Positive Compact Operators Research Articles

A semilinear elliptic eigenvalue problem, I

SynopsisThe nonlinear operator equation (N), , describes both semilinear elliptic boundary value problems (P) and their natural discretisations (Ph). (Here is a positive compact linear operator and [t]+ ≡ (t + |t|)/2 for all t ∊ ℝ.) It is proved that, for q ≧ 0 (q ≢ 0), in a Banach lattice E the equation (N) has an unbounded continuum ℱ of nontrivial solutions (λ, ψ) ∊ ℝ × E bifurcating from infinity at (λ1, ∞) (here is the first (positive) eigenvalue of ). All nontrivial solutions (λ, ψ) have λ ≧ λ1, and if maps a smaller cone Ks, into itself then there is a ℱs ⊂ℝ×Ks with similar properties to ℱ. The existence theory for (N) is applied to problems (P) and (Ph) which are defined on the simply-connected domain Ω. It is shown that the projection of ℱ on the λ-axis is either unbounded if the continuous function q > 0 except on a set of measure zero in Ω, or is bounded if q ≡ 0 on a subdomain of Ω. If is the second eigenvalue for in (Ph) then there is at most one nontrivial solution for each λ satisfying λ1 < λ < λ2; in the corresponding uniqueness result for (P) q is restricted to being strictly positive somewhere on ∂Ω Additional properties for the solutions of (P) and (Ph) are also proved.

Approximation of the eigenvectors of a positive compact operator and estimate of the error

Vengono date formule per calcolare l'errore nella approssimazione degli autovettori di un operatore compatto e positivo mediante il metodo di Rayleigh-Ritz.

A positive compact operator

I give an example of a positive compact operator on L2[0,1] which is not representable by any Lebesgue measurable function on [0,1]2. This example can be adapted to answer a question of H.H.Schaefer (§4 below).

Upper and lower bounds to the eigenfrequencies of elastic frames

A method for the computation of upper and lower bounds to the eigenfrequencies of elastic frames is presented. The upper bounds are obtained by the Rayleigh-Ritz method, with reference to the variational formulation of the eigenvalue problem. The formulation of an equivalent eigenvalue problem for a positive compact operator allows to apply the theory of Orthogonal Invariants to obtain the lower bounds. A numerical example shows the effectiveness of the method.

A fixed point theorem for positivek-set contractions

The abstract theory of positive compact operators (acting in a partially ordered Banach space) has proved to be particularly useful in the theory of integral equations. In a recent paper (2) it was shown that many of the now classical theorems for positive compact operators can be extended to certain classes of non-compact operators. One result, proved in (2, Theorem 5), was a fixed point theorem for compressivek-set contractions (k<l). The main result of this paper (Theorem 3.3) shows that some of the hypotheses of (2, Theorem 5) are unnecessary. We use techniques based on those used by M. A. Krasnoselskii in the proof of Theorem 4.12 in (4), which is the classical fixed point theorem for compressive compact operators, to obtain a complete generalisation of this classical result to thek-set contractions (k< 1). It should be remarked that J. D. Hamilton has extended the same result toA-proper mappings (3, Theorem 1). However apparently it is not known, even in the case when we are dealing with a Π1-space, whetherk-set contractions areA-proper or not.

Weakly Compact Positive Operators on Summable Functions

This paper uses elementary integration and operator techniques combined with some theorems of Nakano to give a simple proof of the well-known Dunford-Pettis theorem for positive operators.

Weakly compact positive operators on summable functions

This paper uses elementary integration and operator techniques combined with some theorems of Nakano to give a simple proof of the well-known Dunford-Pettis theorem for positive operators.