We prove in this article some general steady state bifurcation theorem for a class of nonlinear eigenvalue problems, in the case where algebraic multiplicity of the eigenvalues of the linearized problem is even. These theorems provide an addition to the classical Krasnoselskii and Rabinowitz bifurcation theorems, which require the algebraic multiplicity of the eigenvalues is odd. For this purpose, we prove a spectral theorem for completely continuous fields, which can be considered as a generalized version of the classical Jordan matrix theorem and the Fredholm theorem for compact operators. An application to a system of second order elliptic equations is given as well.

We study in this article dynamic bifurcation of nonlinear evolution equations due to higher order nonlinear terms, focusing on detailed bifurcation behavior of nonlinear evolution equations in the cases where the algebraic multiplicity of the eigenvalues of the linearized problem is one or two.

With every continuous function a on the complex unit circle one can associate a sequence {T_n(a)}^infinity _n=1 of Toeplitz matrices and a sequence {C_n(a)} ^infinity_n=1 of circulant matrices. By employing some advanced results on the finite sections of Toeplitz operators, we prove asymptotic estimates for the central columns of the matrices T_n^-1(a) - C_n ^-1(a)as n goes to infinity. Our results generalize and sharpen recent results by T. Strohmer and by F.-W. Sun, Y. Jiang, and J. S. Baras, who also discussed the relevancy of the problem in signal processing.

We present a finite element method for the electromagnetic scattering from a 2-D cavity embedded in the infinite ground plane. The problem is first discretized in time by the BETA, LAMBDA Newmark time-marching scheme. The resulting semi-discrete problem is well-posed. Error analysis of the fully discrete finite element formulation is performed. Stability criteria of the time-stepping scheme are also established. Numerical experiments demonstrate the accuracy and stability of the method.

The present paper deals with the solution of an inverse problem in the theory of orthogonal polynomials. It was motivated by a characterization result concerning sequences of polynomials orthogonal with respect to a Sobolev inner product when they can be recursively generated in terms of orthogonal polynomial sequences associated with the measure involved in the standard component. More precisely, we obtain the set of pairs of quasi-definite linear functionals such that their corresponding sequences of monic orthogonal polynomials {P_n} and {R_n} are related by a differential expression.

We show that if v is a regular Laguerre-Hahn (resp. semiclassical) linear form,then the linear form u defined by the relation LAMBDA(x - a )u = (x - c)v is also regular and a Laguerre-Hahn (resp. semiclassical) linear form for every complex LAMBDA except for a discrete set of numbers depending on v, a, and c. We give explicitly the coefficients of the three-term recurrence relation, the structure relation of the orthogonal sequence associated with u, and the class of the linear form u knowing that of v. Finally, we apply the above results to some examples.

We apply the generalized quasilinearization method to a second order three-point boundary value problem involving mixed nonlinear boundary conditions and obtain a monotone sequence of approximate solutions converging to the unique solution of the problem possessing a convergence of order k(k greater than or equal to 2).