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Keywords:
solution; resonance; multi-point boundary value problem; higher order differential equation
solution; resonance; multi-point boundary value problem; higher order differential equation
Summary:
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 \[ x^{(n)}(t)=f(t,x(t),x^{\prime }(t),\dots ,x^{(n-1)}(t))+e(t)\,,\quad 0<t<1\,,\qquad \mathrm {{(\ast )}}\] 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].
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