Abstract
In this paper, we obtain the solution of a new generalized reciprocal type functional equation in two variables and investigate its generalized Hyers–Ulam stability in non-Archimedean fields. We also present the pertinent stability results of Hyers–Ulam–Rassias stability, Ulam–Gavruta–Rassias stability and J. M. Rassias stability controlled by the mixed product-sum of powers of norms.
Highlights
A stimulating and famous talk presented by Ulam [31] in 1940, motivated the study of stability problems for various functional equations
He gave wide range of talk before a Mathematical Colloquium at the University of Wisconsin in which he presented a list of unsolved problems
Among those was the following question concerning the stability of homomorphisms
Summary
A stimulating and famous talk presented by Ulam [31] in 1940, motivated the study of stability problems for various functional equations. In the year 1978, Rassias [29] tried to weaken the condition for the Cauchy difference and succeeded in proving what is known to be the Hyers–Ulam–Rassias Stability for the Additive Cauchy Equation This terminology is justified because the theorem of Th. M. In 1982, Rassias [21] gave a further generalization of the result of D.H. Hyers and proved a theorem using weaker conditions controlled by a product of different powers of norms. Hyers and proved a theorem using weaker conditions controlled by a product of different powers of norms His theorem is presented as follows: Theorem 1.3 (Rassias [21]) Let f : X → Y be a mapping from a normed vector space X into a Banach space Y subject to the inequality f (x + y) − f (x) − f (y) ≤ ||x||p||y||p (1.6)
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