Solutions and stability of generalized Kannappan’s and Van Vleck’s functional equations
Abstract
We study the solutions of the integral Kannappan’s and Van Vleck’s functional equations
∫Sf(xyt)dμ(t) + ∫Sf(xσ(y)t)dμ(t) = 2f(x)f(y), x,y∈S;
∫Sf(xσ(y)t)dμ(t) - ∫Sf(xyt)dμ(t) = 2f(x)f(y), x,y∈S,
where S is a semigroup, is an involutive automorphism of S and μ is a linear combination of Dirac measures (δz_i)i∈I, such that for all i∈I, zi is in the center of S. We show that the solutions of these equations are closely related to the solutions of the d’Alembert’s classic functional equation with an involutive automorphism. Furthermore, we obtain the superstability theorems for these functional equations in the general case, where σ is an involutive morphism.
Keywords
Hyers–Ulam stability; semigroup; d’Alembert’s equation; Van Vleck’s equation; Kannappan’s equation; involution; automorphism; multiplicative function; complex measure
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Department of Mathematics, Faculty of Sciences, University Ibn Zohr, Morocco Morocco
Department of Mathematics, Faculty of Sciences, University Ibn Zohr, Morocco Morocco
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