Classical stabilities of multiplicative inverse difference and adjoint functional equations

Abstract

Abstract The aim of this present article is to investigate various classical stability results of the multiplicative inverse difference and adjoint functional equations m d ( r s r + s ) − m d ( 2 r s r + s ) = 1 2 [ m d ( r ) + m d ( s ) ] $$ m_{d} \biggl(\frac{rs}{r+s} \biggr)-m_{d} \biggl( \frac{2rs}{r+s} \biggr)= \frac{1}{2} \bigl[m_{d}(r)+m_{d}(s) \bigr] $$ and m a ( r s r + s ) + m a ( 2 r s r + s ) = 3 2 [ m a ( r ) + m a ( s ) ] $$ m_{a} \biggl(\frac{rs}{r+s} \biggr)+m_{a} \biggl( \frac{2rs}{r+s} \biggr)= \frac{3}{2} \bigl[m_{a}(r)+m_{a}(s) \bigr] $$ in the framework of non-zero real numbers. A proper counter-example is illustrated to prove the failure of the stability results for control cases. The relevance of these functional equations in optics is also discussed.

Keywords

• Multiplicative inverse function
• Reciprocal functional equation
• Ulam–Hyers stability
• Real numbers