Example 6. Find the maximum

$$
|(K \varphi, \varphi)|=\left|\int_{0}^{\pi} \int_{0}^{\pi} K(x, t) \varphi(x) \varphi(t) d x d t\right|
$$

under the condition

$$
(\varphi, \varphi)=\int_{0}^{\pi} \varphi^{2}(x) d x=1
$$

if

$$
K(x, t)=\cos x \cos 2 t+\cos t \cos 2 x+1 .
$$
Solution. Solving the homogeneous integral equation

$$
\varphi(x)=\lambda \int_{0}^{\pi}(\cos x \cos 2 t+\cos t \cos 2 x+1) \varphi(t) d t
$$

as an equation with a degenerate kernel, we find the characteristic numbers $\lambda_{1}=\frac{1}{\pi}$ and $\lambda_{2,3}= \pm \frac{2}{\pi}$ and the corresponding eigenfunctions $\varphi_{1}(x)=C_{1}, \varphi_{2}(x) \stackrel{\pi}{=}$ $C_{2}(\cos x+\cos 2 x), \varphi_{3}(x)=C_{3}(\cos x-\cos 2 x)$, where $C_{1}, C_{2}$ and $C_{3}$ are arbitrary constants.

The smallest characteristic number in absolute value is $\lambda_{1}=\frac{1}{\pi}$, to which the eigenfunction $\varphi_{1}(x)=C_{1}$ corresponds. From the normalization condition $(\varphi, \varphi)=1$ we find $C_{1}= \pm \frac{1}{\sqrt{2 \pi}}$. Therefore,

$$
\max \left|\int_{0}^{\pi} \int_{0}^{\pi}(\cos x \cos 2 t+\cos t \cos 2 x+1) \varphi(x) \varphi(t) d x d t\right|=2 \pi
$$

and it is achieved on the functions $\varphi(x)= \pm \frac{1}{\sqrt{2 \pi}}$.

## Problem for Independent Solution

159. Find the maximum

$$
\left|\int_{a}^{b} \int_{a}^{b} K(x, t) \varphi(x) \varphi(t) d x d t\right|
$$

under the condition that

$$
\int_{a}^{b} \varphi^{2}(x) d x=1
$$

if:
a) $K(x, t)=x t, \quad 0 \leqslant x, t \leqslant 1$;
b) $K(x, t)=x t+x^{2} t^{2},-1 \leqslant x, t \leqslant 1$;
c) $K(x, t)= \begin{cases}(x+1) t, & 0 \leqslant x \leqslant t, \\ (t+1) x, & t \leqslant x \leqslant 1 .\end{cases}$

## § 11. Solution of Homogeneous Integral Equations with a Degenerate Kernel

A homogeneous integral equation with a degenerate kernel

$$
\varphi(x)-\lambda \int_{a}^{b}\left[\sum_{k=1}^{n} a_{k}(x) b_{k}(t)\right] \varphi(t) d t=0
$$

when the parameter $\lambda$ is not its characteristic number (i.e., $\Delta(\lambda) \neq 0$), has a unique zero solution: $\varphi(x) \equiv 0$. If, however, $\lambda$ is a characteristic number (i.e., $\Delta(\lambda)=0$), then in addition to the zero solution, equation (1) has non-zero solutions, which are the eigenfunctions corresponding to this characteristic number. The general solution of the homogeneous equation (1) is obtained as a linear combination of these eigenfunctions.