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# Article

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.
References:
 C. P. Gupta: Solvability of a three-point boundary value problem for a second order ordinary differential equation. Jour. Math. Anal. Appl. 168 (1992), 540–551. DOI 10.1016/0022-247X(92)90179-H | MR 1176010
 C. P. Gupta: A note on a second order three-point boundary value problem. Jour. Math. Anal. Appl. 186 (1994), 277–281. DOI 10.1006/jmaa.1994.1299 | MR 1290657 | Zbl 0805.34017
 C. Gupta, S. Ntouyas and P. Tsamatos: On an $m$-point boundary value problem for second order ordinary differential equations. Nonlinear Analysis 23 (1994), 1427–1436. DOI 10.1016/0362-546X(94)90137-6 | MR 1306681
 C. Gupta, S. Ntouyas and P. Tsamatos: Existence results for $m$-point boundary value problems. Differential Equations and Dynamical Systems 2 (1994), 289–298. MR 1386275
 V. Il’in and E. Moiseev: Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differential Equations 23 (1987), 803–810.
 V. Il’in and E. Moiseev: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Differential Equations 23 (1987), 979–987.
 S. A. Marano: A remark on a second order three-point boundary value problem. Jour. Math. Anal. Appl. 183 (1994), 518–522. DOI 10.1006/jmaa.1994.1158 | MR 1274852 | Zbl 0801.34025
 J. Mawhin: Topological degree methods in nonlinear boundary value problems. “NSF-CBMS Regional Conference Series in Math.” No. 40, Amer. Math. Soc., Providence, RI, 1979. MR 0525202 | Zbl 0414.34025