Stability of nonlinear difference inclusions

The stability of difference inclusions x(k+1) is an element of F(x(k)) is studied, where F(x) = {F(x, gimel) : is an element of Lambda} and the selections F(., gimel) : E -->E assume values in a Banach space E, partially ordered by a cone K. It is assumed that the operators F(.,gimel) are heterotone or pseudoconcave. The main results concern asymptotically stable absorbing sets, and include the case of a single equilibrium point. The results are applied to a number of practical problems.

Stability of linear difference and differential inclusions

Any given n X n matrix A is shown to be a restriction, to the A-invariant subspace, of a nonnegative N x N matrix B of spectral radius p(B) arbitrarily close to p(A). A difference inclusion x(k+1) is an element of Ax(k), where A is a compact set of matrices, is asymptotically stable if and only if A can be extended to a set B of nonnegative matrices B with \ \B \ \ (1) < 1 or \ \B \ \ (infinity) < 1. Similar results are derived for differential inclusions.

The effects of interleukin-10 depletion in vivo on the immune response to Porphyromonas gingivalis in a murine model

Background: The immune response to Porphyromonas gingivalis in the mouse abscess model is known to be dependent upon CD4 T-cell activation and the regulatory role of cytokines. The role of interleukin-10 (IL-10) in this mouse model was examined in vivo. Methods: One-week-old, female BALB/c mice were divided into 4 groups. Groups 1 and 2 were given intraperitoneal (ip) injections of phosphate buffered saline (PBS) weekly for 5 weeks. Group 3 was given an ip injection of rat immunoglobulin. Group 4 was injected with rat anti-IL-10 antibodies. At week 6, group 1 was sham-immunized with PBS, and groups 2, 3, and 4 were injected with P gingivalis lipopolysaccharide (Pg-LPS) weekly for 2 weeks. One week after the final immunization, delayed-type hypersensitivity (DTH) was assessed by footpad swelling to Pg-LPS. The level of serum antibodies to Pg-LPS and IFN-gamma (IFN-gamma) was determined by enzyme-linked immunosorbent assay. Dorsal abscess formation induced by the injection of viable P gingivalis was examined daily for 30 days. Results: The footpad swelling of the anti-IL-10-treated group (group 4) was significantly higher than that of groups 1 to 3. Similarly, the serum IFN-gamma level in group 4 was much higher than that of the other experimental groups. There was no significant difference in serum IgG antibodies to Pg-LPS in any of the experimental groups. However, the level of IgM antibodies in group 4 mice was significantly lower than that in groups 2 and 3. In addition, serum IgG1 was suppressed in group 4 mice, while IgG2a antibodies were raised. However, there was no difference observed between the levels of IgG2b and IgG3 antibodies in any group of mice. The lesions in sham-immunized mice (group 1) persisted for 30 days, and those in group 2 and 3 were undetected by day 18 and 20, respectively. In sharp contrast, lesions in group 4 had healed completely by day 13. Conclusions: This study has shown that IL-10 depletion in vivo in P gingivalis LPS-induced immune response in mice led to an elevated DTH response, an increase in serum IFN-gamma levels, and raised levels of IgG and IgG2a antibodies. Treatment with anti-IL-10 antibodies resulted in suppressed IgG I and IgM responses and a more rapid healing of abscesses than in non-IL-10-depleted mice. These results suggest that IL-10 depletion in Pg-LPS-induced immune response in mice may lead to a Th1-like immune response and provide strong protection against a subsequent challenge with live P gingivalis in an abscess model.

GABAA receptor β3 subunit gene and psychiatric morbidity in a post-traumatic stress disorder population

GABAergic systems have been implicated in the pathogenesis of anxiety, depression and insomnia. These symptoms are part of the core and comorbid psychiatric disturbances in post-traumatic stress disorder (PTSD) In a sample of Caucasian male PTSD patients, dinucleotide repeat polymorphisms of the GABAA receptor beta3 subunit gene were compared to scores on the General Health Questionnaire-28 (GHQ). As the major allele at this gene locus (GABRB3) was GI, the alleles were divided into GI and non-GI groups. On the total score of the GHQ, which comprises the somatic symptoms, anxiety/insomnia, social dysfunction and depression subscales, patients with the GI non-GI genotype had a significantly higher score when compared to either the G1G1 genotype (alpha = 0.01) or the non-GI non-GI genotype (alpha = 0.05). No significant difference was found between the G1G1 and non-Gl non-G1 genotypes. When the GI non-G1 heterozygotes were compared to the combined G1G1 and non-GI non-GI homozygotes, a significantly higher total GHQ score was found in the heterozygotes (P = 0.002). These observations suggest a heterosis effect. Further analysis of GHQ subscale scores showed that heterozygotes compared to the combined homozygotes had higher scores on the somatic symptoms (P = 0.006), anxiety/insomnia (P = 0.003), social dysfunction (P = 0.054) and depression (P = 0.004) subscales. In conclusion, the present study indicates that in a population of PTSD patients, heterozygosity of the GABRB3 major (GI) allele confers higher levels of somatic symptoms, anxiety/insomnia, social dysfunction and depression than found in homozygosity. (C) 2001 Elsevier Science Ireland Ltd. All rights reserved.

Towards a definition of "difference" in osteoarthritis

To assess existing information regarding detectable differences in osteoarthritis (OA), a systematic literature search was conducted up to December 1999. Thirty-three articles were considered methodologically relevant to the definition and categorization of detectable differences in OA. It was determined that the musculoskeletal literature contains a wealth of information that relates to observed changes, much of which is derived from the clinical trials literature, but there have been relatively few methodological studies that have systematically evaluated the nature, categorization, and relevance of the change. Furthermore, most of those that have been published take the perspective of an individual or groups of experts other than that of the patient. This summary of the current literature reveals that the diverse sources of information go part way towards developing an understanding of detectable differences and their importance in the area of OA research and clinical practice. Stakeholders' interests as well as factors that modulate perceptions of importance need to be taken under consideration. In particular, the patient's perspective of the importance of change at an individual level requires further evaluation. This area of clinical research is relatively underdeveloped, but there is considerable opportunity for progress.

Boundary value problems for systems of difference equations associated with systems of second-order ordinary differential equations

We study difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order ordinary differential equations. We formulate conditions which guarantee a priori bounds on first differences of solutions to the discretized problem. We establish existence results for solutions to the discretized boundary value problems subject to nonlinear boundary conditions. We apply our results to show that solutions to the discrete problem converge to solutions of the continuous problem in an aggregate sense. (C) 2002 Elsevier Science Ltd. All rights reserved.

Difference equations associated with fully nonlinear boundary value problems for second order ordinary differential equations

We study the continuous problem y"=f(x,y,y'), xc[0,1], 0=G((y(0),y(1)),(y'(0), y'(1))), and its discrete approximation (y(k+1)-2y(k)+y(k-1))/h(2) =f(t(k), y(k), v(k)), k = 1,..., n-1, 0 = G((y(0), y(n)), (v(1), v(n))), where f and G = (g(0), g(1)) are continuous and fully nonlinear, h = 1/n, v(k) = (y(k) - y(k-1))/h, for k =1,..., n, and t(k) = kh, for k = 0,...,n. We assume there exist strict lower and strict upper solutions and impose additional conditions on f and G which are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. We show that the discrete approximation also has solutions which approximate solutions of the continuous problem and converge to the solution of the continuous problem when it is unique, as the grid size goes to 0. Homotopy methods can be used to compute the solution of the discrete approximation. Our results were motivated by those of Gaines.