Geodesic
Solutions of a multi-point boundary value problem for higher-order differential equations at resonance. (II)
Archivum mathematicum, Tome 41 (2005) no. 2, pp. 209-227.

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In this paper, we are concerned with the existence of solutions of the following multi-point boundary value problem consisting of the higher-order differential equation \[ x^{(n)}(t)=f(t,x(t),x^{\prime }(t),\dots ,x^{(n-1)}(t))+e(t)\,,\quad 01\,,\qquad \mathrm {{(\ast )}}\] and the following multi-point boundary value conditions \begin{align*}{1}{*}{-1} x^{(i)}(0)=0\quad \mbox{for}\quad i=0,1,\dots ,n-3\,,\\ x^{(n-1)}(0)=\alpha x^{(n-1)}(\xi )\,,\quad x^{(n-2)}(1)=\sum _{i=1}^m\beta _ix^{(n-2)}(\eta _i)\,. \tag{**}\end{align*} Sufficient conditions for the existence of at least one solution of the BVP $(\ast )$ and $(\ast \ast )$ at resonance are established. The results obtained generalize and complement those in [13, 14]. This paper is directly motivated by Liu and Yu [J. Pure Appl. Math. 33 (4)(2002), 475–494 and Appl. Math. Comput. 136 (2003), 353–377].
Classification : 34B10, 34B15, 47H11, 47N20
Keywords: solution; resonance; multi-point boundary value problem; higher order differential equation 
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     title = {Solutions of a multi-point boundary value problem for higher-order differential equations at resonance. {(II)}},
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Liu, Yuji; Ge, Weigao. Solutions of a multi-point boundary value problem for higher-order differential equations at resonance. (II). Archivum mathematicum, Tome 41 (2005) no. 2, pp. 209-227. http://geodesic.mathdoc.fr/item/ARM_2005__41_2_a9/