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Zbl 1202.34038
Hao, Xinan; Liu, Lishan; Wu, Yonghong; Sun, Qian
Positive solutions for nonlinear $n$th-order singular eigenvalue problem with nonlocal conditions.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 6, 1653-1662 (2010). ISSN 0362-546X

Summary: The nonlinear $n$th-order singular nonlocal boundary value problem $$\cases u^{(N)}(t)+\lambda a(t)f(t,u(t))=0,\quad t\in(0,1),\\ u(0)=u'(0)=\cdots=u^{(n-2)}(0)=0,\quad u(1)=\displaystyle\int^1_0u(s)\,dA(s),\endcases$$ is considered under some conditions concerning the first eigenvalue corresponding to the relevant linear operator, where $\int^1_0u(s)\,dA(s)$ is given by a Riemann-Stieltjes integral with a signed measure, $a$ may be singular at $t=0$ and/or $t=1$, $f(t,x)$ may also have a singularity at $x=0$. The existence of positive solutions is obtained by means of the fixed point index theory in cones.
MSC 2000:
*34B09 Boundary value problems with an indefinite weight
34B10 Multipoint boundary value problems
34B16 Singular nonlinear boundary value problems
34B18 Positive solutions of nonlinear boundary value problems
47N20 Appl. of operator theory to differential and integral equations

Keywords: positive solution; singular higher order differential equation; nonlocal boundary conditions; fixed point index

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