Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type
Słowa kluczowe
Fixed point index, positive solution, eigenvalue criteriaAbstrakt
New criteria are established for the existence of multiple positive solutions 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 $g \in L^{1}$. These criteria are determined by the relationship between the behaviour of $f(t,u)/u$ as $u$ tends to $0^+$ or $\infty$ and the principal (positive) eigenvalue of the linear Hammerstein integral operator $$ Lu(t)=\int_{0}^1 k(t,s)g(s)u(s)ds. $$ We obtain new results on the existence of multiple positive solutions of a second order differential 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.Pobrania
Opublikowane
2006-03-01
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1.
WEBB, Jeffrey R. L. & LAN, Kunquan Q. Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type. Topological Methods in Nonlinear Analysis [online]. 1 marzec 2006, T. 27, nr 1, s. 91–115. [udostępniono 22.7.2024].
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