In this paper we investigate the functional equation φ( x+y/2) ( ψ _1(x) - ψ _2(y) ) = 0 ( for all x ∈ I_1 and y ∈ I_2 ) where I_1 , I_2 are open intervals of ℝ , J = 1/2( I_1 + I_2 ) moreover ψ _1: I_1 →ℝ , ψ _2: I_2 →ℝ and φ : J →ℝ are unknown functions. We describe the structure of the possible solutions assuming that φ is measurable. In the case when φ is a derivative, we give a complete characterization of the solutions. Furthermore, we present an example of a solution consisting of irregular Darboux functions. This provides the answer to an open problem proposed during the 59th International Symposium on Functional Equations.

Let C ⊆ℝ^n be a convex set. The mapping f : C ⟶ℝ^n is strictly increasing, if ⟨ f(x)-f(y), x-y ⟩ > 0 for all distinct elements x,y ∈ C. Applying classical theorems of finite dimensional convex geometry and convex analysis, we show that the inverse functions has a unique extension f^(-1) : conv (f(C)) ⟶ C such that f^(-1) is monotone, continuous and it acts as a left-inverse of f. As an application we introduce the concept of vector-valued weighted quasi-arithmetic means and discuss their equality problem.

In this paper we describe the solutions of the functional equation F (x+y/2 )+f_1(x)+f_2(y)= G (g_1(x)+g_2(y)) defined on an open subinterval of ℝ . Improving previous results we assume differentiability on each involved function, eliminate a former condition on g'_1 and g'_2 , moreover we determine a brand new family of solutions. We also present a particular member of this class as an example. In order to achieve this, we strengthen known results about certain auxiliary functional equations as well.

A quasisum is a function F: I_1 ×…× I_n ⟶ℝ of the form F ( x_1 , … , x_n ) = g ( f_1(x_1) + … + f_n(x_n) ) ( x_1 ∈ I_1 , … , x_n ∈ I_n ) where n ≥ 2 is an integer and f_k: I_k ⟶ℝ is a continuous, strictly monotone function defined on a nonempty open interval of ℝ (for k = 1 , … , n ), moreover g: f_1(I_1) + … + f_n(I_n) ⟶ℝ is also continuous, strictly monotone. In this paper we will show that if p ∈ℕ and the quasisum F is p-times continuously differentiable then each of the generator functions g , f_1 , … , f_n are p-times continuously differentiable as well. We present applications of our results for p-times continuously differentiable semigroup operations and additively separable utility functions as well.

For a real valued function f, defined on an open interval I, and an arbitrary real number h, we consider the lower and upper limits of2n(f(y+h2n)−f(y)) whenever n tends to infinity and y tends to a fixed element x of I. We consider two families of functions determined by the properties of these limits. The first interesting property is when these lower and upper limits are finite and equal to each other for every real number h and every x in I. The second notable family is determined by the property that both limits are finite and increasing (in a specific sense) with respect to x for every positive number h. These properties are motivated by the families of continuously differentiable functions and convex functions, respectively. However, restrictions of additive mappings belong to these classes as well. Our decomposition theorems establish that these motivating examples generate the whole classes. Namely, every function belonging to the first family can be represented as the sum of a continuously differentiable function and an additive one, while every function taken from the second family turns out to be the sum of a convex function and an additive one. We apply our results in order to give a local and approximate characterization of affine functions and Wright-convex functions, respectively.

In this paper we introduce the concept of translation invariant functions: considering an arbitrary set ∅ S ⊂ℝ^n , the function F : S ⟶ℝ is translation invariant if F(x) = F(y) implies F(x+t)=F(y+t) for any vectors x,y,t ∈ℝ^n such that x ,y , x+t ,y+t ∈ S . In our main results we shall consider an open, connected set ∅ D ⊂ℝ^n . We prove that if F : D ⟶ℝ is a translation invariant, continuous function, then there exists a vector a = (a_1, … , a_n) ∈ℝ^n and a strictly monotone, continuous function f such that F(x_1, … , x_n) = f (a_1 x_1 + … + a_n x_n) holds for all (x_1, … , x_n) ∈ D . Using this result we also show that continuous solutions F : D ⟶ℝ of the system of functional equations F(x_1 , … , x_j + t_j , … , x_n) = Ψ _j (F (x_1 , … , x_j , … , x_n), t_j) (j=1,… ,n) can be represented as the composition of a strictly monotone, continuous function and a linear functional as well. Applying the latter theorem, we give a characterization of Cobb–Douglas type utility functions.

Adenosine A(2A) receptor (A(2A)R)-dependent signaling in macrophages plays a key role in the regulation of inflammation. However, the processes regulating A(2A)R targeting to the cell surface and degradation in macrophages are incompletely understood. For example, the C-terminal domain of the A(2A)R and proteins interacting with it are known to regulate receptor recycling, although it is unclear what role potential A(2A)R-interacting partners have in macrophages. Here, we aimed to identify A(2A)R-interacting partners in macrophages that may effect receptor trafficking and activity. To this end, we performed a yeast two-hybrid screen using the C-terminal tail of A(2A)R as the "bait" and a macrophage expression library as the "prey." We found that the lysosomal protease cathepsin D (CtsD) was a robust hit. The A(2A)R-CtsD interaction was validated in vitro and in cellular models, including RAW 264.7 and mouse peritoneal macrophage (IPM) cells. We also demonstrated that the A(2A)R is a substrate of CtsD and that the blockade of CtsD activity increases the density and cell surface targeting of A(2A)R in macrophages. Conversely, we demonstrate that A(2A)R activation prompts the maturation and enzymatic activity of CtsD in macrophages. In summary, we conclude that CtsD is a novel A(2A)R-interacting partner and thus describe molecular and functional interplay that may be crucial for adenosine-mediated macrophage regulation in inflammatory processes.

The ultrapower T* of an arbitrary ordered set T is introduced as an infinitesimal extension of T. It is obtained as the set of equivalence classes of the sequences in T, where the corresponding relation is generated by a free ultrafilter on the set of natural numbers. It is established that T* always satisfies Cantor’s property, while one can give the necessary and sufficient conditions for T so that T* would be complete or it would fulfill the open completeness property, respectively. Namely, the density of the original set determines the open completeness of the extension, while independently, the completeness of T* is determined by the cardinality of T.