Existence of Solutions
One of the Fredholm theorem's closely related to the Fredholm alternative, concerns the existence of solutions to the inhomogeneous Fredholm equation
Solutions to this equation exist if and only if the function is orthogonal to the complete set of solutions of the corresponding homogeneous adjoint equation:
where is the complex conjugate of and the former is one of the complete set of solutions to
A sufficient condition for this theorem to hold is for to be square integrable on the rectangle .
Read more about this topic: Fredholm's Theorem
Famous quotes containing the words existence of, existence and/or solutions:
“Truth exists. The sole purpose of this proposition is to assert the existence of truth against imbeciles and sceptics.”
—Edward Herbert Of Cherbury, Lord (1583–1648)
“Some say existence like a Pirouot
And Pirouette, forever in one place,
Stands still and dances, but it runs away;
It seriously, sadly, runs away
To fill the abyss’s void with emptiness.”
—Robert Frost (1874–1963)
“Every man is in a state of conflict, owing to his attempt to reconcile himself and his relationship with life to his conception of harmony. This conflict makes his soul a battlefield, where the forces that wish this reconciliation fight those that do not and reject the alternative solutions they offer. Works of art are attempts to fight out this conflict in the imaginative world.”
—Rebecca West (1892–1983)