Fundamental Solutions of Some Linear Operator Equations and Applications

A concept of a fundamental solution is introduced for linear operator equations given in some functional spaces. In the case where this fundamental solution does not exist, the representation of the solution is found by the concept of a generalized fundamental solution, which is introduced for operators with nontrivial and generally infinite-dimensional kernels. The fundamental and generalized fundamental solutions are also investigated for a class of Fredholm-type operator equations. Some applications are given for one-dimensional generally nonlocal hyperbolic problems with trivial, finite- and infinite-dimensional kernels. The fundamental and generalized fundamental solutions of such problems are constructed as particular solutions of a system of integral equations or an integral equation. These fundamental solutions become meaningful in a general case when the coefficients are generally nonsmooth functions satisfying only some conditions such as p-integrablity and boundedness.