Cauchy Difference Research Articles

Generalized Cauchy difference equations. II

The main result is an improvement of previous results on the equation\[f(x)+f(y)−f(x+y)=g[ϕ(x)+ϕ(y)−ϕ(x+y)]f(x)+f(y)-f(x+y)=g[\phi (x)+\phi (y)-\phi (x+y)]\]for a given functionϕ\phi. We find its general solution assuming only continuous differentiability and local nonlinearity ofϕ\phi. We also provide new results about the more general equation\[f(x)+f(y)−f(x+y)=g(H(x,y))f(x)+f(y)-f(x+y)=g(H(x,y))\]for a given functionHH. Previous uniqueness results required strong regularity assumptions on a particular solutionf0,g0f_{0},g_{0}. Here we weaken the assumptions onf0,g0f_{0},g_{0}considerably and find all solutions under slightly stronger regularity assumptions onHH.

Generalized Hyers-Ulam Stability of Quadratic Functional Equations: A Fixed Point Approach

The stability problem of functional equations originated from a question of Ulam 1 concerning the stability of group homomorphisms. Hyers 2 gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki 3 for additive mappings and by Rassias 4 for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias 4 has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability or as Hyers-Ulam-Rassias stability of functional equations. A generalization of the Rassias theorem was obtained by Gavruţa 5 by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The functional equation

On functions with the cauchy difference bounded by a functional. III

We are going to discuss special cases of a conditional functional inequality % MathType!MTEF!2!1!+-% feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaaceiGaa8hEaiabgwQiEjaa-LhacqGHshI3caWFMbac% eaGaa4hkaiaa-HhacaWFRaGaa8xEaiaa+LcaiiGacqqFsislcaWFMb% Gaa4hkaiaa-HhacaGFPaaccaGaeWNeI0Iaa8Nzaiaa+HcacaWF5bGa% a4xkaiabgwMiZkaa+bdaieaacqWESaalcaGFGaGaa4hiaiaa+bcaca% GFGaGaa4hiaiaa-HhacaWFSaGaa8hiaiaa-LhacaWFGaGaeyicI4Sa% a8hiaiaa-HfacaWFSaaaaa!5C02! $$x \bot y \Rightarrow f(x + y) - f(x) - f(y) \geqslant 0, x, y \in X,$$ whereX is a real inner product space. In particular, we will give conditions which force the representationf(x)=c‖x‖2+a(x) for x ∈X, where c ∈ R anda:x→ℝ is an additive functional.

Generalized Cauchy difference functional equations

This article concerns primarily the study of functional equations of the form f(x) + f(y) − f(x + y) = g(H(x, y)), where H is a given function of two (real or complex) variables, and where f and g are unknown functions. The problem of finding general methods to solve equations of this type was posed at the 37th International Symposium on Functional Equations in 1999 by R. Ger and L. Reich. We present results including the following. Let I be a nonvoid real interval or complex region with I + I ⊂ I. Suppose there exists a particular solution (f0, g0) on I such that f0 is analytic and not affine (or, in the real case, continuously differentiable and strictly convex). If g0 and H are analytic (or, in the real case, continuously differentiable), then all solutions of the equation are of the form \(g(u) = A_{1} \circ g_{0} (u) + c,\;f(x) = A_{1} \circ f_{0} (x) + A_{2} (x) + c,\) where A1, A2 are arbitrary additive functions and c is an arbitrary constant. Analogous results hold for functional equations of the form \(f(x) + f(y) - f \circ \theta ^{{ - 1}} (\theta (x) + \theta (y)) = g(H(x,y))\) with invertible smooth θ.

On functions with the Cauchy difference bounded by a functional. Part II

We are going to consider the functional inequality f(x + y) − f(x) − f(y) ≥ ϕ(x, y), x, y ∈ X, where (X, +) is an abelian group, and ϕ : X × X → ℝ and f : X → ℝ are unknown mappings. In particular, we will give conditions which force biadditivity and symmetry of ϕ and the representation f(x) = (1/2)ϕ(x, x) + a(x) for x ∈ X, where a is an additive function. In the present paper, we continue and develop our earlier studies published by the author (2004).

On Functions with the Cauchy Difference Bounded by a Functional

We are going to discuss special cases of a conditional functional inequality $$x \bot y \Rightarrow f(x + y) - f(x) - f(y) \geqslant 0, x, y \in X,$$ whereX is a real inner product space. In particular, we will give conditions which force the representationf(x)=c‖x‖2+a(x) for x ∈X, where c ∈ R anda:x→ℝ is an additive functional.

The cocycle equation on periodic semigroups

The cocycle functional equation \(F(x,y) + F(xy, z) = F(x,yz) + F(y,z)\) has a long and rich history, with important roles in contexts from homological algebra to polyhedral algebra to information theory. This paper deals with the problem of finding explicit forms of symmetric cocycles on periodic semigroups. A semigroup S is periodic if each of its elements has finite order, that is if the cyclic subsemigroup \(\langle a \rangle = \{ a^k \mid k = 1,2,3,\ldots \}\) generated by each element a of S is finite. For several classes of abelian semigroups, including idempotent semigroups, cancellative semigroups, and certain types of topological semigroups, it is known that every symmetric cocycle F is a coboundary (in other words, a Cauchy difference), that is \(F(x,y) = f(x) + f(y) - f(xy).\) We show that in fact every cocycle has this form on periodic semigroups.

On the Cauchy difference on normed spaces

LetX be a real linear normed space, (G, +) be a topological group, andK be a discrete normal subgroup ofG. We prove that if a continuous at a point or measurable (in the sense specified later) functionf:X →G fulfils the condition:f(x +y) -f(x) -f(y) ∈K whenever ‖x‖ = ‖y‖, then, under some additional assumptions onG,K, andX, there esists a continuous additive functionA :X →G such thatf(x) -A(x) ∈K.

On functions having Cauchy differences in some prescribed sets

AssumeE is a real topological vector space,F is a real Banach space,K is a discrete subgroup ofF andC is a symmetric, convex and compact subset ofF such thatK ∩ (6C) = {0}. If a functionh:E → F is continuous at at least one point andh(x + y) − h(x) − h(y) ∈ K + C for allx, y ∈ E, then there exists a continuous linear functiona:E → F such thath(x) − a(x) ∈ K + C for everyx ∈ E.

The Cauchy and Jensen differences on semigroups

The Cauchy and Jensen differences on semigroups

On the Cauchy difference

Assume thatE andF are real topological vector spaces (which are assumed to be Hausdorff) and letK be a countable and discrete subgroup ofF. Supposef: E → F satisfies $$f(x + y) - f(x) - f(y) \in K$$ for allx, y ∈ E. Conditions are established under whichf has the formA + k, whereA: E →F is a continuous linear operator andk takes values inK only. For example, ifF is assumed to be locally convex and iff andE satisfy suitable conditions (such asE being a Baire space andf a Baire function) thenf has the formA + k, whereA : E →F is a continuous linear operator and k(E)\(k(E) \subseteq K\)

On the Cauchy difference

On the Cauchy difference

On Cauchy differences of all orders

This paper deals with the problem of characterizing higher order Cauchy differences of mappings on groups and semigroups. Symmetric, first order Cauchy differencesf(x + y)−f(x)−f(y) for mapsf between groups were characterized by Jessen, Karpf, and Thorup [8] through the use of first partial Cauchy differences. Our results are similar and extend their result to higher order differences. Our results also extend those of Heuvers [6] for mappings between vector spaces over the rationals.