FUNCTIONAL INEQUALITIES IN PARANORMED SPACES

Abstract

Abstract. In this paper, we investigate additive functional in-equalities in paranormed spaces. Furthermore, we prove the Hyers-Ulam stability of additive functional inequalities in paranormedspaces. 1. Introduction and preliminariesThe concept of statistical convergence for sequences of real numberswas introduced by Fast [3] and Steinhaus [26] independently and sincethen several generalizations and applications of this notion have beeninvestigated by various authors (see [5, 14, 16, 17, 25]). This notion wasdeflned in normed spaces by Kolk [15].We recall some basic facts concerning Fr¶ e chet spaces.Definition 1.1. [28] Let X be a vector space. A paranorm P : X ! [0 ;1 ) is a function on X such that(1) P (0) = 0;(2) P ( ix ) = P ( x ) ;(3) P ( x + y ) • P ( x )+ P ( y ) (triangle inequality)(4) If ft n g is a sequence of scalars with t n ! t and fx n g ‰ X with P ( x n ix ) ! 0, then P ( t n x n itx ) ! 0 (continuity of multiplication).The pair ( X;P ) is called a paranormed space if P