On the general solutions of three functional equations

The general solutions of the functional equations F ( p q ) = q α G ( p ) + p α H ( q ) + K ( p ) L ( q ) , F ( p q ) = q β G ( p ) + p α H ( q ) + c K ( p ) K ( q ) and f ( p q ) = q α f ( p ) + p α f ( q ) + c g ( p ) g ( q ) with g ( 1 ) = 0 and c , a given nonzero real constant, are obtained. Here F , G , H , K , L , f and g are real-valued functions each with domain I , the unit closed interval and 1 ≠ α > 0 , α ∈ R ; 1 ≠ β > 0 , β ∈ R .