Theorem 12, we get that . From (121), letting to get Putting , we get that By interchanging the roles of and , we obtain that for each , there exists such that Hence, , and letting , we get the conclusion.

3. Strict Fixed Points and Well Posedness

Firstly, we define the notions of well posedness of a fixed point problem.

Definition 29 [38, 39]. Let be a , , and be a multivalued operator. Then, the fixed point problem is well posed for with respect to if
(a₁) ;
(b₁) If , , and as , then as

Definition 30 [38, 39]. Let be a , , and be a multivalued operator. Then, the fixed point problem is well posed for with respect to if
(a₂)
(b₂) If , , and as , then as

Remark 31. Note that if the fixed point problem is well posed for with respect to and , then the fixed point problem is well posed for with respect to .

Theorem 32. Let be a and be a multivalued operators. Assume that (1)There exist , a continuous, nondecreasing real-valued function on and a real-valued function on satisfying condition such that (N1) and (N2) hold for with ,Then, (a);(b)The fixed point problem is well posed for with respect to

Proof. By Theorem 17, we have that fix . Next, We will prove that . From (N1) and (N2), we get that Since is nondecreasing function, we obtain for all , Let , with ; then , and we have Since , above inequality implies that which is a contradiction. Hence, ; so, .
(b) Let , , such that We claim that where . If (131) is not true, there exists such that for each , we have that On the other hand, from (130), there exists such that for each . Hence, for each , we get Since , so by passing the limit , we obtain as , a contradiction. Consequently, proof is complete by Remark 31.

4. Conclusion

In the theory of set-valued dynamic systems, fixed points and strict fixed points of multivalued operators are essential notions. A rest point of the dynamic system can be read as a fixed point for the multivalued map , whereas a strict fixed point for can be viewed as the system’s endpoint. We have made a contribution in this approach by establishing some basic problems in multivalued fixed point and strict fixed point theory. We have proved several existence and data dependence results for multivalued nonlinear mappings satisfying a new class of contractive conditions via auxiliary functions. The obtained outcomes are backed up by a nontrivial example. The findings add to and expand on some of the most recent results in the literature.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Iram Iqbal 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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