Convergence and Evaluation of Sums of Reciprocal Powers of Eigenvalues of Boundary Value Problems Nonlinear in the Eigenvalue Parameter
Anthony V. Laginestra, William E. Boyce · SIAM Journal on Mathematical Analysis · 1974
In this paper the trace equations \[(1)\qquad \sum_{i = 1}^\infty {\lambda _i^{ - p} = } \int_0^1 {K_p (x,x)dx} \] arising in the Hilbert-Schmidt theory of Fredholm integral equations are extended to integral equations of the form \[(2)\qquad \phi (x) = \lambda \int_0^1 {K(x,y,\lambda )\phi (y)dy,} \] in which the kernel $K(x,y,\lambda )$ is a rather general function of $\lambda $. Attention is focused on equations that correspond to differential boundary value problems via the Green’s function. The underlying boundary value problem may be nonlinear in $\lambda $ in the differential equation, in the boundary conditions, or in both. Three theorems are proved, each of which asserts the convergence of \[(3)\qquad \sum\limits_{i = 1}^\infty {\lambda _i^{ - p} } \]for p sufficiently large, under relatively mild hypotheses on the coefficients appearing in the boundary value problem. Finally, a procedure is established whereby (3) can sometimes be evaluated in terms of the Taylor coefficients of a certain function, but without the necessity for the repeated integration implied by (1).