and Applied Analysis 3 Clearly, every Menger PN-space is probabilistic metric space having a metrizable uniformity on X if supa<1T a, a 1. Definition 1.3. Let X,Λ, T be a Menger PN-space. i A sequence {xn} in X is said to be convergent to x in X if, for every > 0 and λ > 0, there exists positive integer N such that Λxn−x > 1 − λ whenever n ≥ N. ii A sequence {xn} in X is called Cauchy sequence ...

In this paper, we solve the quadratic $alpha$-functional equations $2f(x) + 2f(y) = f(x + y) + alpha^{-2}f(alpha(x-y)); (0.1)$ where $alpha$ is a fixed non-Archimedean number with $alpha^{-2}neq 3$. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the quadratic $alpha$-functional equation (0.1) in non-Archimedean Banach spaces.

The Madelung constant-essentially the Coulomb energy density of a crystal-is usually calculated via Ewald error function expansions or, for the simpler cubic structures, by the 'cosech' series of modern vintage. By considering generalised functional equations for multidimensional zeta functions, we provide explicit expansions for the spatial potential and energy density of three-dimensional per...

in this article differential transformation method (dtms) has been used to solve neutral functional-differential equations with proportional delays. the method can simply be applied to many linear and nonlinear problems and is capable of reducing the size of computational work while still providing the series solution with fast convergence rate. exact solutions can also be obtained from the kno...