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Solvability of functional differential equations with multi-point boundary value problems at resonance. (English) Zbl 1142.34357
Summary: We discuss the following third order functional differential equations ${x}^{\text{'}\text{'}\text{'}\left(t\right)}=f\left(t,x\left(t\right),\left(Fx\right)\left(t\right),{x}^{\text{'}}\left(t\right),\left(G{x}^{\text{'}}\right)\left(t\right),{x}^{\text{'}\text{'}}\left(t\right),\left(H{x}^{\text{'}\text{'}}\right)\left(t\right)\right)$, $t\in \left[0,1\right]$, subject to the boundary conditions $x\left(0\right)=0$, ${x}^{\text{'}\text{'}}\left(0\right)=0$, ${x}^{\text{'}}\left(1\right)={\sum }_{i=1}^{m-2}{\alpha }_{i}{x}^{\text{'}}\left({\eta }_{i}\right)$, where $f:\left[0,1\right]×{ℝ}^{6}\to ℝ$, $F,G,H$ are three operators, ${\alpha }_{i}$ $\left(i=1,\cdots ,m-2\right)\ge 0$, $0<{\eta }_{1}<{\eta }_{2}<\cdots <{\eta }_{m-2}<1$. Under some appropriate conditions, some existence and multiplicity results are given for the problem at resonance by using a priori estimates and the topological degree theory of Mawhin.
##### MSC:
 34K10 Boundary value problems for functional-differential equations 34B10 Nonlocal and multipoint boundary value problems for ODE
##### References:
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