Abstract
AbstractThe general solutions of two functional equations, without imposing any regularity condition on any of the functions appearing, have been obtained. From these general solutions, the Lebesgue measurable solutions have been deduced by assuming the function(s) to be measurable in the Lebesgue sense.
Highlights
For n = 1, 2, . . . , let Γn = (p1, . . . , pn) : pi ≥ 0, i = 1, . . . , n; pi = 1 i=1 be the set of all n-component complete discrete probability distributions with nonnegative elements
Let R denote the set of all real numbers; I = {x ∈ R : 0 ≤ x ≤ 1} = [0, 1], the closed unit interval; ]0, 1[ = {x ∈ R : 0 < x < 1}, the open unit interval and ]0, 1] = {x ∈ R : 0 < x ≤ 1}
The object of this paper is to study the functional equations (FE1)
Summary
F (pq) = qβf (p) + pαf (q), where f : I → R is an unknown function, p, q ∈ I; α and β are fixed positive real exponents which satisfy the following conventions F (pq) = qβf (p) + pαf (q) + g(p)f (q) where f : I → R, g : I → R are unknown functions; p, q ∈ I; α and β are fixed positive real exponents which satisfy the conventions stated in (1.4); and c is a given real constant.
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