General Solutions and Generalized Hyer-Ulam Stability in Banach Spaces: A Direct Method Approach for a System of Functional Equations
Yagachitradevi G. *
Department of Mathematics, Siga College of Management and Computer Science, Villupuram - 605 601, Tamil Nadu, India.
Lakshminarayanan S.
Department of Mathematics, Arignar Anna Government Arts College, Villupuram - 605 602, Tamil Nadu, India.
Ravindiran P.
Department of Mathematics, Arignar Anna Government Arts College, Villupuram - 605 602, Tamil Nadu, India.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we have obtained general solutions and demonstrated the generalized Hyer-Ulam stability for the following system of functional equations in Banach spaces using the direct method:
(i) \(\mathit{h}\)(\(\mathit{u}\)1 + \(\mathit{u}\)2 + \(\mathit{u}\)3) + \(\mathit{h}\)(\(\mathit{u}\)1 + \(\mathit{u}\)2 - \(\mathit{u}\)3) + \(\mathit{h}\)(\(\mathit{u}\)1- \(\mathit{u}\)2 + \(\mathit{u}\)3) +\(\mathit{h}\)(\(\mathit{u}\)1 - \(\mathit{u}\)2 - \(\mathit{u}\)3)
= 4\(\mathit{h}\)(\(\mathit{u}\)1),
(ii) \(\mathit{h}\)(3\(\mathit{u}\)1 + 2\(\mathit{u}\)2 + \(\mathit{u}\)3) + \(\mathit{h}\)(3\(\mathit{u}\)1 + 2\(\mathit{u}\)2 - \(\mathit{u}\)3) + \(\mathit{h}\)(3\(\mathit{u}\)1 - 2\(\mathit{u}\)2 + \(\mathit{u}\)3) +\(\mathit{h}\) (3\(\mathit{u}\)1 - 2\(\mathit{u}\)2 - \(\mathit{u}\)3)
= 12\(\mathit{h}\)(\(\mathit{u}\)1),
(iii) \(\mathit{h}\)(\(\mathit{u}\)1 + 2\(\mathit{u}\)2 + 3\(\mathit{u}\)3) + \(\mathit{h}\)(\(\mathit{u}\)1 + 2\(\mathit{u}\)2 - 3\(\mathit{u}\)3) + \(\mathit{h}\)(\(\mathit{u}\)1 - 2\(\mathit{u}\)2 + 3\(\mathit{u}\)3) +\(\mathit{h}\)(\(\mathit{u}\)1 - 2\(\mathit{u}\)2 - 3\(\mathit{u}\)3)
= 4\(\mathit{h}\)(\(\mathit{u}\)1).
Keywords: Generalized hyers-ulam stability, additive functional equation, ulam stability, banach space