In this paper, the mixed-type linear and Euler-Lagrange-Rassias functional equations introduced by J. M. Rassias is generalized to the following n-dimensional functional equation: f (We prove the general solutions and investigate its generalized Ulam-Gavruta-Rassias stability.

We study the alternative Jensen's functional equation f (x) ± 2f (xy) + f (xy 2 ) = 0 when f is a function from a semigroup or a group to a uniquely divisible abelian group.

It is well known that the concept of Hyers-Ulam-Rassias stability was originated by Th. M. Rassias (1978) and the concept of Ulam-Gavruta-Rassias stability was originated by J. M. Rassias (1982–1989) and by P. Găvruta (1999). In this paper, we give results concerning these two stabilities.

ABSTRACT:Given an integer λ = 2, we establish the general solution of an alternative functional equation of Jensen type on certain groups. First, we give a criterion for the existence of the general solution for the functional equationwhere f is a mapping from a group (G, ·) to a uniquely divisible abelian group (H, +). Then we show that, for λ / ∈ {0, −1, −2}, the above alternative functional equation is equivalent to the classical Jensen's functional equation. We also find the general solution in the case when G is a cyclic group and λ = 2 is an integer.

The partition function of the finite q-state Potts model in the limit q→1 is shown to yield a closed form for the distribution of clusters in the immediate vicinity of the percolation transition. Various important properties of the transition are manifest, including scaling behavior and the emergence of the spanning cluster.