• Title/Summary/Keyword: additive functional inequality

Search Result 23, Processing Time 0.024 seconds

ON AN ADDITIVE FUNCTIONAL INEQUALITY IN NORMED MODULES OVER A $C^*$-ALGEBRA

  • An, Jong-Su
    • The Pure and Applied Mathematics
    • /
    • v.15 no.4
    • /
    • pp.393-400
    • /
    • 2008
  • In this paper, we investigate the following additive functional inequality (0.1) ||f(x)+f(y)+f(z)+f(w)||${\leq}$||f(x+y)+f(z+w)|| in normed modules over a $C^*$-algebra. This is applied to understand homomor-phisms in $C^*$-algebra. Moreover, we prove the generalized Hyers-Ulam stability of the functional inequality (0.2) ||f(x)+f(y)+f(z)f(w)||${\leq}$||f(x+y+z+w)||+${\theta}||x||^p||y||^p||z||^p||w||^p$ in real Banach spaces, where ${\theta}$, p are positive real numbers with $4p{\neq}1$.

  • PDF

THE GENERALIZED HYERS-ULAM STABILITY OF ADDITIVE FUNCTIONAL INEQUALITIES IN NON-ARCHIMEDEAN 2-NORMED SPACE

  • Kim, Chang Il;Park, Se Won
    • Korean Journal of Mathematics
    • /
    • v.22 no.2
    • /
    • pp.339-348
    • /
    • 2014
  • In this paper, we investigate the solution of the following functional inequality $${\parallel}f(x)+f(y)+f(az),\;w{\parallel}{\leq}{\parallel}f(x+y)-f(-az),\;w{\parallel}$$ for some xed non-zero integer a, and prove the generalized Hyers-Ulam stability of it in non-Archimedean 2-normed spaces.

A FIXED POINT APPROACH TO STABILITY OF ADDITIVE FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES

  • Kim, Chang Il;Park, Se Won
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.29 no.3
    • /
    • pp.453-464
    • /
    • 2016
  • In this paper, we investigate the solution of the following functional inequality $$N(f(x)+f(y)+f(z),t){\geq}N(f(x+y+z),mt)$$ for some fixed real number m with $\frac{1}{3}$ < m ${\leq}$ 1 and using the fixed point method, we prove the generalized Hyers-Ulam stability for it in fuzzy Banach spaces.

ADDITIVE ρ-FUNCTIONAL INEQUALITIES IN β-HOMOGENEOUS F-SPACES

  • LEE, HARIN;CHA, JAE YOUNG;CHO, MIN WOO;KWON, MYUNGJUN
    • The Pure and Applied Mathematics
    • /
    • v.23 no.3
    • /
    • pp.319-328
    • /
    • 2016
  • In this paper, we solve the additive ρ-functional inequalities (0.1) ||f(2x-y)+f(y-x)-f(x)|| $\leq$ ||${\rho}(f(x+y)-f(x)-f(y))$||, where ρ is a fixed complex number with |ρ| < 1, and (0.2) ||f(x+y)-f(x)-f(y)|| $\leq$ ||${\rho}(f(2x-y)-f(y-x)-f(x))$||, where ρ is a fixed complex number with |ρ| < $\frac{1}{2}$. Using the direct method, we prove the Hyers-Ulam stability of the additive ρ-functional inequalities (0.1) and (0.2) in β-homogeneous F-spaces.

ADDITIVE ρ-FUNCTIONAL INEQUALITIES

  • LEE, SUNG JIN;LEE, JUNG RYE;SEO, JEONG PIL
    • The Pure and Applied Mathematics
    • /
    • v.23 no.2
    • /
    • pp.155-162
    • /
    • 2016
  • In this paper, we solve the additive ρ-functional inequalities (0.1)${\parallel}f(x+y)+f(x-y)-2f(x){\parallel}$ $\leq$ ${\parallel}{\rho}(2f(\frac{x+y}{2})+f(x-y)-2f(x)){\parallel}$, where ρ is a fixed complex number with |ρ| < 1, and (0.2) ${\parallel}2f(\frac{x+y}{2})+f(x-y)-2f(x)){\parallel}$ $\leq$ ${\parallel}{\rho}f(x+y)+f(x-y)-2f(x){\parallel}$, where ρ is a fixed complex number with |ρ| < 1. Furthermore, we prove the Hyers-Ulam stability of the additive ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces.

APPROXIMATION OF CAUCHY ADDITIVE MAPPINGS

  • Roh, Jai-Ok;Shin, Hui-Joung
    • Bulletin of the Korean Mathematical Society
    • /
    • v.44 no.4
    • /
    • pp.851-860
    • /
    • 2007
  • In this paper, we prove that a function satisfying the following inequality $${\parallel}f(x)+2f(y)+2f(z){\parallel}{\leq}{\parallel}2f(\frac{x}{2}+y+z){\parallel}+{\epsilon}({\parallel}x{\parallel}^r{\cdot}{\parallel}y{\parallel}^r{\cdot}{\parallel}z{\parallel}^r)$$ for all x, y, z ${\in}$ X and for $\epsilon{\geq}0$, is Cauchy additive. Moreover, we will investigate for the stability in Banach spaces.

ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES

  • YUN, SUNGSIK;LEE, JUNG RYE;SHIN, DONG YUN
    • The Pure and Applied Mathematics
    • /
    • v.23 no.3
    • /
    • pp.247-263
    • /
    • 2016
  • Let $M_{1}f(x,y):=\frac{3}{4}f(x+y)-\frac{1}{4}f(-x-y)+\frac{1}{4}f(x-y)+\frac{1}{4}f(y-x)-f(x)-f(y)$, $M_{2}f(x,y):=2f(\frac{x+y}{2})+f(\frac{x-y}{2})+f(\frac{y-x}{2})-f(x)-f(y)$. Using the direct method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities (0.1) $N(M_{1}f(x,y),t){\geq}N({\rho}M_{2}f(x,y),t)$ where ρ is a fixed real number with |ρ| < 1, and (0.2) $N(M_{2}f(x,y),t){\geq}N({\rho}M_{1}f(x,y),t)$ where ρ is a fixed real number with |ρ| < $\frac{1}{2}$.

ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN FUZZY BANACH SPACES

  • LEE, SUNG JIN;SEO, JEONG PIL
    • The Pure and Applied Mathematics
    • /
    • v.23 no.2
    • /
    • pp.163-179
    • /
    • 2016
  • Let $M_1f(x,y):=\frac{3}{4}f(x+y)-\frac{1}{4}f(-x-y)+\frac{1}{4}(x-y)+\frac{1}{4}f(y-x)-f(x)-f(y)$, $M_2f(x,y):=2f(\frac{x+y}{2})+f(\frac{x-y}{2})+f(\frac{y-x}{2})-f(x)-f(y)$ Using the direct method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities (0.1) $N(M_1f(x,y)-{\rho}M_2f(x,y),t){\geq}\frac{t}{t+{\varphi}(x,y)}$ and (0.2) $N(M_2f(x,y)-{\rho}M_1f(x,y),t){\geq}\frac{t}{t+{\varphi}(x,y)}$ in fuzzy Banach spaces, where ρ is a fixed real number with ρ ≠ 1.