HTML

ePub

PDF

1 Introduction and literature reviews

The study for the set-valued dynamics in Banach spaces has been developed in the last decades. The pioneering published papers by Aumann [2] and Debreu [9] were inspired by some problems arising in the control theory and mathematical economics. We refer to the articles by Arrow and Debreu [1], McKenzie [29], and the survey by Hess [18].

The stability of functional equations was first introduced by Ulam [38] in 1940. He proposed the following problem: Given a group G₁, a metric group , and a positive number , does there exist a such that if a mapping satisfies the inequality for all , then there is a homomorphism such that for all . If the answer is positive, we say that the homomorphisms from to are stable. In 1941, Hyers [19] gave a partial solution of Ulam’s problem for the case of approximate additive mappings under the assumption that and are Banach spaces. In 1978, a generalized version of the theorem of Hyers by considering the stability problem with unbounded Cauchy differences was given by Rassias [36]. This phenomenon of stability that was introduced by Rassias [36] is called the Hyers–Ulam–Rassias stability of functional equations.

Theorem 1. Let be a mapping from a normed vector space E into a Banach space E′ subject to the inequality

where ϵ and p are constants with , and . Then there exists a unique additive mapping such that in the case of ,

while in the case of ,

The solution to this problem was obtained by Gajda [13] for , and the problem for was solved by Rassias [36]. Rassias and Semrl [37] proved that the stability does not occur for . The result of the Rassias theorem was generalized by Forti [11] and Gávruta [14], who permitted the Cauchy difference to become arbitrary unbounded.

The stability problems of several functional equations have been extensively investigated by many mathematicians. The results of these kinds of problems have been extensively studied. We refer, for instance, to [6, 12, 15–17, 19, 27] and also [11, 13, 14, 22, 25, 36, 37] and references therein.

A stability problem of Ulam for the cubic functional equation

was established by Jun and Kim [22] for mapping , where is a normed space, and a Banach space. Also, they solved the stability problem of Ulam for the generalized Euler–Lagrange-type cubic functional equation

for fixed integer a with, and

for fixed integers a, b with , and , and the equations being equivalent to (1). Afterwards, referring to [7], Chu et al. extended the cubic functional equation to the following generalized form:

where is an integer, and they also investigated the Hyers–Ulam stability. Moreo ver, in [25], Jung and Chang investigated a generalized Hyers–Ulam–Rassias stability for a cubic functional equation by using the fixed point alternative. The first systematic study of the interative methods in the stability of mappings is due to Isac and Rassias [20]

The stability of the set-valued functional equations has been widely examined by a number of authors (see [8, 21, 30–32]), and the Hyers–Ulam stability of the set-valued functional equations was proved in [21, 26, 28]. Also, there are many interesting stability results concerning this problem (see [8, 23, 24]).

Quite recently, Eshaghi et al. [10] and M. Ramezani et al. [35] introduced the notion of orthogonal sets and gave a real generalization of the Banach fixed point theorem in incomplete metric spaces. The main result of [10] is the following theorem.

Theorem 2.(See [10].) Let be an O-complete orthogonal metric space (not necessarily complete metric space) and . Let be O-continuous, -contraction with Lipschitz constant , and -preserving. Then has a unique fixed point . Also, is a Picard operator, that is, l for all .

For more details about the orthogonal space, we refer the reader to [3–5, 10, 34, 35].

The aim of this paper is to offer a new generalized Hyers–Ulam–Rassias stability result for the functional equation (2) for the set-valued mappings in normed spaces, which are not necessarily Banach spaces, by using the fixed point alternative [10] as in [3]. Examplewise, we present a special case of our results, which is a real extension of the previous results as of this literature.

At first, we recall some basic definitions and our main tools.

Definition 1. (See [10].) Let and be a binary relation. If satisfies the following condition be a binary relation. If satisfies the following condition

then is called an orthogonal relation, and the pair – an orthogonal set (briefly, O- set)

Note that in the above definition, we say thatis an orthogonal element. Also, we say that elements are -comparable either or .

Definition 2. (See [10, 35].) Let be O-set. A sequence is called

(i) an orthogonal sequence (briefly, O-sequence) if

(ii) an strongly orthogonal sequence (briefly, SO-sequence) if

Every SO-sequence is an O-sequence. But the converse is not true in general.

Definition 3. (See [10, 35].) Let be an orthogonal metric space ( is an O-set, and a metric space). is

1. orthogonal complete (briefly, O-complete) if every Cauchy O-sequence is convergent;

2. strongly orthogonal complete (briefly, SO-complete) if every Cauchy SO-sequence is convergent.

It is easy to see that every complete metric space is O-complete and every O-complete metric space is SO-complete. In [3, 35], the authors proved that the converse is not true in general.

Definition 4. (See [10, 35].) Let be an orthogonal metric space. Then is

1. orthogonal continuous (briefly, O-continuous) at if for each O-sequence in implies .

2. strongly orthogonal continuous (briefly, SO-continuous) at if for each SO-sequence in implies.

Also, is O-continuous (SO-continuous) on if is O-continuous (SO-continuous) in each .

It is obvious that every continuous mapping is O-continuous and every O-continuous mapping is SO-continuous, but the converse is not hold in general (see [3, 35]).

Definition 5. (See [3].) Let be an O-set. A mapping is said to be -preserving if whenever and .

Theorem 3.Let be an SO-complete orthogonal metric space (not necessarily complete metric space) and . Let be SO-continuous, -preserving, and -contraction with Lipschitz constant . Then f has a unique fixed point . Also, is a Picard operator, that is, for all .

Proof. The proof of this result uses the same ideas in Theorem 3.11 of [10], and it suffices to replace the O-sequence by SO-sequence.

By the aforementioned results we can conclude that Theorem 3 is a real generalization of Theorem 2. So, in the next steps, we are going to prove the stability of functional equation (2) in the SO-complete normed spaces.

2 An incomplete distance on subsets of a set

Before introducing the main results, we recall some notations and definitions.

Let be a normed space (not necessarily a Banach space), and let be anorthogonal relation on such that is an orthogonal metric space, wheredisthe induced metric by.

We say that is -preserving whenever implies for each . See the next example.

Example 1. Let , and let two relations and on be defined as

It is obvious that an orthogonal element of and is zero. However, is not -preserving. To see this, if , and, then , while . Notice that it is easy to see that is -preserving.

Let be the set of all nonempty, closed, convex, and bounded subsets of . Consider the addition and the scalar multiplication as follows:

where ) and. One can show that

for all and . We consider on pairs of elements in by

where and . Pathak and Shahzad in [33] proved that is a metric on . We define the relation on as

The following proposition can be proved from some properties of the distance .

Proposition 1 .(See[33].) For any and , the following properties hold:

• 

• 

• 

• ;

• , wherefor all positive real number r;

• for each positive real number r.

Given. We define the relation between A and B as follows:

If is an orthogonal element of , then the singleton is an orthogonal element for .

Theorem 4. If is an SO-complete .not necessarily complete. metric space, then with orthogonal relation is SO-complete.

Proof. Let be a Cauchy SO-sequence in . We need to show that converges to some element in .

Let A be the set of limit points of sequences with for all . Our aim is to prove that and converges to A. To see end, let us to divide the proof in the following steps.

Step 1:Ais closed. Let . Definition of A ensures that we can choose the sequence in A converging to a. This leads to for all , there exists in such that for anyand as . Let be a strictly increasing sequence of positive integers such that for any, . We observe that

As, the right-hand of above inequality converges to zero, which implies .

Step 2:A is convex. Let and . Take two sequences and such that for each , and and as . Since for any , is a convex set, then . The closeness of A implies that.

Step 3: A is nonempty. We observe from is a Cauchy sequence that there exists a strictly increasing sequence such that for all and , . Definition of ensures that for each , there exists for which kank . This results show that the sequence is Cauchy.

On the other hand, since is an SO-sequence, it follows that

Therefore, is a Cauchy SO-sequence in . Since is SO-complete and for each , the conclusion follows easily.

Step 4:. Fix . There exists a positive integer such that for all . By definition of and condition (v) of

Proposition 1 we see that for all and , where. Let and be a sequence such that for all i and converges to a. We observe that for all , and the continuity of D implies that for all . This results show that for all.

On the other hand, we can choose a positive integer such that for all and a strictly increasing sequence of positive integers such that and for all .

Assume and . It follows from that there is for whichky . Similarly, for each i, since , then there is for which . We easily see that is a Cauchy sequence. Arguing in the Step 3, we obtain that is an SO-sequence in and so converges to an element . Moreover, for all ,

For large enough numbers of i, , which implies that , and hence, for all .

Now, take , then condition (vi) of Proposition 1 ensures that

for each . This completes the proof of Step 4.

3 New generalized Hyers–Ulam–Rassias stability

Throughout this section, we assume and are two normed spaces. Also, and are the same orthogonal relations on Y and as defined in the previous section, respectively. We consider the relation as-persevering and d as the metric induced by .

Definition 6. Let be a set-valued mapping.

(i) The n-dimensional cubic set-valued functional equation is defined by

for every , where is an integer.

(ii) Every solution of the n-dimensional cubic set-valued functional equation is called an n-dimensional cubic set-valued mapping.

Theorem 5.Let be an integer, and be an SO-complete metric space (not necessarily a complete metric space). Assume that is a set-valued mapping such that and are -comparable for each and , and also, there exist two functions and satisfying the following conditions:

for all and also

(A1)

(A2) For all ,

(A3)For all ,

Then there exist an n-dimensional cubic set-valued mapping and a subset in with card such that for some positive real number , we have

for all . In particular, if , then the mapping is unique.

Proof. We denote by the set

and the generalized metric D on as follows:

Consider the set . Putting in (A2) yields that , and by using (3) we observe that . Hence S is a nonempty set.

Now, let be a function as given by for all . We must show that T is a self-adjoint mapping, that is,. To see this, put and in inequality (3). Since the range of f is convex and applying (A2), we have

and so,

for all . Dividing by 8m in (5), we get

for all . Replacing x by in (6) and applying (A2), we have

for all. This ensures that . On the other hand, if , definition of D conclude that , and the triangle inequality implies that , that is,. Consider

for all . Define the relation on S as the following:

It follows from Theorem 4 that is an SO-complete metric space. Since the relation is -preserving, definition of and imply that T is -preserving. By using the hypothesis we obtain

for all and . From -preserving of and definition of T we get

for all and . This means that

for all . It follows from -preserving of T that

That is, and consequently for all are SO-sequences in S and, respectively. In order to show that the SO-sequence is Cauchy, replacing and multiplying by 8^r in (7) and using (A2) and (A3), we get

all and . Considering

we obtain that

for all and with . Since , letting in the above inequality, we deduce that the sequence is a Cauchy sequence for each . By SO-completeness of we obtain that for every , there exists an element , which is a limit point of . That is, is well defined and given by

for all . On the other hand, since, then there exist and such that for all . Put

It follows from that . Also, if is an arbitrary nonzero point of, then by using (A2) we can easily see that

So, there exists a natural number for which

and this means that belongs to. This implies that card. Now, were place by in definition of . For , we have the following implications

Hence, we see that for all . It follows from that T is a contraction. Consequently, T is an SO-continuous mapping and is a contraction on -comparable elements with Lipschitz constant L. Since is SO-complete and T is also S-preserving, then from Theorem 3 we conclude that T has a unique fixed point and T is a Picard operator. This means that the sequence is convergent to the fixed point of T . It follows from (8) that F is a unique fixed point of T . Moreover,

Therefore,. Relation (7) ensures that inequality (4) holds. Finally, we need to show that F is an n-dimensional cubic set-valued mapping. To this end, let be fixed elements of . Since is a nonnegative and decreasing sequence, then there is for which as . Taking into account (A1), we have , so there exist and such that for all . Consider the positive integer N such

that for . By virtue of (3) we obtain

Therefore, F is an n-dimensional cubic set-valued mapping as desired.

Corollary 1. Let be an integer and . Let Y be a Banach space and be a mapping such that there exists a function satisfying

for all . If there exists a positive real number such that

for all , then there exists a unique n-dimensional cubic mapping , which satisfies the inequality

for all . The mapping F is given by

Proof. For every , define if and only if . It is clear that is an O-set. Moreover, we can consider as a closed subset of ,which d is the metric induced by . Since Y is a Banach space, so is an SO-complete metric space. From definition of follows that

It is enough to pick for all . The result is an immediate consequence of Theorem 5.

Theorem 6. Let be an integer, , and be an SO-complete metric space (not necessarily complete metric space). Suppose that is a set-valued mapping such that and are -comparable for each and , and there exists a function satisfying equation (3) of Theorem 5 and the following property:

(B1) if and only if for all , and is an increasing sequence for all that are not all zero. Also,

is an unbounded sequence for some .

If is a mapping, which satisfies relation (A1) of Theorem 5 and the following conditions:

(B2) For all that are not all zero,

(B3) For every nonzero element x of ,

Then there exist an n-dimensional cubic set-valued mapping and a sub- set in with card such that for some positive real number , we have

or all . Moreover, if , then F is unique.

Proof. By the same reasoning as in the proof of Theorem 5, there exist and such that for each . Set

As a result of (B1), we can easily see that for some , the sequence

Is a decreasing sequence which converges to zero. This concludes that card . By the same argument of Theorem 5 one can show that the mapping defined by for all is a -preserving mapping and is a contraction with Lipschitz constant L on . Define by for all . Replacing by in definition of S0 and applying Theorem 3,weobtain that is a unique fixed point of T. It follows from (6) that and so

and consequently,

That is, inequality (10) holds. To show that the function F is an n-dimensional set-valued mapping on , let be fixed elements of , which are not all zero. Since

is a nonnegative and decreasing sequence, so the rest of the proof is similar to the proof of Theorem 5.

Corollary 2. Let be an integer and Y be a Banach space. Suppose that is a mapping such that there exists a function satisfying conditions (B1) of Theorem 6 and, in addition

for all . If there exists a positive real number such that

For all , then for every ,there exists a unique n-dimensional cubic mapping , which satisfies the inequality

for all . The mapping F is given by

Proof. Take the same metric d and the orthogonal relation of Corollary 1. By the same argument of Corollary 1 one can show that is an SO-complete metric space and and are -comparable for each and. Putting for all and applying Theorem 6, we can easily obtain the results.

Corollary 3. Suppose that Y is a Banach space and and are fixed. Assume that is a function satisfies the functional inequality

for all . Then there exists a unique n-dimensional cubic mapping such that the inequality

holds for all , where , or the inequality

holds for all , where .

Proof. Take the same metric d and the orthogonal relation of Corollary 1. By the same argument of Corollary 1 one can show that is an SO-complete metric space. Moreover, definition of ensures that and . are-comparable for each and . Similarly, and are -comparable for each and .

We define It follows that

for all , where. Set for all . This ensures that and relations (A1) and (A3) of Theorem 5 hold. Applying Theorem 5, we see that inequality (4) holds with,which yields inequality (12). On the other hand, the function φ satisfies properties (B1), (B2) and also

for all , where. Putting for every , it is easily seen that and conditions (A1) and (B3) are hold. Employing Theorem 6, we see that inequality (10) holds with ^. This implies inequality (13).

The next example shows that Theorem 6 is a real extension of Corollary 1.

Example 2. Let be an integer and , and Y be a Banach space. Let be a sequence defined by for all natural number p with . It is easy to see that is a strictly increasing sequence of real numbers. Suppose that is a mapping satisfying

for all . Define a mapping by

and the function ) as

Then the following hold:

(i) For every .

(ii) For every ,

(iii) For every ,

(iv) For every positive real number s, there exist a constant and an n-dimensional cubic mapping such that

for all . with .

Proof. Take the same metric d and the orthogonal relation of Corollary 1. By the same argument of Corollary 1 one can show that is an SO-complete metric space and and ) are-comparable for each and . Let us take and and let p be the smallest natural number such that . Then

We observe that

This follows that there exists , which

Assume that k is the smallest natural number satisfying the above condition. Clearly, and

Now, we suppose that q is the smallest natural number that then . Since. then , and we conclude . This implies that

That is, condition (i) holds. From definition it is easily seen that is a nondecreasing mapping.

Finally, it follows from that for every , there exists such that

for all x with . By the same proof of Theorem 5 we prove (iv).

Notice that there is no such that inequality (9) holds, and hence, the stability of does not imply by Corollary 1.

Now, we observe in the following example that our results go further than the stability on Banach spaces.

Example 3. Let and be given. Consider (the set all of continuous functions on [0,1]) with norm, where 1. Suppose that is a mapping satisfying inequality (11) and the following condition:

Then there exists a unique n-dimensional cubic mapping such that inequality (12) holds for all , where , or inequality (13) holds for all , where .

Proof. Let q be the conjugate of . For all , define

And . .We claim that is an SO-complete metric space. Indeed, let be a Cauchy SO-sequence in Y, and for all . The relation ensures that for all ,

We distinguish two cases.

Case 1. There exists a subsequence of such that a.e. for all k. This implies that .

Case 2. For all sufficiently large . Take such that for all . It follows from (15) that for all , there exists for which . It leads to

for each . As , the right-hand side of the above inequality tends to 0.Therefore, is a Cauchy sequence in . Assume that as . Put It follows that and for all ,

This implies that as . Note that the case for all is in a similar way.

By virtue of (14) and definition of we obtain that and are - comparable elements for each x and . Moreover, putting in (14), we can also see that and are -comparable elements in Y for all and . The rest of the proof is similar to the proof of Corollary 3.

References

1. K.J. Arrow, G.A.C. Debereu, Existence of an equilibrium for a competitive economy, Econometrica, 22:265–290, 1954, https://doi.org/10.2307/1907353

2. R.J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl., 12:1–12, 1965, https://doi.org/10.1016/0022-247X(65)90049-1.

3. H. Baghani, M. Eshaghi, M. Ramezani, Orthogonal sets: The axiom of choice and proof of a fixed point theorem, J. Fixed Point Theory Appl., 18:465–477, 2016, https://doi.org/ 10.1007/s11784-016-0297-9.

4. H. Baghani, M. Ramezani, A fixed point theorem for a new class of set-valued mappings in R-complete (not necessarily complete) metric spaces, Filomat, 31:3875–3884, 2017, https://doi.org/10.2298/FIL1712875B.

5. H. Baghani, M. Ramezani, Coincidence and fixed points for multivalued mappings in incomplete metric spaces with application, Filomat, 33:13–26, 2019, https://doi.org/ 10.2298/FIL1901013B.

6. J. Brzde¸k, K. Cieplin´ski, A fixed point theorem in .-Banach spaces and Ulam stability, J. Math. Anal. Appl., 470(1):632–646, 2019, https://doi.org/10.1016/j.jmaa.2018.10. 028.

7. H.-Y. Chu, D.S. Kang, On the stability of an .-dimensional cubic functional equation, J. Math. Anal. Appl., 325:595–607, 2007, https://doi.org/10.1016/j.jmaa.2006.02. 003

8. H.-Y. Chu, A. Kim, S.K. Yoo, On the stability of the generalized cubic set-valued functional equation, Appl. Math. Lett.,37:7–14, 2014, https://doi.org/10.1016/j.aml. 2014.05.008.

9. G. Debreu, Integration of correspondences, in Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II, Part I, Univ. California Press, Oakland, CA, 1966, pp. 351–372.

10. M. Eshaghi, M. Ramezani, D. la Sen, Y.J. Cho, On orthogonal sets and Banach fixed point theorem, Fixed Point Theory, 18:569–578, 2017, https://doi.org/10.24193/fpt- ro.2017.2.45.

11. G.L. Forti, Hyers–Ulam stability of functional equations in several variables, Aequationes Math., 50:143–190, 1995, https://doi.org/10.1007/BF01831117.

12. R. Fukutaka, M. Onitsuka, Best constant in Hyers–Ulam stability of first-order homogeneous linear differential equations with a periodic coefficient, J. Math. Anal. Appl., 473(2):1432– 1446, 2019, https://doi.org/10.1016/j.jmaa.2019.01.030.

13. Z. Gajda, On isometric mappings, Int. J. Math. Math. Sci., to appear.

14. P. Gávruta, A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184:431–436, 1994, https://doi.org/1999.6546.

15. M. Eshaghi Gordji, H. Habibi, Existence and uniqueness of solutions to a first-order differential equation via fixed point theorem in orthogonal metric space, Facta Univ., Ser. Math. Inf., 34(1): 123–135, 2019, https://doi.org/10.22190/FUMI1901123G.

16. M. Eshaghi Gordji, H. Habibi, Fixed point theory in .-connected orthogonal metric space, Sahand Commun. Math. Anal., 16:35–46, 2019, https://doi.org/10.22130/scma. 2018.72368.289.

17. M. Eshaghi Gordji, H. Habibi, M.B. Sahabi, Orthogonal sets; orthogonal contrac- tions, Asian-Eur. J. Math., 12(3):1950034, 2019, https://doi.org/10.1142/ S1793557119500347.

18. C. Hess, Set-valued integration and set-valued probability theory: An overview, in E. Pap (Ed.), Handbook of Measure Theory, North-Holland, Amsterdam, 2002, https://doi. org/10.1016/B978-044450263-6/50015-4.

19. D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA,

20. G. Isac, T.M. Rassias, Stability of .-additive mappings: Applications to nonlinear analysis, Int. J. Math. Math. Sci., 19:219–228, 1996, https://doi.org/10.1155/ S0161171296000324.

21. S.Y. Jang, C. Park, Y. Cho, Hyers–Ulam stability of a generalized additive set-valued functional equation, J. Inequal. Appl., 101, 2013, https://doi.org/10.1186/1029-242X- 2013-101.

22. K.-W. Jun, H.-M. Kim, The generalized Hyers–Ulam–Rassias stability of a cubic functional equation, J. Math. Anal. Appl., 274:867–878, 2002, https://doi.org/10.1016/ S0022-247X(02)00415-8.

23. S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011, https://doi.org/10.1007/978-1-4419-9637-4

24. S.-M. Jung, D. Popa, Th.M. Rassias, On the stability of the linear functional equation in a single variable on complete metric groups, J. Glob. Optim., 59:165–171, 2014, https:

25. Y.-S. Jung, I.-S. Chang, The stability of a cubic type functional equation with the fixed point alternative, J. Math. Anal. Appl., 306:752–760, 2005, https://doi.org/10.1016/j. jmaa.2004.10.017.

26. H.A. Kenary, H. Rezaei, Y. Gheisari, C. Park, On the stability of set-valued functional equations with the fixed point alternative, Fixed Point Theory Appl., 81, 2012, https://doi.org/ 10.1186/1687-1812-2012-81.

27. S. Khalehoghli, H. Rahimi, M. Eshaghi Gordji, R-topological spaces and SR-topological spaces with their applications, Math. Sci., 14:249–255, 2020, https://doi.org/10. 1007/s40096-020-00338-5.

28. G. Lu, C. Park, Hyers–Ulam stability of additive set-valued functional euqtions, Appl. Math. Lett.,24:1312–1316, 2011, https://doi.org/10.1016/j.aml.2011.02.024.

29. L.W. McKenzie, On the existence of general equilibrium for a competitive market, Econometrica, 27:54–71, 1959, https://doi.org/10.2307/1907777.

30. K. Nikodem, On quadratic set-valued functions, Publ. Math., 30:297–301, 1983, https://doi.org/10.1007/BF02591511.

31. K. Nikodem, On Jensen’s functional equation for set-valued functions, Rad. Mat., .:23–33, 1987, https://doi.org/10.1007/s00025-017-0679-3.

32. K. Nikodem, Set-valued solutions of the Pexider functional equation, Funkc. Ekvacioj, Ser. Int., 31(2):227–231, 1988, https://doi.org/10.1007/s00025-017-0679-3.

33. H.K. Pathak, N. Shahzad, A generalization of Nadler’s fixed point theorem and its application to nonconvex integral inclusions, Topol. Methods Nonlinear Anal., 41:207–227, 2013.

34. M. Ramezani, Orthogonal metric space and convex contractions, Int. J. Nonlinear Anal. Appl., .:127–132, 2015, https://doi.org/10.22075/IJNAA.2015.261.

35. M. Ramezani, H. Baghani, The Meir-Keeler fixed point theorem in incomplete modular spaces with application, J. Fixed Point Theory Appl., 19:2369–2382, 2017, https://doi.org/ 10.1007/s11784-017-0440-2.

36. T.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc., 27(2):297–300, 1978, https://doi.org/10.1090/S0002-9939-1978- 0507327-1.

37. T.M. Rassias, P. Šemrl, On the behavior of mappings which do not satisfy Hyers–Ulam stability, Proc. Am. Math. Soc., 114(4):989–993, 1992, https://doi.org/10.2307/ 2159617.

38. S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1940, https://doi. org/10.4236/ojpp.2012.21010.

IR