A note on a nonlinear functional equation and its application

This paper treats the following type of nonlinear functional equations φ ( x ) = m H ( x , φ [ g ( x ) ] ) H φ ( x , φ [ g ( x ) ] ) , where m is a real number, H ( x , y ) and g ( x ) are given functions, and φ ( x ) is an unknown function. Under certain conditions, we prove that such type of equations admits a unique continuous solution.