Stability of some functional equations related to distance measures—I

The present work aims to study the stability of the following three functional equations: (i) f ( p r , q s ) + f ( p s , q r ) = f ( p , q ) f ( r , s ) , (ii) f ( p r , q s ) + f ( p s , q r ) = f ( p , q ) g ( r , s ) , and (iii) f ( p r , q s ) + f ( p s , q r ) = g ( p , q ) f ( r , s ) for all p , q , r , s ∈ ( 0 , 1 ) . The first functional equation arises in the characterization of symmetrically compositive sum form distance measures.