LAUSR

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.