TY - JOUR
AU - Webb, Jeffrey R. L.
AU - Lan, Kunquan Q.
PY - 2006/03/01
Y2 - 2023/02/04
TI - Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type
JF - Topological Methods in Nonlinear Analysis
JA - TMNA
VL - 27
IS - 1
SE -
DO -
UR - https://apcz.umk.pl/TMNA/article/view/TMNA.2006.003
SP - 91 - 115
AB - New criteria are established for the existence of multiple positivesolutions of a Hammerstein integral equation of the form$$u(t)= \int_{0}^1 k(t,s)g(s)f(s,u(s))ds \equiv Au(t)$$where $k$ can have discontinuities in its second variable and $g \inL^{1}$. These criteria are determined by the relationship between thebehaviour of $f(t,u)/u$ as $u$ tends to $0^+$ or $\infty$ and theprincipal (positive) eigenvalue of the linear Hammerstein integraloperator $$Lu(t)=\int_{0}^1 k(t,s)g(s)u(s)ds. $$We obtain new resultson the existence of multiple positive solutions of a second orderdifferential equation of the form$$u''(t)+g(t)f(t,u(t))=0 \quad\text{a.e. on } [0,1],$$subject to general separated boundary conditions and also to nonlocal$m$-point boundary conditions. Our results are optimal in some cases.This work contains several new ideas, and gives a {\it unified}approach applicable to many BVPs.
ER -