Acta mathematica scientia,Series B ›› 2015, Vol. 35 ›› Issue (6): 1501-1510.doi: 10.1016/S0252-9602(15)30071-0

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QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN BANACH SPACES

Choonkil PARK   

  1. Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Republic of Korea
  • Received:2014-04-30 Online:2015-11-01 Published:2015-11-01
  • Supported by:

    This work was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF- 2012R1A1A2004299).

Abstract:

In this paper, we solve the quadratic ρ-functional inequalities
||f(x + y) + f(x y) 2f(x)2f(y)||≤||ρ(2f((x+y)/2)+2f((x-y)/2)f(x) f(y)||, (0.1)
where ρ is a fixed complex number with |ρ|< 1, and
||2f((x+y)/2)+2f((x-y)/2)f(x) f(y)||≤||ρf(x + y) + f(x y) 2f(x)2f(y)||, (0.2)
where ρ is a fixed complex number with |ρ|< 1/2 .Using the direct method, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces and prove the Hyers-Ulam stability of quadratic -functional equations associated with the quadratic ρ-functional inequalities (0.1)and (0.2) in complex Banach spaces.

Key words: Hyers-Ulam stability, quadratic ρ-functional equation, quadratic ρ-functional inequality, complex Banach space

CLC Number: 

  • 39B62
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