A Fredholm integral equation is a type of integral equation that involves an unknown function under the integral sign, typically expressed in the form $$f(x) = ho + \int_{a}^{b} K(x, y) g(y) dy$$, where $$K(x, y)$$ is a known kernel function and $$g(y)$$ is an unknown function. These equations can be classified into two main types: first kind and second kind, and are crucial in various applications such as physics, engineering, and mathematical modeling. Understanding these equations is vital to solving problems involving linear operators and understanding the concepts of compactness and continuous spectra.
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