Refined stability of additive and quadratic functional equations in modular spaces

Abstract

The purpose of this paper is to obtain refined stability results and alternative stability results for additive and quadratic functional equations using direct method in modular spaces.

Stability of a ramanujan type additive functional equation

“…On the other hand, many authors have investigated the stability using fixed point theorem of quasicontraction mappings in modular spaces without ∆ 2 − condition, which has been introduced by Khamsi [17]. Recently, the stability results of additive functional equations in modular spaces equipped with the Fatou property and ∆ 2 −condition were investigated in H. M. Kim, H. Y. Shin [16] and Sadeghi [34] who used Khamsis fixed point theorem. Also the stability of quadratic functional equations in modular spaces satisfying the Fatou property without using the ∆ 2 −condition was proved by Wongkum, Chaipunya and Kumam [38] .…”

Approximate Cubic Lie Derivations on ρ-Complete Convex Modular Algebras

“…In [15], the authors have proved the generalized Hyers-Ulam stability of quadratic functional equations via the extensive studies of fixed point theory in the framework of modular spaces whose modulars are convex and lower semicontinuous but do not satisfy any relatives of Δ 2 -conditions (see also [17,18]). Recently, the authors [14,19,20] have investigated stability theorems of functional equations in modular spaces without using the Fatou property and Δ 2 -condition. In 2001, J. M. Rassias [21] has introduced to study Hyers-Ulam stability of the following cubic functional equation:…”

Two multi-cubic functional equations and some results on the stability in modular spaces

“…An alternative generalized Hyers-Ulam stability theorem of a modified quadratic functional equation in a modular spaces using 3 -condition without the Fatou property on a modular function was presented in [20]. Furthermore, a refined stability result and alternative stability results for additive and quadratic functional equations using the direct method in modular spaces are given in [21]. For the stability of mixed additive-quadratic-cubic mappings in modular spaces which were recently studied, we refer to [25].…”