The aim of this paper is to investigate the stability problem for the Pexider type (hyperbolic) trigonometric functional equation f(x + y) + f(x + σy) = λg(x)h(y) under the conditions : |f(x + y) + f(x + σy)− λg(x)h(y)| ≤ φ(x), φ(y), and min{φ(x), φ(y)}. As a consequence, we have generalized the results of stability for the cosine(d’Alembert), sine, and the Wilson functional equations by J. Bak...

We further investigate the class of models of a strongly dependent (first order complete) theory T , continuing [Sh 715], [Sh 783] and related works. Those are properties (= classes) somewhat parallel to superstability among stable theory, though are different from it even for stable theories. We show equivalence of some of their definitions, investigate relevant ranks and give some examples, e...

The main purpose of this paper is to establish some new results onthe superstability and stability via a fixed point approach forthe Pexiderized exponential equation, i.e.,$$|f(x+y)-g(x)h(y)|leq psi(x,y),$$where $f$, $g$ and $h$ are three functions from an arbitrarycommutative semigroup $S$ to an arbitrary unitary complex Banachalgebra and also $psi: S^{2}rightarrow [0,infty)$ is afunction. Fur...

We discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module. Let A be a Banach algebra and X a Banach A-module, f : X → X and g : A → A. The mappings Δ1 f,g , Δ2 f,g , Δ3 f,g , and Δ4 f,g are defined and it is proved that if ‖Δ1 f,g x, y, z,w ‖ resp., ‖Δ3 f,g x, y, z,w, α, β ‖ is dominated by φ x, y, z,w , then f is a generalized re...

We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of orbital types) in some cardinal. Assuming the singular cardinal hypothesis (SCH), we prove a full characterization of the (high-enough) stability cardinals, and connect the stability spectrum with th...

We develop a notion of forking for Galois-types in the context of AECs. Under the hypotheses that an AEC K is tame, type-short, and failure of an order-property, we consider Definition 1. Let M0 ≺ N be models from K and A be a set. We say that the Gaois-type of A over M does not fork over M0, written A^ M0 N , iff for all small a ∈ A and all small N− ≺ N , we have that Gaois-type of a over N− i...

S. Shelah [20] has given a description of all countable complete theories T that permit a classification of their models. A necessary condition for T is superstability. A stable (superstable) group is a group interpreted in some stable (superstable) structure. It is equivalent to say that the groups investigated have additional relations and functions such that the whole structure is stable (su...