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Symmetric positive solutions for nonlinear singular fourth-order eigenvalue problems with nonlocal boundary condition. (English) Zbl 1196.34028

By using a fixed point theorem in cones, the authors prove the existence of symmetric positive solutions of the fourth-order nonlocal boundary value problem \[ \begin{cases} u^{(4)}(t)-\lambda h(t)f(t,u,u'')=0,\,\,\,0<t<1,\\ u(0)=u(1)=\displaystyle\int_0^1a(s)u(s)\,ds,\\ u''(0)=u''(1)=\displaystyle\int_0^1b(s)u''(s)\,ds,\end{cases} \]
where \(a,\,b\in L^1[0,1]\), \(\lambda>0\), \(h\) may be singular at \(t=0\) and/or \(t=1\) and \(f(t,x,y)\) may also have singularities at \(x=0\) and/or \(y=0\).

MSC:

34B09 Boundary eigenvalue problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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References:

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