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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 allows to reduce infinite-dimensional equations in Banach spaces to finite-dimensional equations.

Problem setup

Let

$f(x,\lambda )=0$

be the given nonlinear equation, $X,\mathrm{\Lambda },$ and $Y$ are Banach spaces ($\mathrm{\Lambda }$ is the parameter space). $f(x,\lambda )$ is the $C^p$-map from a neighborhood of some point $(x_0,{\lambda }_0)\in X\times \mathrm{\Lambda }$ to $Y$ and the equation is satisfied at this point

$f(x_0,{\lambda }_0)=0.$

For the case when the linear operator $f_x(x,\lambda )$ is invertible, the implicit function theorem assures that there exists a solution $x(\lambda )$ satisfying the equation $f(x(\lambda ),\lambda )=0$ at least locally close to ${\lambda }_0$.

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

Assumptions

One assumes that the operator $f_x(x,\lambda )$ is a Fredholm operator.

$\ker f_x(x_0,{\lambda }_0)=X_1$ and $X_1$ has finite dimension.

The range of this operator $\mathrm{r}\mathrm{a}\mathrm{n}{f}_x(x_0,{\lambda }_0)=Y_1$ has finite co-dimension and is a closed subspace in $Y$.

Without loss of generality, one can assume that $(x_0,{\lambda }_0)=(0,0).$

Lyapunov–Schmidt construction

Let us split $Y$ into the direct product $Y=Y_1\oplus Y_2$, where $\dim Y_2<\mathrm{\infty }$.

Let $Q$ be the projection operator onto $Y_1$.

Let us consider also the direct product $X=X_1\oplus X_2$.

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

$Qf(x,\lambda )=0$

$(I-Q)f(x,\lambda )=0$

Let $x_1\in X_1$ and $x_2\in X_2$, then the first equation

$Qf(x_1+x_2,\lambda )=0$

can be solved with respect to $x_2$ by applying the implicit function theorem to the operator

$Qf(x_1+x_2,\lambda ):X_2\times (X_1\times \mathrm{\Lambda })\to Y_1$

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

Thus, there exists a unique solution $x_2(x_1,\lambda )$ satisfying

$Qf(x_1+x_2(x_1,\lambda ),\lambda )=0$

Now substituting $x_2(x_1,\lambda )$ into the second equation, one obtains the final finite-dimensional equation

$(I-Q)f(x_1+x_2(x_1,\lambda ),\lambda )=0$

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 :$\lambda$

References

• Louis Nirenberg, Topics in nonlinear functional analysis, New York Univ. Lecture Notes, 1974.