In this paper, we apply Schauder's fixed point theorem, the upper and lower solution method, and topological degree theory to establish the existence of unbounded solutions for the following fourth order three-point boundary value problem on a half-line egin{align*} &x''''(t)+q(t) f(t, x(t), x'(t), x''(t),x'''(t))=0, qquad hbox{$tin(0,+infty)$,} &x''(0)=A,qquad x(eta)=B_1,qquad x'(eta)=B_2, qquad x'''(+infty)=C, end{align*} where $etain(0,+infty),$ but fixed, and $fcolon [0,+infty)imes mathbb{R}^4ightarrowmathbb{R}$ satisfies Nagumo's condition. We present easily verifiable sufficient conditions for the existence of at least one solution, and at least three solutions of this problem. We also give two examples to illustrate the importance of our results.
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机译:在本文中,我们采用Schauder不动点定理,上下解方法和拓扑度理论来确定半线 begin {align * }&x''''(t)+ q(t)f(t,x(t),x'(t),x''(t),x'''(t))= 0, qquad hbox {$ t in(0,+ infty)$,} &x''(0)= A, qquad x( eta)= B_1, qquad x'( eta)= B_2, qquad x'''((+ infty)= C, end {align *}其中$ eta in(0,+ infty),$但固定,而$ f 冒号[0,+ infty)次 mathbb {R} ^ 4 rightarrow mathbb {R} $满足Nagumo的条件。对于存在至少一个解决方案和至少三个解决方案的问题,我们提出了易于验证的充分条件。我们还举两个例子来说明我们结果的重要性。
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