Higher regularity of solutions of an iterative functional equation

1. Problem Statement

The paper investigates the existence and regularity of solutions for a class of second-order iterative functional equations of the form:
$$\phi^2(x) = h(\phi(f(x))) + g(x), \quad x \in \mathbb{R}$$
where:

• $\phi$ is the unknown function.
• $\phi^2(x)$ denotes the second iterate $\phi(\phi(x))$.
• $h, f, g$ are given functions.

Context: Previous studies (e.g., by Brillouët-Belluot, Zhang, Tang) had established the existence of continuous or Lipschitz solutions for specific cases of this equation. However, the existence of $C^n$ solutions (functions with continuous derivatives up to order $n$) with bounded derivatives for $n \geq 2$ remained an open challenge, particularly due to the nonlinearity of the iteration operator and the complexity of higher-order derivatives in composite functions.

2. Methodology

The authors employ a combination of advanced functional analysis tools and combinatorial calculus:

• Fiber Contraction Theorem: This is the core fixed-point technique used. The authors construct a "bundle map" $\Gamma$ acting on a product space of function spaces. The theorem allows for the establishment of a unique fixed point (the solution) by proving that the map is a contraction in specific "fibers" (variables) while maintaining continuity in others.
• Faà di Bruno's Formula: To handle the higher-order derivatives of the composite function $\phi(f(x))$, the authors utilize Faà di Bruno's formula. This provides the explicit expression for the $n$-th derivative of a composite function, which is essential for defining the iterative steps for the derivatives of the solution.
• Function Space Construction: The problem is reformulated in Banach spaces of bounded continuous functions ($C^0_b$) and Lipschitz functions ($C^{0,1}_b$). Specifically, they define spaces $C^0_{b, \rho}(R)$ (bounded functions with norm $\leq \rho$) and construct a mapping on the product space:
  $$Y = C^{0,1}_b(R; L) \times C^0_{b, \rho_1}(R) \times \dots \times C^0_{b, \rho_n}(R)$$
  where the components represent the solution $\phi$ and its derivatives $\phi', \dots, \phi^{(n)}$.

3. Key Contributions and Technical Setup

A. Reformulation of the Equation

The original equation is transformed into an equivalent fixed-point problem:
$$\phi(x) = h^{-1}(\phi^2(f^{-1}(x)) - g(f^{-1}(x)))$$
This requires proving that $h$ and $f$ are bijections, which is established using the condition that their derivatives are bounded away from zero (specifically $|h'(x)| \geq m > 1$ and $|f'(x)| \geq \alpha > 0$).

B. Construction of the Bundle Map

The authors define a map $\Gamma$ acting on the tuple $(\phi, \phi_1, \dots, \phi_n)$, where $\phi_k$ represents the $k$-th derivative.
$$\Gamma(\phi, \phi_1, \dots, \phi_n) = (\Lambda(\phi), \Psi_1(\phi, \phi_1), \dots, \Psi_n(\phi, \dots, \phi_n))$$

• $\Lambda(\phi)$: The operator for the function itself.
• $\Psi_k$: The operators for the $k$-th derivative, derived using Faà di Bruno's formula. These operators express $\phi^{(k)}$ in terms of lower-order derivatives and the given functions.

C. Conditions for Existence

The paper establishes rigorous constraints on the constants $m, \alpha, \beta$ (bounds on derivatives of $h, f, g$) to ensure the contraction properties hold:

1. Boundedness of $g$: The derivatives of $g$ up to order $n$ must be uniformly bounded by a sufficiently small constant $\beta$.
2. Lipschitz Constant $L$: The solution $\phi$ must be Lipschitz with a constant $L$ satisfying a specific quadratic inequality involving $m, \alpha,$ and $\beta$ (Equation 3.6).
3. Derivative Bounds $\rho_k$: The bounds for the $k$-th derivative must satisfy a system of inequalities (Equations 3.7–3.9) to ensure the map $\Psi_k$ maps the space into itself and acts as a contraction.

4. Main Results

Theorem 1: Under the assumptions that $h, f, g$ are of class $C^n$, $|h'(x)| \geq m > 1$, $|f'(x)| \geq \alpha > 0$, and the derivatives of $g$ are sufficiently small (bounded by $\beta$), the functional equation admits a bounded $C^n$ solution $\phi$. Furthermore, the derivatives $\phi', \phi'', \dots, \phi^{(n)}$ are all bounded on $\mathbb{R}$.

Key Technical Steps in the Proof:

1. Well-definedness: Proved that the operators $\Lambda$ and $\Psi_k$ map the constructed product space into itself.
2. Contraction:
   • $\Lambda$ is shown to be a contraction on the space of Lipschitz functions.
   • $\Psi_k$ is shown to be a uniform contraction with respect to the $k$-th variable (the $k$-th derivative) while treating lower-order variables as parameters.
3. Continuity: Proved that the maps are continuous with respect to the previous variables, satisfying the conditions of the Fiber Contraction Theorem.
4. Convergence: The sequence of iterates converges to a unique fixed point $(\phi^*, \phi_1^*, \dots, \phi_n^*)$, where $\phi^*$ is the solution and $\phi_k^* = (\phi^*)^{(k)}$.

5. Significance and Impact

• Extension of Regularity: This work significantly extends previous results which were limited to continuous or Lipschitz solutions. It provides a constructive proof for the existence of smooth ($C^n$) solutions for a broad class of iterative equations.
• Handling Nonlinearity: By successfully applying the Fiber Contraction Theorem to higher-order derivatives via Faà di Bruno's formula, the paper overcomes the technical difficulty of the nonlinearity inherent in the iteration operator $\phi^2(x)$.
• Boundedness: The result guarantees not just the existence of smooth solutions, but that these solutions and their derivatives are globally bounded, a crucial property for stability analysis in dynamical systems.
• Applicability: The paper provides a concrete example (Equation 1.8: $\phi^2(x) = 2\phi(3x) + \frac{\sin x}{100}$) demonstrating how the theoretical conditions can be verified for specific functions, validating the practical utility of the theorem.

In summary, Feng and Tang provide a robust theoretical framework for establishing high-order regularity in iterative functional equations, bridging the gap between existence theory and smoothness requirements in nonlinear functional analysis.