In mathematics, the Lyapunov–Schmidt reduction or Lyapunov–Schmidt construction is used to study solutions to nonlinear equations in the case when the implicit function theorem does not work. It permits the reduction of infinite-dimensional equations in Banach spaces to finite-dimensional equations. It is named after Aleksandr Lyapunov and Erhard Schmidt.
Problem setup Let
f ( x , λ ) = 0 {\displaystyle f(x,\lambda )=0\,}
be the given nonlinear equation, X , Λ , {\displaystyle X,\Lambda ,} and Y {\displaystyle Y} are Banach spaces ( Λ {\displaystyle \Lambda } is the parameter space). f ( x , λ ) {\displaystyle f(x,\lambda )} is the
C p {\displaystyle C^{p}} -map from a neighborhood of some point ( x 0 , λ 0 ) ∈ X × Λ {\displaystyle (x_{0},\lambda _{0})\in X\times \Lambda } to
Y {\displaystyle Y} and the equation is satisfied at this point
f ( x 0 , λ 0 ) = 0. {\displaystyle f(x_{0},\lambda _{0})=0.}
For the case when the linear operator f x ( x , λ ) {\displaystyle f_{x}(x,\lambda )} is invertible, the implicit function theorem assures that there exists a solution x ( λ ) {\displaystyle x(\lambda )} satisfying the equation f ( x ( λ ) , λ ) = 0 {\displaystyle f(x(\lambda ),\lambda )=0} at least locally close to λ 0 {\displaystyle \lambda _{0}} . In the opposite case, when the linear operator f x ( x , λ ) {\displaystyle f_{x}(x,\lambda )} is non-invertible, the Lyapunov–Schmidt reduction can be applied in the following way.
Assumptions One assumes that the operator f x ( x , λ ) {\displaystyle f_{x}(x,\lambda )} is a Fredholm operator, ker f x ( x 0 , λ 0 ) = X 1 , {\displaystyle \ker f_{x}(x_{0},\lambda _{0})=X_{1},} and X 1 {\displaystyle X_{1}} has finite dimension. The range of this operator r a n f x ( x 0 , λ 0 ) = Y 1 {\displaystyle \mathrm {ran} f_{x}(x_{0},\lambda _{0})=Y_{1}} has finite co-dimension and is a closed subspace in Y {\displaystyle Y} . Without loss of generality, one can assume that ( x 0 , λ 0 ) = ( 0 , 0 ) . {\displaystyle (x_{0},\lambda _{0})=(0,0).}
Lyapunov–Schmidt construction Let us split X {\displaystyle X} and Y {\displaystyle Y} into the direct sums X = X 1 ⊕ X 2 {\displaystyle X=X_{1}\oplus X_{2}} and Y = Y 1 ⊕ Y 2 {\displaystyle Y=Y_{1}\oplus Y_{2}} , respectively, where dim X 2 , dim Y 2 < ∞ {\displaystyle \dim X_{2},\dim Y_{2}<\infty } . Also, let Q {\displaystyle Q} be the projection operator onto Y 1 {\displaystyle Y_{1}} . Applying the operators Q {\displaystyle Q} and I − Q {\displaystyle I-Q} to the original equation and writing x = x 1 + x 2 {\displaystyle x=x_{1}+x_{2}} , one obtains the equivalent system
… excerpt ends here. Continue reading the full article.
