Lyapunov–Schmidt reduction

Problem setup

Let

be the given nonlinear equation, X , Λ , and Y are Banach spaces ( Λ is the parameter space). f ( x , λ ) is the C p -map from a neighborhood of some point ( x 0 , λ 0 ) ∈ X × Λ to Y and the equation is satisfied at this point

For the case when the linear operator f x ( x , λ ) is invertible, the implicit function theorem assures that there exists a solution x ( λ ) satisfying the equation f ( x ( λ ) , λ ) = 0 at least locally close to λ 0 .

In the opposite case, when the linear operator f x ( x , λ ) is non-invertible, the Lyapunov–Schmidt reduction can be applied in the following way.

Assumptions

One assumes that the operator f x ( x , λ ) is a Fredholm operator.

ker ⁡ f x ( x 0 , λ 0 ) = X 1 and X 1 has finite dimension.

The range of this operator r a n f x ( x 0 , λ 0 ) = Y 1 has finite co-dimension and is a closed subspace in Y .

Without loss of generality, one can assume that ( x 0 , λ 0 ) = ( 0 , 0 ) .

Lyapunov–Schmidt construction

Let us split Y into the direct product Y = Y 1 ⊕ Y 2 , where dim ⁡ Y 2 < ∞ .

Let Q be the projection operator onto Y 1 .

Let us consider also the direct product X = X 1 ⊕ X 2 .

Applying the operators Q and I − Q to the original equation, one obtains the equivalent system

Let x 1 ∈ X 1 and x 2 ∈ X 2 , then the first equation

can be solved with respect to x 2 by applying the implicit function theorem to the operator

(now the conditions of the implicit function theorem are fulfilled).

Thus, there exists a unique solution x 2 ( x 1 , λ ) satisfying

Now substituting x 2 ( x 1 , λ ) into the second equation, one obtains the final finite-dimensional equation

Indeed, the last equation is now finite-dimensional, since the range of ( I − Q ) is finite-dimensional. This equation is now to be solved with respect to x 1 , which is finite-dimensional, and parameters : λ