Abstract
Let ℝ+ and B be the set of positive real numbers and a Banach space, respectively, f, g, h : ℝ+ → B and ψ: ℝ + 2 →ℝ be a nonnegative function of some special forms. Generalizing the stability theorem for a Jensen-type logarithmic functional equation, we prove the Hyers-Ulam stability of the Pexiderized logarithmic functional inequality | | f ( x y ) - g ( x ) - h ( y ) | | ≤ ψ ( x , y ) in restricted domains of the form {(x, y) : xkys≥ d} for fixed k, s ∈ ℝ, d > 0. We also discuss an L∞-version of the Hyers-Ulam stability of the inequality. 2000 MSC: 39B22.
Highlights
The Hyers-Ulam stability problems of functional equations go back to 1940 when Ulam proposed a question concerning the approximate homomorphisms from a group to a metric group
A partial answer was given by Hyers [2,3] under the assumption that the target space of the involved mappings is a Banach space
After the result of Hyers, Aoki [4] and Bourgin [5,6] treated with this problem, there were no other results on this problem until 1978 when Rassias [7] treated again with the inequality of Aoki [4]
Summary
The Hyers-Ulam stability problems of functional equations go back to 1940 when Ulam proposed a question concerning the approximate homomorphisms from a group to a metric group (see [1]). We discuss an L∞-version of the Hyers-Ulam stability of the inequality. We refer the reader to [24,25,26,27,28,29] for some interesting results on functional equations and their Hyers-Ulam stabilities in restricted conditions. In this article, generalizing the result in [8], we consider the Hyers-Ulam stability of the Pexiderized Jensen functional equation
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