Abstract

In this paper we investigate general representations of various classes of generalized inverses for bounded operators over Hilbert and Banach spaces. These representations are expressed by means of the full-rank decomposition of bounded operators and adequately selected operators.

Keywords: Full-rank factorization, reflexive generalized inverses, Moore-Penrose inverse, Drazin inverse.

MSC: 47A05, 15A09

Abstract

We consider a class of nonlinear equations which admit characteristic equations. If the roots of this algebraic equation are real and distinct, the growth and bounds of solutions of the differential equation exist. An example is given which illustrates mentioned problems. An example of equation that is similar illustrates the problem of the existence of unique solutions.

MSC: 34C11

Abstract

In this paper we study the convergence of finite difference schemes to weak solutions of the third boundary value problem for Poisson's equation on the unit square. Using the theory of interpolation of function spaces, we obtain error estimates in a discrete $W^1_2$ Sobolev norm consistent, or ``almost'' consistent, with the smoothness of the data.

Keywords: Third Boundary Value Problems, Finite Difference Schemes, Sobolev Spaces, Interpolation of Function Spaces, Convergence Rate Estimates.

MSC: 65N15, 46B70

Abstract

Asymptotic behaviour of singular values, together with the estimate of the remainder term, of Cauchy operator and the operator of logarithmic potential type, as well as the corresponding trace formula are determined.

Keywords: Cauchy integral operator, operator of logarithmic potential type, singular values, trace formula.

MSC: 47B10, 47G10, 45C05

Abstract

In this paper we consider the problem of global uniform convergence of spectral expansions and their derivatives, $\sum_{n=1}^{\infty}f_nu^{(j)}_n(x)$ $(j=0,1,\dots)$, generated by non-negative self-adjoint extensions of the operator $\Cal L(u)(x) = - u^{\prime\prime}(x) + q(x)u(x)$ with discrete spectrum, for functions from the class $W^{(1)}_2(G)$, where $G$ is a finite interval of the real axis. Two theorems giving conditions on functions $q(x)$, $f(x)$ which are sufficient for the absolute and uniform convergence on $\overline G$ of the mentioned series, are proved. Also, some convergence rate estimates are obtained.

Keywords: Spectral expansion, global uniform convergence, Schrödinger operator.

MSC: 34L10

Abstract

The paper contains a brief exposition of the 50 years of history of the Union of Mathematicians, Physicists (and Astronomers) of Yugoslavia, which was founded on November 12th, 1949 at Bled.

MSC: 01A74