Liapunov Theory for Integral Equations with Singular Kernels and Fractional Differential Equations

This is the first book to present theory, construction, and application of Liapunov functionals for integral equations with singular kernels. The study covers equations with kernels that are either singular, continuous, differentiable, or sums of these types. Basic results on existence, uniqueness, and resolvents are included giving a concise foundation for integral equation theory sufficient to support the Liapunov theory. Many examples are constructed including small kernels, convex kernels, convex kernels with singularities, and sums of such kernels. Fractional differential equations of Caputo type invert to exactly that type of kernel and they are treated throughout the book, including Liapunov functionals for the resolvent equation. Applications yield bounded, asymptotically stable, L^p, periodic, and asymptotically periodic solutions. Fixed point theory is used in conjunction with Liapunov theory. The book is being translated into the Russian language and will eventually become available from the Autonomous Nonprofit Organization, Izhevsk Institute of Computer Science, Universitetskaya, I, Izhevsk, 426034, Russia.