EXISTENCE OF SOLUTIONS FOR FOURTH-ORDER THREE-POINT BOUNDARY VALUE PROBLEMS ON A HALF LINE
Project no. 660 · Status: Kapanmış · 2015
Summary
In this paper, we apply Sch\"{a}uder'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 \begin{eqnarray*} \left. \begin{array}{ll} x''''(t)+q(t) f(t, x(t), x'(t), x''(t),x'''(t))=0,\ \ \hbox{$t\in(0,+\infty)$,} \\ x''(0)=A,\ \ x(\eta)=B_1,\ \ x'(0)=B_2,\ \ x'''(+\infty)=C, \end{array} \right. \end{eqnarray*} where $\eta\in(0,+\infty),$ but fixed, and $f:[0,+\infty)\times \mathbb{R}^4\rightarrow\mathbb{R}$ 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 an example to illustrate the importance of our results.