Composite functional equations (arising in applications) are presented that may be interpreted as extensions of the Gołąb-Schinzel equation and as modifications of d'Alembert's equation. Depending on the type of the considered equation, continuous, and finite rate of growth solutions are discussed. Geometric interpretations are given.

It is shown that Gołąb-Schinzel's equation may be transformed into one of Cauchy's equations by an embedding and limit process concerning the general continuous solution.

In this paper, we solve the functional equation
f₁(pr,qs) + f₂(ps,qr) = g(p,q)h(r,s) (p,q,r,s∈]0,1])
where f₁,f₂,g,h are complex-valued functions defined on ]0,1]. This functional equation is a generalization of a functional equation which was instrumental in the characterization of symmetric divergence of degree α in [3]. This equation arises in the characterization of symmetric weighted divergence of degree α and symmetric inset divergence of degree α.

On considère des fonctions remplissantes la condition (3) et dans le cas particulate l'implication (1). On donne des conditions sous lesquelles ces fonctions sont généralament homogènes.

Let G be an abelian group, let K be the real or complex field, let X be a normed space over ????, and let u∈X such that ‖u‖=1 be given. We assume that there exists a subspace X₁ of X such that
X = Lin(u) ⊕ X₁
and
‖αu+x₁‖ ≥ max(|α|, ‖x₁‖) for α∈K , x₁∈X₁.
Then we prove that the general solution f: G→K of the equation
f(x+y) + f(x-y) + ‖f(x+y)-f(x-y)‖u = 2f(x) + 2f(y)    for x,y∈G
is given by the formula
f(x) = |a(x)|u    for x∈G,
where a: G→ℝ is an additive function.

We consider approximate solutions of the functional equation (1) in the class of functions which satisfy on compact sets the condition (2) with an increasing, subadditive, continuous at zero and vanishing at zero function γ: [0,+∞)→[0.+∞).

Some properties of non-negative measurable solutions of equation (1) are studied. The obtained results are stronger versions of those from [6] and their proofs are shorter and simpler.

It is proved that if
0 < q ≤ (1 - ∛2 + ∛4)/3,
then the zero function is the only solution f: ℝ→ℝ of (1) satisfying (2) and bounded in a neighbourhood of at least one point of the set (3).

The aim of this paper is to prove results on solutions of the Schröder equation (1) defined on cones in Banach spaces and having some properties connected with monotonicity and boundedness.

In the present paper we find a linear operator on a function space, essentially larger than the space of all bounded functions on an amenable semigroup, which behaves like an invariant mean. This leads to an extension of the Hyers-Ulam stability theorem for Cauchy's functional equation in the case of vector-valued mappings defined on amenable semigroups.

We consider a class of approximate solutions of the generalized orthogonality equation in ℝⁿ (n≥2). We prove that this class coincides with the class of solutions of the equation, i.e., the superstability of the generalized orthogonality equation holds.

We deal with functional congruences of the form
f(x+y) - f(x) - f(y) ∈ U + V
where U and V are given sets subjected to satisfy some "separability" conditions essentially weaker than that occurring in [4] which proved to be pretty useful especially while investigating various types of Hyers-Ulam stability problems. The goal is to factorize f into a sum of two functions whose Cauchy differences remain in U and V, respectively, or, at least to obtain an approximation of f by such a sum. An application of the newly established result in that spirit is given. Moreover, a stability result for the celebrated cocycle equation is presented and, finally, the behaviour of mappings whose Cauchy differences fall into a given Hamel basis is described.

Some characterizations of homogeneous polynomials and of polynomials in general are given. This is done not only in the usual case but also for vector spaces over non archimedean valued fields. Moreover stability results in connection with these characterizations are given.

We consider the real solutions of the functional equation
(⚹)       ϕ^m(x) = 1/m ϕ(mx),   ϕ(0) = 0,
where m∈ℕ and ϕ^m denotes the m-th iterate of the unknown function ϕ. We will handle this functional equation for a fixed m, but also for all naturals m, and give a representation of all C²-solutions (even weaker, see Theorem 2.1) of (⚹), but also treat the case of other solutions of this equation. In the introduction we will show the origin of this equation.

We consider for a given formal power series F(x) = ρx + c₂x² + ... , ρ ≠ 0, with complex coefficients and for a given integer N > 1 the functional equation G^N = F where G is again a formal series (G is called an iterative root of F). If ρ is a root of 1 and F is not conjugate to its linear part we derive a criterion for the existence of solutions G and describe the general solution. Representations of the coefficients of G by means of universal polynomials are given, also in the case where ρ is not a root of 1 (where existence is almost trivial). Our main tools are maximal families of commuting series (i.e. the Aczél-Jabotinsky equation of third type) and semicanonical forms.

Let D be a simply connected region on the plane. We prove that a continuous iteration group of homeomorphisms {f^t : t∈ℝ} defined on D is of the form
f^t(x) = ϕ^-1(ϕ(x)+te₁)   for x∈D, t∈ℝ,
where e₁=(1,0) and ϕ is a homeomorphism mapping D onto ℝ, if and only if f¹ is a singularity-free homeomorphism, i.e. f¹ =: f has the property that for every Jordan domain B ⊂ D there exists an integer n₀ such that B∩fⁿ[B] = ⌀ for |n| > n₀, n∈ℤ.

A set-valued function F is called hull-concave if
F(tx + (1-t)y) ⊂ co(tF(x) + (1-t)F(y))
for all x,y from the domain of F and all t∈[0,1]. It is shown that if a hull-concave set-valued function F is defined on an open convex subset D of ℝⁿ and for every x∈D the set clF(x) is convex and bounded, then F is continuous on D. Some other properties of hull-concave set-valued functions are also given.

The functional equation
ϕ(x+y) + ψ(x) + ψ(y) = ψ(x+y) + ϕ(x) + ϕ(y)
is studied for set-valued functions.

In this paper we prove that there exists a biadditive selection f of a biadditive set-valued function F and a continuous selection when F is lower semicontinuous.