We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Green’s functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associate...

Abstract. In this paper, existence and uniqueness theorems for nonlinear fuzzy fractional Fredholm integro-differential equations are investigated. We interpret Fredholm integro-differential equations under fractional generalized Hukuhara derivatives in the Caputo sense, and for this interpretation, we prove existence results in two type of differentiability. Moreover, two examples are given to...

We give several new formulas of the least positive Fredholm eigenvalue for a quasicircle, answering a problem posed recently by Kühnau. During the proof, we show that the Grunsky functionals corresponding to the two complementary domains of the quasi-circle are the same and equal to the reciprocal of the Fredholm eigenvalue.

In this expository article we describe the asymptotics of certain Fredholm determinants which provide solutions to the cylindrical Toda equations, and we explain how these asymptotics are derived. The connection with Fredholm determinants arising in the theory of random matrices, and their asymptotics, are also discussed.

In this article the nonlinear mixed Volterra-Fredholm integral equations are investigated by means of the modified threedimensional block-pulse functions (M3D-BFs). This method converts the nonlinear mixed Volterra-Fredholm integral equations into a nonlinear system of algebraic equations. The illustrative examples are provided to demonstrate the applicability and simplicity of our scheme.

We study the time and temperature dependent correlation functions for an impenetrable bose gas with open boundary conditions. We derive the Fredholm determinant formulae for the correlation functions, by means of the Bethe Ansatz. In the case of time independent ground state, our Fredholm determinant formulae degenerate to the one which have been obtained by the help of fermions [1].

We prove a Fredholm alternative theorem for a class of asymptotically hyperbolic linear diierential diierence equations of mixed type. We also establish the cocycle property and the spectral ow property for such equations, providing an eeective means of calculating the Fredholm index. Such systems can arise from equations which describe traveling waves in a spatial lattice.

In this paper we completely characterize compact Toeplitz operators on the harmonic Bergman space. By using this result we establish the short exact sequences associated with the Toeplitz algebra and the Hankel algebra. We show that the Fredholm index of each Fredholm operator in the Toeplitz algebra or the Hankel algebra is zero.  2002 Elsevier Science (USA). All rights reserved.

The well-known Kantorovich technique based on majorizing sequences is used to analyse the convergence of Newton’s method when it is used to solve nonlinear Fredholm integral equations. In addition, we obtain information about the domains of existence and uniqueness of a solution for these equations. Finally, we illustrate the above with two particular Fredholm integral equations.