Let (G,+) be a locally compact abelian Hausdorff group, 𝓀 is a finite automorphism group of G, κ = card𝒦 and let µ be a regular compactly supported complex-valued Borel measure on G such that $\mu ({\rm{G}}) = {1 \over \kappa }$ . We find the continuous solutions f, g : G → ℂ of the functional equation $$\sum\limits_{k \in {\cal K}} {\sum\limits_{\lambda \in {\cal K}} {\int_{\rm{G}} {{\rm{f}}({\rm{x}} + {\rm{k}} \cdot {\rm{y}} + } \lambda \cdot {\rm{s}}){\rm{d}}\mu ({\rm{s}}) = {\rm{g}}({\rm{y}}) + \kappa {\rm{f}}({\rm{x}}),\,{\rm{x}},{\rm{y}} \in {\rm{G}},} } $$ in terms of k-additive mappings. This equations provides a common generalization of many functional equations (quadratic, Jensen’s, Cauchy equations).

Throughout the paper we work in the following framework and with the following notation and terminology. Let G be a group and S be a semigroup (a set with an associative composition rule). Let σ : G → G be an involutive automorphism, that is an homomorphism such that σ • σ = id. If (G; +) is an abelian group, then the inversion σ(x) := −x is an example of an involutive automorphism. Another example is the complex conjugation map on the multiplicative group of non-zero complex numbers. A function A : G → C is called additive, if it satisfies A(xy) = A(x) + A(y) for all x, y ∈ G. A multiplicative function on S is a map µ : S → C such that µ(xy) = µ(x)µ(y) for all x, y ∈ S. A character on a group G is a homomorphism from G into the multiplicative group of non-zero complex numbers. If S is a topological space, then we let C(S) denote the algebra of continuous functions from S into C.

In the present paper we characterize, in terms of characters, multiplicative functions, the continuous solutions of some functional equations for mappings defined on a monoid and taking their values in a complex Hilbert space with the Hadamard product. In addition, we investigate a superstability result for these equations.

In this paper, we present the hyperstability results of a mixed type cubic-quartic functional equations in ultrametric Banach spaces.

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