mathematics

Hyers–Ulam–Rassias stability

Hyers–Ulam–Rassias stability is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hyers–Ulam–Rassias stability rather than just read about it. In short: The stability problem of functional equations originated from a question of Stanisław Ulam, posed in 1940, concerning the stability of group homomorphisms. In the next year, Donald H.

Key takeaways

Hyers–Ulam–Rassias stability belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hyers–Ulam–Rassias stability to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hyers–Ulam–Rassias stability from memory before moving on to harder problems.

Reference excerpt

The stability problem of functional equations originated from a question of Stanisław Ulam, posed in 1940, concerning the stability of group homomorphisms. In the next year, Donald H. Hyers gave a partial affirmative answer to the question of Ulam in the context of Banach spaces in the case of additive mappings, that was the first significant breakthrough and a step toward more solutions in this area. Since then, a large number of papers have been published in connection with various generalizations of Ulam's problem and Hyers's theorem. In 1978, Themistocles M. Rassias succeeded in extending Hyers's theorem for mappings between Banach spaces by considering an unbounded Cauchy difference subject to a continuity condition upon the mapping. He was the first to prove the stability of the linear mapping. This result of Rassias attracted several mathematicians worldwide who began to be stimulated to investigate the stability problems of functional equations. By regarding a large influence of S. M. Ulam, D. H. Hyers, and Th. M. Rassias on the study of stability problems of functional equations, the stability phenomenon proved by Th. M. Rassias led to the development of what is now known as Hyers–Ulam–Rassias stability of functional equations. For an extensive presentation of the stability of functional equations in the context of Ulam's problem, the interested reader is referred to the books by S.-M. Jung, S. Czerwik, Y.J. Cho, C. Park, Th.M. Rassias and R. Saadati, Y.J. Cho, Th.M. Rassias and R. Saadati, and Pl. Kannappan, as well as to the following papers. In 1950, T. Aoki considered an unbounded Cauchy difference which was generalised later by Rassias to the linear case. This result is known as Hyers–Ulam–Aoki stability of the additive mapping. Aoki (1950) had not considered continuity upon the mapping, whereas Rassias (1978) imposed extra continuity hypothesis which yielded a formally stronger conclusion.

References

See also Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Applicandae Mathematicae, 62(1)(2000), 23-130. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings[link removed], J. Math. Anal. Appl. 184(1994), 431–436. P. Gavruta and L. Gavruta, A new method for the generalized Hyers–Ulam–Rassias stability, Int. J. Nonlinear Anal. Appl. 1(2010), No. 2, 6 pp. J. Chung, Hyers-Ulam-Rassias stability of Cauchy equation in the space of Schwartz distributions[link removed], J. Math. Anal. Appl. 300(2)(2004), 343 – 350. T. Miura, S.-E. Takahasi, and G. Hirasawa, Hyers-Ulam-Rassias stability of Jordan homomorphisms on Banach algebras, J. Inequal. Appl. 4(2005), 435–441. A. Najati and C. Park, Hyers–Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation[link removed], J. Math. Anal. Appl. 335(2007), 763–778. Th. M. Rassias and J. Brzdek (eds.), Functional Equations in Mathematical Analysis, Springer, New York, 2012, ISBN 978-1-4614-0054-7. D. Zhang and J. Wang, On the Hyers-Ulam-Rassias stability of Jensen’s equation, Bull. Korean Math. Soc. 46(4)(2009), 645–656. T. Trif, Hyers-Ulam-Rassias stability of a Jensen type functional equation[link removed], J. Math. Anal. Appl. 250(2000), 579–588. Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer, New York, 2009, ISBN 978-0-387-89491-1. P. K. Sahoo and Pl. Kannappan, Introduction to Functional Equations, CRC Press, Chapman & Hall Book, Florida, 2011, ISBN 978-1-4398-4111-2. W. W. Breckner and T. Trif, Convex Functions and Related Functional Equations. Selected Topics, Cluj University Press, Cluj, 2008. I. A. Vestfrid, Linear approximation of approximately linear functions, Aequationes Math. 66 (2003), 37–77

Worked examples

Example 1 — a first encounter with Hyers–Ulam–Rassias stability

Start with the simplest possible case. Write down what Hyers–Ulam–Rassias stability claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hyers–Ulam–Rassias stability before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hyers–Ulam–Rassias stability ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hyers–Ulam–Rassias stability

In research: Hyers–Ulam–Rassias stability appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hyers–Ulam–Rassias stability in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hyers–Ulam–Rassias stability is common in secondary-school and first-year university syllabi. It links to neighbouring topics Functional analysis, Functional equations, Mathematical analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Hyers–Ulam–Rassias stability outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Hyers–Ulam–Rassias stability” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Hyers–Ulam–Rassias stability in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hyers–Ulam–Rassias stability means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hyers–Ulam–Rassias stability out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hyers–Ulam–Rassias stability in simple terms?

The stability problem of functional equations originated from a question of Stanisław Ulam, posed in 1940, concerning the stability of group homomorphisms. In the next year, Donald H.

Why does Hyers–Ulam–Rassias stability matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hyers–Ulam–Rassias stability?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hyers–Ulam–Rassias stability.