Locally Expansive Solutions for a Class of Iterative Equations

Iterative equations which can be expressed by the following form f n(x) = H(x, f(x), f 2(x),…, f n−1(x)), where n ≥ 2, are investigated. Conditions for the existence of locally expansive C 1 solutions for such equations are given.

As a natural generalization of the problem of iterative roots, iterative equations of the following form are known as polynomial-like iterative equations. Here, ≥ 2 is an integer, ∈ R ( = 1, 2, . . . , ), : → R is a given mapping, and : → is unknown. As mentioned in [9,10], polynomial-like iterative equations are important not only in the study of functional equations but also in the study of dynamical systems. For instance, such equations are encountered in the discussion on transversal homoclinic intersection for diffeomorphisms [11], normal form of dynamical systems [12], and dynamics of a quadratic mapping [13]. Some problems of invariant curves for dynamical systems also lead to such iterative equations [14].
For the case that is linear, where (1) can be written as many results [15][16][17] have been given to present all of its continuous solutions. Conditions that ensure the uniqueness of such solutions are also given by [18,19]. For the case that is nonlinear, the basic problems such as existence, uniqueness, and stability cannot be solved easily.
In 1986, Zhang [20], under the restriction that 1 ̸ = 0, constructed an interesting operator called "structural operator" for (1) and used the fixed point theory in Banach space to get the solutions of (1). Hence, he overcame the difficulties encountered by the formers. By means of this method, Zhang and Si made a series of works concerning these qualitative problems, such as [21][22][23][24]. After that, (1) and other type equations were discussed extensively by employing this idea (see [25][26][27][28][29][30][31] and references therein).
On the other hand, great efforts have been made to solve the "leading coefficient problem" which was raised by [32,33] as an open problem. The essence of solving this problem is to abolish the technical restriction 1 ̸ = 0 and discuss (1) under the more natural assumption ̸ = 0. As mentioned in [34,35], a mapping is said to be locally expansive (resp., locally contractive) at its fixed point 0 , if | ( 0 )| > 1 (resp., 0 < | ( 0 )| < 1). In 2004, Zhang [35] gave positive answers to this problem in local 1 solutions in some cases of coefficients, but this paper only discussed the locally expansive case and the nonhyperbolic case. In 2009, Chen and Zhang [34] gave positive answers to this problem with more combinations between locally expansive mappings and 2 The Scientific World Journal locally contractive ones and combinations between increasing mappings and decreasing ones. The main tools used in the two papers above are Schröder transformation and Schauder fixed point theorem. In 2012, J. M. Chen and L. Chen [36] consider the locally contractive 1 solutions of the iterative equation ( , ( ), . . . , ( )) = ( ), and some results on locally contractive solutions of [34] were generalized. In 2007, Xu and Zhang [37] answered this problem by constructing 0 solutions of (1). Their strategy is to construct the solutions piece by piece via a recursive formula obtained form (1). Following this idea, global increasing and decreasing solutions [38,39] were also investigated.
Motivated by the above results, we will consider the existence of locally expansive 1 solutions for the iterative equation of the following form: where ≥ 2. Some results on locally expansive solutions in [34] are generalized.
The assumption on is where is an interval to be determined.
Assumptions on are Let > 0, > 0, and > 0 be three constants, and define a set The set A( , , ) is nonempty and is a convex compact subset of 1 ([− , ], R). For ∈ R, | | > 1, ∈ A( , , ), and ∈ H, we define two functions as follows: If the solution of (3) can be expressed as ( ) = ( ( −1 ( ))) by the Schröder transformation, where is a constant to be determined, then (3) can be reduced to the following auxiliary equation: If function is a solution of (3), then we can differentiate the equation. In fact, we can get that the derivative (0) is a zero of the following polynomial: (8) We refer to the polynomial (8) as the characteristic polynomial of (3).
Finally, we give a basic lemma.
Proof. If is real and (7) has a local 1 solution with (0) = 0 and (0) ̸ = 0, then by differentiating the equation, we can see that is a root of characteristic polynomial (8).

Proof of Theorems 2-4.
Let be the solution of (7) Therefore, is a locally expansive 1 solution of (3).
Obviously, ( 0 , 1 , 2 ) = 2sin( 0 ) + sin ( 2 ). It is easy to verify that satisfy the assumptions of Theorem 2. This equation has at least one locally expansive increasing 1 solution in a neighborhood of 0.
Example 2. Consider the following equation: Obviously, ( 0 , 1 , 2 ) = −2 sin( 0 ) − sin ( 2 ). It is easy to verify that satisfy the assumptions of Theorem 3. This equation has at least one locally expansive decreasing 1 solution in a neighborhood of 0.