NEW QUADRATIC FUNCTIONAL EQUATION AND ITS (HURS)

Abstract

The primary subject in the stability of differential equations is to answer the question of when is it real that a mapping which roundly satisfies a differential equation must be close to an exact solution of the equation. For this reason, the Hyers-Ulam and Hyers-Ulam Rassias stability of differential equations is fundemantal. Currently, researchers have used various methods (open mapping, direct method, integral factor, fixed point method) to research that the Hyers-Ulam Rassias and Hyers-Ulam stability of differential equations. The direct method has been succesfully apllied for investigate of the Hyers-Ulam Rassias stability of many different functional differential equations. But it does not enough for some important cases. The second most popular method is the fixed point method.
In this study, we make an attemp to establish the Hyers-Ulam Rassias stability (HURS) of a new quadratic type functional equation (QFE)
g({ + + + ) + g({ 􀀀 􀀀 􀀀 ) = 4g({) + g( + ) + g( + + 2) 􀀀 g({ 􀀀 ) 􀀀 g({ + ); by direct method and fixed point method. We consider that this research will contribute to the related literature and it may be useful for authors studying on the Hyers-Ulam Stability of the quadratic functional differential equations.

Keywords

• HURS,
• QFE
• Fixed point method

References

1. [1] F. Skof, \Proprieta locali e approssimazione di operatori," Rend. Sem. Mat. Fis. Milano 53, 113-129 (1983).
2. [2] Lee Y.H, Jung S.M, Rassias T.M. Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation. Journal of Mathematical Inequalities. 2018; 12(1): 43-61.