Bifurcation problems for second order systems. (December 2020)

• Record Type: Journal Article
• Authors: Schmitt, Klaus
• Is Part Of: Nonlinear analysis. Volume 201(2020)
• Publication Date: 2020-12
• Subjects: 34B05 -- 34B15; Systems of differential equations -- Conjugate points -- Bifurcation problems; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.112042
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we consider the following linear system of second order differential equations (0.1) u ′ ′ + A ( t ) u = 0, 0 ≤ t < ∞ where, for each t, A ( t ) is an n × n matrix with real components, and positive with respect to the usual cone K in R n . Conditions are provided in order that the first conjugate point T of t = 0, i.e. the smallest T such that the above equation has a nontrivial solution u : [ 0, T ] → K satisfying the boundary conditions u ( 0 ) = 0 = u ( T ) will be a bifurcation point for higher order perturbations of the equation. The paper is mainly motivated by results in Ahmad and Lazer (1997, 1980); Ahmad and Salazar (1981) and Schmitt (1975); Schmitt and Smith (1978). Some additional new consequences are discussed.