Non-Single-Valley Solutions for p -Order Feigenbaum’s Type Functional Equation f ( φ ( x ) ) = φ p ( f ( x ) )

This work deals with Feigenbaum’s functional equation f ( φ ( x ) ) = φ p ( f ( x ) ) , φ ( 0 ) = 1 , 0 ≤ φ ( x ) ≤ 1 , x ∈ [ 0 , 1 ] , where p ≥ 2 is an integer, φ p is the p -fold iteration of φ , and f ( x ) is a strictly increasing continuous function on [ 0 , 1 ] that satisfies f ( 0 ) = 0 , f ( x ) < x , ( x ∈ ( 0 , 1 ] ) . Using a constructive method, we discuss the existence of non-single-valley continuous solutions of the above equation.