Math Problems Vault

Problem 11.90

Prove that the general solution of the equation

x(t)-\int _{a}^{b}\left(\sum _{i = 1}^{n} p_{i}(t) q_{i}(s)\right) x(s) d s = y(t)

(with respect to the function $x$ , with $y$ known) has the form

x(t) = \sum _{i = 1}^{n} c_{i} p_{i}(t)+y(t)

here $c_{i} \in \mathbb {C}, 1 \leqslant i \leqslant n$ , are solutions of the system of equations

\sum _{j = 1}^{n} a_{i j} c_{j} = b_{i}, \quad i = 1, \ldots , n

where the coefficients $a_{i j}$ and $b_{i}$ depend on $y, p_{i}$ and $q_{i}$ . State the Fredholm alternative in terms of the matrix $a_{i j}$ and the right-hand side $b_{i}$ . Assume the functions $p_{i}$ to be linearly independent.

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Problem 11.91

(a)(b)

In the space $C[0,1]$ , find the solution of the following integral equations for all values of the complex parameters $\lambda , \alpha , \beta$ and $\gamma$ :

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(a)

$x(t)-\lambda \int_{0}^{1} t s x(s) d s = \alpha +\beta t+\gamma t^{2}$ ;

(b)

$x(t)-\lambda \int_{0}^{1}\left(t s+t^{2} s^{2}\right) x(s) d s = \alpha t$ .

Problem 11.92

(a)(b)

In the space $C[0, \pi ]$ , find the solution of the following integral equations for all values of the complex parameters $\lambda , \alpha$ and $\beta$ :

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(a)

$x(t)-\lambda \int_{0}^{\pi } \cos (t-s) x(s) d s = \alpha \sin t+\beta \cos t ;$

(b)

$x(t)-\lambda \int_{0}^{\pi } \cos (t+s) x(s) d s = \alpha \sin t+\beta \cos t$ .

Problem 11.93

Under what conditions on the function $y \in L_{2}[0, \pi ]$ and the parameter $\lambda \in \mathbb {C}$ does the equation

x(t)+\lambda \int _{0}^{\pi } \sin (t-s) x(s) d s = y(t)

have a unique solution in the space $L_{2}[0, \pi ]$ ?

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Problem 11.94

Find all $\beta \in \mathbb {R}$ for which the equation

x(t)+\int _{0}^{1}(1+\alpha t s) x(s) d s = \beta +t^{2}

is solvable in $L_{2}[0,1]$ for every $\alpha \in \mathbb {R}$ .

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Problem 11.95

(a)(b)

In the space $L_{2}[0,1]$ , find the solution of the following integral equations for all values of the complex parameters $\lambda , a$ and $b$ :

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(a)

$x(t)-\lambda \int_{0}^{1} 3 t s x(s) d s = a t+4 t^{2}$ ;

(b)

$x(t)-\lambda \int_{0}^{1} 5 t^{2} s^{2} x(s) d s = 4 t+b t^{2}$ .

Problem 11.96

(a)(b)

In the space $L_{2}[0,1]$ , find the solution of the following integral equations for all values $\lambda \in \mathbb {C}$ :

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(a)

$x(t)-\lambda \int_{0}^{1} \min (t, s) x(s) d s = 0$ ;

(b)

$x(t)+\lambda \int_{0}^{1} \max (t, s) x(s) d s = 0$ .

Problem 11.97

(a)(b)

In the space $L_{2}[0,1]$ , find the solution of the integral equation for all values $\lambda \in \mathbb {C}$ :

x(t)+\lambda \int _{0}^{1}(t s-\min (t, s)) x(s) d s = y(t)

if

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(a)

$y(t) = t$ ;

(b)

$y(t) = \sin 2 \pi t$ .

Problem 11.98

(a)(b)

In the space $L_{2}[0,1]$ , find the solution of the integral equation for all values $\lambda \in \mathbb {C}$ :

x(t)+\lambda \int _{0}^{1}\left(\frac{1}{2}-\left|t-s\right|\right) x(s) d s = y(t)

if

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(a)

$y(t) = \cos \pi t$ ;

(b)

$y(t) = \cos 2 \pi t$ .

Problem 11.99

(a)(b)

In the space $L_{2}[0,1]$ , find the solution of the integral equation for all values $\lambda \in \mathbb {C}$ :

x(t)+\lambda \int _{0}^{1}(t s+\max (t, s)) x(s) d s = y(t)

if

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(a)

$y(t) = \cos \pi t$ ;

(b)

$y(t) = \cos 2 \pi t$ .

Problem 11.100

(a)(b)(c)

Is the Fredholm equation of the first kind

\int _{0}^{t}(t-s) x(s) d s = y(t)

solvable where

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(a)

$y(t) = \sin t$ ;

(b)

$y(t) = \cos t$ ;

(c)

$y(t) = \sin^{2} t$ in the spaces $C[0, \pi ] ; L_{p}[0, \pi ], p \in [1, \infty ]$ ?