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Keywords:
multi-point boundary value problems; four point boundary value problems; Leray-Schauder Continuation theorem; a priori bounds
Summary:
Let $f\colon [0,1]\times \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function satisfying Caratheodory’s conditions and let $e(t)\in L^{1}[0,1]$. Let $\xi _{i}, \tau _{j}\in (0,1)$, $ c_{i},a_{j}\in \mathbb{R}$, all of the $c_{i}$’s, (respectively, $a_{j}$’s) having the same sign, $i=1,2,\ldots ,m-2$, $j=1,2,\ldots ,n-2$, $0 < \xi _{1}<\xi _{2}<\ldots <\xi _{m-2}<1$, $0 < \tau _{1}<\tau _{2}<\ldots <\tau _{n-2}<1$ be given. This paper is concerned with the problem of existence of a solution for the multi-point boundary value problems \begin{align*} x^{\prime\prime}(t)=f(t, x(t),x^{\prime}(t))+e(t),\qquad t\in (0,1)\tag{E} \\ x(0)=\sum\limits_{i=1}^{m-2} c_{i}x^{\prime}(\xi_{i}),\qquad x(1)=\sum\limits_{j=1}^{n-2} a_{j}x(\tau_{j}) \tag{BC$_{mn}$}\end{align*} and \begin{align*} x^{\prime\prime}(t)=f(t, x(t),x^{\prime}(t))+e(t),\qquad t\in (0,1)\tag {E}\\ x(0)=\sum\limits_{i=1}^{m-2} c_{i}x^{\prime}(\xi_{i}),\qquad x^{\prime}(1)=\sum\limits_{j=1}^{n-2} a_{j}x^{\prime}(\tau_{j}), \tag{BC$_{mn}$'} \end{align*} Conditions for the existence of a solution for the above boundary value problems are given using Leray-Schauder Continuation theorem.
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