1. Introduction

This work addresses the Cauchy problem for a second-order nonlinear evolution equation in a Hilbert space, representing an abstract generalization of the Ball integro-differential equation. The equation features self-adjoint positively defined operators in its main part, which may be unbounded. The primary objective is to develop and analyze a three-layer symmetrical semi-discrete scheme for approximating solutions to this problem, with nonlinear terms approximated using integral means.

The equation under consideration generalizes J.M. Ball's beam equation, which itself extended the Kirchhoff-type nonlinear equation for beams originally derived by S. Woinowsky-Krieger. Ball's contribution introduced damping terms to account for both external and internal damping effects. The investigation of Kirchhoff equations began with Bernstein's seminal work and has since been expanded by numerous researchers including Arosio, Panizzi, Berselli, Manfrin, D'Ancona, Spagnolo, Medeiros, Matos, Nishihara, and others.

Previous research has focused on various aspects including well-posedness, global solvability, and existence of low-regularity solutions for Kirchhoff-type equations. The abstract analogue considered in this work benefits from the participation of the square of the main operator in the linear part, which facilitates obtaining necessary a priori estimates.

2. Mathematical Formulation

The Cauchy problem is formulated in a Hilbert space H for the second-order nonlinear evolution equation:

u''(t) + A u(t) + M(||B u(t)||²) u(t) = f(t, u(t), u'(t)), t ∈ (0,T]

with initial conditions:

u(0) = u₀, u'(0) = u₁

where A and B are self-adjoint positively defined operators in H, potentially unbounded, and M is a nonlinear function representing the integral mean approximation. The term ||B u(t)||² denotes the square of the norm of the gradient in the abstract setting.

The operators A and B satisfy certain spectral conditions that ensure the well-posedness of the problem. The nonlinearity M is assumed to be locally Lipschitz continuous and to satisfy appropriate growth conditions to guarantee the existence and uniqueness of solutions.

3. Three-Layer Semi-Discrete Scheme

The proposed three-layer semi-discrete scheme for temporal discretization is given by:

(u^{n+1} - 2u^n + u^{n-1})/τ² + A u^n + M(||B u^n||²) u^n = f(t_n, u^n, (u^{n+1} - u^{n-1})/(2τ))

where τ represents the time step size, u^n approximates u(t_n) at time t_n = nτ, and the nonlinear terms involving the gradient are approximated using integral means.

The scheme is symmetrical and designed to preserve certain energy properties of the continuous problem. The approximation of nonlinear terms using integral means ensures better stability properties compared to straightforward linearization approaches.

The discrete initial conditions are:

u⁰ = u₀, u¹ = u₀ + τ u₁ + (τ²/2)(-A u₀ - M(||B u₀||²) u₀ + f(0, u₀, u₁))

4. Stability Analysis

The stability analysis proceeds in several stages. First, we establish uniform boundedness of the solution to the nonlinear discrete problem and its corresponding difference analogue of the first-order derivative.

Theorem 4.1 (Uniform Boundedness): Under appropriate assumptions on the operators A, B and the nonlinearity M, the solution {u^n} of the nonlinear discrete problem and the difference quotient {(u^{n+1} - u^n)/τ} are uniformly bounded with respect to the discretization parameter τ.

For the corresponding linear discrete problem, we derive high-order a priori estimates using two-variable Chebyshev polynomials. These estimates are crucial for establishing the stability of the nonlinear discrete problem.

Theorem 4.2 (Stability): The three-layer semi-discrete scheme is stable, meaning that small perturbations in the initial data and right-hand side lead to small changes in the numerical solution, with the amplification factor controlled by the discretization parameters.

The proof relies on energy estimates and the careful treatment of the nonlinear terms through the integral mean approximation.

5. Convergence Results

For smooth solutions, we provide error estimates for the approximate solution. The main convergence result is summarized in the following theorem:

Theorem 5.1 (Error Estimate): Assume the exact solution u(t) is sufficiently smooth. Then there exists a constant C > 0, independent of τ, such that the error e^n = u(t_n) - u^n satisfies:

max₀≤n≤N ||e^n|| ≤ C τ²

where N = T/τ is the number of time steps.

The proof utilizes consistency analysis, stability results, and the approximation properties of the integral mean for the nonlinear terms. The second-order accuracy is achieved due to the symmetry of the three-layer scheme and the careful treatment of the nonlinearities.

6. Iteration Method

An iteration method is applied to find an approximate solution for each temporal step. The iterative scheme for solving the nonlinear discrete problem at time step n+1 is given by:

(u^{n+1,k+1} - 2u^n + u^{n-1})/τ² + A u^n + M(||B u^n||²) u^n = f(t_n, u^n, (u^{n+1,k} - u^{n-1})/(2τ))

where k denotes the iteration index.

Theorem 6.1 (Convergence of Iteration Process): Under appropriate conditions on the time step τ and the Lipschitz constant of f, the iteration process converges to the unique solution of the nonlinear discrete problem at each time step.

The proof employs fixed-point arguments and utilizes the stability properties of the linearized operators.

7. Key Insights

8. Conclusion

This work presents a comprehensive analysis of a three-layer semi-discrete scheme for an abstract analogue of the Ball integro-differential equation. The main contributions include:

• Development of a symmetrical three-layer scheme with integral mean approximation for nonlinear terms
• Proof of uniform boundedness for the nonlinear discrete solution and its difference quotient
• Derivation of high-order a priori estimates using Chebyshev polynomials
• Establishment of stability for the nonlinear discrete problem
• Provision of error estimates for smooth solutions
• Proof of convergence for the iteration method used to solve the nonlinear system at each time step

The results demonstrate that the proposed scheme is effective for approximating solutions to this class of nonlinear evolution equations, with maintained stability and second-order accuracy. The abstract framework makes the results applicable to a wide range of concrete problems in mathematical physics described by similar integro-differential equations.

Future research directions include extension to fully discrete schemes, adaptive time-stepping strategies, and applications to specific physical models such as viscoelastic beams and plates.