Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type
Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type
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DOI：
10.12775/tmna.2006.003
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发表时间：
2006-03
影响因子：
0.7
通讯作者：
J. Webb;K. Lan
中科院分区：
文献类型：
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作者：
J. Webb;K. Lan
建立了形如$$u(T)=Int0}^1 k(t，S)g(S)f(S，u(S))ds等价u(T)$$的Hammerstein型积分方程解存在多个正解的新判据，其中$k$在其第二变元中可以有间断，在L^{1}中有$g。这些判据是由$f(t，u)/u的行为与线性Hammerstein积分算子的主(正)本征值之间的关系所决定的，所述行为是当$u趋于$0+$或$inty时，与线性Hammerstein积分算子$$Lu(T)=Int0}^1k(t，S)g(S)u(S)ds的主(正)本征值之间的关系.关于形式为$$u‘’(T)+g(T)f(t，u(T))=0\quad\Text{A.E.On}[0，1]，$$受一般分离边界条件和非局部$m$点边界条件的约束。我们的结果在某些情况下是最优的。这项工作包含了几个新的思想，并给出了一种适用于许多BVP的统一方法。
New criteria are established for the existence of multiple positive solutions of a Hammerstein integral equation of the form $$ 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.