An elliptic equation of order 2m with general nonlocal boundary-value conditions, in a plane bounded domain G with piecewise smooth boundary, is considered. Generalized solutions belonging to the Sobolev space Wm 2 (G) are studied. The Fredholm property of the unbounded operator corresponding to the elliptic equation, acting on L2(G), and defined for functions from the space Wm 2 (G) that satis...

Solvability conditions for linear differential equations are usually formulated in terms of orthogonality of the right-hand side to solutions of the homogeneous adjoint equation. However, if the corresponding operator does not satisfy the Fredholm property such solvability conditions may be not applicable. For this case, we obtain another type of solvability conditions, for ordinary differentia...

For a continuous complex-valued function g on the real line without zeros, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrix-valued symbol G to be semi-Fredholm in terms of mean winding numbers of detG. The matrix function G is assumed to be continuous on the real line, and no other apriori assumptions on it are made.

We introduce a mathematical model for ultrasound modulated optical tomography and present a simple reconstruction scheme for recovering the spatially varying optical absorption coefficient from scanning measurements with narrowly focused ultrasound signals. Computational results for this model show that the reconstruction of sharp features of the absorption coefficient is possible. A formal lin...

The “spectral picture” of a bounded operator on a Banach space consists of its essential spectrum together with a mapping from its holes to the group of integers, obtained by taking the Fredholm index. In this note we abstract this from the Calkin algebra to a general Banach algebra, replacing the integers with the quotient of the group of invertibles by its connected component of the identity.

The principal results of this paper consist of an intrinsic definition of the Evans function in terms of newly introduced generalized matrix-valued Jost solutions for general first-order matrix-valued differential equations on the real line, and a proof of the fact that the Evans function, a finite-dimensional determinant by construction, coincides with a modified Fredholm determinant associate...

We prove some generalizations of the first Fredholm theorem for Hammerstein operator equations in Banach spaces and study the number of their solutions using a projection like method.The linear part is assumed to be either selfadjoint or nonseladjoint while the nonlinearities are such that the corresponding map is (pseudo) A-proper. In particular, the nonlinearities can be either of monotone ty...

We give some results concerning stability in the Fredholm operators and Browder operators set, via the concept of measure of noncompactness. Moreover, we prove some localization results on the essential spectra of bounded operators on Banach space. As application, we describe the essential spectra of weighted shift operators. Finally, we describe the spectra of polynomially compact operators, a...

This paper deals with a class of nonlinear fractional Sturm–Liouville boundary value problems. Each sub equation in the system is partial including second kinds Fredholm integral and p-Laplacian operator, simultaneously. Infinitely many solutions are derived due to perfect involvements calculus theory variational methods some simpler more easily verified assumptions.