On the Solution of an Integral-Functional Equation with a Parameter

For a homogeneous integral-functional equation containing a parameter, we show existence and uniqueness of a compactly supported solution with given value for its integral. The solution is infinitely often differentiable, symmetric with respect to the point , monotonous at both sides of , and satisfies further functional equations. The Fourier series of the periodic continuation is determined. We also investigate spectral properties of the integral equation and find surprising connections between the Laplace transform of the eigenfunction and the eigenfunctions of the adjoint equation, and also directly between different eigenfunctions both in the compact and in the non-compact case. Moreover, asymptotic considerations are made.