A Constructive Lyapunov–Schmidt Reduction for a Semilinear Dirichlet Problem via Higher-Order Functional Spaces

We develop a constructive version of Lyapunov–Schmidt reduction by working in a sequence of higher-order functional spaces. The method is applied to the semilinear Dirichlet problem -u''-λ u+u³=0, u(0)=u(π)=0, which undergoes bifurcation from the trivial branch at the first eigenvalue λ₁=1. At this point the state-space linearization is singular, and the classical Lyapunov–Schmidt reduction only proves the existence of a local branch without providing an explicit construction. Our approach replaces the traditional implicit-function argument by an explicit recursive coefficient formula. At each order, the kernel amplitude is retained as a functional coordinate; the complementary component is solved analytically via the spectral gap; and the scalar bifurcation equation determines the next coefficient. This yields an infinite convergent power series representation of the solution branch: u(a)=∑q=0∞a2q+1w2q+1,λ(a)=1+∑q=1∞μ2qa2q, where (w2q+1,μ2q) are recursively determined by a linear complementary solve and a scalar projection. The construction is purely functional-analytic: it does not rely on elliptic-function closed forms. Convergence is proved by an analytic contraction argument in H²(0,π). Explicit tail estimates, residual bounds, and a posteriori Newton–Kantorovich certificates are provided. The first three nontrivial orders are computed explicitly: λ(a)=1+3/2²-3/32π²a⁴+15/512π³a⁶+O(a⁸). Comparison with the Jacobi elliptic-function solution confirms the coefficients to the computed order. The same quotient, splitting, and recursive structure extends to analytic semilinear equations on smooth bounded domains in Rⁿ in the Sobolev scale Hˢ(Ω)∩ H₀¹(Ω), $s>n/2$. The method converts Lyapunov–Schmidt reduction from an existence theorem into a constructive algorithm with explicit recursion, strict error control, and posterior verification, and it remains valid in higher dimensions where elliptic closed forms are unavailable. --- Keywords Constructive Lyapunov–Schmidt reduction; semilinear Dirichlet problem; bifurcation from simple eigenvalue; higher-order functional spaces; recursive coefficient expansion; analytic convergence; posterior error estimates; spectral gap; Sobolev spaces.