We study weakly perturbed integrodifferential equations with degenerate kernel in Banach spaces and establish conditions for the bifurcation of solutions at point ε = 0. A convergent iterative procedure is proposed determination form series $$ {\sum}_{i=1}^{+\infty }{\varepsilon}^i{z}_i(t) powers ε..

A maximum principle and some a priori estimates of a class of degenerate equations with X-ellipticity in the sense of distributions are established. A local comparison of the generalized Green function with its fundamental solutions is obtained. As an application, by means of the power of the Green function as a kernel function of a local integral, we also derive local Hölder continuity for non...

The aim of this paper is to carry out an improved analysis the convergence Nyström and degenerate kernel methods their superconvergent versions for numerical solution a class linear Fredholm integro-differential equations second kind. By using interpolatory projection at Gauss points onto space (discontinuous) piecewise polynomial functions degree ?r?1, we obtain order 2r methods, while, iterat...

In this paper, the degenerate scale for plate problem is studied. For the continuous model, we use the null-field integral equation, Fourier series and the series expansion in terms of degenerate kernel for fundamental solutions to examine the solvability of BIEM for circular thin plates. Any two of the four boundary integral equations in the plate formulation may be chosen. For the discrete mo...

Degenerate scales usually refer to a size effect which causes non-unique solutions of boundary integral equations for certain type of boundary value problems with a unique solution. They are closely connected to the presence of a logarithmic function in the integral kernel of the single-layer potential operator. The equations of the elasticity theory provide one of the known application fields ...

We consider the eigenvalue problem of certain kind of non-compact linear operators given as the sum of a multiplication and a kernel operator. A degenerate kernel method with piecewise linear interpolation with respect to the second variable is used to approximate eigenvalues. It is shown that the error is of order O(h). A bound for the condition number of this method is obtained. By a numerica...

Abstract A uniform in probability approximation is established for Studentized processes of non degenerate U -statistics of order m ≥ 2 in terms of a standard Wiener process. The classical condition that the second moment of kernel of the underlying U-statistic exists is relaxed to having 5 3 moments. Furthermore, the conditional expectation of the kernel is only assumed to be in the domain of ...

We present moment inequalities for completely degenerate Banach space valued (generalized) U-statistics of arbitrary order. The estimates involve suprema of empirical processes, which in the real valued case can be replaced by simpler norms of the kernel matrix (i.e. norms of some multilinear operators associated with the kernel matrix). As a corollary we derive tail inequalities for U-statisti...

(1) Contain all real and imaginary parts of bounded holomorphic functions. (2) Be describable as \Poisson integrals" over the Bergman-Shilov boundary against a real kernel (the \Poisson" kernel). (3) Be invariant under all bi-holomorphisms of the domain. (4) Be describable as the nullspace HL of a degenerate-elliptic system L of second order di erential operators. (We refer to HL as the space o...