The Stability of Functional Equations for Trigonometric and Similar Functions

  • Donald H. Hyers
  • George Isac
  • Themistocles M. Rassias
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 34)


In this section, we will be interested primarily in the stability of the following equations:

(6.1) \(f\left( {x + y} \right) + f\left( {x - y} \right) = 2f\left( x \right)f\left( y \right)\) (d’Alembert)

(6.2) \(f{\left( {\frac{{x + y}}{2}} \right)^2} = f\left( x \right)f\left( y \right) \) (Lobačevskiĭ)

(6.3a) \(f\left( {x + y} \right)f\left( {x - y} \right) = f{\left( x \right)^2} - {\left( y \right)^2}\) (sine equation)

(6.3b) \(f\left( {x + y} \right) = f\left( x \right)g\left( y \right) + g\left( x \right)f\left( y \right)\) (sine equation)

(6.4) \(f\left( {x + y} \right) = f\left( x \right)f\left( y \right) - g\left( x \right)g\left( y \right)\) (cosine equation)

where f and g may be defined on a group or semigroup with values in a field K which usually is the field of real or complex numbers. Methods of solving such equations are described in the books by Aczél (1966) and Aczél and Dhombres (1989).


Abelian Group Functional Equation Trigonometric Function Vector Space Versus Normed Algebra 
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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Donald H. Hyers
  • George Isac
    • 1
  • Themistocles M. Rassias
    • 2
  1. 1.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada
  2. 2.Department of MathematicsNational Technical University of AthensAthensGreece

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