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Positive solutions for nonlinear $n$th-order singular eigenvalue problem with nonlocal conditions. (English) Zbl 1202.34038
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.

34B09Boundary eigenvalue problems for ODE
34B10Nonlocal and multipoint boundary value problems for ODE
34B16Singular nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
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