Nonlinear Alternative: Application to an Integral Equation

We prove the existence of solutions to an integral equation modeling the infiltration of a fluid in an isotropic homogeneous porous medium. Let us consider the mathematical theory of the infiltration of a fluid from a cylindrical reservoir into an isotropic homogeneous porous medium. Describing this phenomena in the Boussinesq model (see [1]), we are led to the nonlinear integral equation (see also [5]) (1) u2(t) = L(t) + ∫ t 0 K(t− s)u(s)ds, t ∈ R+, where K and L are known smooth functions depending on physical parameters. The unknown function u denotes the height of the percolating fluid above the horizontal base, multiplied by a positive factor. This is the reason why, from the physical point of view, nonnegative solutions of (1) are most interesting. Our analysis of equation (1) is based on the following (see [3, 4]): 1991 Mathematics Subject Classification. 45 G.