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Positive solutions for a nonlinear third order multipoint boundary value problem. (English) Zbl 1209.34032
Summary: We study the existence of positive solutions for the third order $m$-point boundary value problem $$\align & x'''(t)+f(t, x(t), x'(t), x''(0) = 0\text{ for }t\in [0,1],\\ & x(1)=\sum^{m-2}_{i=1}\beta_i x(\xi_i),\quad x'(0)=\sum^{m-2}_{i=1} \alpha_ix'(\xi_i),\\ & x''(0)=0,\endalign$$ where $0<\xi_1<\xi_2 <\cdots <\xi_{m-2}<1$, $$\align & 0\le\alpha_i<1\text{ and }0\le\beta_i<1\text{ for }i=1,2,\dots,m-2,\\ & \sum^{m-2}_{i=1}\alpha_i<1\text{ and }\sum^{m-2}_{i=1}\beta_i< 1\endalign$$ and $f\in C([0,1]\times [0,+\infty)\times\Bbb R^2$, $[0,+\infty))$. By using the Avery-Peterson fixed point theorem, we obtain the existence of three positive solutions. An example illustrates the main results.

34B18Positive solutions of nonlinear boundary value problems for ODE
34B10Nonlocal and multipoint boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
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