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On the existence of $$m$$-point boundary value problem at resonance for higher order differential equation. (English) Zbl 1046.34029
The paper considers an $$m$$-point boundary value problem for a higher-order differential equation of the form $x^{(k)}(t)=f(t,x(t),x'(t),\dots,x^{(k-1)}(t))+e(t), \quad t\in (0,1),$
$x'(0)=0,\;x''(0)=0, \dots, x^{(k-1)}(0)=0, \quad x(1)=\sum_{i=1}^{m-2} a_ix(\xi_i),$ where $$f:[0,1]\times \mathbb{R}^k \to \mathbb{R}$$ and $$e:[0,1]\to \mathbb{R}$$ are continuous functions. Further, $$m\geq 3$$, $$k\geq 2$$ are two integers, $$a_i\in \mathbb{R}$$, $$\xi_i \in (0,1)$$, $$i=1,2,\dots, m-2$$, $$\xi_1 <\xi_2<\dots <\xi_{m-2}$$. The authors study the problem at resonance because they assume that $\sum_{i=1}^{m-2}a_i=1.$ Moreover, they do not need that all $$a_i$$’s, $$1\leq i\leq m-2$$, have the same sign.
The authors prove a new existence result under certain sign and growth conditions imposed on $$f$$. The growth of $$f$$ in some variables can be superlinear. The proofs are based on Mawhin’s continuation theorem. Some examples illustrate the obtained result.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
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