Stability of Jensen functional equation on semigroups

In this paper we introduce a Jensen type functional equation on semigroups and study the Hyers-Ulam stability of this equation. It is proved that every semigroup can be embedded into a semigroup in which the Jensen equation is stable.     

On the hyperstability of {\sigma}-Drygas functional equation on semigroups

In this paper, we obtain hyperstability results for the \({\sigma}\)-Drygas functional equation and the inhomogeneous \({\sigma}\)-Drygas functional equation on semigroups. Namely, we show that a function satisfying the \({\sigma}\)-Drygas equation approximately must be exactly the solution of it.     

A functional equation

The translational functional equation (1) with right-hand side analytic in a finite convex region D is investigated. It is proved that a solution exists which is analytic in a finite convex region G₁ determined by G. A corollary of this result is the existence of analytic solutions of differential-difference equations of finite order and of difference equations of infinite order with constant coefficients.Translated from Matematicheskie Zametki, Vol. 5, No. 6, pp. 733–742, June, 1969.     

A functional equation

Existence theorems for and the determination of continuous solutions, defined on the real axis R, of the functional equationf (t)=A[t,f (at–b),f (at–c)], wherea, b, and c are real parameters, A:R×E×E E is a continuous operator, and E is a Banach space.Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 161–170, February, 1971.The author wishes to thank B. N. Sadovskii for his interest and help in this work.     

Global existence for the periodic dispersive Hunter–Saxton equation

Monatshefte für Mathematik - In this paper, we study an integrable dispersive Hunter–Saxton equation in periodic domain. Firstly, we establish the local well-posedness of the Cauchy...     

A Nash–Moser approach for the Euler–Arnold equations

We study the local well-posedness in the smooth category for a class of Euler equations. A Nash–Moser approach is used to extend, for the case of an invertible elliptic pseudo-differential operator, some results obtained by Escher and Kolev, with the help of some geometric arguments.     

A universal functional equation

It is shown that the S-chains solving Rubel's universal fourth-order differential equation also satisfy a third-order functional equation.

     

A sine functional equation

Aequationes mathematicae -     

A curious functional equation

For Israel Gohberg, outstanding analyst, with affection and admiration.     

On stability of the Cauchy equation on semigroups

Summary We say that Hyers's theorem holds for the class of all complex-valued functions defined on a semigroup (S, +) (not necessarily commutative) if for anyf:S such that the set {f(x + y) – f(x) – f(y): x, y S} is bounded, there exists an additive functiona:S for which the functionf – a is bounded.Recently L. Székelyhidi (C. R. Math. Rep. Acad. Sci. Canada8 (1986) has proved that the validity of Hyers's theorem for the class of complex-valued functions onS implies its validity for functions mappingS into a semi-reflexive locally convex linear topological spaceX. We improve this result by assuming sequential completeness of the spaceX instead of its semi-reflexiveness. Our assumption onX is essentially weaker than that of Székelyhidi. Theorem.Suppose that Hyers's theorem holds for the class of all complex-valued functions on a semigroup (S, +) and let X be a sequentially complete locally convex linear topological (Hausdorff) space. If F: S X is a function for which the mapping (x, y) F(x + y) – F(x) – F(y) is bounded, then there exists an additive function A : S X such that F — A is bounded.     

A stochastic functional equation

Aequationes mathematicae -     

Optimality of the quantified Ingham–Karamata theorem for operator semigroups with general resolvent growth

We prove that a general version of the quantified Ingham–Karamata theorem for $$C_0$$-semigroups is sharp under mild conditions on the resolvent growth, thus generalising the results contained in a recent paper by the same authors. It follows in particular that the well-known Batty–Duyckaerts theorem is optimal even for bounded $$C_0$$-semigroups whose generator has subpolynomial resolvent growth. Our proof is based on an elegant application of the open mapping theorem, which we complement by a crucial technical lemma allowing us to strengthen our earlier results.     

A remark on a logarithmic functional equation

We revisit the logarithmic functional equation of Heuvers and Kannappan [K.J. Heuvers, Pl. Kannappan, A third logarithmic functional equation and Pexider generalizations, Aequationes Math. 70 (2005) 117-121] and give a simple proof of the result and discuss the locally integrable solutions of the equation.     

A note on the D'Alembert functional equation

It is proved that the analog of DAlemberts equation for a group has symmetric solutions which are central functions only in the commutative case. Necessary and sufficient conditions are given for the representability of the solutions in the form of a semisum of well-defined homomorphisms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 182–183, 1974.