Abstract
In this paper, we introduce the notion of -contractive multivalued weakly Picard operators via simulation functions, named as -contractions. We present some related fixed point theorems. We investigate data dependence and strict fixed point results. The well-posedness for such operators is also considered. Moreover, we generalize the results of Moţ and Petruşel. To show the usability of our results, we give some examples and an application to resolve a functional equation arising in dynamical systems.
1. Introduction and Preliminaries
The fundamental theorem concerning the existence of a fixed point for a self-mapping on a metric space is due to Banach [1].
Definition 1. A self-map on is called a contraction if there exists such thatholds for all .
On the basis of the above definition, Banach [1] in 1922 stated the following well-known contraction principle.
Theorem 1. Let be a contraction mapping defined on a complete metric space . Then, possesses a unique fixed point in . Furthermore, for any , we havewith
In 1969, investigating the case for multivalued mappings, Nadler [2] proved that the multivalued contraction mapping possesses at least one fixed point. Before moving towards this new generalization, we recall that , , , and the nonempty, closed, closed and bounded, and compact subsets of a metric space , respectively. For and , considerthe Pompeiu–Hausdorff metric is defined bywhere .
Lemma 1 (see [2]). Suppose and . Then, for , there is so that .
Theorem 2 (see [2]). Let be a multivalued contraction mapping on a complete metric space . Then, possesses a fixed point.
It is obvious that if is a complete metric space, then the pair and is also complete (see e.g., [3–5]).
Definition 2 (see [6]). A mapping is called a multivalued weakly Picard (MWP) operator if, for all and , there exists a sequence in such that(i)(ii), for all (iii) is convergent and its limit is a fixed point of
Popescu [7] defined the notion of -contractive multivalued operators.
Definition 3 (see [7]). A multivalued operator on a complete metric space is called -contractive if , , and such thatwhere
Popescu [7] showed that -contractive multivalued operators are Picard, while in the single-valued case, it was shown that such operator possesses a unique fixed point. Later on, Kamran and Hussain [8] generalized the results of Popescu [7] to a weakly -contractive multivalued operator. The set of fixed points of the mapping is defined as , while the set of strict fixed points is defined as . It is clear that .
Definition 4. (see [9, 10]). Let , where is a metric space, and be a multivalued operator. Then, the fixed point problem is well-posed for appropriate to if(i)(ii)For a sequence in , as ; then, as
Observe that a fixed point problem, which is well-posed for appropriate to , is also well-posed for appropriate to . Moţ and Petruşel in [11] proved the results of strict fixed point sets, well-posedness, and also data dependence of the fixed point sets.
An important class of functions was proposed by Khojasteh and Shukla [12] and was named as the set of simulation functions. Let verify the following conditions: . , for all . If are sequences in such that , then Such is known as a simulation function proposed by Khojasteh et al. Utilizing such broad class of functions, they defined the notion of -contractions. We denote such class of functions by .
Definition 5 (see [12]). A self-map on is said to be -contraction appropriate to if the inequalityis fulfilled.
On the basis of contraction mappings defined above, they gave a version of the contraction principle, which generalizes and unifies several existing fixed point results in the literature.
After this work, studies involving simulation functions have been performed by various researchers (see [13–15] and references therein). Later on, Argoubi et al. [16] reshaped the notion of a simulation function by withdrawing the assertion . We denote such class of functions by .
Example 1 (see [16]). Let be a function defined bywhere . Then, .
Later on, the assertion of a simulation function was replaced with by Roldán-López-de-Hierro et al. [17].
If are sequences in such that and , then
The class of simulation functions fulfilling , and is known as the class of simulation functions in the manner of Roldpez-de-Hierro, and we denote it by . We need the following lemma.
Lemma 2 (see [18]). Let be a metric space and let be a sequence in such thatIf is not a Cauchy sequence in , then there exist and two sequences and of positive integers such that and the following sequences tend to when :
The purpose of this paper is to introduce the notion of weakly multivalued -contractions and to prove some fixed point results. We also discuss examples to illustrate and elaborate these new concepts. After that, we present data dependence, strict fixed point set, and well-posedness results. Following these ideas, we generalize the Moţ and Petruşel result. Moreover, we present an application to functional equations arising in dynamical systems to show the usability of our results.
2. Main Results
We begin with the following definition.
Definition 6. Let be a metric space. A mapping is called a weakly multivalued -contraction with respect to if there are , , and so thatimplieswhere , , and
Example 2. Let and . Define byChoose . For , , we havewhich impliesHence, is a weakly multivalued -contraction.
Theorem 3. Let be a complete metric space and be a weakly multivalued -contraction. Then, is a MWP operator.
Proof. Let , , and be a real number such that . Choose . Then,From (15) and , we havewhich impliesHence,impliesSimilarly, we can get a sequence in such that andSince , this shows that is a Cauchy sequence. is complete, so there exists such thatWe now show that there exists a subsequence of such thatOn the contrary, we assume that there is a positive integer such thatThis impliesBy induction, we obtainRecall thatTaking , we get thatThus, we haveFrom (30) and (33), one writesBy taking , we have , for all . It is a contradiction with respect to equation (30). Therefore, there is a subsequence of such thatwhich impliesNow, from (36) and using together with , we havewhere . So,From this contradiction, we get , that is, . Hence, is a MWP operator.
Example 3. Let and . DefineNow, for , values of , , and for all possible pair of points are given in Table 1.
By choosing , we haveFurthermore, for , we also haveNow, choosing and , we haveHence, is a weakly multivalued -contraction. Thus, by Theorem 3, is a MWP operator.
Remark 1. The weakly multivalued -contraction is the generalization of Popescu notion of -contractions. By using Definition 3 for some and , we havewhich implieswhich implieswhich impliesHence, from equations (44)–(48), it is clear that the notion of -contraction defined by Popescu [7] is failed for Example 3.
Theorem 4. Let be a complete metric space and be a weakly single-valued -contraction operator. Then, possesses a fixed point. Moreover, if and , then possesses a unique fixed point.
Proof. From Theorem 3, possesses a fixed point. Suppose that and . Assume that possesses two distinct fixed points and . Then,From (49) and by using , we haveWe know that . Therefore, we get , which is a contradiction to our assumption. Thus, possesses a unique fixed point.
Theorem 5. Let be a weakly multivalued -contraction from into . Assume that there exist such thatwhich impliesThen, is a Picard operator.
Proof. Without loss of generality choose . Take a real number such that . Let and such that .
Then,By hypothesis, we haveSo,Following similar steps as in Theorem 3, we can easily obtain that . Therefore, a sequence can be constructed in such that and for all That is, is a Cauchy sequence. Completeness of yields that there is so that converges to .
SinceBy taking , we obtainWe haveSuppose now that there is , such thatTherefore,impliesa contradiction. So, there exists a subsequence of such thatSincethus, we haveThis implies thatFrom (66), , and , one obtainswhere . Hence,This contradiction shows that , that is, . Hence, is a Picard operator.
Corollary 1. Let be a weakly single-valued -contraction mapping from into . Assume that there exist so thatimpliesThen, there exists such that .
Proof. One can easily show that, for every , the sequence defined by satisfies the relationship as done in Theorem 3. Thus, the sequence is Cauchy, so there is such that . Following Theorem 5, we can show that for all and there exists a subsequence of such that for all . Therefore, we obtain thatWe deduce that . Thus, we get .
2.1. Data Dependence of the Fixed Point Set
In this section, we study data dependence of the fixed point set for weakly multivalued -contractions.
Theorem 6. Let be a metric space and and be two multivalued operators. Assume that(1) is a weakly -contraction for each (2)There exists a real number such that , Then,(1) for .(2) and are weakly multivalued operators and
Proof. From Theorem 3, is nonempty for . First of all, we will show that the set of fixed point of a weakly multivalued operator is closed. Let be a sequence in such that as . One obtainsThis implies thatNow, from (74), , and , we haveThus, from this contradiction, we deduce that . Since , we have . Hence, . Secondly, from Theorem 3, we get that a weakly multivalued -contractive operator is a MWP operator. Let be a real number and be arbitrary. Then, there exists such that . Next, for , there is such that . Since . So, we haveTherefore, in a similar way, we obtain that the sequence of successive approximations for starting from fulfills the following assertions:Hence, for all and ,Choosing now and letting , we obtain that is a Cauchy sequence in . Then, there exists such that as . We will prove that is a fixed point for . Suppose that there exists a positive number such that . Then, . Since as , we get a contradiction. Hence, there exists a subsequence such that . Thus,Now, from (79), , and , we havewhere . So,It is a contradiction. So, we obtain that , that is, . Hence, .
By taking in (78), we have for each . Then, for , we get . In a similar way, we get that, for each , there exists such that . Hence,Letting completes the proof. Moreover, we get that is a MWP operator for .
2.2. Strict Fixed Point and Well-Posedness
Now, we prove the well-posedness for a weakly multivalued -contractive operator with .
Theorem 7. Let be a complete metric space and be a multivalued operator. Assume that(i) is a weakly multivalued -contractive operator with (ii)Then,(a)(b)The fixed point problem is well-posed appropriate to if
Proof. (a)We will prove that . We suppose that with . Since one obtains By using , we have Hence, we obtain . We know that . So, Thus, , where . It is a contradiction. Hence, we deduce that and .(b)Let , be such that as . We will prove that as . Arguing by contradiction, we suppose that does not converge to 0. Then, there exist and a subsequence such that . If there exists a subsequence of such that it implies By following similar calculations as in Theorem 3, one gets as . Hence, we obtain as , which is a contradiction to our supposition, that is, . Thus, there exists for all . We know that as , so there exists for all . Thus, for all . It is impossible. Therefore, as .
2.3. An Extension of Moţ–Petruşel Theorem
Following Reich [19], Moţ and Petruşel initiated the following concepts.
Definition 7. Let be a metric space and . Then, is called an -KSR multivalued operator if there are so that, for , we have
Theorem 8. Let be a complete metric space and be an -KSR multivalued operator. Then, . Moreover, is a MWP operator.
Now, we prove the generalization of Theorem 8 in the case , by using a weakly multivalued -contractive operator.
Theorem 9. Let be a complete metric space and . Assume that there exist with , so that, for all , we havewhich impliesThen, .
Proof. Let and . Choose be a real number such thatOne can choose , thenwhich impliesThat is,Therefore,Thus, we obtainFrom this way, we can easily construct the sequence in such that andAs , one obtainsThis shows that is a Cauchy sequence. Since is complete, there is so that . Now, we suppose that there exists a subsequence of such thatfor all . Assume on the contrary that there is a positive integer such thatThen, following similar procedure of Theorem 3, we can show that this contrary assumption leads to the contradiction. Therefore, there exists a subsequence of such thatfor all . This implies thatBy solving this inequality, using the definition of a simulation function, we will obtain , that is, . Hence, .
3. An Application
Many authors [20–22] have studied the existence and uniqueness of a solution of functional equations arising in dynamic programming. We make use of Theorem 4 to explore the existence and uniqueness of a solution for a class of functional equations. From now on, and are Banach spaces and , and is the set of real numbers. We denote by the set of all bounded real-valued functions on . The set equipped with the metric,is a complete metric space. Viewing and as the state and decision space, respectively, the problem of dynamic programming reduces to the problem of solving the functional equation , where represents the transformation of the process and corresponds to the optimal return function with an initial functional equation:where and are bounded functions. Let be defined by , where and .
Theorem 10. Suppose that there exist such that, for every and , the inequalityimplieswherewith . Then, the functional equation (106) possesses a bounded solution. Moreover, if and , then such a solution is unique.
Proof. Let be a self-map of . Let be an arbitrary real number and . Pick . We can choose so thatwhere , .
Using the definition of , we obtainIf inequality (107) holds, thenFrom (114) and , one writesThis yields thatIn view of (111), (114), and (117), we haveSimilarly, from (112), (113), and (117), we haveThus, from equations (117) and (118), the following,holds for all and . Hence, we get thatimplieswhereHence, all assertions of Theorem 4 are fulfilled for the mapping ; therefore, we get the required result.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this manuscript.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.