Asymptotic Properties of Eigenvalues of Integral Equations

Source abstract

The eigenvalue problem is considered for the integral equation $$\int _0^a K( {x,y} )\varphi ( y )dy = \lambda \varphi ( X ).$$ The eigenvalues and eigenfunctions are studied as functions of the upper limit a. For small values of a with K smooth, expansions of the solutions in powers of a are obtained, and it is shown that $\lambda ^{( N )} ( a ) = O( {a^{2N = 1} } )$. For real symmetric kernels it is shown that $| {\lambda ^{( N )} ( a )} |$ is an increasing function of a for all values of a. For large values of a with additional conditions on K, expansions of the solutions in powers or fractional powers of $a^{ - 1}$ are obtained.