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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.

34B18Positive solutions of nonlinear boundary value problems for ODE
34B16Singular nonlinear boundary value problems for ODE
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
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