On the functional equationf(x) g(y) = p(x + y) q(x/y)

Aequationes mathematicae -     

On the system of functional equationsf(x + y) = f(x) + f(y) andf(xy) = p (x)f(y) + q(y)f(x)

Aequationes mathematicae -     

The functional differential equation x′(t) = x(x(t))

A classification of the solutions of the functional differential equation x′(t) = x(x(t)) is given and it is proved that every solution either vanishes identically or is strictly monotonie. For monotonically increasing solutions existence and uniqueness of the solution x are proved with the condition x(t₀) = x₀ where (t₀, x₀) is any given pair of reals in some specified subset of $\text{R}$². Every monotonically increasing solution is thus obtained. It is analytic and depends analytically on t₀ and x₀. Only for t₀ = x₀ = 1 is the question of analyticity still open.     

On the functional equation x + f(y + f(x))=y + f(x + f(y))

For an abelian group (G, + ,0) we consider the functional equation $$f : G \to G, x + f(y + f(x)) = y + f(x + f(y)) \quad (\forall x, y \in G), \quad\quad\qquad (1)$$ most times together with the condition $$f(0) = 0.\qquad\qquad\qquad\qquad\qquad (0)$$ Our main question is whether a solution of ${(1) \wedge (0)}$ must be additive, i.e., an endomorphism of G. We shall answer this question in the negative (Example 3.14) Rätz (Aequationes Math 81:300, 2011).     

The Asymptotic Stability of x(n + k) + ax(n) + bx(n-l) = 0

We established necessary and sufficient conditions for the asymptotic stability of the difference equation where the coefficients a and b are real numbers and k and l are nonnegative integers.     

On the boundedness of the solutions of the equation\dddot x + a\ddot x + f(x)\dot x + g(x) = p(t)

Summary In this paper my previous result [1] on the boundedness of solutions of (1.1.1) is fackled by use of a suitably chosen Liapounov function. This fresh approach leads to a more direct proof of the boundedness Theorem and makes for substantial reduction in each of my previous conditions on f and g.     

Equation f(p(x)) = q(f(x)) for given real functions p, q

We investigate functional equations f(p(x)) = q(f(x)) where p and q are given real functions defined on the set ? of all real numbers. For these investigations, we can use methods for constructions of homomorphisms of mono-unary algebras. Our considerations will be confined to functions p, q which are strictly increasing and continuous on ?. In this case, there is a simple characterization for the existence of a solution of the above equation. First, we give such a characterization. Further, we present a construction of any solution of this equation if some exists. This construction is demonstrated in detail and discussed by means of an example.     

Über die Funktionalgleichungf (1 + x) + f (1 + f (x)) = 1

Ohne Zusammenfassung