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Symmetric positive solutions for nonlocal boundary value problems of fourth order. (English) Zbl 1135.34310
Summary: The singular fourth-order nonlocal boundary value problem $$\cases u''''(t)=h(t)f(t,u),\quad & 0<t<1,\\ u(0)=u(1)=\int^1_0p(s)u(s)\,ds,\\ u''(0)= u''(1)=\int^1_0q(s)u(s)\,ds\endcases$$ is considered under some suitable conditions concerning the first eigenvalue of the corresponding linear operator, where $p,q\in L_1[0,1]$, $h:(0,1)\to [0,+\infty)$ is continuous, symmetric on $(0,1)$ and may be singular at $t=0$ and $t=1$, $f:[0,1]\times [0,+\infty)\to [0,+\infty)$ is continuous and $f(\cdot,x)$ is symmetric on $[0,1]$ for all $x\to [0,+\infty)$. The existence of at least one symmetric positive solution is obtained by the application of the fixed point index in cones.

MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B16 Singular nonlinear boundary value problems for ODE 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces
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References:
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