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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 \(x^{\prime\prime\prime(t)}=f(t,x(t),(Fx)(t),x'(t),(Gx')(t),x^{\prime\prime} (t), (Hx^{\prime\prime})(t))\), \(t\in [0,1]\), subject to the boundary conditions \(x(0)=0\), \(x^{\prime\prime}(0)=0\), \(x'(1)=\sum_{i=1}^{m-2} \alpha_i x' (\eta_i)\), where \(f:[0,1] \times\mathbb{R}^6\to \mathbb{R}\), \(F,G,H\) are three operators, \(\alpha_i\) \((i=1,\dots ,m-2) \geq 0\), \(0<\eta_1<\eta_2<\dots <\eta_{m-2} <1\). 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 ordinary differential equations
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