The Hyers–Ulam and Hahn–Banach Theorems and Some Elementary Operations on Relations Motivated by Their Set-Valued Generalizations

Száz, Árpád

doi:10.1007/978-1-4614-3498-6_39

Cited by 4 publications

(7 citation statements)

References 140 publications

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“…These only slightly improve some earlier observations [70,63] of the second author. Therefore, the proofs are usually omitted.…”

Section: Remarksupporting

confidence: 76%

“…Remark 84 By using the notation F = ∞ n=1 F n of [70], with F n (x) = n −1 F (nx), the assertion the above theorem can be briefly expressed by saying that (F * G) is a total relation on X to Y .…”

Section: Remark 83mentioning

confidence: 97%

“…However, by using the notations of [70], or rather [68], the above definition and the following obvious theorem can be more briefly formulated.…”

Section: Some Further Important Homogeneity Properties Of Relationsmentioning

confidence: 98%

“…Moreover, as some useful reformulations of a particular case of Definition 8, we can also easily establish the following theorem which can again be more briefly formulated by using the notations of [70], or rather [68].…”

Section: The Importance Of Quasi-odd Relations and Odd-like Selectionsmentioning

confidence: 98%

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Constructions and Extensions of Free and Controlled Additive Relations

Glavosits

Száz

2014

Handbook of Functional Equations

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By using several auxiliary results on relations and their intersection convolutions, we give some necessary and sufficient conditions in order that a certain additive partial selection relation Φ of a relation F of one group X to another Y could be extended to a total, additive selection relation Ψ of the relation F + Φ(0).The results obtained extend some Hahn-Banach type extension theorems of B. Rodríguez-Salinas, L. Bou, Z. Gajda, A. Smajdor, W. Smajdor, and the second author. Moreover, they can be used to prove some alternate forms of the Hyers-Ulam type selection theorems of Z. Gajda, R. Ger, R. Badora, Zs. Páles, and the second author.Keywords Additive and homogeneous relations · Intersection convolutions of relations · Extensions of additive partial selection relations

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“…These only slightly improve some earlier observations [70,63] of the second author. Therefore, the proofs are usually omitted.…”

Section: Remarksupporting

confidence: 76%

Section: Remark 83mentioning

confidence: 97%

“…However, by using the notations of [70], or rather [68], the above definition and the following obvious theorem can be more briefly formulated.…”

Section: Some Further Important Homogeneity Properties Of Relationsmentioning

confidence: 98%

Section: The Importance Of Quasi-odd Relations and Odd-like Selectionsmentioning

confidence: 98%

See 2 more Smart Citations

Constructions and Extensions of Free and Controlled Additive Relations

Glavosits

Száz

2014

Handbook of Functional Equations

Self Cite

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“…Therefore, some of its generalizations should be derived from those of the Hahn-Banach extension theorems. (For some ideas in this respect, see [22,Chapter 34] and [35,77,36].) However, it is now more important to note that recently Bahyrycz, Páles and Piszczek [9] have proved a new type, asymptotic stability theorem for the Cauchy functional equation by using metric Abelian groups instead of normed spaces.…”

Section: Introductionmentioning

confidence: 99%