Volume 7, Issue 1

In this paper, by defining Lyapunov functionals, we investigate proper sufficient conditions for the uniform stability of the zero solution, and also for the uniform boundedness and uniform ultimate boundedness of all solutions of a certain third-order nonlinear vector delay differential equation of the type X ˙˙˙ + A X ˙˙ + G ( X ˙) + H ( X ( t - r )) = P ( t ).

We consider the Heisenberg motion groups ℍ𝕄 = ℍ n ⋊ K , where ℍ n is the Heisenberg group and K is a compact subgroup of U ( n ) such that ( K ,ℍ n ) is a Gelfand pair. We study the Segal–Bargmann transform on ℍ𝕄 and characterise the Poisson integrals associated to the Laplacian for ℍ𝕄 using Gutzmer's formula. We also prove a Paley–Wiener type theorem involving complexified representations using explicit realisations of some unitary irreducible representations of ℍ𝕄.

In this paper, we obtain the general solution and we study the generalized Hyers–Ulam stability of the mixed type additive-quadratic functional equation f (- x ) + f (2 x - y ) + f (2 y ) + f ( x + y ) - f (- x + y ) - f ( x - y ) - f (- x - y ) = 3 f ( x ) + 3 f ( y ) in a fuzzy normed space.

Let G be a locally compact group and let μ be a regular, compactly supported, complex-valued Borel measure on G . In the present paper, we determine the continuous solutions f , g : G → ℂ of the functional equation ∫ G { f ( x y t ) + f ( x y  -1 t )} d μ( t ) = 2 g ( x ) f ( y ), x , y ∈ G , in terms of characters, additive maps and matrix elements of irreducible, two-dimensional representations of G . Many consequences of this result are presented.

In this paper, we consider the generalized Fourier transform associated with the Jacobi–Cherednik operator on ℝ and we give conditions of quite different kind for a function to have a transform belonging to certain L p -classes.

Let 𝔸 n m be an arbitrary n -dimensional commutative associative algebra over the field of complex numbers with m idempotents. Let e 1 = 1, e 2 , e 3 be elements of 𝔸 n m which are linearly independent over the field of real numbers. We consider monogenic (i.e., continuous and differentiable in the sense of Gateaux) functions of the variable x e 1 + y e 2 + z e 3 , where x , y , z are real, and obtain a constructive description of all mentioned functions by means of holomorphic functions of complex variables. It follows from this description that monogenic functions have Gateaux derivatives of all orders.